International Association of Geodesy Symposia Fernando SansO, Series Editor
International Association of Geodesy Symposia Fernando Sansb, Series Editor
Symposium 101: Global and Regional Geodynamics Symposium 102: Global Positioning System: An Overview Symposium 103: Gravity, Gradiometry, and Gravimetry Symposium 104: Sea SurfaceTopography and the Geoid Symposium 105: Earth Rotation and Coordinate Reference Frames Symposium 106: Determination of the Geoid: Present and Future Symposium 107: Kinematic Systems in Geodesy, Surveying, and Remote Sensing Symposium 108: Application of Geodesy to Engineering Symposium 109: Permanent Satellite Tracking Networks for Geodesy and Geodynamics Symposium 11O: From Mars to Greenland: Charting Gravity with Space and Airborne Instruments Symposium 111: Recent Geodetic and Gravimetric Research in Latin America Symposium 112: Geodesy and Physics of the Earth: Geodetic Contributions to Geodynamics Symposium 113: Gravity and Geoid Symposium 114: Geodetic Theory Today Symposium 115: GPS Trends in Precise Terrestrial, Airborne, and Spaceborne Applications Symposium 116: Global Gravity Field and Ist Temporal Variations Symposium 117: Gravity, Geoid and Marine Geodesy Symposium 118: Advances in Positioning and Reference Frames Symposium 119: Geodesy on the Move Symposium 120: Towards an Integrated Global Geodetic Observation System (IGGOS) Symposium 121: Geodesy Beyond 2000: The Challenges of the First Decade Symposium 122: IV Hotine-Marussi Symposium on Mathematical Geodesy Symposium 123: Gravity, Geoid and Geodynamics 2000 Symposium 124: Vertical Reference Systems Symposium 125: Vistas for Geodesy in the New Millennium Symposium 126: Satellite Altimetry for Geodesy, Geophysics and Oceanography Symposium 127: V Hotine Marussi Symposium on Mathematical Geodesy Symposium 128: A Window on the Future of Geodesy Symposium 129: Gravity, Geoid and Space Missions Symposium 130: Dynamic Planet - Monitoring and Understanding a Dynamic Planet with Geodetic and Oceanographic Tools Symposium 131: Geodetic Deformation Monitoring: From Geophysical to Engineering Roles
Paul Tregoning Chris Rizos
(Eds.)
Dynamic Planet Monitoring and Understanding a Dynamic Planet with Geodetic and OceanographicTools lAG Symposium Cairns,Australia 22-26 August, 2005
With 795 Figures
Springer
Volume Editors
Series Editor
Dr. Paul Tregoning
Prof. Fernando Sans6
Research School of Earth Sciences The Australian National University Canberra ACT 0200 Australia
Polytechnic of Milan D.I.I.A.R.- Surveying Section Piazza Leonardo da Vinci, 32 20133 Milan Italy
Dr. Chris Rizos
University of New South Wales School of Surveying and Spatial Information Systems Sydney NSW 2052 Australia
Library of Congress Control Number: 2006936097 ISSN ISBN-10 ISBN-13
0939-9585 3-540-49349-5 Springer Berlin Heidelberg New York 978-3-540-49349-5 Springer Berlin Heidelberg New York
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32/3141/as - 5 4 3 2 1 0
Contents
Part l Joint IAG/IAPSO Papers ..........................................................
1
E K n u d s e n • O. B. A n d e r s e n . R. Forsberg • H. E F 6 h . A. V. Olesen • A. L. Vest D. Solheim • O. D. O m a n g • R. H i p k i n • A. H u n e g n a w . K. H a i n e s . R. B i n g h a m j._E Drecourt • J. A. Johannessen • H. Drange • F. S i e g i s m u n d • F. H e r n a n d e z G. L a r n i c o l . M.-H. R i o . E Schaeffer
Chapter I
Combining Altimetric/Gravimetric and Ocean Model Mean Dynamic Topography Models in the GOClNA Region . . . . . . . . . . . . . . . . . . . . . . .
3
P. K n u d s e n . C. C. Tscherning
Chapter 2
Error Characteristics of Dynamic Topography Models Derived from Altimetry and GOCEGravimetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
Z. Z h a n g • Y. L u . H. H s u
Chapter 3
Detecting Ocean Currents from Satellite Altimetry, Satellite Gravity and Ocean Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
F. N. Teferle • R. M. Bingley • A. I. Waugh • A. H. D o d s o n • S. D. P. Williams T. E Baker
Chapter 4
Sea Level in the British Isles: Combining Absolute Gravimetry and Continuous GPS to Infer Vertical Land Movements at Tide Gauges ......... 23
Chapter 5
Snow Accumulation and Snowmelt Monitoring in Greenland and Antarctica .. 31
Chapter 6
H.-P. Plag E s t i m a t i n g Recent Global Sea Level C h a n g e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
Chapter 7
I. v. Sakova • G. Meyers • R. C o l e m a n On t h e L o w - F r e q u e n c y Variability in t h e Indian Ocean
..........................
47
Chapter 8
w. B o s c h . R. Savcenko S a t e l l i t e A l t i m e t r y : Multi-Mission Cross Calibration
.............................
51
S. V. N g h i e m • K. Steffen • G. N e u m a n n • R. Huff
D. N. Arabelos • G. Asteriadis • M. E. Contadakis • D. Papazachariou • S. D. Spatalas
Chapter 9
Assessment of Recent Tidal Models in the Mediterranean Sea . . . . . . . . . . . . . . . . . .
57
Chapter 10
s. M. Barbosa • M. J. Fernandes • M. E. Silva Scale-Based Comparison of Sea Level Observations in the North Atlantic from Satellite Altimetry and Tide Gauges . . . . . . . . . . . . . . . . .
63
M. J. Garc~a. B. P. G 6 m e z . F. Raicich. L. R i c k a r d s . E. B r a d s h a w . H.-P. Plag X. Z h a n g • B. L. Bye. E. Isaksen
Chapter 11
European Sea Level Monitoring: Implementation of ESEASQuality Control ..... 67 R. Dalazoana • S. R. C. de Freitas • J. C. B~iez • R. T. Luz
Chapter 12
Chapter 13
Brazilian Vertical Datum Monitoring Vertical Land Movements and Sea Level Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
M. Tervo • M. Poutanen • H. Koivula Tide Gauge Monitoring Using GPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
Y. C h u . J. Li. W. Jiang. X. Z o u . X. X u . C. Fan
Chapter 14
Determination of Inland Lake Level and Its Variations in China from Satellite Altimetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
80
Vl
Contents M. C. M a r t i n . C. L. Villanoy
Chapter 15
Sea Surface Variability of Upwelling Area Northwest of Luzon, Philippines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
84
Y. Fukuda • Y. Hiraoka. K. Doi
Chapter 16
Chapter 17
An Experiment of Precise Gravity Measurements on Ice Sheet, Antarctica . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
M. Amalvict • P. Willis. K. Shibuya Status of DORIS Stations in Antarctica for Precise G e o d e s y . . . . . . . . . . . . . . . . . . . . . .
94
H. H. A. Schotman • E N. A. M. Visser • L. L. A. Vermeersen
Chapter 18
High-Harmonic Gravity Signatures Related to Post-Glacial Rebound ......... 103
Part II Frontiers in the Analysis of Space Geodetic Measurements ................
Chapter 19
113
C. Urschl • G. Beutler. W. Gurtner • U. Hugentobler • S. Schaer GPS/GLONASS Orbit D e t e r m i n a t i o n Based on Combined Microwave and SLR Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
115
Chapter 20
s. Bergstrand • H.-G. Scherneck • M. Lidberg • J. M. Johansson BIFROST: Noise Properties of GPS Time Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
123
Chapter 21
w. Bosch Discrete Crossover Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
131
s. SchGn • H. Kutterer
Chapter 22
A Comparative Analysis of Uncertainty Modelling in GPS Data Analysis ...... 137 P. Willis. E G. Lemoine • L. Soudarin
Chapter 23
Looking for Systematic Error in Scale from Terrestrial Reference Frames Derived from DORISData .......................................................
143
A. Nothnagel • J.-H. C h o . A. Roy. R. Haas
Chapter 24
WVR Calibration Applied to European VLBI Observing Sessions ...............
152
Chapter 25
M. Kriigel • D. A n g e r m a n n Frontiers in t h e C o m b i n a t i o n of Space Geodetic Techniques
..................
158
Modifying the Stochastic Model to Mitigate GPS Systematic Errors in Relative Positioning .........................................................
166
D. B. M. Alves • J. F. G. Monico
Chapter 26
E. M. Souza • J. F. G. Monico
Chapter 27
Chapter 28
GPSAmbiguity Resolution and Validation under Multipath Effects: Improvements Using Wavelets .................................................
172
R. F. Leandro • M. C. Santos An Empirical Stochastic Model for GPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
179
R. E Leandro • C. A. U. Silva. M. C. Santos
Chapter 29
Feeding Neural Network Models with GPS Observations: a Challenging Task . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
186
P. J. Mendes Cerveira • T. Hobiger • R. Weber. H. Schuh
Chapter 30
Spatial Spectral Inversion of the Changing Geometry of the Earth from SOPACGPS Data ..........................................................
194
L. VGlgyesi. L. FGldv~ry. G. Csap6
Chapter 31
Improved Processing Method of UEGN-2002Gravity Network Measurements in Hungary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter 32
J. Nastula • B. Kolaczek • R. Weber. H. Schuh • J. Boehm Spectra of Rapid Oscillations of Earth Rotation P a r a m e t e r s
Chapter 33
E. Wei. J. Liu. C. Shi On t h e Establishing Project of Chinese Surveying and Control Network
D e t e r m i n e d during t h e C0NT02 C a m p a i g n
....................................
for Earth-0rbit Satellite and Deep Space Detection
Chapter 34
...........................
T. K. Yeh. C. S. Chen Constructing a System to Monitor t h e Data Quality of GPS Receivers . . . . . . . . .
202
208
215 222
Contents Part III Gravity Field Determination from a Synthesisof Terrestrial, Satellite, Airborne and Altimetry Measurements ...........................
229
E G. L e m o i n e • S. B. Luthcke • D. D. Rowlands • D. S. C h i n n • S. M. Klosko C. M. Cox
Chapter 35
The Use of Mascons to ResolveTime-Variable Gravity from GRACE ............ 231 R. S c h m i d t • F. F l e c h t n e r • R. K 6 n i g . U1. M e y e r . K.-H. N e u m a y e r • Ch. Reigber M. R o t h a c h e r • S. Petrovic • S.-Y. Z h u . A. G i i n t n e r
Chapter 36
GRACE T i m e - V a r i a b l e Gravity A c c u r a c y A s s e s s m e n t
...........................
237
G. S. Vergos • V. N. Grigoriadis • I. N. Tziavos • M. G. Sideris
Chapter 37
Combination of Multi-Satellite Altimetry Data with CHAMP and GRACEEGMsfor Geoid and Sea Surface Topography Determination ..... 244
Chapter 38
A New Methodology to ProcessAirborne Gravimetry Data: Advances and Problems ........................................................
B. A. Alberts • P. D i t m a r • R. Klees
251
N. Kiihtreiber • H. A. A b d - E l m o t a a l
Chapter 39
Ideal Combination of Deflection Components and Gravity Anomalies for Precise Geoid Computation .................................................
259
D. Blitzkow • A. (2. O. (2. de Matos • J. P. Cintra
Chapter 40
SRTM Evaluation in Brazil and Argentina with Emphasis on the Amazon Region .........................................................
266
J. H u a n g • G. Fotopoulos • M. K. C h e n g • M. V 4 r o n n e a u • M. G. Sideris
Chapter 41
On t h e E s t i m a t i o n of t h e R e g i o n a l Geoid Error in C a n a d a
.....................
272
A. L6cher • K. H. Ilk
Chapter 42
A Validation Procedure for Satellite Orbits and Force Function Models Based on a New Balance Equation Approach ..................................
280
L. Z h u . C. Jekeli
Chapter 43
Combining Gravity and Topographic Data for Local Gradient Modelling ...... 288 G. A u s t e n • W. Keller
Chapter 44
Numerical Implementation of the Gravity Space Approach Proof of Concept ................................................................
296
R. Klees • T. W i t t w e r
Chapter 45
Local Gravity Field M o d e l l i n g w i t h M u l t i - P o l e W a v e l e t s . . . . . . . . . . . . . . . . . . . . . . .
303
G. S. Vergos • V. N. Grigoriadis • G. K a l a m p o u k a s • I. N. Tziavos
Chapter 46
Accuracy Assessment of the SRTM90m DTM over Greece and Its Implications to Geoid Modelling .......................................
309
C. H i r t . G. Seeber
Chapter 47
High-Resolution Local Gravity Field Determination at the Sub-Millimeter Level using a Digital Zenith Camera System ............ 316
Chapter 48
A Data-Adaptive Design of a Spherical Basis Function Network for Gravity Field Modelling .....................................................
R. Klees • T. W i t t w e r
322
K. H. I l k . A. Eicker • T. Mayer-Giirr
Chapter 49
Global Gravity Field Recovery by Merging Regional Focusing Patches: an Integrated Approach ........................................................
329
D. N. Arabelos • C. C. T s c h e r n i n g • M. Veicherts
Chapter 50
Chapter 51
External Calibration of GOCESGG Data with Terrestrial Gravity Data: a Simulation Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
337
J. P. van L o o n . J. Kusche T o w a r d s a n O p t i m a l C o m b i n a t i o n of S a t e l l i t e Data a n d Prior I n f o r m a t i o n
..........................................................
345
A. J/iggi • G. B e u t l e r . H. B o c k . U. H u g e n t o b l e r
Chapter 52
Kinematic and Highly Reduced Dynamic LEO Orbit Determination for Gravity Field Estimation ....................................................
354
VII
viii
Contents
Chapter 53
E Holota On the Combination of Terrestrial Gravity Data with Satellite Gradiometry and Airborne Gravimetry Treated in Terms of Boundary-Value Problems .....
362
Chapter 54
o. Nesvadba • E Holota • R. Klees A Direct Method and its Numerical interpretation in the Determination of the Earth's Gravity Field from Terrestrial Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
370
Chapter 55
u. Marti Comparison of High Precision Geoid Models in Switzerland . . . . . . . . . . . . . . . . . . .
377
Chapter 56
E Migliaccio. M. Reguzzoni. N. Tselfes GOCE: a Full-Gradient Simulated Solution in the Space-Wise Approach . . . . . . .
383
Chapter 57
Sz. R6zsa. Gy. T6th The Determination of the Effect of Topographic Masses on the Second Derivatives of Gravity Potential Using Various Methods . . . . . . .
Chapter 58
391
s. Bajracharya • M. G. Sideris Density Effects on Rudzki, RTM and Airy-Heiskanen Reductions
in Gravimetric Geoid Determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
397
P. Ditmar • X. Liu. R. Klees • R. Tenzer • P. Moore
Chapter 59
Combined Modeling of the Earth's Gravity Field from GRACE and GOCESatellite Observations: a Numerical Study . . . . . . . . . . . . . . . . . . . . . . . . . .
403
R. Tenzer. P. Moore. O. Nesvadba
Chapter 60
Chapter 61
Analytical Solution of Newton's Integral in Terms of Polar Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
410
c. Tocho • (3. Font. M. (3. Sideris A New High-Precision Gravimetric Geoid Model for Argentina ................
416
Gy. T6th. L. VGlgyesi
Chapter 62
Chapter 63
Local Gravity Field Modeling Using Surface Gravity Gradient Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
424
Part IV Earth Processes:Geodynamics,Tides, Crustal Deformation and Temporal Gravity Changes ...............................................
431
M. Amalvict • Y. Rogister. B. Luck. J. Hinderer Absolute Gravity Measurements in the Southern Indian Ocean . . . . . . . . . . . . . . .
433
j. Beavan. L. Wallace. H. Fletcher. A. Douglas
Chapter 64
Slow Slip Events on the Hikurangi Subduction Interface, New Zealand ....... 438
Chapter 65
A Geodetic Measurement of Strain Variation across the Central Southern Alps, New Zealand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
P. H. Denys • M. Denham • C. F. Pearson
445
B. Ducarme • A. P. Venedikov • A. R. de Mesquita • C. A. de Sampaio Franga • D. S. Costa. D. Blitzkow. R. Vieira Diaz. S. R. C. de Freitas
Chapter 66
New Analysis of a 50 Years Tide Gauge Record at CananGia (SP-Brazil) with the VAV Tidal Analysis Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
453
O. Gitlein • L. Timmen
Chapter 67
Chapter 68
Atmospheric Mass Flow Reduction for Terrestrial Absolute Gravimetry in the Fennoscandian Land Uplift Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
461
T. Jahr. (3. Jentzsch • H. Letz. A. Gebauer Tilt Observations around the KTB-Site/Germany: Monitoring and Modelling of Fluid Induced Deformation of the Upper Crust of the Earth ..... 467 W. Jiang. W. Kuang • B. Chao. M. Fang. C. Cox
Chapter 69
Understanding Time-Variable Gravity Due to Core Dynamical Processes with Numerical Geodynamo Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
473
R. Kiamehr
Chapter 70
A New Height Datum for Iran Based on the Combination of the Gravimetric and Geometric Geoid Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
480
Contents R. Klees • E. A. Zapreeva • H. C. Winsemius • H. H. G. Savenije
Chapter 71
Monthly Mean Water Storage Variations by the Combination of GRACE and a Regional Hydrological Model: Application to the Zambezi River ........ 488 A. Koohzare • P. Vanfcek • M. Santos
Chapter 72
The Use of Smooth Piecewise Algebraic Approximation in the Determination of Vertical Crustal Movements in Eastern Canada ...... 496 C. Kroner. T. Jahr. M. Naujoks • A. Weise
Chapter 73
Hydrological Signals in Gravity- Foe or Friend? ...............................
504
Chapter 74
S. M. Kudryavtsev Applications of t h e KSM03 Harmonic D e v e l o p m e n t of t h e Tidal Potential ....
511
Chapter 75
j. Kusche • E. J. O. Schrama • M. J. F. Jansen Continental Hydrology Retrieval from GPS Time Series and GRACE Gravity Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
517
Chapter 76
j. Miiller • M. Neumann-Redlin • F. Jarecki • H. Denker. O. Gitlein Gravity C h a n g e s in Northern Europe As Observed by GRACE . . . . . . . . . . . . . . . . . .
523
Chapter 77
M. Naujoks • T. Jahr. G. Jentzsch • J. H. Kurz. Y. H o f m a n n Investigations a b o u t E a r t h q u a k e Swarm Areas and Processes . . . . . . . . . . . . . . . .
528
K. Nawa. K. Satake • N. Suda. K. Doi. K. Shibuya • T. Sato
Chapter 78
Sea Level and Gravity Variations after the 2004 Sumatra Earthquake Observed at Syowa Station, Antarctica .........................................
536
J. Neumeyer. T. Schmidt • C. Stoeber
Chapter 79
Improved Determination of the Atmospheric Attraction with 3D Air Density Data and Its Reduction on Ground Gravity Measurements .... 541 H.-P. Plag. G. Blewitt • C. Kreemer • W. C. H a m m o n d
Chapter 80
Chapter 81
Solid Earth Deformations Induced by the Sumatra Earthquakes of 2004-2005 ...................................................................
549
I. Prutkin • R. Klees Environmental Effects in Time-Series of Gravity M e a s u r e m e n t s at t h e
Astrometric-Geodetic 0 b s e r v a t o r i u m W e s t e r b o r k (The Netherlands) .. 5 5 7
E. Rangelova • W. van der Wal. M. G. Sideris • P. Wu
Chapter 82
Chapter 83
Numerical Models of the Rates of Change of the Geoid and Orthometric Heights over Canada .........................................
563
S. Rosat Optimal Seismic Source M e c h a n i s m s to Excite t h e Slichter Mode . . . . . . . . . . . . .
571
s. Shimada • T. Kazakami
Chapter 84
Chapter 85
Recent Dynamic Crustal Movements in the Tokai Region, Central Japan, Observed by GPS Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W. Sun. S. O k u b o . G. Fu New Theory for Calculating Strains C h a n g e s Caused by
578
Dislocations
in a Spherically Symmetric Earth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
585
Part V Advances in the Realization of Global and Regional Reference Frames ... 593
Chapter 86
D. A n g e r m a n n • H. Drewes • M. Kriigel • B. Meisel Advances in Terrestrial Reference Frame C o m p u t a t i o n s . . . . . . . . . . . . . . . . . . . . . . .
595
Chapter 87
A. L. Fey The Status and Future of t h e International Celestial Reference Frame . . . . . . . .
603
Chapter 88
R. Ojha. A. L. Fey. D. L. Jauncey. J. E. J. Lovell • K. J. Johnston Is Scintillation t h e Key to a Better Celestial Reference Frame?
610
................
R. Ojha. A. L. Fey. P. Charlot • K. J. Johnston. D. L. Jauncey • J. E. Reynolds A. K. Tzioumis • J. E. J. Lovell • J. F. H. Quick. G. D. Nicolson • S. P. Ellingsen P. M. McCulloch • Y. Koyama
Chapter 89
Improvement and Extension of the International Celestial ReferenceFrame in the Southern Hemisphere ...................................................
616
IX
x
Contents
Chapter 90
J. B e a v a n . G. Blick L i m i t a t i o n s in t h e NZGD2000 D e f o r m a t i o n M o d e l . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
624
A. J o r d a n . P. D e n y s . G. Blick
Chapter 91
Implementing kocalised Deformation Models into a Semi-Dynamic Datum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
631
L. Sanchez
Chapter 92
Definition and Realisation of the SIRGASVertical Reference System within a Globally Unified Height System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
638
R. T. L u z . S. R. C. de Freitas • R. D a l a z o a n a • J. C. B~iez • A. S. Palmeiro
Chapter 93
Tests on Integrating Gravity and Leveling to Realize SIRGASVertical Reference System in Brazil . . . . . . . . . . . . . . . . . . . . . . . . .
646
L. P. S. F o r t e s . S. M. A. C o s t a . M. A. A. L i m a . J. A. Fazan • M. C. Santos
Chapter 94
Accessing the New SIRGAS2000 Reference Frame through a Modernized Brazilian Active Control Network . . . . . . . . . . . . . . . . . . . . . .
653
J. C. B~iez • S. R. C. de Freitas • H. Drewes • R. D a l a z o a n a • R. T. Luz
Chapter 95
Deformations Control for the Chilean Part of the SIRGAS2000 Frame ........ 660 C. C. C h a n g . H. C. H u a n g
Chapter 96
E s t i m a t i o n of H o r i z o n t a l M o v e m e n t F u n c t i o n for G e o d e t i c or M a p p i n g - 0 r i e n t e d M a i n t e n a n c e in t h e T a i w a n Area . . . . . . . . . . . . . . . . . . . . . . . .
665
M. C. Pacino • D. Del Cogliano • G. F o n t . J. M o i r a n o • P. Natalf • E. Laurfa • R. R a m o s • S. M i r a n d a
Chapter 97
Activities Related to the Materialization of a New Vertical System for Argentina . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
671
F. G. Nievinski • M. C. Santos
Chapter 98
An Analysis of Errors Introduced by the Use of Transformation Grids .......... 677 Z. A l t a m i m i • X. Collilieux • C. B o u c h e r
Chapter 99
P r e l i m i n a r y Analysis in View of t h e ITRF2005 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
685
K. Le Bail. M. Feissel-Vernier. J.-J. Valette. W. Z e r h o u n i
Chapter 100 Long Term Consistency of Multi-Technique Terrestrial Reference Frames, a Spectral Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
692
Part Vl
GGOS:Global Geodetic Observing System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 701 H. Drewes
Chapter 101 Science Rationale of the Global Geodetic Observing System (GGOS) .......... 703 H.-P. Plag
Chapter 102 GGOSand Its User Requirements, Linkage, and Outreach . . . . . . . . . . . . . . . . . . . . . 711 M. P e a r l m a n • Z. A l t a m i m i • N. B e c k . R. F o r s b e r g • W. G u r t n e r S. K e n y o n . D. B e h r e n d • F. G. L e m o i n e • C. M a . C. E. N o l l . E. C. Pavlis Z. M a l k i n • A. W. M o o r e • F. H. W e b b • R. E. N e i l a n • J. C. Hies M. R o t h a c h e r . P. Willis
Chapter 103 GGOSWorking Group on Ground Networks and Communications ............ 719 H.-P. P l a g . G. Beutler • R. Forsberg • C. M a . R. Neilan • M. P e a r l m a n B. R i c h t e r . S. Z e r b i n i
Chapter 104 Linking the Global Geodetic Observing System (GGOS) to the Integrated Geodetic Observing Strategy Partnership (IGOS-P) ......... 727 W. Schlfiter • D. B e h r e n d • E. H i m w i c h • A. N o t h n a g e l • A. Niell. A. W h i t n e y
Chapter 105 IVS High Accurate Products for the Maintenance of the Global Reference Frames As Contribution to GGOS . . . . . . . . . . . . . . . . . . . . . 735 M. P e a r l m a n • C. Noll. W. G u r t n e r • R. N o o m e n
Chapter 106 The International Laser Ranging Service and Its Support for GGOS ........... 741 M. P o u t a n e n . P. K n u d s e n . M. Lilje. T. N o r b e c h . H.-P. Plag H.-G. S c h e r n e c k
Chapter 107
T h e Nordic G e o d e t i c O b s e r v i n g S y s t e m (NGOS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
749
Contents A. Niell. A. W h i t n e y . W. P e t r a c h e n k o • W. Schliiter. N. V a n d e n b e r g . H. Hase Y. K o y a m a • C. M a . H. Schuh • G. Tuccari Chapter 108
VLBI2010: a Vision for F u t u r e G e o d e t i c VLBI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
757
J. I h d e . W. S6hne • W. Schwahn • H. Wilmes • H. W z i o n t e k • T. Kliigel • W. Schliiter
Chapter 109 Combination of Different Geodetic Techniques for Signal Detection a Case Study at Fundamental Station Wettzell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
760
Part VII Systemsand Methods for Airborne Mapping, Geophysics and Hazardsand DisasterMonitoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
767
H. K a h m e n . A. E i c h h o r n • M. H a b e r l e r - W e b e r
Chapter 110 A Multi-Scale Monitoring Concept for Landslide Disaster Mitigation .......... 769 H. Kutterer • C. Hesse
Chapter 111
High-Speed Laser Scanning for near Real-Time Monitoring of Structural Deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
776
H. N e u n e r
Chapter 112 A Method for Modelling the Non-Stationary Behaviour of Structures in Deformation Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
782
H. Z. Abidin • H. A n d r e a s • M. G a m a l • M. A. K u s u m a • M. H e n d r a s t o O. K. S u g a n d a • M. A. P u r b a w i n a t a • F. K i m a t a • I. Meilano
Chapter 113 Volcano Deformation Monitoring in Indonesia: Status, Limitations and Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
790
D. A. Grejner-Brzezinska • C. K. T o t h . S. M o a f i p o o r • E. Paska • N. Csanyi
Chapter 114 Vehicle Classification and Traffic Flow Estimation from Airborne Lidar/CCD Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
799
B. de Saint-Jean. J. V e r d u n . H. D u q u e n n e • J. P. Barriot • S. M e l a c h r o i n o s • J. Cali Fine Analysis of Lever Arm Effects in M o v i n g G r a v i m e t r y . . . . . . . . . . . . . . . . . . . . . .
809
Chapter 115
c. K. T o t h . N. C s a n y i . D. A. Grejner-Brzezinska
Chapter 116
I m p r o v i n g LiDAR-Based S u r f a c e R e c o n s t r u c t i o n Using G r o u n d Control . . . . . . .
817
G. w. R o b e r t s . X. M e n g . C. Brown
Chapter 117 The Use of GPS for Disaster Monitoring of Suspension Bridges . . . . . . . . . . . . . . . .
825
Part VIII AtmosphericStudies Using SpaceGeodeticTechniques . . . . . . . . . . . . . . . . . . . . 835 J. B o e h m • E J. M e n d e s Cerveira • H. Schuh • E Tregoning
Chapter 118 The Impact of Mapping Functions for the Neutral Atmosphere Based on Numerical Weather Models in GPS Data Analysis . . . . . . . . . . . . . . . . . . . .
837
G. Hulley • E. C. Pavlis • V. B. M e n d e s
Chapter 119 Validation of Improved Atmospheric Refraction Models for Satellite Laser Ranging (SLR) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
844
T. Tanaka
Chapter 120 Correlation Analyses of Horizontal Gradients of Atmospheric Wet Delay Versus Wind Direction and Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
853
M. A q u i n o • A. D o d s o n • J. Souter • T. M o o r e
Chapter 121
Ionospheric Scintillation Effects on GPS Carrier Phase Positioning Accuracy at Auroral and Sub-Auroral Latitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
859
D. A. G r e j n e r - B r z e z i n s k a • C.-K. H o n g . P. Wielgosz • L. H o t h e m
Chapter 122 The Impact of Severe Ionospheric Conditions on the GPS Hardware in the Southern Polar Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
867
Y. B. Y u a n . D. B. W e n . J. K. O u . X. L. H u o . R. G. Yang. K. F. Z h a n g • R. Grenfel
Chapter 123 Preliminary Research on Imaging the Ionosphere Using CIT and China Permanent GPS Tracking Station Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
876
XI
xII
Contents Part IX Geodesy of the Planets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
885
V. D e h a n t • T. Van H o o l s t
Chapter 124 Gravity, Rotation, and Interior of the Terrestrial Planets from Planetary Geodesy: Example of Mars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
887
G. B a l m i n o • J. C. M a r t y . J. D u r o n • O. K a r a t e k i n
Chapter 125 Mars Long Wavelength Gravity Field Time Variations: a New Solution from MGS Tracking Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
895
J. Miiller • J. G. W i l l i a m s . S. G. T u r y s h e v • P. J. Shelus
Chapter 126 Potential Capabilities of Lunar Laser Ranging for Geodesy and Relativity .... 903
Contributors
A b d - E l m o t a a l , H u s s e i n A. . (Chap. 39)
A r a b e l o s , D. N. • (Chap. 9, 50)
Civil Engineering Department, Faculty of Engineering, MiniaUniversity, Minia 61111, Egypt
Department of Geodesy and Surveying, Aristotle University of Thessaloniki, 54124 Thessaloniki, GreeceThessaloniki, Greece
A b i d i n , H. Z. • (Chap.
113)
A s t e r i a d i s , G. . ( C h a p . 9)
Department of Geodetic Engineering, Institute of Technology Bandung, J1. Ganesha 10, Bandung 40132, Indonesia,
[email protected] Department of Geodesy and Surveying, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece A u s t e n , G. . (Chap. 44)
A l b e r t s , B. A. . (Chap. 38)
Delft Institute of Earth Observation and Space Systems (DEOS), Delft University of Technology, Kluyverweg 1, PO Box 5058, 2600 GB Delft, TheNetherlands A l t a m i m i , Z. • (Chap. 99, lO3)
Institut Geographique National, ENSG/LAREG,6-8 avenue Blaise Pascal, 77455 Marne-la-Vallee, France
Stuttgart University, Geodetic Institute, Geschwister-SchollStr. 24/D, 70174 Stuttgart, Germany B d e z , ]. C . . (Chap. 12, 93, 95)
Geodetic Sciences Graduation Course (CPGCG), Federal University of Parami (UFPR), Curitiba, Brazil; and Department of Surveying, University of Concepci6n, Chile,
[email protected] B a j r a c h a r y a , S. . (Chap. 58)
Alves, D. B. M. . (Chap. 26)
Department of Cartography, Faculty of Science and Technology, S~o Paulo State University, FCT/UNESP, 305 Roberto Simonsen, Pres. Prudente, S~o Paulo, Brazil
Department of Geomatics Engineering, The University of Calgary, 2500 University Drive N. W., Calgary, Alberta, T2N 1N4, Canada,
[email protected], fax: 403-284-1980 B a k e r , T. E . (Chap. 4)
A m a l v i c t , M . . (Chap. 17, 63)
Institut de Physique du Globe de Strasbourg / t~cole et Observatoire des Sciences de la Terre (UMR 7516 CNRS-ULP), 5 rue Ren~ Descartes, 67000, Strasbourg, France,
[email protected]; and National Institute of Polar Research, 1-9-10 Kaga, Itabashi-ku, 173 8515, Tokyo, Japan A n d e r s e n , Ole B. . (Chap.
B a l m i n o , G. . (Chap.
125)
Centre National d'Etudes Spatiales, 18, Avenue Edouard Belin, 31401 Toulouse Cedex 9, France
1)
Danish National Space Center, Juliane Maries Vej 30, 2100 Kobenhavn O, Denmark A n d r e a s , H. . (Chap. 113)
Department of Geodetic Engineering, Institute of Technology Bandung, J1. Ganesha 10, Bandung 40132, Indonesia Angermann,
Proudman Oceanographic Laboratory, Joseph Proudman Building, 6 Brownlow Street, Liverpool L3 5DA, UK
D e t l e f . (Chap. 25, 86)
Deutsches Geod~itisches Forschungsinstitut (DGFI), Marstallplatz 8, 80539 Munich, Germany,
[email protected] B a r b o s a , S. M. . (Chap.
10)
Departamento de Matematica Aplicada, Faculdade de Ci~ncias, Universidade do Porto, Rua do Campo Alegre, 687, 4169007 Porto, Portugal B a r r i o t , J. P. . (Chap.
115)
Centre National d'Etudes Spatiales, 18, Avenue Edouard Belin, 31401 Toulouse Cedex 9, France B e a v a n , ]. • (Chap. 64, 90)
GNS Science, PO Box 30368, Lower Hutt, New Zealand A q u i n o , M. . (Chap.
121)
Institute of Engineering Surveying and Space Geodesy - IESSG, The University of Nottingham, University Park, Nottingham, UK, NG7 2RD
Beck, N. . (Chap. 103)
Geodetic Survey Division - Natural Resources Canada, Ottawa, ON K1A OE9, Canada
XIV
Contributors
B e h r e n d , D. • (Chap.
103, 105)
Bye, B e n t e Lilja . (Chap.
11)
NVI, Inc./NASA Goddard Space Flight Center, Code 697, Greenbelt, MD 20771-0001, USA
Geodesi, Norwegian Mapping Authority, 3507 Honefoss, Norway
Bergstrand, Sten . (Chap. 20)
Cali, J. . (Chap.
Onsala Space Observatory, Chalmers University of Technology, 439 92 Onsala, Sweden,
[email protected] ESGT, 1, Bd. Pythagore, 72000 Le Mans, France
115)
Chang, C. C. . (Chap. 96) Beutler, G. • (Chap. 19, 52, lO4)
Astronomical Institute, University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland, get-
[email protected] Department of Information Management, Yuda University, Miaoli 361, Taiwan, ROC Ckao, B. . (Chap. 69)
B i n g h a m , R o r y . (Chap. 1)
University of Reading, Environmental Systems Science Centre, PO Box 238, RG6 6AL Reading, UK
Space Geodesy Laboratory, NASA Goddard Space Flight Center, Greenbelt, MD 20771 Charlot, P. . (Chap. 89)
Bingley, R. M. . (Chap. 4)
Institute of Engineering Surveying and Space Geodesy, University of Nottingham, University Park, Nottingham NG7 2RD, UK Blewitt, G. . (Chap. 80)
Nevada Bureau of Mines and Geology and Seismological Laboratory, University of Nevada, Reno, Mailstop 178, Reno, NV 89557, USA
Observatoire de Bordeaux (OASU) CNRS/UMR 5804, BP89, 33270 Floirac, France Cken, C. S. . (Chap. 34)
Institute of Geomatics and Disaster Prevention Technology, Ching Yun University, No. 229, ]iansing Rd., ]hongli 320, Taiwan, R.O.C.,
[email protected] Cheng, M. K. . (Chap. 41)
Center for Space Research, University of Texas at Austin, 3925 West Braker Ln. #200, Austin, Texas 78759, USA
Blick, G.. (Chap. 90, 91)
Land Information New Zealand, Private Box 5501, Wellington, New Zealand
Chinn, D. S. . (Chap. 35)
B l i t z k o w , D. • (Chap. 40, 66)
SGT Inc., 7701 Greenbelt Road, Greenbelt, Maryland 20770, USA
Escola Polit&nica da Universidade de S~o Paulo, EPUSP-PTR, Code Postal 61548, CEP:05424-970, S~o Paulo, Brazil,
[email protected]; FAX: 55 11 30915716
Cho, l u n g - h o . (Chap. 24)
Bock, H. . (Chap. 52)
Geodetic Institute of the University of Bonn, Nussallee 17, 53115 Bonn, Germany; on leave from Korean Astronomy and Space Science Institute, 61-1, Whaam-Dong, Youseong-Gu, Taejeon, Rep. of Korea 305-348
Astronomical Institute, University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland B o e h m , ]. • (Chap. 32,
Cku, Y o n g k a i . (Chap.
118)
Institute of Geodesy and Geophysics (IGG), Vienna University of Technology, Gusshausstrasse 27-29, 1040 Vienna, Austria
14)
School of Geodesy and Geomatics, Wuhan University, 129 Luoyu Road, Wuhan, 430079, China Cintra, J. P.. (Chap. 4o)
Bosck, W o l f g a n g . (Chap. 8, 21)
Deutsches Geod/itisches Forschungsinstitut (DGFI), Marstallplatz 8, 80539 Mtinchen, Germany
Escola Polit&nica da Universidade de S~o Paulo, EPUSP-PTR, Code Postal 61548, CEP: 05424-970, S~o Paulo, S~o Paulo, Brazil,
[email protected]; FAX: 55 11 30915716
Boucker, C. . (Chap. 99)
C o l e m a n , R. . (Chap. 7)
Conseil g~n~ral des ponts et chauss~es, tour Pascal B, 92055 La D~fense, France
School of Geography and Environmental Studies, University of Tasmania, Private Bag 78, Hobart, Tasmania, Australia, 7001
B r a d s k a w , E l i z a b e t k . (Chap.
11)
British Oceanographic Data Centre, Joseph Proudman Building, 6 Brownlow St., Liverpool, L3 5DA, UK
Collilieux, X. . (Chap. 99)
Institut Geographique National, ENSG/LAREG,6-8 avenue Blaise Pascal, 77455 Marne-la-Vallee, France
B r o w n , Chris. (Chap. 117)
C o n t a d a k i s , M. E. • (Chap. 9)
School of Engineering and Design, Brunel University West London, Uxbridge, Middlesex, UB8 3PH, UK
Department of Geodesy and Surveying, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece
Contributors
Costa, D. S. • (Chap. 66)
D e n y s , P. H. . (Chap. 65, 91)
Escola. Polit&nica, Universidade de S~o Paulo, Caixa Postal 61548, 05413-001 Silo Paulo, SP, Brasil
School of Surveying, University of Otago, PO Box 56, Dunedin, New Zealand
Costa, S. M. A. • (Chap. 94)
D i t m a r , P. • (Chap. 38, 59)
Directorate of Geosciences, Brazilian Institute of Geography and Statistics, Av. Brasil 15671, Rio de Janeiro, RJ, Brazil, 21241-051
Delft Institute of Earth Observation and Space Systems (DEOS), Delft University of Technology, Kluyverweg 1, PO Box 5058, 2600 GB Delft, TheNetherlands
121)
Cox, C. M. . (Chap. 35, 69)
D o d s o n , A. H. • (Chap. 4,
Raytheon at Space Geodesy Laboratory, NASA Goddard Space Flight Center; and Raytheon ITSS, 1616 McCormick Drive, Upper Marlboro, Maryland 20774, USA
Institute of Engineering Surveying and Space Geodesy - IESSG, University of Nottingham, University Park, Nottingham, NG7 2RD, UK
Csanyi, N o r a . (Chap. 114, 116)
Doi, K o i c h i r o . (Chap. 16, 78)
Department of Civil and Environmental Engineering and Geodetic Science, The Ohio State University, 2070 Neil Avenue, Columbus, Ohio 43210
National Institute of Polar Research, Kaga 1-chome, Itabashi-ku, Tokyo 173-8515, Japan D o u g l a s , A. . (Chap. 64)
Csap6, G. . (Chap. 31)
E6tv6s Lor~ind Geophysical Institute of Hungary, 1145 Budapest, Hungary, Kolumbusz u. 17-23
School of Earth Sciences, Victoria University of Wellington, PO Box 600, Wellington, New Zealand Drange, H e l g e . (Chap. 1)
D a l a z o a n a , R . . (Chap. 12, 93, 95)
Geodetic Sciences Graduation Course, Federal University of Parami (UFPR), Curitiba, Brazil,
[email protected] Nansen Environmental and Remote Sensing Center, Edvard Griegsvei 3a, 5059 Bergen, Norway Drecourt, J e a n - P h i l i p p e . (Chap. 1)
de Freitas, S. R. C. • (Chap. 12, 66, 93, 95)
Geodetic Sciences Graduation Course, Federal University of Parami (UFPR), Curitiba, Brazil,
[email protected] University of Reading, Environmental Systems Science Centre, PO Box 238, RG6 6AL Reading, UK Drewes, H e r m a n n .
de M a t o s , A. C. O. C. • (Chap. 4o)
Escola Polit4cnica da Universidade de S~o Paulo, EPUSP-PTR, Code Postal 61548, CEP: 05424-970, S~o Paulo, S~o Paulo, Brazil,
[email protected], FAX: 55 11 30915716 de M e s q u i t a , A. R. . (Chap. 66)
(Chap. 86, 95,101)
Deutsches Geod~itisches Forschungsinstitut (DGFI), Marstallplatz 8, 80539 Munich, Germany,
[email protected] D u c a r m e , B. • (Chap. 66)
Chercheur qualifi4 FNRS, Observatoire Royal de Belgique, Av. Circulaire 3, 1180, Bruxelles, Belgique
Instituto Oceanogr~ifico da Universidade de S~o Paulo, SP, Brasil D u q u e n n e , H. . (Chap. de S a m p a i o F r a n f a , C. A. . (Chap. 66)
115)
Instituto Oceanogr~ifico da Universidade de S~o Paulo, SP, Brasil
IGN-LAREG, 6/8 av. Blaise Pascal, Champs sur Marne, 77455 Marne la Vall4e Cedex 2, France
D e h a n t , V.. (Chap. 124)
D u r o n , J. . (Chap. 125)
Royal Observatory of Belgium, Av. Circulaire 3, 1180, Brussels, Belgium;
[email protected] Observatoire Royal de Belgique, 3, Avenue Circulaire, 1180 Brussels, Belgium; presently at CNES, Toulouse
Del Cogliano, D. . (Chap. 97)
E i c h h o r n , A. . (Chap.
Facultad de Ciencias Astron6micas y Geof~sicas, Universidad Nacional de La Plata, Paseo del Bosque s/n, (1900) La Plata, Argentina,
[email protected] Institute of Geodesy and Geophysics, Vienna University of Technology, Gusshausstrasse 27-29/E1283, 1040 Vienna, Austria
D e n h a m , M. . (Chap. 65)
Eicker, A. • (Chap. 49)
School of Surveying, University of Otago, PO Box 56 Dunedin, New Zealand
Institute of Theoretical Geodesy, University of Bonn, Nussallee 17, 53115 Bonn, Germany
D e n k e r , H. • (Chap. 76)
Ellingsen, S. P. • (Chap. 89)
Institut ffir Erdmessung, University of Hannover, Schneiderberg 50, 30167 Hannover, Germany
School of Mathematics and Physics, University of Tasmania, Private Bag 37, Hobart, Tasmania 7001, Australia
110)
XV
XVI
Contributors
Fan, C h u n b o . (Chap. 14)
Fu, G u a n g y u . (Chap. 85)
School of Geodesy and Geomatics, Wuhan University, 129 Luoyu Road, Wuhan, 430079, China
Earthquake Research Institute, University of Tokyo, Tokyo, Japan
Fang, M. . (Chap. 69)
Fukuda, Y. . (Chap. 16)
Department of Earth and Space Sciences, Massachusetts Institute of Technology, Cambridge, MA 02136
Graduate School of Science, Kyoto University, Sakyo-ku, Kyoto 606-8502, Japan
Fazan, ]. A. . (Chap. 94)
Gamal, M. . (Chap. 113)
Directorate of Geosciences, Brazilian Institute of Geography and Statistics, Av. Brasil 15671,Rio de Janeiro, RJ, Brazil, 21241-051
Department of Geodetic Engineering, Institute of Technology Bandung, J1. Ganesha 10, Bandung 40132, Indonesia
Feissel-Vernier, M. . (Chap.
1OO)
Observatoire de Paris/SYRTE and Institut G~ographique National/LAREG, 8 Av. Blaise Pascal, Champs sur Marne, 77455 Marne la Vall~e Cedex 2, France Fernandes, M. ].. (Chap.
10)
Departamento de Matematica Aplicada, Faculdade de Ci4ncias, Universidade do Porto, Rua do Campo Alegre, 687, 4169007 Porto, Portugal Fey, A. L. . (Chap. 87, 88, 89)
u. s. Naval Observatory, 3450 Massachusetts Avenue, NW, Washington, DC 20392-5420 USA
Garcia, Maria ]esfis. (Chap. 11)
Instituto Espafiol de Oceanografla, Coraz6n de Marla, 8, 28002 Madrid, Spain Gebauer, A. . (Chap. 68)
Institute of Geosciences, Department of Applied Geophysics, University of Jena, Burgweg 11, 07749 Jena, Germany Gitlein, 0 . . (Chap. 67, 76)
Institut ffir Erdmessung, University of Hannover, Schneiderberg 50, 30167 Hannover, Germany G6mez, Begofia Pdrez. (Chap. n )
puertos del Estado, Area de Medio Fisico y Tecnologla de las Infraestructuras, Avda Parten6n, 10, 28042 Madrid, Spain
Flechtner, F.. (Chap. 36)
Department 1, Geodesy and Remote Sensing, GeoForschungsZentrum Potsdam (GFZ), Telegrafenberg A17, 14473 Potsdam, Germany
Grejner-Brzezinska, Dorota A. • (Chap. 114,116, 122)
Fletcher, H. . (Chap. 64)
Satellite Positioning and Inertial Navigation (SPIN) Lab, Department of Civil and Environmental Engineering and Geodetic Science, The Ohio State University, 470 Hitchcock Hall, Columbus, OH 43210-1275, USA
GNS Science, PO Box 30368, Lower Hutt, New Zealand F6h, H e n n i n g P. • (Chap.
1)
Grenfel, R. . (Chap. 123)
Danish National Space Center, Juliane Maries Vej 30, 2100 Kobenhavn O, Denmark
School of Mathematical and Geospatial Sciences, RMIT University, Australia
F6ldvdry, L. . (Chap. 31)
Grigoriadis, V. N. . (Chap. 37, 46)
Department of Geodesy and Surveying, Budapest University of Technology and Economics; Research Group of Physical Geodesy and Geodynamics of the Hungarian Academy of Sciences, 1521 Budapest, Hungary
Department of Geodesy and Surveying, Aristotle University of Thessaloniki, University Box 440, 541 24, Thessaloniki, Greece, fax: +30 231 0995948 Gfintner, A. • (Chap. 36)
Font, G.. (Chap. 61, 97)
Facultad de Ciencias Astron6micas y Geofisicas,Universidad Nacional de La Plata, Paseo del Bosque s/n, (1900) La Plata, Argentina
Department 5, Geoengineering, Section 5.4, Engineering Hydrology, GeoForschungsZentrum Potsdam (GFZ), Telegrafenberg Haus F, 14473 Potsdam, Germany
Forsberg, Ren~. (Chap. 1, 103, 104)
Gurtner, W.. (Chap. 19, lO3, lO6)
Danish National Space Center, Juliane Maries Vej 30, 2100 Kobenhavn O, Denmark,
[email protected] Astronomical Institute, University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland
Fortes, L. P. S. • (Chap. 94)
Haas, Ri~diger. (Chap. 24)
Directorate of Geosciences, Brazilian Institute of Geography and Statistics, Av. Brasil 15671,Rio de ]aneiro, R], Brazil, 21241-051
Onsala Space Observatory, Chalmers University of Technology, 439 92 Onsala, Sweden
Fotopoulos, G. . (Chap. 41)
Haberler- Weber, M. • (Chap. 11o)
Department of Civil Engineering, University of Toronto, 35 St. George Street, Toronto, ON, M5S1A4, Canada
Institute of Geodesy and Geophysics, Vienna University of Technology, Gusshausstrasse 27-29/E1283, 1040 Vienna, Austria
Contributors
Haines, Keith . (Chap. 1)
Hong, C.-K.. (Chap. 122)
University of Reading, Environmental Systems Science Centre, PO Box 238, RG6 6AL Reading, UK
Satellite Positioning and Inertial Navigation (SPIN) Lab, Department of Civil and Environmental Engineering and Geodetic Science, The Ohio State University, 470 Hitchcock Hall, Columbus, OH 43210-1275, USA
H a m m o n d , W. C. • (Chap. 8o)
Nevada Bureau of Mines and Geology and Seismological Laboratory, University of Nevada, Reno, Mailstop 178, Reno, NV 89557, USA
Hothem, L. . (Chap. 122)
United States Geological Survey, 521 National Center, 12201 Sunrise Valley Dr., Reston, VA 20192 USA
Hase, H. . (Chap. 1o8)
Bundesamt ffir Kartographie und Geod~isie,Observatorio Geod4sico TIGO, Casilla 4036, Correo 3, Concepci6n, Chile Hendrasto, M. . (Chap. 113)
Directorate of Vulcanology and Geological Hazard Mitigation, J1. Diponegoro 57, Bandung, Indonesia
Hsu, Houtse. (Chap. 3)
Institute of Geodesy and Geophysics, Chinese Academy of Sciences, 340 Xudong Street, Wuhan, China, 430077; and Unite Center for Astro-geodynamics Research, CAS, Shanghai, China, 200030 Huang, H. C.. (Chap. 96)
Hernandez, Fabrice. (Chap. 1)
The 401 st Factory, Armaments Bureau, Taichung 402, Taiwan, R.O.C.
Collecte Localisation Satellites, Space Oceanography Division, 8-10 Rue Hermes - Parc Technologique Du Canal, 31526 Ramonville St. Agne, France
Huang, ].. (Chap. 41)
Hesse, Christian. (Chap. 111)
Geodetic Survey Division, CCRS, Natural Resources Canada, 615 Booth Street, Ottawa, Ontario, K1A 0E9, Canada
Geod~itischesInstitut, Universit~itHannover, Nienburger Strasse 1, 30167 Hannover, Germany,
[email protected] Huff, R. . (Chap. 5)
Himwich, E. . (Chap. lO5)
Cooperative Institute for Research in Environmental Sciences, University of Colorado, Campus Box 216, Boulder, CO 803090216, USA
NVI, Inc./NASA Goddard Space Flight Center, Code 697, Greenbelt, MD 20771-0001, USA
Hugentobler, U. • (Chap. 19, 52)
Hinderer, ].. (Chap. 63)
Astronomical Institute, University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland
Institut de Physique du Globe de Strasbourg/l~cole et Observatoire des Sciences de la Terre, 5 rue Ren4 Descartes, 67000, Strasbourg, France
Hulley, G. . (Chap. 119)
Hipkin, Roger. (Chap. 1)
The University of Edinburgh, School of GeoSciences, West Mains Road, EH9 3JW Edinburgh, UK
Joint Center for Earth Systems Technology (JCET), University of Maryland Baltimore County, Baltimore, MD, USA Hunegnaw, Addisu . (Chap. 1)
Hiraoka, Y. . (Chap. 16)
The University of Edinburgh, School of GeoSciences, West Mains Road, EH9 3JW Edinburgh, UK
Geographical Survey Institute, 1, Kitasato, 305-0811, Japan
Huo, X. L. . (Chap. 123)
Hirt, Christian. (Chap. 47)
Institute of Geodesy and Geophysics, Chinese Academy of Sciences, Wuhan 430077, China
Institut ffir Erdmessung, Universit/it Hannover, Schneiderberg 50, 30167 Hannover, Germany,
[email protected]_hannover.de, fax: +495117624006
Ihde, ].. (Chap. lO9)
Hobiger, T.. (Chap. 3o)
Vienna University of Technology, Institute of Geodesy and Geophysics, Research Unit Advanced, Geodesy, Gusshausstrasse 27-29, 1040 Vienna, Austria
Federal Agency for Cartography and Geodesy, Richard-StraussAllee 11, 60598 Frankfurt am Main, Germany Ilk, K. H. . (Chap. 42, 49)
Institute of Theoretical Geodesy, University of Bonn, Nussallee 17, 53115 Bonn, Germany
Hofmann, Y.. (Chap. 77)
ROSEN Technology GmbH, Lingen, Germany
Isaksen, Espen. (Chap. 11)
Geodesi, Norwegian Mapping Authority, 3507 Honefoss, Norway Holota, P.. (Chap. 53, 54)
Research Institute of Geodesy, Topography and Cartography, 25066 Zdiby 98, Praha-wchod, Czech Republic,
[email protected],tel.: +420 323649235,fax: +420 284890056
Ji~ggi, A. . (Chap. 52)
Astronomical Institute, University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland,
[email protected] XVII
XVIII
Contributors
]ahr, T.. (Chap. 68, 73, 77)
K a r a t e k i n , 0 . . (Chap. 125)
Institute of Geosciences, Department of Applied Geophysics, University of Jena, Burgweg 11, 07749 Jena, Germany
Observatoire Royal de Belgique, 3, Avenue Circulaire, 1180 Brussels, Belgium
Jansen, M. J. E . (Chap. 75)
K a z a k a m i , T. . (Chap. 84)
DEOS, TU Delft, Kluyverweg 1, PO Box 5058, 2600 GB Delft, The Netherlands
Solid Earth Research Group, National Research Institute for Earth Science and Disaster Prevention (NIED), 3-1 Tennodai, Tsukuba-Shi, Ibaraki-Ken 305-0006 Japan
Jarecki, F.. (Chap. 76)
Institut for Erdmessung, University of Hannover, Schneiderberg 50, 30167 Hannover, Germany
Keller, W. . (Chap. 44)
Stuttgart University, Geodetic Institute, Geschwister-SchollStr. 24/D, 70174 Stuttgart, Germany
]auncey, D. L. • (Chap. 88, 89)
Australia Telescope National Facility, CSIRO, PO Box 76, Epping, NSW 1710, Australia
K e n y o n , S. . (Chap. 103)
National Geospatial-IntelligenceAgency,Arnold, MO 63010-6238, USA
Jekeli, C. . (Chap. 43)
Division of Geodetic Science, School of Earth Sciences, Ohio State University, 125 South Oval Mall, Columbus, OH 43210
K i a m e h r , R. . (Chap. 70)
Department of Infrastructure, Division of Geodesy, Royal Institute of Technology (KTH), 100-44 Stockholm, Sweden
]entzsch, G. . (Chap. 68, 77)
Institute of Geosciences, Department of Applied Geophysics, University of Jena, Burgweg 11, 07749 Jena, Germany
K i m a t a , F. . (Chap. 113)
Research Center for Seismology and Volcanology and Disaster Mitigation (RCSVDM), Nagoya University, Japan
fiang, W. . (Chap. 69)
Joint Center for Earth Systems Technology,University of Maryland at Baltimore County, 1000 Hilltop Circle, Baltimore, MD 21229, USA
Klees, R. • (Chap. 38, 45, 48, 54, 59, 71, 81)
Delft Institute of Earth Observation and Space Systems (DEOS), Delft University of Technology,Kluyverweg 1, PO Box 5058, 2600 GB Delft, TheNetherlands
fiang, W e i p i n g . (Chap. 14)
School of Geodesy and Geomatics, Wuhan University, 129 Luoyu Road, Wuhan, 430079, China
Klosko, S. M. . (Chap. 35)
SGT Inc., 7701 Greenbelt Road, Greenbelt, Maryland 20770, USA
] o h a n n e s s e n , J o h n n y A. . (Chap. 1)
Nansen Environmental and Remote Sensing Center, Edvard Griegsvei 3a, 5059 Bergen, Norway
Kliigel, T.. (Chap. lO9)
Federal Agency for Cartography and Geodesy, Sackenrieder Strafle 25, 93444 K6tzting, Germany
]ohansson, Jan M. . (Chap. 20)
Onsala Space Observatory, Chalmers University of Technology, 439 92 Onsala, Sweden
K n u d s e n , Per. (Chap. 1, 2, lO7)
Danish National Space Center, Juliane Maries Vej 30, 2100 Kobenhavn 0, Denmark
Johnston, K. ].. (Chap. 88, 89)
u. s. Naval Observatory, 3450 Massachusetts Avenue, NW, Washington, DC 20392-5420, USA
Koivula, H a n n u . (Chap. 13)
Finnish Geodetic Institute, Geodeetinrinne 2, 02430 Masala, Finland
Jordan, A. . (Chap. 91)
Land Information New Zealand, Private Box 5501, Wellington, New Zealand
Kolaczek, B. . (Chap. 32)
Space Research Centre of the PAS, Bartycka 18a, Warsaw, Poland
K a h m e n , H. . (Chap. 110)
KOnig, R. . (Chap. 36)
Institute of Geodesy and Geophysics, Vienna University of Technology, Gusshausstrasse 27-29/E1283, 1040 Vienna, Austria
Department 1, Geodesy and Remote Sensing, GeoForschungsZentrum Potsdam (GFZ), Telegrafenberg A17, 14473 Potsdam, Germany
K a l a m p o u k a s , G. . (Chap. 46)
Koohzare, A z a d e h . (Chap. 72)
Department of Geodesy and Surveying, Aristotle University of Thessaloniki, University Box 440, 541 24, Thessaloniki, Greece, fax: +30 231 0995948
Department of Geodesy and Geomatics Engineering, University of New Brunswick, PO Box 4400, Fredericton NB, Canada E3B 5A3
Contributors
Koyama, Y.. (Chap. 89,
108)
Leandro, R. F. • (Chap. 28, 29)
Kashima Space Research Center, Communications Research Laboratory, 893-1 Hirai, Kashima, Ibaraki 314-8501, Japan
Department of Geodesy and Geomatics Engineering, University of New Brunswick, PO Box 4400, Fredericton, New Brunswick, Canada, E3B 5A3
Kreemer, C.. (Chap. 80) Nevada Bureau of Mines and Geology and Seismological Laboratory, University of Nevada, Reno, Mailstop 178, Reno, NV 89557, USA
Lemoine, F. G. • (Chap. 23, 35, lO3) NASA Goddard Space Flight Center, Code 697, Greenbelt, Maryland 20771, USA
Kroner, C.. (Chap. 73)
Letz, H. . (Chap. 68)
Institute of Geosciences, Friedrich-Schiller-University Jena, Burgweg 11, 07749 Jena, Germany
Institute of Geosciences, Department of Applied Geophysics, University of Jena, Burgweg 11, 07749 Jena, Germany
Krfigel, M a n u e l a . (Chap. 25, 86) Deutsches Geod~itisches Forschungsinstitut (DGFI), Marstallplatz 8, 80539 Munich, Germany, kruegel@dg?.badw.de
Li, J . . ( C h a p . 14) School of Geodesy and Geomatics, Wuhan University, 129 Luoyu Road, Wuhan, 430079, China
Kuang, W. . (Chap. 69) Space Geodesy Laboratory, NASA Goddard Space Flight Center, Greenbelt, MD 20771 Kudryavtsev, S. M. • (Chap. 74) Sternberg Astronomical Institute of Moscow State University, 13 Universitetsky Pr., Moscow, 119992, Russia
Lidberg, M a r t i n . (Chap. 2o) Onsala Space Observatory, Chalmers University of Technology, 439 92 Onsala, Sweden; and Geodesy Division, Lantm~iteriet, 801 82 G~ivle,Sweden Lilje, M i k a e l . (Chap. lO7) National Land Survey of Sweden, 801 82 G~ivle,Sweden
Ki~htreiber, Norbert . (Chap. 39) Institute of Navigation and Satellite Geodesy, TU-Graz, Steyrergasse 30, 8010 Graz, Austria Kurz, ]. H. . (Chap. 77) Institute of Construction Materials, University of Stuttgart, Germany Kusche, ].. (Chap.
51, 75)
DEOS, TU Delft, Kluyverweg 1, PO Box 5058, 2600 GB Delft, The Netherlands Kusuma, M. A. • (Chap.
Department of Geodetic Engineering, Institute of Technology Bandung, J1. Ganesha 10, Bandung 40132, Indonesia
111)
Geodetic Institute, University of Hannover, Nienburger Strasse 1, 30167 Hannover, Germany,
[email protected] Larnicol, Gilles. (Chap.
1)
Collecte Localisation Satellites, Space Oceanography Division, 8-10 Rue Hermes - Parc Technologique Du Canal, 31526 Ramonville St. Agne, France Lauria, E. . (Chap. 97) Instituto Geogr~ifico Militar, Cabildo 391, Buenos Aires, Argentina,
[email protected] Le Bail, K. . (Chap.
Directorate of Geosciences, Brazilian Institute of Geography and Statistics, Av. Brasil 15671, Rio de Janeiro, RJ, Brazil, 21241-051 Liu, ]ingnan . (Chap. 33) President, Wuhan University, Luojia Hill, Wuhan 430072, China; and GPS Engineering Research Center, Wuhan University, 129 Luoyu Road, Wuhan 430079, China Liu, X. . (Chap. 59)
113)
Kutterer, Hansj6rg. (Chap. 22,
Lima, M. A. A. • (Chap. 94)
1OO)
Institut G6ographique National/LAREG and Observatoire de la C6te d'Azur/GEMINI (UMR 6203), 8 Av. Blaise Pascal, Champs sur Marne, 77455 Marne la Vall6e Cedex 2, France
Delft Institute of Earth Observation and Space Systems (DEOS), Delft University of Technology, Kluyverweg 1, 2629 HS Delft, The Netherlands LOcher, A. . (Chap. 42) Institute of Theoretical Geodesy, University of Bonn, Nussallee 17, 53115 Bonn, Germany Lovell, ]. E. ].. (Chap. 88, 89) Australia Telescope National Facility, CSIRO, PO Box 76, Epping, NSW 1710, Australia Lu, Yang. (Chap. 3) Institute of Geodesy and Geophysics, Chinese Academy of Sciences, 340 Xudong Street, Wuhan, China, 430077; and Unite Center for Astro-geodynamics Research, CAS, Shanghai, China, 200030 Luck, B. . (Chap. 63) Institut de Physique du Globe de Strasbourg/t~cole et Observatoire des Sciences de la Terre, 5 rue Ren6 Descartes, 67000, Strasbourg, France
XIX
XX
Contributors
Luthcke, S. B. . (Chap. 35)
Meng, X i a o l i n . (Chap. 117)
NASA Goddard Space Flight Center, Code 697, Greenbelt, Maryland 20771, USA
Institute of Engineering Surveying and Space Geodesy, The University of Nottingham, University Park, Nottingham, NG7 2RD, UK
Luz, R. T. • (Chap. 12, 93, 95)
Department of Geomatics, Geodetic Sciences Graduation Course, Federal University of Paranfi (UFPR), Curitiba, Paranfi, Brazil; and Coordination of Geodesy, Brazilian Institute of Geography and Statistics (IBGE), Brazil,
[email protected] Meyer, Ul. . (Chap. 36)
Department 1, Geodesy and Remote Sensing, GeoForschungsZentrum Potsdam (GFZ), Telegrafenberg A17, 14473 Potsdam, Germany Meyers, G. . (Chap. 7)
Ma, C.. (Chap. 103, 104, 108)
NASA Goddard Space Flight Center, Greenbelt MD 20771-0001, USA,
[email protected] CSIRO Marine and Atmospheric Research, GPO Box 1538, Hobart, Tasmania, Australia, 7001 Migliaccio, F.. (Chap. 56)
Malkin, Z. . (Chap. lO3)
Institute of Applied Astronomy, St. Petersburg, 191187, Russia
DIIAR, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy
Marti, Urs. (Chap. 55)
Miranda, S. . (Chap. 97)
Federal Office of Topography, Seftigenstrasse 264, 3084 Wabern, Switzerland,
[email protected] Facultad de Ciencias Exactas, Hsicas y Naturales, Universidad nacional de San Juan San Juan, Argentina,
[email protected] Martin, M. C.. (Chap. 15)
Marine Science Institute, University of the Philippines, 1101 Diliman, Quezon City, Philippines Mart),, ]. C. . (Chap. 125)
Moafipoor, Shahram . (Chap. 114)
Department of Civil and Environmental Engineering and Geodetic Science, The Ohio State University, 2070 Neil Avenue, Columbus, Ohio 43210
Centre National d'Etudes Spatiales, 18, Avenue Edouard Belin, 31401 Toulouse Cedex 9, France
Moirano, ].. (Chap. 97)
Mayer-Gfirr, T. • (Chap. 49)
Facultad de Ciencias Astron6micas y Geoffsicas, Universidad Nacional de La Plata, Paseo del Bosque s/n, (1900) La Plata, Argentina
Institute of Theoretical Geodesy, University of Bonn, Nussallee 17, 53115 Bonn, Germany McCulloch, P. M. . (Chap. 89)
School of Mathematics and Physics, University of Tasmania, Private Bag 37, Hobart, Tasmania 7001, Australia
Monico, ]. F. G. • (Chap. 26, 27)
Department of Cartography, Faculty of Science and Technology, S~o Paulo State University, FCT/UNESP, 305 Roberto Simonsen, Pres. Prudente, S~o Paulo, Brazil Moore, A. W. . (Chap. lO3)
Meilano, I r w a n . (Chap. 113)
Research Center for Seismology and Volcanology and Disaster Mitigation (RCSVDM), Nagoya University, Japan Meisel, Barbara. (Chap. 86)
Deutsches Geod~itisches Forschungsinstitut (DGFI), Marstallplatz 8, 80539 Munich, Germany
Jet Propulsion Laboratory, California Institute of Technology, Pasadena CA 91109, USA Moore, P. • (Chap. 59, 60)
School of Civil Engineering and Geosciences, University of Newcastle upon Tyne, Newcastle upon Tyne, NE 17RU, UK Moore, T.. (Chap. 121)
Melachroinos, S. . (Chap. 115)
CNES, 18 av. E. Belin, 31401 Toulouse Cedex 9, France
Institute of Engineering Surveying and Space Geodesy - IESSG, The University of Nottingham, University Park, Nottingham, UK, NG7 2RD
Mendes, V. B. . (Chap. 119)
Laboratorio de Tectonofisica e Tectonica Experimental and Departamento de Matematica, Faculdade de Ciencias da Universidade de Lisboa, Lisbon, Portugal
Mfiller, Jiirgen . (Chap. 76,126)
Institut ffir Erdmessung, University of Hannover, Schneiderberg 50, 30167 Hannover, Germany, mueller@ife'uni-hann°ver'de
Mendes Cerveira, P.. ]. • (Chap. 30,118)
Vienna University of Technology, Institute of Geodesy and Geophysics, Research Unit Advanced, Geodesy, Gusshausstrasse 27-29, 1040 Vienna, Austria
Nastula, ].. (Chap. 32)
Space Research Centre of the PAS, Bartycka 18a, Warsaw, Poland
Contributors
Natali, P.. (Chap. 97)
Nievinski, F. G. . (Chap. 98)
Facultad de Ciencias Astron6micas y Geofisicas, Universidad Nacional de La Plata, Paseo del Bosque s/n, (1900) La Plata, Argentina
Department of Geodesy and Geomatics Engineering, University of New Brunswick, PO Box 4400, Fredericton, New Brunswick, Canada, E3B 5A3
Naujoks, M. . (Chap. 73, 77)
Noll, C. F,. . (Chap. lO3, lO6)
Institute of Geosciences, Friedrich-Schiller-University Jena, Burgweg 11, 07749 Jena, Germany
NASA Goddard Space Flight Center, Greenbelt MD 20771-0001, USA
Nawa, K a z u n a r i . (Chap. 78)
N o o m e n , R. . (Chap. 1o6)
Geological Survey of Japan, AIST,AIST Tsukuba Central 7, Higashi 1-1-1, Tsukuba, Ibaraki 305-8567, Japan
Facility of Aerospace Engineering, Delft University of Technology, 2629 HS Delft, The Netherlands
Neilan, R. E. . (Chap. lO3, lO4)
Norbech, Torbjorn. (Chap. lO7)
Jet Propulsion Laboratory, California Institute of Technology, Pasadena CA 91109, USA,
[email protected] Norwegian Mapping Authority, 3507 Honefoss, Norway Nothnagel, A x e l . (Chap. 24, lO5)
Nesvadba, 0 . . (Chap. 54, 60)
Delft Institute of Earth Observation and Space Systems (DEOS), Delft University of Technology, 2629 HS Delft, Kluyverweg 1, The Netherlands; and Research Institute of Geodesy, Topography and Cartography Zdiby 98, Praha v~chod, 25066, The Czech Republic; and Land Survey Office, Pod sldli~t~em 9, Praha 8, 18211, The Czech Republic
Geodetic Institute of the University of Bonn, Nussallee 17, 53115 Bonn, Germany Ojha, R . . (Chap. 88, 89)
Australia Telescope National Facility, CSIRO, PO Box 76, Epping, NSW 1710, Australia Okubo, S h u h e i . (Chap. 85)
N e u m a n n , G. . (Chap. 5)
Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, 300-235, Pasadena, CA 91OO7,USA N e u m a n n - R e d l i n , M. . (Chap. 76)
Institut ffir Erdmessung, University of Hannover, Schneiderberg 50, 30167 Hannover, Germany
Earthquake Research Institute, University of Tokyo, Tokyo, Japan Olesen, A r n e V.. (Chap. 1)
Danish National Space Center, Juliane Maries Vej 30, 2100 Kobenhavn 13, Denmark Omang, Ore D. . (Chap. 1)
Neumayer, K.-H. • (Chap. 36)
Norwegian Mapping Authority, Geodetic Institute, Kartverksveien 21, 3504 Honefoss, Norway
Department 1, Geodesy and Remote Sensing, GeoForschungsZentrum Potsdam (GFZ), Telegrafenberg A17, 14473 Potsdam, Germany
Ou, J. K. . (Chap. 123)
Neumeyer, ].. (Chap. 79)
Institute of Geodesy and Geophysics, Chinese Academy of Sciences, Wuhan 430077, China; and Graduate School of Chinese Academy of Sciences, Beijing, 100039, China
GeoForschungZentrum Potsdam, Dept. Geodesy and Remote Sensing, Telegrafenberg, 14473 Potsdam, Germany
Pacino, M. C. . (Chap. 97) Neuner, H. . (Chap. 112)
Geodetic Institute, University of Hanover, Nienburger Str. 1, 30167 Hanover, Germany
Facultad de Ciencias Exactas, Ingenierla y Agrimensura, Universidad Nacional de Rosario, Av. Pellegrini 250, (2000) Rosario, Argentina, mpacin°@fceia'unr'edu'ar
Nghiem, S. V.. (Chap. 5)
Palmeiro, A. S. . (Chap. 93)
Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, 300-235, Pasadena, CA 91OO7,USA
Geodetic Sciences Graduation Course, Federal University of Parami (UFPR), Curitiba, Brazil, ale-palmeir°@yah°°'c°m'br Papazachariou, D. • (Chap. 9)
Nicolson, G. D. • (Chap. 89)
Hartebeesthoek Radio Astronomy Observatory, PO Box 443, Krugersdorp 1740, South Africa
Department of Geodesy and Surveying, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece Paska, E v a . (Chap. 114)
Niell, A. . (Chap. 105,108)
MIT Haystack Observatory, Off Route 40, Westford, MA 01886-1299, USA
Department of Civil and Environmental Engineering and Geodetic Science, The Ohio State University, 2070 Neil Avenue, Columbus, Ohio 43210, USA
XXI
XXII
Contributors
Pavlis, E. C. • (Chap. lO3,119)
Reigber, Ch. . (Chap. 36)
NASA Goddard Space Flight Center, Greenbelt MD 20771-0001, USA; and Joint Center for Earth Systems Technology (JCET), University of Maryland Baltimore County, Baltimore, MD, USA
Department 1, Geodesy and Remote Sensing, GeoForschungsZentrum Potsdam (GFZ), Telegrafenberg A17, 14473 Potsdam, Germany
P e a r l m a n , M . . (Chap. lO3, lO4, lO6)
R e y n o l d s , ]. E. . (Chap. 89)
Harvard-Smithsonian Center for Astrophysics (CfA), Cambridge, MA 02138, USA, mpearl-
[email protected] Australia Telescope National Facility, CSIRO, PO Box 76, Epping, NSW 1710, Australia
Pearson, C. E • (Chap. 65)
Richter, B. • (Chap. lO4)
National Geodetic Survey, 2300 S Dirksen Parkway, Springfield IL 62764, USA
Bundesamt ffir Kartographie und Geod~isie,Frankfurt a. M., Germany,
[email protected] P e t r a c h e n k o , W. . (Chap. lO8)
Rickards, L e s l e y . (Chap.
Geodetic Survey Division, Natural Resources Canada, Dominion Radio Astrophysical Observatory (DRAO), Box 248, Penticton, B. C., V2A 6K3, Canada
British Oceanographic Data Centre, Joseph Proudman Building, 6 Brownlow St., Liverpool, L3 5DA, UK
11)
Ries, J. C. . (Chap. 103) Petrovic, S. . (Chap. 36)
Department 1, Geodesy and Remote Sensing, GeoForschungsZentrum Potsdam (GFZ), Telegrafenberg A17, 14473 Potsdam, Germany Plag, H a n s - P e t e r . (Chap. 6,11, 80, lO2, lO4, lO7)
Nevada Bureau of Mines and Geology and Seismological Laboratory, University of Nevada, Reno, Mailstop 178, Reno, NV 89557, USA,
[email protected] P o u t a n e n , M a r k k u . (Chap. 13, 107)
Center for Space Research, The University of Texas, Austin TX 78712, USA Rio, M a r i e - H e l e n e . (Chap.
1)
Collecte Localisation Satellites, Space Oceanography Division, 8-10 Rue Hermes - Parc Technologique Du Canal, 31526 Ramonville St. Agne, France Roberts, G e t h i n W y n . (Chap. 117)
Institute of Engineering Surveying and Space Geodesy, The University of Nottingham, University Park, Nottingham, NG7 2RD, UK
Finnish Geodetic Institute, Geodeetinrinne 2, 02430 Masala, Finland Rogister, Y.. (Chap. 63) P r u t k i n , I. . (Chap. 81)
Delft Institute of Earth Observation and Space Systems (DEOS), Delft University of Technology, 2629 HS Delft, Kluyverweg 1, The Netherlands
Institut de Physique du Globe de Strasbourg/l~cole et Observatoire des Sciences de la Terre, 5 rue Ren4 Descartes, 67000, Strasbourg, France Rosat, S. . (Chap. 83)
P u r b a w i n a t a , M. A. . (Chap. 113)
Directorate of Vulcanology and Geological Hazard Mitigation, J1. Diponegoro 57, Bandung, Indonesia
National Astrogeodynamics Observatory, Mizusawa, Iwate, 023-0861 Japan,
[email protected] R o t h a c h e r , M . . (Chap. 36, lO3)
Quick, J. F. H. . (Chap. 89)
Hartebeesthoek Radio Astronomy Observatory, PO Box 443, Krugersdorp 1740, South Africa
Department 1, Geodesy and Remote Sensing, GeoForschungsZentrum Potsdam (GFZ),TelegrafenbergA17, 14473Potsdam, Germany R o w l a n d s , D. D. • (Chap. 35)
Raicich, Fabio . (Chap. 11)
Consiglio Nazionale delle Ricerche, Istituto di Scienze Marine, Viale Romolo Gessi, 2, 34123, Trieste, Italy
NASA Goddard Space Flight Center, Code 697, Greenbelt, Maryland 20771, USA Roy, A l a n . (Chap. 24)
R a m o s , R. . (Chap. 97)
Instituto Geogr~ificoMilitar, Cabildo 391, Buenos Aires, Argentina
Max-Planck-Institute for Radio Astronomy, Auf dem Hfige169, 53121 Bonn, Germany
R a n g e l o v a , E. . (Chap. 82)
R6zsa, Sz. . (Chap. 57)
Department of Geomatics Engineering, University of Calgary, 2500 University Drive NW, Calgary, Alberta, Canada, T2N 1N4
Department of Geodesy and Surveying, Budapest University of Technology and Economics, 1521 Budapest, PO Box 91, Hungary,
[email protected] R e g u z z o n i , M. . (Chap. 56)
Geophysics of the Lithosphere Dept., Italian National Institute of Oceanography and Applied Geophysics (OGS) c/o Politecnico di Milano, Polo Regionale di Como,Via Valleggio,11, 22100 Como, Italy
Saint-Jean, B. d e . (Chap.
115)
IGN-LAREG, 6/8 av. Blaise Pascal, Champs sur Marne, 77455 Marne la Vall4e Cedex 2, France
Contributors
Sakova, L V.. (Chap. 7)
Schrama, E. ]. 0 . . (Chap. 75)
CSIRO Marine and Atmospheric Research, GPO Box 1538, Hobart, Tasmania, Australia, 7001,
[email protected] DEOS, TU Delft, Kluyverweg 1, PO Box 5058, 2600 GB Delft, The Netherlands
Sanchez, Laura. (Chap. 92)
Schuh, 1-1. • (Chap. 3o, 32, lO8,118)
Deutsches Geod~itisches Forschungsinstitut (DGFI), Marstallplatz 8, 80539 Munich, Germany
Vienna University of Technology, Institute of Geodesy and Geophysics, Research Unit Advanced, Geodesy, Gusshausstrasse 27-29, 1040 Vienna, Austria
Santos, Marcelo C. • (Chap. 28, 29, 72, 94, 98)
Department of Geodesy and Geomatics Engineering, University of New Brunswick, PO Box 4400, Fredericton, New Brunswick, Canada, E3B 5A3
Schwahn, W. . (Chap. lO9)
Federal Agency for Cartography and Geodesy, Richard-StraussAllee 11, 60598 Frankfurt am Main, Germany
Satake, Kenji . (Chap. 78)
Seeber, Gfinter. (Chap. 47)
Geological Survey of Japan, AIST,AIST Tsukuba Central 7, Higashi 1-1-1, Tsukuba, Ibaraki 305-8567, Japan
Institut ffir Erdmessung, Universit~it Hannover, Schneiderberg 50, 30167 Hannover, Germany
Sato, Tadahiro. (Chap. 78)
Shelus, Peter ].. (Chap. 126)
National Astronomical Observatory, Hoshigaoka 2-12, Mizusawa, Iwate 023-0861, Japan
University of Texas at Austin, Center for Space Research, 3925 W. Braker Lane, Austin, TX 78759, USA
Savcenko, R o m a n . (Chap. 8)
Shi, Chuang. (Chap. 33)
Deutsches Geod~itisches Forschungsinstitut (DGFI), Marstallplatz 8, 80539 Mfinchen, Germany
GPS Engineering Research Center, Wuhan University, 129 Luoyu Road, Wuhan 430079, China
Savenije, H. H. G. . (Chap. 71)
Shibuya, K a z u o . (Chap. 17, 78)
Department of Water Management, Delft University of Technology, Stevinweg 1, 2628 CN, Delft, The Netherlands
National Institute of Polar Research, 1-9-10 Kaga, Itabashi-ku, 173 8515, Tokyo, Japan
Schaeffer, Philippe. (Chap.
1)
Collecte Localisation Satellites, Space Oceanography Division, 8-10 Rue Hermes - Parc Technologique Du Canal, 31526 Ramonville St. Agne, France
Shimada, S. . (Chap. 84)
Solid Earth Research Group, National Research Institute for Earth Science and Disaster Prevention (NIED), 3-1 Tennodai, Tsukuba-Shi, Ibaraki-Ken 305-0006 Japan,
[email protected], tel.: +81-29-863-7622, fax: +81-29-854-0629
Schaer, S. . (Chap. 19)
Astronomical Institute, University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland
Sideris, M. G.. (Chap. 37, 41, 58, 61, 82)
Department of Geomatics Engineering, The University of Calgary, 2500 University Drive N. W., Calgary, Alberta, T2N 1N4, Canada
Scherneck, Hans-Georg. (Chap. 2o, lO7)
Onsala Space Observatory, Chalmers University of Technology, 439 92 Onsala, Sweden Schliiter, W. . (Chap.
Siegismund, Frank. (Chap. 1)
Nansen Environmental and Remote Sensing Center, Edvard Griegsvei 3a, 5059 Bergen, Norway
105,108,109)
Bundesamt for Kartographie und Geod~isie,Fundamentalstation Wettzell, Sackenrieder Strasse 25, 93444 K6tzting, Germany
Silva, C. A. U. . (Chap. 29)
Department of Civil construction, Federal Technologic Learning Centre, Bel4m, Para, Brazil
Schmidt, T.. (Chap. 36, 79)
Department 1, Geodesy and Remote Sensing, GeoForschungsZentrum Potsdam (GFZ),TelegrafenbergA17,14473 Potsdam, Germany Sch6n, S. . (Chap.
22)
Engineering Geodesy and Measurement Systems, Graz University of Technology (TUG), Steyrergasse 30, 8010 Graz, Austria Schotman, H. H. A. . (Chap.
18)
Delft Institute of Earth Observation and Space Systems (DEOS), Aerospace Engineering,Delft Universityof Technology,Kluyverweg1, 2629 HS Delft,The Netherlands; and SRON Netherlands Institute for Space Research,Sorbonnelaan 2, 3584 CA, Utrecht, The Netherlands
Silva, M. E. • (Chap. lO)
Departamento de Matematica Aplicada, Faculdade de Ci4ncias, Universidade do Porto, Rua do Campo Alegre, 687, 4169007 Porto, Portugal S6hne, W. . (Chap. 109)
Federal Agency for Cartography and Geodesy, Richard-StraussAllee 11, 60598 Frankfurt am Main, Germany Solheim, Dag. (Chap. 1)
Norwegian Mapping Authority, Geodetic Institute, Kartverksveien 21, 3504 Honefoss, Norway
XXIII
XXIV
Contributors
S o u d a r i n , L. . (Chap. 23)
Tocho, C.. (Chap. 61)
Collecte Localisation Satellite, parc technologique du canal, 31526 Ramonville Saint-Agne, France
Facultad de Ciencias Astrondmicas y Geoflsicas, Paseo del Bosque s/n, 1900 La Plata, Argentina,
[email protected] Souter, ].. (Chap. 121)
Toth, Charles K. . (Chap. 114,116)
Institute of Engineering Surveying and Space Geodesy- IESSG, The University of Nottingham, University Park, Nottingham, UK, NG7 2RD
Center for Mapping, The Ohio State University, 1216 Kinnear Road, Columbus, Ohio 43212,
[email protected] T6th, Gy. . (Chap. 57, 62)
Souza, E. M. • (Chap. 27)
Department of Cartography, S~o Paulo State University UNESP, Roberto Simonsen, 350, Pres. Prudente, SP, Brazil Spatalas, S. D. . (Chap. 9)
Department of Geodesy and Surveying, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece Steffen, K. . (Chap. 5)
Cooperative Institute for Research in Environmental Sciences, University of Colorado, Campus Box 216, Boulder, CO 80309-0216, USA
Department of Geodesy and Surveying, Budapest University of Technology and Economics, 1521 Budapest, PO Box 91, Hungary; and Research Group of Physical Geodesy and Geodynamics of the Hungarian Academy of Sciences, 1521 Budapest, Hungary, Mtiegyetem rkp. 3 rregoning, P. . (Chap.
118)
Research School of Earth Sciences, The Australian National University, Canberra, ACT,Australia Tscherning, Carl C h r i s t i a n . (Chap. 2, 50)
Department of Geophysics, University of Copenhagen, Juliane Maries Vej 30, 2100 Copenhagen, Denmark
Stoeber, C. . (Chap. 79)
Institute for Geodesy,Technical University Berlin, Germany Suda, N a o k i . (Chap. 78)
Hiroshima University, Kagamiyama 1-3-1, Higashi-Hiroshima, Hiroshima 739-8526, Japan S u g a n d a , O. K. . (Chap. 113)
Directorate of Vulcanology and Geological Hazard Mitigation, J1. Diponegoro 57, Bandung, Indonesia Sun, W e n k e . (Chap. 85)
Tselfes, N. . (Chap. 56)
DIIAR, Politecnico di Milano, Polo Regionale di Como, Via Valleggio, 11, 22100 Como, Italy Tuccari, G. . (Chap. lO8)
Istituto di Radioastronomia/INAF, Contrada Renna, PO Box 141, Noto (SR), 96017, Italy Turyshev, Slava G. • (Chap.
126)
Jet Propulsion Laboratory, 4800 Oak Grove Drive, Pasadena, CA 91109, USA
Earthquake Research Institute, University of Tokyo, Tokyo, Japan,
[email protected] Tziavos, I. N. • (Chap. 37, 46)
Tanaka, Torao . (Chap. 12o)
Department of Geodesy and Surveying, Aristotle University of Thessaloniki, University Box 440, 541 24, Thessaloniki, Greece, fax: +30 231 0995948
Department of Environmental Science and Technology, Meijo University, 1-501 Shiogamaguchi, Tenpaku-ku, Nagoya 468-8502, Japan Teferle, F. N. . (Chap. 4)
Institute of Engineering Surveying and Space Geodesy, University of Nottingham, University Park, Nottingham NG7 2RD, UK
Tzioumis, A. K. • (Chap. 89)
Australia Telescope National Facility, CSIRO,PO Box 76, Epping, NSW 1710, Australia Urschl, C. • (Chap. 19)
Astronomical Institute, University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland, claudia'urschl@aiub'unibe'ch
Tenzer, R . . (Chap. 59, 6o)
School of Civil Engineering and Geosciences, University of Newcastle upon Tyne, Newcastle upon Tyne, NE17RU,United Kingdom
Valette, J.-J. . (Chap. 1oo)
Collecte Localisation Satellites (CLS), 8-10 rue Hermbs, 31526 Ramonville-Saint-Agne Cedex, France
Tervo, M a a r i a . (Chap. 13)
van der Wal, W. . (Chap. 82)
Finnish Geodetic Institute, Geodeetinrinne 2, 02430 Masala, Finland
Department of Geomatics Engineering, University of Calgary, 2500 University Drive NW, Calgary, Alberta, Canada, T2N 1N4
124)
T i m m e n , L. . (Chap. 67)
Van Hoolst, T.. (Chap.
Institut ftir Erdmessung, University of Hannover, Schneiderberg 50, 30167 Hannover, Germany
Royal Observatory of Belgium, Av. Circulaire 3, 1180 Brussels, Belgium,
[email protected] Contributors
van Loon, ]. P. • (Chap. 51)
V6lgyesi, L . . (Chap. 31, 62)
DEOS, TU Delft, Kluyverweg 1, PO Box 5058, 2600 GB Delft, The Netherlands
Department of Geodesy and Surveying, Budapest University of Technology and Economics; and Research Group of Physical Geodesy and Geodynamics of the Hungarian Academy of Sciences, 1521 Budapest, Hungary, Mfiegyetem rkp. 3
Vandenberg, N. . (Chap. 108)
NVI, Inc./NASA Goddard Space Flight Center, Code 697, Greenbelt, MD 20771-0001, USA
Wallace, L. . (Chap. 64)
GNS Science, PO Box 30368, Lower Hutt, New Zealand Vanicek, P e t r . (Chap. 72)
Department of Geodesy and Geomatics Engineering, University of New Brunswick, PO Box 4400, Fredericton NB, Canada E3B 5A3
Waugh, A. I. . (Chap. 4)
Institute of Engineering Surveying and Space Geodesy, University of Nottingham, University Park, Nottingham NG7 2RD, UK
Veicherts, M. . (Chap. 50)
Webb, F. H. . (Chap. lO3)
Department of Geophysics, University of Copenhagen, Juliane Maries Vej 30, 2100 Copenhagen, Denmark
Jet propulsion Laboratory, California Institute of Technology, Pasadena CA 91109, USA
Venedikov, A. P. • (Chap. 66)
Weber, R . . (Chap. 30, 32)
Geophysical Institute and Central Laboratory on Geodesy, Acad. G. Bonchev Str., Block 3, Sofia 1113
Vienna University of Technology, Institute of Geodesy and Geophysics, Research Unit Advanced, Geodesy, Gusshausstrasse 27-29, 1040 Vienna, Austria
Verdun, ].. (Chap. 115)
IGN-LAREG, 6/8 av. Blaise Pascal, Champs sur Marne, 77455 Marne la Vall~e Cedex 2, France
Wei, E r h u . (Chap. 33)
School of Geodesy and Geomatics, Wuhan University, 129 Luoyu Road, Wuhan 430079, China
Vergos, G. S. . (Chap. 37, 46) Weise, A. . (Chap. 73)
Department of Geodesy and Surveying, Aristotle University of Thessaloniki, University Box 440, 541 24, Thessaloniki, Greece,
[email protected], fax: +30 231 0995948
Institute of Geosciences, Friedrich-Schiller-University Jena, Burgweg 11, 07749 Jena, Germany; and Society for the Advancement of Geosciences Jena, H61derlinweg 6, 07749 Jena
Vermeersen, L. L. A. • (Chap. 18)
Wen, D. B. . (Chap. 123)
Delft Institute of Earth Observation and Space Systems (DEOS), Faculty of Aerospace Engineering, Delft University of Technology, Kluyverweg 1, 2629 HS Delft, The Netherlands
Institute of Geodesy and Geophysics, Chinese Academy of Sciences, Wuhan 430077, China; and Graduate School of Chinese Academy of Sciences, Beijing, 100039, China W h i t n e y , A. . (Chap. 105,108)
V d r o n n e a u , M. . (Chap. 41)
Geodetic Survey Division, CCRS, Natural Resources Canada, 615 Booth Street, Ottawa, Ontario, K1A 0E9, Canada Vest, A n n e L. . (Chap. 1)
Danish National Space Center, Juliane Maries Vej 30, 2100 Kobenhavn 0, Denmark
MIT Haystack Observatory, Off Route 40, Westford, MA 01886-1299, USA Wielgosz, P. . (Chap. 122)
Satellite Positioning and Inertial Navigation (SPIN) Lab, Department of Civil and Environmental Engineering and Geodetic Science, The Ohio State University, 470 Hitchcock Hall, Columbus, OH 43210-1275, USA
Vieira Diaz, R. . (Chap. 66)
Williams, ] a m e s G. . (Chap. 126)
Instituto de Astronomfa y Geodesia (CSIC-UCM), Facultad de Matem~iticas, Plaza de Ciencias, 3, 28040 Madrid, Spain
Jet propulsion Laboratory, 4800 Oak Grove Drive, Pasadena, CA 91109, USA Williams, S. D. P. • (Chap. 4)
Villanoy, C. L. . (Chap. 15)
Marine Science Institute, University of the Philippines, 1101 Diliman, Quezon City, Philippines
Proudman Oceanographic Laboratory, Joseph Proudman Building, 6 Brownlow Street, Liverpool L3 5DA, UK Willis, P... (Chap. 17, 23, lO3)
Visser, P. N. A. M. • (Chap. 18)
Delft Institute of Earth Observation and Space Systems (DEOS), Faculty of Aerospace Engineering, Delft University of Technology, Kluyverweg 1, 2629 HS Delft, The Netherlands
Institut G~ographique National, Direction Technique, 2, avenue Pasteur, BP 68, 94160 Saint-MandG France; and Jet Propulsion Laboratory, California Institute of Technology, MS 238-600, 4800 Oak Grove Drive, Pasadena CA 91109, USA, p ascal'willis@ign'ff
XXV
XXVI
Contributors
W i l m e s , H. . (Chap. lO9)
Z a p r e e v a , E. A. . (Chap. 71)
Federal Agency for Cartography and Geodesy, Richard-Strauss-Allee 11, 60598 Frankfurt am Main, Germany
Delft Institute of Earth Observation and Space Systems (DEOS), Physical and Space Geodesy group, Delft University of Technology, Kluyverweg 1, 2629 HS Delft, The Netherlands
W i n s e m i u s , H. C. • (Chap. 71)
Department of Water Management, Delft University of Technology, Stevinweg 1, 2628 CN Delft, The Netherlands
Z e r b i n i , S. . (Chap. lO4)
Department of Physics, University of Bologna, Bologna, Italy,
[email protected] W i t t w e r , T.. (Chap. 45, 48)
Delft Institute of Earth Observation and Space Systems (DEOS), Delft University of Technology, 2629 HS Delft, Kluyverweg 1, The Netherlands
Z e r h o u n i , W. . (Chap.
1OO)
Centre National des Techniques Spatiales (CNTS) BP 13, Arzew, 31200, Oran, Algeria
Wu, P. . (Chap. 82)
Z h a n g , K. E . (Chap. 123)
Department of Geology and Geophysics, University of Calgary, 2500 University Drive NW, Calgary, Alberta, Canada, T2N 1N4
School of Mathematical and Geospatial Sciences, RMIT University,Australia
W z i o n t e k , H. • (Chap. lO9)
Z h a n g , X i u h u a . (Chap.
Federal Agency for Cartography and Geodesy, Richard-Strauss-Allee 11, 60598 Frankfurt am Main, Germany
Geodesi, Norwegian Mapping Authority, 3507 Honefoss, Norway
11)
Z h a n g , Z i z h a n . (Chap. 3) X u , X i n y u . (Chap. 14)
School of Geodesy and Geomatics, Wuhan University, 129 Luoyu Road, Wuhan, 430079, China
Institute of Geodesy and Geophysics, Chinese Academy of Sciences, 340 Xudong Street, Wuhan, China, 430077; and Graduate School of the Chinese Academy of Sciences, Beijing, China, 100049
Yang, R. G. . (Chap. 123)
Institute of Geodesy and Geophysics, Chinese Academy of Sciences, Wuhan 430077, China; and Graduate School of Chinese Academy of Sciences, Beijing, 100039, China
Z h u , L. . (Chap. 43)
Division of Geodetic Science, School of Earth Sciences, Ohio State University, 125 South Oval Mall, Columbus, OH 43210
Yeh, T. K. . (Chap. 34)
Z h u , S.-Y.. . (Chap. 36)
Institute of Geomatics and Disaster Prevention Technology, Ching Yun University, No. 229, Jiansing Rd., Jhongli 320, Taiwan, R. O. C.,
[email protected] Department 1, Geodesy and Remote Sensing, GeoForschungsZentrum Potsdam (GFZ), Telegrafenberg A17, 14473 Potsdam, Germany
Yuan, Y. B. . (Chap. 123)
Z o u , X i a n c a i . (Chap. 14)
Institute of Geodesy and Geophysics, Chinese Academy of Sciences, Wuhan 430077, China
School of Geodesy and Geomatics, Wuhan University, 129 Luoyu Road, Wuhan, 430079, China
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Part l Joint IAG/IAPSOPapers Chapter 1
Combining Altimetric/Gravimetric and Ocean Model Mean Dynamic Topography Models in the GOCINA Region
Chapter 2
Error Characteristics of Dynamic Topography Models Derived from Altimetry and GOCE Gravimetry
Chapter 3
Detecting Ocean Currents from Satellite Altimetry, Satellite Gravity and Ocean Data
Chapter 4
Sea Level in the British Isles: Combining Absolute Gravimetry and Continuous GPS to Infer Vertical Land Movements at Tide Gauges
Chapter 5
Snow Accumulation and Snowmelt Monitoring in Greenland and Antarctica
Chapter 6
Estimating Recent Global Sea Level Changes
Chapter 7
On the Low-Frequency Variability in the Indian Ocean
Chapter 8
Satellite Altimetry: Multi-Mission Cross Calibration
Chapter 9
Assessment of Recent Tidal Models in the Mediterranean Sea
Chapter 10
Scale-Based Comparison of Sea Level Observations in the North Atlantic from Satellite Altimetry and Tide Gauges
Chapter 11
European Sea Level Monitoring: Implementation of ESEAS Quality Control
Chapter 12
Brazilian Vertical Datum Monitoring Vertical Land Movements and Sea Level Variations
Chapter 13
Tide Gauge Monitoring Using GPS
Chapter 14
Determination of Inland Lake Level and Its Variations in China from Satellite Altimetry
Chapter 15
Sea Surface Variability of Upwelling Area Northwest of Luzon, Philippines
Chapter 16
An Experiment of Precise Gravity Measurements on Ice Sheet, Antarctica
Chapter 17
Status of DORIS Stations in Antarctica for Precise Geodesy
Chapter 18
High-Harmonic Gravity Signatures Related to Post-Glacial Rebound
Chapter I
Combining altimetric/gravimetric and ocean model mean dynamic topography models in the GOCINA region Per Knudsen, Ole B. Andersen, Rend Forsberg, Henning P. F6h, Arne V. Olesen, Anne L. Vest Danish National Space Center, Juliane Maries Vej 30, 2100 Kobenhavn O, Denmark. Dag Solheim, Ove D. Omang, Norwegian Mapping Authority, Geodetic Institute, Kartverksveien 21, 3504 Honefoss, Norway Roger Hipkin, Addisu Hunegnaw, The University of Edinburgh, School of GeoSciences, West Mains Road, EH9 3JW Edinburgh, UK Keith Haines, Rory Bingham, Jean-Philippe Drecourt, University of Reading, Environmental Systems Science Centre, P.O.Box 238, RG6 6AL Reading, UK Johnny A. Johannessen, Helge Drange, Frank Siegismund, Nansen Environmental and Remote Sensing Center, Edvard Griegsvei 3a, 5059 Bergen, Norway Fabrice Hernandez, Gilles Larnicol, Marie-Helene Rio, and Philippe Schaeffer Collecte Localisation Satellites, Space Oceanography Division, 8-10 Rue Hermes - Parc Technologique Du Canal, 31526 Ramonville St. Agne, France
Abstract. Initially, existing mean dynamic topography (MDT) models were collected and reviewed. The models were corrected for the differences in averaging period using the annual anomalies computed from satellite altimetry. Then a composite MDT was derived as the mean value in each grid node together with a standard deviation to represent its error. A new synthetic MDT was obtained from the new mean sea surface (MSS) KMS04 combined with a regional geoid updated using GRACE gravity and gravimetric data from a recent airborne survey. Compared with the composite MDT the synthetic MDT showed very similar results. Then combination methods were tested for the computation of MDT models from gravity data and MSS data. Both a rigorous and an iterative combination method have been tested in the GOCINA region. At this stage, the iterative combination method with its efficient handling of large data sets covering the whole region appears to give the best solution. Naturally, the errors associated with the MDT can be obtained using the rigorous method only.
1 Background The ocean transport through the straits between Greenland and the UK is known to play an important role in the global circulation as well as on the climate in Northern Europe. Warm Gulf Stream water flows into the Nordic seas and feeds the formation of heavy, cold bottom water that returns back into the Atlantic Ocean. The central quantity associated with the ocean circulation and transport is the mean dynamic topography (MDT), which is the difference between the mean sea surface (MSS) and the geoid. The MDT provides the absolute reference surface for the ocean circulation.
A major goal of the EU project GOCINA (Geoid and Ocean Circulation In the North Atlantic) is to determine an accurate mean dynamic topography model in the region between Greenland and the UK. The improved determination of the mean circulation will advance the understanding of the role of the ocean mass and heat transport in climate change.
4
Knudsenet al. $0°W
40'W
30°W
2O ° W
1O=W
O°
[O°E
2O°L
70~N
7 0~N
• ...:..
~0=N
6 0=N
:~0°N
$0°N
50~W
40~W
3O°W
20~W
1O~W
O~
1O~E
2O°~E
Figure 1. The region of interest in the GOCINA project.
Through the first phases of the GOCINA project the main focus has been on the determinations of the geoid the mean sea surface and the mean dynamic topography using the individual techniques and on the assessment of the models in the region of interest (Figure 1).
2 Description of data Existing mean dynamic topography (MDT) models were collected and reviewed. It was decided to compute a composite MDT model using the available models. The Composite MDT was derived as the mean value in each grid node together with a standard deviation to represent its error. To calculate the best possible synthetic mean dynamic topographies a new MSS (KMS04) has been derived from nine years of multi mission altimetric data (1993-2001). The regional geoid has furthermore being updated using GRACE and gravimetric data from a recent airborne survey. The new synthetic mean dynamic topography model has been computed from this geoid model and the MSS model KMS04. Error estimates associated with the synthetic MDT have been derived from the errors of the geoid and the errors of the mean sea surface model. Compared with the Composite MDT the synthetic MDT derived from the mean sea surface and the geoid showed very similar results (see Figure 2). Details on the data processing and how the individual models have been obtained are described below.
were done in a band from Greenland over Iceland and the Faeroe Islands to Norway and Scotland. A total of 84 airborne flight-hours were flown. The airborne gravity measurements were processed and incorporated with the revised marine gravity data. The gravity data have been cross-over adjusted for survey biases. Two different approaches to bias adjust the marine gravity data has been tested. Mean differences between overlapping surveys, both marine and airborne, were estimated and used as observations in a joint least squares collocation estimation of survey biases. 190 individual surveys were inter-compared and 1531 overlapping areas were identified. Surveys considered to be of superior quality were held fixed in the adjustment; these surveys include airborne surveys and recent marine surveys processed together with a few other sources acknowledged for their reliability. The identification of reliable sources, which should be held fixed in the adjustment. The geoid modelling was based on the adjusted gravity. Some altimetric gravity data from the KMS02 model were patched in areas with larger data voids, i.e. more than 20 km to the nearest marine/airborne data point. A newly released GRACE geopotential model from JPL was used for the longer wavelengths of the gravity/geoid field in the geoid modelling. This model was used up to degree and order 120. The residual geoid was determined with the spherical FFT approach used on gravity data reduced for reference field, restoring gridded terrain-corrected RTM anomalies into Faye anomalies prior to FFT. The FFT routine was applied with Wong-Gore kernel modification to degree 90. The reference geoid then subsequently restored. This final geoid is called NAT04 geoid in the sequel (e.g. Forsberg et al., 2004). For the error assessment a standard deviation was assigned to the gravity data based on the nature (terrestrial, marine, airborne) and the history (collector and processor) of the data. The surface, marine and airborne gravity data was used in a rigorous least-squares collocation error estimate. Because least-squares collocation requires the solution of as many linear equations as the number of data, data were thinned prior to applying the method. The satellite altimetric gravity was omitted in the data set, as these data are not "real" gravity values. 1.2 Altimetry and mean sea surface
1.1
Gravity data and geoid
An airborne survey activity was carried out in June 26-July 18 and Aug 7-9 2003. The measurements
A new global mean sea surface has been derived using the best available dataset for the GOCINA region. In deriving this high resolution MSS grid
Chapter 1
•
Combining Altimetric/Gravimetric and Ocean Model Mean DynamicTopography Models in the GOCINARegion
file for the period 1993-2001 with associated quality indication grid on 2 km or 1/30 ° by 1/60 ° resolution the following scientific achievements were obtained. The MSS model is the only available MSS based on 9 years of data (1993-2001) using T/P as reference. All data have been interpolated using least squares collocation taking into account the varying quality and coverage of the data. Global sea level change over the 1993-2001 was taken into account in the computation. A new method has been derived to account for the inter-annual ocean variability (like the major E1-Nifio event in 199798), as well as sea surface trends and pressure effects on the ocean surface. The MSS is available both with and without correction for the atmospheric pressure correction applied to the altimeter range (inv. barometer correction). Furthermore, annual averages of the sea level with respect to the nine years mean sea surface have been computed, so that mean sea surface from other sources and mean dynamic topographies covering different periods of time may be inter-compared. The MSS model is denoted KMS04 (Andersen et al., personal communication) 1.3 Ocean topography
models
and
mean
dynamic
Existing mean dynamic topography (MDT) models were collected and reviewed. It was found that the best currently available products were those developed by GOCINA project partners and identified as CLSv2 and OCCAM, along with the MDT based on the assimilation results available from the UK Met Office FOAM system. It was decided to compute a composite MDT model using the available models (see Table 1).
computed from satellite altimetry as described above. Also, the high resolution models were smoothed to one by one degree grids. The Composite MDT was derived as the mean value in each grid node. Furthermore, at each node the standard deviation was computed to represent the error of the mean value (Bingham et al., personal communication). The composite MDT and its errors are shown in Figure 2.
3
Combining data sources
The MDT may be obtained simply by subtracting the geoid from the MSS as described in the previous section. If a full coverage of both the gravity data and the altimeter data exist in the region, then this simple approach will probably give a nice result. But, if that is not the case, then more advanced methods may be needed. |n the GOCINA region the coverage of gravity data is not very good in the Northern and the Southwestern parts of the region. For the geoid computation described above altimetric anomalies are inserted into data gaps, hereby, combining the two data sources already at this point. To avoid major errors a so-called draping technique is applied when merging the two data sources. When different data types are combined it is important that it is done in a rigorous way and that the full signal/error content is taken into account. Else fatal inconsistencies between different data types may occur. The MSS consists of the geoid and the MDT as expressed in: h =N+~'+n
(1)
where
MDT
Time period
Resolution
CLS v l
1993-1999
l°xl °
CLS v2
1993-1999
l°xl °
h N
ECCO
1992-2001
l°xl °
4
is the mean sea surface height, is the geoid height, is the mean dynamic topography, and
ECMWF
1993-1995
1.4°x 1.4 °
n
is the measurement noise.
FOAM
May02-May03
1/9°x 1/9 °
O C C A M vl
1993-1995
0.25°x0.25 °
O C C A M v2
1993-1995
0.25°x0.25 °
T a b l e 1: T h e i m p o r t a n t f e a t u r e s o f the M D T s u s e d in this
The geoid height is a quantity associated with the anomalous gravity potential 7". Hence, N can be expressed in terms of a linear functional (or as in this case linearized functional according to Bruns' formula) applied on T (), is the normal gravity):
study
The models were corrected for the differences in averaging period using the annual anomalies
N=LN(T) -
T 2"
(3)
5
6
Knudsen et al.
At this point the important link between altimetry and gravimetry can be made, gravity anomalies are associated with T too. They are expressed as
variances. Then the covariance between T in the points P(%)~) and Q(qD() is expressed as oO
Ag = LAg ( T ) = - c9T_ 2 --T Or r
(4)
K(P, Q)= Z °-TTpi (cosg)
(7)
i=2 The gravity data describes in the previous section are used in the combination solution rather that the estimated geoid. Hereby the structure of the original data source, e.g. data distribution and their individual errors, is maintained and represented in the computations. Furthermore, remaining biases in the individual gravity surveys may be taken into account. For the ship data such a bias is considered. The airborne data are considered to be bias free. In principle, the altimeter data should be used as described in Knudsen (1993). However, since the mean sea surface determination described above include a very large number of data and since those data have been combined using a rigorous method, it was decided to use the KMS04 mean sea surface with its associated errors in the combination solution. MDT information from ocean circulation models may be taken into account if reliable error estimates may be derived. Hence, the composite MDT described in section 1.3 can be used in the combination procedures described in this section.
3.1 Rigorous combination
(5)
An estimate of the a-posteriori error covariance between two estimated quantities, x and x/, is obtained using
Cx'x = Cx'x-Crx (C + D)-' Cx'
TT are degree variances and q~ is the
o-i
spherical distance between P and Q. Hence, eq.(7) only depends on the distance between P and Q and neither on their locations nor on their azimuth (i.e. a homogeneous and isotropic kernel). Expressions associated with geoid heights and gravity anomalies are obtained by applying the respective functionals on K(P,Q), e.g. CNv-=LN(LN(K(P,Q))) (more on collocation by Sans6, 1986, Tscherning, 1986). Then
C N N --
Pi (cos N)
(8)
°-/I Pi (c°s ¢/ )
(9)
Pi (cos ~)
(10)
i=2 oO
- Z
2
i=2 oO
CNAg i=2
The kernel associated with the MDT, eq.(7), is expressed in a similar manner as the gravity fields as
The combination of the mean sea surface heights and the gravity data is done rigorously using the optimal estimation technique called least squares collocation (LSC). Here the results in the following expression x - C~ (C + D)-'y
where
(6)
where cxx/is the a-priori (signal) covariance between x and x/ (see e.g. Moritz, 1980). The elements of the covariance matrices of eq.(56) are calculated according to the mathematical model of the observations. In this case, signals associated with the gravity data and the mean sea surface are considered. The covariance values are obtained using the kernel functions. The kernel associated with the gravity field is derived using eq.(5) and some a-priori
o(3
CG- - Z o-~( Pi (cos ~' )
(11)
i 1
Finally, it is shown how covariance functions associated with the geostrophic ocean surface currents are obtained from the sea surface topography. If accelerations and friction terms are neglected and horizontal pressure gradients in the atmosphere are absent, then the components of the surface currents are obtained from the MDT by
y ~4 u - - - --y -, G v = f R c3~b f R cos ~b 0A
(12)
where f=2mosinq) is the Coriolis force coefficient. Expressions associated with the geostrophic surface currents and the MDT depend on the azimuth between the two points P and Q, apQ as (Knudsen, 1991)
Chapter 1
•
Combining Altimetric/Gravimetric and Ocean Model Mean DynamicTopography Models in the GOCINARegion
2 7
C bl bl m
(- COS~pQCOS£ZQp Cll -
fPfQ
sin apQsin aQp
(13)
Cqq)
2
Y
C VV m
UpUQ
(- sin C~pQsin C~QpCll-
(14)
COS O[pQ COS OCQp Cqq)
-7" cosapQ Ct4"
C.4" -
(15)
fe
Cv( =
2" sin CZpQ C
fP
l
(
(
1
6
)
where z (cos p' C ' ( ( - sin 2 ~ C"(4- ) CM - -~__
Cqq
1
(17) (18)
and C l ( - - ~-k- sin ~ C ' ( (
(19)
The modelling of the covariance function associated with the gravity field is described in Knudsen (1987a). This technique has been applied using empirical covariance values calculated from marine gravity data reduced using a hybrid reference model consisting of the GRACE GGM01 model up to degree 90 and EGM96 from degree 91 to degree 360. As degree variance model a Tscheming/Rapp model (Tscherning & Rapp, 1974) was used. This expression has the advantage that the kernel can be evaluated using a closed expression instead of the infinite sum. The model is GRACE oei
-
-
gi
( k~ k~ ] i+1 cr)~S=b• k3+is k]+i s "s
¢
Re C ( (
O~T T m
0.25. The covariance function associated with gravity anomalies has a variance of (11.8 mgal) 2 and a correlation length (which is the distance where the covariance is 50 % of the variance) of 0.15 °. The corresponding geoid height covariance function has a variance of (0.20 m) 2 and a correlation length of 0.26 ° . A determination of a covariance function model associated with the MDT was carried out using an empirical covariance function was determined and a degree variance model chosen. The degree variance model was constructed using 3 rd degree Butterworth filters combined with an exponential factor. Hence, the spectrum of the MDT is assumed to have a decay similar to the geoid spectrum. Then the model was fitted iteratively to the empirical covariance values as described in Knudsen (1993). This resulted in the model:
i = 2 ..... 90
EGM96 _A_
i : 91 .....
360
where b = 6.3 10 .4 m 2, kl = l, k2 = 90, s = ((R5000.0)2/R2) 2. The variance and correlation length are (0.20 m) 2 and 1.3 ° respectively. The variance and correlation length of the current components are (0.16 m/s) 2 and 0.22 ° respectively. 3.2 I t e r a t i v e c o m b i n a t i o n
As in the geoid determination described in section 1.1 approximation methods based on FFT may be applied as an alternative to the rigorous LSC. A socalled local collocation technique may be used to grid data, so that the FFT technique can be used. This procedure is very efficient compared to LSC and does not require inversion of large equation systems. Hence, geoid computations in large regions with many observations may easily be handled using the FFT method. Stokes integral which relates the gravity anomalies to geoid undulations as (Heiskanen and Moritz, 1967)
(20)
( R2R]i+l i = 361 .....
(i - /) (i - 2)(i+ 4) [--~ )
N = S(Ag)
(22)
are used in the iterative combination method. Two metods may be applied for iterative scheme (Hipkin et al., private communication). One method is based on
where A = 1544850 m4/s 4, RB = R - 6.823 km were found in an adjustment. The error degree variances, ei, associated with the EGM96 model was multiplied by
(21)
'(f,
7
8
Knudsenet al.
where the i'th increment is based on the difference between the MSS and the geoid. In each iteration a geoid is computed using a set of gravity data with fill-in gravity values from MSS-MDT. The geoid is iteratively improved as the synthetic fill-in gravity anomalies computed from the MSS-MDT is iteratively improved. The other method is based on applying Stokes integral on the gravity equivalent of the MDT. That is A ( i - S(S -1 ( h - ( i )
-
AN)
(24)
where the gravity equivalent of the i'th increment which is the difference between the synthetic gravity anomalies and the real data tend to zero iteratively. In both cases the i+l 'th MDT is obtained using
(i+1--(i-t-wA(i-t-we((
c
-- (i )
(25)
where weights are introduced to balance the M D T between a freely iterating procedure and an iteration that is constrained to the composite MDT, e.g. within its errors. Also, the composite MDT may be used as initial MDT. The two procedures have their strengths in cases where the coverage of the gravity data is better or poorer than the altimeter data coverage respectively (Zlotnicki, 1984, Knudsen, 1992).
4
Results
Both the rigorous and the iterative combination methods have been tested in the GOCINA region. in the initial tests results were obtained using gravity data and MSS data only to derive ocean model free MDTs to be assimilated into ocean models. For the computation of a MDT from gravity and MSS data using the rigorous method it was nessesary to devide the region into nine sub-regions to limit the number of observations to overcome the task of solving the equation systems. The subregions were constructed so that they have an overlap of about 1 degree to avoid edge effects when the nine solutions later are to be merged. Furthermore, data were selected with a spacing of about 0.1 by 0.2 degrees for the gravity data and a spacing of 0.2 by 0.4 degrees for the MSS data. The estimated MDT and it error estimates are shown in Figure 3. Compared to the MDTs in Figure 2 this MDT show the same general pattern
though much more details are recovered. Those details are probably spurious in terms of ocean circulation are may be caused by edge effects at coastlines and by unrecovered errors in the data. Furthermore the Southeastern sub-region appeared to be biased by long wavelength signals that are not recovered fully within each of the sub-regions. The errors reflect the actual data distribution and range from 4 to 12 cm. Furthermore, the characteristics of the MDT error covariances were studied. That was done at a few locations within the GOCINA region where error covariances were computed using eq. (6). The results showed that the correlation lengths of the error covariance function all were similar and close to 0.3 degrees. Hence, a expression having a fixed correlation length such as CpQ - epeo cov(q,,po)
g/ = epCQ[l+ ~// I.C 0.18 L
(26)
0.18 a
may be a fairly accurate expression for the MDT error covariances. The result of the iterative combination method was obtained using the second approach, eq. (2425) by Hipkin et al. (private communication). The result has been smoothed at 75 km wavelength. The general pattern is very similar compared with the composite MDT in Figure 2. Parts of the spurious details shown in Figure 3 are also present in Figure 4. At this stage, the iterative combination method with its efficient handling of large data sets covering the whole region appears to give the best solution. However, the errors associated with the MDT can be obtained using the rigorous method only.
5 Perspectives The next tasks of the GOCINA project are associated with the merging of all three data streams. These methods will rely on new techniques of data assimilation. The following experiments will examine the mass and heat exchange across the Greenland-Scotland Ridge, considering the Atlantic inflow, the surface outflow in the East Greenland Current, and the overflows. Also the impact on the current running along the continental shelf from the Bay of Biscay to the northern Norwegian Sea will
Chapter 1
•
Combining Altimetric/Gravimetric and Ocean Model Mean DynamicTopography Models in the GOCINARegion
~1'~" LB.),
f~.." ~"
hU'.%
~11',%"
..~ ' .%"
...-..~,...,.
•g l ' Vl
.~P.I' ~'1.
.,.~1' ~lIw"
[ I~ 1~"
IJ'
111 r.
On
4.0" th
.'~," W
~'~
It)'~h
l'
a Ih'/I-.
40"tY
.W~"
110-~h
10" ~t"
(c
LtFIF:
a:t.~
:Ip'N
•li..~.".'~
~'..,;
,~'5"
,an',,v
,1,i',~,"
., r,"l,,,"
In'~,'l
n"
an,'i.:
ql 0.~
,.1.01
0.02
O.~:a
0.~
0.05
0.~
,.1.~
0.~
0.~
O.IQ
nl
0.414)
Ik4~
4". L't
41.15
(I _.'.'.~
Figure 2. Mean dynamic topography models with error estimates. The Composite MDT model is shown top left and its errors are shown lower left. The synthetic MDT is derived using the mean sea surface and the gravimetric geoid model and shown top right with its estimated errors lower right. Gray scale bar associated with the MDTs is found on the right. be analysed. A best possible ocean circulation experiment will be performed, which will also include sea floor pressure data from GRACE based on methods developed in a separate project. This analysis will give invaluable information on the ocean role in climate. Finally, the GOClNA project will support the GOCE mission in two distinct cases, namely (1) to educate and prepare the community in using GOCE data for oceanography including sea level and climate research as well as operational prediction; and (2) to develop methods for generating regional gravity fields and to use them to generate a best possible regional gravity field and geoid model for the North Atlantic that can be used in validation of the GOCE products.
Acknowledgement GOCINA is a shared cost project (contract EVG1CT-2002-00077) co-funded by the Research DG of the European Commission within the RTD activities of a generic nature of the Environment
and Sustainable Development sub-programme of the 5th Framework Programme. More information on the GOCINA project may be found on the Internet at http://www.gocina.dk
References Andersen, O., P. Knudsen, and R. Trimmer, Improved High Resolution Altimetric Gravity Field Mapping (KMS2002 Global Marine Gravity Field. lAG symposia, Vol. 128, Springer Verlag, ISBN 3-540-24055-1,326-331, 2005. Forsberg, R., A. Olesen, A. Vest, D. Solheim, R. Hipkin, O. Omang, P. Knudsen: Gravity Field Improvements in the North Atlantic Region. Proc. GOCE Workshop, ESA-ESRIN, March 2004. Heiskanen, W. A., and Moritz, H.: Physical Geodesy, W. H. Freeman, San Francisco, 1967. Knudsen, P.: Estimation and Modelling of the Local Empirical Covariance Function using gravity and satellite altimeter data. Bulletin G6od6sique, Vol. 61,145-160, 1987.
9
10
Knudsen
et al.
Knudsen, P.: Determination of local empirical covariance functions from residual terrain reduced altimeter data. Reports of the Dep. of Geodetic Science and Surveying no. 395, The Ohio State University, Columbus, 1988. Knudsen, P.: Simultaneous Estimation of the Gravity Field and Sea Surface Topography From Satellite Altimeter Data by Least Squares Collocation. Geophysical Journal International, Vol. 104, No. 2, 307-317, 1991. Knudsen, P.: Estimation of Sea Surface Topography in the Norwegian Sea Using Gravimetry and Geosat Altimetry. Bulletin Gdoddsique, Vol. 66, No. 1, 27-40, 1992. Knudsen, P.: Integration of Altimetry and Gravimetry by Optimal Estimation Techniques. In: R. Rummel and F. Sans6 (Eds.): Satellite Altimetry in Geodesy and Oceanography, Lecture Notes in Earth Sciences, 50, Springer-Verlag, 453-466, 1993. Knudsen P., R. Forsberg, O. Andersen, D. Solheim, R. Hipkin, K. Haines, J. Johannessen & F. Hernandez. The GOCINA Project- An Overview and Status. Proc. Second International GOCE User Workshop "GOCE, The Geoid and Oceanography", ESA-ESRIN, March 2004, ESA SP-569, June 2004. Moritz, H.: Advanced Physical Geodesy. Herbert Wichmann Verlag, Karlsruhe, 1980. Omang, O., R. Forsberg, G. Strukowski: Comparison of New Geoid models and EIGEN2S in the North Atlantic Region. In: F. Sanso (Ed.): A Window on the Future of Geodesy, Sapporo, Japan, IAG Symposium Series 128, Springer Verlag, pp. 306-309, 2003. Sans6, F.: Statistical Methods in Physical Geodesy. In: Stinkel, H.: Mathematical and Numerical Techniques in Physical Geodesy. Lecture Notes in Earth Sciences, Vol. 7, 49-155, Springer-Verlag, 1986.. Tscherning, C.C.: Functional Methods for Gravity Field Approximation. In: Stinkel, H.: Mathematical and Numerical Techniques in Physical Geodesy. Lecture Notes in Earth Sciences, Vol. 7, 3-47, Springer-Verlag, 1986. Tscherning, C.C., and R.H. Rapp: Closed Covariance Expressions for Gravity Anomalies, Geoid Undulations, and Deflections of the Vertical Implied by Anomaly Degree Variances. Report no. 208, Dept. of Geodetic Science and Surveying, The Ohio State University, Columbus, 1974. Wunsch, C., and V. Zlotnicki: The accuracy of altimetric surfaces. Geophys. J. R. astr. Soc., 78, 795-808, 1984.
Zlotnicki, V.: On the Accuracy of Gravimetric Geoids and the Recovery of Oceanographic Signals from Altimetry. Marine Geodesy, Vol. 8, 129-157, 1984.
-{1,41(I -(I, L~ ~10'
0-05
i),,35
(I. LO ¢, I,~
~,OL'I,
m
~
,,,
Figure 3. Mean dynamic topography model with error estimates from rigorous combination of gravity data and mean sea surface data by least squares collocation.
-di.d
-I}.~
-{I.4
-¢.3
-~_2
-{I. 1
i).¢
0.1
¢.2
I}.3
0.4
¢.:~
Figure 4. Mean dynamic topography model from iterative combination method smoothed at 75 km wavelength (Hipkin et al., private communication).
Chapter 2
Error Characteristics of Dynamic Topography Models Derived from Altimetry and GOCE Gravimetry Per Knudsen Danish National Space Center, Juliane Maries Vej 30, 2100 Kobenhavn O, Denmark. Carl Christian Tscherning University of Copenhagen, Juliane Maries Vej 30, 2100 Kobenhavn O, Denmark.
The impact of the GOCE satellite mission on the recovery of the gravity field is analysed for two simulated cases. In the first case the GOCE Level 2 product is used where the gravity field is approximated by spherical harmonic coefficients up to degree and order 200. In the second case synthetic GOCE Level 1B data are used directly in a gravity field determination using least squares collocation. In case two the full spectrum geoid error was improved from 31 cm to 15 cm and the resolution was doubled. To get reliable errors associated with the mean dynamic topography (MDT) a reliable model for the spectral characteristics of the MDT is needed. Such a model was derived reflecting empirically derived properties such as MDT variance and correlation length. Combining the MDT characteristics with the estimated geoid errors in the spectral domain resulted in a-posteori error estimates. In the two cases the MDT errors were improved from 20 cm to 6 cm and 5 cm respectively. For the geostrophic surface current components the errors were improved from 23 cm/s to 18 cm/s and 16 cm/s. Abstract.
1 Background The GOCE (Gravity and Ocean Circulation Experiment) satellite mission by the European Space Agency is planned to improve the knowledge about the Earth gravity field and to improve the modelling of the ocean circulation. The central quantity associated with the ocean circulation and transport is the mean dynamic topography (MDT), which is the difference between the mean sea surface (MSS) and the geoid. The MDT provides the absolute reference surface for the ocean circulation. In the EU project GOCINA (Geoid and Ocean Circulation In the North Atlantic) the use of GOCE
data in oceanographic modelling is prepared by developing methodology for the determination of accurate mean dynamic topography models in the region between Greenland and the UK (Knudsen et al., 2004). The improved determination of the mean circulation will advance the understanding of the role of the ocean mass and heat transport in climate change. The GOCINA project will support the GOCE mission in a distinct case, namely to educate and prepare the community in using GOCE data for oceanography including sea level and climate research as well as operational prediction. In this study, the impact of GOCE on the mapping of the gravity field is studied. Furthermore, its impact on the estimation of the MDT is analysed.
2
GOCE simulations
The simulations of GOCE impact on the gravity field recovery are done using the full spectra of the signals and the errors. Hence, both commission and omission errors are taken into account when, e.g., a spherical harmonic expansion truncated at a certain harmonic degree and order is considered. This is important because the omission error usually is larger that the commission error. Using spherical harmonic functions, the signal and the error covariances associated with the gravity field between points P and Q may be expressed as a sum of Legendre's polynomials multiplied by degree variances as oo
K(P, Q)= 2
crTTPi (cosp')
(1)
i=2 TT
where cri
are degree variances associated with the
anomalous gravity potential field and ~, is the spherical distance between P and Q. Hence, eq.(1) only depends on the distance between P and Q and
12
P. Knudsen. C. C. Tscherning
not on their locations nor on their azimuth (i.e. K(P,Q) is a homogeneous and isotropic kernel). Expressions associated with geoid heights and gravity anomalies are obtained by applying the respective functionals on K(P,Q), e.g. CuN=LN(LN(K(P,Q))) and become as follows (For more information on collocation, see for example Sans6, 1986, Tscherning, 1986) CNN --
Pi (cos~,)
(2)
0-; 7, Pi (cos ~)
(3)
i=2 oo
i=2 oO
-- Z
i-1 or/TTPi(cosp')
(4)
i=2
where N is the geoid, ), is the normal gravity, and Ag is the gravity anomaly. The determination of the degree variances is essential to obtain reliable and useful signal and error covariance functions. For the gravity field it has been accepted that the degree variances tend to zero somewhat faster than i-3 and that the Tscherning-Rapp model may be used as a reliable model (Tscherning & Rapp, 1974, and Knudsen, 1987). This expression has the advantage that the kernel can be evaluated using a closed expression instead of the infinite sum. When a spherical harmonic expansion of the gravity field up to degree and order N has been used as a reference model and has been subtracted from the gravity field related observations, then the error degree variances, ei, associated with the reference model should enter the expression, eq. (1), up to harmonic degree N, so that the degree variances are expressed as i=2
o.TiT
-
-
A
..... N
(5)
(RS] i-'
7;- u(,-3)(,-+-4) kT)
200 that can fulfil the aim of the satellite mission, which is to model the geoid at a resolution of 100 km with an accuracy of 1-2 cm. Based on mission parameters and extensive simulations it has been demonstrated that GOCE will meet those requirements (e.g. Visser et al., private communication). An important outcome of the simulations is a set of error degree variances that may be included as commission errors in other simulations of the GOCE performance. In this case the simulated error degree variances described above are used as commission errors, i.e. the ai, in eq. (5), in the evaluation of the impact of GOCE on the recovery of the gravity field. The unmodelled part of the gravity field remains unknown and will be considered as the omission error. The omission error was modelled using the Tscherning-Rapp model from degree 201 and up. Hence, the full spectrum error of the GOCE level 2 harmonic expansion as an approximation of the gravity field consists of both the commission and the omission errors as expressed in eq. (5). The degree variances are shown in Figure 1. The error covariance function is shown in Figure 2. it has a variance of (0.31 m) 2 and a correlation length (which is the distance where the covariance is 50 % of the variance) of about 0.3 ° . 2.2 GOCE Collocation Product
As part of the simulations of the GOCE performance alternative methods such as least squares collocation have been tested (Tscherning, 2004). In this case a test was carried out using simulated GOCE Level 1B observations in the GOCINA region in the North Atlantic Ocean. Using both the along track and the vertical second order derivatives, least squares collocation was applied to estimate the geoid in the region using the following expression x = C~r (C + D)-ly
(6)
i = N + I .....
where A = 1544850 m4/s 4, RB = R - 6.823 km were found in an adjustment so that agreement with empirical covariance values calculated from marine gravity data was obtained. This procedure is described in Knudsen (1987). 2.1 GOCE Level 2 Product
The standard Level 2 product coming from GOCE is a spherical harmonic expansion to degree and order
where C and D are covariance matrices associated with the signal and the errors of the observations y. x is the estimated quantity. As covariance function used for determination of the values in the C matrix the Tscherning-Rapp model with the same parameters as in eq.(5) was used from harmonic degree 2 and up. The error covariance matrix D is diagonal and contains the error variance of the observations. Then error covariances were estimated using G , x - c x,~-
d(C+D)' x Cx,
(7)
Chapter2 • Error Characteristics of Dynamic Topography Models Derived from Altimetry and GOCEGravimetry
where c J is the a-priori (signal) covariance between x and x/ (see e.g. Moritz, 1980). The results show that the estimated errors range from 10 cm to 15 cm. Note that these are full spectrum errors demonstrating that the approach will give a significant improvement of the geoid. The estimated error covariances show that the errors are associated with scales shorter that half a degree. Hence, the resolution appears to have been doubled. For the subsequent simulations of the GOCE performance in terms of modelling a MDT a degree variance model was obtained by extending the GOCE part of the previous degree variance model to harmonic degree 360 (also shown in Figure 1). This model give an error covariance function (also shown in Figure 2), which has a variance of (0.15 m) 2 and a correlation length of about 0.1 °. Hence, compared to the standard GOCE Level 2 product, a significant improvement may be obtained by using the GOCE data directly in a determination of the gravity field using least squares collocation.
Modelling the Signal Characteristics of the Mean Dynamic Topography
)
(9)
where b, kl, k2, and s are determined so that the variance and the correlation length agree with empirically derived characteristics. Since geostrophic surface currents are associated with the slope of the MDT, it may be shown how covariance functions associated with the geostrophic surface currents can be obtained. If accelerations and friction terms are neglected and horizontal pressure gradients in the atmosphere are absent, then the components of the surface currents are obtained from the MDT by
u-----, vf R c3~b f R c o s ~ b 02
(10)
where f=2mesino is the Coriolis force coefficient. Covariance functions associated with the geostrophic surface currents depend on the azimuth between the two points P and Q, aeQ (Knudsen, 1991). Hence, the respective current components are not isotropic. They are expressed as 2
Cuu To get reliable results of simulations and tests carried out using least squares methods it is important that both the signal and the error characteristics have been taken into account. In least squares collocation that means that the covariance function models should agree with empirically determined characteristics such as the variance and correlation length. In analysis of errors formally estimated using eq.(7), it is very important that those quantities are reliable. That is also the case when MDT errors are analysed. Hence, a model describing the magnitude and the spectral characteristics of the MDT is needed. A kernel function associated with the MDT, may be expressed in a similar manner as the gravity fields
,+,
o~'~'=b • k~+i 3 k]+i 3 "s
Y
/
(- COS~TpQCOS~Qp Cll -
fpfo
(11)
sin C~pQsin C~Qe C qq) 2 C
VV m
7" fpfQ
(- sinc~pQsin~zQp Cll (12) \
COS 6~'pQ COS 6~'Qp Cqq )
-y COS 6~ pQ Cl(
C,,;- =
(13)
fp
Y Cv( -- - - sin apQ Cl(
f~
(14)
where
as oO
(s) i=l where the degree variances in this expression are associated with the MDT, naturally. The degree variance model was constructed using 3 rd degree Butterworth filters combined with an exponential factor (e.g. Knudsen, 1991). Hence, the spectrum of the MDT is assumed to have properties similar in smoothness and infinite extent to the geoid spectrum. That is
C~ -_ ) 7t (cos ~ C';;- - sin2gtC"4-4 -)
(15)
Cqq - 1 R2 C, ~
(16)
are the longitudinal and the transverse components respectively (see also Tscherning, 1993), and C~4 - - z T sin gt
C'
(4
(17)
With those expressions it is possible to estimate geostrophic surface currents using collocation. They
13
14
P. Knudsen. C. C. Tscherning
also give a very important constraint on the modelling of the degree variance model. In Knudsen (1993) the parameters in the degree variance mode, eq.(9), were fitted iteratively to the empirical covariance values. This resulted in the model where b = 6.3 10 -4 m 2, kl = 1, k2 = 90, s = ((R-5000.O)2/R2) 2. The variance and correlation length are (0.20 m) 2 and 1.3 ° respectively. The variance and correlation length of the current components are (0.16 m/s) 2 and 0.22 ° respectively.
Modelling A-posteori Mean Dynamic Topography Error Characteristics Combining the MDT signal degree variances and the geoid error degree variances it may provide information about the a-posteori errors of an MDT that has been estimated using the geoid and a mean sea surface computed from satellite altimetry. Using least squares to estimate the MDT by degree its error degree variance is expressed as 1
To study the properties of the degree variance model in more detail characteristic parameters associated with the MDT and its associated geostrophic surface current components were derived. This was done for block averages of varying block sizes, because those numbers may be compared with output parameters from ocean circulation models with different grid sizes and resolutions. The MDT signal covariance properties were computed rigorously using the series of Legendre's polynomials to which the smoothing operators associated with the running averages have been applied. Then the following expression may be used oo
C~;- - E / 5 ' 2 (s) cr~" Pi (cos ~,,)
(18)
i 1
where the beta factors are the so-called Pellinen operators that depend on the side length, s, of the cells. The covariance functions associated with the MDT and with the geostrophic surface current components (represented by the Cll, eq. (15)) and averaged in cells of ½, 1, and 2 degree were computed. The resulting variances and correlation lengths are summarized in Table 1. it is important to emphasize that those numbers are statistical expected values representing a region as the GOCINA region. They are not representative for the strong Western boundary currents as the Gulf Stream. Table 1. Standard deviations in cm and cm/s and correlation lengths in degrees of MDT and geostrophic surface current components (u,v) (represented by the Cll) as point values and averaged in cells.
(u,v)
MDT
Points ½o x ½o 1o x 1o 2°x2 °
St.dev. 20 19 18 16
C.length 1.3 ° 1.4 ° 1.6 ° 2.1 °
St.dev. 16 13 10 6
C.length 0.22 ° 0.38 ° 0.56 ° 0.85 °
=
1
1
(19)
where the errors of the mean sea surface have been ignored since they are very small compared to the geoid errors. For both GOCE simulations the a-posteori MDT error degree variances were computed. Subsequently, error covariance functions for the MDT and the surface current components were computed and their variances were found. The results are summarized in Table 2. Table 2. A-posteori errors in cm and cm/s of MDT and geostrophic surface current components (u,v) (represented by the Cll) as point values and averaged in cells as estimated using the two GOCE simulations; to harmonic degree 200 and 360 respectively.
(u,O
MDT
Points ½o x ½o l°xl ° 2°x2 °
200 6 5 3 2
360 5 4 3 2
200 12 8 4 1
360 11 6 3 1
By comparing the numbers in Table 1 with the numbers in Table 2 it is obvious that the GOCE satellite mission will have a large impact on the estimation of the MDT. With the Level 2 product the error of point values is brought down substantially from 20 to 6 cm. The current components are associated with shorter wavelengths and moderately improved from 16 to 12 cm/s. For 1 x 1 degree averages however, the current components is improved from 10 to 4 cm/s. The solution obtained using least squares collocation improved the recovery of the geoid substantially. The impact on the estimation of the MDT is not that pronounced, since most of the signal contents in the MDT have a more long wavelength character. However, the MDT is improved at point values and ½ x ½ degree averages by about 20 %.
Chapter 2 • ErrorCharacteristicsof DynamicTopographyModels Derivedfrom Altimetry and GOCEGravimetry
The improvement has a larger impact of the current components. Compared to the standard deviations of the MDT the current components have been improved twice as much almost.
5
Perspectives
The impact of the GOCE satellite mission on the recovery of the gravity field has been analysed for two simulated cases. In the first case the GOCE Level 2 product is used where the gravity field is approximated by spherical harmonic coefficients up to degree and order 200. In the second case synthetic Level 1B GOCE data are used directly in a gravity field determination using least squares collocation. In case two the full spectrum geoid error was improved from 31 cm to 15 cm and the resolution was doubled. The results are important for the future users of GOCE that need the extra accuracy. Then the impact of the improved geoid on the estimation of the MDT is analysed. To get reliable errors associated with the MDT a reliable model for the spectral characteristics of the MDT is needed. Such a model was derived reflecting empirically derived properties such as MDT variance and correlation length. This model, naturally, is purely empirical and more information needs to be collected to verify the reliability of the model characteristics. Combining the MDT characteristics with the estimated geoid errors in the spectral domain resulted in a-posteori error estimates. In the two cases the MDT errors were improved from 20 cm to 6 cm and 5 cm respectively. For the geostrophic surface current components the errors were improved from 23 cm/s to 18 cm/s and 16 cm/s. Those results depend of the MDT a-priori degree variance model and will not be reliable unless that model is reliable. So the results may change accordingly. However, since this a-priori model actually do reflect empirically derived characteristics, they may not change that much.
Acknowledgement GOCINA is a shared cost project (contract EVG1CT-2002-00077) co-funded by the Research DG of the European Commission within the RTD activities of a generic nature of the Environment and Sustainable Development sub-programme of the 5th Framework Programme.
References
Heiskanen, W. A., and Moritz, H.: Physical Geodesy, W. H. Freeman, San Francisco, 1967. Knudsen, P.: Estimation and Modelling of the Local Empirical Covariance Function using gravity and satellite altimeter data. Bulletin Gdoddsique, Vol. 61,145-160, 1987. Knudsen, P.: Simultaneous Estimation of the Gravity Field and Sea Surface Topography From Satellite Altimeter Data by Least Squares Collocation. Geophysical Journal International, Vol. 104, No. 2, 307-317, 1991. Knudsen, P.: Estimation of Sea Surface Topography in the Norwegian Sea Using Gravimetry and Geosat Altimetry. Bulletin Gdoddsique, Vol. 66, No. 1, 27-40, 1992. Knudsen, P.: Integration of Altimetry and Gravimetry by Optimal Estimation Techniques. In: R. Rummel and F. Sans6 (Eds.): Satellite Altimetry in Geodesy and Oceanography, Lecture Notes in Earth Sciences, 50, Springer-Verlag, 453-466, 1993. Knudsen P., R. Forsberg, O. Andersen, D. Solheim, R. Hipkin, K. Haines, J. Johannessen & F. Hernandez. The GOCINA Project - An Overview and Status. Proc. Second international GOCE User Workshop "GOCE, The Geoid and Oceanography", ESA-ESRIN, March 2004, ESA SP-569, June 2004. Moritz, H.: Advanced Physical Geodesy. Herbert Wichmann Verlag, Karlsruhe, 1980. Sans6, F.: Statistical Methods in Physical Geodesy. In: Stinkel, H.: Mathematical and Numerical Techniques in Physical Geodesy. Lecture Notes in Earth Sciences, Vol. 7, 49-155, Springer-Verlag, 1986. Tscherning, C.C.: Functional Methods for Gravity Field Approximation. In: Stinkel, H.: Mathematical and Numerical Techniques in Physical Geodesy. Lecture Notes in Earth Sciences, Vol. 7, 3-47, Springer-Verlag, 1986. Tscherning, C.C.: Computation of covariances of derivatives of the anomalous gravity potential in a rotated reference frame. Manuscripta Geodaetica, Vol. 18, no. 3, pp. 115-123, 1993. Tscherning, C.C.: Simulation results from combination of GOCE SGG and SST data. 2. Int. GOCE user Workshop, ESRIN, ESA SP-569, March, 2004. Tscheming, C.C., and R.H. Rapp: Closed Covariance Expressions for Gravity Anomalies, Geoid Undulations, and Deflections of the Vertical Implied by Anomaly Degree Variances. Report no.
15
16
P. Knudsen. C. C. Tscherning
208, Dept. of Geodetic Science and Surveying, The Ohio State University, Columbus, 1974. Wunsch, C., and V. Zlotnicki: The accuracy of altimetric surfaces. Geophys. J. R. astr. Soc., 78, 795-808, 1984.
Zlotnicki, V.: On the Accuracy of Gravimetric Geoids and the Recovery of Oceanographic Signals from Altimetry. Marine Geodesy, Vol. 8, 129-157, 1984.
10 -1
i 0 -i
°~E.E'
10 .3
10 .3
>
1 0-5
>
0-5
121
D 10 .7
10 .7
_
GOCE - 200
10-9
10-9
. . . . . . . . . . . . . . . . . . . . 10
100
Harm.
1000
10
Figure 1. Geoid error degree variances associated with.GOCE Level 2 harmonic expansion to degree 200 and to degree 360 simulating the collocation solution
100 Harm. deg.
0.04 .
Coll." v = 15 cm 200+: v = 31 cm
~
.
.
.
.
.
.
~ ' ~
oq
.
.
.
.
.
.
.
0.04
E
L
.
.
.
.
.
t
CII: s = 16 cm/s, cl = 0.22 deg : = , = . deg
~
0.03
8 e-
1000
Figure 3. MDT signal degree variance model shown together with the geoid error degree variances associated with.GOCE Level 2 harmonic expansion to degree 200 and to degree 360 simulating the collocation solution
t
0.08
oOcE_200 -360 I
.....
deg.
°12t ...................
_
....
0.02
g
> 0
>
o
8
-0.04 ' 0
0.5
1.0
1.5
o.ol
2.0 -0.01
Lag [sph. degrees]
0
1
2
3
4
lag (deg)
Figure 2. Geoid error covariance functions associated with.GOCE Level 2 harmonic expansion to degree 200 and to degree 360 simulating the collocation solution
Figure 4. Covariance functions of the MDT and the geostrophic surface current components based of the MDT degree variance model shown in Figure 3.
Chapter 3
Detecting Ocean Currents from Satellite Altimetry, Satellite Gravity and Ocean Data Zizhan Zhang 1'2 Yang L u 1'3 Houtse HSU 1'3 1 Institute of Geodesy and Geophysics, Chinese Academy of Sciences, 340 Xudong Street, Wuhan, China, 430077 2 Graduate School of the Chinese Academy of Sciences, Beijing, China, 100049 3 Unite Center for Astro-geodynamics Research, CAS, Shanghai, China, 200030
Abstract: With the preparation and launch of the high accuracy geodetic missions CHAMP, GRACE and GOCE, the geoid models have been improved greatly. This gives us a good chance to combine a high precision sea surface height with improved geoids to estimate the dynamic ocean topography (DOT) and associated surface currents accurately. Combining the geoid model (EIGEN-CG01 C) from CHAMP and GRACE missions with the timeaveraged sea surface model KMS04, we calculate a new high accurate mean DOT, and then compare it with DOT derived from ocean hydrological data. When taking 3km as the reference depth datum, the correlation coefficient is larger than 0.9 and the discrepancy of mean values is very small between the DOT derived from altimetry/geoid and that from WOA01 salinity and temperature data. The surface geostrophic currents from the altimetry/geoid derived DOT map are very close to those from WOA01 derived DOT map. This independent knowledge of surface currents from the altimetry/geoid derived DOT is used in combination with the salinity and temperature data from NOAA's WOA01 to retrieve the ocean currents as a function of depth. At 2000db, comparisons of the retrieved velocities of the deep geostrophic currents with the WOCE current meter observation show that the spread of velocity deviation is approximately normal distribution, while the spread of velocity deviation is ruleless. The reasons caused these discrepancies need further analysis. Keywords. Ocean currents, satellite gravity, satellite altimetry, WOA01
1 Introduction Ocean currents transport mass and heat between different regions of the Earth, so knowledge of the ocean currents is vitally important for Earth sciences (i.e. climate, atmosphere, hydrology etc.), but global
measurements of them, especially the deep ocean currents, are very difficult. Fortunately, ocean currents are very close to the geostrophic balance on time scales longer than a few days. Therefore, the dynamic ocean topography, the deviation of the stationary sea surface from the marine geoid, is a direct measure of the dynamic pressure at the sea surface that leads to the geostrophic part of the surface current velocity (Dobslaw et al., 2004). Previously published dynamic ocean topography models suffered mainly from the poor quality of the marine geoid when derived solely from satellite tracking data, or were biased when using altimetry measurements for geoid determination (Le Traon et al., 2001). During the last two decades, satellite altimetry has offered an abundance of measurement of the sea surface height (SSH), which results in some high accuracy mean SSH models, such as CLS01 (Hernandez et al., 2001), GSFC00 (Wang, 2001). KMS01, KMS04 (Andersen et al., 2004) etc. With the advent of the new models from the dedicated gravity missions CHAMP and GRACE, the quality of the gravity field models based on satellite tracking data significantly increased (Tapley et al., 2003), especially, the long-wavelength components of gravity field models from GRACE missions (see Figure 1). The goals of this paper are that computing a new mean global DOT by combining the gravity field model EIGEN-CG01C from CHAMP and GRACE missions with the KMS04 mean SSH model from multi-satellite altimetry, revealing the relationship between altimetry/geoid derived DOT and WOA01 hydrological data derived DOT, and retrieving the surface geostrophic currents and deep ocean currents. Additionally, the results of currents are compared to observed velocities from WOCE current meters.
2 Data used Several sources of data are used in this work; they are
18
Z. Zhang. Y. Lu. H. Hsu 50
models (GGM01) actually gives very reliable geostrophic currents in the Arctic region (Andersen, 2004). To perform the comparing analysis, hydrological data from WOA01 annual analysis temperature and salinity fields (Conkright et al., 2002) are used to estimate the oceanography dynamic topography and to retrieve deep ocean currents, and WOCE current meter observation is included.
30
.[ "~
0.5 O.1 0.01 0.001 0
60
120
180
240
300
360
Spheri cal H a r m o n i c D e g r e e
Figure l:Error amplitudes as a function of maximum degree spherical harmonic degree of gravity field models (EGM96, GGM01 S, GGM02S, CG01 C, CG03C) in terms of geoid heights
the gravity field model EIGEN-CG01C, the mean sea surface model KMS04, the WOA01 temperature and salinity data and WOCE current meter observation. The newly obtained global mean gravity field model EIGEN-CG01C is a combination of GRACE mission (376 days out of February to May/July to December 2003 and February to July 2004) and CHAMP mission (860 days out of October 2000 to June 2003) data plus altimetry and gravimetry surface data (F6rste et al. 2005). The altimetry and gravity surface data used in CG01C include geoid undulations over the oceans derived from CLS01 altimetric SSH and ECCO simulated sea surface topography (Stammer et al., 2002), and NIMA gravity anomalies data etc. This model is complete to degree/order 360 in terms of spherical harmonic coefficients and resolves wavelengths of 110 km in the geoid and gravity anomaly fields. Compared to pre-CHAMP/GRACE global high-resolution gravity field models, the accuracy could be improved by one order of magnitude to 4 cm and 0.5 mgal in terms of geoid heights and gravity anomalies, respectively, at a spatial resolution of 400 km wavelength. The overall accuracy of the full model is estimated to be 20 cm and 5 mgal, respectively (Reigber et al., 2004). The mean sea surface KMS04 is derived from a combination of altimetry from a total of 5 different satellites and a total of 8 different satellite missions like the T/P, T/P Tandem Mission, ERS 1 ERM+GM, ERS2 ERM, GEOSAT GM, and GFO-ERM data. New data from the JASON and ENVISAT data are used to validate the mean sea surface models. This model covers the period 1993-2001using T/P as reference (Andersen et al., 2004). The resolution of the mean sea surface is 2 minutes equivalent to 4 km at the Equator. For the first time, the mean DOT derived from KMS04 and GRACE derived geoid
3 Primary theory The geoid is defined as the equipotential surface of the gravitational field that best fits the undisturbed ocean. This ocean will be in a state of equilibrium, subject only to the force of gravity, and free from variations with time (Torge, 2001). The difference the time-averaged sea surface H and the geoid undulation N (both related to the same conventional ellipsoid of revolution) is called the sea surface topography ( : (1)
(=H-N
Mostly, the altimetry-based SSH models have higher spatial resolution than the gravity field models, so a filter is needed to fade out the high frequency signals in SSH models. The motion of a fluid can be generally described by the Navier-Stokes equation combining the effects of pressure gradient, internal friction, gravity gradient and the Coriolis-force originating from the rotation of the Earth. For periods longer than one day, and away from oceanic boundaries (the coasts, the seafloor, the sea surface), the effects of internal friction and the Coriolis-force dominate all other accelerating forces can be neglected. Based on the geostrophic equilibrium and hydrostatic balance assumption, in local Cartesian coordinate system, the surface geostrophic current is o f the form (Apel, 1987, Hwang, 2000, Wahr et al., 2002) ~,
f u
= g__a(
f c~x =
(2)
g0( f
~Y
where Vs and Us are the east and north velocity components of the surface geostrophic currents, respectively; g is the local acceleration of gravity, f = 2f~sin ~b, with f2 being the earth's rotational rate (7.292115x104 rad s-i); ( i s the sea surface dynamic height.
Chapter 3 • Detecting Ocean Currents from Satellite Altimetry, Satellite Gravity and Ocean Data
i
-2.0
2.0
i-
-1.6 1.2
- 0.8
0.8
-0.4
0.4
,5,
- -0.0
--0.0
0 - -0.4
-0.4 -0.8 --1.2
i
--1.6
..,_. m ._1
- -0.8
-1.2
i
-50-
-2.0
0
60
120
180
240
300
- -1.6 - -2.0
m
m
(a)
1.6 - 1.2
0
360
Longitude (deg.)
60
120
180
(b)
240
300
360
Longitude (deg.)
Fig.2 (a) Mean DOT estimated by the K M S 0 4 model minus the CG01C geoid. (b) Mean DOT estimated from W O A 0 1 based on 3000db depth datum.
Let the atmosphere pressure at the sea surface equal to zero, the deep ocean geostrophic currents can be presented in the following form (Pond et al., 1978).
l.pS #
- - + V
z
u
=
g
z
s
----F
f.p
(3)
zi s
, -~y
where Vz and Uz are the east and north velocity components of the geostrophic currents at the z depth level. P is the pressure at z depth level. 4 C o m p a r i s o n of D O T s f r o m altimetry, geoid and f r o m W O A 0 1 The mean DOT (see Fig. 2 (a)) estimated by the KMS04 model minus the CG01C geoid is filtered with 400kin Gauss filter in order to reduce the noise signal and signal wavelength shorter than 400km. Figure 2 (b) is the mean DOT derived from WOA01 annual salinity and temperature data based on the reference depth datum 3000db. Comparing figure 2
0.6
'
I
'
I
'
I
'
.,,_, c
o
0.4
c
0.8
%
o
I
E 4-
0.6
1
I
E ,,,
o (1) o c O 'o-
0.2
~
correlationcoefficent
I _
(a)(b), it can be seen that the altimetry/geoid derived DOT map is very similar with WOA01 derived DOT (mean value is removed). Figure 3 (a)(b) shows the correlation coefficients and deviation between means of altimetry/geoid derived DOT and WOA01 derived DOT based on different reference depth datum in global area and in a test region with bounds of (El70 ° ~ 210 °, N0 ° ~ 40°). From figure 3 (a), one can see that, above 3kin, the correlation coefficients is 0.89 and deviation between the mean values of altimetry/geoid derived DOT and WOA01 derived DOT is o n l y - 0 . 1 c m (altimetry/geoid derived DOT minus WOA01 derived DOT). In the test region, when taking the reference depth shallower than 3km, the correlation coefficients is larger than 0.9 and deviation of mean values of altimetry/geoid derived DOT and WOA01 derived DOT is zero at 3.5kin deep level (see Figure 3 (b)). Both figure 3 (a) and (b) show that the correlation coefficients decrease quickly and deviation mean increase obviously as the reference depth datum becomes deeper than 3km. Several reasons can be used to explain this matter. One is
mOOT AG " mOOT WOA01
0,6
o~ (1) O o ,-
.9 ....,
~
0.4
0
%
E o
0.4
car) t-O~
i 0
(a)
I
1000
i
I
i
I
2000 3000 depth m
i
I
4000
i
0.2 5000
%
% -
o E
0.8
%
%
0.2
o.6 N O o
oo
c
0
0.4 %%%
c (1)
-"-
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-*--, C .o_
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o -0.2 -f.
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_
-0.4
I
0 (b)
%%
m~OrAG-m~orwoAo~ I
'
1000
I
-'
I
2000
-
3000
I
.o .,,_,
_
0.2 o
i
4000
I
0
5000
depth m
Fig. 3 Correlation coefficients and difference of the mean values of DOTs from altimetry/geoid and W O A 0 1 based on different reference depth datums. (a) (N82 ° ~$82 °, 0 ° ~ 360°), mDOT-AGnotes the mean value of DOT derived from altimetry/geoid. (b) (NO ° ~40 °, E170 ° ~ 210 °), mDOT-WOA01notes the mean value of DOT derived from W O A 0 1 salinity & temperature data.
19
20
Z. Zhang. Y. Lu. H. Hsu
I --40
I --20
I --15
--10
--5
I
"
0
5
I 15
10
I 20
I
I" cm,"s 40
-20
-12
,
I
'
i
,
,
-8
-4
0
4
8
12
~ cm,,~
Fig. 4 Surface geostrophic currents from altimetry (KMS04), geoid (CG01C) and WOA01 map on 3000db depth Left-zonal geostrophic currents, positive currents are toward the east. Right-Meridional geostrophic currents, positive currents are toward the north. (a) (top) from DOT derived from altimetry, geoid. (b) (bottom) from WOA01D poor quality or sparsity with ocean salinity and temperature data, as the reference depth deeper than 3km. The other is no enough quality with altimetry data in high latitude areas, near polar area, and geoid. So we consider that the reference depth datum at 3000db is reasonable for oceanography to determine the sea surface dynamic height.
5 Retrieval of surface geostrophic currents Tapley et al. (2003) computed geostrophic currents with an earlier GRACE geoid model. Similar to that study, we compute the currents with this DOT and show the results in Figure 4. From figure 4, it can be seen that both altimetry/geoid derived DOT and WOA01 derived DOT (except near equator N2 ° $2 °) show all major surface geostrophic currents (such as Kuroshio Extension, Gulf Stream, the
..r-I
.~
'
I ~ ' I
'
I
Antarctic Circumpolar Current (ACC)), especially those in the tropics. The locations and magnitudes are very similar to each other. The altimetry/geoid derived DOT maps show better details closer to land due to the depth required for the WOA01 maps. While the ACC from altimetry/geoid derived DOT is stronger than from WOA01 derived DOT, which may be due to sparsity of salt and temperature data or no effective experience density formula in this region. Figure 5 plots the correlation coefficients and the RMS of the difference of zonal components and meridional components from altimetry/geoid derived DOT and that from WOA01 derived DOT based on different depth datums. We can see that the correlation coefficients of two current components from altimetry/geoid derived DOT and WOA01 derived DOT are very larger and the RMS of the
1
' ~t20
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..-
I
'
I
'
I
'
I
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'
correlatio n coefficent
=
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16 o
(D u~
_
¢~ 0.6
correlation c
o o
12 fE
/
o
- - - - - - RMS of the difference
E
or)
.o,. _,e0.4 t,o
0.2
t
0 (a)
ItJ s
i
i 1000
i 2000 3000 depth m
-, 8 rv~
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--4
i
i 4000
t
'-
(19 o
0.8
-
-
. . . .
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16
of the difference
/
¢~ 0.6
/
o o
/
E
-,__....
o
,-,-,----,
0 0
(b)
¢..)
t
s J
,'- 0.2
5000
12"~ 8
_~°"-'¢0.4
0
o
f
1000
2000
3000
4000
co
n,' 4
0 5000
depth m
Fig. 5 Correlation coefficient and RMS of the difference of zonal and meridional surface geostrophic currents from the altimetry/geoid DOT and from the WOA01 DOT maps as a function of depth. (a) Zonal component (b) Meridional component
Chapter 3 • Detecting Ocean Currents from Satellite Altimetry, Satellite Gravity and Ocean Data
. . . . . . .
__-.
__
I
i
..&--l.
-
-
-
I
I
j--; I
,
-12
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-8
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-4
[
,
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--I--
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6
I
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-9
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-6
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_
I
_
,
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3
,
I' cm/s
6
9
Fig. 6 Deep geostrophic currents from altimetry (KMS04), geoid (CG01C) and WOA01 salinity & temperature data Left column-zonal geostrophic currents, positive currents are toward the east. Right column-meridional geostrophic currents, positive currents are toward the north. (a) (top) 1000db (b) (middle) 2000db (c) (bottom) 3000db difference of them is very small when the reference depth datums are shallower 3km than deeper, which indicates the reference depth datum on 3km is optimal. More important, the high correlation indicates that accuracy of the geostrophic currents derived from space measurements has been improved greatly. 6 Retrieval
of deep
geostrophic
currents
Figure 6 shows the deep geostrophic currents retrieved from altimetry/geoid derived DOT and
25
i
i
I
i
I
i _1
i
I
i
I
WOA01 salinity and temperature data at l km, 2km, and 3km depth levels, respectively. From figure 6, we see that deep currents grow weaker with depth, and the changes between them are not obvious, because the velocities of deep currents are very weaker, majority of them no more than 2 cm/s. Figure 7 shows the frequency distribution about the deviation between retrieval velocity from altimetry/ geoid & WOA01 and W O C E current meter observation at 2000db. We collect all W O C E current
25
i
2O
20 -
=~ 15
o 10 5 0
,
-:30 (a)
5 ,
-20
|, ||
-10
0
||,
10
Velocity deviation (cm/sec)
,
20
0 -180
,
30 (b)
-120
-60
0
60
120
180
Direction deviation (deg.)
Fig. 7 Frequency distribution about the deviation between retrieved velocity from altimetry, geoid & WOA01 and the WOCE current meter observations on 2000db. (a) (left) velocity deviation (b) (right) direction deviation
21
22
Z. Zhang. Y. Lu. H. Hsu
meter data at 2000db in global area, and choose the surveying time period longer than one year, from Jan. 1992 to Dec. 1995, to compare with the velocity and direction retrieved from altimetry/geoid and salinity and temperature data. The total number of valid points is 32. We interpolate the altimetry/geoid derived point velocity and direction to the current meter position. The statistic discrepancy about velocity deviation and direction deviation at 2000db are illustrated in Figure7 (a) (b). From figure 7(a), it can be see that the spread of the difference of velocity is approximately normal distribution and the average difference is about 1.4cm/sec. While the spread of the direction deviation at 2000db is ruleless, their differences of direction, beyond 68% of the total number, are larger than 45 ° . It is difficult to explain what caused these discrepancy, because we do not know which, sea surface height errors, geoid errors or uncertainty of salinity and temperature data, has m u c h effect on these discrepancy. Additionally, the current meters provide all components of deep ocean currents, while retrieved currents are only the geostrophic components.
7 Conclusion and further w o r k The correlation is very high and the discrepancy of means is much small between altimetry/geoid derived D O T and WOA01 derived DOT, when WOA01 takes 3km as the reference depth datum. The surface geostrophic currents derived from altimetry and satellite gravimetry recovered earth gravity model (CG01C) are very close to currents from WOA01 based on 3km depth datum, which indicates that accuracy of the geostrophic currents derived from space measurements has been improved greatly. The deep geostrophic currents change unobviously as depth increasing. The retrieved deep ocean currents have discrepancy with W O C E current meter observations. The reasons need to be further analysed.
Acknowledgements The authors are grateful to Dr. Don Chambers and the reviewers of this paper for their helpful recommendations. The data used in this work were obtained from GeoForschungsZentrum Potsdam, NOAA, CSR of Texas University and Ole B. Andersen etc. in Danmark. This work was supported by the National Natural Science Foundation of China (Grant No. 40374007, 40234039), Foundations of Chinese Academy of Science (Grant N o s . KZCX2-SW-T1, KZCX3-SW-132), Foundation of marine 863 (Grant No. 2002AA639280) and the Informatization Construction of Knowledge Innovation Projects of the Chinese Academy of Sciences "Supercomputing Environment Construction and Application" (Grant No. INF 105-SCE).
References Andersen O. B., A. L. Vest, and R Knudsen (2004). KMS04 mean sea surface and inter-annual sea level variability. EGU meeting poster, April 2004, Nice, France Andersen O. B (2004). Sea Level Determination from satellite altimetry-Recent development. Observing and Understanding Sea Level Variations, 36 Apel, J. R (1987). Principles of Ocean Physics, pp. 634, Academic, San Diego Calif. Conkright M. E., R. A. Locarnini and H. E. Garcia et al (2002). WORLD OCEAN ATLAS 2001:Objective Analyses, Data Statistics, and Figures CD-ROM Documentation. National Oceanographic Data Center Internal Report 17 Dobslaw H., R Schwintzer and F. Barthelmes, et al (2004). Geostrophic Ocean Surface Velocities from TOPEX Altimetry, and CHAMP and GRACE Satellite Gravity Models. Scientific Technical Report, GeoForschungsZentrum F6rste C, F. Flechtner and R. Schmidt et al. (2005). A new high-resolution global gravity field model derived from combination of grace and champ mission and altimetry/gravimetry surface gravity data. European Geosciences Union, General Assembly 2005, Vienna, Austria, 24-29, April, 2005 Gruber T and P. Steigenberger (2002). Impact of new Gravity Field Missions for Sea Surface Topography determination, in: I.N. Tziavos (Ed.), Gravity and Geoid, 3rd Meeting of the International Gravity and Geoid Commission (IGGC), Univ. Of Thessaloniki, Greece, 320-325 Hernandez F. and P. Schaeffer (2001). The CLSO1 Mean Sea Surface." A validation with the GSFCO0 surface, in press. CLS Ramonville St Agne, France Hwang C. and Chen S. (2000). Circulation and eddies over the South China Sea derived from TPOEX/Poseidon altimetry. Journal of Geophysical Research, 105(C10): 23943~23965 Le Traon P. Y., P. Schaeffer and S. Guinehut, et al. (2001). Mean Ocean Dynamic Topography from GOCE and Altimetry, Int. GOCE Workshop, Noordwijk Pond, S. and L. Pickard (1978). Introductory Dynamic Oceanography. Pergamon Press. First Edition. Roemmich, D., and C. Wunsch, (1982). On combining satellite altimetry with hydrographic data, Journal Marine Research, 40, 605-619. Reigber C, Schwintzerl P and Stubenvolll R, et al. A high resolution global gravity field model combining CHAMP and GRACE satellite mission and surface gravity data. Joint CHAMP~GRACE Science Meeting, Solid Earth Abstracts, 2004, 17 Tapley B. D, D. P. Chambers and S. Bettadpur, et al. (2003). large-scale ocean circulation from the GRACE GGM01 geoid. Geophysical Research Letter, 30:22, 2163 Torge W (2001). Geodesy, de Gryter, Berlin Wahr J. M., S. R. Jayne and F. O. Bryan, (2002). A method of inferring changes in deep ocean currents from satellite measurements of time-variable gravity. Journal of Geophysical Research, vol. 107, C12:3218 Wang Y. M. (2001). GSFC00 mean sea surface, gravity anomaly, and vertical gravity gradient from satellite altimeter data. Journal of Geophysical Research, vol. 106 (C12): 31167-31174
Chapter 4
Sea Level in the British Isles" Combining Absolute Gravimetry and Continuous GPS to Infer Vertical Land Movements at Tide G a u g e s F.N. Teferle, R.M. Bingley, A.I. Waugh, A.H. Dodson Institute of Engineering Surveying and Space Geodesy University of Nottingham, University Park, Nottingham NG7 2RD, UK. S.D.R Williams, T.F. Baker Proudman Oceanographic Laboratory, Joseph Proudman Building, 6 Brownlow Street, Liverpool L3 5DA, UK.
Abstract.
The current terrestrial reference frame, current global GPS products and current precise GPS processing techniques, limit the determination of accurate, long-term, vertical station velocities from continuous GPS measurements on a global scale. Several authors have reported biases in their vertical station velocities determined from continuous GPS when compared to alternative geodetic methods. It has been argued that until these problems have been resolved, the study of relative land and sea level rates on regional scales is the only way to investigate vertical land movements at tide gauges co-located with continuous GPS. In the UK, we have been operating a network of continuous GPS and absolute gravimetry stations for the purpose of determining vertical land movements at tide gauges for almost ten years. This network consists of ten continuous GPS stations and three absolute gravimetry stations, all of which are either co-located or close to tide gauges. In this paper, we compare vertical land movements obtained from both geodetic methods with estimates of vertical land movements from high quality, independent geological and geophysical evidence, and derive a GPS-specific bias for which the estimates of vertical land movements from all continuous GPS stations are corrected. Based on recently published mean sea level trends by the Permanent Service for Mean Sea Level, we estimate a change in sea level, de-coupled from vertical land movements, for the British Isles.
Keywords. absolute gravimetry, continuous GPS, tide gauge, sea level, British Isles
1
Introduction
Recent studies of 20th century sea level from combined tide gauge and satellite altimetry measurements showed a global averaged rise in sea level of 1.7 to 1.8 mm/yr (Church et al., 2004; Holgate and Woodworth, 2004; White et al., 2005; Church and White, 2006). Furthermore, it was identified that sea level was not rising at the same rate everywhere, but was, e.g. slower in the east Atlantic and North Sea than in the west Atlantic (Church et al., 2004; Holgate and Woodworth, 2004). |n all such studies, evidence from tide gauges is used extensively, which must be de-coupled from the specific vertical land movements (VLM) occurring at each of the tide gauge sites. By using continuous GPS (CGPS) at or close to tide gauges (CGPS@TG) (e.g. Teferle et al., 2002a; Caccamise et al., 2005), estimates of VLM can be obtained from vertical station velocity estimates derived from height time series and used to correct the tide gauge records. However, it is now clear that biases related to, e.g. the terrestrial reference frame (e.g. Herring, 2001; Dong et al., 2002, 2003), the relative modelling of the phase centers at satellite and receiver antennas (e.g. Ge et al., 2005; Schmid et al., 2005) and the neglect of higherorder ionospheric terms (e.g. Fritsche et al., 2005), all affect vertical station velocity estimates in some form or another. These biases currently prevent estimates of VLM for CGPS@TG stations to be obtained with the accuracy required for sea level studies. As it is generally believed that these biases cancel out over small regions, Caccamise et al. (2005) argued that, currently, investigations into VLM and changes in sea level at CGPS@TG stations can only be carried out in a relative sense over regional, and not global, scales.
24
F.N. Teferle • R. M. Bingley. A. I. Waugh • A. H. Dodson • S. D. P. Williams. T. F. Baker
Absolute gravimetry (AG) measurements, however, do not suffer from the above biases, hence, vertical station velocity estimates from this technique provide accurate estimates of VLM, independent of the problems associated with the GPS. Excellent previous results (e.g. Williams et al., 2001; Lambert et al., 2006) confirm the ability of AG to deliver accurate estimates of VLM, and as this paper shows, have enabled an assessment of the vertical station velocity estimates based purely on CGPS and a demonstration of how CGPS and AG may be combined to obtain better estimates of the desired VLM. Since the mid 1990s the Institute of Engineering Surveying and Space Geodesy (IESSG) and the Proudman Oceanographic Laboratory (POL) have been using AG and CGPS to measure the VLM at or close to tide gauges in the UK. Teferle et al. (2002a,b) reported of the establishment of five CGPS@TG stations between 1997 and 1999 and produced the first vertical station velocity estimates for these. Since then, the IESSG and POL have set up five more CGPS@TG stations in the UK. Williams et al. (2001) discussed the establishment of three AG sites close to tide gauges in the UK in 1995 and 1996 and reported of their initial results. This paper presents an update from Teferle et al. (2006) and was prompted by new estimates of VLM and new mean sea level (MSL) trends from the Permanent Service for Mean Sea Level (PSMSL) for the CGPS@TG stations used.
2
CGPS
Measurements
and
Analysis
During the period from 1997 to 2005, the IESSG and POL established CGPS stations at ten tide gauges in the UK, namely Sheerness, Newlyn, Aberdeen, Liverpool, Lowestoft, North Shields, Portsmouth, Lerwick, Stornoway and Dover. All of these CGPS stations were established such that the GPS antennas were sited as close as possible to the tide gauge, i.e. within a few meters of the tide gauge itself. As the CGPS@TG stations Lerwick, Stornoway and Dover have only been established in late 2005, no data from these were used in this study. At Lerwick, data from a CGPS station, initially established in 1998 for the purpose of estimating integrated precipitable water vapour, are available for investigating VLM. This additional CGPS station is located within 5 km of the newly established CGPS@TG station and the GPS antenna is mounted on a survey monument connected to bedrock. Although, this means that this station is not monitoring
the VLM at the tide gauge directly, it is monitoring the underlying geophysical movements in the area (Teferle et al., 2002a). As the observation timespan for the CGPS@TG station Lerwick is still too short to give reasonable results and by assuming that there are no differential VLM between the two CGPS stations within the area, this study only uses data from the CGPS station 5 km from the tide gauge in Lerwick. An earlier study (Sanli and Blewitt, 2001) claimed to detect uplift of the tide gauge in North Shields by analysing episodic and six months of CGPS data collected over a period of only 2.5 years up to early 2000. In contrast, this analysis uses the same data as Sanli and Blewitt (2001) plus data from the period since the installation of the CGPS receiver in 2001. Furthermore, Teferle et al. (2003) showed that GPS data from North Shields are affected by severe radio frequency interference, and that potentially a much longer observation period will be needed in order to form any conclusions on the VLM at that tide gauge with confidence. Therefore, the results for North Shields and Portsmouth, which currently only has an observation timespan of 3.1years, are still considered as preliminary. To complement the eight CGPS stations of Aberdeen, Lerwick, Liverpool, Lowestoft, Newlyn, Portsmouth and Sheerness, the CGPS@TG station Brest in northern France, has also been used in this study, making a total of nine CGPS stations at or close to tide gauges (see Fig 1). The processing for all of the CGPS stations shown in Fig. 1 has been carried out using the GPS Analysis Software (Stewart et al., 2002), which was developed at the IESSG and uses the double-difference observable. The results presented in this study are based on 24-hour, dual frequency GPS data for the period up to November 2004. All CGPS data were processed along with data from IGS stations at Kootwijk, Onsala, Villafranca and Wettzell, and consistent coordinate time series for each of the CGPS stations were obtained in the ITRF2000 (Altamimi et al., 2002). More details of the GPS processing methodology applied can be obtained from Teferle (2003). In order to obtain accurate vertical station velocity estimates and realistic uncertainties from the CGPS results, it is important to understand the timecorrelated (coloured) noise content of height time series, as the often quoted statistical uncertainties, assuming time-uncorrelated (white) noise, can lead to largely optimistic error bounds (e.g. Langbein and Johnson, 1997; Zhang et al., 1997; Mao et al., 1999; Williams et al., 2004). By using Maximum-
Chapter 4 • Sea Level in the British Isles: Combining Absolute Gravimetry and Continuous GPS to Infer Vertical Land Movements at Tide Gauges
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For the British Isles, a number of alternative, high quality and independent evidence of VLM have been published. These data include, estimates based o n : geological information (Shennan and Horton, 2002), denoted as GEO; the negative of the difference between the MSL trend measured by the tide gauge at each site (Woodworth et al., 1999; PSMSL, 2005) and the average sea level rise for Northern Europe of 1.5 mm/yr (Holgate and Woodworth, 2004), denoted as -(MSL-GSL); glacial isostatic adjustment models, e.g. Lambeck and Johnston (1995), Peltier (2001), denoted as GIA(L) and GIA(P) respectively. Table 1 and Fig. 4 show the independent estimates along with the values obtained from CGPS and AG. From Table 1, it is clear that there is generally good agreement between the estimates of VLM based on geology, tide gauges and GIA models, at all sites except Brest and Lerwick. Taking the estimates based on AG into account, these would seem to suggest that the anomaly at Lerwick is in the tide
Table 1" Vertical land movements comparison. All figures shown are in mm/yr. Station CGPS Sheerness 0.2-+-0.4 Lerwick 0.1±0.7 Newlyn 0.7±0.7 Aberdeen 1.4±0.7 Liverpool 1.2±0.7 Lowestoft 0.14-0.6 Brest 1.6+0.8 North Shields 1.2±0.7 Portsmouth 0.5±0.5 Station -(MSL-GSL) Sheerness -0.1 Lerwick 2.3 Newlyn -0.2 Aberdeen 0.6 Liverpool 0.1 Lowestoft - 1.0 Brest 0.5 North Shields -0.4 Portsmouth -0.3 afrom Woodworth et al. (1999)
AG -1.14-1.1 -0.5±0.9
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GEO -0.7 -11 07 -0 2 -0 6 - 0 2a 02 -0 6 GIA(P) -02 -05 -03 06 04 -04 -03 04 -01
Chapter 4 • Sea Level in the British Isles: Combining Absolute Gravimetry and Continuous GPS to Infer Vertical Land Movements at Tide Gauges
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Fig. 4" Vertical land movement (VLM) estimates for nine CGPS stations close to or at tide gauge sites in the UK and northern France compared to alternative evidence for VLM. l-or uncertainties are shown where available. From Fig. 4, however, it is also clear that the CGPS estimates of VLM are systematically offset from the AG estimates of VLM and from the estimates of VLM from the other independent evidence. It would appear that the AG estimates of VLM are closely aligned to the estimates of VLM from geology and G|A models. Whereas, the CGPS estimates are in the range of 0.7 to 1.8 mm/yr greater than the geology estimates, 0.6 to 2.5 mm/yr greater than the GIA(L) estimates, and 0.4 to 1.9 mm/yr greater than the GIA(P) estimates. Similar offsets of CGPS estimates of VLM from independent evidence have also been reported by Prawirodirdjo and Bock (2004) and MacMillan (2004). From the analysis of a global CGPS network, Prawirodirdjo and Bock (2004) showed average offsets between CGPS and GIA(P) of 1.1 mm/yr for stations in North America, and 1.7 mm/yr for stations in northern Europe. Separately, in a comparison of VLBI and CGPS at 24 globally distributed sites, MacMillan (2004) showed the CGPS estimates
to be on average 1.5 mm/yr greater than the VLBI estimates. In considering the nature of the offsets apparent in our CGPS estimates, it is worth noting that the various sources of independent evidence have their own reference frames, e.g. the GIA(L) and GIA(P) estimates are based on GIA models for the last 10,000 years, and are referred to a Centre of mass of the Solid Earth reference frame (Blewitt, 2003; Dong et al., 2003), and the geology estimates are based on changes in sea level for the last 10,000 years, assuming no net global melting for the last 3,000 to 4,000 years. The ITRF2000 reference frame, to which our CGPS solutions are aligned, has an origin that is defined as the Center of Mass of the Earth Syswm (Blewitt, 2003; Dong et al., 2003), based on Satellite Laser Ranging. However, the CGPS estimates given in this paper are effectively referenced to their own realization of the ITRF2000 reference frame and, hence, their own definition of the centre of mass
27
28
F.N. Teferle • R. M. Bingley. A. I. Waugh • A. H. Dodson • S. D. R Williams. T. F. Baker
3
(CM), which depends mostly on the sub-set of 1GS stations constrained in ITRF2000 and partly on the fact that the IGS final orbit is in its own reference frame that is not exactly in ITRF2000. Hence, the CGPS estimates of VLM could be offset from the truth because the reference frame of the regional network solutions is not identical to ITRF2000, and because the origin of ITRF2000 varies with respect to the true CM (Blewitt, 2003; Dong et al., 2002, 2003).
5
Combining CGPS and AG and Computing a Sea Level Rise
Using weighted least-squares and data for Newlyn and Lerwick it is possible to compute an offset of 1.2+0.4mm/yr between the VLM estimates based on CGPS and those based on AG. This positive offset is consistent with that found in the comparisons between CGPS and the other independent evidence, both in this paper and e.g. Prawirodirdjo and Bock (2004) and MacMillan (2004). Therefore, the combination of CGPS and AG for tide gauges in the UK has been effected by aligning the CGPS estimates of VLM to the AG estimates. The MSL trends and the AG-aligned CGPS estimates of VLM have been used to compute an estimate of sea level rise for the British Isles. Fig. 5 shows the MSL trends compared to the negative of the AG-aligned CGPS estimates of VLM. As stated previously, the Lerwick tide gauge measurements only began in the 1960s and show a fall of mean sea level, which appears to be an anomaly specific to this tide gauge. Considering Sheerness, Newlyn, Aberdeen, Liverpool, Lowestoft, Brest, North Shields and Portsmouth, a sea level rise of 1.3 +0.3 mm/yr is obtained. Furthermore, the exclusion of data for Portsmouth and Lowestoft, the tide gauges with the shortest tide gauge records, does not change this estimate of 1.3 mm/yr.
6
Conclusions
This paper provides details of research that is ongoing in relation to the use of CGPS and AG for measuring vertical land movements (VLM) at tide gauges in the UK. Results, from CGPS time series dating back to 1997 and AG time series dating back to 1995/6, have been used to demonstrate the complementarity of these two techniques, and a series of AG-aligned CGPS estimates of VLM have been computed for nine tide gauges. An initial comparison between these estimates of VLM and changes in MSL observed by the tide gauges, suggests a change in sea level, de-coupled from VLM, as a rise of
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Fig. 5: MSL trends for seven tide gauges compared with the negative of the vertical station velocity estimates (E/S or emergence/submergence rates) based on AG-aligned CGPS. In this figure, SHEE is Sheerness, LERW is Lerwick, NEWL is Newlyn, ABER is Aberdeen, LIVE is Liverpool, LOWE is Lowestoft, BRST is Brest, NSTG is North Shields and PMTG is Portsmouth. Data points on the dashed line would imply a sea level rise of 1.0 mm/yr.
1.3 mm/yr for the British Isles. Clearly, the statistical significance of such results cannot be assured as yet, due to the level of the uncertainties in the CGPS and AG time series. However, these should reduce as the time series are extended into the future. It has recently been argued that studies of VLM using CGPS data and sea level changes at tide gauges are best performed on regional scales and more importantly only in a relative sense. However, the alignment procedure demonstrated in this paper has been shown to have the potential for determining sitespecific VLM at tide gauges, by using a combination of CGPS and AG. Furthermore, the procedure enables multiple CGPS stations to be deployed without the need for simultaneous AG measurements at each site.
Acknowledgements This research was funded by the Department for Environment, Food and Rural Affairs (Defra) R&D Projects FD2301 and FD2319. The CGPS data were taken from BIGF - British Isles GPS archive Facility (www.bigf.ac.uk) funded by the Natural Environment Research Council (NERC). The authors would
Chapter 4 • Sea Level in the British Isles: Combining Absolute Gravimetry and Continuous GPS to Infer Vertical Land Movements at Tide Gauges
also like to thank the two anonymous reviewers for their helpful comments.
References Altamimi, Z., P. Sillard, and C. Boucher (2002), ITRF2000: A new release of the International Terrestrial Reference Frame for Earth science applications. J. Geophys. Res., 107 (B 10), 2214, 10.1029/2001JB000561. Blewitt, G. (2003), Self-consistency in reference frames, geocenter definition, and suface loading of the solid Earth. J. Geophys. Res., 108 (B2), 2103, 10.1029/2002JB002082.
Ge, M., G. Gendt, G. Dick, F.P. Zhang, and C. Reigber (2005), Impact of GPS satellite antenna offsets on scale changes in global network solutions. Geophys. Res. Lett., 32, L06310, 10.1029/2004GL022224. Herring, T.A. (2001), Vertical reference frame for sea level monitoring. EOS Trans. AGU, 82 (47), Fall Meet. Suppl., Abstract G31D-07. Holgate, S. and P.L. Woodworth (2004), Evidence for enhanced coastal sea level rise during the 1990s. Geophys. Res. Lett., 31, L07305, 10.1029/2004GL019626.
Boucher, C., Z. Altamimi, P. Sillard, and M. Feissel-Vernier (2004), The ITRF2000, international Earth Rotation Service (IERS). Technical Note 31, Verlag des Bundesamts ftir Kartographie und Geod/isie, Frankfurt am Main.
Lambeck, K. and P.J. Johnston (1995), Land subsidence and sea-level change: Contributions from the melting of the last great ice sheets and the isostatic adjustment of the Earth. In: Land Subsidence, F.B.J. Barends, F.J.J. Brouwer, and F.H. Schr6der (eds.), 3-18, Balkema, Rotterdam.
Caccamise ||, D.J., M.A. Merrifield, M. Bevis, J. Foster, Y.L. Firing, M.S. Schenewerk, F.W. Taylor, and D.A. Thomas (2005), Sea level rise at Honolulu and Hilo, Hawaii: GPS estimates of differential land motion. Geophys. Res. Lett., 32, L03607,
Lambert, A., N. Courtier, and T.S. James (2006), Long-term monitoring by absolute gravimetry: Tides to postglacial rebound. J. Geodyn., 41,307317, 10.1016/j.jog.2005.08.032.
10.1029/2004GL021380. Church, J.A., N.J. White, R. Coleman, K. Lambeck, and J.X. Mitrovica (2004), Estimates of the Regional Distribution of Sea Level Rise over the 1950-2000 Period. J. Clim., 17, 2609-2625. Church, J.A. and N.J. White (2006), A 20th century acceleration in global sea-level rise. Geophys. Res. Lett., 33, L01602, 10.1029/2005GL024826. Dong, D., P. Fang, Y. Bock, M.K. Cheng, and S. Miyazaki (2002), Anatomy of apparent seasonal variations from GPS derived site position time series. J. Geophys. Res., 107(B4), 10.1029/2001JB000573. Dong, D., T. Yunck, and M. Heflin (2003), Origin of the International Terrestrial Reference Frame. J. Geophys. Res., 108(B4), 2200, 10.1029/2002JB002035. Fritsche, M., R. Dietrich, A. Kn6fel, A. Rfilke, S. Vey, M. Rothacher, and P. Steigenberger (2005), Impact of higher-order ionospheric terms on GPS estimates. Geophys. Res. Lett., 32, L23311,
10.1029/2005GL024342.
Langbein, J. and H.O. Johnson (1997), Correlated errors in geodetic time series: Implications for time-dependent deformation. J. Geophys. Res., 102 (B1), 591-603, 10.1029/96JB02945. MacMillan, D.S. (2004), Rate Difference Between VLBI and GPS Reference Frame Scales. Eos Trans. AGU, 85 (47), Fall Meet. Suppl., Abstract G21B-05. Mao, A., C.G.A. Harrison, and T.H. Dixon (1999), Noise in GPS coordinate time series. J. Geophys. Res., 104 (B2), 2797-2818. Niebauer, T.M., G.S. Sasegawa, J.E. Faller, R. Hilt, and F. Klopping (1995), A new generation of absolute gravimeters. Metrologia, 32, 159-180. Peltier, W.R. (2001), ICE4G (VM2) glacial isostatic adjustment corrections. In: Sea Level Rise History and Consequences. International Geophysics Series, 75, Academic Press, San Diego, 65-96. Penna, N.T. and M.R Stewart (2003), Aliased tidal signatures in continuous GPS height time series. Geophys. Res. Lett., 30(23), B08401, 10.1029/2004JB003390.
29
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E N. Teferle • R. M. Bingley. A. I. Waugh • A. H. Dodson • S. D. P. Williams. T. E Baker
Prawirodirdjo, L. and Y. Bock (2004), Instantaneous global plate motion model from 12 years of continuous GPS observations. J. Geophys. Res., 109 (8), B08405, 10.1029/2003JB002944. PSMSL (2005), Table of MSL secular trends derived from PSMSL RLR data [online]. Liverpool: Permanent Service for Mean Sea Level (PSMSL). Available at: < U R L : h t t p : / / w w w . p o l . a c . uk/psmsl/datainfo/rlr.trends> [Accessed 16 August 2005]. Sanli, D.U. and G. Blewitt (2001), Geocentric sea level trend using GPS and > 100-year tide gauge record on a postglacial rebound nodal line. J. Geophys. Res., 106 (B 1), 713-719, 10.1029/2000JB900348. Schmid, R., M. Rothacher, D. Thaller, and R Steigenberger (2005), Absolute phase center corrections of satellite and receiver antennas. GPS Sol., 9,283293, 10.1007/s 10291-005-0134-x. Sella, G.F., T.H. Dixon, and A. Mao (2002), REVEL: A model for recent plate velocities from space geodesy. J. Geophys. Res., 107 (B4), 10.1029/2000JB000033. Shennan, I. and B. Horton (2002), Holocene landand sea-level changes in Great Britain, J. Quaternary Sci., 17 (5-6), 511-526. Stewart, M.R, G.H. Ffoulkes-Jones, W.Y. Ochieng, P.J. Shardlow, N.T. Penna, and R.M. Bingley (2002), GAS: GPS Analysis Software version 2.4 user manual. IESSG, University of Nottingham, Nottingham, U.K. Teferle, F.N., R.M. Bingley, A.H. Dodson, and T.F. Baker (2002a), Application of the dual-CGPS concept to monitoring vertical land movements at tide gauges. Phys. Chem. Earth, 27, 1401-1406. Teferle, F.N., R.M. Bingley, A.H. Dodson, N.T. Penna, and T.F. Baker (2002b), Using GPS to separate crustal movements and sea level changes at tide gauges in the UK. In: H. Drewes, A.H. Dodson, L.P.S. Fortes, L. Sanchez and P. Sandoval (eds), Vertical Reference Systems. International Association of Geodesy Symposia, 124, SpringerVerlag, Heidelberg Berlin, 264-269. Teferle, F.N. (2003), Strategies for long-term monitoring of tide gauges with GPS. PhD thesis, University of Nottingham, [Available at: < URL : http ://etheses .nottingham. ac. uk>].
Teferle, F.N., R.M. Bingley, A.H. Dodson, P. Apostolidis, and G. Staton (2003), RF Interference and Multipath Effects at Continuous GPS Installations for Long-term Monitoring of Tide Gauges in UK Harbours. In: Proc. 16th Tech. Meeting of the Sat.
Div. of the Inst. of Navigation, ION GPS/GNSS 2003, Portland, Oregon, 9-12 September 2003, pp. 12. Teferle, F.N., R.M. Bingley, S.D.R Williams, T.F. Baker, and A.H. Dodson (2006), Using continuous GPS and absolute gravity to separate vertical land movements and changes in sea level at tide gauges in the UK, Phil. Trans. R. Soc. A, 364, 1841, 10.1098/rsta.2006.1746. Van Camp, M., S.D.R Williams, and O. Francis (2005), Uncertainty of absolute gravity measurements. J. Geophys. Res., 110, B05406, 10.1029/2004JB003497. White, N.J., J.A. Church, and J.M. Gregory (2005), Coastal and global averaged sea level rise for 1950 to 2000. Geophys. Res. Lett., 32, L01601, 10.1029/2004GL021391. Williams, S.D.P. (2003), The effect ofcoloured noise on the uncertainties of rates estimated from geodetic time series. J. Geod., 76 (9-10), 483-494. Williams, S.D.P., T.F. Baker, and G. Jeffries (2001), Absolute gravity measurements at UK tide gauges. Geophys. Res. Lett., 28 (12), 2317-2320, 10.1029/2000GL012438. Williams, S.D.R, Y. Bock, P. Fang, P. Jamason, R.M. Nikolaidis, L. Prawirodirdjo, M. Miller, and D.J. Johnson (2004), Error analysis of continuous GPS position time series. J. Geophys. Res., 109(B3), B03412, 10.1029/2003JB002741. Woodworth, P.L., M.N. Tsimplis, R.A. Flather, and i. Shennan (1999), A review of the trends observed in British Isles mean sea level data measured by tide gauges. Geophys. J. Int. 136, 651-670. Zhang, J., Y. Bock, H.O. Johnson, P. Fang, S.D.P. Williams, J. Genrich, S. Wdowinski, and J. Behr (1997), Southern California Permanent GPS Geodetic Array: Error analysis of daily position estimates and site velocities. J. Geophys. Res., 102(B8), 18035-18055, 10.1029/97JB01380.
Chapter 5
Snow Accumulation and Snowmelt Monitoring in Greenland and Antarctica S. V. Nghiem 1, K. Steffen 2, G. Neumann 1, and R. Huff2 ~Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, 300-235, Pasadena, CA 91007, U.S.A. 2Cooperative Institute for Research in Environmental Sciences, University of Colorado, Campus Box 216, Boulder, CO 80309-0216, U.S.A.
Abstract. Snow deposition, accumulation, and melt on an ice sheet are key components of mass balance. Innovative algorithms using satellite scatterometer data have been developed to monitor snowmelt, ice layer extent, and snow accumulation on Greenland with verifications using in-situ data from the Greenland Climate Network (GC-Net). QuikSCAT/SeaWinds Scatterometer (QSCAT) has collected data over Greenland and Antarctic two times per day since July 1999. QSCAT data show the shortest melt season in 2004, verified by GCNet data at ETH/CU Camp, and detect peculiar snowmelt during wintertime in Greenland in 2005. QSCAT results reveal a record increase in the snow accumulation rate on the Greenland ice sheet including the west flank in January-March 2005 with an estimate of 565 km 3 of total snow accumulation volume. The record snow anomaly is verified by GC-Net snow measurements, showing the largest snow accumulation rate in the first half of 2005 ever recorded in the past decade since the inception of the GC-Net. The QSCAT algorithms developed for Greenland are adapted for Antarctica. QSCAT results show strong melt in 2002 and prolonged melt in 2005 at McMurdo. New extensive ice layers, created by refreezing of melt water in the firn layer, were identified by QSCAT along the Antarctica Walgreen, Bakutis, and Hobbs coasts extending well inland in 2005. Extensive regions of ice layering, evidence of preceding strong melt occurrence, were also found over the Rockefeller Plateau and along the Ross Ice Shelf adjacent to Queen Maud Mountains in 2005.
Keywords. QuikSCAT, ICESat, ice mass balance, snowmelt, snow accumulation, sea level, Greenland, Antarctica
1 Introduction The possibility of future sea level rise necessitates a precise knowledge of the mass balance of the large ice sheets in Greenland and Antarctica. Understanding the ice-sheet melt characteristics is critical to the assessment of ice sheet mass balance and the interpretation of mass balance observations. Because of the positive albedo feedback associated with snow melt and the fact that wet snow absorbs as much as three times more incident solar energy than dry snow (Steffen (1995)), ice sheet melt characteristics play a major role in the energy and mass exchanges at the ice sheet surface. Moreover, surface melt can act to enhance the flow of outlet glaciers through crevasse overdeepening (Robin, (1974), van der Veen (1998)) and is believed to have contributed to the very rapid thinning of a number of outlet glaciers in Eastern Greenland (Krabill et al. (1999)) due to increase of ice velocity (Zwally et al. (2002)). As a result, not only is ice sheet melt directly tied to ablation through surface runoff, but is also indirectly linked to fresh water discharge in forms of ice bergs through the potential increase in ice velocity of tide water glaciers. The northern parts of the Greenland ice sheet and most of Antarctica have low accumulation rates. Consequently, it takes a long time for the temperature signal of the currently warming climate to penetrate into these ice masses; hence, the effects are unlikely to be very large over a century timescale. However, in areas with increased melt water percolation as observed over large regions of the Greenland ice sheet and some low-lying coastal regions of Antarctica, including the Peninsula, the effect of latent heat release is likely to cause a faster response in the thermal regime. In a warming climate, we would expect an increase in vapor flux convergence that may deposit more snow at higher-
32
S.V. N g h i e m • K. Steffen • G. N e u m a n n • R. Huff
altitude regions. These regions are extensive and any change in snow accumulation is crucial to the total mass balance. Furthermore, monitoring the Greenland and Antarctic ice sheet melt characteristics and snow accumulation can facilitate the assessment and interpretation of satellite altimeter data (e.g., ICEsat) in time and in space.
2 Approach 2.1 Snowmelt Algorithm Detection of surface melt at large spatial scales is most effectively accomplished through the use of wide-swath satellite microwave data. There are several approaches to detect and map different snowmelt conditions on ice sheets. Passive microwave data have a clear melt signature due to strong microwave absorption by wet snow during melt onset (M~itzler and Happi (1989)). As such, wet-snow emission approaches the black body behaviour. This change in emission characteristics is detectable by most microwave sensors at frequencies above 10 GHz. (Ulaby et al. (1986)). Passive microwave data provide a long-term record since 1979 for snowmelt assessment (Mote and Anderson (1995), Abdalati and Steffen (1995)). In particular, the cross-polarized gradient ratio (XPGR) algorithm was developed by Abdalati and Steffen (1995) for consistent snowmelt detection. Here, we use an active microwave algorithm for snowmelt mapping based on diurnal change in QSCAT backscatter data (Nghiem et al. (2001), Steffen et al. (2004)). Melt areas are detected based on diurnal change in QSCAT backscatter caused by diurnal differential wetness in melting snow. The advantages of the QSCAT algorithm include: high sensitivity to snowmelt allowing daily delineation of snowmelt and refreezing, applicability to all areas and facies of the ice sheet, and independence from absolute sensor calibration (avoiding inconsistency due to long-term drift in sensor gain factor). Results show that QSCAT can detect early melt and more extensive melt areas. QSCAT melt results correspond to locations where ice surface albedo switches to a low value due to melt, while XPGR results are consistently coincident with melt areas undergoing 10 days to two weeks or longer of melting detected by QSCAT (Steffen et al. (2004)). Thus, for the first time, QSCAT independently
verifies the consistency for which the XPGR algorithm is designed. We use the QSCAT algorithm to obtain updated snowmelt results including 2005.
2.2 Snow Accumulation Algorithm Snow accumulation (SA) rate has been estimated with a C-band synthetic aperture radar in the dry snow zone in Greenland (Munk et al. (2003)). A new method has been developed to measure SA in the percolation zone using QSCAT (Nghiem et al. (2005)), and the results are verified with in-situ data from the GC-Net. The QSCAT algorithm for SA measurement (Nghiem et al. (2005)) relies on the attenuation decreasing QSCAT backscatter as snow accumulates on an ice layer formed during a past melt season. An ice layer consists of large scatterers such as clumps of coalesced ice grains, icicles, ice columns, or ice lenses formed by the percolation of melt water that refreezes in the firn layer. Such scatterers in the ice layer can dominate radar backscatter (Jezek et al. (1993)), which is attenuated by the overlaying SA. GC-Net data are used to determine the attenuation coefficient and QSCAT data measure the backscatter decrease, which is inverted into SA with the GC-Net derived coefficient (Nghiem et al. (2005)). The QSCAT algorithm is applicable in the percolation zone where the last (or previous year's) melt created an ice layer. It is noted that the strong and extensive melt in 2002 (Steffen et al. (2004), Nghiem et al. (2005)) significantly increases the region of validity for the QSCAT algorithm well into the traditional dry-snow zone, defined by Benson (1962), while reducing the validity region of the C-band radar approach. We will use the QSCAT algorithm to derive new SA results over Greenland, and adapt the algorithm to obtain initial SA results around McMurdo in Antarctica.
2.3 Verification Approach We will verify our results using in-situ measurements from networks of climate and meteorological stations. Over Greenland, we utilize data from the GC-Net, established in 1995 to monitor climatological and glaciological parameters at various locations on the ice sheet (Steffen and Box (2001)). GC-Net currently consists of 18
Chapter 5 • Snow Accumulation and Snowmelt Monitoring in Greenland and Antarctica
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descending (des) orbit passes, and (c) air temperature at 2 m above the surface.
automatic weather stations distributed over the entire Greenland ice sheet. The GC-Net objectives are to measure hourly, daily, annual and interannual variability in accumulation rate, surface climatology and surface energy balance parameters at selected locations on the ice sheet, and to monitor nearsurface snow temperatures at the automated weather station (AWS) locations for the assessment of snow densification, accumulation, and metamorphosis. The National Oceanic and Atmospheric Administration (NOAA) National Climatic Data Center (NCDC) maintains the Global Summary Of the Day (GSOD) dataset consisting of meteorological data measured by weather stations in the World Meteorological Organization/Global Telecom-munications System network. Data from NCDC/GSOD include snow depth, precipitation, temperature, humidity, dew point, pressure, and other meteorological data. NCDC/GSOD includes more than 50 stations in Antarctica. To compare satellite and station measurements, we have developed the Special Satellite Station Processing (SSSP) at the Jet Propulsion Laboratory. SSSP uses a mask of stations with a selectable radius up to 60 km around each station, and an effective pointer
algorithm is specifically designed to collect and collocate massive time-series satellite data together with in-situ station data. We apply the SSSP to selected stations from GC-Net and NCDC/GSOD to obtain the results presented in this paper. 3 3.1
Results Greenland
Recent results from QSCAT data revealed the most extensive melt in 2002 when melt areas penetrated well into the traditional dry-snow zone, especially in the northeast region of the Greenland ice sheet (Steffen et al. (2004)). The 2002 melt zone almost connected the west and east coast across the central region at about latitude 75°N (Figure 5 in Nghiem et al. (2005)). The 2002 anomalous melt also created a new significant ice layer in vast areas in the dry snow zone on both the east and west sides of the ice sheet. The QSCAT SA algorithm captures an extreme snowfall in mid-April 2003 when half a meter of snow deposited on the southeast side of the ice sheet in a single day (Nghiem et al. (2005)). Moreover, the semi-annual SA rate from October
33
34
S.V. Nghiem • K. Steffen • G. Neumann • R. Huff
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Fig. 2 Snow accumulation rates obtained by semi-annual linear regression from GC-Net data at the AWS locations: (a) NASA-U at 73.83°N and 49.5°W, (b) Crawford Point 1 at 69.88°N and 46.97°W, and (c) DYE-2 at 66.48°N and 46.28°W. Light-grey bars are for January-June and dark-grey bars are for July-December. QuikSCAT snow accumulation rate is estimated over an area with a radius of 25 km around each station. The left panel shows the map of the GC-Net station locations.
2002 to March 2003 doubled the normal amount in a southeast region, while the SA on the southwest side is less than other years (Nghiem et al. (2005)). This SA anomaly significantly impacts the regional mass balance in the 2002-2003 season. New results from the latest QSCAT data detected peculiar melt in wintertime on the west flank of the ice sheet. The SSSP provides QSCAT satellite backscatter and GC-Net measurement time-series at the ETH/CU Camp AWS (69.57°N, 49.32°W) in 2004-2005 in Figure 1, which extends the result by Steffen et al. (2004) to six years of data since the start of QSCAT. Compared to the past melt season lengths in 2000-2003 (Steffen et al. (2004)), the melt season length in 2004 (26 May to 23 August) of 90 days is the shortest. In 2005, the melt onset occurred on 16 May, followed by a period of partial re-freezing and then a strong melt on 9 June 2005. QSCAT melt results are verified with the AWS temperature data (Figure l c). Moreover, results at ETH/CU AWS capture a peculiar winter melt event in February 2005 (Figure 1), observed for the first time by QSCAT in the last six years. The associated QSCAT diurnal signature on 19 February 2005 (Figure lb) has negative value less than -2 dB indicating a warmer condition in the early morning
(-6:20 am) compared to that in the afternoon (-6:20 pm). This winter melt event was caused by a strong katabatic storm with hourly mean wind speeds exceeding 25 m s-1. The 2005 wintertime melt was also measured by QSCAT and verified by GC-Net temperature data at JAR l, JAR 2, and JAR 3 AWS locations, all located on the west flank of the Greenland ice sheet in a line from the ETH/CU station towards the coast. The west flank underwent a significant anomaly in snow accumulation in 2005. QSCAT results at NASA-U AWS location indicated an SA rate of 1.6 m/year from January to June 2005, which is the same SA rate measured by the in-situ instrument for snow height monitoring at the NASA-U AWS location. In general, SA measured by QSCAT is within 10% of SA measured by GC-Net AWS (Nghiem et al. (2005)). The anomalous SA of 1.6 m/year in January-June 2005 is double the longterm average of 0.8 m/year at NASA-U. This record amount of SA is the highest observed in the full data record at NASA-U over the past decade (Figure 2). At Crawford Point 1, the AWS SA rate of 3.1 m/year for January-June 2005 is the highest record in the decadal dataset, which is significantly larger than the second record of 2.3 m/year in the
Chapter 5
2003
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•
Snow Accumulation and Snowmelt Monitoring in Greenland and Antarctica
2004
0.75
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Fig. 3 Snow accumulation maps from QSCAT snow-accumulation algorithm for the period from 1 January to 31 March in 2003 (left panel), in 2004 (center), and in 2005 (right). The grey scale is for snow depth in meter.
first half of 1999. At DYE-2, the AWS SA rates for January-June 2005 is the largest during the first half of all year since 1996, and among the top four values of all semi-annual SA rates (Figure 2). QSCAT SA rates are close to the AWS values at both Crawford Point 1 and DYE-2, but the QSCAT SA is below the AWS value at Crawford 1 and above the AWS value at DYE-2. Furthermore, another confirmation of the record snow in 2005 is provided by measurements at the NCDC/GSOD Station 042200 (Aasiaat and Egedesminde, 68.7°N, 52.85°W, west coast of Greenland), which show the record snow depth of 0.8 m and the highest value of mean snow depth over winter and spring seasons in 2005 throughout the entire data time-series since 1994. Note that NASA-U, Crawford Point 1, and DYE2 mostly line up from north to south extending about one third of the length of the Greenland ice sheet. However, GC-Net measurements represent localized conditions at each station location. Without extensive spatial data, it is not possible to connect point measurements from the three local locations (NASA-U, Crawford Point 1, and DYE-2)
to extrapolate over a large region with a high confidence level. Here, QSCAT enables the continuous spatial mapping of SA to reveal the enormous region of the SA anomaly in 2005, in particular over the west flank of the Greenland ice sheet. Figure 3 presents SA maps of areas that have more than 0.3 m of SA for a 3-month period from January to March in 2003, 2004, and 2005. These QSCAT maps of SA reveal: (1) the extensive snow accumulation anomaly in 2005, (2) large spatial variability of SA in each year, and (3) large interannual variability in SA during January-March of 2003-2005. These characteristics of SA are also supported by GC-Net results in Figure 2. Results in Figure 3 also show more SA in southeast Greenland that is consistent with the general pattern of snow climatology in Greenland (Bales et al., (2001)). Integrated over the full surface area, QSCAT results yield a January-March SA volume of: 196 km 3 for 2003, 280 km 3 for 2004, and 565 km 3 for 2005. The results are conservative estimates accounting only for areas that have SA larger than 0.3 m (QCAT SA algorithm can be improved in the future for more accurate results over areas with low
35
36
S.V. N g h i e m • K. Steffen • G. N e u m a n n • R. Huff
SA rate). Assuming a bulk density of 330 kg m-3 (Krabill et al. (2004)), the corresponding snow accumulation mass is 6.47x1013 kg, 9.24x1013 kg, and 18.6x 1013 kg in the first quarter of 2003, 2004, and 2005, respectively. Compared to the estimate of total Greenland glacier discharge by calving (Reeh et al. (1999)) of 263 km 3 per year, equivalent to a mass of 5.92x 1013 kg per quarter or a potential sea level rise of 0.16 mm per quarter, the SA in the first quarter of 2005 is more than 3 times larger. 3.2 A n t a r c t i c a
We adapt the QSCAT algorithms to observe snowmelt, ice layer extent, and snow accumulation in Antarctica. Snowmelt is detected when the absolute value of the diurnal backscatter difference is larger than 1 dB and refreezing when the diurnal change decrease to below 1 dB from the melting conditions. At McMurdo, QSCAT data show no detectable snowmelt in the austral summer seasons of 1999-2000 and 2000-2001, and then melting occurred consecutively during the last 4 summer seasons. The strongest melt was in the austral summer of 2001-2002, which preceded the anomalous melt in Greenland in 2002. Air temperature measured at McMurdo (data from NCDC/GSOD Station 896640) was as high as 10.5°C (51 F) on 30 December 2001, corresponding to the strong melt indicated by a sharp drop in QSCAT backscatter. This strong melt created a significant ice layer identified by a large increase in backscatter after the last melt date in January 2002.
on 16 January 2005 (AZ'm=40 days) and is by far the longest one observed by QSCAT since the launch of the satellite mission (Table 2). These melt results are consistent with NCDC/GSOD temperature data measured at McMurdo. The sharp increase in backscatter (-2dB) at McMurdo in January 2005 indicates the formation of a new significant ice layer (Nghiem et al. (2005)) caused by the prolonged melt. In east Antarctica, QSCAT detected the first melt ever recorded at NCDC/GSOD Station 893320 (Elizabeth, 82.62°S, 137.08°W, 549 m in elevation) in January 2005 over the entire six-year record of QSCAT data since July 1999. This melt event was so strong that QSCAT backscatter decreased by more than 13 dB (a decreasing factor of 20 times). This result is verified by data at Station 893320 with a record air temperature of 5.3°C in the 11-year dataset.
hts Coast
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Table 2. Timing of snowmelt onset and freeze-up at McMurdo (77.85°S, 166.67°E) in Antarctica detected by QSCAT data from July 1999 to June 2005. There was no melt before December 2001 detectable with QSCAT data.
AZ'm
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Fig. 4 QSCAT map of areal extent (grey areas) of ice layer formed by the 2004-2005 melt season in Antarctica. The grey scale represents the backscatter increase in dB.
days days days days
Dec 2001 Jan 2003 Jan 2004 Dec 2004
2002 2003 2004 2005
In Table 2, the melt season length A'rm(Nghiem et al., (2005)) is defined as the time period between the first and the last melt date, within which multiple melt events can occur. The most recent melt season started on 7 December 2004 and ended
1.0
I 1.5
Figure 4 shows QSCAT ice-layer extent over west Antarctica created by the refreezing of melt water in the firn layer. The ice layer is detected by the strong increase in bi-weekly backscatter signatures before (October) and after (February of the next year) the melt season. QSCAT results show
Chapter 5
extensive melt along the Antarctica Walgreen, Bakutis, and Hobbs coasts extending well inland in 2005 (Figure 4). Extensive melt regions are also detected by QSCAT over the Rockefeller Plateau and along the Ross Ice Shelf adjacent to the Queen Maud Mountains in 2005 (Figure 4). Assuming the same attenuation coefficient derived for Greenland, QSCAT data give a snow accumulation depth of 1.24 m over the McMurdo area during the 325 days between 3 February 2002 and 25 December 2002. This is equivalent to a snow accumulation rate of 1.4 m/year for the period of February-December 2002, which was a heavy snow season after the unusually warm summer. However, the precision of this result is uncertain, and a future field experiment is necessary to calibrate an accurate snow attenuation coefficient for Antarctica snow accumulation inversion and to validate QSCAT SA inversion results. In this regard, QSCAT data can be used to select appropriate locations with different snow accumulation depths for such a field campaign.
• Snow
Accumulation and Snowmelt Monitoring in Greenland and Antarctica
The combination of GC-Net and QSCAT results provides an alternative for independent verification of altimetry results. GC-Net time-series data, consisting of thousands of data points in time, are used to calibrate QSCAT results over a long time period to determine QSCAT accuracy. Since QSCAT has large and frequent coverage in time and space (two times per day over Greenland and Antarctica), calibrated QSCAT results can be compared with altimeter results integrated over the QSCAT pixel in which altimeter data are collected. Since this is a statistical method over a large spatial extent (25 km by 25 km for a QSCAT pixel over large spatial coverage), effects of uncertainties in altimetry data location are minimized while the capability to collocate data in time and space are maximized for the comparison and verification of results. Once validated, altimeter results can provide high-resolution observations at a sub-pixel scale relative to scatterometer measurements. Moreover, a laser altimeter can measure snow height regardless of surface melt conditions when the scatterometer approach is not applicable.
4 Combination with Altimetry 5 Summary and Conclusions Snowmelt and snow accumulation results derived from QSCAT data will facilitate the interpretation of satellite laser altimetry and radar altimetry data. First, melt water formed at the surface may percolate into the snowpack and refreeze to form ice lenses and glands. This densification mechanism can decrease the ice surface height without loss of ice mass. The densification can result in surface lowering, which over a prolonged period of warming will become important for mass balance estimates from satellite altimeters. Frequent QSCAT snow accumulation maps over extensive areas will greatly complement altimeter data that are limited in spatial and temporal coverage. A satellite laser altimeter collects data points along surface ground tracks, and a satellite radar altimeter acquires data with a very narrow beam measuring surface profiles. Ascending and descending orbit passes provide crossover points on the surface where data can be compared for accuracy assessment. Because of the narrow surface profiles, a limited number of crossover points, and uncertainties in data locations, it is difficult to have altimetry data collocated with accurate in-situ or field measurements for result validation in time and in space.
This paper addresses two important components of mass balance using QSCAT data: snowmelt and snow accumulation in Greenland and Antarctica. The innovative scatterometry approach can create a new dataset of snow accumulation to compare with glacier discharge. Such data are continuous and extensive in space and in time, over 6 years so far and extending into a decadal time-series. In Greenland, QSCAT results show the record melt in 2002, prolonged melt in 2003, and peculiar melt in winter 2005. Snow accumulation maps are obtained over Greenland in the first quarter of each year in 2003-2005. The record SA in JanuaryMarch 2005 supplied a massive amount of total snow mass compared to total Greenland glacier discharge by calving over the quarterly period. The record SA results are verified with GC-Net and NCDC/GSOD data. The 2005 snow anomaly highlights the importance of SA in the total estimate of mass balance and sea level change. For major catchment basins of the Greenland ice sheet (Thomas et al. (2001)), QSCAT SA can provide the crucial input to assess mass balance of each basin. In Antarctica, QSCAT detected the strongest melt in the austral summer of 2001-2002 and the longest
37
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S.V. Nghiem • K. Steffen • G. Neumann • R. Huff
melt season in 2004-2005 at M c M u r d o . Both the strong melt and the long melt created significant ice layer due to the refreezing o f melt water over the area around M c M u r d o . The first map of ice layer extent in Antarctica is derived from Q S C A T data. This map reveals the strong impact o f s n o w m e l t in extensive regions from coastal zones extending well in land. The first initial estimate o f snow a c c u m u l a t i o n is obtained over the area around McMurdo assuming the Greenland snow attenuation coefficients. H o w e v e r , field c a m p a i g n and in-situ m e a s u r e m e n t s are necessary to calibrate and validate Q S C A T results for snow accumulation. To derive results over full annual cycles, each pixel in G r e e n l a n d or Antarctica should be treated with the full time-series Q S C A T data to m o n i t o r melt and re-freezing so that the SA algorithm can be applied to the freezing conditions, over which the backscatter is stable. Effects due to a z i m u t h angle and seasonal temperature should be accurately accounted for improving the performance o f the SA algorithm. M e t h o d s to c o m b i n e satellite altimeter and scatterometer data should be developed to obtain more precise results over extensive coverage. Accurate estimates o f s n o w m e l t and snow a c c u m u l a t i o n will advance the understanding and determination of mass balance and sea level change.
Acknowledgment. The
research carried out at the Jet Propulsion Laboratory, California Institute of T e c h n o l o g y , was supported by the National Aeronautics and Space Administration (NASA). The research carried out at the Cooperative Institute for Research in E n v i r o n m e n t a l Sciences, University of Colorado, was also supported by N A S A .
References
Abdalati, W., and K. Steffen (1995). Passive microwavederived snow melt regions on the Greenland ice sheet, Geophys. Res. Lett., Vol. 22, pp. 787-790. Bales, R. C., J. R. McConnell, E. Mosley-Thompson, and B. Csatho (2001). Accumulation over the Greenland ice sheet from historical and recent records, J. Geophys. Res., Vol. 106, pp. 33813-33825. Benson, C. S. (1962). Stratigraphic Studies in the Snow and Firn on the Greenland Ice Sheet, Res. Report 70, Snow, Ice and Permafrost Research Establishment, Corps Eng., US Army, 1962. Jezek, K. C., M. R. Drinkwater, J. P. Crawford, and R. Kwok (1993). Analysis of synthetic aperture radar data collected over the southwestern Greenland ice sheet, J. Glac., Vol. 39, No. 131, pp. 119-132.
Krabill, W., E. Frederick, S. Manizade, C. Martin, J. Sonntag, R. Swift, R. Thomas, W. Wright, J. Yungel (1999). Rapid thinning of parts of the southern Greenland ice sheet, Science. Vol. 283, pp. 1522-1524. Krabill, W., E. Hanna, P. Huybrechts, W. Abdalati, J. Cappelen, B. Csatho, E. Frederick, S. Manizade, C. Martin, J. Sonntag, R. Swift, R. Thomas, and J. Yungel (2004). Greenland Ice Sheet: increased coastal thinning, Geophys. Res. Lett., Vol. 31, L24402, doi:10.1029/ 2004GL021533. Mfitzler, C. H., and R. Huppi (1989). Review of signature studies for microwave remote sensing of snowpacks, Adv. In Space Res., Vol. 9, pp. 253-265. Mote T. L., and M. Anderson (1995). Variations in snowpack melt on the Greenland ice sheet based on passivemicrowave measurements. J. Glaciol., Vol. 41, pp. 51-60. Munk, J., K. C. Jezek, R. R. Foster, and S. P. Gogineni, An accumulation map for the Greenland dry-snow facies derived from spaceborne radar (2003). J. Geophys. Res., Vol. 108, No. D9, art.4280. Nghiem, S. V., K. Steffen, R. Kwok, and W.Y. Tsai, Detection of snow melt regions on the Greenland ice sheet using diurnal backscatter change (2001). J. Glac., Vol. 47, No. 159, 539-547. Nghiem, S. V., K. Steffen, G. Neumann, an R. Huff, Mapping of ice layer extent and snow accumulation in the percolation zone of the Greenland ice sheet (2005). J. Geophys. Res., Vol. 1 1 0 , F03017, doi: 10.1029/ 2004JF000234. Reeh, N., C. Mayer, H. Miller, H. H. Thomsen, and A. Weidick (1999). Present and past climate control on ~ord glaciations in Greenland: implications for IRD-deposition in the sea, Geophys. Res. Lett., Vol. 26, pp. 1039-1042. Robin, G. de Q., Depth of water-filled crevasses that are closely spaced (1974). J. Glaciol., Vol. 13, pp. 543. Steffen, K. (1995). Surface energy exchange during the onset of melt at the equilibrium line altitude of the Greenland ice sheet, Ann. Glaciol., Vol. 21, pp. 13-18. Steffen, K, and J. E. Box (2001). Surface climatology of the Greenland ice sheet: Greenland climate network 19951999, J. Geophys. Res., Vol. 106, pp. 33065 - 33982. Steffen, K., S. V. Nghiem, R. Huff, and G. Neumann (2004). The melt anomaly of 2002 on the Greenland ice sheet from active and passive microwave satellite observations, Geophys. Res. Lett., Vol. 31, L20402, doi:10.1029/ 2004GL020444. Thomas, R., B. Csatho, C. Davis, C. Kim, W. Krabill, S. Manizade, J. McConnell, and J. Songtag (2001). Mass balance of higher-elevation pasrts of the Greenland ice sheet, J. Geophys. Res., Vol. 126, pp. 33707-33716. Ulaby, F.T, R.K. Moore, and A.K. Fung (1986). Microwave Remote Sensing. Vol. 3, From Theory to Applications, 1120 pp., Artech House Inc., Norwood, MA. van der Veen, C.J., (1998). Fracture mechanics approach to penetration of surface crevasses on glaciers. Cold Regions Sci. Tech., Vol. 27, pp. 31-47. Zwally, H.J.W. Abdalati, T. Herring, K. Larsen, J. Saba, and K. Steffen (2002). Surface melt-induced acceleration of Greenland ice-sheet flow, Science, Vol. 297, pp. 218-222.
Chapter 6
Estimating Recent Global Sea Level Changes H.-P. Plag Nevada Bureau of Mines and Geology and Seismological Laboratory, University of Nevada, Reno, Mailstop 178, Reno, NV 89557, USA, e-mail:
[email protected].
Abstract An empirical model for sea level trends over several decades is set up such that it is consistent with the global pattern of Local Sea Level (LSL) trends observed by the global network of tide gauges. The forcing factors taken into account are steric sea level variations, present-day ice load changes, and post-glacial rebound. Model parameters are determined in a least squares fit of the model to the LSL trends. The model allows the determination of the contribution of each factor to the global average LSL trend. Here we compare the solutions for two different LSL trend sets, namely one determined without and one with taking into account local atmospheric forcing at the tide gauges (denoted here as T1 and T2). From the globally given model, the global average trend over the last 50 years in LSL is found to be of the order of 1.05 + 0.75 mm/yr and 1.20 + 0.70 mm/yr for T1 and T2, respectively. For T1, the contribution of the Antarctic and Greenland ice sheets to the global average are 0.39 + 0.11 mm/yr and 0.10 + 0.05 mm/yr, respectively and for T2 0.31 + 0.16 mm/yr and 0.16 -+- 0.03 mm/yr, respectively. Using T1, the contribution from steric change is clearly identified and found to be at least 0.2 mm/yr with the most likely value being close to 0.35 mm/yr. For T2, there is no correlation between the spatial pattern of the observed LSL trends and the steric sea level trends, and the steric contribution to the global average turns out to be equal to zero. This result indicates a very high correlation between the local atmospheric forcing and the thermosteric sea level changes, which may be the result of a feedback of temperature changes in the upper layer of the ocean into the air pressure and wind field over the ocean. Keywords: global sea level rise, local sea level trends, ice sheets changes
1
Introduction
The mass balance of the global water cycle is of paramount interest for understanding, predicting and mitigating the impact of climate change. Understanding the sea level changes over the last decades and century is a prerequisite for quantifying climaterelated changes in the oceans volume and mass, as
well as for establishing future sea level scenarios. Church et al. (2001) emphasize the considerable uncertainties in the mass balance of the ocean and, in consequence, the global sea level. In particular, the contribution of the large ice sheets to current sea level changes is rather uncertain. The global mass and volume of the ocean are two absolute quantities characterizing the ocean as a reservoir in the global hydrological cycle. Changes in these quantities are directly related to changes in the hydrological cycle and therefore to climate change. Local Sea Level (LSL), which is defined here as the (absolute) vertical distance between the surface of the ocean and the surface of the solid Earth, depends on the distribution of the ocean water in a given topography of the Earth surface. Thus, LSL depends on many different factors, such as the Earth's topography, the (timevariable) geoid, changes of the Earth's rotation, atmospheric circulation, heat and salinity distribution in the ocean, ocean circulation, past and present mass movements in the Earth system, the visco-elastic properties of the Earth's interior, sedimentation, and even anthropogenic subsidence due to groundwater, gas, and oil extraction. At coastal locations, LSL is measured relative to a benchmark on land, which, if properly chosen, follows the vertical motion of the land around the tide gauge, including the ocean bottom below it. Over the last thirty years a number of studies have utilized the unique sea level data set provided by the Permanent Service f o r Mean Sea Level (PSMSL) for the determination of a global sea level rise (see Church et al., 2001, for a review). The global trends estimated in these studies range from + 1 to ÷2.5 mm/yr. This relatively wide range mainly is due to the selection criteria used by the different researchers to select subsets of tide gauges as well as the methodology to determine a global trend. However, the link between LSL changes and changes in the global ocean mass and volume is complex and all forcing factors result in spatially highly variable trends. Taking into account that the global network of tide gauges only samples a small fraction ofthe ocean's surface, any estimate of a global rise not taking into account the spatial variability of the different contribution is bound to be biased. In order to account for the spatial variability of the forcing, Plag (2006) derived a LSL balance equation that accounts for each forcing factor individually.
40
H.-P. Plag
Using an approximate LSL equation which accounts for the contribution due to ocean temperature changes, post-glacial rebound, and the present-day changes in Greenland and Antarctica, he determined a global average rise in LSL of 1.05 + 0.75 mm/yr. Here we extend this approach and study how the local atmospheric forcing affects the estimates of the global average. In the next section, we briefly introduce the LSL equation and discuss the spatial fingerprint of the main forcing factors. In Section 3 we summarize the database used by Plag (2006), which is here complemented with an atmospheric dataset. Then, in Section 4 we consider the effect of the local forcing on the global grid of observed LSL trends, in Section 5, we introduce the regression model and in Section 6 discuss the effect of the local atmospheric forcing on the global estimates by comparing the results of Plag (2006), which do not account for the local atmospheric forcing, to the results obtained here after the local atmospheric forcing has been removed.
3
The database
and methodology
Time series of monthly mean values for LSL are taken from the PSMSL data base (Woodworth & Player, 2003), which contains records from more than 1950 tide gauges, i.e. a major fraction of the global tide gauge data. Those records, for which the history of a local reference can be established are compiled into a subset denoted as Revised Local Reference (RLR) datasets, and these records can be used confidently to determine local LSL trends. Most of the PSMSL records are restricted to the time window of approximately 1950 to 2000. Thus, only in that time interval a spatially sufficient picture of the pattern of LSL trends can be expected. Considering that the steric sea level changes are given for the time window 1950 to 1998 (see below) we choose this time window as a compromise between highest accuracy for the local secular LSL trends and the optimal spatial coverage. For each tide gauge, Plag (2006) determined a secular trend by fitting the model function 2
2
L o c a l Sea Level B a l a n c e
g(t) - a + bt + E Ai sin(wit + 0i)
(2)
i=1
For the discussion of secular trends, we approximate the monthly mean LSL hM(:g, t) at a tide gauge located at a point E on the Earth surface as a sum of several factors, namely
h(:< t) =
t) +
t) + A ( i , t) +
I(i, t) + a(i, t) + T(I, t) + p(1)(t
- to) + V o ( 1 ) ( t - to) +
6v( , t)
where t is time, to an arbitrary time origin, and where we have considered the following contributions to LSL changes: S: steric changes, C: changes in ocean currents, A: changes in atmospheric circulation, I: changes in the mass of large ice sheets, G: changes in the mass of glaciers, T: changes in the terrestrial hydrosphere, P: post-glacial rebound (assuming a timeindependent velocity), V0: tectonic vertical land motion (assuming a time-independent velocity) 6V: nonlinear vertical land motion. (Plag, 2006, see there for a detailed discussion of these factors). The factors that contribute to secular trends with a fingerprint exhibiting large spatial variations on regional to global scales are the post-glacial rebound signal, the steric signal and the present-day contribution from the two large ice sheets in Antarctica and Greenland (Plag, 2006). Moreover, changes in atmospheric circulation are also likely to have regional scales. The database for these factors are discussed in the next Section.
to the series of monthly mean sea levels, where t is time, a is an offset and b the constant secular LSL trend. Ai and ~bi are the amplitude and phase, respectively, of an annual and semiannual constituent, in the fit, the parameters a and b and the amplitudes of the sine and cosine terms of the annual and semi-annual constituents are determined simultaneously. Here we use an alternative equation to determine the LSL trends, i.e. 2
3
g(t) - a + bt + E Ai sin(wit + Oi) + E diai (3) i=1
i=1
where o-i, i = 1, 2, 3, are the relevant components of the atmospheric stress tensor on the sea surface, and di are the respective regression coefficients, which we determine together with the other parameters in the least squares fit to the LSL records. The component of the atmospheric stress tensor perpendicular to the sea surface is the air pressure p. The horizontal components are taken to be proportional to the wind stress components, i.e.
a2 ~ WEV/W~ + w'~ ~
+
(4)
(5)
where WE and WN are the east and north components of the wind vector, respectively. We denote the sets of
Chapter 6 • Estimating Recent Global Sea Level Changes
LSL trends determined with eq. (2) and eq. (3) as T1 and T2, respectively, Monthly mean values of the air pressure and the wind stress components are computed from the ERA40 reanalysis data provided by the European Center for Medium Range Weather Forecast (ECMWF). The ERA40 dataset has a spatial and temporal resolution of 2.5°x2.5 ° and 6 hours, respectively. Monthly means of or2 and era are computed as averages of the six-hourly values of these quantities. With respect to thermosteric sea level changes, post-glacial rebound signal and the fingerprints of the two large ice sheets in Antarctica and Greenland, we use the same data base as Plag (2006). Thermosteric sea level variations are computed from observations of the subsurface temperature field. Currently, two global datasets are available, namely Levitus et al. (2000)and Ishii et al. (2003). The two datasets are, to a large extent, based on the same observations; however, different analysis schemes are used to create the gridded datasets. As pointed out by Plag (2006), the sea level trends derived from these two datasets display considerable differences, with the former having more short wavelength variations and a larger range of local trends. The computed sea level changes depend on the depth interval used for the integration. For the Levitus et al. (2000) dataset, steric sea levels are available for 500 m and 3000 m (denoted as L500 and L3000). For the Ishii et al. (2003) dataset, sea levels are only available for 500 m (denoted here as I500). All grids have a spatial resolution of 1°. For the L500 and L3000 datasets, which are given as annual means, local trends were determined by a least squares fit of a polynomial of degree 1 to the data for each grid point. I500 is given as monthly means, and the model function (2) was used instead. In large parts of the ocean, the thermocline depth is much deeper in the ocean than 500 m. Therefore, L500 and I500 are likely to underestimate the thermosteric sea level variations in these areas. The large uncertainties in the mass changes of the two large ice sheets in Antarctica and Greenland over the last five decades do not allow to use a spatial fingerprint deduced from observations to represent the contribution of these two ice sheets to LSL. Therefore, based on the static elastic sea level equation (Farrell & Clark, 1976), Plag & Jfittner (2001) determined for each ice sheet a fingerprint function for a constant, unit trend over the complete area of the ice sheet. The resuiting Antarctic fingerprint has a distinct zonal component, while the Greenland fingerprint shows more variations with longitude, particularly in the northern hemisphere (Plag, 2006).
The present-day LSL fingerprint of the post-glacial rebound signal (PGS) is fairly well predicted by geophysical models. Plag (2006) used a suite of models (Milne et al., 1999) to study the effect of the uncertainties of the predicted PGS fingerprint in LSL on the global results. Here we also use the same set of models but will not discuss in detail the sensitivity of the global results to the PGS model. The thermosteric contribution is only given on a 1o by 1° grid. Using a regression based on individual LSL trends at tide gauges would require interpolation and in most cases extrapolation of the steric signal from nearby grid points to the exact tide gauge location. Based on a detailed sensitivity study, Plag (2006) chose to create a gridded dataset of LSL trends instead, assigning a weighted average of all available LSL trends to each grid cell with tide gauges. The sensitivity study also indicated that a grid resolution of 2 ° x2 ° was a reasonable compromise between the accuracy of the individual LSL trends assigned to the grids and the spatial resolution required to capture the main features of the spatial fingerprint of e.g. the postglacial signal. This 2 ° grid as defined by the available tide gauges only covers about 3.1% of the global ocean surface. Similar 2 ° grids were created for all available forcing factors by averaging the 1° grids.
4
Effect of the local atmospheric forcing on LSL
At many stations, a large amount (up to 90%) of the variability of monthly LSL is explained by the regression model according to eq. (3), with a large fraction of the model coming from air pressure and to a lesser extent the two wind stress components. In many areas, the regression coefficient for air pressure is close to the equilibrium value of ~ - 1 0 mm/HPa (Fig. 1). However, close to the equator, where air pressure variations are small, much larger values than that are found. The relative importance of the local atmospheric forcing depends mainly on the latitude and the coastal geometry. At stations at latitudes outside the -+-25° band the combined local atmospheric forcing explains most of the intraseasonal variations, while significant interannual to decadal variations are more pronounced in the residuals than in the observations (e.g. Halifax, Cuxhaven, San Francisco, and Sydney). At most stations the main atmospheric contribution is due to air pressure, but at Cuxhaven in the German Bight, the east component of the windstress is by far the dominating contribution. For the stations on the west coasts of the Americas, the E1 Nifios events are visible in sea level. At
41
42
H.-P. Plag 40
40
L~.
IN
÷+ .
20
÷
$
20
÷÷
i
+++
0
-20
-20
-40
-20
÷ ÷+ +
+.
÷+÷
-40
+
+
+! ÷
-60
-80 -90
-60
-30
0
30
60
90
÷
-40
2
-60
-90
-60
0
-30
Latitude
30
60
90
-90
-60
-30
0
30
60
90
Latitude
Latitude
Fig. 1 • Regression coefficients for the local atmospheric forcing. P" air pressure, E: east component of wind stress, N" north component of wind stress. The horizontal line in the left diagram indicates the equilibrium value of ~ - 1 0 mm/HPa. Station
HALIFAX
Station
40
Station 20
0
-20
0 -40
SANFRANCISCO
60
-60
6o
4o
0
0
-60 i00 0
-60
E
4 0
-40
-4
60
4000
~
-1-J I
-6o
-i00
-400
-180
0
/
OOo
0
-80
-140
140
180
0
-160 200
0
-140
0
-180
-180
-200
-200
-200 1970
Station
1982
1994
2006
1958
1970
Station
CUXHAVEN2
IW N
1982
1994
2006
600 0
COCHIN WILLINGDONI
30
-40
1.20
40 0
9
60
-9
-6O
-180
-27
-180
.
.
.
0 -140
7o ° . . . . . . . . . . . . . . . . . . . . . .
-210
60
6O0 0
160
-6O
-6OO
0
-180
6OO
-160
2O0
0
2006
-40
60
.
1994
SYDNEY- FORTDENISON
0
-90
-60
. 0 .
1982
40
•1 . 2 0
-140
1970
Station
0.00
-600
~%
1958
-30
o
0
-60O 1958
SANTACRUZ
-200 1970
1982
1994
2006
1958
-180 1970
1982
Year
1994
Year
2006
1958
1970
1982
1994
2006
Year
Fig. 2: Regression model for selected tide gauges. For each station, we show O" observations, M: regression model, R: residual, as well as the individual contributions of AP" air pressure, WE: east component of windstress and WN" north component of windstress. All parameters are given in mm. San Francisco, the high sea levels during the largest E1 Nifios are not modeled by the local atmospheric forcing, while at Santa Cruz in the Galapagos Islands, the local forcing captures a large fraction of these high sea levels very well. However, at Santa Cruz, the regression coefficient for air pressure is as large as - 7 2 mm/HPa, which indicates that there air pressure may
be correlated with other factors influencing sea level. At many stations close to the equator, the rather small air pressure variations are dominated by a (likewise) small seasonal signal, which is also present in sea level (e.g. Cochin at the southwest coast of India). Comparing the LSL grid derived from T1 to the grid derived here from the T2 LSL trends, we see
Chapter 6 • Estimating Recent Global Sea Level Changes
0 90
30 i
60
60 i
90 i
120 i
........
150 i
.,j
180 i
".~
210 i
240 i
270 i
-.2
300 i
330 i
360 90
60
.....
30
-30
-30
-60 ~
-90
--
/ 0
0
-60
-90
i
i
i
i
i
i
i
i
i
i
i
30
60
90
120
150
180
210
240
270
300
330
-12 901
-
30 i
60 i
-8
90 i
-5 -3 - 1 0 1
120 i
150
180 i
3
5
210 i
8
360
12
240 i
270 i
300 i
330 i
360 I
I
90 6o 3o
•
.
-.,
0 3o -60
-30
~ ~ - ~ - - - ~
-90
--
-
-60
-90
i
i
i
i
i
i
i
i
i
i
i
0
30
60
90
120
150
180
210
240
270
300
330
360
0 90 I
30 I
60 I
300 I
330 I
360
-12
-8
90 I
-5 -3 - 1 0 1
120 I
150
180 I
3
5
210 I
8
12
240 I
270 I
....::
30
~30
" '
"
90
Fig. 3: Effect of local atmospheric forcing on the secular LSL trend pattern. Upper diagram: LSL trend grid derived by Plag (2006) for T1. Middle: LSL trend grid determined in the present study for T2. Lower Diagram: atmospheric contribution to LSL trends computed as the difference between T2 and T 1. All scales are in mm/yr. In the computation of the grid values, only local LSL trends within 4-12 mm/yr are used and the LSL trends in a grid cell are averaged using the length of the records as weights. Minimum record length is 10 years.
30
~30
~)~
-60
-60
-90 0
i
i
i
30
60
90 -5
-4
120 -3
-2
i
i
i
i
i
i
i
150
180
210
240
270
300
330
-1
0
1
I
I
I
2
3
4
that the spatial pattern shows some differences particularly in the Pacific (Fig. 3). The effect of the atmospheric forcing on the long-term LSL trends depends on the geographical region. For example, along the west coast of North America, the atmospheric contribution in general is positive, exceeding 2 mm/yr in some areas. For most of the North Atlantic, the con-
-90 360
5
tribution is also positive, while over the western Pacific, large negative contributions are found reaching -5 mm/yr in some areas. For the Mediterranean stations, the atmospheric contribution is negative (particularly in the Adriatic and the eastern Mediterranean, where it reaches values up to -0.8 mm/yr). The results for the Mediterranean are confirmed by Tsimplis et al.
43
44
H.-P. Plag
(2006) on the basis of a hydrodynamical model run.
5
The regression model
The regression model set up by Plag (2006) for the gridded LSL trends is given by K
(6) j=l N
where i is the index of the grid cell, bi is the modeled LSL trend, --I/ij) the LSL trend due to a unit mass change in ice sheet [(J), I£ the number of individual ice sheet fingerprints included in the regression, Pi the predicted PGS, Si the thermosteric LSL trend, and c a mean global LSL trend introduced to collect all unaccounted contributions. The c~j are unknown mean mass trends of the ice sheets, which are determined as a results of the regression analysis. /3 is introduced to account for any scale error in the PGS predictions. With the introduction of /3, we preserve the predicted PGS fingerprint but we allow for adjustments in the amplitude. The same is true for the thermosteric effect, where we have introduce a scale factor ~,. In the following, we have to distinguish between the regression grid, which is defined by the fact that all factors used in a regression are available, and the global (ocean) grid, which is defined as far as possible for the complete ocean surface. Since all models are available globally, the regression grids are determined as the set of grid cells where both the steric sea level trends and the observed LSL trends are given. The global grids are defined by the steric grid used. The regression grids cover typically 2% of the ocean surface, while the steric grids reach up to approximately 95%. Any regression analysis is hampered by the presence of high covariance of the forcing factors. Using T 1, the fingerprints of the Antarctic and Greenland ice sheet are significantly anti-correlated (Table 1). Moreover, the Greenland ice sheet fingerprint is weakly correlated with the present-day PGS fingerprint. For the steric fingerprints, the correlation with the T1 LSL trends are of the order of 0.25 for the L500 and I500 datasets, while the correlation is less than 0.2 for the L3000 dataset (Table 1). Using the T2 trends does not change the cross correlation between LSL and most of the factors (Table 1). However, there is no correlation between the steric sea level heights and the T2 trends, in particular, the cross correlation coefficients for I500 and L500 are 0.00 while for L3000 a non-significant value of 0.04
is found. Taking into account the local atmospheric forcing completely removes the correlation between the LSL and steric trend patterns. This rather surprising results has a profound effect on the results of the regression analysis. It indicates that the steric signal sensed by the tide gauges is mainly due to changes in the upper ocean layer, which appear to be highly correlated with local atmospheric forcing. Thus, a warming of the upper layer and an increase in sea surface temperature will tend to lower the air pressure above the warming area. Both processes will lead to an increase of sea level, which is then fully absorbed in the separate removal of the contribution due to local atmospheric forcing. Nevertheless, in the next section, we will compare selected results to those obtained by Plag (2006) in order to assess the effect of the local atmospheric forcing on the global estimates.
6
Results
In Table 2, results of the regression analysis for a few selected combinations of fingerprints are given for the T1 and T2 LSL trend grids. The actual values of the regression coefficients for T1 are discussed in detail in Plag (2006). Here it is only mentioned that/3 is in the range of 0.4 to 1.5, depending on the PGS model, while -y ranges from approximately 0.4 for L3000 to approximately 1.2 for I500, depending slightly on the PGS model used. For T2,/3 is generally lower, while -y is close to zero for all three steric data sets (see below). For all regressions, the constant c is of the order of 0.5 mm/yr and 0.75 mm/yr for the T1 and T2 LSL trends, respectively. Thus, the unexplained average trend at the tide gauges is between 50 and 80% of the average sea level rise. Moreover, the assumption that c determined for the (small) tide gauge grid can be extrapolated over the whole ocean is uncertain, since most contributions included in eq. (1) are spatially highly variable. Compared to the T 1 results, the T2 results explain a smaller fraction of the spatial pattern in the LSL trends (Table 3). They show a smaller contribution from the Antarctic ice sheet and a larger one from the Greenland ice sheet. The PGS contribution is slightly lower or unchanged, while no contribution comes from the steric forcing. A larger fraction of the LSL trends remains unexplained. For the T1 solutions, the steric signal based on I500, L500 and L3000 contributes significantly to the global LSL pattern. For the L500 dataset, the explained fraction of the variance is the highest and the
Chapter 6 • Estimating Recent Global Sea Level Changes
Table 1: Correlation matrix for the fingerprints of the forcing factors and LSL. Columns are: A, G: fingerprints of the Antarctic, and Greenland ice sheets, respectively, P 1: post-glacial rebound model (ice history is ICE-3G, upper and lower mantle visocities are 1 • 1021 and 2 • 1021 Pas, respectively, and lithosphere thickness is 120 km, see Milne et al. (1999)), I500, L500, L3000: thermosteric fingerprints, and LSL: LSL trend pattern. Matrix is for cross correlation on the regression grid. Upper and lower matrix are for T1 and T2, respectively. For a detailed discussion, see Plag (2006).
A
G
P1
i500
L500
L3000
LSL
A G P1 I500 L500 L3000 LSL
1.000 -0.492 -0.118 -0.026 -0.004 -0.120 -0.097
-0.492 1.000 -0.187 0.100 -0.023 -0.001 -0.074
-0.118 -0.187 1.000 0.014 -0.006 0.023 0.317
-0.026 0.100 0.014 1.000 0.826 0.675 0.248
-0.004 -0.023 -0.006 0.826 1.000 0.832 0.247
-0.120 -0.001 0.023 0.675 0.832 1.000 0.186
-0.097 -0.074 0.317 0.248 0.247 0.186 1.000
A G P1 i500 L500 L3000 LSL
1.000 -0.493 -0.136 -0.030 0.008 -0.102 -0.088
-0.493 1.000 -0.169 0.106 -0.024 0.001 -0.101
-0.136 -0.169 1.000 0.014 -0.007 0.023 0.273
-0.030 0.106 0.014 1.000 0.820 0.676 0.000
0.008 -0.024 -0.007 0.820 1.000 0.827 -0.002
-0.102 0.001 0.023 0.676 0.827 1.000 0.036
-0.088 -0.101 0.273 0.000 -0.002 0.036 1.000
Table 2: Selected results of the regression analysis. The column M (Model) indicates the LSL trend set used and gives the
factors included in the regression function (6). Parameters are as in eq. (6). The other columns are as follows: Emod: mean of modeled RSL trends, (weighted by area); V: fraction of the variance in % explained by the regression model. For each solution, the upper and lower lines give the regression results in mm/yr equivalent contribution to the mean over the (small) regression grid and the (near-global) complete steric grids, respectively (see Plag, 2006, for a more detailed discussion of the global model). P2 is similar to P1 (see Table 1), except for a lower mantle viscosity of 4.75- 1021 Pas. The contributions are given with 95% confidence limits. Bared quantities are spatial averages over the respective grid. M TI" A,G,P2,I500,c
0.61 0.27 T2: A,G,P2,I500,c 0.42 0.19 T 1" A,G,P2,L500,c 0.54 0.24 T2: A,G,P2,L500,c 0.44 0.20 TI: A,G,P2,L3000,c 0.41 0.18 T2: A,G,P2,L3000,c 0.40 0.18
OZA" /-A 4- 0.37 4- 0.16 4- 0.31 4- 0.14 + 0.37 4- 0.16 4- 0.32 4- 0.15 4- 0.37 4- 0.16 4- 0.32 4- 0.14
0.04 0.09 0.07 0.14 0.02 0.05 0.06 0.14 0.02 0.05 0.06 0.13
CtG" /-G 4- 0.03 4- 0.07 4- 0.03 4- 0.06 + 0.02 4- 0.05 4- 0.03 4- 0.07 4- 0.02 4- 0.05 4- 0.03 4- 0.07
-0.09 -0.04 -0.07 -0.04 -0.12 -0.05 -0.09 -0.04 -0.12 -0.05 -0.09 -0.04
regression coefficient is close to one. B a s e d on that, Plag (2006) c o n c l u d e d that the L 5 0 0 is m o s t consistent with the spatial pattern o f the T1 L S L trends. H o w ever, in the o p e n ocean, the L 3 0 0 0 set m a y be better, and for L 3 0 0 0 the global average is 0.3 5 ram/yr. Consequently, the estimates given in Table 2 for the steric contribution are likely to be at the lower end. H o w ever, for the T2 L S L trends, the regression coefficients for all three data sets are very close to zero, leaving no r o o m for a steric contribution. For all regressions, the global average values are lower than the averages obtained for the regression grid, and they are at the very low end of values gen-
ft. /5 4- 0.02 0.08 4- 0.01 0.06 4- 0.02 0.00 4- 0.01 0.00 4- 0.02 0.20 4- 0.01 0.20 4- 0.02 - 0 . 0 2 4- 0.01 - 0 . 0 2 4- 0.02 0.14 0.14 4- 0.01 4- 0.02 0.01 4- 0.01 0.01
444444444444-
7" ~' 0.02 0.02 0.02 0.02 0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04
0.46 0.46 0.73 0.73 0.50 0.50 0.80 0.80 0.65 0.65 0.76 0.76
c Emod V 1 10 13.10 4- 0.40 0.83 4- 0.40 1 15 9.08 4- 0.35 1.02 4- 0.35 + 0.40 1 14 13.79 0.94 4- 0.40 1 19 9.79 4- 0.35 1.07 4- 0.35 1 10 12.43 4- 0.40 4- 0.40 0.97 1 15 9.57 4- 0.35 1.04 4- 0.35
Table 3" Comparison of T1 and T2 results. The contributions are to global average.
Factor/parameter Variance explained (%) Antarctica (mm/yr) Greenland (mm/yr) Thermosteric (mm/yr) Unexplained (mm/yr) Global average (mm/yr)
T1 11.3 to 13.8 0.394-0.11 0.104-0.05 0.30-t-0.10 0.354-0.40 0.904-0.75
T2 8.5 to 10.1 0.314-0.13 0.164-0.03 0.004-0.01 0.454-0.35 1.004-0.70
erally reported for the global sea level rise over the last 50 to 100 years. The basic a s s u m p t i o n s for the extrap-
45
46
H.-P. Plag
olation of the models from the regression grid to the global grid are: (1) the regression model is appropriately representing the long spatial wave length in sea level trends, and (2) there are no other open ocean contributions to global sea level rise not sensed by the tide gauges, that would affect the extrapolation of e. Plag (2006) argued that the results for the steric contributions using the L500 and I500 datasets are likely to be minimum estimates, while the global average of the L3000 dataset indicates that the actual steric contribution may be larger by 0.1 to 0.2 mm/yr. Therefore, he considered a global sea level rise value of 1.05 -+- 0.75 mm/yr to be more likely. For the T2 results, it is likely that the steric contribution has been absorbed by the regression of local atmospheric forcing. Therefore, if we add a similar contribution to compensate for a bias of the steric contribution, the global average LSL trend for the T2 grid is of the order of 1.20 -+- 0.70 mm/yr.
7
Conclusions
The comparison of the regression results for the T1 and T2 LSL trends reveals a high correlation between spatial patterns of the LSL trends locally attributed to atmospheric forcing and the thermosteric contribution. This correlation may be due to a feedback from sea surface temperature changes to the regional air pressure and wind fields. The regression results for both LSL trend datasets show that the observed spatial pattern of L S L trends is compatible with melting of both the Greenland and Antarctic ice sheets. In fact, the results assign a high significance to this melting. Based on the regression results for T1, the steric contribution to the global LSL average trend is at least 0.20 + 0.04 mm/yr but more likely to be larger. However, a part of that signal may actually be due to a correlated effect of atmospheric forcing on sea level. The results presented here underline the potential of the fingerprint method to extract useful information from the sea level observations provided by the global network of tide gauges. Potential biases of the regression results and particularly the extrapolation to the global ocean surface are discussed by Plag (2006) and can result from (1) long-term changes in the Greenland and Antarctic ice sheets, (2) errors in the PGS predictions, and (3) unaccounted factors with fingerprints having large spatial variations. Here we have identified an additional error source, which results from the correlation of atmospheric forcing and steric changes. In order to reduce the biases, a more comprehensive
regression model needs to be set up, that takes into account the different forcing factors in one common fit.
Acknowledgements The author would like to thank J. Hunter for a thorough review, J.X.Mitrovica for the provision of the post-glacial rebound models, and Anny Cazenave for the provision of the steric sea levels computed from the Levitus and ishii datasets. The tide gauge data was taken from the database maintained by the Permanent Service f o r Mean Sea Level. Without the long-lasting work of the PSMSL, this study would not have been possible. Part of this work was supported by a NASA grant in the frame of the Interdisciplinary Science Program.
References Church, J. A., Gregory, J. M., Huybrechts, P., Kuhn, M., Lambeck, K., Nhuan, M. T., Qin, D., & Woodworth, P. L., 2001. Changes in sea level, in Climate Change 2001: The Scientific Basis. Contribution of Working Group I to the Third Assessment Report of the Intergovernmental Panel on Climate Change, edited by J. T. Houghton, Y. Ding, D. J. Griggs, M. Noguer, P. J. van der Linden, X. Dai, K. Maskell, & C. A. Johnson, pp. 639-693, Cambridge University Press, Cambridge. Farrell, W. E. & Clark, J. A., 1976. On postglacial sea level, Geophys. J. R. Astron. Soc., 46, 647-667. Ishii, M., Kimoto, M., & Kachi, M., 2003. Historical ocean subsurface temperature analysis with error estimates, Monthly Weather Rev., 131, 51-73. Levitus, S., Stephens, C., Antonov, J., & Boyer, T., 2000. Yearly and year-season upper ocean temperature anomaly field, 1948-1998, Tech. rep., U.S. Gov. Printing Office, Washington, D.C. Milne, G. A., Mitrovica, J. X., & Davis, J. L., 1999. Nearfield hydro-isostasy: the implementation of a revised sea-level equation, Geophys. J. Int., 139, 464482. Plag, H.-P., 2006. Recent relative sea level trends: an attempt to quantify the forcing factors, Phil. Trans. Roy. Soc. London, In press. Plag, H.-P. & Jfittner, H.-U., 2001. Inversion of global tide gauge data for present-day ice load changes, in Proceed. Second Int. Syrup. on Environmental research in the Arctic and Fifth Ny-~lesund Scientific Seminar, edited by T. Yamanouchi, no. Special Issue, No. 54 in Memoirs of the National Institute of Polar Research, pp. 301-317. Tsimplis, M. N., ilvarez-Fanjul, E., Gomis, D., FenoglioMarc, L., & P~rez, B., 2006. Mediterranean sea level trends: Separating the meteorological and steric effects, GRL, In press. Woodworth, P. & Player, R., 2003. The Permanent Service for Mean Sea Level: an update to the 21st century, J. Coastal Research, 19, 287-295.
Chapter 7
On the low-frequency variability in the Indian Ocean I.V. Sakova, G. Meyers CSIRO Marine and Atmospheric Research, GPO Box 1538, Hobart, Tasmania, Australia, 7001 e-mail: Irina.
[email protected] R. Coleman School of Geography & Environmental Studies, University of Tasmania, Private Bag 78, Hobart, Tasmania, Australia, 7001
Abstract. This paper presents the results of an investigation of low-frequency variability in the Indian Ocean (IO) primarily using satellite altimeter sea surface height (SSH) observations and expendable bathy-thermograph (XBT) data. We found that in most regions of the IO the low-frequency part of the SSH spectra (corresponding to signals with periods of a few months or longer) is concentrated in four frequency bands separated by substantial spectral gaps. These bands correspond to periods of approximately 6 months, 12 months, 18-20 months and more than 30 months (hereafter referred to as the 30-month band). For both 18-20-month and 30month bands the spectral density shows a dipolelike pattern with some degree of similarity; however, analysis of the spatial-temporal evolution of these signals suggests that the 18-20-month signal is an internal mode of the IO, while that of the 30month component propagates from the Pacific Ocean, in particular with a much stronger signal during the period from 1998 to 2003. Keywords. Indian Ocean, Indian Ocean Dipole, Satellite Altimetry, spectral analysis 1 Introduction Today more than ever the oceanographic community enjoys the availability of extensive observational data sets including satellite altimetry (Fu and Cazenave 2001; Born 2003). The TOPEX/Poseidon and Jason-1 satellite altimeter data cover a time interval of more than 13 years. Such a long time interval opens new opportunities for analysis of the low-frequency processes in the ocean with characteristic time scales of up to several years. In this paper we present results of an investigation of the low-frequency variability in the IO by applying spectral analysis methods to the satellite altimeter and XBT data sets. The existence of annual and semi-annual processes in the IO is well known; they have been sub-
jects of numerous studies (e.g., Clarke and Liu 1993). In the last decade there has also been a growing interest in studying of the interannual variability in the tropical Indian Ocean (TIO) (Perigaud and Delecluse 1993; Masumoto and Meyers 1998; Feng and Meyers 2003), particularly after the discovery of the Indian Ocean Dipole (IOD) mode (e.g., Saji et al. 1999). The interannual variability of sea surface temperature (SST) is also well studied (e.g., Behera et al. 2000), with only several studies of subsurface variability of the TIO (Tourre and White 1995; Meyers 1996; Murtugudde and Busalacchi 1999; Schiller et al. 2000; Rao et al. 2002; Feng and Meyers 2003). Rao et al. (2002) found that the dominant modes of interannual variability in the IO do not show co-variability between the surface and subsurface. Using empirical orthogonal functions (EOF) for analysis of SSH satellite data for the period 1993-1999, as well as ocean model output, the authors found two dominant modes of the interannual subsurface variability of the TIO. The first mode was found to be governed by the IOD, while the second mode "shows the interesting quasibiennial tendency". In another study, Feng and Meyers (2003) found a 2-year time-scale of the upper-ocean evolution in the TIO near the Java coast that is unique to the IOD (that is, not directly forced by an atmospheric teleconnection from the Pacific). This conclusion was based on the timing of the temperature anomalies, associated with strength of upwelling of Java, when "cold anomaly during 1994 and 1997 [was] followed by warm anomaly during 1995/96 and 1998". In this work, we study the low-frequency variability of the IO by using spectral analysis methods. Spectral analysis enables investigation of spatial and temporal characteristics of the low-frequency processes in the whole IO basin in a systematic way by identifying principal frequencies, calculating spatial distributions of the power spectrum density, and reconstructing the spatial-temporal evolution of the main variability modes.
48
I.V. Sakova. G. Meyers • R. Coleman
2 Data and methods Two data sets were used in this study: gridded (1 degree) SSH weekly-averaged data for the period from October 1992 to August 2004 collected by the ERS/Envisat/TOPEX/Jason- 1.satellites (http://www.j ason. oce anob s. com/html/donn ee s/pro d uits/msla_uk.html) and expendable bathythermograph (XPT) temperature data for the period from January 1989 to December 2002 (http://www.marine. c siro. au/~pigot/REPORT/overv iew.html) (Meyers and Pigot 1999). The spectral analysis of these data sets was conducted by using Discrete Fourier Transform (DFT) method (e.g. Emery and Thomson 1997). The XBT data had irregular time sampling; therefore it was interpolated to a monthly grid. The time series contained 168 monthly entries. For analyzing twodimensional SSH fields, DFT was applied to the time series of SSH at each grid point. The resulting gridded spectra were then either presented as power spectral density maps or used to select a particular frequency band and conduct Inverse Discrete Fourier Transform (IDFT) to investigate the temporal dynamics of the corresponding process. Each time series contained 618 weekly entries.
3 Analysis 3.1 XBT data analysis The broad application of spectral analysis to the SSH data for the whole IO was initially motivated by results of spectral analysis of upwelling offshore of Indonesia using XBT data. Figure l a shows a typical temperature profile near the Sumatra-Java coast obtained by averaging XBT data in a rectangle with opposite comers at 7.0°S, 104.0°E and 8.0°S, 106.0°E. One can see a number of strong upward displacements of the thermocline associated with upwelling that result in the cold water appearing at the sea surface, the most noticeable being during the 1994 and 1997 IOD events, and possibly in 1991. There are also frequent minor displacements that can be easily seen from the shape of the 20°C (D20) isothermal surface shown by the white line. It is well known that the thermocline in this region heaves with a periodicity of two times a year, which is related to the development of the Wyrtki jet (Clarke and Liu 1993), and some of these displacements can be associated with this semi-annual process. It is difficult to make other suggestions about a possible decomposition of the movement of the D20 surface from Figure l a until conducting a DFT of this signal.
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Figure 1. (a) XBT temperature data (°C) at 6-7°S, 104-106°E, where white line - 20°C isotherm; (b) Power spectrum for depth of the 20 ° C isotherm, where psd is power spectral density, and cpm is cycle per month.
Figure lb shows the power spectrum of the depth of the D20 surface. Interestingly, the spectrum contains a number of well-separated maxima. As expected, one can easily identify the semi-annual component. There are also two surprisingly clear and strong low-frequency maxima corresponding to periods of approximately 18.7 months and 34 months. These low-frequency maxima rise far above the "noise" level in the spectrum and do carry much more energy than the 6-month component. Because of the long associated time scales, such processes may be expected to manifest themselves not only near Java coast but at larger spatial scales. This motivated us to conduct a broader spectral analysis of the low-frequency variability in the whole IO.
3.2 SSH data analysis Figure 2 shows power spectra of SSH in a number of different locations in the IO. All these spectra have one common feature: they contain a few rather strong and well separated maxima. Apart from the semi-annual and annual components, there are also two inter-annual variability modes already encountered in the spectrum of D20: those corresponding to periods of approximately 18-20 and more than 30 months. Overall, in different parts of the ocean, the power spectra of SSH may lack one or another of the identified four standalone maxima, but in all cases, all well-separated narrow maxima arising in power spectra do belong to one of these four frequency bands.
Chapter 7 • On the Low-Frequency Variability in the Indian Ocean
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suggestive of Rossby waves, while the maximum on the equator in the eastern TIO is suggestive of Kelvin waves. It is well known that the semi-annual signal is generated by wind forcing in the central basin (Birol and Morrow 2001); the remote Kelvin and Rossby responses are then seen in the SSH signal. The figure also shows strong signal along the coast in the Bay of Bengal and in the Arabian Sea close to the Gulf of Aden (Shankar et al. 2002). a
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Figure 3 presents statistics (mean, lower quartile and upper quartile values) for the power spectral density in a number of regions in the IO. Compared to Figure 2, the outlines of the spectra are smoother, which points to the possibility that the involved variability modes are localised, reaching maxima at some locations and being insignificant at others. Nevertheless, even with the values of the spectral maxima being smoothed by the spatial averaging, the maxima of 6-, 12-, 18- and 30-month frequency bands can still be clearly seen for most of the regions.
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Period (months)
Figure 3. Statistics, spectral analysis of SSH, where psd is normalised power spectral density.
Following in Figure 4 are the plots of the spatial distribution of the spectral density for 6-, 12-, 18and 30-month bands. Figure 4a shows the spatial distribution of spectral density for the semi-annual signal. It shows strong signals in upwelling regions offshore of Indonesia between approximately 5 N and 8 S and offshore of Africa between approximately 7°N and 5°S. The U-shaped structure in the western TIO with the maxima off the equator is
The spatial distribution of the 1-year signal in Figure 4b has maxima in the Red Sea, Arabian Sea, western part of the Bay of Bengal and south-east of Indonesia. The maximum in the Arabian Sea is obviously caused by strong monsoon winds in this region. The maximum south of the equator is due to wind-forced, annual Rossby waves (Masumoto and Meyers, 1998). Figure 4c shows the spatial distribution of spectral density of the 18-20 month signal. It contains strong signals offshore of Indonesia and in the Bay of Bengal. The two new interesting features are the
49
50
I.V. Sakova. G. tvleyers • R. Coleman
strong maximum in the central IO, in between approximately 5°S and 15°S and 60°E and 95°E and zonally elongated narrow maximum at approximately 23°S and between 70°E and 98°E. The spatial distribution of the 30-month signal in Figure 4d contains two major maxima: offshore of Indonesia and in the central western IO. The location of the latter maximum is different to that for the 18-20-month signal, being wider and more westward. Analysis of the spatial distributions of the power spectral densities can provide important clues to the physics of the corresponding variability modes; however, the power spectrums contain no phase information. To study temporal variability of signals in different frequency bands, we conduct IDFT of the filtered signals and plot series of snapshots of the reconstructed signal in the given frequency band (not shown)
4 Discussion and Summary Using frequency analysis of satellite SSH data, two main frequency bands on the interannual time scale in the IO were identified. These bands correspond to the signals with periods of 18-20 and more than 30 months. While the patterns of spatial density of these signals do have similar dipole-like features, analysis of their temporal evolution (not shown) points at the different physical backgrounds of the corresponding variability modes. It is quite possible that the 18-20-month signal arising in our spectral analysis corresponds to the two-year time scale of variability in the TIO found by Rao et al. (2002) and Feng and Meyers (2003). Concerning the 30-month signal, its temporal dynamics suggests propagation of the signal from the Pacific in 1998-2003. The question of whether the signal in this frequency band correlates with ENSO and/or IOD has to be addressed in the future, including studying of the correlation with surface wind data and spectral analysis of signals over longer time intervals than in the current study. Overall the use of the spectral analysis of the SSH satellite observations allowed us to find strong well separated low-frequency spectral maximums in the spectrums of the signals and to investigate the spatial distribution and temporal dynamics of the corresponding variability modes. We believe that this information gives a strong quantitative basis for subsequent investigations of the underlying complex physical processes in the ocean-atmosphere system.
Acknowledgments The first author is supported by a joint CSIROUTAS PhD scholarship in quantitative marine science (QMS) and a top-up CSIRO PhD stipend (funded from Wealth from Oceans National Research Flagship)
References Behera, S. K., P. S. Salvekar, et al. (2000) Simulation of interannual SST variability in the tropical Indian Ocean. Journal of Climate vol. 13, no. 19: 3487-3499. Birol, F. and R. Morrow (2001) Source of the baroclinic waves in the southeast Indian Ocean. Journal of Geophysical Research 106(C5): 9145-9160. Born, G. H. (2003) Jason-1 calibration/validation. Special issue. Source: Marine Geodesy; 26(3-4): 129-421. Clarke, A. J. and X. Liu (1993) Observations and dynamics of semiannual and annual sea levels near the eastern equatorial Indian Ocean boundary. Journal of Physical Oceanography 23(2): 386-399. Emery, W. J. and R. E. Thomson (1997). Data analysis methods in physical oceanography, Pergamon. Eeng, M. and G. Meyers (2003) Interannual variability in the tropical Indian Ocean: a two-year time-scale of Indian Ocean dipole. Deep-Sea Research Part Ii-Topical Studies in Oceanography 50(12-13): 2263-2284. Fu, L. and A. Cazenave (2001) Satellite altimetry and Earth sciences. A handbook of techniques and applications. International Geophysics Series, 69. Academic Press, San Diego. Masumoto, Y. and G. Meyers (1998) Forced Rossby waves in the southern tropical Indian Ocean. Journal of Geophysical Research-Oceans 103(C 12): 27589-27602. Meyers, G. (1996) Variation of indonesian throughflow and the E1 Nino Southern Oscillation. Journal of Geophysical Research-Oceans 101 (C5): 12255-12263. Meyers, G. and L. Pigot (1999) Analysis of frequently repeated XBT lines in the Indian Ocean. CSIRO Marine Laboratories Report, Hobart, Australia: 43pp. Murtugudde, R. and A. J. Busalacchi (1999) Interannual variability of the dynamics and thermodynamics of the tropical Indian Ocean. Journal of Climate 12(8): 2300-2326. Perigaud, C. and P. Delecluse (1993) interannual sea level variations in the tropical Indian Ocean from Geosat and shallow water simulations. Journal of Physical Oceanography vol.23, no.9: 1916-1934. Rao, S. A., S. K. Behera, et al. (2002) Interannual subsurface variability in the tropical Indian Ocean with a special emphasis on the Indian Ocean Dipole. Deep-Sea Research Part IiTopical Studies in Oceanography 49(7-8): 1549-1572. Saji, N. H., B. N. Goswami, et al. (1999) A dipole mode in the tropical Indian Ocean. Nature 401(6751): 360-363. Schiller, A., J. S. Godfrey, et al. (2000) Interannual dynamics and thermodynamics of the Indo-Pacific oceans. Journal of Physical Oceanography 30(5): 987-1012. Shankar, D., P. N. Vinayachandran, et al. (2002) The monsoon currents in the north Indian Ocean. Progress in Oceanography 52(1): 63-120. Tourre, Y. M. and W. B. White (1995) ENSO signals in global upper-ocean temperature. Journal of Physical Oceanography
Chapter 8
Satellite Altimetry: Multi-Mission Cross Calibration Wolfgang Bosch and Roman Savcenko Deutsches GeoEitisches Forschungsinstitut (DGFI), Marstallplatz 8, 80539 Mtinchen, Germany
A b s t r a c t . Multi-mission satellite altimetry provides
a unique opportunity to perform an innovative, utmost rigorous cross-calibration of all satellites operating simultaneously. Data from TOPEX/Poseidon, ERS-2, GFO, and Jasonl is used to c o m p u t e - in all combinations - nearly simultaneous single and dual satellite crossovers that are unaffected by sea level variability. The total set of crossovers provides a rather dense sampling of the orbits of all satellites and realizes a rigid network with high redundancy to o b t a i n - by a discrete crossover analysis technique - a reliable estimate of the radial error components. The analysis is performed for a sequence of 10 day periods (the cycle 96-457 of TOPEX) with 3 days overlap to neighbouring periods. For all satellites the error estimates for the radial component exhibit significant geographical pattern - even for TOPEX. The error components are also taken to calculate time series of relative range biases and geocentre offsets between all altimeter missions. Satellite altimetry, cross calibration, crossover analysis, least squares
Keywords.
1 Introduction Crossover analysis is a powerful approach to estimate errors and improve observations of satellite altimetry. The orbit error of early altimeter satellites has been essentially improved this way (Schrama 1989). Crossover analysis helps to identify time tag errors and to estimate improved correction models (Chambers 2003). Since the contemporary operation of TOPEX/Poseidon and ERS-1 (later on ERS-2) crossover analysis has become an important tool to cross calibrate altimeter missions with different sampling characteristic (Le Traon & Dibarboure 1999). For several reasons, TOPEX/Poseidon played a dominant role in cross calibration: Due to three independent tracking systems (Laser, GPS and DONS), the first two frequency altimeter sensor (allowing in-situ corrections for the ionosphere) and the mean
orbit height of 1300 km (reducing non-gravitational errors) it was reasonable to consider TOPEX/Poseidon as the most precise altimeter mission. Consequently, TOPEX/Poseidon has been used as reference for the cross calibration of different altimeter missions. Le Traon & Ogor (1998) use crossover differences of ERS and between ERS and TOPEX to adjust the ERS orbits to TOPEX. The TOPEX orbits are not changed. Other altimeter systems (e.g. GFO and ENVISAT) are treated the same way. During the several month tandem phase both, TOPEX and Jasonl observed the same ground track with only 1 minute delay in order to allow an utmost precise cross calibration. For consistency the orbits of Jason l and TOPEX were computed with JGM3. Figure 1 shows, however, systematic differences between TOPEX and Jason l. New, GPS based orbits computed with the GRACE gravity fields confirm significant geographical orbit errors for TOPEX. Thus, there is no longer a justification for the prominent role of TOPEX. In the present paper, the orbits of TOPEX are subject to error estimates and improvements just as all other missions. Crossover differences between all contemporary altimeter missions increasing considerably the redundancy in the crossover network and opens the challenge to identify also systematic errors of the TOPEX mission. In this paper a common analysis of TOPEX, Poseidon (treated independent of TOPEX), ERS-2, GFO, and Jasonl is performed. The extended mission of TOPEX (with shifted ground tracks) is treated separately and is further on identified as T/P-EM.
,
,
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,
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Fig. 1. Mean sea surface height differences between TOPEX and Jason l during their tandem phase (TOPEX cycles 344 - 365) reveal systematic errors.
52
W. Bosch. R. Savcenko
2 Discrete Crossover Analysis In the present investigation a discrete crossover analysis (DCA) is performed (Bosch 2006). It does not estimate parameters of any functional error model but solves only for the radial errors at the "observed" crossover locations. To ensure a certain degree of smoothness for adjacent errors a least squares minimization is applied to both, the crossover differences and adjacent differences of the radial errors. The DCA is realized by minimizing mink~=0 I D l x - I ; 1 2
(1)
where the vector d compiles the n observed crossover differences. The vector x keeps the 2n unknown error components which are assumed to be ordered with respect to time. Then, matrix
D--
1
-1
0
......
0
0
1
-1
0
.--
0
0
• ..
0
1
-1
0
0
......
0
1
-1
:
accounts for the consecutive differences and matrix
n
1
0
...
0
-1
0
0
1
-1
0
•..
0
and the Jasonl data (AVISO & PODACC 2003) were taken from the ftp server of JPL. E R S - 1 0 P R data has not been included, because it is based on different models and software versions which could affect the cross calibration in an unpredictable way. ENVISAT mission data preparation is ongoing. This data will be included in any further analysis. For the computation of sea surface heights the standard corrections have been applied with the following modifications: - for all missions ocean tide corrections have been computed from FES2004 (Letellier et al. 2004)) - for the sea state bias correction for TOPEX, side B the model of Chambers et al. (2003) was used - the pole tide for ERS-2 was added with the same mean pole as used for TOPEX/Poseidon - all sea surface heights were transformed to the TOPEX reference ellipsoid - a common time reference epoch and the TAI time scale was introduced for all missions The orbit data, however, was not changed, although improved orbits are available (e.g. for ERS-2 as provided by the DEOS group). The orbits of TOPEX and Jasonl are based on the same gravity field model, JGM3. Orbits for ERS-2 are based on the precise orbits generated by the "German ERS Processing and Archiving Facility", DPAF. GFO precise orbits are generated by GSFC. No altimeter range bias was a p p l i e d - it will result from the crossover analysis.
m
4 Crossover Computation 0
•••
0
1
0
-1
relates the radial errors to the crossover differences. The linear system (1) has a rank defect of one because the sum of all columns of the coefficient matrix is a null vector• This rank defect is overcome by a single constraint, a linear combination of unknowns x. For n crossover points there are 2n unknown error components to estimate• Thus, the normal equation system is in general huge, but s p a r s e - it is therefore solved by the iterative "Conjugate Gradient Projection" algorithm. More details on the DCA are described in Bosch (2006, this volume).
3 Altimeter Data TOPEX/Poseidon data has been taken from the "Geophysical Data Record" (GDR-M), Version C, distributed by AVISO (1996). The "Ocean Product Record" (OPR) for ERS-2 were delivered by CERSAT. GFO data were kindly provided by NOAA's Laboratory for Satellite Altimetry (NOAA 2002).
Single satellite crossovers are performed as intersections of ascending (A) and descending (D) passes. For satellites with different inclination four types of crossovers are possible: AA, DD, AD and DA. In favour of a large intersection angle only two of these combinations were computed, if both satellites have prograde (or retrograde as ERS-2 and GFO) orbits, DA and AD crossings are favourable. If orbits are different, prograde and retrograde (e.g. TOPEX and ERS-2), only AA and DD crossings were generated. Crossover points were excluded if the absolute value of the crossover difference exceeds 100 cm or if the standard deviation was larger than 10 cm. Finally, crossovers are used only if their time difference is less than three days to ensure that crossover differences are as little as possible affected by sea level variability. Single satellite crossovers with short time difference are concentrated at high latitudes, the distribution of nearly simultaneous dual satellite crossovers is much better and leads to a dense sampling of the orbits of all satellites involved.
Chapter 8 • Satellite Altimetry: Multi-Mission Cross Calibration
5 The A n a l y s i s R e s u l t s
mm2
The crossover analysis was performed for the TOPEX cycles 096-456. Every 9.9156 day period was extended by three days at the beginning and the end such that an overall period of 16 days was analysed by DCA. For this period all single and dual satellite crossovers between all missions and all cycles were computed. A weighting with cosd0 was applied to the crossovers. For two missions there are about 20000 crossovers. With four simultaneously observing altimeters, the number of crossovers may reach 1 0 0 0 0 0 - in spite of limiting the crossover time difference to three days. For every 16 day period the solution was regularized by forcing the sum of all TOPEX error components to zero. The solution was iterated once. Crossovers in the second run were removed, if the absolute value of the residual crossover difference exceeds three times the posteriori unit weight standard deviation a 0. The number of rejected crossovers is always below 1%. Differences between the two errors estimates within the overlapping periods were found to be at most 1 - 2 mm. This justifies to consider further on only those errors that belong to the central cycle period. Concatenating all theses errors a temporal sequence of discrete error components is created for all altimeters. The estimation of empirical autocovariance functions for each mission is straightforward and shown in Figure 2. The overall crossover statistic, illustrated in Figure 3 shows the gain obtained by the DCA. The gain is measured by the ratio of rms values for the crossover differences before and after the analysis. Up to cycle 266 rms values of about are reduced by 1 - 2 cm only.
100
120 140
160
Autocovariance
~K 0
1000
2000
3000
4000
5000
6000
8000
9000
10000
When Jasonl enters the DCA (at TOPEX cycle 344), the gain becomes significant: rms values of about 11 cm are reduced to about 7cm[ A few rather high rms values are due to orbit anomalies of GFO.
6 Relative R a n g e B i a s e s and C e n t r e - o f Origin Shifts It is of particular interest to investigate how the errors are geographically distributed. Systematic error patterns may be explained by inconsistencies in the centre-of-origin implied by the satellite orbit. Non vanishing mean values of the errors are caused by relative range biases between (uncalibrated) altimeter missions, but may also indicate scale differences between different tracking systems used to compute the orbits. To estimate relative range biases and centre-of-origin shifts the model x~ + Vx, -- A r + A x cos ~ sin X + & y cos ~ cos X + & z sin c~ was fitted by least squares to the errors x i of every cycle and for every mission. A weighting with cosqb was applied to account for an increasing number of
180 200 220 240 260 280
300 320 340 360 380 400 420 440 460 •
m
0.20
i
0.16
m
•
0.20
• •
0.16 eo
• Oo 0.12
~
~
' " °q"" ° "~ ! ' . ~ '
°•
•
,,~,
.
"~" "%~';'%~o, - - , . ' ~ ' V ' ~ , ~ % ~ , ~
oa,,,
-~,~
~,~,~.-
~---~
-~,0---
"-"~-
%~~o
0.12
"~~°%~o~%~",~~-~,~-~0.08 o
0.08
7000
Fig. 2 Empirical Autocovariance function for the radial error components of all missions
o_
0.04
0.04
ERS-2
GFO
Jason1
Fig. 3 Crossover rms-values before (black dots) and after (circels) analysis. Up to cycle 266 (with TOPEX and ERS-2 only) the analysis gain is moderate: the rms-values (on average about 9cm) is reduced by 1-2cm. From cycle 344 on (all four missions) the analysis gain is significant: the rms-values of about 11cm is reduced to about 7cm. Large rms-values are due to significant orbit errors. E.g., GFO had a momentum wheel anomaly at cycle 118 (corresponds to TOPEX cycle 409/410).
53
54
W. Bosch. R. Savcenko 1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
20
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a} rel,~tive ~ a n a e biase: Ar [cnnl
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.
.
240 1999 .
.
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........,;,._Zt..~ ,~.-;,. ...... ~-,=._..-~."
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Fig. 2. Fiducial stations
300=
320"
Chapter 12 • Brazilian Vertical Datum Monitoring - Vertical Land Movements and Sea Level Variations
Maximum observation strategy (OBS-MAX) was used to create the baselines. The sampling rate used was 30 sec., with an elevation cut off angle of 5 °. QIF (quasi ionosphere free) strategy was used for Ambiguities Resolution with Niell (Dry and Wet) troposphere modeling (Hugentobler et al., 2004). The Global Ocean Tide M o d e l - GOT00.2 was applied for ocean loading correction. The combination of the daily solutions (NEQ) provided a final solution for each campaign.
2.2 GPS Processing Results Table 2 presents the estimated positions and standard deviations (~s) for IMBI station in each campaign. The processing software gives precisions better than 1.5mm for the coordinates; this is very optimistic, needing more studies in order to verify if these values are realistic. Table 2. Geocentric Coordinates - IMBI Station 1997 Campaign (ITRF2000 X (m)= 3 714 672.4278 c~(mm) = 0.6 Y (m)=-4 221 791.4718 c~(mm) = 0.8 Z (m) = -2 999 637.9143 (mm) = 0.5 2000 Campaign (ITRF2000 X (m)= 3 714 672.4277 c~(mm) = 0.4 Y (m)=-4 221 791.4845 c~(mm) = 0.5 Z (m)=-2 999 637.8821 c~(mm) = 0.3 2005 Campaign (ITRF2000 X (m)= 3 714 672.4333 c~(mm) = 1.4 Y (m) = -4 221 791.5173 c~(mm) = 1.2 Z (m)=-2 999 637.8391 c~(mm) = 0.8 2005 Campaign (ITRF2000 X (m)= 3 714 672.4390 (mm) = 0.4 Y (m)=-4 221 791.5062 c~(mm) = 0.5 Z (m)=-2 999 637.8215 c~(mm) = 0.3
epoch 1997.4) qb=-28 ° 14' 11.809025" X =-48 ° 39' 21.882063" h-- 11.8546m epoch 2000.4) qb=-28 ° 14' 11.807957" )~ =-48 ° 39' 21.882371" h = 11.8477m epoch 2005.1) qb=-28 ° 14' 11.806292" )~ = -48 ° 39' 21.883015" h = 11.8523m epoch 2005.5) qb=-28 ° 14' 11.805857" )~ =-48 ° 39' 21.882588" h = 11.8399m
Estimated and derived velocities from SIRGAS velocity model for IMBI station are indicated on Table 3. There is a small tendency of subsidence at IMBI station. The estimated velocities are similar to
the ones derived from SIRGAS velocity model (Drewes and Heidbach, 2005). The processing software gives precisions better than 0.1mm/a for the estimated velocities. Table 3. Velocities - IMBI station Estimated velocities Vy (m/a) VE (m/a) Vup (m/a) Vx (m/a) Vy (m/a) Vz (m/a)
0.0118 -0.0019 -0.0002 0.0021 -0.0053 0.0105
Velocities derived from SIRGAS velocity model VLAT(m/a) VLONG(m/a) Vx (m/a) Vv (m/a) Vz (m/a)
0.0124 -0.0028 0.0018 -0.0062 0.0109
3 Integrating Temporal Time Series of Sea Level Data Usually, the sea level time series in one station were obtained by different tide gauges. Then, it is necessary to relate all the measurements to a common reference to have a coherent sea level time series. This relationship is done based in a network of BMs periodically leveled and associated to the tide gauge. The necessary information to integrate temporal time series of sea level data are: the sea level values; the tide gauge datum definition; the BMs description and the leveling surveys performed. Regarding the datum definition and the sea level values, BVD was established by the MSL with observations between the years 1949 and 1957. Nowadays the periods with available sea level data are: a) 1949 to 1969 monthly and annual means at Permanent Service for Mean Sea Level (PSMSL) database; b) 1969 to 1987 graphic records not found yet; c) 1987 to 1992 graphic records recovered, which digitizing is being made at UFPR's Geodetic Instrumentation Laboratory; d) 1992 to 1998 tide gauge operation seems to be interrupted; e) since 1998 graphic records being digitized at IBGE's (Brazilian Institute of Geography and Statistics) Coordination of Geodesy and f) since 2001 data from digital sensors are available. The process of graphic records digitizing is not an easy task and it is time consuming; efforts are being made in order to automate this process. Concerning the leveling surveys performed at Imbituba is important to say that the United States Coast and Geodetic Survey (USCGS) performed periodical leveling between 1948 and 1971. IBGE performed leveling surveys in 1946, 1980, 1986,
73
74
R. Dalazoana • S. R. C. de Freitas. J. C. B~ez. R. T. Luz
1995, and annually since 2001. The temporal control of the original BMs implanted at the harbor's area during the period of BVD definition is a great problem due to the destruction of the marks and due to the lack of repeated leveling surveys. From the original BMs (approximately eight), only one (BM 3M) was not destroyed until July 15 th, 2005. This BM was of great importance in the integration between old surveys and newer ones. With the 2005 campaigns it was possible to perform leveling surveys on almost all the actual BMs located within the harbor area. The leveling process includes BMs located below the pier, the tide staff, the stilling well and two new BMs established on July 2005; it was performed in order to make an adjustment of the observed height differences. Height of BM 3M was estimated from USCGS reports and was kept fixed (6.555m). Table 4 gives an overview of the adjusted leveled heights of the existing BMs. Table
4. Adjusted Leveled Heights
BMs UFPR2 3012Z 3010A CBD3A 3010B 4X |MB| PORT3 3012X UFPR1
AdjustedHeight (m) 1.829 2.098 6.160 6.140 9.480 8.642 10.508 6.025 2.008 6.889
StandardDeviation (mm) 0.6 0.6 0.9 0.2 1.4 1.4 1.4 0.9 0.6 0.4
The recovering, digitizing and integration of sea level time series are still under development and would be enhanced. Initial results, presented by Dalazoana et al. (2004), based on the integration of sea level data from PSMSL database and sea level data from the digital sensor, had shown relative increasing of about 2 mm/year in the local MSL. With the evolution of graphic records digitizing and GPS processing results, it will be possible to have an estimative of the absolute sea level trend at BVD.
4 Final Remarks Monitoring tide gauges by GPS positioning can give some information about possible vertical crustal movements allowing the separation between apparent and true sea level changes. Precise geocentric positioning associated with leveling gives the
link of the tide gauge zero point to a geocentric reference system. Initial results had shown relative increasing in the local MSL at Imbituba harbor. Due to the periods with no digital sea level data efforts are been done in order to automate the process of graphic record digitalization. There is a small subsidence at IMBI station indicated by the GPS data. Estimated velocities are very similar to the velocities derived from SIRGAS velocity model. The ideal situation in order to estimate velocities, mainly the vertical component, is to have several years of GPS data. There is a project to install very soon a permanent dual frequency GPS receiver at Imbituba, and the collected data will be very important in future studies.
Acknowledges The authors would like to thank CAPES (Brazilian Public Agency of Foment in Education) and CNPq (National Council of Research) by the financial support through scholarship and support to projects. IBGE for historical leveling and tide gauge data. Companhia Docas de Imbituba and Imunizadora Imbituba for operational support.
References Bevis, M., W. Scherer, and M. Merrifield (2002). Technical Issues and Recommendations Related to the Installation of Continuous GPS Stations at Tide Gauges. Marine Geodesy, v. 25, n. 1-2. Cordini, J. (1998). Estudo dos Aspectos Geodindmicos no Datum da Retie Altimdtrica do SGB. Curitiba. 159 f. Tese (Doutorado em Ciencias Geoddsicas)- Setor de Ciencias da Terra, Universidade Federal do Paranfi. Dalazoana, R., R. T. Luz, S. R. C. de Freitas, and J. C. Baez. (2004). First Studies to Estimate Temporal Variations of the Sea Level at the Brazilian Vertical Datum. In: IA G International Symposium - Gravity, Geoid And Space Missions (GGSM2004). Porto, Portugal, 30th August to 3rd
September 2004. CD-ROM Proceedings. Drewes, H. and O. Heidbach. (2005). Deformation of the South American Crust Estimated from Finite Element and Collocation Methods. IAG Symposia, Springer 2004, vol.
128. Hugentobler, U., R. Dach, and P. Fridez (2004). Bernese GPS Software. Version 5.0, University of Bern. 388p. IBGE. Funda~o Instituto Brasileiro de Geografia e Estatistica. (2002). S I R G A S - Boletim Informativo n. 7. Access" 4 jun. 2003. IBGE. Fundagfio Instituto Brasileiro de Geografia e Estatistica. (2003). Coordenadas SIRGAS 2000. Access" 4 jun. 2003.
Chapter 13
Tide gauge monitoring using GPS Maaria Tervo, Markku Poutanen, Hannu Koivula Finnish Geodetic Institute, Geodeetinrinne 2, 02430 Masala, Finland
Abstract. In this work we have studied accuracy of GPS for tide gauge stability monitoring as well as possibilities for observing the absolute sea level rise of the Baltic Sea with GPS and tide gauge time series. Our determination give the average sea level rise for the long, up to 120 years, time series 1.9 _+ 1.0 mm/year, and when corrected for the geoid rise, 1.6 _+ 1.0 m m / y e a r . Rates of recent years are even eight times higher. The possibility to detect minor changes in tide gauge benchmark height with GPS may be limited by GPS-related errors, which can be up to several mm even in a 10 km baseline.
Benchmark stability can be monitored also with GPS. GPS offers better temporal coverage, ultimately continuous tracking in real time, and the data collection and analysis can be automated. However, there are periodic and random temporal variations, especially in the vertical component, which may limit the resolving power. These variations have been studied for example in Mao et al. (1999) and Williams et al. (2004). The major component of land movements in Fennoscandia is the postglacial rebound, which has been studied also earlier with GPS (see e.g. Johansson et al., 2002).
K e y w o r d s : Sea level rise, land uplift, Baltic Sea, GPS, tide gauge, stability monitoring
70"
20"
15"
25"
30" 70"
1 Introduction Sea level monitoring is an important part of oceanography and climate investigation. Information of the sea level can be used for forecasts of climate change and marine resources as well as for natural hazard mitigation and improvement of use and protection of coastal areas. The sea level can be observed from the coasts by tide gauges or from space with different satellite borne radars. Tide gauges are spatially limited, while they are of limited number and situated on the coasts, but they provide the longest continuous sea level time series. Satellite data are spatially more evenly distributed but they can be temporally limited and their time series are much shorter. Both data are needed to complete each other. There are several studies combining the methods for shorter and longer time periods (e.g. Church et al. 2004, White et al., 2005, Holgate and Woodworth, 2004). Tide gauges measure the sea level relative to a benchmark. The problem with the tide gauges, besides their spatial limitations, is their vertical movements. Movements of these benchmarks and thus movements of tide gauges and the ground around them have been conventionally observed by levelling. Because levelling is done mostly once a year, or even less frequently, it is not very fast or accurate method if sudden movements happen.
~) Sodankyli /
\ ~ e m i
65"
Oulu
Oulu
65" Rom
Vaasa
Rauma &Turk~
60"
Olkiluo~ Hamina
'Tuorla
60"
Degerby
~0.
Helsinki Hanko . ~
~.
~'0.
Fig. 1. The GPS stations of Finnish permanent GPS network FinnRef are marked with spots and the Finnish tide gauges with triangles.
76
M. Tervo • M. Poutanen • H. Koivula
a)
b)
Earth's surface )
Earth's surface
Sob s
Sea surface
/II\ ~
S
--
Sea surface
bill1 ', ~ _n ~zxS ___Ho ', 8-1. . . . . .
N1No
\
.....
Ellipsoid
Ellipsoid
Fig. 2. Reference surfaces and heights (a) and their changes (b). Sobs is the observed sea level height, S is the absolute sea level height, H is the orthometric height of a benchmark, h is the height above the ellipsoid, N is the geoidal height. Subscripts 0 and 1 in b) denote epochs 0 and 1. Vertical changes are not in scale.
2 Sea level change
or from ellipsoidal heights using geoid height
We have calculated absolute sea level change rates for the Baltic Sea using time series from Finnish tide gauges and Finnish permanent GPS stations (Figure 1), (Tervo, 2004, Poutanen e t a l . , 2004). We chose six tide gauge - GPS station pairs that were less than 30 km from each other. We also assumed that the Finnish bedrock is stable enough for the GPS land uplift rates to be valid also for the closest tide gauge. By combining these two series we get the absolute sea level rise, i.e. the change relative to the mass centre of the Earth. The GPS data was achieved from the Finnish permanent GPS network, FinnRef, and the tide gauge data was provided by the Finnish Institute of Marine Research (K. Kahma, private communication, 2004). Tide gauges measure the sea level relative to a benchmark; this is the observed sea level height Sobs. The height of the benchmark, and thus the ground in the vicinity of the tide gauge, can be observed either by levelling or with GPS. Levelling gives the orthometric height H and GPS the ellipsoidal height h (Figure 2a). The height of the geoid N is the difference between orthometric and ellipsoidal heights (N = h - H). The absolute sea level height S is the difference between the orthometric height and the observed sea level height. The surfaces change in time (Figure 2b) and changes in their heights can be observed. The deformation of the crust AH between epochs 0 and 1 can be calculated from orthometric heights
z~-H
1 -H
o
(1)
- (h, - h o ) - (N 1 - N O) - a h - AN.
(2)
The observed sea level height is the height between the benchmark and the sea level (3)
Sob s - H - S ,
so the observed sea level change is Sobsl -- Sobs2 =
(H 1-
S 1 ) --
=(H,-Ho)-(S
~o~
= aH
(H 0 - S0)
(4)
, - S o)
- AS.
(5)
The absolute sea level change becomes AS = AH
- ASob s
(6)
and combining this with Eq. (2) gives the equation to be used with ellipsoidal heights AS = Ah - AN - ASob,"
(7)
The observed sea level change contains components of crustal deformation, sea surface topography changes and geoid changes. The crustal deformation can be calculated from GPS observations assuming the long-term stability of the reference frame. This requirement is not yet fully achieved in sub-cm level in the current ITRF realisations, or the frames used in GPS satellite orbit computations. The geoid rise values were calculated using the uplift rates from GPS time series and an empirical relation between the land uplift and geoid rise in Fennoscandia. The relation was derived by Ekman and M/ikinen, (1996) using repeated precise levellings, tide gauge records and gravity data. They obtained the geoid rise to be 6% of the land uplift.
Chapter
Vaasa
o
Ob
o
Tide Gauge
Monitoring
Using
GPS
Table 1. The absolute sea level rise rates of the Finnish tide gauges (mm/yr). 'Long time series' is the longest existing time series for each tide gauge and the ' Short ' covers years 1996-2002. 'With uplift' values are corrected for the land uplift and 'With geoid' also for the geoid rise derived from a model.
GPS
o CO
c_
13 •
Long time series
Short time series
Length (yr)
With Uplift
With ge0id
With uplift
With Ge0id
Helsinki
123
22
1.9
16.3
16.0
Hamina
74
14
1.2
15.5
15.3
Turku
80
26
2.3
16.5
16.1
O
I
I
I
I
1997
I
I
1999
I
I
2001
2003
Year
Rauma
69
33
2.9
17.0
16.6
Vaasa
119
16
1.1
16.7
16.2
Oulu
112
05
0.1
14.5
14.1
Vaasa tide gauge 1996 - 2002 LO
To obtain the absolute sea level change rates for the Baltic Sea, we calculated A S using Eq. (7). In Figure 3 one can see examples of the GPS and tide gauge time series used in the calculation. The GPS time series are relative to the Mets~hovi GPS station. Land uplift rates agree well with the previous results, e.g. Mfikinen et al., 2003.
--
E O ¢(.O3
il v
o
,.,?-
"1o LO
"T
i
I
1996
1998
I
I
I
2000
I
I
2002
Year
Vaasa
tide gauge
1883
- 2001
o o LO OO
E E
o
o_
¢(.(D
° o
-
-1o o
I
Lo
1880
I
I
1920
I
1960
I
I
2000
Year
Fig. 3. Time series for Vaasa GPS station and for Vaasa tide gauge. Note, that the scale of the GPS time series is 10 cm and the scales of the tide gauge time series are 3 m. Spikes in GPS time series are due to the snow on the antenna radome in winter time.
The tide gauge time series are given for two different time periods, the shorter being years 19962002 and the longer one the longest time series existing for each tide gauge. In Vaasa (63 ° 06' N; 21 ° 34' E, Fig. 3) the tide gauge observations have started in year 1883. The trends for the six tide gauges corrected with land uplift from GPS can be seen in Table 1 (Tervo, 2004). To compute the absolute rates in sea level change, one should use as long tide gauge time series as possible. The average of the long time series gives for the sea level rise 1.9 _ 1.0 m m / y e a r (varying b e t w e e n 0.5 - 3.3 mm/year), and when corrected for the geoid rise, 1.6 _ 1.0 m m / y e a r (varying between 0.1 - 2.9 mm/year). Globally the sea level rise is observed to be 1 - 2 m m / y e a r (Church and Gregory, 2001, Church et al., 2004). There are also regional differences in the sea level rise rates, as Church et al. (2004) present in their study. Our results for the Baltic Sea agree with the previous studies but the scatter is too large to make any further conclusions. One can see the change in observed rates in the short time series. It means that the sea level rise in the Baltic Sea is now different than what it has been before the year 1990. Possible explanations are discussed in (Johansson et al., 2003), but the reason for the change is not yet satisfactory explained.
77
78
M. Tervo • M. Poutanen • H. Koivula
3 Stability and accuracy of the GPS solution
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,
i
20
,
,
i
25
,
,
30
i
,
,
35
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METS - MASA (QIF) 0.015
-g o
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.
.
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.
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. 25
.
.
. . 30
.
. 35
40
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Figure 4. Results of the GPS test. Two top rows: Vectors METS-MASB computed with two softwares. Solid line is the actual change in height between METS and MASB. Third figure: Observed height change using only the 3 m vector MASA-MASB. Bottom: Observed drift METS-MASA. There has been no actual change in height between these stations. There is more than 5 mm change during the period of one month.
To simulate the accuracy of a footprint GPS installation to detect minor vertical m o v e m e n t s , we made a test where two identical antennas were 3 m from each other, and a reference station was about 10 k m away. In a footprint technique, the stability of a marker is controlled with a network of reference stations 10 - 15 k m from the site. In our test we m o v e d one test antenna in vertical direction, and the second one was u n m o v e d during the test. The GPS data were processed with two different softwares (Pinnacle and Bernese) to find out the difference in results b e t w e e n different softwares, and to confirm that observed features are not artefacts from the data processing. In both cases a standard troposphere and P C V models were used. W e made a total of 18 sessions, 24 h each, in J a n u a r y - February 2004. The reference station is called M E T S , the fixed test antenna is M A S A and the m o v i n g antenna is M A S B . In Fig. 4. we have three different solutions for M A S B , two c o m p u t e d from M E T S (about 10 km vector) with two different softwares, and one using the 3 m baseline M A S A - M A S B . In the figures, the "ground truth", i.e. the k n o w n vertical shift of M A S B , is shown with a solid line. Height of M A S B was changed 5 - 15 m m during the test. Sudden changes of a few m m in height are hardly visible in some cases in our GPS solution when the 10 k m vector M E T S M A S B is used. However, if we process the data of the 3 m vector, 0.2 m m changes m a y b e c o m e visible. It means that even with the footprint technique sudden sub-cm shifts m a y r e m a i n undetectable in episodic measurements. In continuous m e a s u r e m e n t s the change will b e c o m e visible in long time series, but even in that case, periodic changes m a y degrade the resolving p o w e r (Poutanen et al., 2005). Most notable thing is the observed drift in height M E T S - M A S A . Both antennas were fixed, and we have tried to exclude external reasons for the drift. M E T S is an IGS station, and data are used also for our p e r m a n e n t network analysis. The same pattern in M E T S - M A S A is visible in the results of two independent softwares, thus excluding the software based reason. There are several possible cause for the drift, including e n v i r o n m e n t a l effects (like snow) and crustal loading, as discussed in P o u t a n e n et al., 2005. This data set does not allow us to make any detailed conlusions but longer time series are needed to analyse the temporal variation in this vector.
Chapter 13 • Tide Gauge Monitoring Using GPS
The same drift is visible also in M E T S - M A S B vector. If we c o m p u t e separately M E T S - M A S A and M E T S - M A S B and from these the height difference M A S A - M A S B , the a g r e e m e n t with the ground truth b e c o m e s better than directly f r o m the vector M E T S - M A S B .
4 Conclusion T h e r e are m a n y applications of GPS in tide gauge monitoring. A m o n g the most important ones is the possibility to calculate rates of absolute sea level rise. C o m b i n i n g tide g a u g e s and GPS stations world wide w o u l d give a significant contribution to the sea level monitoring, giving it a well-defined reference frame. W e calculated absolute sea level rates for the Baltic Sea using GPS and tide gauge time series. The rate was found to be b e t w e e n 0.1 - 2.9 m m / y e a r , a v e r a g e being 1.6 m m / y e a r . The results agree with the global rate, though the scatter and uncertainty of the trend are large. R e c e n t rates differ significantly, but these can be a d d r e s s e d to local t e m p o r a l variation and they do not represent the l o n g - t e r m trend. This stresses the i m p o r t a n c e of uninterrupted l o n g - t e r m tide g a u g e time series. W e have s h o w n that GPS m a y be used for controlling the stability of the tide with a s u b - m m accuracy w h e n the baseline is very short. D i s a d v a n t a g e of the short baseline is that the reference antenna m a y m o v e together with the tide g a u g e b e n c h m a r k a n t e n n a if there are local deformations. This is a v o i d e d by locating the reference antenna further a w a y from the tide gauge. In this case the G P S - r e l a t e d errors, which can be up to several m m over a b a s e l i n e of 10 k m limit the accuracy obtained. One should find a suitable c o m b i n a t i o n b e t w e e n the desired accuracy and the distance b e t w e e n the reference antenna and tide gauge.
Acknowledgements This study was partially s u p p o r t e d by Finnish F u n d i n g A g e n c y for T e c h n o l o g y and I n n o v a t i o n T E K E S (decision n u m b e r 40414/04).
References Church, J.A. and J.M. Gregory (Co-ordinating lead authors), 2001, Changes in sea level, In: Climate Change 2001, The Scientific Basis, Contribution of working group I to the third assessment report of the IPCC, Cambridge University Press, p. 639 - 694 Church, J. A., N. J. White, R. Coleman, K. Lambeck, and J. X. Mitrovica, 2004: Estimates of the regional distribution of sea-level rise over the 1950 to 2000 period. Journal of Climate, 17(13), 2609-2625 Ekman M. and J. M~ikinen, 1996, Mean sea suface topography in the Baltic Sea and its transition area to the North Sea: A geodetic solution and comparisons with oceanographic models, Journal of Geophysical Research, 101:11993 - 11999 Holgate S.J. and P.L. Woodworth, 2004, Evidence for enhanced coastal sea level rise during the 1990s, Geophysical Research Letters, 31, L07305 Johansson, J.M., J.L. Davis, H.-G. Scherneck, G.A. Milne, M. Vermeer, J.X. Mitrovica, R.A. Bennett, B. Jonsson, G. Elgered, P. Elosegui, H. Koivula, M. Poutanen, B.O. R6nn~ing and I.I. Shapiro, 2002, Continuous GPS measurements of postglacial adjustment in Fennoscandia, 1. Geodetic results, Journal of Geophysical Research, 107:2157 Johansson, M., K. Kahma and H. Boman, 2003, An improved estimate for the long-term mean sea level on the Finnish coast, Geophysica, 39:51-73 Mao, A., C.G.A. Harrison and T.H. Dixon, 1999, Noise in GPS coordinate time series, Journal of Geophysical Research, 104; 2797 - 2816 M~ikinen, J., H. Koivula, M.Poutanen and V. Saaranen, 2003, Vertical velocities in Finland from permanent GPS networks and repeated precise levelling, Journal of Geodynamics, 38:443-456 Poutanen, M., J. Jokela, M. Ollikainen, H. Koivula, M. Bilker, H. Virtanen, 2005, Scale variation in GPS time series. In: A Window on the Future Geodesy (Ed. F. Sans6). lAG Symposia vol. 128, Springer Verlag, Berlin, pp. 15-20. Poutanen, M., H. Koivula, M. Tervo, K. Kahma, M. Ollikainen and H. Virtanen, 2004, GPS time series and sea level, In: Celebrating a decade of the international GPS s e r v i c e - Proceedings, Meindl, M. (ed.) Astronomical Institute, University of Berne, CD Tervo, M., 2004, Benefits of combining tide gauges and GPS stations, Master's thesis, University of Helsinki, 45 p. White, N.J., J.A. Church and J.M. Gregory, 2005, Coastal and global averaged sea level rise for 1950 to 2000, Geophysical Research Letters, 32:L01601 Williams, S.D.P., Y. Bock, P. Fang P. Jamason, R. M. Nikolaidis, L. Prawirodirdjo, M. Miller and D. J. Johnson 2004, Error analysis of continuous GPS position time series, Journal of Geophysical Research, 109:B03412.
79
Chapter 14
Determination of Inland Lake Level and Variations in China from Satellite Altimetry*
Its
Yonghai Chu, J. Li, Weiping Jiang, Xiancai Zou, Xinyu Xu, Chunbo Fan School of Geodesy and Geomatics, Wuhan University, 129 Luoyu Road, Wuhan, 430079, China Email: yh_chu@ 163.com
Abstract. With the rapid development and the widely application of satellite altimetry, it provides an efficient tool and a well-validated technique for real-time and continual monitoring of the inland lake level. However, only few lakes in China are covered with historical altimeters' GDRs. In this study, we investigate the level fluctuations of Poyang and Dongting (Yangtze River area), Qinghai (west China) and Hulun (northeast china) from altimetry and analyze the connections between level changes and the local climate.
Keywords: Satellite Altimetry, Inland Lake, Lake level Variation in China
basin waters from T/P satellite altimetry. Mercier (2002) studied the connections between 12 lakes of Africa and the climate of Indian Ocean using T/P data (1993-1999). Jekeli & Dumrongchai (2003) monitored the vertical datum with satellite altimetry and water-level gauge data on large lakes. Zhang (2002) has studied the Great Lakes level changes with the T/P data (1992-2002). Hwang (2005) used T/P to investigate and analyze the links between lake level variations and ENSO over six lakes in China. In this paper, four lakes level changes are presented using 2 yr. of Envisatl GDRs and 13 yr. of time series of Hulun are shown based on T/P and Jason 1 data.
2 The Chinese Inland Lakes 1 Introduction Altimeters were primarily designed to study the ocean, their ability to investigate marine geoid, gravity anomaly; mean sea surface height (Jiang, 2001) has already been applied since 1970's.This technology is limited inland for the inherent quality of altimeter. However, it has been demonstrated that altimetry has an advantage over other tools in the research of inland waters. Birkett (1994, 1995) evaluated the potentialities of this technology over continental waters and monitored several climatically sensitive lakes level with Geosat and analysed the total error budget of T/P for lake height and obtained 24 lakes level changes including Hulun Lake of China with T/P altimeter data. Morris & Gill (1994) evaluated the T/P altimeter over the Great Lakes based on the previous 40 cycle's data. Ponchaut & Cazenave (1998) investigated the relationships between level fluctuations and climatic environment about four lakes of Africa and three lakes of North America. Birkett (1999) studied the influence of the Indian Ocean climate on the lakes of east Africa. Oliveira (2001) presented the temporal variations of Amazon
There are many lakes in China, but the distributions are very wide and uneven. Moreover, the construction is also varying. Poyang, Dongting, Tai, Hongze and Chao are the five largest fresh lakes which are located at southeast of China, a subsiding region of crust, and regulate the water quantity of Yangtze River. Contrarily, western china, for instance Tibet, is an intense upthrust region. The altitude of lakes there is very high and the type of the lakes is usually closed. The lake supplies depend on the dissolution of ice or snow and rainfall precipitation, such as Qinghai and Bositeng Lake. We selected four lakes as mentioned earlier to study (Fig.l, Tab. 1). T/P and Jasonl data (13 years) are used in Hulun Lake, and Envisat 1 data (2 years) are used in others.
3 Data Analysis We did not employ the same methods of data processing as used in open oceans when using altimetric data to monitor the level changes of inland lakes for satellite altimetry was developed and optimized for open sea.
* This study is funded by the Natural Science Foundation Committee of China under grants 40274004 and 40304001
Chapter
14 • Determination of
Inland Lake Level and Its Variations in China from Satellite Altimetry
Table. 1 Features of 4 studied lakes in China (Hulun: T/P and Jasonl a=6378136.3m, other: Envisatl a=6378137m) Borders Latitude Longitude 28.75-29.75 115.75-116.75 28.50-29.50 112.00-113.00 36.50-37.50 99.50-101.00 48.50-49.50 116.90-117.90
Lake Poyang Dongting Qinghai Hulun
Province
Mean Surface areanr~'2~t,,~..J ellipsoid height (m)
Points Pass
Type
Jiangxi
3960
8.18
144
163,621,980
Open
Hunan
2740
10.76
44
694,879
Open
Qinghai
4635
3149.30
254
094,479,552
Open
Inner Mongolia
2315
533.94
1816
27,36(TP-Jasonl)
Open
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'
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To retain the maximum data, the following basic requirements must be considered: 1) data in lake; 2) altitude above reference ellipsoid is valid; 3) valid range. If data quantities are very rich, 4) number of 1Hz ranges points; 5) altimeter echo type or radiometer surface type; 6) correction items valid. This paper only considers items 1)-3).Certainly, abnormal values may exist in the data, artificial elimination is also a very important step to get the final LLH with respect to the reference ellipsoid: LLH=
23.0" ~
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43.0
,
,
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48.5 113.0"
11 0" 11k5" '
Fig 1 Geographical locations of 4 studied lakes and data distribution (Hulun: T/P and Jasonl, other: Envisat1)
3.1 Distance Corrections
The GDRs (AVISO, ENVISAT) provide many corrections in open sea. Generally, some corrections are not suitable for studying in inland waters, such as troposphere and ionosphere corrections are usually unavailable in the GDRs. Inverted barometer correction is often ignored for the small lake area. The instrumental bias has been applied in 1Hz range data. In this paper, ionosphere, troposphere (dry and wet), solid earth tide, ocean tide, pole tide, sea state biases are used. Among these corrections, the dry troposphere is from model, but the wet troposphere is from radiometer correction. The wet model is used if it is available when the radiometer correction is invalid. The dualfrequency altimeter correction is replaced by DORIS. But it is also used if it is available when the DORIS is invalid.
Ral t
-
-
R - AR
(1)
Where L L H is lake level height, Ra~, is altimeter altitude with respect to the reference ellipsoid, R is the range from altimeter to lake's surface and AR is the range corrections mentioned above. The instantaneous LLH is split into geoid height above the reference ellipsoid and lake surface height above the geoid (Ponchaut, 1998). We average all LLH represent the mean lake level (MLLH). To obtain the monthly average LLH, we also average the LLH month to month and mark symbol as L L H M . Finally, the monthly level variations d L L H are expressed: dLLH
= LLH M - MLLH
(2)
With month number as x-axis and d L L H as y-axis, the graphs of variations plot are shown in Fig2-3.
4 Analysis of Lake Level Variation and Climate 4.1 Poyang Lake Poyang Lake is the largest fresh-water lake that can regulate or control Yangtze River. Its level is affected by five rivers and Yangtze River. The major flood period of five rivers is from Apr to Jun, so the level is influenced by the water discharge of these five rivers, nevertheless, the level is still lower. Although the water quantities of the five rivers
81
82
Y. Chu. J. Li. W. Jiang • X. Zou. X. Xu. C. Fan
Poyang, Dongting and qinghai lake
> -2.5 Time (month) &
Poyang
--
Dongting - - O -
Qinghai ]
Fig. 2 Monthly Mean Level Variation of Poyang, Dongting and Qinghai Lake (2002.11-2005.2)
Hulun Lake
• _°o: v--4
Month Fig. 3 Monthly Mean Level Variation of Hulun Lake (1992.10-2004.11), blue triangle: Jasonl data
decrease from Jul to Sep which is the major flood period of Yangtze River, the level is still steadily rising and keeps high and comes to a head in Jul for the reverse flow of the water. Fig.2 (triangle symbol) shows the seasonal changes of the level.
4.2 Dongting Lake Dongting Lake is located at the middle reaches of Yangtze River. There are four rivers that flow into Dongting from the south and four outfalls which link to Yangtze River from the north. The lake water flows into Yangtze River at Yueyang. Dongting seems to be a natural big reservoir which contains water from the four rivers and gorges or disgorges Yangtze River and regulates flood. So the level feature is affected not only by rainfall, surface runoff and lake conservation, but also by the situations of Yangtze River. The advent of flood season of the four rivers starts from Apr and cause the level to rise (Fig.2 square symbol)). During the raininess season in Yangtze River catchment's area from Jun to Sep, The level of Dongting rises continuously. Generally speaking, the highest water level appears in Jul. When the water intake is less than the water output after Sep, the level declines slowly and arrives at the bottom because it is lowwater season from Sep of previous year to Mar of the ensuing year.
4.3 Qinghai Lake Qinghai Lake, the largest inland lake and the largest saltwater lake in China, belongs to a high and cold, semiarid climate. Although there are more than 40 rivers that flow into Qinghai Lake, the water of its is supplied by surface runoff and precipitations. From Fig.2 (circular symbol), we can say that the level keeps balance. Even so, the level displays a seasonal feature during 2002-2004. In spite of its coldness with its surface freezing in winter, the level rises rapidly from Nov and keeps high level to next Apr for rainfall increasing and evaporation decreasing in spring. However, from Apr to Nov, the amplitude of variation is about 0.5m, less than 0.9m in winter. On the whole, the amount of water in Qinghai keeps balance during 2002-2004. We investigated the environment around Qinghai and found that the policy of reusing farmland for planting grass since 2001 contributes to the rising level due to the conservation of soil and water.
4.4 Hulun Lake Hulun Lake, the fifth largest one in China, located at Inner Mongolia region where the continental climate is very obvious for Great Xingan Mountains
Chapter 14 • Determination of Inland Lake Level and Its Variations in China from Satellite Altimetry
shielding off moist air from ocean in east and the Mongolia Plateau stands at the west. Fig 3 is a 13year time series of level changes from 1992 to 2004 based on T/P and Jason l. From 1992 to 1999, the interannual fluctuations almost kept balance, but it decreased linearly from 2000 to 2004. Flake ice occurs in the last ten-day period of Oct and the total lake surface freezes up at the beginning of Nov, so the level is low in Jan to Mar, and high levels appear from Jul to Sep. In recent years, no regulation and control measures are put into practice, the environment and water supplies around Hulun and its catchment's basin are being destroyed by human activities and natural factors. All these aggravate the decline of level from 2000 to 2004 about 2.5m.
5 Conclusions Water level fluctuations of inland lakes are related to regional to global scale climatic changes. Water level fluctuations reflect variations in evaporation and precipitation over lake area and its catchments area. To study the correlations between level variation and climate, the aridity index, water balance equation or the fundamental equation of lake water balance can be used. The meteorological parameters in these methods include lake area, precipitation, evaporation rates, river runoff, discharge rates, groundwater inflow or outflow. If satellite altimetry data are combined with local hydrologic data, the comparison of lake level derived from altimetry and from the in-situ lake gauge records could be a valid verification method (Hwang, 2005), but unfortunately, no available gauge data can be obtained, the comparison can not be made in the paper. Even so, the results from this study also validate the widely use of satellite altimetry in inland lakes or rivers. If possible, Geosat, ERS1/2, T/P, Jasonl and Envisatl data could be used together to get multi-altimetry and long time series numerical results. Waveform retracking technology also could help improve the accuracy of altimeter range over inland lakes (Chu, 2004, 2005).
References AVISO and PODAAC User Handbook (2003). IGDR and GDR Jason Products SMM-MU-M5-OP-13184-CN (AVISO), Edition 2.0.
AVISO/Altimetry (1996). AVISO user hand book for merged Topex/Poseidon products. AVI-NT-02-101, Edition 3.0. Birkett, C.M., 1 9 9 4 , Radar altimetry: a new conception monitoring lake level changes, EOS, Trans., AGU 75 ( 2 4 ) , 273-275 Birkett, C.M (1995). The Contribution of Topex/Poseidon to the Global Monitoring of Climatically Sensitive lakes, J. Geophys.Res. 100 (C12), 25179- 25204. Birkett, C.M (1998), Contribution of TOPEX NASA radar altimeter to global monitoring of large rivers and wetlands. Water Resour. Res, 34(5), 1223-1239 Birkett, C.M., Murtugudde, R., Allan, T(1999), Indian Ocean climate event brings floods to east Africa's lake and the Sudd Marsh. Geophys. Res. Lett. 26, 1031-1034 Cheinway Hwang, Ming-fong Peng, Jingsheng Ning, Jia Luo, Chung-Hsiung Sui (2005), Lake Level Variation in China from TOPEX/POSEIDON Altimetry: Data Quality Assessment and Links to Precipitation and ENSO, Geophys.J. int, 161, p 1- 11. Chu Yonghai, Li Jiancheng, Zhang Yan, Xu Xinyu, Fan Chunbo, Zou Xiancai (2005), Analysis and Investigation of Waveform Retracking about ENVISAT[J],Journal of Geodesy and Geodynamics, Vol. 25(1). Chu Yonghai, Li Jiancheng, Jiang Weiping Zhang Yan (2005). Monitoring the Hulun Lake Level and Its Fluctuation with Jason-1 Altimetric Data, Journal of Geodesy and Geodynamics, Vol. 25(4), p 11-16. Chu Yonghai(2004). The Theory and Technology of Waveform about Satellite Altimetry, Wuhan University, master dissertation. ENVISAT RA2/MWR Product Handbook (2002), European Space Agency, issue 1.0 Jekeli C. Dumrongchai (2003). On Monitoring a Vertical Datum with Satellite Altimetry and Water-level Gauge Data on Large Lakes, Journal of Geodesy, vol.77, p447453 Mercier, F., Cazenave, A., Maheu, C., (2002). Interannual Lake Level Fluctuations (1993-1999) in Africa from Topex/Poseidon: Connections with Ocean-atmosphere Interactions over the Indian Ocean, Global and Planetary Change, and vol.32:141-163 Morris, CS., Gill, SK. (1994). Evaluation of the TOPEX/POSEIDON Altimeter System over the Great Lakes. J Geophys Res 99(C12):24527-24539 Oliveria Campos, Mercier, F., Maheu, C., Cochnneau, G., Kosuth, P., Blitzkow, D., Cazenave, A., (2001). Temporal Variations of River Basin Waters from Topex-Poseidon Satellite Altimetry. Application to the Amazon basin [J], Earth and Planetary Sciences, v333:633-643 Ponchaut F., Cazenave A (1998). Continental Lake Level Variations from Topex/Poseidon (1993-1996), Comptes Rendus De IAcademie des Sciences Series IIA Earth and Planetary Science, vol.326, 13-20 Weiping Jiang (2001). The Application of Satellite Altimetry in Geodesy, Wuhan University, doctoral dissertation Zhang Ke,(2002). Monitoring Continental Lake Level Variations by using Satellite Altimeter Data, Wuhan University, master dissertation.
83
Chapter 15
Sea Surface Variability of Upwelling Area Northwest of Luzon, Philippines M.C.Martin, C.L. Villanoy Marine Science Institute University of the Philippines, 1101 Diliman, Quezon City, Philippines
Abstract. The upwelling events during winter at the northwest tip of Luzon are among the sea surface features occurring in the South China Sea. The variability of sea surface in the upwelling area was determined using 9-year satellite altimeter data from TOPEX/ Poseidon and ERS 1 & 2. Using Empirical Orthogonal Function (EOF) decomposition the variability of the area was resolved. Two modes were derived representing the sea level anomaly variations of the upwelling area that peak during the monsoon periods. The sea level anomaly associated with the upwelling event coincided with the positive wind stress curl, which has been suggested to be due to the topographic steering of the wind that can be initiated by the northeast wind at the tip of Luzon. ENSO events also appear to modulate the timing of the development and decay of upwelling in the northwest of Luzon. E1 Nifio events initiate early development while La Nifia episodes delay the occurrence of upwelling events in the area. The intensity or strength of E1 Nifio and La Nifia episodes seems to influence the extent/magnitude of the upwelling area. Keywords. Sea level anomaly (SLA), upwelling, sea surface variability, ENSO, South China Sea
1 Introduction Upwelling, the movement of nutrient-rich deep water to the surface, is considered one of the most important processes in the coastal ocean (Tomczak, 1996) because of the role it plays in biological productivity and, consequently, to fisheries. The upwelling area northwest of Luzon, as part of the South China Sea (SCS) (Figure 1), has likewise been subjected to the seasonal variability occurring in the region. The SCS as part of the East monsoon system (Wyrtki, 1961) is dominated by monsoon winds. The northeast monsoon winds prevail from November to March and southwest monsoon winds blow from May to September (Pickard and Emery, 1982).
t on v
l, 100
105
110
115
120
125
Longitude (°E)
Fig 1. Map of the Philippines. A box highlights the upwelling area northwest of Luzon. The occurrence of upwelling area off Luzon in winter was first documented by Shaw et al (1996). Evidence of this feature was based on in-situ data of temperature, salinity and dissolved oxygen concentration they had gathered during a cruise in 1990. Udarbe-Walker and Villanoy (2001) analyzed the three-dimensional thermal structure and described the extent and timing of upwelling using historical temperature data. In this study, the spatial and temporal variations of upwelling in the area northwest of Luzon were determined using the 9year data set from merged T/P and ERS 1-2 satellite altimeter data. The likely evolution of upwelling as modulated by interannual variations are also presented.
2 Methods The sea level anomaly fields between 11 ° to 22 ° N and 112 ° to 122°E (Figure 1) were extracted from the global sea surface topography maps of the merged TOPEX/POSEIDON (T/P) and ERS-1/2 altimetry data. These 0.25 ° x 0.25 ° gridded SLA maps produced by the CLS Space Oceanography
Chapter 15
Division as part of the Environment and Climate EU AGORA (ENV4-CT9560113) and DUACS (ENV44-T96-0357), were obtained from the ftp://ftp.cls.fr/pub/oceano/AVISO/MSLA site. The 10-day sea level anomaly (SLA) data from October 1992 to September 2001 were grouped on a yearly basis. Time series data with missing values were linearly interpolated and each SLA was Ztransformed to ensure equal weights for each data point. The nine-year sea level anomaly data were then subjected to Empirical Orthogonal Function (EOF) analysis (Emery and Thomson, 1997) to describe the temporal and spatial variability of the upwelling area northwest of Luzon. Further, QUICKSCAT wind data (1999-2002) were examined to determine the possible influence of winds stress curl on upwelling.
•
Sea Surface Variability of Upwelling Area Northwest of Luzon, Philippines MODE A
MODE B
,~.~.:..z~;~. - : " " ' ; , . , . : - - _ ~ . . . ~'"::'//1 . .j . . i "--- ~ -
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Decomposition of sea surface height variability in the upwelling area northwest of Luzon derived two modes that together accounted for 33% of the total variance. Mode A, which explained 2 1 % variance, represents seasonal/annual variation. The spatial and temporal distributions for this mode are shown in Figures 2 and 3. The spatial distribution shows an elongated band of low sea level in the northeastsouthwest orientation located northwest of Luzon that encloses three eddy-like structures within it (Figure 2). The two eddy-like features situated west of Luzon (between 15°-18.5 ° N, 116.5°-119°E) and at the centre of SCS (14.5 °- 16.5°N, 113 °- 115.5°E) are consistent with the two of the three dynamically active areas described in Ho et al (2000a). This likewise conforms to the cyclonic circulation measured using drifters during the experiment conducted by National Taiwan University in winter of 1993-1994 (Ho et al., 2000b). As positive amplitudes indicate low sea level, the positive peaks that were generally observed during December of 1993-1996 and 1998-1999, thus, demonstrated that lowest sea level occurred during this month. The occurrences of highest sea levels, on the other hand, were distributed as the negative peaks in June (1994, 1996, and 2000), July (1993, 1995 and 1999) and May (1997, 1998 and 2001). Periods that corresponded to Mode A positive amplitudes included the months from November to February while the negative amplitude months were from May to September.
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ence of errors in these models (compare Velicogna and Wahr (2002)). If we assume that all other processes have been removed error-free, then the only uncertainty in this hypothetical case is the ice-load history. In the next section we will show that it is still possible to extract information on the properties of CLVZs from GOCE data in the presence of uncertainties in the ice-load history.
250
harmonic degree
Fig. 8. Differences in CLVZ induced perturbations in gravity anomaly degree amplitudes due to different properties of the CLVZ and different ice-load histories, compared with the performance of GOCE and GGM02S have a specific form, which is related to the different behavior for a thicker CLVZ and a higher viscosity CLVZ. Further on we will deduce from these curves spectral signatures for different properties of the CLVZ. From Figure 8 we conclude that in principle GOCE data will add constraints to estimates of the ice-load history in the presence of a CLVZ (with known parameters) and about properties of a CLVZ (if the iceload history is known). In practice, things are more complicated, because the measured gravity signal consists of large number of contributions, see Section 1. If we consider the gravity field as given by GGM02S, in the spectral range where we expect the largest amplitude (from degree 40 to 90, compare Figure 9), we see no direct relation with the modeled gravity anomalies induced by a CLVZ (Figure 5). Short-scale features are visible, but the amplitudes are much larger than predicted by our PGR model. This means that we have to remove other geophysical signals from the measured gravity field. If models for these geophysical signals are available, the question is if the information on CLVZs is still recoverable in the pres-
4.3
Spectral
Signatures
If we regard the differences between the ANU and I3G ice-load histories as realistic uncertainties in the ice-load history, then it is already clear from Figure 9 that it will be difficult to extract information from GOCE data on the properties of the CLVZ. This can be illustrated more clearly using the degree correlation coefficient, defined as (Mitrovica and Peltier, 1989):
Pn --
~-~nm=° Cnm-Dnm
(12)
with C ~ , D ~ different sets of spherical harmonic coefficients. In Figure 10 we have plotted the degree correlation coefficient between gravity anomaly values computed with our reference model (CLVZ from Table 1 and ice-model ANU) and different test model values. If our test model is the same as our standard model, then the degree correlation coefficient will be equal to one (not plotted). If we use in our test model the I3G iceload history, then the correlation between the reference and test model is very poor (p(Z~gANU~ z~gI3G)). If we know the ice-load history, then we can clearly distinguish between a model with a thicker CLVZ
Chapter 18 • High-Harmonic Gravity Signatures Related to Post-Glacial Rebound
(p(Agtt2, Agt2o), from degree 70) or a higher viscosity CLVZ (p(Agv,8, Agv~9), from degree 100). If we do not know the ice-load history, and correlate for example our reference model with a test model with different parameters of the CLVZ and the I3G iceload history, then the degree correlation coefficient will be poor due to the bad correlation for different ice-load histories (not plotted), and we cannot constrain the properties of the CVLZ. This means that we cannot extract information on the properties of the CLVZ in the presence of uncertainties in the ice-load history. A large part of this uncertainty can be removed by normalizing the degree amplitudes with the degree amplitudes of the ice-load history at LGM. This is because the computed freeair gravity anomaly perturbations are a convolution of the temporal and spatial impulse response of the earth (i.e. the time-dependent Love numbers for a certain CLVZ) and the time- and space-dependent input sequence (i.e. the ice-load history). If we consider the response at a certain time interval and assume the ice-load is constant (which is obviously not the case), then the spatial spectrum (i.e. the degree amplitudes) of the impulse response is equal to the ratio of the spectrum of the output (i.e. the gravity anomalies) and the ice-load. The assumption of constant ice load is approximated by using the ice load at LGM, which is justified by the long period of glaciation compared to deglaciation. In Schotman and Vermeersen (2005) we have shown that the computed normalized degree amplitudes closely resemble the time-dependent Love numbers for a certain CLVZ as a function of harmonic degree. In practice, we do not know the real ice-load history, but we can estimate which ice-load history best fits the gravity anomaly perturbations by correlating the spatial spectrum of the gravity anomaly perturbations with the spectrum of the ice-load history (at LGM). We see from Figure 11 that the correlation is always significantly better for the ice model that generated the anomaly perturbations. Moreover we see that the ANU ice model correlates up to higher degree with the corresponding gravity anomalies than the I3G ice model, mainly because the I3G model has less power in the high harmonics. In Figure 12 we show perturbation degree amplitudes computed with I3G and ANU, normalized by the dimensionless degree amplitudes of the I3G and ANU ice height at LGM, respectively. We can see that the curves are very close, except above degree 70, where the correlation of the perturbations computed from I3G and the ice heights of I3G drop very fast, see Figure 11. If we consider the degree correlations from Figure 11 as a measure for the quality of the normalized degree amplitudes, we can compute a best estimate from the two normalized degree amplitude curves in a weighted least squares (WLSQ) sense, with weights determined by the degree correlations. If we follow the same procedure for different prop-
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and Outlook
We have shown that gravity anomaly perturbations induced by crustal low-viscosity zones (CLVZs) are above the expected GOCE performance up to harmonic degree 140 and above the realized GRACE
109
110
H.H.A. Schotman • P. N. A. M. Visser • L. L. A. Vermeersen
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Acknowledgements
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Fig. 13. Spectral signature for different properties of the CLVZ performance (GGM02S) up to degree 90. For an asthenospheric LVZ (ALVZ) the gravity anomaly perturbations are above the expected G O C E performance and G G M 0 2 S up to degree 70 and 60, respectively. G O C E is thus especially useful for detecting CLVZs. It was found that G O C E is also sensitive to changes in theological properties of a CLVZ, which means that in principle G O C E should be able to constrain the theology. G O C E is however also sensitive to uncertainties in the ice-load history, though a large part of this uncertainty can be removed by manipulating the data in the spectral domain to obtain spectral signatures for different CLVZs. Note that this study is valid for the limiting case of a laterally h o m o g e n e o u s earth, which is clearly not realistic everywhere with regard to the presence and properties of CLVZs. In practice, it will be difficult to extract information from satellite gravity data. F r o m a filtered version of G G M 0 2 S we have seen that a large n u m b e r of geophysical signals is present in the gravity field. We therefore need to remove as well as possible all geophysical signals in the frequency range that we are interested in, and use some form of spatio-spectral filtering to isolate the relevant signal (see e.g. Simons and Hager (1997)). In future studies, we will show if
We thank Jerry Mitrovica, an anonymous reviewer and Mark Drinkwater for their constructive comments, Kurt Lambeck and co-workers (ANU, Canberra) for their global ice sheet model and Radboud Koop (SRON, Utrecht) for discussions.
References
Di Donato, G., J.X. Mitrovica, R. Sabadini, and L.L.A. Vermeersen (2000). The influence of a ductile crustal zone on glacial isostatic adjustment; geodetic observables along the U.S. East Coast, Geophys. Res. Lett., 27, pp. 3017-3020. Dziewonski, A.M. and D.L. Anderson (1981). Preliminary Reference Earth Model, Phys. Earth Planet. Inter, 25, pp. 297-356. Farrell, W.E. (1972). Deformation of the earth by surface loads, Rev. Geophys. Space Phys., 10, pp. 761-797. Hart, S.-C., C.K. Shum, E Ditmar, E Visser, C. van Beelen and E.J.O. Schrama (2006) Aliasing effect of high frequency mass variations on GOCE recovery of the earth's gravity field, J. Geodyn., 41, pp. 69-76. Heiskanen, W. and H. Moritz (1967). Physical Geodesy, W.H. Freeman and Co., San Francisco, 364 pp. Johnston, F'. and K. Lambeck (1999). Postglacial rebound and sea level contributions to changes in the geoid and the earth's rotation axis, Geophys. J. Int., 136, pp. 537558. Kendall, R., J.X. Mitrovica and R. Sabadini (2003). Lithospheric thickness inferred from Australian post-glacial sea-level change: The influence of a ductile crustal zone, Geophys. Res. Lett., 30, pp. 1461-1464. Klemann, V. and D. Wolf (1999). Implications of a ductile crustal layer for the deformation caused by the Fennoscandian ice sheet, Geophys. J. Int., 139, pp. 216226. Lambeck, K., C. Smither and P. Johnston (1998). Sea-level change, glacial rebound and mantle viscosity of northern Europe, Geophys. J. Int., 134, pp. 102-144. Lambeck, K., A. Purcell, P. Johnston, M. Nakada, Y. Yokoyama (2003). Water-load definition in the glaciohydro-isostatic sea-level equation, Quat. Sci. Rev., 22, p. 309-318. Longman, I.M. (1963). A Green's function for determining the deformation of the earth under surface mass loads, 2. Computations and numerical results, J. Geophys. Res., 68, pp. 485-496. Mikhailov, V., S. Tikhotsky, M. Diament, I. Panet, V. Ballu (2004). Can tectonic processes be recovered from new gravity satellite data?, Earth Planet. Sci. Lett., 228, 10.1016/j.epsl. 2004.09.035. Mitrovica, J.X. and G.A. Milne (2003). On post-glacial sea level: I. General theory, Geophys. J. Int., 154, pp. 253267.
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Chapter 18 • High-Harmonic Gravity Signatures Related to Post-Glacial Rebound
Mitrovica, J.X. and W.R. Peltier (1989). Pleistocene deglaciation and the global gravity field, J. Geophys. Res., 94, pp. 13,651-13,671. Peltier, W.R. (1974). The impulse response of a Maxwell earth, Rev. Geophys. Space Phys., 12, pp. 649-669. Pollitz, F.F. (2003). Transient rheology of the uppermost mantle beneath the Mojave Desert, California, Earth Planet. Sci. Lett., 215, pp. 89-104. Ranalli, G. and D. Murphy (1987). Rheological stratification of the lithosphere, Tectonophys., 132, pp. 281-295. Schotman, H.H.A. and L.L.A. Vermeersen (2005). Sensitivity of glacial isostatic adjustment models with shallow low-viscosity earth layers to the ice-load history in relation to the performance of GOCE and GRACE, Earth Planet. Sci. Lett., 236, 10.1016/j.epsl.2005.04.008. Siegert, M.J. and J.A. Dowdeswell (2004). Numerical reconstructions of the Eurasian Ice Sheet and climate during the Late Weichselian, Quat. Sci. Rev., 23, pp. 12731283. Simons, M. and B.H. Hager (1997). Localization of the gravity field and the signature of glacial rebound, Nature, 390, pp. 500-504. Stein, S. and M. Wysession (2003). Introduction to Seismology, Earthquakes, and Earth Structure, Blackwell Publishing, Oxford, 498 pp. Tapley, B., J. Ries, S. Bettadpur, D. Chambers, M. Cheng, F. Condi, B. Gunter, Z. Kang, R Nagel, R. Pastor, T. Pekker, S. Poole and F. Wang (2005). GGM02 - An improved earth gravity field model from GRACE, jr. Geodesy, 10.1007/s00190-005-0480-z. Tushingham, A.M. and W.R. Peltier (1991). ICE3G: A new global model of late Pleistocene deglaciation based upon geophysical predications of postglacial relative sea level change, J. Geophys. Res., 96, pp. 4497-4523.
van der Wal, W., H.H.A. Schotman and L.L.A. Vermeersen (2004). Geoid heights due to a crustal low viscosity zone in glacial isostatic adjustment modeling; a sensitivity analysis for GOCE, Geophys. Res. Lett., 31, 10.1029/2003 GL019139. Velicogna, I. and J. Wahr (2002). Postglacial rebound and earth's viscosity structure from GRACE, Jr. Geophys. Res., 107, 10.1029/2001JB001735. Vermeersen, L.L.A. (2003). The potential of GOCE in constraining the structure of the crust and lithosphere from post-glacial rebound, Space Sci. Rev., 108, pp. 105-113. Vermeersen, L.L.A. and R. Sabadini (1997). A new class of stratified visco-elastic models by analytical techniques, Geophys. J. Int., 139, pp. 530-571. Visser, RN.A.M., R. Rummel, G. Balmino, H. Stinkel, J. Johannessen, M. Aguirre, RL. Woodworth, C. Le Provost, C.C. Tscherning and R. Sabadini (2002). The European earth explorer mission GOCE: Impact for the geosciences, In: Ice Sheets, Sea Level and the Dynamic Earth, J.X. Mitrovica and L.L.A. Vermeersen (eds), AGU Geodynamics Series, 29, AGU, Washington DC, pp. 95-107. Wahr, J., M. Molenaar and F. Bryan (1998). Time variability of the earth's gravity field: Hydrological and oceanic effects and their possible detection using GRACE, J. Geophys. Res., 103, pp. 30,205-30,229. Watts, A.B. and E.B. Burov (2003). Lithospheric strength and its relationship to the elastic and seismogenic layer thickness, Earth Planet. Sci. Lett., 213, pp. 113-131. Wu, R and W.R. Peltier (1982). Viscous gravitational relaxation, Geophys. J. R. Astron. Soc., 70, pp. 435-485. Wu, R, H. Wang and H. Schotman (2005) Postglacial induced surface motions, sea-levels and geoid rates on a spherical, self-gravitating, laterally heterogeneous earth, J. Geodyn., 39, pp. 127-142.
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Part II Frontiers in the Analysis of Space Geodetic Measurements Chapter 19
GPS/GLONASS Orbit Determination Based on Combined Microwave and SLR Data Analysis
Chapter 20
BIFROST: Noise Properties of GPS Time Series
Chapter 21
Discrete Crossover Analysis
Chapter 22
A Comparative Analysis of Uncertainty Modelling in GPS Data Analysis
Chapter 23
Looking for Systematic Error in Scale from Terrestrial Reference Frames Derived from DORIS Data
Chapter 24
WVR Calibration Applied to European VLBI Observing Sessions
Chapter 25
Frontiers in the Combination of Space Geodetic Techniques
Chapter 26
Modifying the Stochastic Model to Mitigate GPS Systematic Errors in Relative Positioning
Chapter 27
GPS Ambiguity Resolution and Validation under Multipath Effects: Improvements Using Wavelets
Chapter 28
An Empirical Stochastic Model for Gps
Chapter 29
Feeding Neural Network Models with GPS Observations: a Challenging Task
Chapter 30
Spatial Spectral Inversion of the Changing Geometry of the Earth from SOPAC GPS Data
Chapter 31
Improved Processing Method of UEGN-2002 Gravity Network Measurements in Hungary
Chapter 32
Spectra of Rapid Oscillations of Earth Rotation Parameters Determined during the CONT02 Campaign
Chapter 33
On the Establishing Project of Chinese Surveying and Control Network for Earth-Orbit Satellite and Deep Space Detection
Chapter 34
Constructing a System to Monitor the Data Quality of GPS Receivers
Chapter 19
GPS/GLONASS orbit determination based on combined microwave and SLR data analysis C. Urschl, G. Beutler, W. Gurtner, U. Hugentobler, S. Schaer Astronomical Institute, University of Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland email:
[email protected] Abstract. The combination of space-geodetic techniques is considered as an important tool for improving the accuracy and consistency of the resulting geodetic products. For GNSS satellites, tracking data is regularly collected by both the microwave and the SLR observation technique. In this study, we investigate the impact of combined analysis of microwave and SLR observations on precise orbit determination of GNSS satellites. Combined orbits are generated for the two GPS satellites equipped with Laser retroreflector arrays and for three GLONASS satellites that are currently observed by the ILRS network. The combination is done at the observation level, implying that all parameters common to both techniques are derived from both observation types. Several experimental orbits are determined using different observation weights. As the well-known 5 cm-bias between SLR measurements and GPS microwave orbits is unexplained, SLR range biases as well as satellite retroreflector offsets are estimated in addition to the orbital parameters. The different orbit solutions are then compared in order to determine whether and to which extent the SLR measurements influence a microwave orbit primarily derived from microwave observations. Key words. GNSS orbit determination, Multitechnique combination, GPS, SLR
1 Introduction Different space-geodetic techniques contribute to the generation of the International Earth Rotation and Reference Systems Service (IERS) products. Therefore, the IERS shows an increasing interest in intertechnique combinations, in particular to improve the accuracy and consistency of the IERS products, the International Terrestrial Reference Frame (ITRF), the international Celestial Reference Frame (ICRF), and the Earth Orientation Parameters (EOP). Efforts are underway to develop methods for combining observations and products from individual space-geodetic techniques (Rothacher, 2002; Ray et al., 2004). Sys-
tematic biases between the individual technique solutions are topics of current investigations. Within the scope of an ongoing project "Combined analysis of the major satellite-geodetic observables", this paper presents the estimation of combined GNSS (Global Navigation Satellite System, consisting of GPS and GLONASS) satellite orbits using measurements of two space-geodetic techniques, microwave and SLR measurements. First experiments were carried out by Zhu et. al (1997), who determined two 1-day arc orbits using microwave and SLR tracking data for two days in 1995. Previous studies on validating microwave orbits with SLR observations have shown a constant bias of about 5.5 cm between the range measurements and the GPS microwave orbit (see, e.g., Appleby and Otsubo, 2000; Springer, 2000; Urschl et al., 2005). This bias is not yet understood. We wanted to know whether the bias disappears by including SLR measurements into the orbit determination process, or whether the orbit dynamics is strong enough not to absorb the bias. SLR range biases as well as satellite reflector offsets that might be responsible for the mean biases between SLR and microwave observation are estimated in addition to the orbital parameters. We analyze the resulting orbit to see whether the SLR observations help to improve the combined orbit. The combination strategy is outlined in Section 2. Section 3 describes the data set used. Section 4 introduces three combination experiments and the corresponding experimental orbit solutions. The analysis and comparison of the different combined orbit solutions is done in Section 5. Section 6 derives conclusions from the combination experiments and Section 7 summarizes the results.
2 Combination strategy The weighted least-squares method is used for the parameter estimation process. We apply a combination strategy at the observation level using microwave (double difference phase, ionosphere-free linear combination) and SLR range observations. Technically, the combination is done on the normal equation level by stacking the technique-specific nor-
116
C. Urschl • G. Beutler. W. Gurtner • U. Hugentobler. S. Schaer
mal equations. Parameters common to both observation types are derived from the observations of both techniques. These common parameters are orbit parameters, Earth orientation parameters (EOP), geocenter coordinates and coordinates of collocated sites constrained with local ties. As this study focuses on the estimation of common orbit parameters only, we constrain all other parameters to highly accurate apriori values. The geocenter is constrained to the origin of the ITRF2000. Earth orientation parameters are constrained to the weekly EOPs, derived from the CODE (Center for Orbit Determination in Europe) final orbit solution. For the datum definition, the coordinates of the Laser tracking sites and the sites with microwave receivers are constrained to their ITRF2000 estimates. We compute two types of combined orbits, namely 1-day and 3-day arcs, using a development version of the Bernese GPS Software V5.0 (Hugentobler et al., 2005). For 3-day arcs the orbit dynamics is a stronger constraint than for 1-day arcs, affecting the impact of the SLR observations on the resulting orbit. The CODE orbit model is adopted. The six orbital elements, nine dynamical orbit parameters (radiation pressure parameters) and one stochastic pulse at noon and at the day boundaries for the 3-day arcs are estimated.
3 Data set Combined orbits are computed for five GNSS satellites, the two GPS satellites equipped with Laser retroreflectors and three of the GLONASS satellites currently tracked by the ILRS (International Laser Ranging Service) network. Table 1 characterizes the satellites used. The microwave phase measurements from 157 GNSS sites as well as the Laser ranging measurements from 13 SLR sites are used to obtain a time series of 41 days in 2004 (DoY 305-345). Figure 1 shows the global distribution of the used SLR (triangle) and GNSS (circles) sites. Most of the sparsely distributed SLR sites tracking GNSS satellites are located on the northern hemisphere. Only 27 of the 157 sites with microwave receivers track both GPS and GLONASS satellites with the majority (i.e. 130) tracking only GPS satellites.
Table 1. Denotation of the used GNSS satellites Satellite system RINEX ILRS COSPAR GPS G05 35 1993-054A G06 36 1994-016A GLONASS R03 87 2001-053B R22 89 2002-060A R24 84 2000-063B
l
Fig. 1. Geographic location of SLR and GNSS sites; triangle, SLR; circle, GNSS The number of observations available for the parameter estimation process differs greatly for the two techniques and for the two satellite systems, GPS and GLONASS. About 20,000 microwave measurements for each GPS satellite and about 3,000 for each GLONASS satellite are used to generate a combined orbit (1-day arc), considering a sampling rate of 180 seconds. The SLR measurements are so called normal point data, formed by averaging the individual range measurements over 5-min intervals. On a daily average, only 5-20 normal points for each GPS satellite and about 10 to 40 normal points for each GLONASS satellite are available from the selected sites during the considered time interval. Figure 2 shows the greatly varying number of normal points for each satellite over the analyzed time interval.
4 Combination experiments We perform the three following combination experiments.
Experiment I The first experiment is a sort of "quick-look" experiment to study the impact of SLR observations on the combined orbits without attempting to model the 5 cm-bias between SLR and microwave observations. We address the question whether the microwave observation model is able to absorb this bias by, e.g., adapting phase ambiguities or receiver clocks, or whether the orbit dynamics is strong enough not to absorb the bias. The former would result in orbit scaling, while the latter would induce an orbit deformation, if the weight of the SLR observations is increased.
Experiment 2 The second experiment is complementary to the first one. Daily SLR range biases for each station and each satellite are estimated in addition to the orbital parameters. Thus, only the pass-specific SLR informa-
Chapter 19
• GPS/GLONASS Orbit Determination
Based on Combined Microwave and SLR Data Analysis
,._ G05 (D
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---+-+..........+. .......i......+ . .......+-' . --------"++...........+--".. ......i.........+. ............+---"+--+' . ---"--+.........-'-+i ----+--+--+.....+--"?-'+..........................10 ° and often lower; antenna monuments are, with few exceptions temperature regulated to suppress thermally related vertical variations and mounted on unweathered crystalline bedrock. Data with 30 s sampling interval have been used to obtain daily position estimates and these position time series constitute the data set used for interpretation in this study. The primary reason for reprocessing has been optimal station velocity estimation, and the variables in the different solutions have been chosen independently. Thorough presentations of the networks, processing strategies etc. have been given by Scherneck et al. (2002), Johansson et al. (2002), and Lidberg et al. (2005). Previous publications have applied different editing strategies to the data before interpreting the results; the current analysis is applied to the original data time series. Data availability for the series, significant gaps, etc are shown in Fig. 2.
3 Method •
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2004
F i g u r e 2. The 35 stations' data availability for the two solutions. The time spans for JGR02 (1993-08 - 200005) and JG05 (1996-01 - 2004-05) are indicated with the arrows above the data set.
We aim to characterize the noise in the different solutions and ran an editing loop that removed outliers larger than 5cy in the time series. This removal is conservative compared to the 3c~ in the edited solution of Johansson et al. (2002), and removed less than 1% from the total data set, compared to e.g. the 1-4% in Nikolaidis (2002) and 4.8% in Beavan (2005). We also note that the stations that are most affected by snow still use close to every observation, in contrast to earlier studies (e.g. Scherneck et al., 2002). When data gaps are present, Langbein and Johnson (1997) suggested adding white noise to their EDM interpolate, and was able to match a value that corresponded to
125
126
S. Bergstrand • H.-G. Scherneck. M. Lidberg. J. M. Johansson 0.15
,
,
,
,
,
1
,
v , - ,i, Cu,
(5)
0.1
is used to remove the common mode from the time series so that
0.05
G'- G-VlV~G
(6)
-0.05
1997
1998
1999
2000
2001
2002
2003
2004
Figure 3. The effect of the rate and periodic fit removal from the original time series for the vertical component for station KUUS. From top to bottom: raw data series, linear trend and seasonal component fit, and residual after quasi-deterministic component removal. Vertical scale in meters. presumed instrumental noise. Previous authors followed this suggestion; with the higher continuity of GPS measurements compared to EDM, the minor importance of the high frequency end for GIA evaluation and the unknown structure of GPS noise in mind we refrain from mixing observed and synthetic data,. We therefore exclude voids in the time series for the linear fit and include them with a zero value in the following unbiased autocovariance estimate in the spectral evaluation.
3.1 Empirical orthogonal function (EOF) We assume that the noise in general is stationary but with a considerable superimposed seasonal variation. Before the EOF evaluation, we remove a rate and seasonal component with 4 sinusoidal harmonics • from the data through least squares, weighted with the formal uncertainty of each observation. The fit is made to the model:
The common mode reduced G' is then used for the rest of the evaluation.
3.2 Spectral fit Mao et al. (1999) used a boxcar window on the time series autocovariance function, and judged from shown power spectra, so did Zhang et al. (1997), Calais (1999) and Williams et al. (2004). On average our time series are longer, and we apply a 4year Kaiser taper window with shape parameter 37r to the time series' autocovariance functions before evaluating the power spectrum with an ordinary fast Fourier transform (FFT). The Kaiser window yields a power spectrum with less variance compared to that of the boxcar at the expense of spectral resolution (Harris, 1978). In order to get uncorrelated estimates of the frequency powers, we pick independent frequencies co with an interval based on the main lobe width ratio ,o of the Kaiser and implicit full length boxcar windows in the Fourier domain. We then notice from Fig. 4 that our power spectra exhibit little or generally no sign of a white noise floor at high frequencies before we make a ic fit to the spectrum. The starting solution is taken from a
r(t)=e(t)+a+bt+Zsysin(O~/)+Zc:cos(O~/)(3) J
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
i
J
where r(t) is the original observation (including tectonic plate motion and GIA), c(t) is the residual kept for the evaluation and and s are coefficients which are disregarded for the rest of this study. The effect of the removal of these quasideterministic components is shown in Fig. 3 for one of the most affected stations. An EOF of the network can be obtained by a numerical search of the eigenvectors that account for the largest variances of the observations. The task is to solve the eigenvalue problem:
a,b,c
G T G u = )~2u
(4)
where G is a matrix consisting of the observed residual time series z~(t) in each column and u~...n are the eigenvectors that represent the common mode. The temporal eigenvector
1 g-6 .
10-I
.
.
.
.
.
.
.
I
i
lO o
.
.
.
.
.
.
II .
.
i
lO I
i
,
lO z
Figure 4. The effect of windowing on power spectrum evaluations. The upper power spectrum is the FFT of the windowed autocovariance function of the SUND station time series Up component. The lower power spectrum is obtained with the implicit boxcar window. The lines at the bottom indicate the independent frequencies based on the lobe width ratio that are used to obtain the spectral fit of the windowed function. The straight line fitted to the upper power spectrum is the initial crude fit.
Chapter 20 • BIFROST:Noise Properties of GPS Time Series
simple linear fit of P versus co in the log-log diagram N
362- Z[logPob~, i - OogPoi-Klogoi/fl)]
2 (7)
j=,o,2,o...
to get a model p(0)_P0.(co/f2) -~ that is conformant with equation (1). After windowing, the power spectrum estimate is zZ-distributed around the expected value pr{VP(c°) } 2 H2 < / 9 > - Xv (/9)
(8)
where v is the degree of freedom of the estimate (Jenkins and Watts, 1968, ch.6.4.2). In order to fit a model to the observations we assume that the model is identical to the expected spectrum, or H 2 _~0cob _ H2(~o,b)
(9)
However, for a least-squares fit the power spectrum random values must be transformed in order to have the property of Gaussian deviates N(/s, ~,p), where /s - X/2"f(~-)
z¢=0.5 (though JGR02 east appear slightly whiter before the EOF removal). This difference could possibly be attributed to the geometry of the global net. The determination of the spectral index for the Up component isn't affected much by the EOF for JGR02. Strong candidates to the explanation are the large data gaps in the Finnish time series, but further investigation is needed to outrule other possible causes. For JG05 whose time series are more complete than JGR02, the spectral index of the vertical component was reduced from 0.8 to 0.7. However, for the extreme cases of high spectral index the EOF failed to whiten the noise. This indicates that this noise is site specific and not a fundamental part of the system. The obtained degrees of freedom v for the spectral fit of the two solutions were larger than 6 for JGR02 and larger than 8 for JG05. We feel that this is representative of an adequate trade-off between spectral resolution and variance in our power spectra.
(10)
The sought-for transform is
F(x)- H .N / /~,I;z.2
(ll)
We take the square root of the model power spectrum and search min {2"2 -
1.2
[F(P((Di))-I](Po(n+I),b(n+I))] 2t (12) J iteratively (we actually use a relaxed version of the iteration, where the next starting value is between the n'th and n+l'st solution). In each iteration the transformation F employs the most recent update of the model parameters.
b)
I
0.8 -1I
.7 •
°
'75-
.
Results
The results of the EOF removal and spectral index estimation are shown in Fig. 5. Except the east components of JGR02 and the vertical of JG05, the EOF doesn't significantly change the spectral index estimate but certainly aids in the accuracy of the determination for the horizontal components' spectral indices. North components are adequately described with z¢=0.6, and East components with
I
I
5-
"
i
I I
I I
'~ I I I I _1_ _1_
0.6 I
0.4
I .L •
i
i
-J-
JGR62 4
-
Y
,,
_
I I I I I I I.I_-I-
I I I
f___l-lI _1_
•
JG05
0.2
Figure 5. Boxplot results of the North, East and Up components from the analysis of the two solutions. From left to right for each component: the crude results before the EOF removal, the crude results after the EOF removal, and the results after the transformed fit. The horizontal lines of each box show the median and 1 IQR, the whiskers extend to a maximum of 2 IQR from the median and spectral indices outside this range are shown as dots. Box notches graph a robust estimate of the uncertainty about the mean.
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S. Bergstrand • H.-G. Scherneck. M. Lidberg. J. M. Johansson 5
Discussion
JGR02 and JG05 have an overlapping period from 1996 to 2000, and we don't consider this time span long enough to exclude data outside the period. A direct comparison between the different time series' results in Fig. 5 may therefore result in conclusions of little significance, especially as the majority of Finnish stations lack data in the early years. It is possible that higher order harmonics would improve the fit to the time series and whiten the time series in Fig. 3. However, for stations less affected then KUUS, the risk of fitting nonparsimonious parameters is just as evident. With the high conformity of the network as well as constraints provided by independent observations and model fit to our station velocities, we feel that the chosen parameterization of the linear fit is representative of the factors that are considered important for the network as a whole. Compared to other methods, e.g. baselines (cf. Bergstrand, 2005) the EOF method is a tool to reduce the common mode noise from a set of GPS time series that requires a small amount of computational code. Another asset is that we find more realistic observation uncertainties on individual stations to constrain the GIA earth and ice models. Results are also more easily compared to those of other independent techniques. Langbein and Johnson (1997) used an 1100 sample Hann window on their EDM series and noted that the removal of a secular trend from data removes significant amounts of energy from the low frequency part of the spectra and that ~c estimates therefore are biased low. This could be complemented with a reasoning of leakage between spectral bins; through our subsampling strategy based on the implicit main lobe width relation to that of the full length boxcar, we only use the straight part of the spectrum and are not affected by the trend removal. This approach reduces the redundancy of the spectral fit slightly, but enhances the structure of the spectral power relation and still leaves enough degrees of freedom to fit fairly complicated models if desired. We also sacrifice the very low frequencies that could be resolved with a rectangular window as we consider these biased after the secular motion reduction in equation (3), anyway.
It appears from Fig. 5 that the devised power fit has little influence on the inferred parameters. However, we investigated the crude first order linear regression power fit for the two solutions (Fig. 6) and found offsets to the ideal that are equivalent to the observed shift in Fig. 5. We also notice that for ~c>l.0 in JGR02 and ~c>1.2 in JG05, the crude fit started to diverge but that preliminary tests on the transformed fit for 2500 sample long synthetic data series up to x=2.0 show excellent results. We are thus confident that the obtained results are representative of the time series actual noise content for the chosen model. An advantage of the scheme is that it is reasonably fast. The first crude fits to the power spectra for the whole network are obtained within 5 minutes and with the relaxed iteration the final results are obtained in less than 30 minutes on an ordinary PC without optimizing the underlying MATLAB script. We are thus provided with a tool that makes a comparison between e.g. different outlier removal strategies and their impact on velocity uncertainties feasible. Davis et al. (2003) asked whether the estimate of velocity from GPS is fundamentally limited by one or more error sources and if so, what these error 1.4
I
1.2
0.8
0.6
0.4
0.2
0
i
i
i
i
i
0.2
0.4
0.6
0.8
1
Figure 6. Simulation of the crude power fit for synthetic time series. (JGR02 top, JG05, bottom). Straight lines represent the ideal fit, dots the mean of 1000 formal tc series with gaps equivalent to the original data sets. The gray uncertainty band is the ensemble median _+ the maximum standard deviation. The larger bandwidth for JGR02 is solely attributed to the ROMU station. The offset from the ideal is comparable to that of the crude to normalized fit observed in Fig. 5.
Chapter 20 • BIFROST:Noise Properties of GPS Time Series
sources are. Zhang et al. (1997) could not distinguish between an a priori chosen model with a white noise floor and one where ~c was estimated along with the noise amplitudes. Mao et al. (1999) found a better fit with an a priori chosen model then with estimated ~c, but at the same time failed to fit three out of ten synthetic runs with a white and flicker noise combination (Mao et al., 1999, Table 4). Williams et al. (2004) used a larger data set and also advocated a white noise floor in the power spectra and searched for a dominant error source in the time series. One parameter of the user segment that has received considerable attention in geodetic time series evaluation is the monument stability (Langbein and Johnson (1997), Johnson and Agnew (2000), Williams et al. (2004), Beavan (2005)). Combrinck (2000) gave an introduction to space geodetic monumentation and we adhere to his holistic approach, i.e. that monument stability is more than a strictly geodetic issue. Our stations are essentially invariant to the physical parameters listed by Moore (2004, section 2.3, online) and there is no conspicuous reason to assume dominant monument noise in our network. We are primarily interested in the true station velocity uncertainties and content with parameters that allow us to scale the formal uncertainties of the observations (e.g. as suggested by Williams, 2003). In BIFROST, the (easily identified!) winter time snow coverage of the northern stations' antennas reddens the noise. Should we be able to completely remove the snow e.g. with hot air fans, the noise for these stations would probably whiten and our velocity estimate uncertainties hence be reduced. Although of minor importance for the GIA evaluation, we notice that we are generally unable to observe any white noise floor in the higher end of our power spectra. Beavan (2005, section 5.1) used two different MLE packages and with the Williams et al. (2004) package also noticed a recurring absence of white noise in the time series from the PositioNZ network. This observation could possibly be used for evaluation in areas where shorter term movements are targeted and monument stability in not-well-consolidated sediment might be an impediment to the observations, e.g. the Southern California Integrated Geodetic Network (SCIGN), cf. Williams et al. (2004) and Beavan (2005). Given the complexity of the error sources, we question whether an a priori assumption of white noise
properly addresses the noise process in GPS time series before the contributions from individual sources have been duly separated. As we don't observe any white floor in our power spectra, there is evidence that sampling should be more frequent than once per day in order to get smaller error in parameter uncertainties. Given the complexity of the overall uncertainty, we feel that this is a valid result. The next higher order candidate to address could be a semidiurnal tide related signal resolved with a bihourly time series. However, with daily sampling interval strategies, the questions by Davis et al. (2003) will probably remain unanswered. 6
Conclusion
We investigated two different GPS time series ensembles from the BIFROST network and used an EOF approach to remove the common mode noise. Our time series have reached a length where it is useful to apply a tapering window to the autocovariance functions before we estimate the power spectra using FFT. Power law models appear to accurately represent the noise process in the BIFROST time series. The spectral indices are in the range 0.5--0.7 for the North, East and Up components and determined with excellent precision in the horizontal. We are able to determine the spectral power relationship with a simple least squares analysis in the log-log diagram by using uncorrelated spectral bins, but prefer the devised normalizing transformation since it performs better for a wider range of power spectral indices and yields more accurate results. Winter-time snow coverage of antennas still affects some of our position time series, but we now have an unbiased parameter that can be used to scale the individual stations formal velocity uncertainties and still use more than 99% of the observed data. There is little evidence for a white noise floor in our power spectrum evaluations, and this indicates that a higher sampling frequency than once per day will get smaller errors in parameter uncertainties. This also leads us to the conclusion that a priori assumptions of noise structure may be misguiding in a search for dominant error sources. Acknowledgments. The authors would like to thank Simon Williams for thorough and constructive reviews of this manuscript as well as an earlier version, and John Beavan for the review of the early version. We also appreciate the effort put in by the SWEPOS staff and our colleagues at the Finnish Geodetic Institute for running and maintaining the networks.
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References Altamimi Z., P. Sillard and C. Boucher (2002). ITRF 2000: A New Release of the International Terrestrial Reference Frame for earth Science Applications. J. Geophys. Res., 107(B10), 2214, doi:10.1029/2001JB000561. Beavan J. (2005). Noise properties of continuous GPS data from concrete pillar geodetic monuments in New Zealand and comparison with data from U.S. deep drilled monuments, J. Geophys. Res., 110, B08410, doi: 10.1029/2005JB003642. Bergstrand S., H.-G. Scherneck, G. A. Milne and J. M. Johansson (2005). Upper mantle viscosity from continuous GPS baselines in Fennoscandia. J. Geodyn., 39, 91--109. Calais E. (1999). Continuous GPS measurements across the western alps, 1996-1998, Geophys. J. Int., 138,221--230. Combrinck L. (2000). Local Surveys of VLBI Telescopes. In: Int. VLBI Service for Geodesy and Astrometry 2000 Gen. Meeting Proc., N. R. Vandenberg and K. D. Bayer (eds.), NASA/CP-2000-209893. Davis J. L., R. A. Bennett, and B. R Wernicke (2003). Assessment of GPS velocity accuracy for the Basin and Range Geodetic Network (BARGEN), Geophys. Res. Lett., 30(7), 1411, doi: 10.1029/2003GL016961. Ekman M. (1996). A consistent map of the postglacial uplift of Fennoscandia, Terra Nova, 8, 158--165. Harris, F. J. (1978). On the use of windows for harmonic analysis with the discrete Fourier transform, Proc. IEEE, vol. 66, 51-83, Jaldehag R.T.K., J. M. Johansson, J. L. Davis, P. E16segui (1996). Geodesy using the Swedish Permanent GPS Network: Effects of snow accumulation on estimates of site positions. Geophys. Res. Lett., 26, 1601-- 1604. Jenkins G. M.and D. G. Watts (1968). Spectral analysis and its applications, Holden-Day, San Francisco. Johansson J. M., J. L. Davis, H. -G. Schemeck, G. A. Milne, M. Vermeer, J. X. Mitrovica, R. A. Bennett, B. Jonsson, G. Elgered, P. E16segui, H. Koivula, M. Poutanen, B. O. ROnn~ing and I. I. Shapiro (2002). Continuous GPS measurements of postglacial adjustment in Fennoscandia: 1. Geodetic results. J. Geophys. Res., 107(B8), 2157, doi: 10.1029/2001JB000400. Johnson H. and D. C. Agnew (2000). Correlated noise in geodetic time series, U.S. Geol. Surv. Final Tech. Rep.,
FTR-1434-HQ-97-GR-03155. Lambeck K., C. Smither and P. Johnston (1998). Sea-level change, glacial rebound and mantle viscosity for northern Europe, Geophys. J. Int., 134(1), doi: 10.1046/j. 1365-
246x.1998.00541.x. Langbein J. O. and H. Johnson (1997). Correlated errors in geodetic time series: Implications for time dependent deformation. J. Geophys. Res., 102, 591--603. Lidberg M., J. M. Johansson, H.-G. Scherneck and J. Davis (2005). An improved and extended GPS derived velocity field of the postglacial adjustment in Fennoscandia, submitted. Mandelbrot B. (1983). The Fractal Geometry of Nature, W.H. Freeman, New York. Mandelbrot B. and J. Van Ness (1968). Fractional Brownian motions, fractional noises and applications. SIAM Rev, 1O, 422--439. Milne G. A., J. L. Davis, J. X. Mitrovica, H. -G. Scherneck, J. M. Johansson and M. Vermeer (2001). Space Geodetic constraints on glacial isostatic adjustment in Fennoscandia. Science, 291,2381--2385. Milne G. A., J. X. Mitrovica, H. -G. Schemeck, J. L. Davis, J. M. Johansson, H. Koivula and M. Vermeer (2004). Continuous GPS measurements of postglacial adjustment in Fennoscandia: 2. Modeling results. J. Geophys. Res., 109, B02412, doi:10.1029/2003JB002619. Moore A. W. (2004). IGS Site Guidelines,
http.'//igscb.jpl.nasa.gov/network/guidelines/guidelines.htm 1, cited: 2005-11-01. Nelder J. A. and R. Mead (1965). A simplex Method for Function Minimization, Comput. J., 7, 308--313. Segall P. and J. L. Davis (1997). GPS applications for geodynamics and earthquake studies. Ann. Rev. Earth Planet. Sci., 25, 301--336. Schemeck H.-G., J. M. Johansson, G. Elgered, J. L. Davis, B. Jonsson, G. Hedling, H. Koivula, M. Ollikainen, M. Poutanen, M. Vermeer, J. X. Mitrovica and G. A. Milne (2002). Observing the Three-Dimensional Deformation of Fennoscandia. In: Ice Sheets, Sea Level and the Dynamic Earth. J. X Mitrovica. and B. L. A. Vermeersen (eds.), Geodynamics Series vol. 29, AGU Washington, D.C. Turcotte D. L. (1997). Fractals and Chaos in Geology and Geophysics, 2nd ed., Cambridge University Press, Cambridge. Williams S. D. P. (2003). The effect of coloured noise on the uncertainties of rates estimated from geodetic time series. J. Geodesy, 76, 483--494. Williams S. D. P., Y. Bock, P. Fang, P. Jamason, R. M. Nikolaidis, L. Prawirodirdjo, M. Miller, and D. J. Johnson (2004). Error analysis of continuous GPS position time series. J. Geophys. Res., 109, B03412, doi: 10.1029/2003JB002741.
Chapter 21
Discrete Crossover Analysis Wolfgang Bosch Deutsches Geodfitisches Forschungsinstitut (DGFI), Marstallplatz 8, 80539 Mfinchen, Germany
Abstract. Crossover analysis is performed to improve observations on intersecting profiles. The most prominent application are sea surface heights observed by satellite altimetry. We discuss different versions of a discrete crossover analysis, which do not apply an analytic function of the error model (like polynomials, splines, or Fourier series), but estimate the errors at crossover locations only. In order to achieve for the error a certain degree of smoothness we suggest a least squares approach, minimizing both, crossover differences and consecutive differences. The weighting scheme can be modified to balance the influence of crossovers and consecutive differences. The normal equations are large but sparse and can be solved by iterative algorithms. The approach is flexible and can be applied to global as well as regional crossover analysis and has been demonstrated for multi-mission crossover analysis.
Keywords. crossover differences, least squares, satellite altimetry
1 Introduction Crossover analysis is known as a powerful approach to estimate errors and improve observations along intersecting profiles. It takes advantage of the redundancy given by the two (more or less) independent observations taken at the intersection (the crossover) of the two profiles. Gravity measurements along ship routes can be analysed in this way. The most prominent example, however, are the sea surface height observations of satellite altimetry taken along the intersecting satellite ground tracks. The application of satellite altimetry will be often referred to in order to illustrate the theoretical concept developed below. In general the error of profiled observations is described by an error function modelling the error as a function of time or profile length. Low degree polynomials are used, for example, for short profiles or ground tracks. For extended profiles or sequential ground tracks, as they appear e.g. in satellite
altimetry, observation errors have been modelled by splines or Fourier series. As soon as the parameters of such models are estimated the radial error can be computed by evaluating the error function at any point of the profile. In general, functional error models impose some assumption about the error characteristic. A Fourier series, for example, pre-defines the frequencies that are supposed to explain the dominant error and estimate amplitudes and phases for these frequenciess. For other frequenciess Fourier series are 'blind'. In the present investigation, we do not predefine any functional error model, but consider exclusively the (radial) errors xi and xj for the two observations (the sea surface heights) taken at the crossing of two intersecting profiles (ground tracks). These consideration leads to a discrete form of crossover adjustment subsequently referred to as DCA. The two observations are expected to be identical, their differences reflect radial errors and are subject to a minimization. Minimization of crossover differences alone is the most trivial adjustment procedure - leading, however, as shown below to discontinuous results. In order to achieve a certain degree of smoothness for the radial errors we consider in addition - as a substitution for any analytical model - the consecutive differences between radial errors. The weighted minimization of consecutive differences was first introduced by Cloutier (1983). He used the crossover differences as constraint and did not account for observation errors. We therefore extend the method in order to simultaneously minimize crossover and consecutive differences. It will be shown that the method is flexible and can account for prior knowledge of observation errors and the statistical property of radial errors. First a few remarks are given to crossover differences. Then, the basic concept of discrete error modelling is explained. The notation is used to shortly present Cloutiers method. Section 5 describes the suggested approach to minimize both, consecutive and crossover differences. Finally, numerical aspects of the DCA are discussed.
132
W. Bosch
2 Crossover Differences
X'--(Xl,X2,...,X2n )
The crossing of two intersecting profiles is called a crossover. In satellite altimetry the intersecting profiles are the ground tracks o f the satellite. Ground tracks are the orthogonal projection of the satellite orbit to a mean Earth ellipsoid. At a crossover the height is given twice, once by the observations of the ascending ground track and the second time by the observation along the descending ground track. In general, there is no direct observation at the crossover and a least squares interpolation procedure is applied to obtain 'observations' at the intersection (see Figure 1). Low degree polynomials can be used, for example, to interpolate the heights along the observed profiles. If the crossover location is surrounded by a sufficient number of observation points (e.g. six) and the polynomial degree remains small (e.g. three) there is no risk of oscillating polynomials.
(2)
then the linear relationship between 'observed' crossover differences and radial error components is d = Ax
(3)
with the (n×2n) coefficient matrix 1
0
...
0
1
-1
0 .
"i"
• ..
0
1
0
A --
.
.
-1
0
0 .
(4)
-1
In each row, matrix A has only two non-vanishing elements, namely '+ 1' and '- 1 '. Moreover, every column of A has only a single non-vanishing element. It follows A A ~ = 2I
(5)
where I is the identy matrix. In addition A~A = I - P
(6)
where P is a permutation matrix with a single nonvanishing element in every column and row. Permutation matrices are orthogonal pip = pp1=
I
which implies p-1 = pI Fig. 1 Observations at intersecting profiles are to be interpolated to the crossover location (red)
The difference of the two interpolated 'observations' is called crossover difference. The least squares interpolation is used to associate an error estimate to the crossover differences.
Equation (3) is an underdetermined system, which has no unique solution. If the number of unknowns in x is larger than the number of observations in d a minimum norm solution is given by A
x--A+d with the pseudo inverse A + -- A t ( A A t ) -1
(8)
With equations (7), (8), and (5) it follows
3 Discrete Crossover Analysis (DCA) For a set of n crossover events the observed crossover differences A d k, k = 1 .... n shall be compiled to the vector dt - ( / X d l , / X d 2 , . . . , /Xd~)
(7)
(1)
For every crossover observation we then consider errors for both, the ascending and the descending ground track. Thus, there are 2n radial error components xi(t ), i - 1, ... 2n which shall be sequentially ordered in time such that ti < ts+l. If the error components are compiled in the vector
^ 1 x -- - A ~ d (9) 2 Thus, the minimum norm solution consists in equally distributing crossover differences to the radial error components of the two intersecting ground tracks. In general, the computation of crossover differences is accomplished by error estimates, derived from the least squares interpolation procedure. These error estimates can be used to weight the crossover differences. Assuming uncorrelated crossover differences the weights can be compiled into the diagonal matrix
Chapter 21 • Discrete Crossover Analysis
0
...
above. For the consecutive differences the weights
0
"°"
0
1~ 2
"''
0
•
"'°
i
0
...
o __
w
0
which can be used to make the linear system (3) homogeneous by left multiplying with W d1/2 l/2AAtW
**d
/2
__
2W d
The solution (9) does not change, however, because I1/2)+ (AW
Id
1/2d_
The minimum norm solution of system (3) implies that consecutive error components are adjusted independent of each other such that the time series of the errors may exhibit significant offsets from one error to the next. This property can be hardly accepted, because the actual orbit of a satellite obeys a certain degree of smoothness: the gravity field at satellite height is continuous, an analytic function and the satellite is always moving at some distance to the attracting masses of the Earth. As the discrete crossover analysis abandons any functional error model the smoothness of the errors has to be introduced by some other means. The following section describes a method to ensure smoothness by minimizing the differences between consecutive errors.
4 The method of Cloutier Already in 1983 Cloutier introduced an approach to minimize the variations of consecutive e r r o r s - under the constraint that the crossover equation (1) is fulfilled. The method is summarized with the notation used here - as a foundation for the extensions to be described in section 5. With n crossover events there are 2n error components and 2n-1 differences between consecutive errors. The latter can be expressed by the matrix-vector product D x where 1
0
-1
1
(11)
w~
0
-1
......
0
0
--.
0
are introduced in order to impose smaller variations between error components close in time compared to error components separated by a larger period• Weights can be derived from an empirical auto-covariance function o r - as a first guess - may be set, for example, by the covariance function ~.
w,-
2 C0.5
~-.
w,(At,)-
t2
2
Co.5 + A
where the constant c0. 5 defines the 'correlation length', the time difference where the covariance function is decreased to the value 0.5-wi(0 ). Using these notations, the cost function, introduced by Cloutier, may be written as m i n 21 IDx!-',~v - - rain -1x D ' ~:A,, d=Ax 2
'W
Dx
(12)
As shown in detail in the appendix Cloutiers approach leads to a rank deficient system• A single additional constraint, e.g. a linear combination of the error components k/x - - c is necessary and sufficient to regularize the system• The standard procedure for least squares problems leads - as shown in the a p p e n d i x - to the solution x -- Q-~AZ(AQ-~AZ)-I d
(13)
with Q - T + kk I
(14)
and Wm
-w~
0
.-•
-wi T
m
w~ + w 2
2
-w
0 o
".
.
0
0
--W 2
W2 @1473
".
•
",
•
"•
--WH
0
...
0
-w,
w,
(10)
(15)
is a matrix with 2n-1 rows and 2n columns and x is the vector of error components already introduced
Equation (13) is the solution for the 2n radial errors at the crossover location. Note, left multiplication of (13) with A immediately shows that indeed the crossover equation (3) is fullfiled.
U
m
0
.-.
0
......
0
1
-1
0
0
1
-1
133
134
W. Bosch
5 Minimizing consecutive and crossover differences
f~ -- x~M~WM x _ 2 x M ~ W b + b 1 W b + 2kk~x
Cloutier's method introduces the crossover equations (3) as a constraint. This, however, appears not appropriate and too restrictive: crossover differences are derived from observed quantities which are affected by observation errors. Therefore crossover differences should be treated as observations and introduced in the crossover analysis as quantities that get a residual in order to account for errors and to make the overall system more consistent. Consequently, the following section develops a modification to the method of Cloutier. It essentially consist in minimizing both, consecutive differences and the crossover differences. In contrast to the cost function (12) the following minimum principle shall be applied m i n l M x -- b 2w
(16)
where
with the necessary conditions for a minimum leads to the normal equation system MIWM
k
k ~
0
The coefficient matrix
N2 --
MIWM kI
k 0
is regular and has the inverse N~ ~ -
Qj1- a~ss' OLS !
o
with Q2 - M I W M + k k l the summation vector
1
1]
and the constant is composed of matrix D, equation (10), and matrix A, equation (4). M is now a matrix with 3n-1 rows and 2n columns. The vector
o~ -- ( k ' s ) - ' The product M I W M - D I W D + A/Wd A
bI:l
T + AIWo A
consist of a null-vector of length 2n-l, which may be considered as pseudo-observations of the consecutive differences, followed by the vector d which keeps, compare equation (1), all crossover differences. The weight matrix
w~
0
W-
0
w.
compiles the diagonal weight matrices W v and W d for the consecutive differences and the crossover differences respectively. Correlation between consecutive and crossover differences are not considered. Matrix M has - similar as in Cloutiers approach - a rank defect of one: the sum of all columns results in a null vector such that one of the columns can be represented as a linear combination of the other columns. In order to overcome the rank defect, a single constraint, e.g. k'x -- 0
can be introduced. The Lagrange function
(17)
is composed of the tri-diagonal part T, already defined by equation (15) above and the sparse matrix product A'WjA which may be d e c o m p o s e d - similar to equation (6) - to a diagonal part and a scaled permutation matrix. Elaborating the explicit solution for the 2n radial error components, and considering that s ' M = 0 and M W ' b = A'Wdd one obtains x -- Q 2 1 A I W j d
(18)
with Q2 - T + A I W d A + k k I
(19)
6 Numerical Aspects The system, equations (18) and (19) can become very large, because the size of the normal equation is twice the number of crossovers. In Bosch (2005) the DCA is used to perform a common adjustment
Chapter 21
of all single- and dual-satellite crossovers between 2-4 contemporary altimeter missions. Up to 100000 nearly simultaneous crossover events are considered for a ten-day period with three day overlap to neighbouring periods. Thus the systems to be solved have about 200000 unknowns. A strict solution for systems of this size is difficult, if not impossible. The only way to solve such systems is to apply an iterative procedure such as the conjugate gradient projection (CGP) algorithm as described, for example, by Golub & van Loan (1983). This iterative solution algorithm is applied here - without any preconditioning. Fortunately, the matrix Q of the normal equations (18) to be solved has a rather simple structure. The first component of the matrix Q (see equation (19)) is a weighted Gauss transform of matrix D and has tri-diagonal structure. The second component of Q, the Gauss transform of matrix A, is a sparse matrix with a non-zero diagonal and as many non-zero offdiagonal elements as there are crossovers. If the constraint (17) is taken to fix a single unknown, then the product kk' contributes only to the diagonal. This way the structure of Q can be kept tri-diagonal plus sparse as shown in Figure 2. The CGP algorithm requires to perform repeated products between matrix Q and a vector. The computations can be considerably speed up if the specific structure of Q is taken into account and the product of the tri-diagonal matrix and the remaining sparse matrix is treated separately. We therefore modified the CGP algorithm accordingly with the effect that the iterative solution becomes very fast (systems with 200000 unknowns were solved within a few second computation time on a state-of-the-art PC) and has very low storage requirements.
N.
4000
•
eUe Q
00•
000
•
Appendix According to section 4 Cloutiers approach is based on the cost function,
1] Dx12v
min d=Ax 2
min -1x 'D' W v D X 0=Ax 2
-
(12)
minimizing consecutive differences under the constraint that the crossover differences are fulfilled. The structure of D, equation 10, is as simple as the structure of A, equation 4" in every row there are only two non-vanishing elements, namely +1 and -1. It follows that the product T = D'WvD is a symmetric, tri-didagonal matrix
T
w~
-w~
0
--W 1
W 1 -~ W 2
--W 2
".
0
--W 2
W 2 ---~W 3
".
•
o
...
0
m
0
...
,
-%
0
0 -w~ % (15)
with size 2nx2n. With the Lagrange function -- l x ' T x + A ' ( A x - d) --+ min 2 a vector A with n Lagrange factors is introduced. The two necessary conditions to minimize ~, c~/c~x=0 and c~/c~A=0 lead to the normal equation system [i
Ot]'[A] - [0d]
(20)
This system cannot be solved without further conditions, because it is obvious that T has no full rank: the sum of all rows (columns) is a null-vector and the 2n columns (rows) are not linear independent. The rank of T is, however, 2n-1. Thus, a single additional condition as, for example, k~x--c is necessary and sufficient. The extended Lagrange function is then
•
•00
CrossoverAnalysis
• Discrete
-- -1 x ' T x + A ' ( A x - d) + X o ( k ' x - c) -+ min 2
•
with the additional Lagrange factor )~o and leads to the modified normal equation system
"||, 0@•
•
"ii
Fig. 2 The structure o f the normal equation matrix Q is composed o f a tri-diagonal matrix, and a sparse structure with as many @ d i a g o n a l non-zero elements as there are crossovers.
T
k
Al
x
~ ;0 00 0 ")"° A --
0
(21)
135
136
W. Bosch
The upper left submatrix
A -- - ( A Q - 1 A ' ) - 1 d
NI[ T
(22)
For the remaining solve-for parameters it follows Nix o -- Ao(AQ-~A~)-I d +
is now regular and possesses the inverse
[
such that
o']
X Xo z
with the summation vector s'-[1
X0 -- NI-' (Ao ( A Q - ' A / ) - ' d + Co) Q-1At (AQ-lAl)-I d
1]
1--.
Co
c~s~A~
the constant
Considering again As = 0 and taking into account that N~c0 vanishes it follows )~0 = 0. Finally, we get the solution, already shown in section 4
oL -- (k's) -1 and the matrix.
1..
Q -- T + k k '
(23)
t
I;'] I
x
x o --
0 ,
Co - -
)k 0
References
C
BOSCH, W.: Simultaneous crossover adjustment for
the system (21) is transformed to N~ Ao
A o . xo _ 0
A
To solve the regularized system - first for the Lagrange factors A - we get -1
/
- A o N 1 AoA
-
d-
AoN
-1
1 e,
•
(13)
with Q and T as defined above, by equations (23) and (15) respectively.
With equation (22) and the short hand notations A o --
x -- Q - 1 A t ( A Q - 1 A I ) - I d
•
-1
l
Because As = 0, the coefficient matrix A0N 1 A 0 -1 l -I reduces to A Q A and the product A0N ~ c 0 becomes a null vector• The solution for the Lagrange factors A is therefore
contemporary altimeter mission. ESA Scientific Publications SP 572, ESA ESTEC, 2004 BoscH, W.: Satellite Altimetry: Multi-Mission Cross Calibration. In: Rizos, Ch. et al. (Eds.) Dynamic Planet 2005. l a G Symposia, Vol. 13?, Springer, Berlin (this volume) CLOUTmR, J.R., A Technique for Reducing Low-Frequency, Time-Dependent Errors in network-Type Surveys. J. Geophys. Res., Vol.88 (B 1), 659-663, 1983 GOLUB, G•E. AND C.F. VAN LOAN, Matrix Computation, John Hopkins Press, 1983
Chapter 22
A comparative analysis of uncertainty modelling in G PS data analysis S. Sch6n Engineering Geodesy and Measurement Systems, Graz University of Technology (TUG), Steyrergasse 30, A8010 Graz, Austria H. Kutterer Geodetic Institute, University of Hannover, Nienburger Strasse 1 D-30167 Hannover, Germany
Abstract. A thorough assessment and mathematical treatment of all relevant errors in GPS data processing and analysis are essential for the further use and interpretation of the processing results. In this study two mathematical approaches for the error handling are studied in a comparative way. A probabilistic approach is based on the construction of a fully populated variance-covariance matrix of zero difference phase observations by introducing the uncertainty measures of the respective influence parameters in terms of standard deviations. A deterministic approach interprets these uncertainty measures as error bands and uses formalisms from interval mathematics. Both approaches are applied to a simulated EUREF sub-network. The deterministic approach yields more realistic results, in particular with respect to the dependence of the uncertainty measures on the baseline length.
ters). However, remaining systematic errors may persist. In the following we focus on the uncertainty due to remaining systematic errors in GPS results. Two approaches are studied, a probabilistic and a deterministic one. The common starting point is the wellknown GPS phase observation equation (HofmannWellenhof et al., 2001)
+ri
"}-
ri
or-
v
i
Or- C
(4
'or-
'Jr-
where L~ denotes the metric phase observation, ,o~ the geometric distance between the receiver i and the satellite k. The ionospheric phase advance is denoted by I~, the tropospheric delay by T~, the multipath by M/~, satellite clock errors by 6 ~ , receiver clock errors by 4 , and the velocity of light
Keywords. GPS, systematic errors, correlations, imprecision, interval mathematics
in vacuum by c. The ambiguity parameter is given by N,~ , the wavelength of the carrier wave by 2.
1 Motivation
Eq. (1) is valid for both GPS carrier wave frequencies L 1 and L2 with differences mostly for the ionospheric phase advance and the multipath effects.
In GPS data processing all errors have to be considered which are relevant for the proper interpretation of the obtained results. Various approaches exist for modelling and reducing the arising uncertainty. For the random component advanced weighting schemes can be applied (e.g. Han and Rizos, 1995; Hartinger and Brunner, 1999; Brunner et al. 1999; Wieser, 2002). Fully populated variance covariance matrices can be constructed by means of variance covariance component estimation (Wang et al., 2002; Tiberius and Kenselaar, 2003) or time series analysis (Howind, 2005). For the systematic component different reduction methods are used such as double differencing, linear combination, or model extension (e.g. estimation of tropospheric parame-
Eq. (1) depends on a multitude of parameters like the temperature, air pressure, etc. (influence parameters). As the values of these parameters are imprecise in general, systematic effects remain. In this paper two approaches for the assessment of uncertainty due to these effects are compared and discussed, a probabilistic one and a deterministic one. Both use a forward modelling strategy. The probabilistic one is based on the construction of an additional variance-covariance matrix (vcm) of the zero difference (ZD) phase observations by introducing the influence parameters as random variables (Jfiger and Leinen, 1992; Schwieger, 1996). The deterministic approach interprets these uncertainty measures as error bands and uses formalisms
138
S.Sch6n• H.Kutterer
from interval mathematics (Sch6n 2003, Sch6n and Kutterer 2003, 2005). In the next section both approaches are developed. Then a sub network of 8 EUREF GPS stations is simulated in order to study the behavior of both uncertainty measures. Finally, the obtained results are compared and discussed.
matrices of the partial derivatives are given by F 1 and F 2 . As only the frequency dependent ionospheric parts of these matrices are different this approach leads to high positive correlations between the L1 and L2 ZD observations. The e x t e n d e d v c m of the ZD £ZD is then obtained as
2 Mathematical
£ ~ - £ ~0 + £ v~ .
Concept
2.1 Modelling of influence p a r a m e t e r s
The initial random uncertainty component can be described by the vcm
Mz)z)12zz) l l M r ~-"DD --
,
, 22 M T DD
assuming uncorrelated ZD observations for both frequencies and an elevation depending weighting. In order to describe the systematic component, the impact of the influence parameters on the GPS phase observations is derived by means of a sensitivity analysis. Therefore, Eq. (1) is linearized with respect to the influence parameters. For L1 and L2 phase observations, respectively, this yields a linear relation between differential changes dLk of the phase observations on one side and the partial derivatives contained in the Jacobi matrix F and the differential ds of all considered influence parameters on the other side: dL~ - V ds. (3) The main difference between the probabilistic and deterministic approaches lies in the interpretation of ds and the associated mathematical handling.
- £DD121 £DD.22 with the DD operation described by the matrix M vv. Eq. (6) can be also represented by an initial term and an additional term encountering the uncertainty (covariances) for remaining systematic errors (
-E°z),l 1 --
vcm
/~-" ZD,11
£v
ZD,21
"~
£,
ZD,12 _
FI diag(s)2
F, r
(4)
ZD,22
of the ZD observations describes the uncertainty due to remaining systematic errors. Here, Z' ZD,11 "~'ZD,22
are
the fully populated vcm (in [m2])
,
)
r
-~- ~[] 'DD,11
M DD ( ~-"ZD,22 o -4- '~ ,ZD,22 ) M D Dr
~Z~DD,22 --
"
(7)
0 v -- ~"DD,22 + ~ DD,22
12o~,12 -
12
'o~,12
The ionosphere-free linear combination L3 of the DD L1 and L2 observations can be introduced in a similar way. The associated vcm reads XDD,Ls - MLS12DDM[S
(8)
with M Ls-(
If the differentials of the influence parameters are modelled as random variables, their expectation values are considered as zero. Here, a diagonal vcm is introduced. Hence, the law of variance propagation has to be applied to Eq. (3). When both frequencies L1 and L2 are studied, the fully populated
0
£DD,~ -- MDD £zD,~I + E zD,~ MDD
2.2 Probabilistic a p p r o a c h
\
(6)
e diag [ O-L2
and
MDD~ZO
+
12zv -
£'zD-
Mz)z)12zz),12Mz)z)
DD
MDD]F-"ZD,21MrDD
0
(5)
In order to reduce systematic errors in GPS phase observations double differences (DD) are built as
a,
""
a,
~, / part of L1 DI~ observations
a 2 ""
a2 )
v / part of L2 DI~ observations
and the frequency dependent coefficients f,2 j], a2
.1
al
f2
./~ .if2 .2/] "
Again the vcm can be split into two terms. The second term describes the impact of the probabilistic formulation of additional uncertainty due to remaining systematic errors on the initial n x n vcm
(
,
+ a~X DD,11 nt- 2al
a2~2,DD,12 -~- a ~ £
,
DD,22
)
(9)
_£0
DD,L3 -Jr-~ VDD,L3
Finally, the u × u vcm of the estimated coordinates is obtained as
of the L1 and L2 ZD, respectively, and 12'ZD,12 denotes their covariance matrix. The corresponding
12~ - (ArI2~,LsA) 1
(10)
Chapter 22
assuming a least-squares estimation in the GaussMarkov model (Koch, 1999) with the n x u design matrix A .
•A
Comparative Analysisof Uncertainty Modelling in GPS Data Analysis
tion of Oct 1st 2004. The common reference station of the baselines was POTS which was fixed during the adjustment. The baseline length and ellipsoidal heights of the stations are given in Table 1.
2.3 Deterministic approach o
In the deterministic approach, the uncertainty measures of the influence parameters are interpreted as deterministic error bands and modelled in terms of interval radii s r . These intervals enclose all possi-
o
60 ° ~___~o~ E 1~ E 2o°E
E
.:: NSA (~::-
ble values of the influence parameters. Hence, maximum effects are considered. Methods from interval mathematics (see, e.g., Alefeld and Herzberger, 1983) are applied for the mathematical handling. Here, only the final equation is given. Details on all important derivations can be found in Sch6n (2003) and Sch6n and Kutterer (2003, 2005). The direct application of interval methods to the estimated coordinates leads to the interval radii
2c ~
\
,
Fig. 1: EUREF sub-network with 8 stations
xF
I/A r (E°D,L3
A
°
(I~DD,L3) -1
(11)
Table 1. Overview of baseline lengths and ellipsoidal heights
station
where l e I denotes the element-by-element absolute values of the matrix coefficients. Besides these absolute values which change the matrix structure significantly, there is a formal analogy of Eq. (11) with the measure of external reliability. However, the concepts are different: the minimum detectable biases needed for external reliability are obtained through hypothesis tests, the s r used here are given deterministic values for the forward modelling.
2.4 Relationship between both approaches Both approaches use forward modelling and start with the same linearization. But the interpretation of the uncertainty of the influence parameters is different. In the probabilistic approach (PROB) the total differential ds is associated with the standard deviations % of the influence parameters which leads to a fully populated vcm. In the deterministic approach (DE7) the total differential ds leads to interval radii s r . In order to reflect maximum effects, the relation s r = 3% is set for reasons of comparability in the following.
3 Simulation studies For the comparison of the two approaches a simulation study for an 8 station EUREF sub-network (Fig. 1) was carried out using the satellite distribu-
POTS BOR1 WTZR ONSA GRAZ RIGA GRAS MATE
baseline length [km] with respect to POTS 273.3 360.1 562.9 615.9 870.5 1060.7 1330.6
ellipsoidal height [m] 144.4 124.4 666.0 45.6 538.3 34.7 1.319.3 535.6
Table 2. Overview of the magnitude of the influence parameters
Parameter
sr =
3or
Orbit errors in alongtrack, crosstrack and radial dir. Carrier frequency stability
_+10-12 f0, f0: n o m .
Clock errors
_+10 -12 [S]
Ionosphere -VTEC-representativity -Model constants -Mapping function in terms of the zenith angle Troposphere -Temperature -Pressure -Partial pressure of water vapour -Model constants -Mapping function in terms of the zenith an~le Antenna offsets and variations in a local system Clock stability
+0,1 [m]
frequency
+1% VTEC = +10 -15 [el/m2] +0,5 of last digit +1 ["]
+1 [°C] +1 [mbar] +1 [mbar] +0,5 of last digit +1 ["] +2 [mm] _+5.10-9 f0
139
140
S. Sch6n • H. Kutterer
An 8h session was simulated using a cut-off elevation angle of 15 ° and a sampling rate of 6 min. In addition to the coordinates, tropospheric parameters were estimated with a temporal resolution of 2h using the Niell mapping function. Table 2 shows the applied magnitude of influence parameters. Two scenarios were considered with different vcm of the L3 DD; see Eq. (9). In the scenario S1 only the matrix I:vv,L 3 ° is considered, in the scenario $2
Fig. 3 shows the interval radii obtained by Eq. (11), i.e. the deterministic measure of additional uncertainty due to remaining systematic errors. Obviously, the interval radii increase significantly with the baseline length, especially for the north and height component. Second the interval radii of the height component are greater than those of the horizontal components (up to seven times for short baselines and approximately three times for long baselines). BOR1WTZR ONSAGRAZ
the whole matrix I2DD,L3 is used.
IOt. .N.o.~. h.' ...........
Fig. 2 shows for both scenarios the standard deviations of the north, east and height components of the 7 EUREF stations, respectively. The stations are sorted by increasing baseline length. Three aspects are of interest. First, the standard deviations are rather independent of the baseline length; for numerical values see Table 3. Second, the magnitude of the height component is four to five times the one of the horizontal components. Third, the standard deviation obtained under consideration of additional uncertainty due to remaining systematic errors by the stochastic approach ($2; grey+black) is greater than in the standard case (S1," grey). On average, these changes are very small and certainly below a typically expected magnitude. BOR1WTZR ONSAGRAZ
RIGA
ss°
"°2["Ei"~ ~~ "o
~
~1 . . . . . . . . . . . . . . . . . . . . . . . . . . .
GRAS
MATE
I Iscenar'° 1Z°o L3 "~cenari° 2 ZDD'i3f ~ ~---
~' 10 East.
RIGA
i .... ' ...............
GRAS
' ...........
MATE
'. ..............
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
~. 5I o ............................................................. I-1~ ~ ~ ~ ~ Up
0200
..........
'
400
'
600
800
Baseline length
......
'
1000 [km]
......... '
1200
I 1400
Fig. 3: Interval radii (DET) for the coordinate components of 7 EUREF stations [mm] obtainedfrom Eq (11).
In Fig. 2 and Fig. 3 PROB and DET are depicted with different scales. Their comparison shows that the magnitudes of both uncertainty measures for the horizontal components are similar for short baselines. For the vertical component, the interval radii are more than twice the standard deviation. Due to the strong dependency of the interval radii on the baseline length they dominate the standard deviations for long baselines in all components.
4 Discussion
200
400
600
800
1000 Baseline length [km]
1200
1400
Fig. 2: Comparison of the standard deviations of the coordinate components of 7 EUREF stations for both scenarios [mm]. Grey." PROB for S1, Grey+Black: PROB for $2.
Table 3. Comparison of standard deviations and interval radii for the shortest considered baseline POTS-BOR1 (min. values) and the longest baseline POTS-MATE (max. values) in [mm]
N E U
standard deviation S1($2) min [mm] max[mm] 1.48 (1.49) 1.59 (1.86) 1.12 (1.14) 1.18 (1.30) 7.24 (7.50) 8.21 (8.99)
interval radius S1 min [mm] max [mm] 1.16 12.36 2.17 6.5 14.68 26.15
In order to explain these results the composition of the vcm of the observations is analysed. Here, the consideration of two epochs is sufficient. For the initial ZD observations, four types of correlations are generated which have different impact: (1) Station specific effects (PCV, receiver clocks) lead to strong correlations between observations at one particular station. (2) Satellite specific effects like orbit and satellites clock errors correlate observations to the same satellite over all stations. (3) Tropospheric and ionospheric effects correlate all observations. (4) The additional consideration of an initial, diagonal vcm decorrelates the observations.
Chapter 22 • A Comparative Analysis of Uncertainty Modelling in GPS Data Analysis
Table 4 shows the magnitudes of the contributing matrices in terms of their Frobenius norm. It is obvious and consistent with the analysis of real data that the receiver clock error dominates the uncertainty, cf. fig. 5. In addition, the ZD on L1 and L2 are highly correlated. Table 4. Magnitude of the vcm and contributing parts in log 10 Contribution
total initial vcm orbit satellite clock troposphere ionosphere station specific receiver clock
Frobenius norm of L 1 ZD 1.0166e7 135 4.70e4 0.4 9.89e3 451.1 40.6 1.01 e7
Frobenius norm of L 1 DD 596 527 137 0 118 5 0 0
Frobenius norm of L3 DD 5.4e3 5.4e3 678 0 770 26 0 0
one particular epoch and between successive epochs are obvious. Figures 6 and 7 show the resulting vcm and correlation matrix after double differencing. Since the partial derivations with respect to the receiver and satellite clocks errors are time invariant, their contributions to the vcm are cancelled out, cf. 3 rd column of Table 4. In addition, the time variable parts of, e.g., ionosphere, troposphere and orbit errors are reduced. Hence, the contribution of the initial vcm is increased during the double differencing and finally dominates the resulting vcm. If only the additional vcm is considered (see Eq. (4)), L1 and L2 observations are still highly correlated.
BOR1
•
•
..
,, • , i
L1 epoch1
i
L1 epoch2
L2 epoch1
i L2 epoch2
Fig. 6: Magnitude [logl0] of the elements of the vcm 100
,
BOF~I '~'-..:"-..:'...'-.."- "-.,' epoch1
WTZR ['..,""%"-..."-..,"-., "-.."~
epoch2
Fig. 4: Magnitude [logl0] of the elements of the vcm of L1 ZD
O.SA.'-"-.?"-.X"..."-. GRAZ ;"-..'.. "..%,,,,"-. % %R,GA"-.::"-.."-..'..... . i-"
80
GRAS "" ,'%" ," . . . . ""~'-, MAT E -'%,%, "%, %., BOR1 WTZR ONSA GRAZ RIGA GRAS MATE BOR1 WTZR ONSA GRAZ RIGA GRAS MATE BOR1 WTZR ONSA GRAZ RIGA GRAS MATE
60 40
20
mm
0
",. " , " , ",, " . " . "m "mm " . "~, ~'m
) -20
-40 _
....3", ?-,",~.":"-i"-."-.i ..... I
I
I
L1 epoch1
L1 epoch2
L2 epoch1
Fig. 7: Correlation matrix associated with
epoch1
.~"
%"'-3"~
L2 epoch2
-60
-80 -100
~-"DD'Cf. Eq. (6)
epoch2
Fig. 5: Correlation matrix of L1 ZD associated with
~2ZD,cf.
Eq. (5)
Figures 4 and 5 show the vcm and the corresponding correlation matrix of original L1 ZD observations. The very high correlations at each station at
Finally, the additional application of the L3 ionosphere-free linear combination reduces the magnitudes further; see Fig. 8. This is due to the high correlation between L 1 and L2 DD. The correlation structure of the vcm 12DD,L3 of DD L3 differs only slightly from the one obtained from uncorrelated
141
142
S. Sch6n • H. Kutterer
Z D ( XDD,L3 0 )" Hence, the final standard deviations of the resulting coordinates are close to those obtained without considering remaining systematic errors.
ters. Those studies should be carried out for different baseline lengths, geographic regions, etc. This will also allow to better quantify the magnitudes of uncertainty of the estimated coordinates.
Born % %. "". %'. "% ""- ""-
References
...'-...'-%'-.. "-.. "... ".. ' .= =%==%===% =%= %% =%= =.
-., ,.. .?-.?-.i"..i"-..',?-.."-. _-,-.., "-.. %. "%'%'_--...%..'%.~-,.%...'-%.%,
=%= ==== %% =%= ======% "~=% ==%%==%%==%%==%%== "-.._"%"-. .. -.. %. "; •.% ===.=%=% %% "% ". .. ======== =====~== ==m===
BOR1
_
%=
_
•
% "-% % %% == "%% =% " ~ " . .== %%= . ==%•-..."-...%% "-...'%.. "%."...~ •
%.',...',... "....'.... %..'% I
L3epoch1 L3epoch2 Fig. 8: Correlation matrix associated with ~DD,L3 ' cf. Eq. (9)
5 Conclusions and outlook In this paper, both the probabilistic and deterministic approaches to treat uncertainty caused by remaining systematic errors are compared. For the probabilistic approach two mechanisms are relevant. The uncertainty measure is directly related to and combined with the initial vcm. This initial vcm is increased during the transformation of ZD to L3 DD while the additional vcm is reduced. Considering the intrinsic quadratic error propagation this leads to results (extended vcm) which are very close to those obtained when uncorrelated ZD were used. In general these results are considered to be too optimistic. The deterministic approach provides an adequate and alternative way for the handling of uncertainty caused by remaining systematic errors. Due to its intrinsic linear propagation the uncertainty measures are less optimistic.
The present study indicates the reduced adequacy of a purely probabilistic uncertainty concept. However, the findings have to be verified through the analysis of real GPS data. So future work has to focus on the determination of both, the actual correlation patterns in GPS data as well as the maximum magnitudes (interval radii) of the influence parame-
Alefeld G. and Herzberger J. (1983): Introduction to Interval Computations. Academic Press, New York. Brunner F.K., Hartinger H., and Troyer L. (1999): GPS signal diffraction modelling: the stochastic SIGMA-D model, Journal of Geodesy, 73:259-267. Han S. and Rizos C. (1995) Standardization of the variancecovariance matrix for GPS rapid static positioning. Geomatics Research Australasia 62:37-54 Hartinger H. and Brunner F.K. (1999): Variances of GPS Phase Observations: The SIGMA-e Model. GPS Sol. 2/4:35-43. Hofmann-Wellenhof B., Lichtenegger H., and Collins J. (2001): GPS Theory and Practice, 5 ed, Springer Wien New York. Howind J. (2005): Analyse des stochastischen Modells von Trdgerphasenbeobachtungen. Dissertation. DGK C 584. Mtinchen. J~iger R. and Leinen S. (1992): Spectral Analysis of GPSNetworks and Processing Strategies due to Random and Systematic Errors. In: Defence Mapping Agency and Ohio State University (Eds.) Proceedings of the Sixth International Symposium on Satellite Positioning, Columbus/Ohio (USA), 16-20 March. Vol. (2), p. 530-539. Koch K.-R. (1999): Parameter Estimation and Hypothesis Testing in Linear Models (2d Ed.). Springer, Berlin. Sch6n S. (2003): Analyse und Optimierung geoddtischer Messanordnungen unter besonderer Beriicksichtigung des Intervallansatzes. Dissertation. DGK C 567. Mtinchen. Sch6n S. and Kutterer H. (2003): Imprecision in Geodetic Observations- Case Study GPS Monitoring Network. In: Stiros S. and Pytharouli S. (Eds.): Proceedings of the 1 lth: FIG Inter national Symposium on Deformation Measurements. Geodesy and Geodetic Applications Lab., Patras University. Publication No.2: 471-478. Sch6n S. and Kutterer H. (2005): Realistic uncertainty measures for GPS observations. In: Sanso F. (Ed.): A Window on the Future, Proceedings of the 36 th IAG General Assembly, 23 rd IUGG General Assembly, Sapporo, Japan, 2003, IAG Symposia series, No. 128, 54-59. Schwieger V. (1996): An Approach to Determine Correlations between GPS monitored epochs. Proceedings of the 8th International Symposium on Deformation Measurements, Hong Kong, 17-26 Tiberius C. and Kenselaar F. (2003): Variance Component Estimation and Precise GPS Positioning: Case Study. Journal of Surveying Engineering 129(1 ): 11-18. Wang J., Satirapod C., and Rizos C. (2002): Stochastic assessment of GPS carrier phase measurements for precise static relative positioning, Journal of Geodesy 76:95-104. Wieser A. (2002): Robust and fuzzy techniques for parameter estimation and quality assessment in GPS. Dissertation. Shaker-Verlag, Aachen. Acknowledgement: The first author is recipient of a Feodor Lynen fellowship. He gratefully acknowledges his host Fritz K. Brunner and the Alexander-von-Humboldt-Foundation for their support.
Chapter 23
Looking for systematic error in scale from terrestrial reference frames derived from DORIS data P. Willis Institut G6ographique National, Direction Technique, 2, avenue Pasteur, BP 68, 94160 Saint-Mande, France Jet Propulsion Laboratory, California Institute of Technology, MS 238-600, 4800 Oak Grove Drive, Pasadena CA 91109, USA F.G. Lemoine
Goddard Space Flight Center, Code 697, Greenbelt MD 20771, USA L. Soudarin Collecte Localisation Satellite, parc technologique du canal, 31526 Ramonville Saint-Agne, France
Abstract
1 Introduction
The long-term stability of the scale of Terrestrial Reference Frames is directly linked with station height determination and is critical for several scientific studies, such as global mean sea level rise or ocean circulation with consequences to global warming studies. In recent International Terrestrial Reference Frame (ITRF) solutions, the DORIS technique was not considered able to provide any useful information on scale (derived from VLBI and SLR). We have analyzed three different DORIS time series of coordinates (GSFC, IGN/JPL, LEGOS/CLS) performed independently using different software packages. In the long-term, we show that the DORIS technique, due to its very stable and geographically distributed network, has extremely good stability (70 deg) 2.7 cm 3.6 cm 3,2 cm
Typically over 50000 laser residuals were tested for ENVISAT in 2004 (56484 for GSFC, 51623 for IGN/JPL and 47693 for LEGOS/CLS). In order to better test the radial component a subcategory was also analyzed, selecting only SLR residuals at high elevation (over 70 degrees). This subset already comprises a lot of data points: 1347 for GSFC, 1248 for IGN/JPL and 1141 for LEGOS/CLS. The number of points is slightly different for each group, because there is an edit criterion imposed at 3-hours at start that affects the three groups in a different manner. Furthermore, some groups did not submit all possible arcs and, usually arcs around maneuvers are not considered at all. It can be seen that GSFC and LEGOS/CLS provide better results in all cases and especially when all residuals are considered (high and low
DORIS Data
elevation together). However, in the case of the IGN/JPL solution, the Laser residual test could be altered by the constant along-track offset previously detected. High elevation Laser residuals would not be affected by a timing error but all other SLR residuals would be. These results also suggest that we should extend this study by comparing single satellite orbits from all groups for all satellites, compare them internally (one against another) and test them using other source of information when available (SLR residuals, altimeter cross-over). Conclusions In order to investigate the stability of the DORIS-derived terrestrial reference frame scale factor, we have analyzed weekly station coordinates and daily orbits obtained by three different groups using three different software and analysis strategies (GSFC, IGN/JPL and LEGOS/CLS). Results show that the TRF scale derived by all groups are significantly affected by satellitedependent biases, even if the IGN/JPL solution seems to be less affected. Typically single satellite TRF solutions can show biases in scale up to almost 10 ppb. However, multi-satellite DORIS solutions show a better agreement with ITRF2000 (typically up to 5 ppb in bias). Seasonal signals are also superimposed in DORIS results with a typical amplitude of 0.5 ppb. However, DORIS provides an excellent long-term stability for scale monitoring (typically 0.05 ppb/yr). Some differences in the analysis strategies were noted and presently it seems that the way the center of phase correction is applied by the ACs could be a possible explanation for differences in results. Preliminary tests on ENVISAT orbit show a 10 cm mis-modeling bias in the along-track component that could also be linked with timing issues. In order to better understand these systematic errors, future tests are needed in which all groups try to use the same analysis strategy. With the recent creation of the International DORIS Service, we hope that future Analysis Centers will join in to perform these tests and discuss these difficult issues. Acknowledgments
Part of this research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration (NASA).
149
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References
Altamimi, Z., P. Sillard, C. Boucher (2002). ITRF2000, A new release of the International Terrestrial Reference Frame for earth science applications. Journal of Geophysical Research, Solid Earth, vol. 107(B10), 2214. Altamimi Z., C. Boucher, P. Willis (2005). Terrestrial Reference Frame requirements within GGOS, Journal of Geodynamics, Vol.
40(4-5), pp. 363-374 Biancale R., G. Balmino, J.M. Lemoine, et al. (2000). A new global Earth's gravity field model from satellite orbit perturbations, GRIM5-S1, Geophysical Research Letters, Vol. 27(22), pp. 3611-3614. Cazenave A., R.S. Nerem (2004). Present-day sea level change, Observations and causes, Reviews of Geophysics, Vol. 42(3), RG3001. Chao B.F., and A. Au (1991), Temporal variation of the Earth's low-degree field caused by atmospheric mass redistribution1980-1988, Journal of Geophysical Research, Solid Earth, Vol. 96(B4), pp. 6569-6575. Cr6taux J.F., L. Soudarin, A. Cazenave, F. Bouille (1998), Present-day Tectonic Plate Motions and Crustal Deformations from the DORIS Space System. Journal of Geophysical Research, Solid Earth vol. 103(B12), pp. 30167-30181. Cr6taux J.F., L. Soudarin, F.J.M. Davidson, M.C. Gennero, M. Berge-Nguyen, A. Cazenave (2002), Seasonal and interannual geocenter motion from SLR and DORIS measurements, comparison with surface loading data, Journal of Geophysical Research, Solid Earth, Vol. 107(B12), 2374. Feissel-Vernier M., J.J. Valette, K. Le Bail, L. Soudarin (2005). Impact of the GRACE gravity fields models on IDS products,
International DORIS Service Report. Gerasimenko M.D., A.G. Kolomiete, M. Kasahara, J.F. Cr6taux, L. Soudarin (2005). Establishment of the three-dimensional kinematic reference frame by VLBI and DORIS global data, Far Eastern
Mathematical Journal. Lemoine F.G., et al. (1998). The development of the joint NASA/GSFC and the National Imagery and mapping Agency (NIMA) geopotential models, EGM96, in NASA, TP1998-206861, pp. 1-575. Lemoine J.-M., H. Capdeville (submitted). Correction model for the SAA effect on Jason/DORIS data. Journal of Geodesy
Luthcke S.B., N.P. Zelensky, D.D. Rowlands, F.G. Lemoine, T.A. Williams (2003). The 1centimeter orbit, Jason-1 precision orbit determination using GPS, SLR, DORIS, and altimeter data, Marine Geodesy, Vol. 26(3-4), pp. 399-421. Meisel B, D. Angermann, M. Krugel, H. Drewes, M. Gerstl, R. Kelm, H. Muller, W. Seemuller, V. Tesmer (2005). Refined approaches for terrestrial reference frame computations, Advances in Space Research, Vol. 36(3), pp. 350-357. Morel L., and P. Willis (2005). Terrestrial reference frame effects on mean sea level rise determined by TOPEX/Poseidon, Advances in Space Research, Vol. 36(3), pp. 358-368. Rummel R., H. Drewes, G. Beutler (2002). Integrated Global Observing System IGGOS, A candidate IAG Project, In Proc.
International
Association
of
Geodesy,
Vol. 125, pp. 135-143. Sillard P., and C. Boucher (2001). A review of algebraic constraints in Terrestrial Reference Frame datum definition, Journal of Geodesy, Vol. 75(2-3), pp. 63-73. Soudarin L., J.F. Cr6taux, A. Cazenave (1999). Vertical Crustal Motions from the DORIS space-geodesy system, Geophysical Research Letters, Vol. 26(9), pp. 1207-1210. Tapley B.D., S. Bettadpur, M. Watkins, C. Reigber (2004), the Gravity Recovery and Climate Experiment, Mission overview and early results, Geophysical Research Letters, Vol. 31 (9), L09607. Tavernier G., H. Fagard, M. Feissel-Vernier, F. Lemoine, C. Noll, J.C. Ries, L. Soudarin, P. Willis (2005). The International DORIS Service, IDS. Advances in Space Research, Vol. 36(3), pp. 333-341. Tregoning P., T. van Dam (2005). Effects of atmospheric pressure loading and sevenparameter transformations on estimates of geocenter motion and station heights from space geodetic observations, Journal of Geophysical Research, Solid Earth, Vol. 110(B3), B03408. Webb F., and J. Zumberge (Eds.) (1995). An introduction to Gipsy/Oasis II, Report Jet Propulsion Laboratory, Pasadena, USA, JPLM D- 11088. Willis P., and M. Heflin (2004), External validation of the GRACE GGM01C Gravity Field using GPS and DORIS positioning results, Geophysical Research Letters, Vol. 31(13), L13616.
Chapter 23 • Looking for Systematic Error in Scale from Terrestrial Reference Frames Derived from DORIS Data
Willis P., and J.C. Ries (2005), Defining a DORIS core network for Jason-1 precise orbit determination based on ITRF2000, methods and realization. Journal of Geodesy, Vol. 79(6-7), pp. 370-378. Willis P., B. Haines, J.-P. Berthias, P. Sengenes, J.L. Le Mouel (2004), Behavior of the DORIS/Jason oscillator over the South Atlantic Anomaly. Comptes Rendus Geoscience, Vol. 336(9), pp. 839-846.
Willis P., C. Boucher, H. Fagard, Z. Altamimi (2005a). Geodetic applications of the DORIS system at the French Institut Geographique National. Comptes Rendus Geoscience, Vol. 337(7), pp. 653-662. Willis P., Y.E. Bar-Sever, G. Tavernier (2005b), DORIS as a potential part of a Global Geodetic Observing System. Journal of Geodynamics, Vol. 40(4-5), pp. 494-501.
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Chapter 24
WVR calibration applied to European VLBI observing sessions Axel Nothnagel, Jung-ho Cho 1 Geodetic Institute of the University of Bonn, Nussallee 17, D-53115 Bonn, Germany 1on leave from Korean Astronomy and Space Science Institute, 61-1, Whaam-Dong, Youseong-Gu, Taejeon, Rep. of Korea 305-348 Alan Roy Max-Planck-Institute for Radio Astronomy, Auf dem Hiigel 69, D-53121 Bonn, Germany Riidiger Haas Onsala Space Observatory, Chalmers University of Technology, SE-439 92 Onsala, Sweden
Abstract. From a conceptual point of view water vapour radiometers (WVRs) are ideal instruments for a direct determination of the water vapour content in the atmosphere above a space geodetic observing site. For various reasons the application of WVRs for a direct correction of geodetic Very Long Baseline Interferometry (VLBI) observations for the wet component of atmospheric refraction has not been brought to a stage where it could be employed routinely. The installation of a new type of WVR at the 100 m Effelsberg radio telescope in Germany opens up new possibilities for a direct correction of the VLBI observations using line-ofsight WVR observations. The WVR has been in operation during two geodetic VLBI sessions of a purely European network. In addition, some WVR data at Onsala have been revisited to embark on a project which aims at applying wet delay corrections from WVRs to VLBI sessions of the European geodetic VLBI network for the Effelsberg, Madrid, Onsala and Wettzell sites. In this paper we present the analysis and results of a few examples of Effelsberg and Onsala observing sessions.
Keywords. Water Vapour Radiometers (WVR), atmospheric refraction, Interferometry (VLBI)
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1 Introduction The rapid variations of water vapour in the atmosphere, both in time and space, belong to the list of factors which limit further progress of
improving the accuracy of space geodetic results based on radio frequency observations. A direct determination of the excess delay caused by water vapour in the atmosphere (wet delay) can be carried out through measurements of the sky brightness temperatures using water vapour radiometers (WVRs) (e.g. Elgered et al. 1982). However, routine operations of WVRs for a direct correction of space geodetic observations like Very Long Baseline Interferometry (VLBI) or GPS have not been achieved due to various unsolved problems. Since early 2003 a new instrument has been in operation at the 100 m radio telescope of the MaxPlanck-Institute for Radio Astronomy at Effelsberg, Germany, which has been constructed with an improved architecture overcoming a number of shortcomings of existing WVRs (Roy et al. 2003). This WVR is installed on the prime focus cabin of the telescope and operates along the line-of-sight taking brightness temperature measurements by sweeping the full frequency band between 18.3 GHz and 26.0 GHz in 25 steps of 900 MHz each. An example spectrum is shown in Fig 1. The radiometer was employed to carry out observations during two geodetic VLBI sessions of the European geodetic VLBI network (Campbell and Nothnagel, 2000) in December 2004 and March 2005. During this period, radiometers were operated at the VLBI telescopes at Madrid Deep Space Complex (Spain), Wettzell Geodetic Fundamental Station (Germany) and Onsala Space Observatory (Sweden). This favourable constellation of four sites equipped with WVR triggered the idea of investigating the current status and future prospects for regular use of WVR data in VLB! data analysis.
Chapter 24 • WVR Calibration Applied to European VLBI Observing Sessions
Here, we put special emphasis on the results of the first Effelsberg observations. In addition, a few older sessions of Onsala are revisited in order to discuss general problems with WVR calibrations on the basis of a case study. Effelsberg WVR Spectrtlnl Measured 2005mar22 17:32:47 UT
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observations for instrumental effects. The gain was measured inserting an ambient temperature load and assuming the sky as a cold load. Since the sky is not perfectly cold, its temperature was measured using a tipping-curve. Furthermore, a water vapour saturation correction has been applied to the Effelsberg data applying the linearization method according to Claflin et al. (1978). As a result, the correction of the brightness temperature can be determined with a linear relationship between the brightness temperatures and the air mass values (Elgered 1993).
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The final aim of reliably applying WVR corrections to VLBI observations will serve the purpose of eliminating the need for estimating zenith wet delay (ZWD) parameters in the VLBI analysis process. This in turn will lead to the benefit that VLBI observations need not be driven to the lowest possible elevation. Today, low elevation observations have to be scheduled for no other reason than to estimate the wet delay parameter as good as possible. If the need for estimating wet delay parameters disappears, so does the need for low elevation observations, at least for those below 7 ° or 8 °. Ultimately, a reduced number of unknown parameters will stabilize the adjustment process and with it the quality of the results.
2 Effelsberg WVR Data Analysis 2.1 Radiometer calibration Water vapour radiometers measure the sky brightness temperatures along their line-of-sight. The first step in WVR data analysis is, thus, the calibration of the raw brightness temperature
The next step is the inversion process to generate delay corrections for the VLBI observables from brightness temperatures. To separate the continuum of liquid water from the water vapour line emission for the determination of the wet delays, a frequencysquared baseline and a van Vleck-Weisskopf water vapour line profile were fitted to the 25 individual frequency channels of the radiometer output, following Tahmoush and Rogers (2000). In addition, for comparison wet delays were also computed applying the Resch (1983) model adapted for different sets of frequency pairs. Ideally, radiosonde observations should be used to support the determination of the conversion coefficients. However, in the absence of radiosonde information standard model coefficients like the ones developed by Resch for the North American continent may serve as a suitable starting point. The first WVR observations with the new Effelsberg radiometer were made during the Euro74 VLBI session on December 14 and 15, 2004. Unfortunately, a few channels of the WVR malfunctioned during the session which happened to take place just at the beginning of its operation. In order to investigate the implications of missing channels the Resch (1983) model was first applied in a slightly modified version with two healthy channels only. In addition, we also applied the Tahmoush and Rogers (2000) model regardless of the missing channels. Figure 2 depicts the resulting zenith wet delays for Effelsberg during this session for both types of inversions. Here, it becomes obvious how important it is that the full spectrum be sampled and what effect the inversion process has because the resulting wet delays are rather different (see Fig. 2 and 3 bottom parts). Since the WVR did
153
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not work to its expectations during the Euro-74 session; further analysis of this data set was abandoned.
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The difference of the Resch and the Tahmoush and Rogers models (Fig. 3 bottom part) show a similar or even larger level of disagreement as in Fig. 2. It is largest where the disagreement between the wet delay estimates and the WVR inversion has a maximum as well. The other features of the WVR inversions in Figure 3 (top part), the apparent positive outliers, originate from clouds due to additional water vapour they contain and perhaps due to imperfect removal of the continuum emission from liquid water droplets in the clouds. This has been verified with a video camera viewing the same line-of-sight as the radiometer. During stability tests the video pictures showed that the passage of clouds through the radiometer beam produced increases, sometimes dramatic, in the spectral baseline level.
20
Time (hour) Fig. 2 Top part: Zenith wet delays derived from WVR at
Effelsberg in the Euro-74 session on Dec. 14 and 15, 2004. Middle part: Median filtered curve of WVR observations and VLBI estimates (both offset by -100 mm for better identification). Bottom part: Differences of raw observations with Tahmoush/Rogers model minus modified Resch model (offset by-200 mm)
The second geodetic VLBI session with participation of Effelsberg on March 22 and 23, 2005 (Euro-75) produced much better WVR results as may be interpreted from the agreement of the wet delay estimates and the WVR inversion (Fig. 3 middle part) however with a noticeable deviation at around 5.00 UT.
3 0 n s a l a WVR Data Analysis At Onsala the WVR operates at two frequencies (20.7 GHz and 31.4 GHz) and the observations are calibrated and inverted to delay corrections using the Johansson algorithm (Johansson et al. 1993). This algorithm is based on empirical coefficients which have been determined with parallel observations by radiosondes. The sky is regularly sampled with a special observing schedule in order to produce a time series of zenith wet delays. In the analysis, a gradient model has been implemented according to Davis et al. (1993). In 1991 the calibration loads for the reference temperatures were changed and in 2001 the WVR underwent a major upgrade.
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In state-of-the-art VLBI data analyses the hydrostatic component of refraction of each individual observation is corrected by applying a hydrostatic zenith delay (e.g. Davis et al. 1985) which is transformed onto the respective elevation using a hydrostatic mapping function (e.g. Niell 2001). The wet component is estimated applying a similar mapping function tailored for the wet path delay. In order to model time-dependent variations either a stochastic process is assumed and a filter adjustment such as the Kalman filter is applied or the variations are considered to be of a less variable type and the model consists of piece-wise linear elements.
Chapter 24 • WVR Calibration Applied to European VLBI Observing Sessions
Instead of estimating the wet component, WVR corrections can be applied. Zenith wet delay corrections have to be mapped to the respective elevation of the observations. If WVR corrections for the wet component are available, it should not be necessary any more to estimate the wet component. Likewise, estimates of the wet component with WVR corrections applied to the VLBI data should be zero within their error margins provided that hydrostatic and ionospheric refraction effects are modelled correctly. So far, a small number of VLBI sessions observed with the European geodetic VLBI network has been processed in this way to provide the basis for a case study. In order to discuss the quality of the WVR corrections, zenith wet delays have been estimated in three different ways employing the Calc/Solve VLBI analysis software with piece-wise linear functions of one-hour segment lengths:
a)
b)
c)
standard solutions have been computed by correcting the hydrostatic component with surface meteorological data and estimating zenith wet delay parameters in the usual way with rate changes at every hour (see Fig. 4, standard). in the second type of solutions we corrected the hydrostatic part in the usual way and also corrected the wet part with WVR wet delays applying the Resch model and still estimated the zenith wet delays as in case a). the third type of solutions differs from case b) only in that the Tahmoush and Rogers model is applied.
5 Results In Figure 4 the zenith wet delay results of the standard solution (a) give a very good indication of the variations which are to be expected during the 24-hour observing period. Curves (b) and (c) depict the residual zenith wet delay estimates if WVR corrections are applied beforehand. Calibrating the VLBI observations with WVR delay corrections and estimating zenith wet delays at the same time in the analysis process is a method used only within this investigation and should normally not be necessary. If the WVR delay corrections describe the wet refraction effect completely and no other systematic effects are absorbed in estimates of the wet delays, the residuals should be zero throughout. It should be emphasised that this assumption is only valid if the estimates of the zenith wet delays truly represent just the water vapour refraction effect and not any additional contamination with a similar signature. In the case of the Effelsberg observations in the Euro-75 session the residual wet delays after WVR calibration are still significant for both inversion models (Fig. 4, lines b and c) at the level of abut 15 20 mm RMS, with no model appearing superior to the other. -
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This type of analysis with its graphical representation in figures 4, 5 and 6 have the advantage that the effects of the least squares estimation with the correlations between all parameters (e.g. also for the clock effects) are correctly transferred to the wet delay estimates. This would not be the case when the observations had not been corrected for the wet delays beforehand. In all sessions analysed here Wettzell was kept fixed at its ITRF2000 coordinates and all other site coordinates were estimated on a session by session basis without constraints.
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A.L. Nothnagel.J. Cho.A. Roy. R. Haas
always have to be positive. There are three possible causes for this phenomenon: the WVR data in its present state over-calibrates the wet component at this time of the observations, or the hydrostatic correction was insufficient, which is less likely, or the wet delay estimates compensate other unmodelled errors. Since these cases occur predominantly at the period of the most rapid changes in the water vapour content it is conceivable that the conversion from line-of-sight data to zenith path delays introduced an additional uncertainty which we will investigate further.
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In addition to the sessions with Effelsberg we would like to discuss two other VLBI observing sessions, Euro-22 on December 28/29, 1994 and Euro-72 on July 13/14, 2004, in which we applied and investigated Onsala WVR delay corrections. They were processed in the same way as in the case of Effelsberg except that here only the Johansson et al. (1993) inversion model was applied. The zenith wet delays estimated after WVR corrections were applied to the VLBI data in the Euro-22 session are predominantly negative (Fig. 5) whilst those of the Euro-72 session are very close to zero (Fig. 6). From the latter fact we deduce that the WVR wet delay corrections very well match the respective estimates, the only exceptions being those data points where the standard wet delay estimates show obvious deviations.
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Time (hour) Fig. 5 VLBI-derived ZWD at Onsala for the Euro-22 session employing two different analysis approaches: a) top line: standard solution, b) bottom line: applying WVR corrections with the Johansson et al. (1993)model.
Without trying to further interpret these interim results we now look directly at the baseline repeatability. As a representative example we just look at the baseline Wettzell - Onsala here without using Wettzell WVR data. When applying WVR calibration at Onsala instead of estimating the zenith wet delay there is no noticeable change for the baseline length determined with the Euro-72 session (Fig. 7 and 8). This neutral behaviour of the Euro-72 session can clearly be attributed to the fact that the WVR wet delays match the estimated zenith wet delays as can bee seen in Figure 6 and little change should thus be expected. However, this situation does not imply that this agreement is the desired standard case because we then would not need any WVR observations.
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.-~)3 .~,
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Fig. 7 0 n s a l a - Wettzell baseline length results with standard solution (i.e. wet delays estimated).
Chapter 24 W VR ~ l u t i on, W ettzell-On sala
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Fig. 80nsala - Wettzell baseline length results with WVR wet delay calibration applied at Onsala for Euro-22 and Euro-72 sessions.
• WVR
Calibration Applied to EuropeanVLBI Observing Sessions
spectral channels to allow a more refined separation of cloud liquid water and water vapour components. Depending on the ongoing performance tests this technology is well placed to play a leading role for future installations. The brightness temperature results are of a state-of-the-art quality. Currently, the inversion algorithm is being adapted to geodetic requirements of mm-level accuracy and a robust data flow is being developed. Similar activities for data of Wettzell, Madrid and Onsala will prepare the grounds for a better usage of W V R monitoring in VLBI data analysis.
References In the case of the Euro-22 session the baseline length Onsala - Wettzell changes significantly if W V R corrections are applied at Onsala instead of estimating zenith wet delays. This is not unexpected since the application of W V R corrections with simultaneous wet delay estimation had produced a significant non-zero wet delay estimate (Fig. 5). It is not necessary here to look at statistical evidence but from the visual inspection of the baseline length graph alone it may already be concluded that the change has happened to the better.
6 Conclusions This study has shown that those VLBI sessions where the VLBI estimates and the W V R measurements disagree significantly are most promising for further insights into the problem of water vapour refraction effects in geodetic VLBI. The question of why the wet delay estimates are so different from W V R delay corrections in some of the sessions is still open. With four radiometers available at VLBI observatories in Europe there is ample scope for analysis of VLBI data where both ends of a baseline are equipped with WVRs. However, each VLBI session has to be analyzed in detail individually in order to figure out possible reasons for the discrepancies between estimates and measurements. The new water vapour radiometer at Effelsberg provides the lowest thermal noise of any radiometer used so far for geodetic applications. It offers many
Campbell J., A. Nothnagel (2000). European VLBI for Crustal Dynamics. J. of Geodynamics, 30, pp. 321-326 Claflin E.S., S.C. Wu and G.M. Resch (1978). Microwave Radiometer Measurement of Water Vapor Path Delay: Data Reduction Techniques. DSN Progress Report 42-48, pp. 22-30 Davis J., T.A. Herring, I.I. Shapiro, A.E.E. Rogers, G. Elgered (1985). Geodesy by Radio Interferometry: Effects of atmospheric modelling errors on estimated baseline lengths. Radio Science, 20, pp. 1593-1607 Davis J., G. Elgered, A.E. Niell, C. Kuehn (1993). Groundbased measurement of gradients in the "wet" radio refractive index of air. Radio Science, 28, pp. 1003-1018 Elgered G., B.O. R6nn/ing, J.I.H. Askne (1982). Measurements of atmospheric water vapor with microwave radiometry. Radio Science, 17, No. 5, pp. 1258-1264 Elgered, G. (1993). Tropospheric radio path delay from ground-based microwave radiometry, In: Atmospheric remote sensing by microwave radiometry, Chap. 5, Wiley, New-York Johansson, J.M., G. Elgered, J i . Davis (1993). Wet Path Delay Algorithm for Use with Microwave Radiometer Data. Contribution of Space Geodesy to Geodynamics: Technology, Vol. 25, AGU, Washington, D.C., pp. 81-98 Niell, A.E. (2001). Preliminary evaluation of atmospheric mapping functions based on numerical weather models. Phys. Chem. Earth, 26, pp. 475-480 Resch G.M. (1983). Inversion Algorithms for Water Vapor Radiometers Operating at 20.7 and 31.4 GHz. TDA Progress Report 42-76, pp. 12-26 Roy, A i . , U. Teuber and R. Keller (2003). Tropospheric Delay Measurement at Effelsberg with Water-Vapour Radiometry.Proc. 16th Working Meeting on European VLBI for Geodesy and Astrometry, May 9th-10th 2003, Leipzig, Germany, Edited by: V. Thorandt, BKG Leipzig, pp. 53-59 Tahmoush, D.A. and A.E.E. Rogers (2000). Correcting atmospheric path variations in millimetre wavelength very long baseline Interferometry using a scanning water vapor spectrometer. Radio Science, 35-5, pp. 1241-1251
157
Chapter 25
Frontiers in the combination of space geodetic techniques Manuela Kriigel and Detlef Angermann Deutsches GeodStisches Forschungsinstitut (DGFI), Marstallplatz 8, D-80539 Munich, Germany e-mail: kruegel~dgfi.badw.de
Abstract. Co-location sites are one of the key elements in the combination of the reference frames of different space geodetic techniques. They are fundamental for a consistent datum realization of the combined networks. This paper deals with a new strategy for the selection and implementation of local tie information and demonstrates how the ERP as common parameters of the different geodetic space techniques can be used for local tie validation. As one result of the new strategy it is shown, that only a small set of very consistent co-location sites should be used for the combination. Additionally, the capability of a combination using only the lengths of the local tie vectors is investigated. The strengths and weaknesses of this method are characterized. Keywords: Combination methods, space geodetic techniques, local ties, Earth Rotation Parameters
1
Introduction
Rigorous combination of space geodetic techniques has become a more and more expedient method to use the individual strengths of the different techniques for the combined product on the one side and to reduce the influence of their weaknesses on the other side. One iraportant and also complex part in the combination is the connection of the station networks of the different techniques into a unique reference frame. Due to the fact, that no stations are observed by two techniques the combination can only be done by introducing terrestrial measurements performed at so-called co-location sites. The recent global International Terrestrial Reference Frame (ITRF) of the International Earth Rotation and Reference Systems Service (IERS) are realized in this way (Altamimi et al., 2002; Boucher et al., 2004). In 2005 a new realization of the ITRF, namely the ITRF2004, will be com-
puted and published by the IERS. As one of the ITRS Combination Centres, DGFI is involved in this work. Within the TRF realization the selection of terrestrial measurements at co-location sites (so-called local ties) is an important step of work. There are partly large discrepancies between coordinate differences derived from space techniques and the local tie measurements, so that not all available information should be used for the combination (Angermann et al., 2004, Kriigel and Angermann, 2005). Additionally, in most cases no co-variance information is available for the local station networks. Hence a careful selection of suitable local ties is necessary. While previous ITRS realizations only coiltain station coordinates and velocities, now additional Earth Rotation Parameters (ERP) are included. The ERP provide new possibilities for the validation of co-location sites, which are investigated for the co-locations between a GPS and a VLBI network. As common parameters to both techniques the ERP are very suitable to identify rotations between the station networks caused by terrestrial measurements introduced within the combination. While Ray and Altamimi (2005) analysed co-location sites after combining the ERP we used the uncombined ERP series for local tie validation. In addition, simulation studies are performed to demonstrate a combination strategy using only vector lengths instead of local tie vectors in three components (3D).
2
Input data
For the investigations the data of a global GPS and a VLBI network are used. For GPS six years (1999-2004, 302 weeks) of the time series of weekly data are available, which are provided by the IGS for the IERS Combination Pilot Project (Rothacher et al., 2005) in SINEX format. These are combined GPS solutions based on individual solutions of up to 10 IGS analysis centers
Chapter 25 • Frontiersin the Combination of Space GeodeticTechniques
and basically come up to the official IGS time series described by Ferland (2004). In the case of VLBI the D G F I time series of normal equations from 1984 to 2004 is used. T h e y comprehend about 2580 sessions of 24 hours of observing time. The normal equations contain station coordinates and ERP. T h e y are combined to technique specific multi-year solutions using the full variance co-variance information. A detailed description of the combination strategy is given in Meisel et al. (2005). We concentrate on three years (2002-2004) of ERP, since this is sufficient for the purpose of the study and the a m o u n t of p a r a m e t e r s becomes very large using the complete E O P time series. For the GPS and VLBI station networks 26 colocations do exist, which are listed in Table 1. All local ties are provided in the I T R F 2 0 0 0 d a t u m .
A second set of 17 local ties was selected, showing residuals smaller t h a n 2.2 cm in positions and 4.5 m m / a in velocities (Ny Alesund, Onsala, Medicina, Noto, Madrid, Wettzell, Tsukuba, Hartebeesthoek, Algonquin Park, Kokee Park, Westford, N o r t h Liberty, M a u n a
Table 1: Residuals of positions resulting from the 14 p a r a m e t e r H e l m e r t - t r a n s f o r m a t i o n between the VLBI and the GPS networks. Stations with a residual of larger t h a n 8 m m in one c o m p o n e n t are not used for the transformation. T h e y are identified by a *
positions [mm]
3
Selection
of co-location
sites
The analysis of the co-location sites for VLBI and GPS was performed using a 14 p a r a m e t e r H e l m e r t - t r a n s f o r m a t i o n between the GPS and the VLBI network. Therefore, the local tie vector was added to the VLBI station coordinates to obtain identical stations for the transformation. Only stations with residuals smaller t h a n 8 m m in each c o m p o n e n t are used for the transformation. Based on the residuals of the transformation (shown in Table 1 and Table 2) a set of high-quality local ties was identified. T h e residuals show a good agreement (smaller t h a n 8 ram) for 9 stations: Ny Alesund, Onsala, Wettzell, T s u k u b a , Hartebeesthoek, Santiago, Westford, M a u n a Kea and Saint Croix. The residuals of the velocities are smaller t h a n 2.5 m m / a . The thresholds of 8 m m and 2.5 m m / a were found empirically with the precondition t h a t at least a set of 6-7 sites with good ties should be selected. The estimated Helmertt r a n s f o r m a t i o n p a r a m e t e r s differ significant from zero within the 3 sigma significance level. T h e y are not given here, because they are not interpretable, since the technique solutions have only a loose d a t u m . The global distribution of the 9 stations is shown in Fig. 1.
station Ny Alesund Onsala Medicina Noto (NOTO)* Noto (NOT1)* Madrid* Yebes Wettzell Tsukuba Hartrao Algonquin Goldstone* Fairbanks* Kokee* Westford* Fort D avis Pietown* North Liberty* Mauna Kea Fortaleza* Santiago Saint Croix Tidbinbilla* Hobart* Syowa* O'Higgins*
(1) (2) (3) (4) (5)
north -0.2 -0.5 7.1 18.1 17.6 0.1 -4.4 3.9 2.4 1.8 3.0 11.3 13.8 2.1 3.3 5.6 7. 7 0.7 1.1 4.2 -2.7 -0.3 3.3 9.4 11.6 -1.9
east -3.0 -1.4 6.2 9.9 5.6 4.5 -2.3 -1.8 -0.7 -1.2 4.3 3.4 2.8 0.9 -8.3 2.3 10.2 1.4 0.2 3.3 -0.4 -2.2 -5.6 -14.9 -3.4 10.8
up
Source
2.4 -6.8
(2) (1)
-2.9
(3)
1.2 lo.3 -16.2 ~.6 -2.4 2.8 4.1 -4.8
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22.8
(1)
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-1.o
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-o.o
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IGN d a t a b a s e ( ftp: / /lar eg. e ns g. i gn. fr / p u b / itrf/ie r s. e cc ) Steinforth et al. 2003 Sarti et al. 2004 Matsuzaka et al. 2004 J o h n s t o n et al. 2004
159
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M. Kr~igel • D. Angermann
Figure 1" VLBI network (triangles" 9 selected stations with good co-location to GPS)
Table 2" Residuals of velocities resulting from a 14 parameter Helmert-transformation between VLBI and GPS. Stations identified by a * are not used for the transformation (see caption of Tab.l)
Kea, Santiago, Saint Croix, Tidbinbilla, Hobart). The criteria for the definition of the thresholds was to select a number of at least five additional sites to the 9 selected above. Generally, it can be seen that the largest discrepancies occur in the height component (see Table 1). This is consistent with the fact, that in particular for GPS and VLBI the station heights do have a higher uncertainty compared to the horizontal components. In Thaller et al. (2005) comparable results are shown for the 14-day continuous VLBI campaign CONT02. So far only the local tie vectors with seperate components are taken into account. In Fig. 2 the corresponding length of the difference vector ld between the local tie derived from terrestrial measurements and from the space techniques after the Helmert-transformation: ld -- v / d x 2 + @ 2 + dz 2
with:
velocities [mm/a] station Ny Alesund Onsala Medicina Noto (NOTO)* Noto (NOT1)* Madrid* Yebes Wettzell Tsukuba Hartrao Algonquin Goldstone* Fairbanks* Kokee* Westford* Fort Davis Pietown* North Liberty* Mauna Kea Fortaleza* Santiago Saint Croix Tidbinbilla* Hobart* Syowa* O'Higgins*
dx dy -dz -
-
I north 0.5 -0.0 0.0 -0.0 0.2 -1.4 -0.1 -0.0 -0.2 -1.7 -0.4 -0.3 0.1 -0.6 -1.3 0.2 1.2 -0.6 0.9 -2.0 -0.5 -0.1 0.6 1.1 -3.5 2.2
east -0.2 0.0 0.0 -0.3 -0.3 0.1 0.1 0.4 0.5 0.9 -0.7 -0.7 -0.4 -0.1 -0.9 -0.8 0.6 -0.6 -0.8 -1.1 1.7 -1.1 0.2 0.8 1.5 -0.0
up 1.1 -0.6 -0.9 1.6 1.2 0.1 4.5 -0.1 -1.5 -1.7 0.6 0.9 2.0 -1.8 -2.0 0.2 1.5 1.0 -1.0 -1.1 1.5 -2.0 1.8 2.6 -7.1 2.2
(1)
Z2kXspac e -- A Xterr A y s p a c e -- A Y t e r r A Zspace -- A Zterr
is compared to the difference between the vector lengths: dl
i
2 2 2 A X s p a c e -+- A Y s p a c e -+- A Z s p a c e
I A x ~ + / x y~
+ Az~ ).
(2)
The second quantity dl represents the difference of "local ties" containing only information about the distances between the stations. This means, that the datum definition (orientation) in ITRF2000 and the information about the network geometry are removed from the local tie. It is clear, that the difference between the vector lengths is smaller in all cases, but for some stations it is extremly smaller, e.g. for Hobart. Possible reasons could be: • For most co-locations the horizontal extension of the local network is much larger than the vertical. Hence a discrepancy between space techniques and terrestrial measurements in the height component has a much smaller effect on the distance between the stations as on the difference vector. For example, for a local network with a horizontal extension of 100 m, a discrepancy in the
Chapter 25
•
Frontiers in the Combination of Space Geodetic Techniques
Table 3: Solution types 20~15-~ ~10-
i
,I, i i . I o_LLi I l i L L I Ii ..ILl • differenceofvectors
• differenceofdistances
Figure 2: Differences between space geodetic solutions (VLBI and GPS) and local tie measurements. Compared are the length of the difference vector and the difference of the vector lengths. Only stations with a length of the difference vectot smaller than 2.2cm are displayed.
height component of 10 cm affects the vector length by only 0.04 mm. • Secondly, a possible rotation of the local network with respect to the space technique solutions is eliminated because no datum information is included in the tie information any more. Thus, it can be concluded that the use of distances between stations instead of local ties in three components minimizes the problems in height component and eliminates errors in the orientation. However, the information content, which can be used to connect the networks is reduced compared to the conventional 3D case and more co-location sites are necessary to ensure stable combination results.
4
Analysis of co-locations using ERP
In the section above sets of 9 and 17 local ties are identified on the basis of the results of a Helmerttransformation. The ERP provide additionally information to validate local ties. Based on the same GPS and VLBI input data we analysed and compared these sets of local tie information with respect to the following two criteria: • The consistency of the combined reference frame should be maximized. • Secondly, the deformation of the networks due to the combination should be minimized.
solution discription type SING stand-alone VLBI and GPS solution DIST vector lengths of all 26 VLBI-GPS local ties 1YD3 3D local ties of 17 sites 9D3 9 good 3D local ties (Ny Alesund, Onsala, Wettzell, Tsukuba, Hartrao, Westford, Santiago, Mauna Kea, St. Croix)
To investigate the first criteria the station networks are combined but not the EOP. The offsets between xpole and ypole values from VLBI and GPS are a measure of the achieved consistency (unless there are systematics in the ERP estimation of one of the techniques). To quarttify the deformation (second criteria) of the combined network caused by the local ties, the RMS of the transformation between stand-alone technique solutions and the combined solution was estimated. Different solution types (see Table 3) were iraplemented and analysed with respect to the two criteria. The combinations are performed introducing the local tie information as pseudo observations with a priori standard deviations of 0.1 ram, 0.3 ram, 1.0 mm and 3.0 ram. In the case of standalone technique solutions the datum for stations and velocities was defined for GPS by no-netrotation (NNR) conditions w.r.t ITRF2000 and not-net-rotation and not-net-translation (NNT) conditions w.r.t. ITRF2000 for VLBI. For all combined solutions NNR conditions for station positions and velocities were applied using only GPS stations. The weighting factors between both techniques within the combination are 1.0. The results are displayed in Fig. 3, which include the offset in the pole values (upper part) and the RMS values estimated within the transformation (lower part). The RMS values are only displayed for the transformations of the VLBI part of the combined solutions to the stand-alone VLBI solution, because due to a much larger number and the better global distribution of stations the GPS network is much less deformed.
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M. Kr~igel • D. Angermann
The most stable results can be achieved using a set of only a few good and well distributed ties. The RMS values for the solution type 9 D 3 are nearly independent from the chosen a priori standard deviation. This proofs the high consistency of the selected ties. The offset in the pole values are also smaller than those of the other solution types. 150 100 so 0 -50
-1 O0 -150
E -ff ~5
~0 rr
E
E
E
E
~
~
~
E
~
~
E
E
a priori standard deviation of local ties
• •
x-pole offset (VLBI-GPS) RMS of positions
• •
y-pole offset (VLBI-GPS) RMS of velocities
Figure 3: Accuracy evaluation of different solution types for the combination of VLBI and GPS. Top figure: pole offsets. The standard deviations of the pole offsets are lower than 5 #as. Lower figure: RMS values of Helmert-transformation. They are estimated including all stations and all coordinate components. Note, that for better perceptibility the RMS values for the velocities are multiplied with a factor 10. If 17 instead of 9 local ties (solution type: 17D3) are used, the RMS values and thus the deformation of the network increases significantly with decreasing a priori standard deviations. The pole offsets increase as well. This indicates that the quality of the local tie information is obviously more important than the quantity of co-location sites. A combination introducing only the lengths of the local tie vectors (solution type: D I S T ) gives RMS values and pole offsets in the same order of magnitude as the combinations using local ties in three components. It demonstrates that this solution type works in principle. If smaller a pri-
ori standard deviations are used, the RMS values increase, whereas the pole offsets are getting smaller. It is obvious that the connection of the VLBI and GPS networks is too weak for an a priori standard deviation of the length of local tie vectors of 3 ram. It has to be noted, that the implementation of vector lengths requires a set of well distributed co-locations and local networks with different relative positions of the GPS and the VLBI stations. Assuming, for example, that all co-locations are orientated identically, i.e., the VLBI stations are located in the west direction of the GPS stations, the use of length information only, will lead to singularities in the translation of the VLBI network. Fortunately, the relative positions of GPS and VLBI stations at the 26 co-location sites are very different. The results in Fig. 3 have shown, that there are large variations in pole offsets between GPS and VLBI between different solution types. This demonstrates the high dependency of the (realized) datum from the selected type of datum definition. If both technique solutions are solved separately, the d a t u m was introduced by NNR conditions w.r.t. ITRF2000 using a set of good and well distributed stations. The offsets in xand y-pole between both solutions a r e - 9 0 and 143 # a s respectively. Whereas, after the combination introducing 9 very good ties with an a priori standard deviation of 0.1 m m and defining the datum by NNR conditions using only GPS stations, the offsets become 4 and 35 # a s as shown in Fig. 4. Thus, it can be recognized, that offsets of 100 to 200 # a s between GPS and VLBI, as they are often seen in comparisons of both techniques, may result from inconsistencies of the underlying reference frames. 5
Simulation
studies
As shown in the previous section, it is generally possible to connect the station networks of the different techniques by introducing the lengths of the local tie vectors. At present, 26 co-location sites for GPS and VLBI with local tie information are available. It has been demonstrated that a combination based on the lengths of these 26 local tie vectors is not an approximate alternative compared to the introduction of 3D local tie information, because the information used for the connection is comparatively weak. In the following we investigate the situation for
Chapter 25 • Frontiers in the Combination of Space Geodetic Techniques L
1.0
J
GPS I
0.5 E
Z o.o 0 13_ x -0.5
GPS II VLBI
-1.0 2002
2003
2004
2005
Years
Figure 5: Simulated station distribution at colocation sites
xpole
RMScps RMSvLBI Offset
0.06 m a s 0.18 m a s 0.004 m a s
1.0
0.5
j 0.0 0 13_ >-0.5
-1.0 2002
r
1
2003
2004
2005
Years
ypole RMSGps RMSVLBI Offset
0.05 m a s 0.17 m a s 0.035 m a s
Figure 4- GPS (grey) and VLBI (black) x- and ypole time series w.r.t IERS-C04 for solution type 9 d 3 (or= 0.1ram). For RMS computation the offset to IERS-C04 is removed.
tation. To get a stand-alone VLBI solution it was solved by adding N N R and N N T conditions w.r.t ITRF2000. The simulated GPS network was created by setting up exactly the same dat u m information as for the VLBI solution. For the combination studies the GPS normal equation with full d a t u m information and the d a t u m free VLBI normal equation were used. The resulting transformation parameters of a 14 p a r a m e t e r Helmert-transformation between the combined solution and the stand-alone VLBI solution are a measure of the consistency of the combined frame. For these studies no discrepancies between the local tie information and the networks of the space techniques do exist and no deformation of the network occur within the combination. The local tie information was introduced with a priori s t a n d a r d deviations ranging from 1 to 100 mm. The transformation parameters of the different solution types are compared in Fig. 7.
a VLBI and a GPS network, assuming t h a t all VLBI stations are co-located to GPS. For this purpose a GPS network was simulated which contains 2 GPS stations per VLBI site. The network was created in such a way, t h a t the GPS stations form a rectangular triangle together with the VLBI station, as shown in Fig. 5. The VLBI network is a real network based on the d a t a described above. Figure 6 shows the network configuration. The 17 stations identified by a triangle are stations with a good co-location to GPS in reality. One solution type was performed connecting the terrestrial networks using only the length of local tie vectors. For comparison, solutions with local tie information in three components are solved. Altogether 6 different solution types (see Table 4) are compared.
Figure 6: Global VLBI network used for simulation studies. Stations signed with a triangle have good co-locations to GPS in reality.
The VLBI normal equation was free of dat u m information concerning translation and ro-
Using an a priori s t a n d a r d deviation of 1 m m for the local tie information the transformation
163
164
M. KriJgel • D. Angermann I
Table 4: Solution types for simulation studies solution type all ties 17 ties all dist 17 dist all dist x2
17 dist x2
description 3D local ties at all 56 VLBI sites 3D local ties at 17 VLBI sites the length of one local ties vector at all 56 VLBI sites the length of one local tie vector at 17 VLBI sites the lengths of two local ties vectors at all 56 VLBI sites the lengths of two local tie vectors at 17 VLBI sites
o
......
I
¢¢=¢CC"" go
~;o
a priori standard deviation of local ties/lengths [mm] 0
1
L.
-10
-20
-30 '0 a pdod standard deviation of local tie~engths [mm]
100
0
-10
-20
-30
parameters become zero for all solution types. That means, that the combined network is really consistent in these cases. By increasing the a priori standard deviations the connection of the VLBI and GPS network is getting weaker and weaker. Thereby, solution types with a small change of the transformation parameters are more stable than those with a higher change.
i 50 a pdod standard deviation of local ties/lengths [ram]
1 O0
I
"'
"= == ~
I
~
-10
-~
"zz.
-20
~
~
.
-30
50
1 O0
a pdon standard deviation of local ties/lengths [mm]
The investigations show, that for the current global VLBI network a combination based on only vector lengths of local ties at all VLBI sites provides a larger consistency of the combined network, than the use of 17 3D local ties. This can be explained by the amount and the global distribution of information used for the connection. If two vector lengths to GPS stations are introduced per co-location site, as expected, a further significant stabilization is achieved. Additionally, the elimination of possible orientation errors of the ties and the reduction of discrepancies in the height component argue for an application of vector lengths. The usage of free norreal equations of local networks (without any information about the orientation) instead of local ties in ITRF2000 datum also eliminates possible errors in orientation, however, the impact of the discrepancies in the height component does still exist. It can be eliminated by introducing only the north and east component of the tie. But at present only for a few sites (e.g. Ny Alesund) free normal equations of the local measurements are available.
I
-10
~
~
~
~
-20
-30
5'0
1~o
a priori standard deviation of local ties/lengths [mm]
-10
-20
-30
s'o
loo
a pdori standard deviation of local ties/lengths [mm]
all ties (#168) all dist x2 (#112) all dlst (#56)
-
-- 17 ties (#51) -- 17 dist ](2 (#34) 17 dlst (#17)
Figure 7: Transformation parameters (translation and rotation) of the combined solution w.r.t. the VLBI stand-alone technique solution for different combination strategies and different a priori standard deviations of the local ties or vector lengths. The number in paranthesis gives the sum of used local tie information.
Chapter 25 • Frontiers in the Combination of Space Geodetic Techniques
6
Conclusions and discussion
In this paper new strategies for the selection and implementation of local ties based on time series of space geodetic solutions containing station positions and E a r t h Rotation P a r a m e t e r s (ERP) have been discussed. Main criteria for the selection of co-location sites and local ties were a high consistency in the d a t u m realisation of the combined networks (i.e., a small offset between the E R P estimates of different techniques) and a minimal deformation of the networks due to the combination. The investigations have shown t h a t the most stable results arc obtained if only a subset of high-quality co-locations is used instead of all available co-locations. The results obtained from the different solution types for the combination of VLBI and GPS indicate, t h a t the offsets between the E R P estimates of both techniques are sensitive w.r.t, the d a t u m definition. By using only 9 co-locations the offset becomes almost zero. However, the sets of 9 co-locations between GPS and VLBI, which were identified here are only valid for this study. Using other input data other constellations of co-location sites may be selected. An alternative m e t h o d for local tie implementation was addressed, which uses only the length of local tie vectors instead of 3D components. A clear advantage is, t h a t the influence of possible errors in the height component and in the orientation of the local networks is significantly reduced or even elimated. Simulation studies for the connection of GPS and VLBI networks have shown, t h a t this approach provides good and stable results, if the co-location sites are well distributed. Additionally, it was shown, t h a t the combination gets more stable if two GPS antennas per site are installed. Besides this, such a constellation would enable the permanent mutual monitoring of the GPS stations to observe the effects of station events (e.g. an antenna change) with millimeter accuracy. Acknowledgements. A significant part of this work is supported by the G e r m a n Bundesministerium fiir Bildung und Forschung within the Geotechnologien project (03F0425C).
References Altamimi Z., Sillard P., Boucher C. (2002) ITRF2000: A new release of the International Terrestrial Reference Frame for earth science applications. J Geophys Res 107 (B7), 2214, doi: 10.1029/2001JB000561.
Angermann D., Drewes H., Krfigel M., Meisel B., Gerstl M., Kelm R., Miiller H., Seemiiller W., Tesmer V. (2004) ITRS Combination Center at DGFI: A terrestrial reference frame realization 2003. Deutsche Geod/itische Kommission, Reihe B, Heft Nr. 313. Boucher C., Altamimi Z., Sillard P., Feissel-Vernier M.(2004) The ITRF2000, IERS ITRS Centre, IERS Technical Note No.31, Verlag des Bundesamtes fib Karthographie und Geodgsie, Frankfurt am Main Ferland, R., Reference Frame Working Group technical report, in I G S 2001-2002 Technical Reports, Jet Propulsion Labratory Publication, Pasadena, California, 2004 Johnston, G., Dawson J. (2004) The 2002 Mount Pleasant (Hobart) radio telescope local tie survey, Geosience Australia, Record 2004/21, Canberra, Australia Kr/igel M., Angermann D. (2005) Analysis of local ties from multi-year solutions of different techniques. Proc. of the IERS Workshop on site co-location, Richter B, Dick W, Schwegmann W (Eds), IERS Technical Note No. 33, 32 37, Bundesamt fSr Kartographie und GeodS~sie, Frankfurt am Main. Matsuzaka S., Masaki Y., Tsuji H., Takashima K., Tsumtsumi T., Ishimoto Y., Machida M., Wada H., Kurihara S. (2004) V L B I co-location results in Japan In: Vandenberg N. and Bayer K.: International VLBI Service for Geodesy and Astrometry 2004 General Meeting Proceedings, NASA/CP2004-212255:138-142 Meisel B., D. Angermann, M. Kriigel, H. Drewes, M. Gerstl, R. Kelm, H. Miiller, W. Seemiiller, V. riles_ mer (2005) Refined apporaches for terrestrial reference frame computations Adv. Space Res 33(6) : 350-357, DOI: 10.1016/j.asr.2005.04.057 Ray J., Altamimi Z., (2005) Evaluation of co-location ties relating the V L B I and the GPS frames. Journal of Geodesy 79(4-5):189-195 doi: 10.1007/s00190005-0456-z
Rothacher M., Dill. R, Thaller D. (2005) I E R S Analysis Coordination. Observation of the Earth System from Space, Rummel, Reigber, Rothacher, BGdecker, Schreiber, Flury (Eds), Springer Verlag, in press. Sarti P., Sillard P., Vittuary L. (2004) Surveying colocated space-geodetic technique instruments for TRF computation, Journal of Geophysical Research, doi: 10.1007/s00190-004-0387-0 Steinforth, C., R. Haas, M. Lidberg, A. Nothnagel (2003) Stability of Space Geodetic Reference Points at Ny-Alesund and their Excentricity Vectors. In: W. Schwegmann and V. Thorandt: Proceedings of the 16th Working Meeting on European VLBI for Geodesy and Astrometry, Bundesamt ffir Kartographie und Geod~sie, Leipzig, Germany.
Thaller D., Dill R. Kriigel M., Steigenberger P., Rothacher M., Tesmer V. (2005) C O N T 0 2 Analysis and Combination of Long E O P Time Series. Observation of the Earth System from Space, Rummel, Reigber, Rothacher, BGdecker, Schreiber, Flury (Eds), Springer Verlag, in press.
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Chapter 26
Modifying the Stochastic Model to Mitigate GPS Systematic Errors in Relative Positioning D.B.M. Alves, J.F.G. Monico Department of Cartography, Faculty of Science and Technology Silo Paulo State University, FCT/UNESP, 305 Roberto Simonsen, Pres. Prudente, Silo Paulo, Brazil
Abstract.
The GPS observables are subject to several errors. Among them, the systematic ones have great impact, because they degrade the accuracy of the accomplished positioning. These errors are those related, mainly, to GPS satellites orbits, multipath and atmospheric effects. Lately, a method has been suggested to mitigate these errors: the semiparametric model and the penalised least squares technique (PLS). In this method, the errors are modeled as functions varying smoothly in time. It is like to change the stochastic model, in which the errors functions are incorporated, the results obtained are similar to those in which the functional model is changed. As a result, the ambiguities and the station coordinates are estimated with better reliability and accuracy than the conventional least square method (CLS). In general, the solution requires a shorter data interval, minimizing costs. The method performance was analyzed in two experiments, using data from single frequency receivers. The first one was accomplished with a short baseline, where the main error was the multipath. In the second experiment, a baseline of 102 km was used. In this case, the predominant errors were due to the ionosphere and troposphere refraction. In the first experiment, using 5 minutes of data collection, the largest coordinates discrepancies in relation to the ground truth reached 1.6 cm and 3.3 cm in h coordinate for PLS and the CLS, respectively, in the second one, also using 5 minutes of data, the discrepancies were 27 cm in h for the PLS and 175 cm in h for the CLS. In these tests, it was also possible to verify a considerable improvement in the ambiguities resolution using the PLS in relation to the CLS, with a reduced data collection time interval.
Keywords. Functional and Stochastic Penalised least squares, Systematic errors
models,
1 Introduction The GPS is a satellite-based radio navigation system providing precise three-dimensional position, navigation, and time information to suitably equipped users. The system is continuously available to a user anywhere in the world at any time, and is independent of the meteorological conditions (Seeber (2003)). But the GPS observables, like all other observables involved in the measurement processes, are subject to random, outliers and systematic errors. The random errors are inevitable, being, therefore, considered an inherent property of the observations. Outliers should be eliminated through the quality control process. A procedure extensively used in the navigation field is denominated Detection, Identification and Adaptation (DIA) (Teunissen (1998b)). Systematic errors can be modeled or eliminated by appropriate observation techniques. These errors can have a significant effect on GPS observables. So, this is a critical problem for high precision GPS positioning applications. In medium and long baselines, the major systematic error sources are the ionosphere and troposphere refraction and the GPS satellites orbit errors. But, for short baselines, the multipath is more relevant (Alves (2004b)). Recently, the semiparametric model and the penalised least squares technique have been proposed to mitigate these systematic errors, using single frequency receiver data (Jia (2000); Jia et al. (2001)). This method uses a natural cubic spline, whose smoothness is determined by a smoothing parameter, computed by using the generalized cross validation to model the errors as a function which varies smoothly in time. The systematic errors functions, ambiguities and station coordinates are estimated simultaneously. It will be shown that it is equivalent to change the stochastic model, in which
Chapter 26 • Modifying the Stochastic Model to Mitigate GPS Systematic Errors in Relative Positioning
the error functions (functional model) are incorporated in the stochastic model. As a result, the ambiguities and the station coordinates are estimated with better reliability and accuracy than the conventional least square method. Besides, the solution may require a shorter data collection interval, minimizing costs (Alves (2004a)). In this paper, the penalised least squares and the semiparametric model were used to mitigate the systematic errors in GPS relative positioning, using single frequency receiver data. Two experiments were accomplished. In first one, with a short baseline (-2 km), the multipath was the main error source. But, in the other one, with medium baseline lengths (-102 km), the atmosphere refraction and orbital errors were predominant. The theoretical revision, results and analyses are presented in this paper. 2 Systematic
Errors
Systematic errors on GPS signals can reduce significantly the precision and reliability of GPS relative positioning. However, for short baselines, errors can be reduced by double differencing, as for example, receiver and satellite clocks, satellites orbits and atmosphere refraction. But multipath is not reduced. Multipath is a phenomenon in which a signal arrives to the receiver antenna for multiple ways, due the reflection (Braasch (1991)). This effect is different for each station and depends of the antenna receiver localization. It is caused mainly due signal reflections in surfaces close to the receiver, such as constructions, cars, trees, hills, etc. Secondary effects are caused by reflections in the own satellite and during the signal propagation (Hofmann-Wellenhof (1997)). Multipath introduces significant errors in code and carrier measurements. So, several techniques have been developed to mitigate it. These techniques include the use of special antennas, several antennas arrangement, antenna localization strategies, software techniques, etc (Souza and Monico (2004)). But, these techniques are, in general, very difficult or expensive. In medium and, principally, in long baselines, the atmosphere conditions are poorly correlated. So, the effects caused in the rover and base stations are different. Therefore, the double differences don't
mitigate all errors caused by atmosphere effects and satellites orbit. In relation to atmosphere effects, the errors caused by troposphere and ionosphere can be compared. But the variability of ionosphere errors is larger than the troposphere, and it is also more difficult to model. The ionosphere errors vary from few meters to dozens of meters, while in the troposphere the zenith errors are usually between two and three meters (Klobuchar (1996)). The propagation delay in the ionosphere depends on the electron content along the signal path and on the frequency used. So, the GPS users can use double frequency receivers to take advantage of this property. But single frequency users don't have this possibility (Klobuchar (1996)). Besides, the ionospheric effect also varies with geographic localization and time (Seeber (2003)). The contribution of the ionosphere to the differential positioning error budget is estimated to be 1 to 2 parts per million (ppm). However, gradients up to 10 (15) ppm in the auroral region and up to 40 ppm in the equatorial region have already been reported (Fortes (2002)). The tropospheric delay is independent of the frequency. Usually, the wet and hidrostatic components express the troposphere influence in GPS measurement. The wet component depends on the distribution of the water vapor in the atmosphere, and is harder to model. But, it comprises just 10% of the total tropospheric refraction. The hidrostatic component is precisely described by familiar models (Seeber (2003)). The contribution of the troposphere to the differential positioning error budget varies typically from 0.2 to 0.4 ppm, after applying a model. However, before applying a model, the contribution of the wet component varies from 1 to 4 ppm, which depends strongly on the satellite elevation angle (Fortes (2002)). Discrepancies between the ephemerides available to the user and the actual orbit are propagated into the determined positions of the user antenna (Seeber (2003)). Information about the GPS satellites orbits can be obtained by broadcast or precise ephemerides. The broadcast ephemerides are transmitted by GPS satellites in the navigation files, and your accuracy is about 2 m (IGS, (2005)). The precise ephemerides are provided by International GNSS Service (IGS), and your accuracy is about 10 cm in real time (IGS, (2005)).
167
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D.B.M.Alves.J. F.G.Monico 3 Semiparametric Model and Penalised Least Squares Technique In a semiparametric model the computed variables are divided into two parts: the parametric part and the nonparametric one. Usually, the parametric part is of interest to the users. In the GPS case, the parametric part can be the site coordinates and the carrier phase ambiguities. The nonparametric part can represent a combination of any error functions that vary smoothly with time (Jia et al. (2001)). A vector semiparametric model can be expressed as:
y, = A , x + M , g ( 6 ) + g ,
i=1,2 ..... n,
where E -1 is the weigh matrix to the observations, czj is the smoothing parameter (Alves (2004b)) and g~(t) is the second-order derivative of the jth function with respect to time. Equation (3) defines the penalised quadratic form. The first part of equation (3) is the least squares residual quadratic form, and the second one is the roughness penalty term. The second part represents the roughness of the functions. The roughness penalty term can be expressed as (Fessler (1991)):
(1)
where Yi • ~ m are the observations (carrier phase or pseudorange double differences); A i e [Rm~ is the design matrix; x • [P-,P is the parameters vector including the carrier phase ambiguities and site coordinates; Mi • J R "Xq is the incidence matrix (Green and Silverman ( 1 9 9 4 ) ) ; g ( t ~ ) • l R q are the systematic error functions; ti is the time index;ci •JR m are the random errors; n is the number of epochs; m is the number of observations per epoch; q is the number of error functions; and p is the number of estimated parameters. Equation (1) contains m*n observations and q*n+p unknowns. So, two cases can be considered for this situation. First, if the number of unknowns is larger than the number of the observations. In this case, equation (1) can not be solved by using the traditional least squares technique. Second, even if the number of the unknowns is less than the number of the observations, equation (1) can not provide a stable solution when the traditional least squares technique is used, due the number of unknowns to be larger than the usual, because the error functions (g(ti)) also have to be computed. in order to obtain a reliable resolution, an additional constraint must be added. This constraint is called roughness penalty term, giving as:
(4)
j=l
where Q and R are the matrices related in Fessler (1991) and Alves (2004b); Iq is an q xq identity matrix; D(c~) = diag(a~l, ..., %); @denotes the Kronecker product. So, substituting equation (4) into equation (3), and minimizing in relation to x and g, the following equations are obtained:
(5)
®z -') ®
® ,,, Xe
(e ®
(6)
®
where In is an n x n identity A = (A 1 A,)r, y = ( y , .., y,)r, ~-, , '"''
=
'"
matrix, ~_ ,
----'''----
n'
"',gl ,g2,'",g2 , " ' , g , , ' " ,
Equations (5) and (6) can be solved using a direct method (Green e Silverman (1994)), substituting equation (6) into equation (5). So, after accomplishing some mathematical manipulations, the final result is given as: (7)
a Ign(/~)2d[.
(2)
Thus, the penalised least squares technique is used. So, the function that will be minimized is:
(y,- A,x- u,g(,.))~Z;'
(y, - A,x - U,g(,,
))+
i-1
~ c ~ / I(g;f(t))2dt-min, j 1
where As = SA is the design matrix smoothed, y~. = Sy is the observaction vector smoothed, and S is the smoothing matrix, givin as:
(3)
(s)
Chapter 26 • Modifying the StochasticModel to Mitigate GPS SystematicErrorsin Relative Positioning
The equation (7) can be written by:
x = [AT ((In ® Z-1)- (In ® Z-1)S )A 1
(9)
It can be seen in equation (9) that the original weight matrix ( I n ® Z -1 ) was changed by another weight matrix ((In ® Z -1)- (I n ® Z-')S)o This happens because the functional model was augmented with extra parameters. An equivalent result could be obtained if the stochastic model was changed instead. This equivalence introduces great flexibility into estimation algorithms, with a wide variety of geodetic applications (Blewitt (1998)).
4 Experiments
and
Analysis
In order to test the penalised least square technique and the semiparametric model, data from two baselines, UEPP-TAKI (~2 km) and UEPP-QUIN (~ 102 km), were used. The data were colleted on 13 and 20 September 2003 to the first baseline, and on 16 and 17 July 2003 to the other one. It was used a Trimble 4600 LS and an Asthech ZXII receivers in the rover stations, which are of single and double frequency respectively. But just data collected with the single frequency receiver were processed. The double frequency receiver data was used to compute the considered "ground truth" coordinates. In all experiments, UEPP station was used as the base station. UEPP is a permanent station from the Brazilian Continuous GPS Network (Rede Brasileira de Monitoramento Continuo - RBMC), located in Silo Paulo state, Brazil (Fortes (1997)). It collects data continuously with a Trimble 4000 SSI (double frequency) receiver. Researchers from the Faculty of Science and Technology (FCT/UNESP) in Brazil have been developing a GPS data processing software, called GPSeq (Machado and Monico (2002)). It was implemented with the conventional least squares method for single frequency data, where phase and pseudoranges double difference are the basic observables. The penalised least squares technique and the semiparametric model were implemented in the GPSeq software. So, in this paper, the results obtained by penalised least squares (PLS) were compared with the conventional least squares (CLS). In all experiments the data were collected with sampling rate of 15 seconds and a cut off elevation
angle of 15 °. During the data processing, the original sampling rate was maintained. Broadcast ephemerides were used to compute the satellite positions and no tropospheric and ionospheric models were used. For solving the ambiguities, the Least Squares Ambiguity Decorrelation Adjustment (LAMBDA) method was used. This method first decorrelates the ambiguities and next computes integer least-squares estimates for the ambiguities in a highly efficient way (Teunissen (1998a)). The ambiguity resolution validation was performed by using the ratio test (Teunissen (1998a)). In this test it is computed the ratio of the second best norm of the residual by the first one. The threshold value of the ratio test was 1.5, accordingly to Jia et al. (2001). Besides, the satellite that had the large elevation angle was chosen as base satellite. The first experiment had as main goal to verify if the penalised least squares can mitigate the multipath effects on GPS signals. So, a short baseline was used, UEPP-TAKI, where the atmosphere effects and the orbital errors are reduced by double differencing. Therefore, initially, the data were collected near a reflector surface (a cart), that was 6 m far from the receivers and 1.3 m from the ground. Its dimensions were 13 m and 2.5 m of length and width respectively. Later on, the cart was removed and another data collection was accomplished in the same place and considering the same sideral time interval. This second data collection was realized just for computing the "ground truth". For testing the proposed method, four sub-sets of data were processed; two sub-sets of 5 min (M1 and M2 sessions) and two of 10 min (M3 and M4 sessions), all with 8 satellites. The resultant coordinates discrepancies (AE2 + AN 2 + Ah 2)~ in relation to the ground truth coordinates, between PLS and CLS are shown in Figure 1.
0,035 0,03 0,025 "t~ 0,02 ~. 0,015
~,
-=
[] CLS 131
O,Ol
----
o 0,005
;~
0. M1
M2
M3
Hh
[] PLS
M4
Sessions
Fig. 1. Resultant coordinates discrepancies in relation to the ground truth for 5 and 10 min of data processing in the UEPP-TAKI baseline
169
170
D.B.M. Alves. J. F. G. Monico
From Figure 1 one can observe that the coordinates discrepancies estimated from the PLS are better than those from the CLS. The largest coordinate discrepancies were in h component, whose values are shown in table 1.
resultant coordinates discrepancies in relation to the ground truth coordinates, between PLS and CLS methods.
2 1,5
Table 1. Largest coordinates discrepancies for the UEPPTAKI baseline Sub-Set Method Session Discrepancies(mm) M1 11 CLS 5 min M2 33 M1 9.3 PLS M2 16 M3 18 CLS M4 12 10rain M3 12 PLS M4 7.7
._B Q
[] CLS
I!
0,5
7
0 Q1
Q2
[] P L S
, Q3
Q4
Sessions
Fig. 3. Resultant coordinates discrepancies in relation to the ground truth for 5 and 10 min of data processing in the UEPP-QUIN baseline (~ 102 km)
The largest coordinates discrepancies are shown in table 2. Table 1 shows that the largest coordinates discrepancies are also related with CLS method. In relation to the ambiguity resolution, Figure 2 shows the values for the ratio test. It is possible to verify that, in all cases, the values obtained for the PLS are larger than that of the CLS. This indicates that the PLS solution may be more reliable than CLS.
Table 2. Largest coordinates discrepancies for the UEPPQUIN baseline Sub-Set 5 rain
10min
18
~o
~.
Method Session Discrepancies(cm) CLS Q1 175 in h Q2 166 in h PLS Q1 20 in h Q2 27 inh CLS Q3 150 in E Q4 92 in h PLS Q3 15 in E Q4 14 in E
[] C L S 9
[] PLS
3 0 M1
M2
M3
M4
Sessions
Fig. 2. Ratio test for 5 and 10 min of data processing in the UEPP-TAKI baseline
The other experiment had the goal of verifying if the PLS could be used to mitigate the atmospheric effects and the orbital errors for a medium baseline. So, the UEPP-QU1N (~102 km) baseline was used. In this baseline four sub-sets of data, two with 5 min (Q 1 and Q2) and two with 10 minutes (Q3 and Q4), were considered. Besides, these data were collected at about 2 pm (local time), when the ionospheric effects are more relevant. Figure 3 shows the
In these experiment, the PLS coordinates quality is also better than the CLS. This indicates the efficiency that the proposed method may provide for mitigating systematic errors. In relation to the ambiguity reliability, Figure 4 shows the ratio test values.
2,5 2 I-
1,5
0,5 0,
I I Q2
Q3
II II Q4
Sessions
Fig. 4. Ratio test for 5 and 10 min of data processing in the UEPP-QUIN baseline
Chapter 26 • Modifying the StochasticModel to Mitigate GPS SystematicErrorsin Relative Positioning
From Figure 4 it can be verified that, again, the values of the ratio test are larger in the PLS method than in CLS. This indicates, as stated before, that the PLS solution may be more reliable than CLS. However, due to the long baseline length together to the short occupation time, it may imply some wrong ambiguities. 5 Conclusions
In this paper, the semiparametric model and the penalised least squares technique were used to mitigate systematic errors on GPS signals. The fundamental of the method was presented. It was shown that changing the functional model is equivalent to change the stochastic model instead. In order to test the advantages of the method, two experiments were realized. While in the first experiment multipath was the main error source, in the other one, the atmosphere refraction and orbital errors were predominant. Results showed that in all experiments the coordinates discrepancies obtained by penalised least squares were always smaller than those obtained with conventional least squares. Additionally, the ratio test values were always larger in the penalised least squares method than in the conventional one. This indicates that the ambiguity solution by this method may be more reliable. These preliminary results have shown that the proposed method may be quite efficient in mitigating systematic errors. Therefore, more experiments must be realized in the future to effectivey proof this statement. They are been carried out now to better qunatify the results Acknowledgments
The authors would like to thank FAPESP for the financial support (01/11858-9 process) for the first autor and to CNPq and Fundunesp for funding the participation in the Dynamic Planet 2005 conference. References
Alves, D. B. M. (2004a). M~todo dos Minimos Quadrados corn Penalidades: Aplicaq~o no posicionamento relativo GPS. 133f. Dissertagfio (Mestrado em Ci~ncias
Cartogrfificas) - Faculdade de Ci6ncias e Tecnologia, Universidade Estadual Paulista, Presidente Prudente. Alves, D. B. M. (2004b). Using Cubic Splines to Mitigate Systematic Errors in GPS Relative Positioning. In: Proc. of ION GNSS 2004, Long Beach, California. Blewitt, G. (1998). GPS Data Processing Metodology. In: Teunissen, P. J. G.; Kleusberg, A. GPS for Geodesy. Berlin: Springer Verlage, pp.231-270. Braasch, M. S. (1991). A Signal Model for GPS, Navigation. vol. 37, no. 4, pp. 363-377. Fessler, J. A. (1999). Nonparametric Fixed-Interval Smoothing With Vector Splines. In: Proc. IEEE Transactions on Signal Processin, pp. 852-859. Fortes, L. P. S. (1997). OperacionalizaqCto da Rede Brasileira de Monitoramento Continuo do Sistema GPS (RBMC). 152f. Disserta~fio (Mestrado em Ci6ncias em Sistemas e Computa~o)- Instituto Militar de Engenharia (IME), Rio de Janeiro. Fortes, L. P. S. (2002). Optimising the Use of GPS MultiReference Stations for Kinematic Positioning. 2002. 355f. Thesis (PhD)- University of Calgary, Calgary. Green, P. J. and B. W. Silverman (1994). Nonparametric Regression and Generalized Linear Models': a roughness penalty approach. 1.ed. London: Chapman & Hall. Hofmann-Wellenhof, B. et al. (1997). GPS Theory and Practice. Wien: Spring-Verlage. 326p. Jia, M. (2000). Mitigation of Systematic Errors of GPS Positioning Using Vector Semiparametric Models. In: Proc. oflON GPS 2000, Salt Lake City, pp. 1938-1947. Jia et al. (2001). Mitigation of Ionospheric Errors by Penalised Least Squares Technique for High Precision Medium Distance GPS Positioning. In: Proc. of KIS 2001, Banff, Canada. Klobuchar, J. (1996). A. Ionospheric Effects on GPS. In: Parkinson, B. W. and J. J. Spilker. Global Positioning System: Theory and Applications. Cambridge: American institute of Aeronautics and Astronautics, pp.485-515. Machado, W. C. and J. F. G. Monico (2002). Utiliza~o do software GPSeq na solu~fio rfipida das ambigfiidades GPS no posicionamento relativo cinemfitico de bases curtas. In: Pesquisa em Geocidncias, Porto Alegre, pp.89-99. Seeber, G. (2003). Satellite Geodesy: Foundations, Methods, and Applications. Berlin, New York: Walter de Gruyter. Souza, E. M. and J. F. G. Monico (2004). Wavelet Shirinkage: High frequency multipath reduction from GPS relative positioning. GPS Solutions, vol. 8, no. 3, pp. 152159. Teunissen, P. J. G. (1998a). GPS Carrier Phase Ambiguity Fixing Concepts. In: Teunissen, P. J. G.; Kleusberg, A. GPS for Geodesy. Berlin: Springer Verlage, pp.271-318. Teunissen, P. J. G. (1998b). Quality Control and GPS. In: Teunissen, P. J. G.; Kleusberg, A. GPS for Geodesy. Berlin: Springer Verlage, 1998b, pp.271-318.
171
Chapter 27
GPS Ambiguity Resolution and Validation Under
Multipath Effects: Improvements using Wavelets E.M. Souza, J.F.G. Monico Department of Cartography S~o Paulo State University- UNESP, Roberto Simonsen, 350, Pres. Prudente, SP, Brazil
Abstract. Integer carrier phase ambiguity resolution is the key to rapid and high-precision global navigation satellite system (GNSS) positioning and navigation. As important as the integer ambiguity estimation, it is the validation of the solution, because, even when one uses an optimal, or close to optimal, integer ambiguity estimator, unacceptable integer solution can still be obtained. This can happen, for example, when the data are degraded by multipath effects, which affect the real-valued float ambiguity solution, conducting to an incorrect integer (fixed) ambiguity solution. Thus, it is important to use a statistic test that has a correct theoretical and probabilistic base, which has became possible by using the Ratio Test Integer Aperture (RTIA) estimator. The properties and underlying concept of this statistic test are shortly described. An experiment was performed using data with and without multipath. Reflector objects were placed surrounding the receiver antenna aiming to cause multipath. A method based on multiresolution analysis by wavelet transform is used to reduce the multipath of the GPS double difference (DDs) observations. So, the objective of this paper is to compare the ambiguity resolution and validation using data from these two situations: data with multipath and with multipath reduced by wavelets. Additionally, the accuracy of the estimated coordinates is also assessed by comparing with the "ground truth" coordinates, which were estimated using data without multipath effects. The success and fail probabilities of the RTIA were, in general, coherent and showed the efficiency and the reliability of this statistic test. After multipath mitigation, ambiguity resolution becomes more reliable and the coordinates more precise.
Keywords. Ambiguity, multipath, wavelets, Ratio Test Integer Aperture (RTIA)
1 Introduction Rapid and high precision relative positioning using GNSS requires the use of the very precise carrier phase measurements. However, the carrier phases are ambiguous by an unknown number of cycles. The double difference (DD) ambiguities are known to be integer-valued, and this knowledge has been exploited for the development of integer ambiguity resolution algorithms (Verhagen 2005). The estimation process consists of three steps. Firstly, a least-squares adjustment is applied in order to obtain the float solution. All unknown parameters are estimated as real-valued. In the second step, the integer constraints on the ambiguities are considered. Thus, the float ambiguities will be mapped to integer values. In relation to the map, different choices are possible. The simplest is to round the float ambiguities to the nearest integer values or to conditionally round so that the correlation between the ambiguities is taken into account. The optimal choice is to use the integer least-squares estimator (LAMBDA Method), which maximizes the probability of correct integer estimation. After fixing the ambiguities to their integer values, the last step is to adjust the remaining unknown parameters considering their correlation with the ambiguities (Verhagen 2005). However, as important as the integer ambiguity estimation, it is their validation, because, even when one uses an optimal, or close to optimal, integer ambiguity estimator, one can still come up with an unacceptable integer solution. This can happen for example when the data are degraded by multipath, which can affect the real-valued float ambiguity solution, and it can lead to an incorrect integer (fixed) ambiguity solution. Thus, it is important to use a statistic test that has a firm theoretical and probabilistic footing, which is possible by using the Ratio Test Integer Aperture (RTIA) estimator (Teunissen and Verhagen 2004).
Chapter 27 • GPS Ambiguity Resolution and Validation under Multipath Effects: Improvements Using Wavelets
In order to analyze the performance of ambiguity resolution and validation, an experiment was carried out using data with and without multipath effects. Reflector objects were placed surrounding the receiver antenna aiming to cause multipath. A method based on Multi-Resolution Analysis (MRA) by wavelet transform is used to reduce multipath of the GPS DD observations. So, the objective of this paper is to compare the ambiguity resolution and validation using data from these two situations: data with multipath and with multipath reduced by wavelets. Furthermore, the accuracy of the coordinates is analyzed by comparing with the "ground truth" coordinates estimated using data not affected by multipath.
Teunissen (1998a) introduced a new class of ambiguity estimators, and it is called the class of |nteger Aperture (IA) estimators. It is defined by dropping the condition that no gaps are allowed between the pull-in regions, so that all pull-in regions should not cover the complete space R""
U f~ -f~'
where f2 c R"
(3a)
ZE Z n
s,,,(a
s
,(n ) - e . v . . z
f~ -z+f~0,
z"...
z
Vze Z n
where ~ c R ~ is called the aperture space. Thus, the integer aperture estimator (Teunissen and Verhagen 2004) is given by
(3b) (3c) m
N
2 Integer Estimation and Validation In this section a brief introduction related to integer estimation and validation is presented based on the works developed at the Delft University of Technology. In the integer estimation, different real-valued ambiguity vectors N are mapped to the same integer vector. So, a subset &oR ~ can be assigned to each integer vector z~ Z" (Teunissen 1998a):
&={xe R"lz=S(x)},ze z".
(1)
The subset S~ has all real-valued float ambiguity vectors that will be transformed to the same integer vector z, and it is called the pull-in region of z. An integer estimator N is said to be admissible if its pull-in regions S~ satisfies (Teunissen 1998a)
Us Z- R ~
(2a)
zE Z n
Int(S )c~ Int(S Z) - •, Vu, z • Z n , u :/: z
S Z - z + S O, V z • Z "
(2b) (2c)
where Int denotes the interior of the subset. The first condition means that all real-valued vectors will be transformed to an integer vector. For the second condition, the float solution is mapped to just one integer vector. The last condition is related to the translational invariance. An admissible integer estimator is the Integer Least-Squares (ILS) estimator, which is used in the LAMBDA (Least squares AMBiquity Decorrelation Adjustment) (Tiberius and de Jonge 1995; Teunissen 1998b).
Therefore, for the last equation, when ~r~ f~ the ambiguity will be fixed using one of the admissible integer estimators, otherwise the float solution should be maintained. This means i)
Ne ~N
success" correct integer estimation;
ii) N ~ ~ \ ~ N fail: incorrect integer estimation. Consequently, the corresponding success (Ps) and fail (Pj) probabilities or rates are given by (4a)
P~ - P 0 V - N ) - ff:~ (x)dx
(4b) (4c)
where f ~ ( x )
is the probability density function of
_ £r_ ~ and f~ (x) of the float ambiguities. However, for a user it is important that the fail rate is below a certain limit. The approach of integer aperture estimation allows choosing a threshold for the fail rate, and then determines the size of the aperture pull-in regions such that the fail rate will be equal to or below this threshold. So, applying this approach means that the ambiguity estimate is validated using an appropriate criterion (Teunissen and Verhagen 2004). Thus, an IA estimator is the RTIA, which is the inverse of the popular Ratio test: Accept N if: ~ - ~, 2Q~ ~ _ ~ 2
<s,,
o"'":-""~,P,~,~;~" ..... ~ . % L ................. "(3
\ :~k:4",;"~*.e~ ~ . . ~ ~ . ;....... 7 ;. ~ ,~ ~ . - . ~ (.9 1 0 -3 -................................... .."5- ~-&',~ -" o~o ~ 7 " ~
.................... ...........................
1 0 .4 ~
a
b
i
X 1 0 -~ . . . . . . . . .
l
i
t
y
of o c e a n i c
./2/"~s.variati°ns from. i.........t w o d f f f e r e n t b a kgroundmodels ...... 50
1 O0
150
Spherical harmonic degree
Fig. 1 Degree and error degree amplitudes of 16 monthly GRACE-only gravity models generated at GFZ Potsdam. The dashed lines represent the error degree amplitudes from the formal errors of spherical harmonic coefficients of the 16 monthly fields. For two state-of-the-art models describing oceanic short-term mass variations the difference degree amplitudes are plotted in Fig.1. It can be seen that the resulting curve for the model difference amplitudes is significantly larger than the formal errors up to degrees 15 - 20. Assuming that the existing model differences are a realistic measure for the uncertainty in the used background model, the comparison with the formal errors amplitudes reveals the underestimation of the true error level for this contribution. Similar results can be obtained for other time-variable signals such as atmospheric mass variations or ocean tides (not shown). Since these contributions add up to the errors in the shortterm oceanic mass variations the actual error level may even be larger. In this way the formal errors must be regarded as too optimistic, at least in the long wavelength part of the solved gravity spectrum.
Chapter 36 • GRACE Time-Variable Gravity Accuracy Assessment
3 Calibration Approaches Since no global data sets of comparable strength and homogeneity exist, a calibration based on GRACE-internal comparisons is considered. It has however to be noted that these comparisons are less internal than the variance-covariance matrices, which refer to internal consistency of single monthly solutions. The approach is to use differences between monthly GRACE-only solutions that are separated by 12 months assuming that the model differences are a representative measure for the true model error. The idea is to cancel out the dominant hydrological signal present in the monthly models, provided that the hydrological signal patterns are equivalent from year to year. The final goal is to derive degree-dependent scaling factors that can be applied degree-wise to the formal variance-covariance matrices. To this end the model differences are transformed into the spectral domain by computing degree difference amplitudes to be compared to the formal error degree amplitudes. In detail the following steps are carried out: From the recovered monthly solutions difference amplitudes are computed for the months February to May in 2003 and 2004. The difference July 2003 - 2004 is not considered because of the degraded ground track coverage in 07/2004. For the resulting difference amplitudes the average (arithmetic mean per degree) is calculated and compared to the GRACE baseline accuracy. It turns out that the averaged difference amplitudes are about 40 times the GRACE baseline (see Fig. 2(a)), which is consistent with a value found by Wahr et al. (2004) based on monthly GRACE-only models generated by the Center of Space Research (CSR), Austin/Texas. Using the 40 times the GRACE baseline accuracy as calibrated mean accuracy of the monthly solutions the scatter of the monthly solutions is added to this curve by applying degree-dependent scaling factors of the formal errors relative to the mean formal error of the monthly solutions. Instead of the 16 fields only 14 monthly models respectively their formal errors are used. The models for June and July 2004 are excluded because of the degraded ground track coverage during these months. The result for the calibrated error degree amplitudes per month is shown in Fig.2 (b). Comparing these calibrated, monthly error degree amplitudes degree-wise to their formal error degree amplitudes gives the degree-dependent scaling factors which are then applied to the
formal variance-covariance to obtain the calibrated variance-covariance matrices. Finally, the calibrated variance-covariance matrices are used for error propagation of gridded gravity functionals respectively surface mass anomalies in the space domain.
-
For comparison, an alternative approach in the space domain is considered. In this second approach, the true error of the monthly GRACE-only models is to be represented by the residual gravity respectively surface mass variability of time series of global grids from GRACE, where in each grid point a constant, a linear trend and a sinusoidal signal with annual frequency are removed. The resulting residual grids can then be compared to the gridded errors of gravity respectively surface mass based on error propagation from the calibrated variance-covariance matrices derived from the 1 st approach.
10 ~-......~ ...........................::.................geold.,sign.a.l.::o.f..t.6............
I
~ m ° n t h l Y
s°liti°ns ~
E'10-1 ~ .......................................................... ~ -u 0
"
~
:
t
......................................... i......................
1 -=~-GR-Ac.E..base,.in.e..x-io~~,7 l :
(.9 10 .3
..............
.................................
-
10 .2 0
50 100 Spherical harmonic degree
150
10~ i~_. '......................................................................................................................... ~ (b) 10 ° ~ .....~
-
...........................i geoid.signal.lof..t6.........~ ~ ~ m ° n t h l Y s ° l ! t i ~ j .
-1-
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
i
i
"E' 10 -[.....caiibrated~0ii~month ' ~ ~~
'
~
i
u
.....
~ o - ~ ; ....................................~ ~ . , : - " ~ , + , ~
............
-
-0
!GRACEb a s e l i ~ ~ " ' ; ' @ ~ b ~ "
10
L:..: ~ " ' : :
~ ~W"
10
I
......~
G
5 _x.._ i
0
i
i
i
......................................................................
of 16~monthly solutions2003, 2004
: i
i
i
i
I
i
50 1 O0 Spherical harmonic degree
i
i
i
150
Fig. 2 (a) Difference degree amplitudes of monthly solutions that are 12 months apart and their mean. These mean difference degree amplitudes are about 40 times the GRACE baseline. (b) The calibrated errror degree amplitudes of the monthly solutions using 40 times the GRACE baseline as mean calibrated error amplitudes and adding the scatter of the formal error amplitudes around their formal mean error amplitudes to the mean calibrated error amplitudes.
239
240
R. Schmidt. F. Flechtner. R. K6nig • U. L. Meyer. K-H. Neumayer. Ch. Reigber. M. Rothacher. S. Petrovic • S-Y. Zhu. A.G~intner
To this end, global grids ( l ° x l °) of surface mass variability in terms of the thickness of an equivalent mass of water for the monthly models with respect to their mean are computed. Only the m o n t h l y gravity models of 2003 and their mean is used in the following. The reason for this limitation is to cover almost exactly an annum period on the one hand and to exclude the degraded fields June and July 2004 on the other. The grids are filtered using the Gaussian averaging filter for different radii of 1500, 1000, 750 and 500 km, respectively. In each grid c o m p a r t m e n t a bias and linear trend plus an annual sine is fitted to the time series of data points and removed. Finally, the R M S of the residuals per grid point is taken as the uncertainty of the monthly solutions.
Figures 3 (a) and 4 (a) show the spatial distribution obtained from error propagation using the resulting calibrated variance-covariance matrix for
(a)
[cm] 0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
i
(b)
4 Comparison of Approaches Table 1 lists the global weighted RMS (cosine latitude weighing) of the global grids of the errors of surface mass variability from error propagation based on the mean calibrated error level scaled to 40 times the baseline and for residual R M S variability from approach 2. The surface mass variability is expressed in terms of the thickness of an equivalent mass of water. Table 1. Global weighted RMS of global grids of errors of
surface mass variability from error propagation based on the mean calibrated error level scaled to 20/40 times the GRACE baseline and for residual RMS variability from approach 2. Surface mass in terms of the thickness of an equivalent mass of water. Unit is centimeter. Mean formal error level scaled to
[cm] 0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
Fig. 3 (a) Spatial distribution of errors of surface mass variability (thickness of equivalent mass of water, unit cm) from error propagation using the complete, calibrated variance-covariance matrix for month 08/2003 (mean level scaled to 20 times the GRACE baseline). Gaussian averages for a filter radius of 750 km. (b) Spatial distribution of absolute values of residual surface mass variability for 08/2003 after removal of a best-fitting constant plus a linear trend plus an annual sinusoid per grid point. Same units and Gaussian filtering applied as in Fig. 3 (a).
Spherical harmonic degree, half-wavelength X/2 in [km]
GRACE baseline X 40 X 20
13 1500 1.3 0.9
20 1000 2.1 1.5
27 750 4.1 2.1
40 500 7.4 3.7
Approach 2
0.9
1.4
2.1
4.0
The comparison shows that the global error level obtained from the calibrated variance-covariance matrix is significantly larger than the level from approach 2 over all spatial wavelengths. This may indicate that a global scaling of the mean error level of the monthly solutions by a value of 40 is too conservative. Systematically reducing this global scaling factor and re-computing the error propagation, a value of 20 times the G R A C E baseline is obtained for the mean calibrated error level of the m o n t h l y models which eventually gives a better agreement b e t w e e n the two approaches (cf. Tab. 1). Consequently calibrated variance-covariances relative to this level are determined.
(a)
[cm] o.o
0.5
1.o
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
(b)
[cm] 0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
Fig. 4 (a) and (b) as in Fig. 3 (a), (b) but Gaussian averages for a filter radius of 500 kin.
Chapter 36 • GRACE Time-Variable Gravity
August 2003 as an example. The full matrix including the covariances has been used for error propagation and Gaussian averages of 750 km (Fig. 3 (a)) and 500 km (Fig. 4 (a)) radius are computed. The displayed maps show meridional oriented patterns that are typical for the propagated errors of the monthly models: largest errors are obtained in the equator region, smallest errors are given towards the poles. This kind of distribution obviously correlates with the ground track coverage, where the sampling along the G R A C E orbit increases in direction of the poles due to the high inclination of the orbital plane (i ~- 89.5 o). Towards the equator larger unsampled areas occur, because of the evolution of the orbit in longitudinal direction. The spatial distribution of the residual variability from approach 2 is shown in Fig. 3 (b) and 4 (b). Again meridional oriented patterns are visible, h o w e v e r the given features differ significantly in space and amplitude from the maps of error propagation (cf. Fig. 3 (a) and 4 (a)). Over continents such deviations can be explained to some extent from non-reduced hydrological signals, e.g. the variability in the Niger region in Africa. However, other signals, in particular of the oceans, cannot be explained by non-reduced hydrology, but represent obviously aliasing effects from various sources like deficiencies in the applied background models for time-variable gravity, but also possible inefficiencies in the data processing like e.g. an imperfect orbit parametrization. Table 2 confirms that on the global scale, represented by the weighted RMS, the two approaches give comparable results. However, spurious signals of large amplitude (cf. m a x i m u m values in Tab.2) caused by aliasing are not covered by the calibrated variance-covariance matrices. Table 2. Statistics of grids from Fig. 3 (a) - Fig. 4 (b). Units cm.
Grid Fig. 3 (a) Fig. 3 (b) Fig. 4 (a)
Min. 0.6 0 0.6
Max. 1.7 7.8 3.3
Fi~. 4 (b)
0
11.1
WRMS 1.3 1.6 2.4 3.0
AccuracyAssessment
G a n g e s (1572 000 km 2) 200 '-~ 150
"
f
"
~_1oo ............................................... ! ........................
0
~
-50
,~-100 -150 0
2
4
6 8 10 12 14 16 Months since 01/2003
18
20
(a)
18
20
(b)
G a n g e s (1572 000 km 2) 20O
_100
~0
0
~
-50
,~.-100 -150
0
2
4
6 8 10 12 14 16 Months since 01/2003
Fig. 5 (a) Time series of basin averages of surface mass for the Ganges river derived from GRACE and the two hydrology models WGHM (D611 et al., 2003) and LaD (Milly and Shmakin, 2002) using the method of Swenson and Wahr (2002). The acceptable satellite model error is set to 2 cm, the model variances are calibrated relative to 40 times the GRACE baseline accuracy. Lines represent the annual-varying part derived from the individual time series. (b) same as (a), but model variances calibrated with respect to 20 times the GRACE baseline accuracy. Lena (2461 000 km 2) 80 :
60 40
20 F-
$
0 -20 -40
,~
-60
-80 0
2
4
6
8
10
12
14
16
18
20
Months since 01/2003
5 Accuracy Assessment of Basin Averages of Surface Mass Variability As an additional test to assess the value of the calibrated variance-covariance matrices time series of surface mass variability for selected river basin, are derived from the 16 monthly G R A C E - o n l y gravity models and monthly output from two state-of-theart hydrology models. The method described in Swenson and W a h r (2002) is applied to derive the basin averages, that are constrained by the given satellite model errors respectively a user-defined value for the acceptable satellite model error (see Swenson and W a h r (2002) for details). In this study
(a)
Lena (2461 000 km 2) 80 E
60
m
40
e--~ .o e--
20
~-
.................................................................
0
....
~: -20 K
~
~
~
_
L
I
,
,i i,
11
t'~O3-40 [_:..........~ J Z " q - ~ "
::
.\... /.."'.. ERSII2, ENVI~AT ............. ~-~. ........ ~ '~. /
"..
Table 2. Statistics of differences between ENVISAT and the gra-
vimetric geoid model. Unit: [m]. cycles
max
15 16 20 25 26 15 - 34 stacked & cross. adi. 15-34 stacked & cross. adj.15-34
3
Mean
'
min
mean
0.912 1.388 0.705 0.639 1.292 1.555
-1.026 -1.071 -1.126 -1.104 -1.052 -1.326
-0.358 -0.243 -0.339 -0.430 -0.236 -0.312
_+0.323 +0.374 _+0.267 +0.263 +0.321 -+0.348
0.694
-1.016
-0.426
_+0.226
0.712
-0.697
0.000
+0.129
Sea Surface
Model
'...:
'..
".
'~ :'
:-.
' .... , ,
".TIPT r a c k s
".":
std
Estimation
After the validation of the ENVISAT and JASON-1 data and the conclusion that the latter will not be used, a database of all available altimetric observations for the area under study was created to determine the MSS/geoid model. The altimetric SSHs came from: (i) the GEOSAT-GM mission (25402 SSHs), (ii) the ERS 1-ERM mission phases c and g (34323 SSHs), (iii) the ERS1-GM mission (14901 SSHs), (iv) the ERS2ERM mission (30991 SSHs), (v) nine years of the T/P mission (136864 SSHs), and (vi) the already used ENVISAT mission (26072 SSHs). Therefore, a total num-
i// Fig. 2" Distribution of ERS 1-ERM, ERS2, ENVISAT, ERS 1-GM, GEOSAT-GM and T/P data.
Consequently, a remove-compute-restore method has been followed to determine the final MSS model, i.e., the altimetric SSHs were referenced to an earth geopotential model, gridded and then the GM contribution was restored to construct the final MSS. During the remove step and in order to assess the improvement that the latest CHAMP and GRACE EGMs offer, two models have been employed, namely the traditional EGM96 (Lemoine et al. 1998) and EIGEN-CG03C (F6rste et al. 2005). The latter is the latest combination model by GFZ using CHAMP and GRACE data and is complete to degree and order 360. Table 3 summarizes the statistics of the SSHs before and after the reduction to the EGMs. From that table it is clear that EGM96 provides a (marginally) better reduction of the data by about 2 cm in terms of the std and 6 cm in terms of the range compared to EIGEN-CD03c. On the other hand the latter
Chapter 37 • Combinationof Multi-Satellite Altimetry Data with CHAMPand GRACEEGMsfor Geoid and Sea SurfaceTopographyDetermination
reduces the mean by 9 cm more than EGM96. Therefore, no clear conclusion can be drawn for their performance other than that the models are comparable.
Table 3. Statistics SSHs before and after the reduction to EGM96 and EIGEN-CG03c. Unit: [m]. max
SSHs
40.401 SSUsred EGM96 1.461 S SHsred EIGEN-CG03c 1.627
min
-0.358 -2.367 -2.264
mean
14.541 -0.266 -0.175
std
+8.665 +0.340 +0.358
To grid the data and generate the reduced MSS/marine geoid mesh at l ' x l ' resolution three methods have been identified, i.e., conventional least squares, splines in tension and least squares collocation. From the analysis performed and the comparisons with global MSS models it was concluded that the LSC solution provided superior results by about +7-11 cm (in terms of the std of the differences) compared to the other methods, something in line with an earlier study (Tziavos et al. 2004), where these algorithms were also tested. Due to the limited space available, only the results from LSC will be reported herein. Table 4 presents the gridded reduced MSS heights as well as the final MSS models referenced to EGM96 and EIGEN-CG03c. The EGM96 MSS model is also depicted in Fig. 3.
of high-importance in terms of the absolute MSS/marine geoid error. In a next step, the computed differences were minimized using Eq. 1 to remove any bias and tilts between the KMS and the compute MSS models. This resulted in smaller std values at the level of +14.2, + 15.3, + 10.7, and + 11.4 cm for the differences between KMS01, KMS04 and the EGM96 and EIGEN-CG03c MSS models, respectively. Therefore, even after the minimization of the differences the MSS referenced to EGM96 outperformed the EIGEN-CG03c model even by a slight margin. In Tziavos et al. (2004) MSS models for the same area were developed and also compared to the KMS models giving an overall best std at the +17 cm (after bias and tilt fit). So, it can be concluded that the MSS models developed in this study are about 5 cm more accurate than the previous ones. Finally, the estimated MSS models agree better by almost +9 cm with KMS04 compared to KMS01, which gives evidence that the latest KMS MSS model is indeed an improved version of its predecessor. The differences between KMS04 and the referenced to EGM96 MSS model (see Fig. 4) are almost zero in marine areas, while they reach their minimum and maximum values close to the coastline, where both models suffer due to the inherent problems of satellite altimetry in such areas.
T a b l e 5. Statistics of the differences between the KMS and estimated MSS models. Unit: [m]. max
max
M S sred EGM96 MsSred EIGEN-CG03c M S S EGM96
MSSEI~EN-cC°3c
0.940 1.060 40.139 39.851
min
-1.720 -1.660 0.596 0.714
mean
-0.370 -0.280 19.828 19.828
std
+0.350 +0.360 +10.839 +10.839
The validation of the estimated MSS models was performed through comparisons with the latest KMS (Danish Survey and Cadastre) MSSs, namely KMS01 (Andersen and Knudsen 1998) and KMS04 (Andersen et al. 2003). Table 5 summarizes the results of the comparisons for both MSS models developed, while Fig. 4 depicts the differences between KMS04 and the referenced to EGM96 MSS model. From that table it becomes once again evident that the two EGMs give almost the same results, but EGM96 outperforms EIGENCG03c by +1 cm in terms of the std and 50 cm in terms of the range, even though KMS04 is referenced to GGM01C ( C H A M P - G R A C E combination EGM). This is an indication that EGM96 can be regarded as a dominant geopotential model and is still not outperformed by the new EGMs. Of course this is true for the present study (relative accuracy), the data used, the area under study and may not repeat in other regions. Furthermore, EIGEN-CG03c gives a much smaller cumulative geoid error (30 cm compared to 42 cm for EGM96), which is
4
min
KMS01-MSSEl~ENcG°3c KMS04 - MSSEGM96 KMS04-MSSEloENcG°3c
1.194 1.363 1.199 1.179
Sea Surface
Topography
K M S 0 1 - M S S wGM96
Table 4. Statistics of the reduced MSS heights and the final MSS models referenced to EGM96 and EIGEN-CG03c. Unit: [m].
mean
-0.959 -1.109 -0.339 -0.919
0.207 0.208 0.127 0.126
std
+0.194 +0.201 +0.117 4-0.122
Estimation
For the determination of the quasi-stationary (QSST) model the estimated MSS model referenced to EGM96 was combined with the gravimetric geoid available for the area under study. The latter was estimated from airborne (Olesen et al. 2003), shipborne and land gravity data (Vergos et al. 2005). To derive a first QSST model, the differences between the MSS and the gravimetric geoid were formed as QSST = MSS air - N grav
(2)
where the gravity anomalies used to determine the gravimetric geoid are free-air reduced, i.e., reduced from the sea surface to the geoid, and the MSS heights refer to the sea surface. The statistical characteristics of this preliminary QSST are given in Table 6, from which it is evident that the QSST estimated presents some unreasonably large variations in the area (3.5 m) and reaches a maximum of almost 2 m. Therefore it is clear that blunders are present in the estimated field. Finally, noisy features are evident, thus low-pass filtering (LPF) was needed in order to reduce these effects.
247
248
G.S. Vergos. V. N. Grigoriadis. I. N. Tziavos • M. G. Sideris
20'
22' I
24"
I
I
I
2S"
,
0
2e'
I
r
r
r
r
20
24
28
32
36
30' I m
40
Fig. 3" The final referenced to EGM96 MSS model.
]m i
i
i
,
i
i
,
i
,
,
,
i
,
i
Fig. 4: Differences between KMS04 and the final referenced to EGM96 MSS model. Table 6. Statistics of the preliminary, before and after the 3cy test, and final QSST model and its differences with MDT04. Unit: [m]. F ( o ) )
max
QSST QSST (after 3cy) QSST (after 3~ and LPF) MDT04 - QSST
1.678 0.950 0.657 0.386
rain
mean
=
0)4_1_0)
4
(3)
std
-1.448 -0.125 +0.318 -0.953 -0.131 +0.287 -0.510 0 . 0 1 4 +0.238 -0.265 0 . 0 0 0 4-0.072
For the detection and removal of blunders, a simple 3cy test was performed, i.e., points with a QSST value larger than 3 times the a of the preliminary field were removed. The statistics of the QSST model after this test are given in Table 6 as well. To low-pass filter the preliminary QSST model, a collocation-type of filter (Wiener filtering) was used, assuming the presence of white noise in the QSST field. Furthermore, it was assumed that Kaula's rule for the decay of the geoid power spectrum holds, i.e., that the geoid height power spectral density decays like k -4 where k is the radial wavenumber. These resulted in the flowing filtering function
where co is the radial frequency, ( t ) = 4 U 2 -]-V 2 , and coc the cut-off frequency. To filter the wanted field, the desired cut-off frequency needs to be selected. The latter relates to the final resolution of the filtered field and the reduction of the noise in the data. Thus, a trade-off is necessary, since higher resolution means more noise will pass the filter, while higher noise reduction means lower resolution of the final model. A high resolution is vital in the determination of regional to local QSST models, since if a high value cannot be achieved then a so-derived local model has little to offer compared to a global solution. It can be clearly seen, that the disadvantage of Wiener filtering is that the selection of the cut-off frequency is based on the spectral characteristics of the field only, while its spatial characteristics are not taken into account. Furthermore, the selection of the cut-off fie-
Chapter 37 • Combination of Multi-Satellite Altimetry Data with CHAMPand GRACEEGMsfor Geoid and Sea Surface Topography Determination
quency is based on solely objective criteria. Thus, a trial and error process, based on maximum noise reduction with minimum signal loss, is needed to determine the desired cut-off frequency. Various cut-off frequencies have been tested corresponding to wavelengths of 5, 10, 20, 40, 60, 100, 110 and 120 km and finally a wavelength of 100 km (about 1o or harmonic degree 180) was selected since it offered the minimum signal loss with maximum noise reduction. Wavelengths shorter than 100 km left too much noise in the field, while those larger than 100 km were reducing not only the noise but some spatial characteristics of the QSST as well. If a longer wavelength was selected, then, and if the area was significantly larger (e.g. the entire Mediterranean Sea) it would have been possible to identify larger-scale QSST features and distinguish them from smaller ones. The problem in this case is that for the rest of the Mediterranean Sea only few ship tracks with gravimetric observations are available, therefore, a gravimetric geoid model cannot be determined at least at such high resolution (1'). The answer in such cases for geoid modeling is the combination of shipborne gravity data with satellite altimetry, but such a combination model cannot be used for QSST modeling (at least in the present context) due to the high correlation with the MSS model. The final QSST field after filtering is shown in Fig. 5 (top), while the statistics are given in Table 6 (last row). From the aforementioned figure it can be seen that the noise present in the preliminary model is reduced significantly, while blunders cannot be identified. For validation purposes the estimated QSST model was compared with a Mean Dynamic Topography (MDT04) model estimated for the entire Mediterranean Sea from an analysis of satellite altimetry and oceanographic data (Rio 2004). The latter was given as a grid of mean QSST values of 3.75'x3.75' resolution in both latitude and longitude. The statistics of the differences between the MDT and the estimated QSST models are given in Table 6 (last row). From the comparison it can be concluded that the two models agree very well to each other (std at the +7 cm level only). The maximum and minimum values of the differences are found close to land areas only, where both models are inadequate. This comparison gives evidence that the estimated QSST model is in good agreement with existing regional oceanographic MDT models. Furthermore, it is a welcoming fact, which supports the appropriateness of the proposed methodology for the determination of a geodetic QSST model. 5
Conclusions
A first validation of the ENVISAT and JASON-1 data in the eastern Mediterranean Sea has been performed, from which it was found that the former provide accurate results comparable to the other altimetric missions, while the latter are of lower accuracy compared to T/P
and present extensive gaps. The latter can be attributed to the radiometric correction problems in the JASON-1 data. The MSS models developed present very good agreement with the corresponding KMS01 and KMS04 ones, with their smallest difference being at the + 11 cm level. Compared to earlier results achieved by the authors, the newly developed MSS is of higher resolution (1' comparing to 5') and accuracy (+ 11 cm comparing to +17 cm) and presents an improved version. Furthermore, it can be concluded that at least in the present stage EGM96 is still comparable to the EIGEN/GRACE type of EGMS, but of course not in terms of the cumulative geoid error. Finally, the estimated QSST model provided very encouraging when compared to an oceanographic MDT model, with its differences only at the +7 cm. This is a tremendous improvement, since it can be used for local/regional geoid and gravity field modeling in the area, due to the inappropriateness of global MDT models in closed sea areas.
Acknowledgement
This research was funded from the Greek Secretariat for Research and Technology in the frame of the 3rd Community Support Program (Opp. Supp. Progr. 2000 - 2006), Measure 4.3, Action 4.3.6, Sub-Action 4.3.6.1 (International Scientific and Technological Co-operation with non-EU countries), bilateral co-operation between Greece and Canada.
References
Andritsanos VD, Vergos GS, Tziavos IN, Pavlis EC and Mertikas SP. (2001) A High Resolution Geoid for the Establishment of the Gavdos Multi-Satellite Calibration Site. In: Sideris MG (ed) Proc of International Association of Geodesy Symposia "Gravity Geoid and Geodynamics 2000", Vol. 123. Springer Verlag Berlin Heidelberg, pp 347-354. Andersen OB, Knudsen P (1998) Global gravity field from ERS1 and Geosat geodetic mission altimetry. J Geophys Res 103(C4): 8129-8137. Andersen OB, Vest AL, Knudsen P (2003) Altimetric Mean Sea Surfaces and inter-annual ocean variability. 2003 JASON-1 Sciency Working Team Meeting, Aries, France. AVISO (1998) AVISO User Handbook- Corrected Sea Surface Heights (CORSSHs), AVI-NT-011-311-CN, Ed 3.1. AVISO (2003) AVISO & PoDaac User Handbook-IGDR & GDR Jason-1 Products, SMM-MU-M5-OP-13184-CN, Ed. 2.0. Cazenave A, Schaeffer P, Berge M, Brosier C, Dominh K, Genero MC (1996) High-resolution mean sea surface computed with altimeter data of ERS 1 (geodetic mission) and TOPEX/POSEION. Geophys J Int 125: 696-704. ESA (2004) ENVISAT RA2/MWR Handbook, Issue 1.2. F6rste C, Flechtner C, Schmidt R, Meyer R, Stubenvoll R, Barthelmes F, K6nig R, Neumayer KH, Rothacher M, Reigber Ch, Biancale R, Bruinsma S, Lemoine J.-M, Raimondo JC (2005) A New High Resolution Global Gravity Field Model Derived From Combination of GRACE and CHAMP Mission and Altimetry/Gravimetry Surface Gravity Data. EGU General Assembly 2005, Vienna, Austria, April 24-29.
249
250
G.S. Vergos. V. N. Grigoriadis. I. N. Tziavos • M. G. Sideris
I
I
I
I
I 0.4
0,6
,
Im
.°'3 .o2 .~, 0'o o11 o12 0'. . . . . . . Fig. 5: The final (top) QSST model and its comparison with the oceanographic model.
Hwang C, Kao EC, Parsons B (1998) Global derivation of marine gravity anomalies from Seasat, Geosat, ERS1 and TOPEX/POSEIDON altimeter data. Geophys J Int 134: 449459. Lemoine FG, Kenyon SC, Factor JK, Trimmer RG, Pavlis NK, Chinn DS, Cox C, Klosko SM, Luthcke SB, Torrence MH, Wang YM, Williamson RG, Pavlis EC, Rapp RH, Olson TR (1998) The development of the join NASA GSFC and NIMA geopotential model EGM96, NASA Technical Paper, 1998 206861. National Oceanographic and Atmospheric Administration NOAA (1997) The GEOSAT-GM Altimeter JGM-3 GDRs. Olesen AV, Tziavos IN, Forsberg R (2003) New Airborne Gravity Data Around Crete - First results from the CAATER Campaign. In: Tziavos IN (ed) Proc of the 3rd Meeting of the Gravity and Geoid Commission "Gravity and Geoid 2002", pp 4044. Rio M.-H. (2004) A Mean Dynamic Topography of the Mediter-
ranean Sea Estimated from the Combined use of Altimetry, InSitu Measurements and a General Circulation Model. Geoph Res Let Vol. 6, 03626. Tziavos IN, Sideris MG, Forsberg R (1998) Combined satellite altimetry and shipborne gravimetry data processing. Mar Geod 21: 299-317. Tziavos IN, Vergos GS, Kotzev V, Pashova L (2004) Mean Sea Level and Sea Level Variation Studies in the Black Sea and the Aegean. Presented at the Gravity Geoid and Space Missions 2004 (GGSM2004) conference, August 30 - September 3, Porto, Portugal (accepted for publication to the conference proceedings). Vergos GS, Tziavos IN, Andritsanos VD (2005) On the Determination of Marine Geoid Models by Least Squares Collocation and Spectral Methods Using Heterogeneous Data. In: Sans6 F (ed) Proc of International Association of Geodesy Symposia "A Window on the Future of Geodesy", Vol. 128. SpringerVerlag Berlin Heidelberg, pp 332-337.
Chapter 38
A new methodology to process airborne gravimetry data: advances and problems B.A. Alberts, R Ditmar and R. Klees Delft Institute of Earth Observation and Space Systems (DEOS), Delft University of Technology, Kluyverweg 1, PO box 5058, 2600 GB Delft, The Netherlands.
Abstract. In the framework of the development of a new methodology for the processing of airborne gravity data, we discuss the problem of edge effects and the treatment of long-wavelength errors in the local geoid solution. The new methodology combines several pre-processing steps, such as filtering and cross-over adjustment, with the parameter estimation. The base functions used in the analytical representation of the disturbing potential are the fundamental solutions of Laplace's equation in Cartesian coordinates. This implicitly assumes periodicity in the gravity data, which does not hold in practice. The limitation to the local area introduces highly oscillating distortions in the adjusted gravity disturbances, which are mainly located along the boundary of the area. These distortions cause long-wavelength errors in the geoid over the whole area. We investigate five approaches to reduce these effects, using a simulated data set. Among these methods, least-squares prediction used to extend the data set gives the best gravity field solution in terms of gravity disturbances. However, the solution still suffers from long-wavelength geoid errors, which partially reflects the non-uniqueness of local geoid determination from airborne gravity data. Therefore, two alternative methods are investigated, which aim to solve this problem. We show that it is possible to slightly reduce the long-wavelength errors, in particular at the center of the area.
Keywords: Airborne gravimetry, regional gravity field determination, edge effects
1
Introduction
For many applications in gravity field modeling, the resolution of satellite-only models derived from current and future satellite missions will not be sufficient. The most suitable technique to determine the short-wavelength information is airborne gravimetry because it can provide gravity observations in a fast
and efficient way. In Alberts et al. (2005) a new methodology is proposed for the processing of airborne gravity measurements, aimed at the computation of gravity field functionals, e.g. geoid heights or gravity anomalies. It is based on a spectral representation of the gravity field. The gravitational potential is parameterized as a linear combination of harmonic functions, which are fundamental solutions of Laplace's equation in Cartesian coordinates. The parameters of this representation are estimated using least-squares techniques. The methodology uses a frequency-dependent data weighting strategy, similar to the one developed by Klees and Ditmar (2004) for the processing of CHAMR GRACE and GOCE data. The base functions used in the representation of the gravity potential are periodic in the horizontal directions. As a consequence, the gravity signal is assumed to be periodic as well, which does not hold in practice. Inequality at the opposite boundaries of the computation area result in strong oscillations that propagate inside the area. Furthermore, when we compute geoid heights the results get distorted by long-wavelength errors, which affect the whole computation area. An additional factor that increases these low-frequency errors is that the gravity signal outside the computation area is neglected. In the first part of this paper five methods that may be used to reduce edge effects are discussed and compared in terms of gravity disturbance and disturbing potential errors. First, the computation area may be extended at flight level, by computing gravity disturbance values using either a taper technique or leastsquares prediction. Especially the latter method reduces edge effects significantly, but may not be applicable for the processing of data contaminated by colored noise. Second, the set of base functions can be extended in such a way that the requirement of periodicity is evaded. The last part of the paper deals with the reduction of the low-frequency errors, that distort the computed geoid heights. In order to solve this problem we corn-
252
B.A. Alberts
• P. D i t m a r
• R.
Klees
pare two approaches, both of which make use of prior long-wavelength information added to the functional model.
2
Representation of the gravity field
2.1 Model description For the representation of the disturbing potential we use a linear combination of harmonic functions, that are the fundamental solutions of Laplace equation in Cartesian coordinates (Alberts et al., 2005): L
with C O S ~2~zx ~
~
1> __ 0
sin 2,~lllx 1 < 0
COS
(Y) --
2rcmy du
~
sin 2~l,~ly dv ,
Ax)TQy-l(y-
//Z >
0
fr~
.................... Fig. 4 Longitudinal profiles-block N02W063.
3° °
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Fig. 6 Longitudinal profiles - block S03W063.
407.0t 370.01
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.
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-69.9
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Fig. 5 Longitudinal profiles - block N00W070.
-99.1
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.
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Fig. 7 Longitudinal profiles - block S06W072.
.
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269
270
D. Blit zkow • A. C. O. C. de M a t o s . J. P. Cintra
8O.O
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River
( L a t . : - o o o 10' 3 3 " ) 60.0
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Fig. 12 Latitudinal profiles - block S04W064.
Fig. 8 Longitudinal profiles - block SO1W063.
..... |,1,|, ~ l
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Fig. 9 Longitudinal profiles - block S02W064.
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Fig. 13 Longitudinal profiles - block S03W056
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4. Conclusions
t
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Negro River
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-1.0
In general, when compared with the BMs heights, SRTM shows differences compatible with the precision specified for the model. In Tables 1 and 2, r e g a r d i n g SRTM, the differences greater than 50 m are due whether to the uncertainty in BM coordinates or the quality of the model. The position of the BM has to be improved. Most BMs in the Amazon region, although in a reduced number and with a located distribution, present a satisfactory behavior. In the considered region profiles (Figures 4 to 13) show that the models with 5' spacing are not representative of the t o p o g r a p h y . The a n a l y s i s also s h o w s that GTOPO30 and GLOBE coincide to each other in many of these blocks, the same occurring with T E R R A I N B A S E and JGP95E. The reason is that the two pairs of models use the same information source. The R A D A R signal theoretically reflects on the water. Therefore, the implied model should provide values of the height referred to that surface. Nevertheless, due to the SRTM vertical accuracy, the height over the rivers oscillates around a mean value. In the regions of umbriferous forest SRTM
Chapter 40 • SRTM Evaluation in Brazil and Argentina with Emphasis on the Amazon Region
30" profiles, show vertical offset around 30 meters. This fact suggests that radar wave is reflecting on the canopy and not on the ground.
5. Acknowledgements W e a c k n o w l e d g e to the B r a z i l i a n Institute o f G e o g r a p h y and Statistics ( I B G E ) , the M i l i t a r y Geographic Institute (IGM) of Argentina and Maria Cristina Pacino of the Rosfirio National University, for m a k i n g a v a i l a b l e the i n f o r m a t i o n o f the levelling network.
References Arabelos, D. (2000). Intercomparisons of the global DTMs ETOPO5, TerrainBase and JGP95E. Physics and Chemistry of the Earth Part A, 25(1), pp. 89-93. Hasting, D.A., and P.K. Dunbar (1999). Global Land Onekilometer Base Elevation (GLOBE) Digital Elevation Model, Documentation, Volume 1.0. Key to Geophysical Records Documentation (KGRD) 34. National Oceanic and Atmospheric Administration, National Geophysical Data Center, 325 Broadway, Boulder, Colorado 80303, U.S.A. Hensley, S., R. Munjy, P. Rosen (2001). Interferometric synthetic aperture radar. In: Maune, D. F. (Ed.). Digital elevation model techonoligies applications." the D E M users manual. Bethesda, Maryland: ASPRS (The Imaging & Geospatial Information Society), cap. 6, pp. 142-206. Hutchinson, M.F. (1989). A new procedure for gridding elevation and stream line data with automatic removal of spurious pits. Journal of Hydrology, 106, pp. 211-232. JPL (2004). SRTM - The Mission to Map the World. Jet Propulsion Laboratory, California Inst. of Techn., http://www2.jpl.nasa.gov/srtm/index.html.
Johnson, C.P., P.A.M. Berry, and R.D. Hilton (2001). Report on A CE g e n e r a t i o n , Leicester, UK, http:// www.cse.dmu.ac.uk/geomatics/ace/ACE_report.pdf. Lemoine, F.G., N.K. Pavlis, S.C. Kenyon, R.H. Rapp, E.C. Pavlis, and B.F. Chao (1998a). New high-resolution modle developed for Earth' gravitational field, EOS, Transactions, AGU, 79, 9, March 3, No 113, 117-118, 1998. Lemoine, F.G., S.C. Kenyon, J.K. Factor, R.G. Trimmer, N.K. Pavlis, D.S. Chinn, C.M. Cox, S.M. Klosko, S.B. Luthcke, M.H. Torrence, Y.M. Wang, R.G. Williamson, E.C. Pavlis, R.H. Rapp and T.R. Olson (1998b). The development of the joint NASA GSFC and the National Imagery and Mapping Agency (NIMA) geopotential model E G M 9 6 , NASA/TP-1998-206861. National Aeronautics and Space Administration, Maryland, USA. Matos, A.C.O.C. (2005). lmplementa¢~to de modelos digitais de terreno para aplica¢Oes na Orea de GeodOsia e Geofisica na AmOrica do Sul. PhD thesis - Escola Politdcnica, Universidade de Silo Paulo, Silo Paulo, 355 p. NOAA (1988). Data Announcement 88-MGG-02, Digital relief of the Surface of the Earth. NOAA, National Geophysical Data Center, Boulder, CO. http ://www.ngdc.noaa. gov/mgg/global/etopo5.html. NOAA (1995). TerrainBase Global Digital Terrain Model, Version 1.0, NOAA, National Geophysical Data Center, Boulder, CO. http://www.ngdc.noaa.gov/seg/fliers/se1104.shtml. NOAA (2001). 2-minute Gridded Global Relief Data (ETOP02). NOAA, National Geophysical Data Center, Boulder,CO. http ://www.ngdc.noaa. gov/mgg/fliers/01mgg04.html. Saley, J., and N.K. Pavlis (2002). The development and evaluation of the global digital terrain model DTM2002, 3rd Meeting of the International Gravity and Geoid Commission, Thessaloniki, Greece. U.S. Geological Survey (1997). GTOP030 Global 30 Arc Second Elevation Data Set, at http:/!edcwww.cr.usgs.gov/landdaac/gtopo30/gtopo30.ht ml.
271
Chapter 41
On the Estimation of the Regional Geoid Error in Canada J. Huang (l~, G. Fotopoulos (2), M. K. Cheng (3), M. Vdronneau (1), M. G. Sideris (4~ (1) Geodetic Survey Division, CCRS, Natural Resources Canada, 615 Booth Street, Ottawa, Ontario, K1A 0E9, Canada (2) Dept. of Civil Engineering, University of Toronto, 35 St. George Street, Toronto, ON, M5S 1A4, Canada (3) Center for Space Research, University of Texas at Austin, 3925 West Braker Ln. #200, Austin, Texas 78759, USA (4) Dept. of Geomatics Engineering, 2500 University Drive N.W. Calgary, Alberta T2N 1N4 Canada The geoid errors for the Canadian Gravimetric Geoid 2005 (CGG05) model are estimated from the error information of the satellite and terrestrial gravity data. Calibration is conducted through the application of variance component estimation (VCE) with GPS-leveling data and their associated covariance matrices. Preliminary results suggest that the error of the geoid heights is generally smaller than 6 cm in Canada, with a range from 6 cm to 31 cm for the Western Cordillera area. Overall, the average error of the CGG05 model is estimated at 5.5 cm.
(CGG05) model (includes the satellite gravity model, terrestrial gravity data, gravity reduction methods and the far-zone contribution error), and (ii) to develop an adequate method for the estimation of the gravimetric geoid error from these error sources. In particular, the iterative almost unbiased estimation (IAUE) scheme is implemented to validate/calibrate the geoid error using existing GPS-leveling data across Canada.
Keywords. Geoid, gravity, error estimation
The CGG05 model was computed using the degreebanded Stokes integral, which is described in Huang and Vdronneau (2005). In order to illustrate how errors propagate into the geoid model, the formula for the geoid determination is simplified as
Abstract.
1 Introduction With the increased use of GPS-based positioning, the demand for directly converting ellipsoidal heights to heights referred to a regional vertical datum with sufficient accuracy is also increasing. For Canada, the use of a gravimetric geoid as the national height reference surface is currently under study. The recent revolutionary development of space gravimetry (i.e., CHAMP, GRACE and the upcoming GOCE mission) offers the opportunity to pursue such a dramatic change in the local vertical datum. However, before a vertical datum based on a gravimetric geoid is adopted, as opposed to one based on conventional leveling networks, it is important to conduct a reliable assessment of the systematic and stochastic errors of the geoid model. This is a challenging task as there is limited or poor information regarding the quality of the terrestrial gravity data, difficulty in quantifying the gravity reduction and interpolation errors, as well as only approximate estimates of the errors associated with the satellite models. The purpose of this paper is twofold, namely (i) to provide a detailed examination of all error sources of the Canadian Gravimetric Geoid 2005
2 Methodology 2.1 Estimation of Gravimetric Geoid Error
N - N S C + R iSDB(V)(Agm_AgSC)d.O,+Fy (1) 4~7 ~40' where the first term on the right hand side of eq. (1) represents the geoid components below spherical harmonic degree L+ 1 from a satellite model (SG), R is the mean radius of the Earth, 7 is the normal gravity on the reference ellipsoid, and Ag denotes the gravity anomalies. The degree-banded Stokes kernel can be expressed as mTG 2 n + l S DB(~1/) -- Z Pn (COS~)
n--L+1 n - 1
It is used as a band-pass filter to compute the geoid components from degree L+I to mT6. The upper limit mTG is dependent on the terrestrial gravity (TG) data spacing. The Stokes integration is performed within a spherical cap limited to a spherical angular distance of 6 arc-degrees. This implies that the band-pass filtering is incomplete and renders aliasing geoid errors that account for an RMS of approximately 2 cm over Canada.
Chapter 41
Finally, FN is the far-zone contribution outside the Stokes integration. It can be evaluated from a combined global spherical harmonic gravity (GGM) model up to its maximum degree (preferably larger than degree 200):
R ~ DB CG 7_. Qn gn 27 n=L+l
FN
•
on the Estimation of the Regional Geoid Error in Canada
terrestrial gravity data in Canada, which inherently contains a combination of errors originating from several sources, including gravity measurements, height measurements at the gravity points, topographic reduction, interpolation of gravity values, digital elevation models (DEM) and actual topographical density distribution. 2.2 Calibration of the Geoid Error
where QDB(V0) -- fo SDB(V)Pn (COSV) sin v d v The geoid error is primarily comprised of errors from the satellite-only gravity model, the combined global gravity models, and the terrestrial gravity data. The satellite gravity signal usually dominates the low-degree part of the geoid components in a combined model while the terrestrial gravity data complete the GGM for higher degrees and orders (Sideris and Schwarz, 1987). For regional geoid determination, the lower limit, L, must be selected according to the quality of the satellite data. Empirical tests show that it should not exceed 30 for GGMs prior to the CHAMP/GRACE missions, if a decimeter-level accurate geoid is sought. A simplified expression for the geoid error is given by: V N = VSG + VTG + VCG
where _ V SG
R (2 ~
n=2
n- 1
Geoid heights can also be determined at co-located GPS and leveling stations, which provides an independent external means to validate and calibrate the gravimetric geoid model and its precision. The discrepancies between the GPS/levelingderived and gravimetric geoid heights can (predominantly) be attributed to a combination of systematic and random errors in the ellipsoidal heights (h), the orthometric heights (H), and the gravimetric geoid heights (N), as discussed in Kotsakis and Sideris (1999). The following general linear functional model was used for the combined (multi-data) least-squares adjustment of the heterogeneous height data: Ax + Bv + w - 0, E{v}- 0
(6)
B-[I
(7)
where
(2)
+ QDB n
V--
-I VH
-I VSG
w-h-H-N R ~SDB (v)eTGAO , VTG = 4rc7 R
mcG ,--. DB CG
- 2_t~ v c~ = - 2"7
~
-I VTG
-I] VCG
(8)
(9)
(4)
(5)
n=L+l
Vso, vm, and Vco represent the geoid errors from the satellite model, the terrestrial gravity data, and the combined model, respectively. Given the covariance (CV) matrices for each of these three types of errors, the geoid standard deviation (std) can be estimated based on eqs. (3) to (5) via error propagation. In our case, the geoid std may be evaluated only approximately because the CV matrices for the satellite and combined harmonic gravity models are approximate. Furthermore, only approximate error values are available for the
The deterministic term, Ax, introduced in eq. (6) represents the parametric model used to approximately model the systematic errors inherent in and among all three types of heights. The selection procedure for the type of model and assessing its validity has been discussed extensively in Fotopoulos (2003). For this particular case a simple four-parameter model was found to be sufficient and therefore incorporated for all calculations. Individual variance components are estimated using the adjustment model in eq. (6) to (9) and the a-priori CV matrices for each of the data types (see Fotopoulos, 2003 for the detailed procedure). This procedure was followed in this study in order to achieve a more realistic estimate of the geoid model
273
274
J. Huang • G. Fotopoulos • M. K. Cheng • M. Vl:ronneau • M. G. Sideris
error that incorporates five individual variance components for the ellipsoidal heights and the orthometric heights at the GPS-leveling benchmarks,
denoted
by
crh2
and
2 crH,
respectively. The geoid height errors are separated for the satellite gravity model, terrestrial gravity data, and the combined gravity model, denoted by craG, cr2G and O'~G, respectively.
The CGG05 geoid model is validated using 430 colocated GPS-leveling stations with a distribution as depicted in Figure 1. The computed h-H-N residuals for the 430 stations plotted as a function of longitude and latitude are shown in Figure 2. The overall standard deviation of these residuals is 10.2 cm. A negative mean value of-40 cm indicates that the zero-height point of the leveling network is approximately 40 cm lower than the CGG05 geoid model.
3 Gravity and GPS-leveling data .
The lower degrees (2 to 90) of the GRACE-based GGM02C model (Tapley et al., 2005) are used for the determination of the long wavelength components of the geoid while the higher degrees (91 to 200) determine the far-zone contribution of the Stokes integration. EGM96 (Lemoine et al., 1998) is used to extend the GGM02C up to degree and order 360. The local residual terrestrial gravity data, i.e., ground, airborne, shipboard (including satellite altimetry-derived), are used to compute the geoid components above degree 90. These terrestrial data are the same as those used for the CGG2000 model (V6ronneau, 2002). The latest model, CGG05, is a high-resolution geoid model for North America with a geographical spacing of 2 minutes of arc along latitudes and longitudes. Its reference ellipsoid is GRSS0 and the reference frame is ITRF (no specific realization). Canadian GPS surveys after year 1994 were used in a least-squares adjustment to compute the ellipsoidal heights with respect to the GRS80 reference ellipsoid and their associated variances/covariances (Craymer and Lapelle, 2004; pers. comm.). The reference frame is ITRF97. The geopotential numbers are determined from a minimally constrained least-squares adjustment (via Helmert-blocking) of the geodetic leveling observations after year 1981. The single fixed station is a benchmark located along the StLawrence River in Rimouski, Qudbec, which is the same constraint used for the North American Vertical Datum of 1988 (NAVD88). Gravity values are interpolated at each benchmark from local measurements and converted to mean values along the plumbline (from geoid to topography) by correcting for the variable terrain. The variances and covariance within each Helmert block are implemented for the calibration of the geoid error, while the correlation between neighboring Helmert blocks have been omitted at this stage.
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The observed north-south and east-west slopes evident in Figure 2 result in a standard deviation of about 8 cm and are caused by systematic errors attributed to all three types of heights. In general, the uncertainty in the location of the ITRF97 reference frame geocenter can generate systematic errors of a few centimeters in the ellipsoidal heights (Altamimi et al., 2002), while the GRACE data may introduce systematic errors of less than a few
Chapter 41 • on the Estimation of the Regional Geoid Error in Canada
centimeters in the low-degree components of the gravimetric geoid heights. However, the leveling data are most likely the major source of systematic errors that accumulate over the 6000 km separation between the east and west coasts. Current knowledge about the mathematical and physical characteristics of the systematic errors in the leveling data is limited and therefore it is difficult to accurately model and correct for these discrepancies.
4 Geoid Error from the GGM02C Model The error model of the GGM02C model propagates into the low-degree components and far-zone contribution of the Canadian geoid model. The error information of the GGM02C model is provided in terms of error coefficients obtained from the diagonal elements of the covariance matrix. Figures 3a and 3b show the low-degree geoid error estimates from the diagonal-only terms and from the fully-populated covariance matrix of the GGM02C model, respectively. In this case, the use of diagonal-only elements does not provide sufficient information for the estimation of the geoid errors.
0.1
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The far-zone contribution error was found to be closely approximated by a constant of 1.6 cm according to the diagonal CV matrix and 0.7 cm to 1.8 cm if the fully-populated form of the CV matrix is utilized. Again, it was determined that the fullypopulated covariance matrix is needed to evaluate the far-zone contribution error. Initial covariance matrices of the low-degree and far-zone contribution components of the geoid heights at the 430 GPS-leveling stations have been estimated from the CV matrix of the harmonic coefficients of GGM02C.
5 Geoid Error from the Terrestrial Data The terrestrial gravity error is comprised of measurement, datum, data reduction and interpolation errors. A fully populated covariance matrix for this data is not available; however, the standard deviation at each gravity station can be estimated from the measurement and elevation standard deviations. By neglecting the covariance between any two gravity stations, and the datum and interpolation errors, the initial geoid error standard deviations can be estimated via simple error propagation of Stokes integration (Li and Sideris, 1994). Figure 4 depicts the computed geoid error standard deviations based on the terrestrial gravity data, which provides an average error of 1.5 cm across Canada and a maximum of 15.7 cm in the western region. These values are most likely too optimistic due to the obvious omissions mentioned previously. However, since this is the best information currently available, these values were used to construct the initial CV matrix for the terrestrial gravity component at the 430 GPSlevelling points.
L20'E
6 Estimation of Variance Components
o.!
Io, I~'W
0.0
0.3
O'
¢~'W
1.0
e",O"E 1.5
L"O' I-] 2.0
b~,From E,,IIIC..,V,'n~lTi~
Figure 3. GGM02C geoid error estimates for degrees 2 to 90 based on (a) diagonal-only CV matrix and (b) fully populated CV matrix
2.5
Given the initial covariance matrices corresponding to the h, H, Ns~, NTG and Ncc data, the geoid errors estimated from the GGM02C model and from the terrestrial data can be verified and calibrated at the GPS-leveling stations as per the procedure described in section 2.2. Figure 5 shows the mean covariances with respect to the spherical distance computed from the initial CV matrices for the (a) ellipsoidal heights, (b) orthometric heights, (c) satellite and (d) terrestrial geoid errors at the 430 GPS-leveling stations, respectively.
275
276
J. Huang • G. Fotopoulos • M. K. Cheng • M. V[~ronneau • M. G. Sideris
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Figure 5. Initial mean covariance (in mm 2) via spherical distance for (a) ellipsoidal heights, (b) orthometric heights, (c) satellite geoid heights, and (d) residual terrestrial geoid heights
Chapter 41
The ellipsoidal heights are wealdy correlated across the GPS-leveling network with a range in standard deviations from 0.2 cm to 7.6 cm and an average value of 1.3 cm. In general, these values obtained directly from the results of the post-processing software tend to be on the optimistic side. The orthometric heights are strongly spatially correlated within each of the Helmert blocks. The corresponding standard deviations increase as the distance with respect to the 'fixed' station in Rimouski increases. These values reach a maximum of approximately 9 cm with a mean standard deviation of 5.4 cm. The low-degree geoid components from the GRACE model values are correlated because they are evaluated from the same set of spherical harmonic coefficients. Standard deviations for this data vary from 0.5 cm to 1.4 cm with a mean of 0.9 cm. The covariance matrix for NcG is similar to that of Nso (Figure 5c) and therefore not shown. The residual geoid components evaluated from the degree-banded Stokes integral exhibit a high correlation only when common data have been used between any two computational points. In this study, the integration cap radius is 6 arc-degrees, which indicates a correlation for any two computational points located within a spherical angular distance of less than 6 arc-degrees. The standard deviations of the residual components of the geoid model are evaluated from the errors of the gravity anomaly data and range from 0.3 cm to 8.8 cm with a mean value of 1.4 cm. Using the iterative almost unbiased variance component estimation scheme (Horn and Horn, 1975; Fotopoulos, 2003) and the a-priori CV matrices described above, five individual variance components were estimated. These values are tabulated in Table 1 for two scenarios, namely (i) diagonal-only CV matrices and (ii) fully-populated CV matrices (where available). The sensitivity of the estimated variance factors to the a-priori covariance information is evident from the differences between the estimated variance factors in each scenario. As expected, if only the diagonal information of the matrices is used as an approximation, the computed variance components are (in general) low for data where correlation is evident (e.g., orthometric heights). The estimates in Table 1 suggest that the a-priori CV matrices corresponding to the ellipsoidal, orthometric and geoid heights are too optimistic,
•
on the
Estimation of
the Regional Geoid Error in Canada
with final estimated variance components suggesting a re-scaling of the a-priori CV matrices of more than 3. The result for the far-zone contribution is less conclusive. In all cases, the number of iterations remained constant at approximately 70. Table 1. Estimated variance factors using fullypopulated and diagonal a-priori covariance matrices (n is the number of iterations)
CV
"2 (3"h
@2 H
(~2 SO
(~2 TO
C~co
"2
n
Diagonal
2.24
0.03
7.71
2.69
6.00
72
Full
9.09
5 . 8 5 3.19
3.61
0.01
69
7 Total Geoid Error The total geoid error is finally computed from the scaled CV matrices (after variance component estimation) of the three estimated components corresponding to SG, TG and CG. Assuming that the variance factors in Table 1 are applicable for non-GPS/leveling points (albeit a bold assumption), the total calibrated geoid error for CGG05 is illustrated in Figure 6 on a 2' x 2' grid. This assumption will be further tested using additional data that was not implemented in this study. The calibrated geoid error ranges from 1 cm to 32 cm, with a mean error of 5.5 cm across the entire Canadian landmass. The mountainous areas of western Canada (up to Alaska) exhibit the largest errors due to sparse gravity anomaly error information (i.e., 2' in the mountains is insufficient). The geoid error in central Canada is generally smaller than 6 cm, with the hilly regions in eastern Canada showing slightly larger geoid errors. In particular, the geoid errors in the Fox Basin and Ungava Bay are significantly larger than those of the surrounding regions, due to the lack of terrestrial gravity data.
8 Discussion of Future Work The progress made in this study represents a significant step forward to achieving realistic error estimates for the Canadian gravimetric geoid model. However, it should be stated that the total geoid errors shown in Figure 6 are preliminary and refinements are ongoing. In particular, major improvements are expected on three fronts, namely (i) the inclusion of additional GPS-leveling data for a regional calibration based on the geographical
277
278
J. Huang • G. Fotopoulos • M. K. Cheng • M. V[~ronneau • M. G. S i d e r i s
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Figure 6. Regional geoid error map for Canada estimated using GPS-leveling data
heterogeneity of the data, (ii) the incorporation of the correlation between the terrestrial gravity data and (iii) the verification of the reliability of the estimated variance components, through an external validation process. Further tests will also be conducted to re-evaluate the suitability of the deterministic term introduced in the functional model for the systematic errors (i.e., type of parametric model).
Acknowledgements The authors thank Dr. Mike Craymer and Dr. Joe Henton for their constructive comments. Appreciation is also extended to Dr. Robert Tenzer and an anonymous reviewer for their critical review.
References Altamimi Z, P Sillard, and C Boucher (2002) ITRF2000: A new release of the International terrestrial Reference Frame for earth science applications, J. Geophys. Res. 107(B10), 2214, doi: 10.1029/2001 JB00056
Fotopoulos G (2003) An analysis on the optimal combination of geoid, orthometric and ellipsoidal height data. PhD Thesis, University of Calgary, Dept. of Geomatics Engineering, No. 20185, Calgary, Canada Horn SD and RA Horn (1975) Comparison of estimators of heteroscedastic variances in linear models. Journal of the American Statistical Association, 70( 132):872-879 Huang J and M V6ronneau (2005) Applications of downward continuation in gravimetric geoid modeling - case studies in Western Canada, Journal of Geodesy, 79:135-145 Kotsakis C and MG Sideris (1999) On the adjustment of combined GPS/leveling/geoid networks. Journal of Geodesy, 73:412-421 Lemoine FG, SC Kenyon, JK Factor, RG Trimmer, NK Pavlis, DS Chinn, CM Cox, SM Klosko, SB Luthcke, MH Torrence, YM Wang, RG Williamson, EC Pavlis, RH Rapp, TR Olson (1998) The development of the joint NASA GSFC and NIMA geopotential model EGM96. NASA/TP1998-206861 GSFC, Greenbelt, MD
Chapter 41 • on the Estimation of the Regional Geoid Error in Canada
Li Y and MG Sideris (1994) Minimization and estimation of geoid undulation errors. Bulletin GOodOsique, 68(4):201-219 Sideris MG and KP Schwarz (1987) Improvement of medium and short wavelength features of geopotential solutions by local gravity data, Bolletino di Geodesia e Scienze Affini, 3:207-221 Tapley B, J Ries, S Bettadpur, D Chambers, M
Cheng, F Condi, B Gunter, Z Kang, P Nagel, R Pastor, T Pekker, S Poole, and F Wang (2005) GGM02 - An Improved Earth Gravity Field Model from GRACE, Journal of Geodesy, 79:467-478 Vdronneau M. (2002) Canadian gravimetric geoid model of 2000 (CGG2000). Report of the Geodetic Survey Division, Natural Resources Canada, Ottawa, ON (available by request)
279
Chapter 42
A Validation Procedure for Satellite Orbits and Force Function Models Based on a New Balance
Equation Approach A. L6cher, K.H. Ilk Institute of Theoretical Geodesy, University of Bonn, Nussallee 17, D-53115 Bonn, Germany
Abstract. Since the availability of CHAMP and GRACE data, the energy approach has become an important tool for the recovery of the gravity field based on continuously observed satellite orbits. Up to now, only the total energy of the satellite's threedimensional motion has been considered which is known as the Jacobi integral if formulated in a constantly rotating Earth-fixed reference frame. Beside this, additional energy integrals can be found for the components of the satellite's motion and various combinations hereof, starting from the three scalar components of Newton' s equation of motion. Furthermore, integrals of motion based on the linear momentum and the angular momentum can be formulated which show even better mathematical characteristics than the Jacobi integral for the determination of the gravity field. Therefore, this new approach seems to be appropriate to validate the consistency of gravity field models and precisely observed satellite orbits and to improve, subsequently, these gravity field models. The advantages and critical aspects of this approach are investigated in this paper. First results with real data were presented using kinematic CHAMP orbits.
Keywords. CHAMP, GRACE, integrals of motion, energy integral, Jacobi integral, balance equations, gravity field recovery, validation
1 Introduction The continuously observed orbits of low flying satellites (LEO - Low Earth O r b i t e r ) b y Global Navigation Satellite Systems (GNSS) as GPS or GLONASS and in future Galileo suggest a paradigm shift of gravity field determination techniques. Instead of the analysis of accumulated orbit perturbations of artificial satellites caused by the inhomogeneous structure of the gravity field the local in-situ analysis techniques gain more and
more importance. These new gravity field determination concepts can be divided in three groups. All of them require densely tracked orbits of the low flying satellites with high accuracy by a GNSS (CHAMP type satellites) or precisely measured inter-satellite functionals between two satellites (GRACE-type twin satellites). A first possibility is based on the formulation of Newton's equation of (relative) motion as integral equation of Volterra or Fredholm type. A second approach uses the equation of motion directly and a third intermediate technique exploits the balance equations of classical theoretical mechanics. The latter technique has been applied yet by using the energy balance principle in form of the so-called Jacobi integral. For an overview of modem techniques of gravity field determination with artificial satellites cf. Schneider (2002). Besides the use of the energy integral approach for gravity field recovery tasks there is also another area of application which has been demonstrated by Ilk and L6cher (2003) and L6cher and Ilk (2005): Because of the balance principle of the energy integral it represents a sort of "absolute" criterion for the proof of consistency of observed orbit and the dynamical model of the satellite's motion along the orbit. If the various energy constituents do not sum up to a constant then either the orbit is incorrect or the force function models are wrong or imperfect. The size of the constant is only of secondary importance, while the structure of the deviations from the constant may give hints to specific force function or orbit determination deficiencies. This is based on the fact that the energy exchange relations caused by the various force function components show typical properties which can be used to separate the different sources of inconsistency, especially those who show specific patterns of deviations in the space and the time domain. It is obvious that validation and gravity field improvement cannot be separated rigorously because any systematic inconsistency of observations and reference model can be used as well
Chapter 42
• A
Validation Procedure for Satellite Orbits and Force Function Models Based on a New Balance Equation Approach
to improve the parameters of the force function models. In this respect the proposal made in the following can be used in a first step as validation procedure to proof the consistency between observations and reference force function model and in a second step to improve those parameters of the reference model which have obviously led to the inconsistencies. This paper addresses alternative integrals of motion for gravity field improvement and validation tasks in comparison to the use of the Jacobi integral. Obviously it has been overlooked in the past that there are numerous alternative balance equations which can be used for the gravity field recovery as well. It will be shown that these alternative balance equations seem to have partly much better properties for validation and gravity field improvements than the frequently used Jacobi integral. In Sect. 1, the integrals of motion are reviewed, then some alternative integrals of motion are presented and the dependencies between the various balance equations are presented. In Sect. 2 selected balance equations such as the balance equation of linear and angular momentum, the Jacobi integral and the coordinate related energy integrals are discussed in the time domain. Sect. 4 includes a discussion of these balance equations in the space domain. In Sect. 5 the important question is investigated, in which way insufficiently modeled surface force constituents acting on the satellites are mapped into the gravity field parameters. Sect. 6 concludes this paper.
time interval[t0,t ] . A similar relation for the angular momentum can be derived. Vector multiplication of Newton's equation of motion with R , MRxR-RxK =0, (2.3) and integration over the time interval [to,t ] results in, t
L - [ R x K dt - L o •
(2.4)
to
This relation connects the angular momentum at time t, L = MR x R , with the angular momentum at time to, L0, and the torque, integrated within the time interval [t0,t ] . The energy balance equation has been derived following a similar procedure: It follows by scalar multiplication with R , MR.R-R.K = 0, (2.5) and after integration over the time interval [t0,t],
_1M R ~ 2
t
(2.6)
fR. K dt - E. d [0
This relation represents the energy integral along the satellite's orbit. Based on this equation the wellknown Jacobi integral can be derived easily if K are conservative forces and referred to a constantly rotating Earth fixed reference system. We realize that the general procedure to derive balance equations based on Newton's equation of motion consists in formulating the basic relation as follows,
f(M,R,R,
ft)-g(M,R,R,K)-O,
(2.7)
and integrating over the time interval [t0,t], t
F(M,R,R)-Ig(M,R,R,K)dt-C.
2 Integrals of motion 2.1 Some classical balance equations We start from the well-known Newton's equation of motion, which was derived from the balance equation of linear momentum by considering a constant mass M:
MR-K =0. (2.1) The quantity R is the acceleration of the satellite and K the force function acting on the satellite. Integration over a time interval [t0,t ] starting from an initial time t o results in t
P - JK dt - P0,
(2.8)
to
(2.2)
The left hand side represents the "kinetic" term, the right hand side the force function integral. In the next section we will derive various alternative energy balance equations following this generalized procedure.
2.2 Alternative energy balance relations We can derive "energy balance equations in the coordinate directions" if we begin with MR i - K i = 0 , for i e {x,y,z}
(2.9)
and multiply these equations with
to
which connects the linear momentum at a time l, P = M R , with the initial linear momentum at time 10, P0, and the integrated force function within the
MRik i - Ki/~ i = 0.
(2.10)
By integration over the time interval [t0,t ] we receive three energy balance relations of the form:
281
282
A.L6cher• K.H. Ilk t .
.
.
0E 0R = P0.
.
(2.17)
to
These formulae represent the energy integrals of the three-dimensional motion of the satellite in the coordinate directions e~, i E {x,y,z} . Another three energy balance relations can be derived if we start from Mki - K~ = 0,
(2.12)
for i E {x, y.,z}, and multiply these equations crosswise with R/ and /~/ for i, j ~ {x, y,z} resulting in the equations
Mk, k~ - K~k/ - o
(2.13)
Mkyk~ -Kyk~ - O,
(2.14)
and respectively. The integrals over the time interval [t0,t ] for all combinations of the sums of(2.13) and (2.14) represent three "energy balance equations in the coordinate surfaces",
The sum of the three energy constants in the coordinate directions (2.11) corresponds to the three-dimensional energy constant in Eq. (2.6), on the one hand, ~x + ~ + ~ - E , (2.1 s) while the derivatives of the energy components Eii with respect to the velocity components R i correspond to the respective components of the linear momentum, on the other hand, c~Eii a/~ = ~ " (2.19) The latter components of the linear momentum can be derived as well by differentiation of the "energy constants in the coordinate surfaces (ei,%)" with respect to the velocities /~/, ~E c~/~ = P" (2.20) .1
M[~[~/ - i(Ki[~/ + Kj[~i )dt - E!j .
(2.15)
tO
These balance equations are the energy integrals of the projections of the three-dimensional motion of the satellite onto the coordinate surfaces (ei, % ). Finally, a "balance equation of the momentum volume" can be derived in a similar way as before, resulting in
Mkk x
y
k
z
-
t
t0
(2.16) it should be pointed out that these 'projected' energies can be formulated in any reference flame, most easily in an inertial or an Earth-fixed reference flame, but also in a body-fixed reference flame. Details of formulating balance equations and integrals of motion related to the gravity field determination are treated by Schneider (2005), such as sensitivity aspects of balance equations, etc. 2.3 The d e p e n d e n c i e s between the balance equations
Except the balance equation for the angular momentum, the balance equations derived in the last sections are not independent. The differentiation of the energy integral (2.6) with respect to the velocity R results in the balance equation for the linear momentum (2.2)
Finally, the "energy constants in the coordinate surfaces (e~,% ) " can be derived by differentiating the "energy constant of the momentum volume" with respect to the velocities / ~ , c~E!Jk = P .
(2.21)
Despite these dependencies the various balance equations show specific characteristics if they are applied for validation and gravity field determination tasks. This will be demonstrated in the next section. Because of lack of space only the balance equations of the linear and angular momentum, the Jacobi integral and the "energy balance equations in the coordinate directions" are treated in the following. 3 Analysis
in t h e t i m e d o m a i n
The balance equations can be evaluated along the orbits. The graphs in these cases reflect the consistency of the force function and the orbit in the time domain. The computation steps consist essentially in a differentiation of the observed orbit on the one hand and in an integration of the force functions on the other hand (Fig. 1). We apply the procedure as shown in Fig. 1 to a half-revolution arc of CHAMP crossing the Himalaya region starting from the polar area in the North and ending in Antarctica in the South. The orbit of CHAMP has been derived either by integration based on the gravity field model EI-
Chapter 42 • A Validation Procedure for Satellite Orbits and Force Function Models Based on a New Balance Equation Approach
GEN-GRACE01S (Reigber et al. 2003) and considered error-free or determined by a precise kinematic orbit determination procedure (Svehla and Rothacher, 2003). The former case will be labeled in the following as "simulated", the latter one as "kinematic". The gravity field model used in the balance equations is EGM96 (Lemoine et al. 1998). Because of the different gravity field models used in the balance equations on the one hand (EGM96) and as a basis for the orbit determination on the other hand (EIGEN-GRACE01S), inconsistencies of gravity field model and orbit must occur in the balance equations as deviations from the specific constants.
coincides with the reality only approximately can be recognized. The graphs on top of Fig. 2 and Fig. 3 and labeled by "simulated" are based on different integrals of motion but should reflect essentially the same sort of inconsistencies. The same should be true for the bottom graphs and labeled "kinematic". It is interesting to note that the different integrals of motion show different sensitivities for the inconsistency features. Also the different coordinates of the specific integrals of motion show different characteristics.
0.002
kinematicorbit I
transformation
into inertial system
Earth orientation parameters
Iq1
-0.002 0.004
with polynomials/
of positions
R~
,,kinetic" terms
~_,~_ ~ ~ kinematicorbit with velocities
integrand of the force function integral quadrature of the integrand
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quadrature ith polynomials I
motionintegral
Fig. 1" Computation scheme for evaluating the balance equations along the orbits. The graphs in Fig. 2 and Fig. 3 reflect the inconsistencies of the force function and the ephemeris of a half-revolution arc as deviations from the respective constants of the balance equation for the linear momentum and the energy balance in x,y,z, respectively. The top graphs show the "simulated" case, the bottom graphs are derived by using a "kinematic" orbit. Both graphs show similar but slightly different features. The top graphs in Fig. 2 and Fig. 3 reflect only the inconsistencies caused by the different gravity field models EGM96 and EIGENGRACE01S. The bottom graphs contain additional noise originating from the noisy observations and additional forces acting on the satellite which are not considered in the integrals of motion. Therefore, the graphs labeled by "simulated" and "kinematic" for the specific integrals of motion show slightly different inconsistencies; only the most pronounced features caused by the fact that EGM96
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4 Analysis in the space domain If the inconsistencies of force function and orbit in the time domain are referred at every time to its subsatellite point then we get an impression of the inconsistencies in the space domain. The Jacobi integral directly reflects these inconsistencies in the
283
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A. LScher. K. H. Ilk
space domain in terms of the gravity potential. This is not the case for the alternative integrals of motion as derived in Sect. 2. In this case the inconsistencies have to be expressed in terms of gravity potential inconsistencies at satellite altitude along the satellite orbits. These inconsistencies can be modeled by spherical harmonics. The corresponding potential coefficients can be considered as corrections to the parameters of the inconsistent gravity field model as well. Again, it becomes clear that the validation procedure corresponds to a global gravity field improvement procedure.
ity field. As reference field we select again EGM96. If we identify the real gravity field with EIGENGRACE01S then the graphics can be compared with Fig. 4. The inconsistencies are plotted along a kinematic 30-days orbit of CHAMP. Fig. 5 shows the inconsistencies of the linear and angular momentum, of the Jacobi integral and of the energy balances in x , y , z - all of them represented by spherical harmonic expansions of the gravity potential. The residual patterns in the time domain and in the space domain can be used to discriminate different causes for the inconsistencies.
Fig. 4" Potential differences of gravity field models EGM96 minus EIGEN-GRACE01S along the satellite orbit in m 2 / s 2 .
Fig. 5: Deviations of the integrals of motion from constants transformed to gravity potential inconsistencies along the satellite orbit in m 2 / s 2 . -150
-100
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-150
-100
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angular momentum Fig. 4 shows the potential differences of the gravity field model EGM96 and the gravity field model EIGEN-GRACE01S along the satellite orbit. Fig. 5 illustrates the inconsistencies as seen in the various balance equations as a consequence of the differences between a reference model and the real grav-
-150
-100
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100
150
-50
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Jacobi integral
- 150
- 1O0
energy balance in x,y,z A careful look at Fig. 5 shows interesting differences in the inconsistency patterns of the various balance equations. We recognize the stripe pattern as well as the rough features in case of the Jacobi integral compared to, e.g., the linear and the angular momentum.
Chapter 42 • A Validation Procedure for Satellite Orbits and Force Function Models Based on a New Balance Equation Approach
The triangle plots of the formal errors of the coefficients describing the potential inconsistencies in Fig. 5 are shown in Fig. 6. In case of the Jacobi integral the stripe pattern in the space domain (Fig. 5) correlates clearly with the formal errors of the spherical harmonic coefficients. These errors increase with increasing degrees and orders. The situation is completely different in case of the linear momentum and also in case of the angular momentum.
But also the formal errors of the coefficients of the inconsistencies of the energy balances in the coordinates x,y,z show a completely different behaviour than the formal errors of the inconsistencies based on the Jacobi integral. This is remarkable in so far as the Jacobi integral represents nothing else as the sum of the balance equations in the coordinates x,y,z. Again, the Jacobi integral seems to be the worst choice for gravity field validation and gravity improvement tasks. order
order
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Fig. 6: Formal errors of the spherical harmonic coefficients, corresponding to the inconsistencies of the integrals of motion. 4E-008 -
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:
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20
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60
Fig. 7: Degree RMS of the formal errors of spherical harmonic coefficients.
0
20
40
Fig. 8" Difference degree RMS with respect to EIGEN-GRACE01S ("true error").
60
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A. LScher. K. H. Ilk
The graphs in Fig. 7 and Fig. 8 show the true and the formal error degree variances- and confirm the results discussed before. The formal but also the true error degree variances for the inconsistencies based on the Jacobi integral are the worst of all integrals of motion considered here. The best result can be achieved with the angular momentum balance, but also the energy balances in the coordinates x,y,z show a remarkable good result. Again, this is interesting in so far as the sum of all three coordinates corresponds to the Jacobi integral.
100
used. The model of the surface forces has been based on the measured accelerations of CHAMP, filtered and approximated by spline functions. Then the surface force model has been falsified by an amount of 10% of the total surface forces. The question is whether the surface forces are shifted into the gravity field parameter corrections and which part of these forces can be recognized after the recovery procedure when the integrals of motion are inspected in the time domain.
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5 Energy shifts: a simulation
example
An interesting question for the validation of force function models and precisely observed orbits is the question how time dependent orbit effects, e.g., insufficiently modeled or measured surface forces are shifted into the gravity field parameters during the gravity field recovery procedure. A simulation experiment shall demonstrate the consequences of not correctly modeled dissipative force function constituents for the gravity field recovery. Surface forces cause a transfer from kinetic energy to heat energy. If the total of heat will be assimilated into the gravity field parameters then it is very difficult to detect these effects by the integrals of motion. The simulated reality is approximated by the gravity field model EIGEN-GRACE01S and a surface force model. For the gravity field recovery the reference gravity field model EGM96 has been
Fig. 9: Gravity field inconsistency shown as potential defects along the satellite orbit in m 2 / s 2 for the various balance equations.
The Fig. 9 shows the gravity field inconsistencies as total potential effects in m 2 / s 2 . The "observations" used in the gravity recovery process are the deviations of the different balance equations from the respective constants. Then the complete spectrum of coefficients of a spherical harmonic expansion complete up to degree n=60 have been recovered for all four cases. The differences between the recovered potential coefficients and the true coefficients reflect the gravity field inconsistencies. It is remarkable that all integrals of motion show different gravity field recovery properties and the procedure based on the Jacobi integral is by far the worst one.
Chapter 42 • A Validation Procedure for Satellite Orbits and Force Function Models Based on a New Balance Equation Approach
Only a minor part of the dissipative surface forces has been transferred to the gravity potential coefficients; the main part can be detected if the integrals of motion are determined with the improved gravity field model. Fig. 10 shows the effects which are caused by these dissipative forces in terms of total energy. In real applications this would be a clear hint that still model misconceptions exist in the mathematical-physical model of the gravity field recovery procedure. 0.0
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temporal and local disturbances which have to be investigated in the future.
Acknowledgement.
We appreciate the anonymous reviewer and J. Kusche for their helpful comments and suggestions. We are grateful to Prof. Dr. M. Schneider for his continuous partnership in discussing various problems related to this research. Our special thanks go to D. Svehla and M. Rothacher for placing the kinematic orbits of CHAMP at our disposal. The support of BMBF (Bundesministerium f'tir Bildung und Forschung) and DFG (Deutsche Forschungs-Gemeinschaft) of the GEOTECHNOLOGIEN programme is gratefully acknowledged. References
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Fig. 10: Energy contribution shifted from 10% of the surface forces into the motion balance constants illustrated in the time domain. 6
Conclusions
In this paper alternative integrals of motion have been presented which show interesting properties both for gravity field recovery as well as for validation tasks. It is surprising that the Jacobi integral, exclusively used as balance equation approach within the last years for gravity field recovery task in case of densely and precisely observed low satellite orbits, is by far the worst choice under various other possibilities. The balance equations of the linear momentum, of the angular momentum or of the coordinate energy balances show much better properties for gravity field recovery as well as for validation of the consistency of force functions and observed orbits. There are alternative balance equations mentioned in Sect. 2 but additional ones which should be investigated in detail. It seems that the proposal of using modified integrals of motion for gravity field recovery and validation tasks is very encouraging. Nevertheless there are some important open questions as the detailed separation of
Ilk KH, L6cher A (2003) The Use of Energy Balance Relations for Validation of Gravity Field Models and Orbit Determination Results, F. Sans6 (ed.) A Window on the Future of Geodesy, IUGG General Assembly 2003, Sapporo, Japan, International Association of Geodesy Symposia, Vol. 128, pp. 494-499, Springer Lemoine FG, Kenyon SC, Factor JK, Trimmer RG, Pavlis NK, Chinn DS, Cox CM, Klosko SM, Luthcke SB, Torrence MH, Wang YM, Williamson RG, Pavlis EC, Rapp RH, Olson TR (1998) The development of the joint NASA GSFC and the National Imagery and Mapping Agency (NIMA) geopotential model EGM96, NASA/TP- 1998-206861, Goddard Space Flight Center, Greenbelt, MD L6cher A, Ilk KH (2005) Energy Balance Relations for Validation of Gravity Field Models and Orbit Determinations Applied to the Results of the CHAMP Mission, in C. Reigber, H. Lfihr, P. Schwintzer, J. Wickert (Eds.): Earth Observation with CHAMP, Results from Three Years in Orbit, pp. 53-58, Springer Mayer-Giirr T, Ilk KH, Eicker A, Feuchtinger M (2005) ITG-CHAMP01: A CHAMP Gravity Field Model from Short Kinematic Arcs of a One-Year Observation Period, Journal of Geodesy (2005) 78:462-480 Reigber C, Schmidt R, Flechtner F, K6nig R, Meyer U, Neumayer KH, Schwintzer P, Zhu SY (2003) First EIGEN Gravity Field Model based on GRACE Mission Data Only, in preparation for GRL Schneider M (2002) Zur Methodik der Gravitationsfeldbestimmung mit Erdsatelliten, Schriftenreihe IAPG/ FESG, 15, Institut ffir Astronomische und Physikalische Geodfisie, Mtinchen, 3-934205-14-3 Schneider M (2005) Beitr/ige zur Gravitationsfeldbestimmung mit Erdsatelliten, Schriftenreihe IAPG/ FESG, 21, Institut ftir Astronomische und Physikalische Geod/isie, Mfinchen, 3-934205-20-8 Svehla D, Rothacher M (2003) Kinematic and reduced dynamic precise orbit determination of low-Earth orbiters, Adv. Geosciences 1:47-56
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Chapter 43
Combining Gravity and Topographic Data for Local Gradient Modeling L. Zhu, C. Jekeli Division of Geodetic Science School of Earth Sciences Ohio State University, 125 South Oval Mall, Columbus, OH
Abstract. The local modeling of gravity gradients
supports gradiometry systems by predicting observables whose residuals then yield new geophysical and geodetic information in the survey area. Gradients are usually modeled using local digital terrain elevation data (DTED). We supplement this with available (lower-resolution) gravity anomaly data for the longer-wavelength features and analyze the total model in terms of its spectral content over a local region such as the San Andreas Fault. New spherical models of transforming gravity anomalies to all components of the gradient tensor using Green's functions are developed and consistently combined with forward models of terrain elevation from very dense Shuttle Radar Topography Mission (SRTM) measurements. The modeling is applied to the case of airborne gradiometry at about 400 m over moderately rough terrain and yields an upper bound on the power spectral density to be expected in this case. Such gradient modeling should contribute to the design of appropriate filters in the processing of airborne gradiometric data. Keywords. gravity gradients, Stokes integral
1
DEM
modeling,
Introduction
The importance of gravity gradient modeling becomes more significant as instrumentation, particularly in airborne applications, becomes more accurate. A good gradient model derived from existing data sources aids in the pre-processing of airborne gradient data, as well as in detecting and interpreting residual density anomalies sought out by an airborne gradiometric survey. Gravity gradients can be modeled theoretically either from surface gravity or topographic data, or a combination of both. Some early studies in obtaining gradients from gravity data include those
43210
of Agarwal and Lal (1972) and Gunn (1975). More recent methods relied on Fourier Transform relationships among various derivatives of the geopotential field (e.g., Mickus and Hinojosa, 2001). Most modeling of gravity gradients, however, is based on topographic data since the gradients reflect primarily the near structures of the Earth's mass. A recent review of the various associated methods was given by (Jekeli and Zhu, 2005); see also references therein. Fortuitously, topographic data usually have much higher resolution than gravity data and thus topographically-derived gradient models inherit the resolution more nearly appropriate to this type of gravitational quantity. Nevertheless, depending on their resolution, gravity data grids also provide information on gradients at corresponding wavelengths. In fact, they more accurately reflect lateral density variation that may be missing in the usual topographically-derived model that is usually based on a constant mass density. We develop a unified approach to modeling gradients at aircraft altitude from a combination of surface gravity and terrain elevation data, and consider the power spectral densities at different altitudes. The objectives of the analysis are 1) to verify the significant attenuation of the gradient signal with altitude, regardless of the high resolution of the data; 2) to study the dependence of the model at aircraft altitude on typical density variability in the topographic masses; and 3) to develop a validation and calibration approach for airborne gradiometry.
2 2.1
Theory Basic Definitions
Before introducing our approach to modeling the gravity gradient, let us review some basic definitions. Let W be the gravity potential; then the gravity vector is given by
Chapter 4 3
rOW" Ox~ g-
g2
- VW -
(l)
3x2
g3
3W
v Ox3 where we have introduced a local Cartesian coordinate system, (x~,x2,x 3 ), with axes pointing north, east and down. The gravity gradient tensor is given by the second derivatives of the gravity potential:
F-
VV~Ig - Vg ~ -
where,
3gj/OXj
/ 3g 1
3g 2
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Ox,
3x,
3x,
3g~
3g 2
3g 3
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OX2
ON2
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3g 2
3g 3
3x3
3x3
(2)
is called the inline gradient, and
Ogi/Oxj the cross gradient. The gradient disturbance is the difference between the total
•
CombiningGravity and TopographicData for Local Gradient Modeling
where GM is Newton's gravitational constant times Earth's total mass, R is a mean Earth radius, Cnm are the harmonic coefficients, and Y,,,n are spherical harmonic functions. Stokes's integral solves (in spherical approximation) the boundary-value problem for the disturbing potential (Heiskanen and Moritz, 1967), and thus,
R I I A (O,/~, ) 02 1-'j~ Ag - ~ g ' OxjOxk S (r, gt)dcr cy
(4)
where cr represents the unit sphere, Ag is the gravity anomaly on the geoid, S is the generalized Stokes function, and g is the spherical distance between integration and evaluation points. Finally, we apply Newton's density integral in terms of local Cartesian coordinates to model the gradients due to local topographic masses:
a ~ ~v~ dv, E/~ - Cp ax/Ox, Ix- x
(5)
gravity gradient and a normal gradient, as implied, e.g., by the standard Geodetic Reference System of 1980 (GRS80). In the following, we use the same symbol, /-'j~, to denote a component of the gradient disturbance tensor.
Here, the volume element is d v - dxldx'2dx; , and the density, p , is assumed constant. The second-order local derivatives in terms of the spherical derivatives are given by 02 1 0 1 02 _ _ z ____-71-____ OXl2 r Or r 2 O02
2.2
32 m cot0 3 1 3 1 32 -Jr-m m -Jr-_ _ 0X22 r 2 0 0 r Or r 2 sin 2 0 022
Individual Models
Three types of models can be used to represent the gradient disturbances: a global spherical harmonic model derived from an existing set of geopotential coefficients; the solution to a boundary value problem in which gravity anomalies constitute boundary values; and Newton's density integral. In each case, we start with the corresponding expression for the disturbing potential and simply obtain second-order derivatives in a local coordinate system. For the global spherical harmonic model, e.g., EGM96 (Lemoine et al., 1998), given in spherical coordinates, ( 0 , , t , r ) , being co-latitude, longitude, and radial distance, we have:
O2
O2
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32
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sin 0 02
1 O r 2 00 1
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32
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The Combined Model
The basic principle that consistently combines these various representations of the gradients is the --~-'~2 32 ( ( R / n + l / remove/restore technique, often used to combine C k EGM96 G M Cnm --Ynm (0,,,~) models of the disturbing potential from different R n=2 m=-n {)XJ~)Xk sources. We use Stokes's integral as the basic model and include the global model and the effect of (3)
289
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L. Zhu. C. Jekeli
local terrain by first removing these from the gravity anomaly. In order to keep the residual gravity anomalies small, the terrain effect is "removed" in the form of the Helmert condensation, approximated by the usual terrain correction (resulting in the Faye anomaly). However, here we must be careful to note that the corresponding restored gradient is not simply the gradient due to the residual terrain, but that due to the restored Helmert condensation. The final formula for the gradient is given by
which the gradients were computed (elevation of 1100 m and 1290 m, respectively), about 400 m above the mean along-track terrain in each case.
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and c is the usual terrain correction. /-'jk Jte,~ain is the difference of gradient effects due to the residual terrain and the Helmert condensation layer on the geoid:
/--'jk [terrain--1--'jk[res.terr.--1--'jk[layer
3
11,
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Examples and Discussion
3.1 Data We chose an area in California over the San Andreas Fault, being the test site for a recent airborne gradiometric survey. Digital terrain data were available from USGS in the form of 30 m gridded values, derived from the Shuttle Radar Topography Mission (SRTM) conducted by N G A and NASA. A total of 286 free-air gravity anomalies obtained from NGA and approximately evenly scattered in this 22.55'
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Chapter 43 • Combining Gravity and Topographic Data for Local Gradient Modeling .14 I:1:1
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Fig. 6 Gradients from high resolution residual gravity data.
293
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~" iii
10 3
,--10 i
i
i
2
4
6
i
0
8
10 7 E ~ O ~
"iii
10 10
~
E -~
(..) •,,-,('N
10
O ~
3
i
i
i
2
4
6
6
8
s 3
8
0
i
i
i
2
4
6
i
i
i
_
_
8
10 7
s
E ~ O ~-~ Pq
3
"" 10 LU 01
"iii
m i
0
4
~ 10 iii ~ 01
10 7 10
2
10 7
,-" 101
E
03
,--10
0
4
0s
O
O
i
10 10
s 3
01
i
2 4 6 0 2 4 6 Frequency [cy/km] Frequency [cy/km] Fig. 7 Spectra of gradients along test profile derived from high-resolution gravity data.
Conclusions
We have developed a consistent approach to modeling gravity gradients from multiple sources, including a global model, regional gravity anomaly data, and local terrain data. Stokes's integral serves as the basis for a remove/restore procedure. High-resolution data, typically topographic data (but also high-resolution gravity data) contribute the most to the total gradient signals. Clearly, as it is well known, the upward continuation acts as a strong low-pass filter. High-frequency variations in the density model do not affect the high-frequency spectrum of the gradients at altitude. Knowing the cut-off of the upward continuation filter of the signal, airborne gradiometer systems can be better tuned with appropriate processing filters to remove high frequency errors due to aircraft dynamics and other systematic sources. For example, airborne gradiometer observations over this test area (400 m above the mean terrain) with spectral content significantly greater than indicated by the model at
8
frequencies higher than 2 cy/km could be suspected as being erroneous. The appropriate filter to use in airborne gradiometry, of course, would also take the instrument noise into account, but the models considered here provide at least an upper bound on the type of signal to be expected.
Acknowledgments This work was supported with funding through contracts with the National GeospatialIntelligence Agency, contract nos. NMA401-02-1-2005 and HM1582-05-1-2009.
References Agarwal, B. N. P. and Lal, T. (1972): A generalized method of computing second derivative of gravity field. Geophysical Prospecting, 20, 385-394. Gunn, P.J. (1975): Linear transformations of gravity and magnetic fields. Geophysical Prospecting, 23, 300-312.
Chapter 4 3
Heiskanen, W.A. and H. Moritz (1967): Physical Geodesy. W.H. Freeman and Co., San Francisco. Jekeli, C. and L. Zhu (2005): Comparison of methods to model the gravitational gradients from topographic data bases. Geophysical Journal International, in press. Lemoine, F.G., Kenyon S.C., Factor J.K., Trimmer R.G., Pavlis N.K., Chinn D.S., Cox C.M., Klosko S.M., Luthcke S.B., Torrence M.H., Wang Y.M., Williamson, R.G., Pavlis E.C., Rapp R.H., Olson T.R. (1998): The development of
•
CombiningGravity and TopographicData for Local Gradient Modeling
the joint NASA GSFC and the National Imagery and Mapping Agency (NIMA) geopotential model EGM96, NASA Technical Paper NASA/TP-1998-206861, Goddard Space Flight Center, Greenbelt. Mickus, K.L. and J.H. Hinojosa (2001): The complete gravity gradient tensor derived from the vertical component of gravity: a Fourier transform technique. Journal of Applied Geophysics, 46, 159-174.
295
Chapter44
Numerical implementation of the gravity space approach - proof of concept G. Austen, W. Keller Stuttgart University, Geodetic Institute, Geschwister-Scholl-Str. 24/D, 70174 Stuttgart, Germany
Keywords. gravity space, geodetic boundary value problem, telluroid
Abstract
unknown surface OG to the unit sphere or. We are looking for a function V : R 3 \ G ~ R and a diffeomorphism ~ with the following properties aVtx)
=
0,
V aa The classical geodetic boundary value problem is a linear free boundary value problem, which implies considerable mathematical difficulties for the investigation of its existence and uniqueness properties. In 1977 F. Sans?) found a breakthrough by transforming the problem into the gravity space, using Legendre transformation. Nevertheless, the transformed problem still suffers from a singularity at the origin. W. Keller proposed in 1987 a modified contact transformation, which provides a boundary value problem free of singularities. Despite its conceptual advantages the gravity space problem was not yet implemented numerically. The paper aims at a study of this numerical implementation in the global case. It gives indication that gravity field determination can also successfully be carried out in the gravity space.
1
Gravity space theory
vv
oc
xca
~\G
f og
(4)
g o ~,
(5)
with (7 being the closure of G. From its mathematical structure (3)-(5) is a socalled free boundary value problem with a high complexity both in the theory of existence and uniqueness as well as in the numerical implementation. In 1977 F. Sansb (Sansb, F. 1977) found a brilliant approach to transform the problem (3)(5) into a boundary value problem with a fixed boundary by introducing new coordinates (6)
:= V V t x )
and a new potential ~, the so-called adjoint potential by ~(~) = x ~ V V ( x ) -
V(x),
x = x(~).
~
-
({TV{~)- ~) {=g
f:cr
--+
R
(1)
g:cr
-+
IRa .
(2)
o
(s)
f.
(9)
Equations (8) and (9) constitute a nonlinear boundary value problem with a Monge-Amp6re type of differential equation and a linear boundary condition. Unfortunately the asymptotic behaviour of the gravitational potential v~
1
,
Ix
-~
x
leads to the asymptotic behaviour The concatenations (f o 9~)(q):= f ( g ( q ) ) and (g o g ) ( q ) : = g(g(q)) map the data given at the
(7)
In the new coordinates ( the adjoint potential ~) solves the following boundary value problem t~(v~) ~ -(t~v~)
The classical geodetic boundary value problem can be formulated as follows: Let G c R 3 be a bounded, open, simply connected domain. Its boundary OG is supposed to be diffeomorphic to the unit sphere or. The diffeomorphism is denoted by g : OG ~ o-. On cr two functions f, g are defined
(3)
(~o)
Chapter44 • NumericalImplementationof the GravitySpaceApproach- Proofof Concept
of the adjoint potential. Equation (11) describes the so-called singularity of the gravity space approach at the origin: The adjoint potential is not differentiable at the origin but it has to solve a differential equation of second order at exactly that place, since (8),(9) is an interior problem. The singularity reduces the mathematical beauty of the gravity space approach and has its origin in the fact that the transformation (6) maps the point at infinity to the origin. Since the concept of differentiability has no meaning at the point at infinity, it is not a surprise that in the new gravity coordinates the differentiability is lost at the origin. The singularity can only be avoided by a similar gravity space transformation, which leaves the point at infinity as a fixed point.
2
Regular
gravity
In (Keller, W. 1987) a gravity space transformation was given, which does not generate a singularity. Similar to equations (6) to (9) the modified transformation is described by OV P~ " - Ox~ '
i = 1,2,3
(12)
& : = - ( G M ) ~ / 2P,ii = 1,2,3 ilpll~/~, , := p i x i - V
-GM (
-
ai~,Vkz
:-
and
&G )
C~ik'Ykj = 5ij
9~
(13) (14)
T h e o r e m 1 Let the coefficients flij~ be defined by
=
R e m a r k : In the geometry space two versions of the Molodensky problem are considered: The so-called vectorial Molodensky problem, where the complete gravity vector g = VV is used as boundary data and the so-called scalar Molodensky problem, where only the modulus g = IIVVII enters the problem as boundary data. From the observational point of view the modulus 9 is much easier to access than the complete gravity vector g. Hence, in the geometry space the scalar Molodensky problem is the prevailing form of dealing with geodetic boundary value problems. The gravity space approach introduced by F. Sansb is an example of a so-called contacttransformation. Contact transformations were first used by F. Klein (Klein, F. 1891) and are based on the idea to consider the solutionmanifold of a differential equation as the envelope of its tangential spaces and to transform the equation for the manifold into an equivalent equation for its tangential spaces. The latter then is hopefully easier to treat than the original one. Since the quantity, which describes the tangential plane of a manifold V is the gradient VV the complete gradient and not only its modulus has to be known in order to apply a contact-transformation. Therefore a contacttransformation (or in other words a gravity space transformation) for the scalar Molodensky problem cannot be found, despite the fact that the scalar formulation is much closer to reality.
space
transformation
(16)
(97 i rn
(17)
®k
and let • denote the following m a t r i x :
a2~
Then
AV=0
~=~
1
d~t+(t~+~-[t~+l
2)
-o
(19)
and
w
- f ~(--~
-~)
-f (20)
holds.
Equations (19) and (20) again establish a nonlinear boundary value problem with a fixed boundary. But this time for tile adjoint potential ~/: the asymptotic relation
W~
1
~,
~1-~o~
(2,)
holds, which means that tile modified gravity space approach is free from any singularity.
297
298
6. Austen• W. Keller 3
Linearised
problem
Nevertheless, the singularity-free boundary value problem (19), (20) is still vastly complicated. Since with the normal potential u :-
GM
(22)
Ilxll
we have a first approximation of the actual gravitational potential U ~ V, we can expect that the adjoint normal potential ~0, given by :=-2--
GM
(23)
I111
is a first approximation of the actual adjoint potential ~. The linearisation of (19), (20) at ~0 leads to the following boundary value problem for T := ~/: -- ~0: Theorem
~- := ~ -
2 The adjoint disturbing potential ~0 solves the following boundary value
problem
-([
w
+
- Vloo
- Ulo
- av
(25)
where E is the gravimetric telluroid defined by the following equation: g = VV(x)=
x •
oc,
•
(26)
Basically, the gravimetric telluroid E is the image of the Earth's surface OG under the mapping (13). In (Keller, W. 1987)it is shown, that for a spherical normal potential U this deftnition of the boundary surface E is equivalent to the implicit definition by (26). It is remarkable that the linearised problem (24), (25) is mathematically of the same structure as the linearised Molodensky problem. Only potential and gravity have changed their places: In the linearised Molodensky problem the boundary surface is defined by a p o t e n t i a l relation V(x) = U(~); in the linearised gravity space approach tile boundary surface is defined by a g r a v i t y relation VV(x) = VU(~). On the other hand, in the linearised Molodensky problem the boundary values are g r a v i t y anomalies, while in the linearised gravity space approach the boundary values are p o t e n t i a l disturbances. Due to Remark:
the identical mathematical structure all algorithms and procedures developed for tile Moledensky problem can be carried over to the linearised gravity space approach. The consideration of the linearised gravity space approach instead of the usual Molodensky problem has two conceptual advantages and two disadvantages: 1. The boundary data 6V in the gravity space approach are smoother than the boundary data Ag of the Molodensky problem. This is conceptual understandable since both field quantitles are measured at tile Earth's surface but the potential is smoother than its gradient. The fact that in the gravity space approach the incasured potential values are mapped to the gravimetric telluroid does not alter their smoothness. Besides this, for a global simulation study this greater smoothness of potential data will be numerically proved. 2. The potential data 6V are related to spiritleveling and are therefore available with a higher density than the gravity measurement related gravity anomalies Ag of the Molodensky problem. Even more: On the oceans (besides the small influence of dynamical topography) the potential V is constant. Hence, the boundary values on the oceans are continuously available. 3. The only disadvantage of the gravity-space formulation is that the necessary gravity vector g = VV is not available but only its modulus 9 -- ]]VVII. This results ill a horizontal uncertainty of the telluroid E smaller than 1500 m or, expressed as a relative deviation, of 2 . 1 0 -4. On tile other hand, the boundary surface E will never actually been used as computation surface. Instead of that all data given on E will be harmonically continued to a sphere. The deviation between the telluroid E and the sphere S enters the harmonic continuation formula only as a correction parameter and has not to be very precise. This means, a deviation error of 2.10 -4 is tolerable for the purpose of harmonic continuation. 4. The normal potential U approximates the actual potential V only with an accuracy of 10 -3. This leads to the unfavourable situation that the telluroid undulations and the potential anomalies 6V get rather large. This means that for the continuation of the boundary values 5V from the telluroid E to the computation surface much more care than in the usual Molodensky case has to be taken. The numerical studies will show that a continuation with a sufficient accuracy is
Chapter 44 • Numerical Implementation of the Gravity Space Approach - Proof of Concept
possible, but on a computationally high price. A linearisation at an ellipsoidal normal potential would very much simplify the continuation process. But at the time being it is not clear whether or not a new linearisation point would preserve the Molodensky-type structure of the linearised problem. Despite those conceptual advantages and the strong similarity to the well known Molodensky problem the gravity space approach was never implemented numerically. With the global sireulation study presented in the following the feasibility of the numerical implementation will be shown.
4
Proof of concept
For the global study the TUG87 model (Wieser, M. 1987) up to degree and order 180 was used as a model for the Earth's topography. As a model for the gravitational field of the Earth the GPM98B model (Wenzel, H.-G. 1998) up to degree and order 720 was applied. On a ~ , j . AA grid, which was designed to meet the requirements of the subsequent Gaussian integration step, the gravitational vector
gij
:=
~TvGPM98b(~gi,j" A,k, hTUG87(qDi,j" /k)~)) (27)
on the Earth' surface was computed. Solving the equations g~j = V U ( ~ j ) (28)
The boundary surface E is shown in Figure 1 by plotting the deviations of the gravimetric telluroid wrt. to a best-fitting ellipsoid of revolution E r . The differences between E and E z (Figure 1) vary f r o m - 1 km to 5 kin, which reflects both the spatial variation of the potential V and the variation of the topographic heights h, though the prevailing influence on E is clearly the topography. The corresponding parameters of E r , i.e. semi-major axis a s and flattening f r , were determined by least-squares methods from rz~j -- a~(1 - fz sin 2 ~i), where rx~j is known from
rr~-
~j
,
~j~E
(30)
and represents the geocentric distance of the gravimetric telluroid points. The following parameters were estimated: a s = 6372993m and e~ - 2 f z - f~ - 0.001599. This means the boundary surface in gravity space features a significant lower flattening than in geometry space (cf. e ~ c M - 0.006694), which leads, on the one hand, to separations of the gravimetric telluroid and the physical Earth surface of up to 10 kin, as a result of using a spherical normal potential as linearisation point. On the other hand a reduced flattening can be of benefit for the continuation process of the boundary data from the gravimetric telluroid to the computational sphere, as will be discussed in the next paragraph.
by Newton's method for (ij, the gravimetric telluroid E was determined point-wise. [m] 5250 1000 700 400 200 0 -200 -400 -650 -850 -1050
Figure 1. Gravimetric telluroid
(29)
299
300
G. Austen • W. Keller
Next, p o t e n t i a l d i s t u r b a n c e s (~V = V[oa- U[oa were c o m p u t e d at the E a r t h ' s surface cOG. U n d e r the m a p p i n g of (13) t h e y represent the b o u n d a r y values on E in gravity space. G r a v i t y anomalies on a reference ellipsoid were d e t e r m i n e d for comparison. Figures 2 and 3 display the Fourier spectra of b o t h d a t a types, confirming t h a t potential d i s t u r b a n c e s are indeed m u c h s m o o t h e r t h a n gravity anomalies. G r a v i t y anomalies on the E a r t h ' s surface would even be m u c h rougher.
2d FFT of potential
T a b l e 1. Geoid accuracies after tion.
..........
i ................................... ~.................. i .................................... ' ..................
0 . 1 5 ......... i ................. i ................. !.................. i .................. ~................. i ..................
disturbances
.............~...............i................ii................i...............................i........................
......i.........~........~........~........i........,........i........ ............i...............i................~................!...............................~.........................
o.1 .......... ~.................. :'................. ~.................. ~.................. ~................. ~..................
0.05 .......
o-
~ ................. ~................. ~.................. i .................. ! ................. !..................
~
............i...............i................',................~...............~................~.........................
.................
[l/rad]
[llrad]
F i g u r e 2. Fourier spectrum of potential disturbances on the gravimetric telluroid
2d 0.12
.........................
" ................. ~ .................
i ..................
FFT
of
gravity anomalies
! ........................................................................................................
~...............................
~........................
0.1-
0.05 .......
~................. "................. T
................ ~.................. ~.................. ~.............................................
0.05 ..........i.................~................. ..-................. i .................. ~.................. ::..................
0.04 ..........
i................ i................ i ............... i.......................................
........... ii ............... !................ i................ i ............... i........................................
...........i ...............i................i................i...............................~................~....... i .................
i .................
~ .................
[l/rad]
! ..................
~..................
2 nd
itera-
no. of
RMS
mean
std
max
rain
iteration
[cm]
[cm]
[cm]
[cm]
[cm]
1
3.3
1.9
+2.8
50.7
-42.9
2
1.5
0.8
±1.2
36.7
-41.9
6Vn-FIIs
-- 6rnls
-]-- c
T E c E
......i.........i........~ '........i................~........~........ i ................. ~...................................................... ~................. ~..................
and
In a following step the p o t e n t i a l anomalies on the gravimetric telluroid were iteratively u p w a r d continued to an enclosing Brillouin sphere (R = aEGM = 6378136.3m) using collocation in a remove-compute-restore mode
............................................................. i ................................................................................................................... i......................................................... O.Z5 . . . . . . .
1 st
~..................
[l/rad]
F i g u r e 3. Fourier spectrum of gravity anomalies on a reference ellipsoid (a -- 6378136.3m, e 2 = 0.006694)
E-1
(6V
--
6Vn)lE
(31) In e q u a t i o n (31) S denotes the Brillouin sphere, C r r the auto covariance m a t r i x of the ~ V ~Vn values on the gravimetric telluroid E and C s z the cross-covariance between the ~ V (~Vn values on the gravimetric telluroid E and on the Brillouin sphere S. T h e iteration was s t a r t e d with 5Vo c o m p u t e d from the E I G E N G R A C E 0 2 S model, a satellite-only spherical harmonic model complete up to degree and order 150. T h e auto- and cross-covariances were derived from the E G M 9 6 model above degree and order 150. Two iterations were carried out. T h e deviations between the continued p o t e n t i a l d i s t u r b a n c e s after each iteration step and the exact values comp u t e d from G P M 9 8 B are displayed in Figures 4 and 5. After the second iteration the continuation errors are below 0.3m2s -2 in t e r m s of potential values or 3 cm in t e r m s of resulting geoid u n d u l a t i o n errors. Errors above are the e x t r e m e values! T h e vast m a j o r i t y of errors is two orders of m a g n i t u d e smaller, giving a root m e a n square (RMS) value of 1.5 cm (compare Table 1). Keeping in m i n d t h a t the c o n t i n u a t i o n was p e r f o r m e d over a distance of up to 10 km the o b t a i n e d accuracy is stunning. W i t h a c o m b i n a t i o n of F F T a n d Gaussian q u a d r a t u r e the spherical h a r m o n i c coefficients v~,~ of the continued p o t e n t i a l d i s t u r b a n c e s ~V were c o m p u t e d (2~
n
Z Z n=2 m=--n
(32)
Chapter 44 • Numerical I m p l e m e n t a t i o n of the Gravity Space Approach - Proof of Concept
[m2/s 2] 5.0 1.0 0.5 0.3 0.1 0.0 -0.1 -0.3 -0.5 -1.0 -4.0
F i g u r e 4. Upward continuation error after first iteration [m2/s 2] 5.00 0.30
60 °
-
.
.
.
.
.
.
.
.
60"
0°
0.10 0.20
0.05 0 o ~,
0o
0.00
~,~.~
-0.05
-30 °
303°
-0.10 -0.20 -0.30 -4.00
F i g u r e 5. Upward continuation error after second iteration This is done by representing the definition of the spherical h a r m o n i c coefficients v~.~ as an i t e r a t e d integral
Vnm
1 f 6v. Y~dS 47cR2 Js
4~_R~, ×
P~(cos 0)
sin rn,k
/o
~v(o,~)
sin OR2dAdO.
(33)
T h e inner integral is evaluated by the t r a p e z i a n q u a d r a t u r e rule
A closer look to the e q u a t i o n (34) shows t h a t the quantities an.~ (0), bn.~ (0) are exactly the discrete Fourier coefficients of 6V(O,A). This m e a n s instead of using the t r a p e z i a n rule these quantities can be c o m p u t e d more efficiently by F F T . T h e r e m a i n i n g outer integral 1 vn.~ = 47cR2
/0
P ~.~ (cos 0)
{
a~.~ (0) bn.~ (0)
}
sin 0R 2dO
(35)
can be t r a n s f o r m e d to
Vnm = 47C
1
P~(~)
b,~(~rcco~)
d~ (36)
and evaluated by a G a u s s i a n q u a d r a t u r e formula N/2
a~.~ (0) b~m (0)
_ 7r
2:r
--ff E-ff-V(O'i--N )
cos rn~ 7 s i n m i 727r
i=0
(34)
v~.~ -- 47c E
j=0
wjP~.~(xj)
anm(arccosxj) bn~(arccosxj)
(37)
301
302
6.
Austen • W.
Keller
with wj and xj being the weights and the nodes of the Gaussian quadrature formula respectively. The (pj = a r c s i n z j , i . AA) grid was designed in such a way that (besides rounding errors) the spherical harmonic analysis is exact up to degree and order 383. (The maximum degree of exact spherical harmonic analysis depends upon the largest available set of nodes and weights for the Gaussian quadrature, which is, if taken from Strout/Secrest, 383). Therefore, the only error contained in the spherical harmonic coefficients v~,~ stems from the downward continuation error and reaches a RMS value of 1.5 cm for the geoid undulations. Since outside the topographic masses
and
1 o% 2 ~l 0 ~ - - T - - 6 V
(39)
holds, the coefficients 7-~ of the spherical harmonic expansion of the adjoint disturbing potential
- Z 0 is the regularization parameter. However, this is not useful, due to the significant overlap of the spectrum of the multi-pole wavelets at various levels (see Figure 1). Instead, we propose the following procedure. We start with the coarsest level j - J,~i~ and estimate the wavelet coefficients at that level, {/3&,,~,~ " n - 1 . . . Nj,,,~}, by minimization of the quadratic functional, Eq. (6). The residuals of the least-squares adjustment are taken as observations to estimate the wavelet coefficients at level J,~i~ + 1. This procedure is continued until the coefficients of the m a x i m u m level J , ~ x have been estimated. Note that the noise covariance matrix of the residuals at level j is the noise covariance matrix of the observations at level j + 1. In the test case mentioned here, the initial covariance matrix C was used for all levels. This introduces small errors in the estimated coefficients, which were negligible. If the data-adaptive algorithm of section 4 is applied, the choice of J,~i~ is up to the user, and J , ~ x is fixed automatically. The coefficients of level J,~i~ represent the solution at the coarsest scale. The coefficients of level j represent information, which is not
•
Local Gravity Field Modelling with Multi-Pole Wavelets
included in the coefficients of the levels J,~i~J~,i~ + j - 1. Since with increasing level tile number of basis functions increases and the bandwidth decreases, the coefficients of higher levels represent short-scale information in the gravity field. Correspondingly, higher levels represent detail information not included in the coarser levels. Therefore, it is justified to call Eq. (1) a multi-scale representation.
4
Choice of the multi-pole wavelet centres and bandwidths
To estimate the wavelet coefficients, we first have to fix the centres and the bandwidths of the multi-pole wavelets. Both are important design criteria to get a good local gravity field model. For a fixed order m, the bandwidth of the multipole wavelet is solely determined by the scale parameter a. Moreover, the scale parameter fixes the depth of the multi-pole wavelet according to Eq. (4). Therefore, fixing the centre and the bandwidth of a multi-pole wavelet means that 3 parameters have to be determined: (i) the location on the unit sphere (2 parameters) and (ii) the scale parameter. Klees and Wittwer (2005) have proposed a data-adaptive network design (DAND) strategy to select the centres and tile bandwidths of radial basis functions automatically using information about data distribution, data quality, and signal variation. We generalize this approach to the multi-scale situation. To keep the derivations concise, we only address the aspects related to the multi-scale situation. Aspects, identical with the single-scale version are skipped; the reader is referred to (Klees and Wittwer, 2005). We start with a template network of points on the unit sphere and compute the region of influence (ROI) of a network point, which is a spherical cap with radius A 2
(1 -
Xo
/
(7)
centered at the network point. A is the size of the area projected onto tile unit sphere. Each level knows its own template network and ROIradius f r o 1 . In the simulations, section 5, we use equal-angular template networks. The generation starts with a square on the unit sphere, bounded by meridians and parallels, and covering the data area. Then, the square is successively subdivided in 4j-1 smaller squares, where j denotes the level. In that way we obtain a
305
306
R. K l e e s • T.
Wittwer
template network of points on the unit sphere for each level j = J , ~ i n . . . Jmax. The template network of level j has 4j-1 points. When the template networks have been generated, we proceed with the coarsest level j = d,~in. We select a candidate parameter p , compute the bandwidth cry = p - ¢ R o I , j and the scale parameter aj, select the centres {yj,~ : n = 1 . . . Nj} (cf. Klees and Wittwer, 2005), and estimate the coefficients ( f l j , n : n = 1 . . . N j } by least-squares. This procedure is repeated until the m a x i m u m level J,~ax is reached. Thereafter, we compute the Generalized Cross Validation functional for the chosen parameter p. The value Pg~v t h a t rainimizes the GCV functional is the optimal p. In this way, we obtain the centres of the multi-pole wavelets in 3D-space for each level j. The solution {flj,n, j = d,~i~ . . . g m a x , n = 1 . . . N j } obtained with the value Pgcv is the optimal solution. If regularization is applied (i.e. for a non-zero c~ in Eq. (6)), a second iteration loop inside the GCV loop is required. The optimal regularization parameter is found using Variance Component Estimation techniques (see Koch and Kusche, 2002).
258"
36 °
• ~
o
34"
34"
32 °
32"
30"
30 °
258 °
260"
Ao
-~o
262 °
o
, mGal
do
~o
Figure 2" Gravity disturbances over the test area. The RMS signal variation is 6.2 regal, the range is - 3 0 to 45 regal. 258
°
±:'i~':"-:-=::i-'.~
260
°
• "i..: .......• .}:: _ ' : ' : i .
262
°
i .-2~.v. "; L:5"i'a.':
--:-i:i-i::~i:.i-L-=-:f: ::. i -- -:4. .-:..-: -.: :.~i :-::-.--:ii:.-:.-::i~ 36 °
Simulations
To investigate the performance of the multi-pole wavelet approach pursued in sections 3 and 4, we did extensive numerical test with simulated terrestrial and airborne gravity data sets. The results to be presented refer to an airborne gravity data set. The area is located in the US, North to the Gulf of Mexico. The size is about 6 x 6 degrees. The gravity disturbances at a flight altitude of 2 km range from - 3 0 reGal to 45 reGal; the RMS signal is about 6.2 regal. The frequency content of the data is limited to degrees 121 to 1800. Therefore, the parameter L in Eq. (2) is set equal to 120. Figure 3 shows the gravity signal over the test area. We generated 5453 gravity disturbances, h e t e r o g e n e o u s l y d i s t r i b u t e d above the area at an altitude of 2 kin, see Figure 3. The data are corrupted with white noise with a standard deviation of 1.5 reGal. All computations are done with multi-pole wavelets of order m = 3. The lowest level is J,~in = 5.
6
262"
36"
ae°
5
260"
Results and discussion
We have computed the level 5-7 multi-scale solution using the approach described in the sections
34 ° i :i -i-.iiiL i iK i::::_.:-:__:_.-:i;2:=-
34 °
32 °
30 °
30 ° 258 °
260 °
262 °
Figure 3: Data distribution over the test area. The data set consists of 5453 observations. Note the heterogeneous distribution with significant gaps in certain areas. 3 and 4. The template networks comprise 256 (level 5), 1024 (level 6), and 4096 (level 7) centres. The DAND procedure outlined in section 4 selects 244 (level 5), 599 (level 6), and 467 centres, if
Chapter 45 • Local Gravity Field Modelling with Multi-Pole Wavelets
level
~ROI
bandwidth
depth
[degree I 0.2128 0.1064 0.0532
[km]
[kml
28.39 14.19 7.10
83.56 46.40 27.73
Table 1: Spherical radius of the region of influence ~Rof, bandwidth, and depth of tile multipole wavelets for levels 5-7 as selected by the multi-scale DAND procedure (cf. section 4). The optimal parameter Pgcv is 1.2.
a threshold of 3o = 4.5 reGal is used. This is a reduction of about 75%. Compared with a level-7 single-scale solution, the reduction is still significant: 1672 basis functions vs. 1319 basis functions (i.e. about 21%). Table 6 shows for the multi-pole wavelets of level 5-7 the results of the multi-scale DAND procedure. Figure 4 shows the estimated residual disturbing potential at ground level for the levels 5, 6, and 7. The level-5 multi-scale solution represents the solution at the coarsest scale. The solution at level 6 represents detail information, which is not included in the solution at level 5. The solution at level 7 represents detail information that is not included in the sum of the solutions at level 5 and 6. The DAND procedure places multi-pole wavelets only in areas with sufficiently high residual signals. Figure 5 shows a histogram of the true errors at ground level in terms of potential values. The RMS error is 0.27 m2/s ~. The low number of basis functions compared to the number of observations is also an indicator of the numerical efficiency of the pursued multiscale approach. The number of basis functions is 76% less than the number of observations. A Least-Squares Collocation solution with tile covariance function of Forsberg (1987) gives comparable results.
7
Conclusions
The following conclusions are drawn from the conducted numerical experiment. They are confirmed by other experiments with simulated terrestrial and airborne gravimetry data.
The pursued multi-scale approach is wellsuited for regional gravity field modelling from terrestrial and airborne gravity data. It is easy to implement and the numerical complexity is low. The quality of the obtained gravity field model is comparable with Least-Squares Collocation, although the number of basis functions is a factor 4-5 less than the number of observations. The multi-scale version of the DAND algorithm leads to a significant reduction of the number of basis functions at each level. The efficiency gain is maximum for heterogeneous data distributions. The performance of the approach for the regional inversion of satellite data has to be investigated. A particular problem to be addressed is the effect of colored observation noise on the performance of the DAND algorithm.
References Forsberg R (1987) A new covariance model for inertial gravimetry and gradiometry. J Geophys Res 92 (B2), 1305-310. Freeden W, Gervens T, Schreiner M (1998) Constructive approximation on the sphere. Clarendon Press, Oxford. Holschneider M, Chambodut A, Mandea M (2003) From global to regional analysis of the magnetic field on the sphere using wavelet frames. Physics of the Earth and Planetary Interiors 135, 107-124. Klees R, Wittwer T (2005) A data adaptive design of a spherical basis function network for gravity field modelling. Proceedings Dynamic Planet 2005, Cairns, Australia. Koch KR, Kusche J (2002) Regularization of geopotential determination from satellite data by variance components. Journal of Geodesy 76, 259-268. Schmidt M, Fabert O, Shum CK, Hart SC (2004) Gravity field determination using multiresolution techniques. Proceedings Second International GOCE User Workshop, ESA-ESRIN, Frascati, Italy.
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Chapter 46
Accuracy assessment of the SRTM 90m DTM over Greece and its implications to geoid modelling G.S. Vergos[], V.N. Grigoriadis, G. Kalampoukas, I.N. Tziavos Department of Geodesy and Surveying, Aristotle University of Thessaloniki, University Box 440, 541 24, Thessaloniki, Greece, Fax: +30 231 0995948, E-mail:
[email protected].
Abstract. With the realization of the Shuttle Radar Topographic Mission (SRTM) and the distribution of the 3 arcsec (90 m) data over Europe, a high-resolution digital terrain model (DTM) became available for Greece. Until today, high-resolution DTMs for Greece were generated by the Hellenic Military Geographic Service (HMGS) only and present variable resolutions with the finest one being set to 100 m. These DTMs were usually determined by digitizing topographic maps and are thus of variable and sometimes unknown accuracy. When a high-resolution, e.g., 0.5 - 1 arcmin geoid is needed, it is absolutely necessary to employ a very high resolution DTM to compute the terrain effects to gravity and the indirect effect to the geoid. If this information is not available and a coarser DTM is used, then the topographic effects computed are aliased, due to the insufficient resolution of the topographic data used. The scope of this work is twofold. First, a validation and accuracy assessment of the SRTM 90 m DTM over Greece is performed through comparisons with existing global models, like GLOBE and GTOPO30, as well as with the Greek 450 m DTMs delivered by HMGS. Whenever a misrepresentation of the topography is identified in the SRTM data, it is "corrected" using the local 100 m DTM. This processes resulted in an improved SRTM DTM called SRTMG, which was then used to determine terrain and RTM effects to gravity field quantities. Then, all available DTMs were used to compute terrain effects on both gravity anomalies and geoid heights at variable spatial resolutions. From the results acquired, the performance of the SRTMG model with respect to geoid modeling was assessed and conclusions on the effect of the DTM resolution were drawn. K e y w o r d s . SRTM, terrain effects, gravity field, geoid modelling.
1
Introduction
Digital Terrain Models (DTMs) play a crucial role in gravity field related studies, since they provide the highfrequency content of the gravity field spectrum. This is due to the high correlation of the short wavelength gravimetric features with the topography. According to Schwarz (1984) about 2% and 34% of the geoid height and gravity anomaly spectrum, respectively, are contained in the high frequencies (harmonic degrees 3 6 0 36000), where terrain effects play a significant role.
Furthermore, modem-day gravity field and geoid approximation is based heavily on the well-known remove-compute-restore (rcr) procedure, during which the terrain data are used to smooth gravity and geoid height observations to aid field gridding, transformations and predictions and avoid aliasing effects (Forsberg and Tscherning 1981; Forsberg 1985; Forsberg and Solheim 1988; Tziavos et al 1988, 1992; Vergos et al. 2005). With the advent and continuous launch of new altimetric and gravity field satellite missions and the collection and availability of new and higher in resolution data, it has become apparent that high-quality and highresolution DTMs should be available for geoid and gravity field approximation. In several countries around the world high-resolution local DTMs are not available due to confidentiality reasons, since they are most commonly generated by the respective geodetic/cartographic military agencies. Furthermore, the DTMs available are usually not homogeneous, since they are derived by a (simple in most cases) merging of available height data. On the other hand, in 2000 the Shuttle Radar Topographic Mission (SRTM) was launched on-board space shuttle Endeavour and collected a wealth of data of the Earth's topography in global scale and with homogeneous coverage. This resulted in the release of a global 3" (roughly 90 m) SRTM DTM by NASA and the National GeospatialIntelligence Agency (NGA). Thus, it was obvious that such a global DTM would offer a great aid in local, regional and global gravity field and geoid determinations, since it could be used to fill-in gaps and densify local and regional/continental DTMs. The first main goal of the present contribution is the validation of the SRTM 90 m DTM over Greece through comparisons with a national DTM generated by the Hellenic Military Geographic Service (HMGS). Their differences are analyzed and a new corrected SRTM DTM called SRTMG05 is generated for the area under study. The second main objective is the evaluation of the generated SRTMG05 DTM, against the national DTM and other global models, for gravity field and geoid determination. This is achieved by estimating the contribution of all models to gravity anomalies and geoid heights through a number of terrain reduction techniques.
2
Digital Terrain Models and Area
For the evaluation of the SRTM DTM over Greece a national DTM for the area under study and the GLOBE
310
G.S. Vergos.V. N. Grigoriadis• G. Kalampoukas.I. N. Tziavos
and ETOPO2 global DTMs were used. The SRTM mission took place in Feb 11-20, 2000 on-board the space shuttle Endeavour. Its main instrumentation was a spaceborne imaging radar modified with a mast like the one used in the International Space station and an additional antenna, so that a 60 m long interferometer could be formed. The SRTM data coverage ranges between 60 ° north to 54 ° south and covers about 80% of the Earth's total landmasses. Bamler (1999) and Farr and Kobrick (2000) should be consulted for more information on the SRTM mission and data. The data used in the present study come from the released "research grade" SRTM 90 m dataset, which means that they were unedited so could contain blunders and voids (gaps). Furthermore, no special processing of the data has been done so that in many cases the measured heights represent what was captured from the radar and not the real elevation. The latter, also known as roof effect, is especially evident over dense forests and populated areas with high trees and high buildings, respectively. The SRTM data for the area under study, bounded between 39 ° < q) < 40.5 ° and 21 ° < )v < 22.5 °, were downloaded from the corresponding US Geological Survey (USGS) ftp site (USGS 2005) and consisted of a total number of 3243601 elevations. Their statistics are presented in Table 1 while Fig. 1 depicts the SRTM 90 m DTM. The total number of undefined elevations (black dots in Fig. 1) in the area was 43199 representing roughly a 1.4% of the total dataset. They were mainly located over river basins and sea areas as well as over the Pindos range stretching from the north-west to the south-west corner of the area. The SRTM data are referenced to the EGM96 global geopotential model, the horizontal datum is WGS84 and their accuracy is at the 16 m level. The local D T M obtained from HMGS, being identified with the same name herein, had a 15" (-450 m) resolution and was generated from the digitization of 1"50,000 topographic maps (HMGS 2005). This is the standard set of heights available in Greece for surveying and engineering applications, it covers the entire country and the heights provided have a formal vertical accuracy of 20 m. A denser 100 m resolution D T M is also available from H M G S but its status is declared as confidential and is not available to the public. In the first part of this study, the 15" HMGS D T M will serve as the ground truth data set against which SRTM will be compared in order to develop a corrected SRTM DTM. Then, the corrected SRTM DTM will be used as reference in the investigation of the D T M effects on the gravity field and the geoid. The statistics of H M G S are presented in Table 1 as well. Moreover, the G L O B E and ETOPO2 DTMs were considered as well to investigate the performance of other, than SRTM, global models in the area under study. GLOBE (GLOBE 2005) is a 30" global D T M generated from a mosaic of vector and raster data sources. Its horizontal datum is WGS84, it refers to the mean sea level and has a formal accuracy for Greece at the 30 m level. Finally, the ETOPO2 D T M (ETOPO
2005) is a global model of 2' (about 3.7 km) resolution generated by assimilating a number of other DTMs and digital depth models (DDM). The models used in the computation of ETOPO2 were GLOBE, ETOPO5, DBDB5, D B D V and the Sandwell and Smith DDM. Using these DTMs and DDMs, ETOPO2 was constructed by regridding them to 2 arcmin resolution by bicubic spline interpolation. Its horizontal datum is WGS84 also and it refers to the mean sea level. The statistics of both GLOBE and ETOPO2 are also listed in Table 1. Gaps in the DTMs over marine areas were replaced by zeros.
Table
1. Statistics of the DTMs and their differences. Unit: [m]. DTMs
max
SRTM HMGS SRTMG05 GLOBE ETOPO2 SRTM-HMGS SRTMG05-HMGS SRTM-SRTMG05
min
2884.00 -23.00 2734.41 -5.68 2884.00 0.00 2710.00 1.00 2552.00 -95.00 653.54 -407.59 5 9 8 . 2 6 -406.94 33.35 -60.54
mean
704.20 716.32 707.16 703.75 690.09 -11.87 -11.73 0.00
std
_+455.04 _+456.33 _+457.77 _+457.41 _+460.28 +70.43 +70.13 +0.94
The first step of the present work was the validation of the SRTM D T M against H M G S and its correction in places where voids in the data existed. Table 1 presents the statistics of the differences between SRTM and HMGS, which have a mean value o f - 1 1 m only and a standard deviation (std) of ___70 m. Taking into account that the topography in the area under study varies significantly, it can be concluded that SRTM provides very good results. The absence of a significant bias between the two models can be attributed to the fact that no roof effect is present in the data, at least in the area under study. Therefore, the only processing done to construct a "corrected" SRTM dataset was to fill-in existing voids with heights predicted from HMGS using spline interpolation. This resulted in the so-called SRTMG05 (SRTM Greece 2005) DTM with the statistical characteristics presented in Table 1. SRTMG05 presents a smaller range of differences with HMGS compared to SRTM by about 60 m. For both models, the larger differences with HMGS are located over the Pindos mountain range and the smallest ones over the plain of Thessaly (central and central-east part of the area). Finally, some large differences can be found approximately at q0=39.5 ° and )v=21.5 ° where the Valia Calda national park, a densely forest-covered area, is located. Nevertheless, the comparisons against the national DTM give significant evidence that the SRTM dataset gives a realistic picture of the topography of the area under study. DTM oid
effects
on the gravity
field and ge-
To investigate the performance of the SRTM D T M and its implications to gravity field and geoid modelling, all
Chapter 46 • AccuracyAssessment of the SRTM90m DTM over Greece and Its Implications to Geoid Modelling
21" 00'
Fig. 1:
21" 30'
22" Off
22" 30'
The original (left) SRTM and the corrected (right) SRTMG05 90 m DTMs. Gaps in data are shown as black dots.
available DTMs (SRTMG05, HMGS, GLOBE and ETOPO2) have been used to estimate various types of topographic corrections on both gravity anomalies and geoid heights. Furthermore, aliasing effects on terrain corrections, i.e., the loss of detail when using coarser in resolution DTMs was studied. This was achieved by constructing lower resolution SRTMG05 DTMs at resolutions of 15", 30", 1', 2' and 5'. The topographic effects on gravity and the geoid computed were (a) full topographic effects, i.e., the combined effect of the Bouguer and terrain corrections, (b) terrain correction (TC) effects, (c) residual terrain model effects and (d) isostatic effects using the Airy model. Furthermore, indirect effects on the geoid have been computed estimating all three terms. The effects from all models were estimated and then compared on a l ' x l ' grid for the area under study, which corresponds to cases that a geoid and/or gravity field model of that resolution is needed. Such a high resolution l'x 1' geoid model is clearly within reach nowadays in the presence of new gravity-field related data. Due to the limited space available no formulations are given since the evaluation of topographical effects is well documented. A very detailed analysis can be found in Forsberg (1984), Heiskanen and Moritz (1967) and Tziavos (1992). The indirect effect on the geoid is explicitly described in Wichiencharoen (1982). Tables 2, 3, 4 and 5 present the statistics of the estimated full topographic effects, terrain corrections, RTM and isostatic effects on gravity from the available DTMs, respectively. From these tables it is evident that the performance of SRTMG05 is directly comparable to that of the HMGS DTM. Their difference in the computed full topographic effect is at the +6.5 mGal level in
terms of the std and ranges between -33 to +36 mGal. This is a very encouraging result, since it shows clearly that the SRTM DTM is indeed accurate and does not introduce any errors when used in gravity field and geoid determination. The same conclusions hold for the computation of the other topographic effects on gravity, since the std of the differences between SRTMG05 and HMGS is +3.3 mGal in the terrain corrections and +6.6 mGal for the RTM and isostatic effects. On the other hand, the differences almost double in magnitude when comparing the topographic effects computed from GLOBE with those derived from either SRTMG05 or HMGS. For example the std of the differences between the TC effects on gravity computed from SRTM and GLOBE are at the +6.5 mGal and reach the +13 mGal on the rest of the effects computed. Moreover, the range of the differences increases from about 60 mGal to 120 mGal. This is evidence that indeed SRTMG05 manages to depict more detail of the topography in the area under study, while GLOBE's
Table
2. Full topographic effects on gravity. Unit: [mGal]. DTMs
max
min
mean
SRTM3" SRTM15" SRTM30" SRTMI' SRTM2' SRTM5' HMGS GLOBE ETOPO2'
244.47 248.29 246.42 250.87 238.54 209.52 257.96 252.79 249.30
- 1.86 -1.56 -2.07 -1.69 -1.16 -9.55 - 1.37 -3.57 -3.05
72.88 72.57 71.63 74.64 75.07 75.88 75.37 73.64 73.78
std
+46.93 4-46.56 4-45.99 4-45.56 +45.30 4-44.37 +46.96 4-46.97 +47.39
311
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G.S. Vergos. V. N. Grigoriadis • G. Kalampoukas. I. N. Tziavos Table
3. Terrain corrections on gravity. Unit: [mGal]. DTMs
SRTM3" SRTM15" SRTM30" SRTMI' SRTM2' SRTM5' HMGS GLOBE ETOPO2'
Table
min
mean
47.00 52.54 52.71 56.84 68.89 125.61 50.56 70.87 134.24
0.04 0.03 0.08 0.02 0.01 -0.02 0.01 0.02 0.03
6.58 7.14 8.64 8.09 6.36 9.37 5.47 8.24 12.49
std
+5.92 +6.38 +7.55 +7.92 +8.17 +13.41 +5.91 +9.26 +16.04
4. Residual terrain model effects on gravity. Unit: [mGal]. DTMs
SRTM3" SRTM15" SRTM30" SRTMI' SRTM2' SRTM5' HMGS GLOBE ETOPO2'
Table
max
max
183.03 183.51 180.91 185.15 174.35 141.38 191.79 183.57 190.82
min
-91.38 -91.41 -88.63 -88.77 -83.48 -82.10 -84.68 -101.90 -87.52
mean
-3.93 -4.26 -5.16 -2.27 -1.76 -1.28 -2.81 -3.42 -1.39
std
+37.76 +37.39 +36.77 +38.04 +36.57 +33.08 +37.51 +38.31 +38.94
5. Isostatic effects on gravity. Unit: [mGal]. DTMs
SRTM3" SRTM15" SRTM30" SRTMI' SRTM2' SRTM5' HMGS GLOBE ETOPO2'
max
215.27 217.41 214.70 218.81 207.26 175.09 225.95 221.73 217.49
rain
-35.88 -34.54 -36.28 -34.57 -36.17 -35.61 -34.67 -33.60 -33.20
mean
37.63 37.23 36.21 39.01 39.09 38.36 39.42 38.59 38.31
std
+43.62 +43.25 +42.66 +44.21 +43.12 +40.40 +43.64 +43.73 +44.13
resolution is inadequate compared to the high-resolution D T M s available. The results achieved from E T O P O 2 are disappointing, since the differences between its topographic effects and the ones computed by either SRTMG05 or H M G S are at the +25 mGal in terms o f the std and reach 230 mGal in terms of the range. This is a clear indication that in the presence of the SRTMG05 elevation data, D T M s of the E T O P O 2 class should not be used for geoid and gravity field modeling anymore. Comparing the topographic effects from the SRTMG05 DTMs at the generated resolutions (15", 30", 1', 2' and 5') to those from the original 3" D T M some aliasing effects are clear. The m a x i m u m and std values in the TC effects increase gradually as moving from the dense to the coarser resolutions. In the TC effects on gravity anomalies, the std of the differences between the 3" and the rest o f the SRTMG05 models increases from +2.2 mGal for the 15", to +3.7, +3.8, +6 and +10.7 mGal for the 30", 1', 2' and 5' models. For
the R T M effects the corresponding std of the differences is at the +3.2, +4.7, +7.5, +12 and +19 mGal level. The same trends hold for the rest o f the topographic effects as well. Therefore it can be concluded that aliasing occurs when using coarser resolution D T M s for gravity field and geoid modeling. Fig. 2 depicts the TC and R T M effects on gravity as computed from the 3" SRTMG05 model. Fig. 3 (left) presents the differences between the R T M effects computed from SRTMG05 and HMGS. The same topographic effects have been computed for geoid heights as well, considering the case when the restore step is reached in the rcr procedure and the effects of the topography previously removed from the gravity data have to be restored to the estimated residual geoid heights. Tables 6 and 7 present the TC and R T M effects on geoid heights computed from the available DTMs. Once again the 3" SRTMG05 model agrees very well with H M G S with the std of the differences being at the +3.5, +2.7, +2.1 and +1.6 cm level for the computed full topographic, isostatic, TC and R T M effect, respectively. The corresponding range o f the differences is at the 20, 17, 16 and 12 cm level showing once again the very good agreement between the two models. The differences between SRTMG05 and G L O B E are again slightly larger and reach the +5.9 cm in terms of the std and the 40 cm in terms of the range for the computed R T M effects on geoid heights. For E T O P O 2 the differences with SRTMG05 have a std at the +9.4, +8.5, +8.9 and +9.2 cm level for the computed full topographic, isostatic, TC and R T M effect, respectively. Fig. 3 (right) depicts the differences in the R T M effects on geoid heights between the 3" SRTMG05 and HMGS. Table
6. TC effects on geoid heights. Unit: [m]. DTMs
SRTM3" SRTM15" SRTM30" SRTMl' SRTM2' SRTM5' HMGS GLOBE ETOPO2'
Table
max
rain
0.723 0.792 0.930 0.878 0.907 1.146 0.627 0.932 1.357
0.257 0.285 0.344 0.124 0.259 0.382 0.206 0.300 0.467
mean
0.484 0.525 0.635 0.624 0.458 0.677 0.403 0.607 0.919
std
+0.101 +0.114 +0.129 +0.127 +0.132 +0.099 +0.090 +0.126 +0.177
7. RTM effects on geoid heights. Unit: [m]. DTMs
SRTM3" SRTM15" SRTM30" SRTMI' SRTM2' SRTM5' HMGS GLOBE ETOPO2'
max
1.184 l.179 1.174 1.158 1.102 0.972 1.154 1.011 1.273
rain
-0.638 -0.639 -0.640 -0.643 -0.647 -0.802 -0.634 -0.655 -0.685
mean
0.060 0.058 0.055 0.049 0.038 0.027 0.055 0.010 0.071
std
+0.340 +0.340 +0.340 +0.340 +0.339 +0.335 +0.337 +0.336 +0.386
Chapter 46 • Accuracy Assessment of the SRTM 90m DTM over Greece and Its Implications to Geoid Modelling 21" 00' ~'~0
~o' °
o
>
a9 ° ~ '
rib 21" oo'
21" 30'
22" 00'
Fig. 2: TC (left) and RTM (right) effects on gravity anomalies from SRTMG05 (C.I. 5 mGal and 20 mGal respectively). 3' 40°~
39" 00'
39" OC
Fig. 3: Differences of TC effects on gravity (left) and of RTM effects on the geoid (right) between SRTMG05 and HMGS. Investigating the aliasing effects on the geoid from the use of coarser resolution DTMs, the same conclusions were reached. The differences between the TC effects on geoid heights from the original 3" SRTMG05 model and the 15", 30", 1', 2' and 5' DTMs was at the +1, +2.6, +10.1, +10.8 and +12 cm level, respectively. Therefore it can be concluded that an error of that
amount is introduced in geoid determination when coarser resolution DTMs are used. The final computation performed was to estimate the indirect effect (the first three terms of the expansion) on the geoid from all available DTMs. Table 8 presents the statistics of the results acquired. Once again, SRTMG05
313
314
G.S. Vergos. V. N. Grigoriadis • G. Kalampoukas. I. N. Tziavos
agrees very well with HMGS since their difference is again at the few cm level. The G L O B E model behaves much better than ETOPO2, even though the latter resulted from just a re-gridding of the former. So, the bad performance of ETOPO2 cannot be attributed to its coarser resolution alone. Aliasing is evident in the computed indirect effects on the geoid which are more pronounced when reaching the 5' resolution. A noticing fact is the very large std of the computed indirect effect from the 5' SRTMG05 model (+85 cm) when it is only +3 cm for the original 3" DTM. This is a very good example of the error introduced in geoid determination when a low-resolution DTM is used.
Table
8. Indirect effects on geoid heights. Unit: [m]. DTMs
SRTM3" SRTM15" SRTM30" SRTM 1' SRTM2' SRTM5' HMGS GLOBE ETOPO2'
4
max
0.121 0.227 0.146 0.115 0.435 0.934 0.207 0.156 0.811
min
-0.185 -0.220 -0.218 -0.286 -0.710 -4.010 -0.218 -0.219 -0.782
mean
-0.009 -0.012 -0.029 -0.078 -0.243 -1.417 -0.012 -0.029 -0.239
std
4-0.033 4-0.038 4-0.036 +0.053 4-0.151 ±0.848 +0.037 +0.036 +0.161
Conclusions
The SRTM 90 m DTM was evaluated over Greece through comparisons with a national DTM and the GLOBE and ETOPO2 global models. A corrected SRTM DTM called SRTMG05 was constructed by filling-in voids in the original dataset with interpolated values from the HMGS DTM. From the results acquired it can be concluded that the SRTM DTM is very accurate, at least in the area under study, since the topographic effects on both gravity anomalies and geoid heights are very close, if not identical, to those estimated using the national model. The std of the differences of the computed topographic effects between SRTMG05 and HMGS are at the +3.3 - +6.6 mGal and the +1.6 - +3.5 cm level for gravity anomalies and geoid heights, respectively. These results are comparable to those acquired in Germany by Denker (2004) and in Switzerland by Marti (2004), proving that the 3" SRTM DTM is indeed a valuable model and can be used for gravity field and geoid determination even at national scales at the absence of higher-resolution national models. From the results acquired for G L O B E and ETOPO2 it can be concluded that the former is indeed a good model, at least for its time, and provided a very useful set of elevation data for global geoid determinations. But, in view of the SRTM data sets, it is of little use, since it introduces an error of +6.5 - + 13 mGal and +3.5 - +5.9 cm to gravity field and geoid determination, re-
spectively. The corresponding results for ETOPO2 are far more disappointing, since the error is at the +14 +25 mGal and +8.5 - +9.5 cm. Finally, from the study on the aliasing effects introduced in gravity field and geoid determination by using coarser resolution DTMs, it can be concluded that a DTM with at least a 15" resolution should be used. In this case the error introduced in geoid heights does not exceed +1 cm. If coarser resolution is used, then errors up to +12 cm can be introduced. The use of the 15" resolution SRTMG05 model introduced an error of +2.5 mGal in gravity anomalies, which reached the +10 mGal for the 5' model. Therefore, the use of a DTM with resolution lower than 15" is prohibitive, if a highaccuracy geoid determination is needed. The next goal is to extent the present study in a larger part of the country to validate and investigate the SRTM performance nationwide. Acknowledgement
This research was funded from (a) the Greek Secretariat for Research and Technology in the frame of (a) the 3rd Community Support Program (Opp. Supp. Progr. 2000 - 2006), Measure 4.3, Action 4.3.6, Sub-Action 4.3.6.1 (international Scientific and Technological Co-operation with non-EU countries), bilateral cooperation between Greece and Canada and (b) the Ministry of Education under the O.P. Education II program x,
-
(4)
~z=l
which can be solved by minimization of the quadratic functional e(x) = (1- Ax)TC-~(1- Ax)-
axTx,
(5)
where 1 is the vector of reduced observations, A is the design matrix, x is the vector of unknown SBF coefficients, and a is tile regularization parameter.
3
The data-adaptive (DAN D) strategy
network
design
The performance of a SBF network critically depends upon the chosen centres of the SBFs, or, in other words, upon the design of the sampling network. In multi-scale modelling, various subdivision schemes are used, generating hierarchical or non-hierarchical grids of points. Examples are the well-known geographical grids, subdivisions of the faces of icosahedrons projected on the sphere, and homogeneous point distributions on the sphere like the Reuter grids (e.g. Freeden et al. 1998). Alternatively, the basis functions are located below the data points at some depth, a strategy, which is often used in pointmass modelling (e.g. Heikkinen 1981, Marchenko et al. 2001). However, placing the basis functions below a subset of the data points is clearly an unsatisfactory method for building a sampling network. The resulting network often performs poorly or has a large size. On the other hand, simply using a sampling network of centres is also not the method of choice; it may cause overfit or instabilities. Finally, the choice of the centres of the SBF network cannot be considered independently of the choice of the bandwidth of the SBFs. Both
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aspects are inter-related and have to be addressed jointly to get a high-performance network. In the following we will propose a dataadaptive network design (DAND) strategy, which selects the SBFs and their bandwidths automatically from the data. The network design starts with a template network, generated by any subdivision scheme. As first step, we remove a basis function if there are less than q _> 1 data points within its region of influence (ROI). The latter is defined as a spherical cap of radius ~Rof, which is only a function of the point density of the template network. The spherical radius of the ROI is determined according to A
2~(1
-
~os~o~)
-
Xov~'
(s)
where A is the size of the area under investiagation projected onto the unit sphere, and Novs is the number of data points in that area. If at least q data points are within the region of influence, the SBF is selected and the q nearest data points are removed from the index list of data points. Then, the next SBF is considered. This process is continued until no data points are left in the index list. As second step, the bandwidth of the SBF is selected (for the definition of the bandwidth, see (Narcowich and Ward 1996)). For the multi-pole wavelets, to be introduced in section 4, the bandwidth is a non-linear function of the distance of the multi-poles from the origin of the co-ordinate system, i.e. from the depths below the surface. The deeper the multi-poles are located, the larger the bandwidth is. In the DAND algorithm, the bandwidth is adapted optimally to the data and the sampling network density. For that reason, we write the bandwidth as
and determine the optimal factor p using Generalized Cross Validation techniques (cf. Golub et al. 1979, Klees and Wittwer, 2005). The first two steps of the data-adaptive network design guarantee that more SBFs are located in data-rich areas, while data-poor areas are represented by fewer SBFs. Li (1996) uses the concept of a region of influence and a proper selection of the bandwidth for the global modelling of temperature fields from a sparse set
of points heterogeneously distributed over the sphere. However, the number of SBFs in a certain area should not be driven by the number of data points, but by the signal variation in that area. If too many SBFs are placed in an area with smooth signal, they mostly model noise. Therefore, in the third step of the data-adaptive network strategy, we look at the signal at the data points. If there are only data points within the ROI that have a signal smaller than some threshold, the SBF is removed from the network. The threshold may be a function of the expected noise variance at the data points, provided this information is available. Otherwise, the user may specify its own threshold value. The third step guarantees that mostly signal above the threshold is modelled, whereas signal and noise below this level are not modelled; this reduces the risk of overfitting. Then, the overall estimation procedure looks as follows: The estimation process starts with the choice of the template network. Next, we compute the ROI of the template network, Eq. (6). Then, we select a candidate parameter p and compute the bandwidth cr of the SBFs, according to Eq. (7). From the bandwidth, we determine the depths of the SBFs by a Newton iteration scheme. The horizontal positions of the SBFs are given by the template network. Next, the SBFs are selected using the ROI criterion. Thereafter, the potential coefficients are estimated by minimizing the quadratic functional, Eq. (5). Thereafter, the GCV functional is computed. This procedure is repeated until the parameter p has been found that minimizes the GCV functional. The corresponding solution represents the residual disturbing potential. If regularization is applied (i.e. for a non-zero c~ in Eq. (5)), a second iteration loop inside the GCV loop is required. The optimal regularization parameter is found using Variance Component Estimation techniques (see Koch and Kusche 2002). The DAND algorithm has been generalized to multi-scale problems. For more details, the reader is referred to (Klees and Wittwer, 2005). 4
Simulations
Our approach to local gravity field modelling is independent of the choice of the SBFs. The re-
Chapter 48 • A D a t a - A d a p t i v e Design of a Spherical Basis Function N e t w o r k for Gravity Field Modelling
sults, to be presented later, have been obtained with the so-called multi-pole wavelets introduced in (Holschneider et al. 2003)"
258 °
260 °
262 °
36"
36"
¢(~, ~) l+1 l----L+1
P~(~)
4rrR 2
(8)
• ~,
m is the order of the multi-pole wavelet. The parameter a determines the distance of the multipole wavelet w.r.t, the origin of the spherical coordinate system according to y __ /~ ~ - - a !),
l) - -
Y I'Y
°
34"
34"
32"
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(9)
For a fixed m, the scale parameter a determines the shape and position of the spectrum of the wavelet. The larger a, the deeper the basis function is located below the surface, and the more the spectrum is concentrated at lower frequencies. With decreasing a, the spectrum covers higher frequencies. Correspondingly, the centre of the basis function moves towards the surface, which is equivalent with a better space localization. We did extensive numerical test with simulated terrestrial and airborne gravity data. The results to be presented later refer to an airborne gravity data set. The area is located in the US, North to the Gulf of Mexico. The size is about 6 x 6 degrees. The gravity disturbances at a flight altitude of 2 km range from - 3 0 regal to 45 regal; the RMS signal is about 6.2 regal. The frequency content of the data is limited to degrees 121 to 1800. Therefore, the parameter L in Eq. (8) is set equal to 120. Figure 2 shows the gravity signal over the test area. We generated 5453 gravity disturbances, heterogeneously distributed at an altitude of 2 km above the area, see Figure 2. The data have been corrupted with white noise with a standard deviation of 1.5 regal. Multi-pole wavelets of order m = 3 have been used in all simulations. In section 5, the results for two different template networks are presented: (i) an equal angular grid at level 7 with 4096 points and (ii) the 5453 data points themselves. Once the SBF coefficients have been estimated, gravity disturbances and disturbing potential values are computed at flight level and at ground level. The estimation at flight level has been done at the data points and at a set of control points. The latter are located in between the data points.
30"
30" 258"
260"
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Figure 1: Gravity disturbances over the test area. The RMS signal variation is 6.2 regal, the range is - 3 0 to 45 regal. 258 °
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258 °
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Figure 2" Data distribution over the test area. The data set consists of 5453 observations.
5
Results and discussion
Five different scenarios have been investigated" • Scenario 1: An equal angular grid is used as template network; the DAND algorithm is applied with a threshold equal to zero; no regularization.
325
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R. Klees • I. Wittwer
Scenario 2: The SBFs are placed on an equal angular grid at level Y, which consists of 4096 points. This point density is sufficient to resolve the spectral content in the data. Regularization is applied. • Scenario 3: An equal angular grid is used as template network; the DAND algorithm is applied with a 3~-threshold, where cr = 1.5 r e g a l is the data noise level; no regularization. • Scenario 4: The data points are used as ternplate network; the DAND algorithm is applied with a 3or-threshold, where cr = 1.5 r e g a l is the data noise level; no regularization. • Scenario 5: A SBF is placed below every data point; regularization is applied. Figure 3 shows the RMS error in the gravity disturbances computed at the data points and at the control point at flight altitude for the various scenarios. First of all, we see that the RMS error at the control points is the smallest if the DAND algorithm is applied (scenarios 3 and 4). The RMS error at the data points, however, is larger for scenarios 3 and 4 compared with scenarios 1,2, and 5, which do not use the DAND algorithm. This can be explained by the presence of 'overfitting' for scenarios 1, 2, and 5. Therefore, one important conclusion is that the DAND algorithm avoids or at least reduces 'overfitting'. A second important conclusion can be drawn from a comparison of the results for scenario 3 and 4. They differ w.r.t, the template network used in the DAND algorithm. In scenario 3, an equal angular grid is used as a ternplate network, whereas scenario 4 uses the data points as a template network. Obviously, if the DAND algorithm is applied, the choice of the template network does not matter, i.e. almost the same results are obtained with various ternplate networks. T h a t is exactly what one would expect from a good algorithm. Remarkable is that the DAND algorithm reduces the number of SBFs significantly. Scenarios 3 and 4 use only 1672 and 1584 SBFs, respectively, whereas 5453 SBFs are needed when placing a SBF below every data point, although the latter provides a significantly larger RMS error at the control points. Moreover, if the DAND algorithm is used, no regularization is necessary. If, however, a SBF is placed below every data point, we found in all
2,5 2 o~
L9 E
1 0,5 0 1
2
3
4
5
Figure 3: Gravity disturbance errors at flight altitude at the data points and the control points for various scenarios. 1: equal angular ternplate network, DAND applied, no regularization, zero threshold; 2 : 4 0 9 6 SBFs on an equal angular grid, regularization applied; 3: equal angular template network, DAND applied, no regularization, 3or-threshold; 4: data points as ternplate network, DAND applied, no regularization, 3a-threshold; 5: one SBF below every data point, regularization applied. The data noise is 1or = 1.5 mGal.
simulations that no physically meaningful solution is obtained without regularization. Figure 4 shows the RMS error in the residual disturbing potential computed at a set of control points at ground level. The conclusions drawn from the results shown in Figure 3 are fully supported. In particular, the DAND algorithm provides the best solution with the smallest number of SBFs. A geometrical explanation for getting suboptimal results if a SBF is placed below every data point is given by the factors p, which determine the depths of the SBFs (cf. Eq. (7)). In that case, we obtain an optimal value p = 2.4. This is much smaller than the value of p obtained when DAND is applied (p = 3.4 for an equal angular grid as template network and p = 3.0 if the data points act as template network). A smaller value of p means that the SBFs are placed more shallow. This can give small residuals, but a worse fit between the data points and at the ground level. Figure 5 shows the performance of the DAND algorithm for various threshold values varying between 0 (i.e. no threshold) to 4or. The RMS error is the smallest for a threshold value of 1or, which corresponds to the data noise level. As one can expect, the RMS error increases with increasing threshold, whereas the number of SBFs to be used decreases. Important for practical ap-
Chapter 48
• A
Data-Adaptive Design of a Spherical Basis Function Network for Gravity Field Modelling
0,4
0,3
0,35 0,3 ' ~ 0 2, 5 % 0,2 E ''0,15 0,1 0,05
0
0,25 ~~ ¢/)
0,2
¢,1
0,15 0,1 1
2
3
4
5
0,05
0 0
Figure 4: Disturbing potential errors at ground level for various scenarios. 1: equal angular ternplate network, DAND applied, no regularization, zero threshold; 2:4096 SBFs on an equal angular grid, regularization applied; 3: equal angular template network, DAND applied, no regularization, 3or-threshold; 4: data points as ternplate network, DAND applied, no regularization, 3or-threshold; 5: one SBF below every data point, regularization applied. The data noise is l a = 1.5 mGal. plications is that the RMS error increases quite moderately, whereas the number of SBFs to be used decreases significantly. For instance, if a threshold of 1or is used, 2876 SBFs are selected by the DAND algorithm. A threshold of 3a reduces the number of SBFs to 1672, a reduction by about 42 %. The RMS error, however, increases at the same time from 0.21 m 2/s 2 to 0.23 m2/s 2, i.e. by only about 10%.
6
Conclusions
The following conclusions are drawn from the conducted numerical experiments. They are confirmed by other experiments with simulated terrestrial and airborne gravimetry data and for other choices of radial basis functions. • The DAND algorithm reduces the number of SBFs significantly. The ratio of the number of SBFs to the number of observations is typically about 30%. This is much less than for Least Squares Collocation or classical point mass modelling. • The DAND algorithm can make regularization superfluous. In all experiments with terrestrial and airborne data, regularization was not necessary, contrary to the use of point masses below the data points. • When DAND is applied, the location of the
1G
2a
3c
4G
Figure 5: Disturbing potential errors at ground level for various thresholds: 1or = 1.5 regal corresponds to the standard deviation of the data noise. DAND algorithm is applied with an equal angular template network. No regularization is used. The number of SBFs used is (from left to right) 3368, 2876, 2231, 1672, and 1207.
SBFs does not matter. That is, any template network can be used including the set of data points. The quality of the solution when the DAND algorithm is applied is superior to the solution without DAND algorithm. The DAND algorithm is a generic tool and has widespread applications in interpolation and approximation of data on the sphere. Experiments with simulated satellite data and real terrestrial, airborne and satellite data are going on and will be reported elsewhere.
References Eicker A, Mayer-Guerr T, Ilk KH (2004) Global gravity field solutions from GRACE SST data and regional refinements by GOCE SGG observations. Proceedings IAG International Symposium Gravity, Geoid and Space Missions (GGSM2004), Porto, Portugal. Freeden W, Gervens T, Schreiner M (1998) Constructive approximation on the sphere. Clarendon Press, Oxford. Golub GH, Heath M, Wahba G (1979) Generalized cross-validation as a method for choosing a good ridge parameter. Technometrics 21, 215-223. Heikkinen M (1981) Solving the shape of the Earth by using digital density models. Finnish
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Geodetic Institute, Report 81:2, Helsinki, Finland. Holsehneider M, Chambodut A, Mandea M (2003) From global to regional analysis of the magnetic field on the sphere using wavelet frames. Physics of the Earth and Planetary Interiors 135, 107-124. Klees R, Wittwer T (2005) Local gravity field modelling with multi-pole wavelets. Proceedings Dynamic Planet 2005, Cairns, Australia. Koch KR, Kusche J (2002) Regularization of geopotential determination from satellite data by variance components. Journal of Geodesy 76, 259-268. Li TH (1996) Multiscale representation and analysis of spherical data by spherical wavelets.
SIAM J. Sci. Comput. 21,924-953. Marchenko AN, Barthelmes F, Meyer U, Schwintzer P (2001) Regional geoid determination: an application to airborne gravity data in the Skagerrak. Scientific Technical Report 01/07, GeoForschungsZentrum, Potsdam, Germany. Nareowich F J, Ward, JD (1996) Nonstationary wavelets on the m-sphere for scattered data. Appl. Comp. Harm. Anal. 3, 324-336. Schmidt M, Fabert O, Shum CK, Han SC (2004) Gravity field determination using multiresolution techniques. Proceedings Second International GOCE User Workshop, ESAESRIN, Fraseati, Italy.
Chapter 49
Global gravity field recovery by merging regional focusing patches: an integrated approach K.H. Ilk, A. Eicker, T. Mayer-Gfirr Institute of Theoretical Geodesy, University of Bonn, Nussallee 17, D-53115 Bonn, Germany
Abstract. Usually the gravity potential is modelled
by a spherical harmonic expansion. Simulation tests and real-data investigations based on POD (precise orbit determination) and SST (satellite-to-satellite tracking) data demonstrated that the heterogeneity of the gravity field cannot be properly taken into account by base functions with global support. It is preferable to model the gravity field only up to a moderate safely determinable spherical harmonic degree without any regularization to cover the long and medium wavelengths characteristics; the specific detailed features tailored to the individual gravity field characteristics in areas of rough gravity field signal can be modelled additionally by space localizing base functions. In a final step, a spherical harmonic expansion up to a maximum degree, only limited by the most detailed structures of the gravity field, can be derived based on a Gauss-Legendre-Quadrature procedure. This last step can be performed without stability problems and without losing the regional details of the gravity field. The proposed integrated gravity field recovery approach integrates consistently a regional gravity field zoom-in into a global gravity field solution. The technique has been applied to the determination of gravity field models based on SST data of GRACE. Keywords. CHAMP, GRACE, global gravity field recovery, regional gravity field recovery
1 Introduction As a result of the dedicated space-borne gravity field missions CHAMP (Challenging Minisatellite P a y l o a d - Reigber et al. 1999) in orbit since 2000 and especially GRACE (Gravity Recovery And Climate Experiment- Tapley et al. 2004), in orbit since 2002, a breakthrough in accuracy and resolution of gravity field models has been
achieved. Subsequent solutions by using the observations collected over a period of time of, e.g., one month, enable the derivation of time dependencies of the gravity field parameters. The innovative character of these missions lies in the continuous and precise observations of the low flying satellites by the Global Positioning System (GPS) and the highly precise line-of-sight range and range-rate Kband measurements between the twin-satellites of the GRACE mission. In addition, the surface forces acting on these satellites are measured and can be considered properly during the recovery procedure. As a result of this mission, the presently best combination static model, EIGEN-CG03C, has been derived from 376 days of GRACE observations and three years of CHAMP (F6rste et al. 2005) as well as monthly snap-shots of the gravity field, showing clearly temporal variations of the gravity field closely correlated to the hydrological water cycle. Another comparable precise GRACE gravity field model is GGM02C, based on the analysis of 363 days of GRACE in-flight data (Tapley et al., 2005). it is represented by a spherical harmonic expansion up to degree 200 and constrained with terrestrial gravity information. The gravity field recovery based on the new types of satellite gravity observations poses a challenge in many respects. One of the key problems seems to be related to the representation of the gravity field by an appropriate set of base functions and of the associated gravity field parameters. The established and approved way is to model the gravity field by spherical harmonics up to a certain maximum degree. This degree is limited by the significance of the gravity field signal in the observations and its ratio to the observation noise. Because of the inhomogeneous gravity field of the Earth the signal content varies in the space domain. The gravity field in regions with rough gravity field features should be modelled up to a higher degree than within regions with smooth gravity field features. But the gravity
330
K.H.Ilk. A. Eicker.T.Mayer-GOrr field cannot be resolved globally up to this degree because corresponding gravity signals in the SSTobservations do not exist over the predominant parts of the world. Therefore a spherical harmonic gravity field model up to this maximum degree results in instabilities; the missing high-resolution signal in the observations of the predominant part of the world acts similar as the polar gaps. The application of the frequently applied Tikhonov regularization leads to a mean damping of the global gravity field features with the consequence that the high frequent gravity field signal in the observations is lost again in some geographical regions. An alternative integrated approach is to determine a global gravity field solution with high long and medium wavelength accuracy and improve this global solution in regions with characteristic gravity field features by an adapted regional recovery procedure. The global solution is parameterized by spherical harmonic coefficients up to a moderate and safely determinable degree and the regional solutions are represented by space localizing base functions, e.g., by spherical splines. This procedure provides several advantages. The regional approach allows exploiting the individual signal content in the observations and a tailored regularization for regions with different gravity field characteristics. In addition, the resolution of the gravity field determination can be chosen for each region individually according to the spectral content of the signal in the specific region. Furthermore, the regional approach has the advantage of dealing with regions with different data coverage more easily, if no data at all is available the regional refinement can be skipped. For regions with sparse data coverage a coarser parameterization can be selected• Several regional solutions with global coverage can be merged by means of numerical quadrature methods to obtain a global solution, in principle, up to an arbitrary degree, only limited by the signal content of the gravity observations. Due to the regionally adapted strategy this method provides better results than calculating a spherical harmonic solution by recovering the potential coefficients directly.
the success of this integrated approach in case of selected examples related to the GRACE mission. Sect. 4 concludes this article with a summary and some perspectives for a future work. 2 Computation
procedure
2.1 Setup of the mathematical models If precise intersatellite functionals as line-of-sight ranges or range-rate measurements are available, as in case of the GRACE mission, the mathematical model can be based on Newton's equation of relative motion, i;,2(t) = g(t; r~2,r~,/'1,/'2;x). (1) This differential equation can be formulated as a boundary value problem, r,2 (t) - (1- r) r,2,A + rr,2,8 -
-T2 I K(r'r')g(t;r12'rl'rl'r2;x)dr" z-'=0
with the integral kernel,
(3) the normalized time variable, r -
t-t A
with T - t~ - tA, t E [tA,t~ ], T as well as the boundary values
r,~,~ := r,~(t~),
r,~,~ := r,~(t~),
t~ < t~.
(4)
(5)
Differentiation with respect to the time results in the relative velocity between both satellites, •
1
) dK(:,:') g(r"
- T ~':0
dr
(6) • d r .' ,rl2,r 1, /'l,/'2,x)
The mathematical model for range observations follows by projecting the relative vector to the lineof-sight connection in combination with Eq. (2), r~2(r) = e,2 (r). r12(r). (7) Analogously, the mathematical model for range-rate measurements in combination with Eq. (6) reads as follows,
~,~(~) = e,~ (~). i~,~(~). Sect. 2 of this article reviews various aspects of the computation procedure for processing the observations of the new gravity satellite missions. It is our concern to point out the importance of combining a regional gravity field zoom-in with a global gravity field recovery to assure the consistency of both views of the gravity field. Sect. 3 demonstrates
(2)
(s)
In both equations, e~2 is the unit vector in the lineof-sight direction of both twin-satellites. This vector is lonown with high accuracy, assuming that the satellite positions are measured with an accuracy of a few cm and taking into account the distance of approximately 200km between the two satellites. The
Chapter 49 • Global Gravity Field Recovery by Merging Regional Focusing Patches:an Integrated Approach
specific force function for the relative motion according to Eq. (1) can be separated into various parts, g(t;r12,r1,i'1,i'2; x) = gd (/;rl,r2,i'l,i'2) +
The anomalous potential T(r, Ax) reads for a global
-~-V ~(12)E (t; 1"12 , r 1 ; x 0 ) + V' T(12)E (t; 1"12 , r l ; A x ) ,
gravity field recovery,
(9)
The quantity gj is the disturbance part, which represents the non-conservative disturbing forces and V V~12)e is the reference part, modeled by the tidal potential of the Earth (E) acting on the satellites 1 and 2, V~(12)E (/~;rl2,rl ; Xo) -- V (V(F, + r , 2 ) - V ( r , ) ) ,
(10)
and represents the long-wavelength gravity field features. The anomalous part, VT(,2)E (t;r,2,r,; Ax) - V ( r ( r , + r,2)- r ( r , ) ) ,
(11)
Q~ (0,2) = p m(COSO)COSm2,
(13)
S .... (,9,2)- P['(cos,9)sinm2.
F
n=2 m=0
(]4)
with the corrections Acre,As" • Ax to the reference potential coefficients cn,,,, s,, • x 0 . In case of a regional recovery the anomalous potential T(r) is modeled by parameters of space localizing base functions, I
models the high frequent refinements, parameterized either by corrections Ax to the global gravity field parameters x 0 or by parameters Ax of a linear approximation with space localizing base functions, modeling the regional gravity field refinements. Details can be found in Mayer-Gfirr et al. (2005a). In case of the analysis of observation sets of months to years, the observation equations are formulated in space domain by dividing the total orbit in short pieces of arcs with a length of approximately 30 minutes. The length of the arcs is not critical at all and can be adapted to the uniformity of the data flow. Because of the fact, that a bias for each of the three components of the accelerometer measurements along a short arc will be determined, the arc length should not be too small to get a safe redundancy and not too long to avoid accumulated not modelled disturbances. When the normal equations of all arcs are merged, for every short arc a variance factor has to be determined by an iterative computation procedure, to take the (possibly) varying accuracy of the short arcs into account (cf. Koch and Kusche (2003) for the variance component estimation, and Mayer-Giirr et al. (2005a) for the iterative computation procedure).
2.2 Gravity field representation The reference potential can be formulated in the usual way as follows,
i=l
with the unknown field parameters a i arranged in a column matrix Ax := (ai, i = 1,...,I) r and the base functions, N...... f R -~n+l
The coefficients k are the degree variances of the (difference) gravity field spectrum to be determined, m=0
with the fully normalized potential coefficients A~ .... A~,,. Re is the mean equator radius of the Earth, r the distance of a field point from the geocentre and P (r,%,) are the Legendre polynomials depending on the spherical distance between a field point P and the nodal points Qi of the set of base functions located at the nodal points i. The maximum degree NMaX in Eq. (16) should correspond to the envisaged maximum resolution expected for the regional recovery. With the definition in (16) the base functions ¢~(r,ro, ) can be interpreted as isotropic and homogeneous harmonic spline functions (Freeden et al., 1998). The nodal points are defined on a grid generated by a uniform subdivision of an icosahedron of twenty equal-area spherical triangles. In this way the global pattern of spline nodal points Qi shows approximately uniform nodal point distances. Details can be found e.g. in Eicker et al. (2005).
2.3 Combination of normal equations F
n 0m=0
(12) with the surface spherical harmonics,
For the analysis of CHAMP or GRACE observations not only the gravity field parameters have to be estimated, but also arc-related parameters as for ex-
331
332
K.H. Ilk. A. Eicker. T. Mayer-G~irr
ample the two boundary position vectors of each arc and additional bias parameters to take into account residual surface force effects. These parameters sum up to thousands of additional unknowns for an analysis period of one month in case of short arcs with a mean arc length of approximately 30 minutes. To reduce the size of the normal equation matrices, the arc-related parameters are eliminated before the arcs are merged to the complete system of normal equations. Every short arc builds a (reduced) partial normal equation. To combine the normal equation matrices for the short arcs, separate variance factors for each arc have to be determined, to consider the variable precision of the range and range-rate observations. Furthermore, due to the fact that the gravity field recovery process is improperly posed, an additional regularization factor and a regularization matrix can be introduced into the gravity field recovery procedure. For details of the iterative combination scheme combined with a variance component estimation and the computation of the regularization factor refer to Mayer-Gfirr et al. (2005a).
2.4 Merging of regional refinement patches For many applications it is useful to derive a global gravity field model by spherical harmonics without losing the details of a regional zoom-in. This can be performed by a direct stable computation step. In our approach the coefficients of the spherical harmonic expansion are calculated by means of the Gauss-Legendre-Quadrature (cf. e.g. Stroud and Secrest, 1966). This method is also referred to as Neumann's method, as described in Sneeuw (1994) among different other quadrature methods,
s
t
- GM4Jr ~ TkP'' (cos 0~) ~ sin(m2~
w~,
(~8)
with the area weights,
T~ is the gravitational potential at the K nodes of the numerical quadrature, P~ are associated Legendre functions and P¢+I are the first derivatives of the Legendre polynomials of degree N + 1, where N is the maximum degree to be determined. This method requires the data points to be located at a specific grid, called Gauss-Legendre-Grid. From the regional spline solutions the gravitational potential can be calculated at the nodes of this grid
without loss of accuracy. It has equiangular spacing along circles of latitude; along the meridian the nodes are located at the zeros of the Legendre polynomials of degree N + 1. This quadrature method based on this specific grid has the advantage of maintaining the orthogonality of the Legendre functions despite the discretization procedure, which allows an exact calculation of the potential coefficients. The resolution of the grid has to be adapted to the envisaged maximum degree of the spherical harmonic expansion and should be limited only by the signal content of the gravity signals in the observables.
Results of regional gravity field refinements The complete recovery procedure consists of three steps which can be applied independently as well: • Least squares global gravity field recovery based on a spherical harmonic expansion up to a moderate degree without any regularization to provide a basis for further refinements, • Least squares regional refinements of the gravity field by spherical splines as space localizing base functions, adapted to the specific gravity field features with respect to nodal point distribution and base function characteristics, if possible covering the globe, • Determination of a global gravity field model by merging the regional refinement solutions and deriving potential coefficients by a numerical quadrature technique without loosing the regional details. We want to point out that a combination of data sets of different origin is simply possible, either based on satellite, airborne or terrestrial gravity information. This can be done, in principle, at every step of the 3step procedure. The global gravity field determination of the first step is performed independently from the subsequent regional refinement computations and no correlations between the parameters have been taken into account yet. The following recovery results refer to the K-band range-rate measurements of the GRACE twin satellite mission for the months September 2003 and July 2003, respectively. The observations are corrected for the tides caused by the Sun, the Moon and the planets. The ephemeris are taken from the JPL405 data set. Effects originating from the deformation of the Earth caused by these tides are modelled following the IERS 2003 conventions. Ocean tides are
Chapter49
• Global Gravity Field Recovery by Merging Regional Focusing Patches: an Integrated
computed from the FES2004 model. Effects of high frequency atmosphere and ocean mass redistributions are removed prior to the processing by the GFZ AOD de-aliasing products. The 30-days-orbit has been split into 1500 short arcs of approximately 30 minutes arc length. For each arc the coordinates of the boundary vectors have been determined as well as an accelerometer bias (Mayer-Gfirr et al., 2flflSh~_
-50 -40 -30 -20 -10
0
10
20
30
40
Approach
has been applied. The arc-related parameters are eliminated before merging the normal equations for each short arc to the total system of normal equations as outlined in Sect. 2.3. Fig. 1 shows the differences of this monthly solution and the EIGENCG03C, here taken up to a maximal degree n=140, representing a slightly higher resolution as our onemonth solution. Fig. 2 demonstrates that a maximum degree of n=140 results only in a slight improvement from 40,7cm to 37,9cm in terms of RMS. This means that a higher resolution of the global model based on spherical harmonics cannot significantly improve the result.
50
[cm]
RMS: 40.70 cm
avg: 29.69cm
Max: 392.05cm
Fig. 1: Differences of direct monthly solution ITGGRACE-2003-09-d (n-120) minus EIGEN-CG03C (n=140).
-50 -40 -30 -20 -10
RMS: 37.90 cm
0
10
[crn] avg: 30.03 cm
20
30
40
50
Max: 196.88 cm
Fig. 2: Differences of direct monthly solution ITGGRACE-2003-09-d (n = 140) minus EIGEN-CG03C (n=140).
In the first step, a global spherical harmonic solution up to degree n=120 beginning from degree n=2 has been determined directly for the month September 2003 from the GRACE range-rate measurements, in the following designated as gravity field model ITG-GRACE-2003-09-d ("d" means direct spherical harmonic solution). The mathematical model (8) with (6) based on a pure spherical harmonic gravity field representation according to (9) and (10) with the spherical harmonic model (12)
Fig. 3: Regional refinement patches covering the complete surface of the Earth with additional frame around a recovery area. For the regional refinement solutions the same mathematical model as used for the global solution and formulated in (8) with (6) has been applied except for the gravity field representation. Based upon the global spherical harmonic solution up to degree n-120 the additional gravity field refinements are represented according to (9) with (11) represented by spherical spline functions according to (15). The number of nodal points I as well as the maximum degree NMax of (16) has been chosen identical for all patches in the present case. To avoid geographical truncation effects at the region boundaries, gravity field parameters defined in an additional strip of 10 ° width around the specific regions have to be taken into account. The regional patches are shown in Fig. 3. This is of course not an ideal adaptation to the different gravity field characteristics. Here the pole regions north of 60 ° and south of-60 ° are separate patches. The nodal points are located at a regular grid with a mean distance between the nodal points of approximately 140kin. A global gravity field model represented by spherical harmonics can be calculated again by a proper application of the Gauss - Legendre - Quadrature method based on the global coverage with regional refinements.
333
334
K.H. Ilk. A. Eicker. T. Mayer-GiJrr
E r~
8 .~
0.1
> tL',(j~o~.'~-~.,ru~e,.c~.an m~.-...l'e'J ~'
,~,,,~W'~"
"
0.01
o.ool
60 °
70 °
80 °
90 °
110 o
120 ° 90 °
,
;
-1;0-80-E~0-4~0-2'0
100 o 2'0
4~0
6'0
80
I
100 o
110 o
120 °
100-100-80-60-40-20
140 °
150 °
I
0 20 [cm]
[cm]
RMS: 56,07 cm Avg: 39,42 cm Max: 355,78 cm
130 °
. . . . . . . . . . 40
6-i
60
80
RMS: 4,62 cm Avg: 3,60 crn Max: 19,27 cm r
I°
90 ° .
100 o .
-10
-8
.
.
-6
110 o .
-4
.
-2
120 ° .
.
.
.
0 2 [cm]
4
6
8
I 10
Fig. 4: Regional refinements of adjacent patches with differences (based on the direct global solution ITG-GRACE-2003-09-d, n=120). Fig. 4 demonstrates the excellent matching of two regional refinements with a common overlapping recovery strip. It should be pointed out that both residual fields are computed independently with individually determined regularization parameters. The bottom graph at the left-hand side has been plotted with the same scale as the two regional refinements at the top of Fig. 4, while the graph at the righthand shows the same overlapping region with a magnified scale.
d~-ecl~ 0
100
RMS: 56,58 crn Avg: 41,41 cm Max: 317,50 cm
ii
0.0682 & 0 0001 10
20
:243
40
50
h~"rr~n~: ~ 60
70
80
n-=140,:~ ~,,~i.,,lanz,i,-,,on.', 90
100
110
t20
t30
t40
spherical harmomc degcee
Fig. 6: Difference degree variances of monthly solutions (September 2003) versus EIGEN-CG03C. Fig. 5 shows the differences of the merged monthly solution ITG-GRACE-2003-09 minus EIGEN-CG03C, both models up to a spherical harmonic degree of n=140. The comparison with Fig. 2 demonstrates a significant improvement, especially in the higher degrees from n=120 upwards. Fig. 6 confirms this result by the corresponding difference degree variances related to EIGEN-CG03C. It should be pointed out that the slightly better results of the direct solution in the long and medium wavelength part of the gravity field spectrum are very small and maybe not really significant. The resolution of the regional refinements corresponds to a spherical harmonic degree of approximately 140. In the following we will compare our merged September 2003 solution to various models by restricting the upper degree to n=120. The statistics of the geoid height differences between the merged monthly solution ITG-GRACE-2003-09 and the GFZ model CG03C are: RMS=15,52cm, avg=12,01cm and Max=74,82. The corresponding values for the differences with respect to the CSR model GGM02C are slightly smaller and read: RMS=13,50cm, avg=10,59cm and Max=72,95.
E
a~ ,~-
0.1
C
'~ -50
-40
-30
-20
-10
0
10
20
30
40
50
0.001
._
[ore] RMS: 23.83 cm
avg: 18.28 cm
Max: 135.49 cm
Fig. 5: Differences of merged monthly solution ITG-GRACE-2003-09 (n=140) minus EIGENCG03C (n = 140).
I~
10
20
30
40 50 ~ 70 813 9'0 ~1~n¢4~1 h B r r n o ~ i ¢ ~%=~ree
tOO
11(I
120
1~1
t40
Fig. 7: Difference degree variances of monthly solutions (September 2003) versus EIGEN-CG03C.
Chapter49 • GlobalGravityField Recoveryby Merging RegionalFocusing Patches: an Integrated Approach
The difference degree variances of our merged solution ITG-GRACE-2003-09 together with the corresponding GFZ and CSR solutions for September 2003 are displayed with respect to the GFZ model EIGEN-CG03C in Fig. 7 and with respect to the CSR gravity field model GGM02C in Fig. 8. It is remarkable that our monthly model has a significant better coincidence with the superior models EIGEN-CG03C and CSR-GGM02C from degree n-70 upwards than the corresponding GFZ and CSR monthly models. In the long and medium frequencies the situation is slightly different; here the GFZ and the CSR monthly models show a better coincidence with the corresponding combined models of these organisations, albeit the differences are very small if the logarithmic scaling is properly taken into account.
i •
ences of the corresponding merged monthly solutions GFZ-GRACE-2003-09 minus GFZ-GRACE2003-07 provided by the GFZ Potsdam and Fig. 11 the differences of these monthly solutions by CSR Austin. We detect a remarkable good fit in the pole regions for our models in contrast to the GFZ- and C S R - solutions. But also the RMS, the average and maximum values are significantly better for our solutions than the GFZ and CSR solutions. The large deviations along the +/-60°-latitudes cannot be explained yet, but they occur also in the other solutions.
0.1
0,01
-50
gleb~~ Io"~,~"k-~,,.~,.,:,n.
0.0t71
-40
-30
-20
RMS: 22.68 cm "V V
-
10
2~
0
10
20
[cm] avg: 17.07 cm
30
40
50
Max: 134.00 cm
r...SR..G~r_..E.,¢~_I~I
'~ 0.~1 i)
-10
30
40
50 i~} 70 80 90 sphe~cal hann~ic degree
100
110
120
130
t40
Fig. 10: Differences of monthly solutions GFZGRACE-2003-09 minus GFZ-GRACE-2003-07.
Fig. 8: Difference degree variances of various monthly solutions (September 2003) versus CSRGGM02C.
-50
-40
-30
-20
RMS: 28.14 cm
-50
-40
-30
-20
RMS: 14.64 cm
-10
0
10
[cm] avg: 10.93 cm
20
30
40
50
-10
0
10
avg: ~1r[~7 cm
20
30
40
50
Max: 152.12 cm
Fig. 11: Differences of monthly solutions CSRGRACE-2003-09 minus CSR-GRACE-2003-07.
Max: 98.54 cm
Fig. 9: Differences of monthly solutions ITGGRACE-2003-09 minus ITG-GRACE-2003-07. The inner consistency of the monthly solutions can be checked, especially in the high-frequent spectral range, if two monthly solutions are compared. Fig. 9 shows the differences of the merged monthly solutions ITG-GRACE-2003-09 minus ITG -GRACE-2003-07. Fi~. 10 displays the differ-
4 Conclusions
The gravity field determination approach by merging regional recovery patches integrates a regional zoom-in with a global gravity field recovery. The achievement of this proposal has been tested based on GRACE low-low satellite-to-satellite data. All tests demonstrated the advantage of our approach
335
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K.H. Ilk. A. Eicker. T. Mayer-G~irr
compared to deriving a global gravity field solution in terms of spherical harmonics directly. The reasons for the superior quality of our procedure are versatile: The tailored calculation of a regularization parameter for each region allows a tailored filtering according to the individual gravity field features. The modelling of these structures can be performed very effectively by tailored space localizing base functions as e.g. spherical splines as applied in our approach, but others as spherical wavelets can be applied as well. To utilize the manifold advantages of spherical harmonics as base functions, our approach allows the calculation of the potential coefficients as well with the advantage that the quadrature procedure does not limit the resolution to an upper degree. A consequence is that no stability problems occur and no regional details of the gravity field are lost. Further advantages are the following: the method is modest in terms of computation costs, as the complete problem is split up into much smaller problems. This procedure enables the computation of a global gravity field model up to an arbitrary resolution on a single PC. It is very simple to include additional types of gravity field information in the computation scheme; combination of terrestrial date causes no problem and the weighting of this information can be determined within the frame of the variance component estimation step. Further improvements are expected with respect to the refinement of the regularization strategy to enable smoother transition zones between the zoom-in-regions and by tailoring the zoom-in areas more accurately to the characteristics of the gravity field in the specific regions. Furthermore, a more precise selection of the base functions and the nodal point distribution adapted to the roughness of the gravity field, possibly combined with a multiresolution strategy, seems to be possible. Additional points of future research are a careful investigation of the aliasing effects originating from the patching of several regional solutions and a homogenisation of the regional solutions to avoid long wavelength errors. A rigorous variance-covariance computation of the final result based on the three computation steps as outlined in Sect. 3 is one additional aspect of a forthcoming research.
Acknowledgement.
We appreciate the anonymous reviewers for their helpful comments and suggestions. The support of BMBF (Bundesministerium fiir Bildung und Forschung) and DFG (Deutsche
Forschungs-Gemeinschaft) of the GEOTECHNOLOGIEN programme is gratefully acknowledged.
References Eicker A, Mayer-Gfirr T, Ilk, KH (2005) Global Gravity Field Solutions Based on a Simulation Scenario of GRACE SST Data and Regional Refinements by GOCE SGG Observations, in C. Jekeli, et al.: Gravity, Geoid and Space Missions- GGSM2004, Porto, Portugal, lAG International Symposium, International Association of Geodesy Symposia, Vol. 129, pp. 6671, Springer F6rste C, Flechtner F, Schmidt R, Meyer U, Stubenvoll R, Barthelmes F, Neumayer KH, Rothacher M, Reigber C, Biancale R, Bruinsma S, Lemoine JM, Raimondo JC (2005) A New High Resolution Global Gravity Field Model Derived From Combination of GRACE and CHAMP Mission and Altimetry/Gravimetry Surface Gravity Data, Poster presented at EGU General Assembly 2005, Vienna, Austria, 24-29, April 2005 Freeden W, Gervens T, Schreiner M (1998) Constructive Approximatiuon on the Sphere, Oxford University Press, Oxford, Ilk KH, Mayer-Gfirr T, Eicker A, Feuchtinger M, (2004) The Regional Refinement of Global Gravity Field Models from Kinematical Orbits, New Satellite Mission Results for the Geopotential Fields und Their Variations, Proceedings Joint CHAMP/GRACE Science Meeting, GFZ Potsdam, July 6-8 Koch KR, Kusche J (2003) Regularization of geopotential determination from satellite data by variance components, Journal of Geodesy 76(5):259-268 Mayer-Gfirr T, Ilk KH, Eicker A, Feuchtinger M (2005a) ITG-CHAMP01: A CHAMP Gravity Field Model from Short Kinematical Arcs of a One-Year Observation Period, Journal of Geodesy (2005) 78: 462-480, Springer-Verlag Mayer-Gfirr T, Eicker A, Ilk KH (2005b) Gravity field recovery from GRACE-SST data of short arcs, in: R. Rummel, et al. Observation of the Earth System from Space (in preparation), Springer Reigber C, Schwintzer P, Lfihr H (1999) The CHAMP geopotential mission, Boll. Geof. Teor. Appl., 40:285289 Sneeuw N (1994) Global Spherical Harmonic Analysis by Least Squares and Numerical Quadrature Methods in Historical Perspective, Geophys. J. Int. 118:707-716 Stroud AH, Secrest D (1966) Gaussian Quadrature Formulas, Prentice-Hall, Englewood Cliffs, N.J. Tapley BD, Bettadpur S, Watkins M, Reigber Ch (2004) The gravity recovery and climate experiment: mission overview and early results., Geophys Res Lett 31, L09607: doil 0.1029/2004GL019920 Tapley BD, Ries J, Bettadpur S, Chambers D, Cheng M, Condi F, Gunter B, Kang Z, Nagel P, Pastor R, Pekker T, Poole S, Wang F Ch (2005) GGM02- An improved Earth gravity field model from GRACE, Journal of Geodesy (2005) 79(8): 467-478, Springer-Verlag
Chapter 50
External calibration of GOCE SGG data with terrestrial gravity data: A simulation study D. N. Arabelos Department of Geodesy and Surveying, Aristotle University of Thessaloniki, GR-54 124 Thessaloniki, Greece C.C. Tscherning, M. Veicherts Department of Geophysics, University of Copenhagen, Juliane Maries Vej 30, DK-2100 Copenhagen, Denmark
Abstract.
Terrestrial gravity anomalies selected from three extended continental regions having a smooth gravity field were used in order to determine the appropriate size of the area for gravity data collection as well as the required data-sampling for calibration of the GOCE satellite gravity gradient (SGG) data. Using Least Square Collocation (LSC), prediction of gravity gradient components was carried out at points on a realistic orbit. Based on the mean error estimation it was shown that up to 80% of the signal of the gravity gradient components, as it is expressed through the covariance function of the terrestrial gravity data, can be recovered in the case of an optimal size of the collection area and of the optimum resolution of the data. These optimal conditions e.g. for the Australian gravity field, correspond to an 10 ° × 12 ° area extend and a 5 f data-sampling. It was also numerically demonstrated that it is possible to calibrate the GOCE SGG data for systematic errors such as bias and tilt.
Keywords. Satellite Gravity gradiometer data, External calibration, Systematic error parameters
1
Introduction
In the last decade numerous investigations were published, concerning the calibration of GOCE satellite gravity gradiometer data, (e.g. Bouman et al., 2004; Wolf & Denker, 2005). The use of Least Squares Collocation (LSC) as a element of the space-wise approach methods has also been discussed in a number of papers (e.g. Tscherning, 2005). The aim of this work was to determine the size of required areas with terrestrial gravity data, as well as the required resolution and accuracy of the gravity data needed for calibration when LSC is used. The aim is to detect possible systematic errors in the GOCE SSG data. The "simple" LSC method was used for the tests concerning the size of the area and the resolution of the
data, while the parametric LSC was used for the tests concerning the detection of systematic errors. The terrestrial gravity data sets used in this study are described in details in section 2. In the first part of this study errors of gravity gradients were computed at points on the realistic orbit of the lAG SC7 simulated data set with the gradients in a reference frame aligned with the velocity vector and the z-axis which lie in the plane formed by this vector and the position vector. In the last part of the study points on a similar orbit were used, but with the gradients given in a more realistic instrument reference frame provided by ESA (R. Floberghagen, private communication 2004). The precision of the calibration will be directly proportional to the gravity field standard deviation for example expressed as the standard deviation of gravity anomalies from which the contribution of a reference field have been subtracted. The areas studied here are therefore areas with a very smooth gravity field. We have used EGM96 to degree 360 (Lemoine et al., 1998) for the reduction of gravity anomalies, in order to smooth as far as possible the gravity anomalies used in all test areas. Then, using the reduced gravity anomalies we have predicted gradient values at points on tracks crossing the areas, further on called control points. Since real GOCE data are not yet available we have used the error estimates given by collocation (eq. 66 in Moritz, 1978), instead of the statistics of the differences between predicted and control values. More specifically, we relate the mean collocation error, depending on the choice of the covariance function, with the formal standard deviation of the gravity gradient components, depending also on the covariance function used, in order to draw conclusions about the optimal size of the area and the resolution of the data needed for the calibration. The mean collocation error is computed as the mean error of the collocation error estimates over all predicted
338
D.N. Arabelos • C. C. Tscherning • M. Veicherts
points. Note, however, that the error in the middle of the area typically is 90% of the mean error.
free-air gravity field is shown in Fig. 1. The corresponding statistics is shown in Table 1.
It will be numerically shown in the next, that in all test areas, using LSC and terrestrial gravity anomalies, up to 80% of the formal standard deviation of the gravity gradient signal can be recovered, in the case of a high data accuracy, size of the area of collection of terrestrial gravity anomalies and of the datasampling. Furthermore, it will be also numerically shown that in this way it is possible to calibrate the G O C E SGG data for systematic errors such as bias and tilt. Here we have used that the expected accuracy of 1 s sampled data in the measurement bandwith will have an error equal to or above 7 mE.
2. Surface gravity anomalies from Australia (further on called region B)
In the computations it was attempted to keep the data-sampling and the size of the terrestrial data collections areas constant for the corresponding experiments from area to area.
2
Gravity data used
For reasons discussed in earlier work (e.g. Arabelos & Tscherning, 1998), for the requirements of the calibration, the terrestrial data have to be collected from areas with possible smooth free air gravity anomaly field. A further reason for this is to avoid topographic reductions to smooth the gravity field, due to errors that could be introduced to gravity anomalies from erroneous altitudes and density hypotheses.
The 2004 edition of the Australian National Gravity Database contains over 1,200,000 point data values in the area bounded by - 4 8 ° _< q0 _< - 8 °, 108 ° _< )~ _< 162 °. This data was made available by Geoscience Australia. The data set covering the continental Australia and the surrounding ocean (1,117,054 point values) was reduced to E G M 9 6 up to degree 360 was removed from the gravity anomalies. The statistics of the original and reduced free-air gravity anomalies is shown in Table 1. As it is shown in Fig. 2 the gravity field is very smooth in the central Australia and consequently, appropriate for the calibration requirements. For this reason, point gravity anomalies were selected from the area bounded by - 3 2 ° _< q0 _< - 2 0 °, 124 ° _< )~ _< 144 ° .
236 °
240 °
244 °
248 °
252 °
68 °
68 °
66 °
66 °
Another requirement was to collect data from extended regions in different geographic latitudes due to the dependence of the distribution of the G O C E data on the latitude.
64 °
64 °
For all these reasons, data from the Canadian plains, Australia and Scandinavia were used.
62 °
62 °
60 °
60 °
58 °
58 °
56 °
56 °
The terrestrial gravity data have errors which we consider random. However, we are aware that systematic errors are present in the data, especially due to height datum problems. Assuming an error of 1 m in the height datum, the correlated noise of the gravity anomalies equals to 0.3 mgal. The calibration procedure may very well take such error correlations into account, if they are known.
1. Terrestrial free-air gravity anomalies from the Canadian plains (further on called region A) This data set was already described in (Arabelos & Tscherning, 1998). In the present paper the reduced values, i.e. the free-air gravity anomalies after the removal of the contribution of the geopotential model E G M 9 6 up to degree 360, were used within the area bounded by 56 ° _< q0 _< 68o,236 ° _< )~ _< 254 °. The
236 °
240 °
244 °
248 °
252 °
~
mGal
-35-30-25-20-15-10
-5
0
5
10 15
Gravity Figure 1o The free-air anomaly-EGM96 gravity fi eld in the Canadian plains (simplified)
Chapter50 • ExternalCalibrationof GOCESGGData with TerrestrialGravityData:a SimulationStudy
Table 1. Statistics of the free-air gravity anomalies used in the regions A, B and C. Unit is mGal Region A, 14,177 point values Observations EGM96 Difference Mean -10.768 -10.678 -0.090 Standard Dev. 22.419 17.497 13.418 Max. Value 133.000 51.044 114.168 Minimum value -81.100 -72.210 -124.857 Region B, 1,117,054 point values Mean 4.901 5.158 -0.258 Standard Dev. 24.504 22.504 12.102 Max. Value 248.602 94.223 219.875 Minimum value -211.327 -104.930 -194.732 Region C, 62,126 terrestrial values Mean -8.466 -8.118 -0.349 Standard Dev. 18.285 16.229 8.789 Max. Value 71.740 36.066 76.805 Minimum value -84.021 -77.351 -47.250 Region C, 4,778 air-borne values Mean -18.443 -19.735 1.292 Standard Dev. 19.834 18.209 10.024 Max. Value 29.660 21.235 35.085 Minimum value -80.540 -68.107 -31.675
F r o m Table 1 it is s h o w n that the r e d u c e d to E G M 9 6 free-air gravity a n o m a l i e s present very similar statistical characteristics. This is m o r e evident f r o m the shape of the c o r r e s p o n d i n g covariance functions (see Fig. 4). For the reasons discussed in section 1 the formal standard deviation of the control values used in the n u m e r i c a l e x p e r i m e n t s of section 3 is s h o w n in Table 2 for the three test regions. This formal standard deviation for each area is based on the c o r r e s p o n d i n g covariance function of Fig. 4. 112°
Try
Txz Tyy Tyz Tzz
Region A 0.0042 0.0049 0.0050 0.0042 0.0050 0.0072
Region B 0.0035 0.0041 0.0042 0.0035 0.0042 0.0060
Region C 0.0047 0.0054 0.0056 0.0047 0.0056 0.0079
3. Gravity anomalies from Scandinavian (further on called region C) Terrestrial as well as air-borne gravity a n o m a l y data sets were m a d e available by R. Forsberg, R K n u d sen and G. Strykovski (private c o m m u n i c a t i o n ) . T h e terrestrial data set (62,126) cover the area 53.99 ° _< q0 _< 64 °, 11.97 ° _< ~, _< 30.02 °. After the r e d u c t i o n to E G M 9 6 we obtain present the statistics given in Table 1. T h e air-borne data set cover the area 54.58 ° < q0 _< 60.12 °, 12.01 ° _< )v _< 26.86 °. T h e statistics of the r e d u c e d to E G M 9 6 air-borne gravity a n o m a l i e s (4778 point values) is s h o w n in Table 1. In the collocation e x p e r i m e n t s both data sets were used jointly with c o m m o n accuracy equal to 2 m G a l . T h e free-air gravity field r e d u c e d to E G M 9 6 is shown in Fig. 3.
128°
136°
144°
152°
160°
_12°
_12°
_18°
_18°
_24°
_24°
_30°
_30°
_36°
_36°
_42°
_42° 112°
Table 2. Signal standard deviation of data at the control points used in the numerical experiments. (E)
Txx
120°
120°
128°
136°
144°
152°
160° mGal
-25-20-15-10-5
0
5
l0 15 20 25 30
Gravity
Figure 2. The free-air anomaly-EGM96 gravity field in Australia (simplifi ed)
3
Numerical Experiments
3.1 T e s t s c o n c e r n i n g t h e r e q u i r e d s i z e of t h e a r e a a n d t h e r e s o l u t i o n of t h e d a t a T h e e x p e r i m e n t s c o n c e r n r e c o v e r y of 5 s sampling noise-free simulated G O C E data p r o v i d e d by IAG, along 1 m o n t h realistic orbit (250 km), using terrestrial gravity anomalies. For the d e t e r m i n a t i o n of the required size of the area for terrestrial data collection in all cases 10 arcmin d a t a - s a m p l i n g was used and prediction e x p e r i m e n t s were carried out in four areas with different size. For the d e t e r m i n a t i o n of the required data-sampling, the prediction e x p e r i m e n t s were carried using data with 5, 7.5, 10, 15 and 20 arcmin, sampling in areas with constant size. In all c o m p u t a t i o n s the G R A V S O F T p r o g r a m s (Tscherning et al., 1992) E M P C O V , C O V F I T and G E O C O L were used.
339
340
D.N. Arabelos • C. C. Tscherning • M. Veicherts
15 ° I
mm
mm
20 ° mm
mm
25 ° mm
mm
mm
30 ° mm
mm
60°
60°
ror estimation concerning all gradient c o m p o n e n t s is shown in Fig. 5. This i m p r o v e m e n t is more significant in the case of Tzz (39%). The mean error of 0.0027 EU correspond to a 37% of the formal standard deviation of Tzz (see Table 2 Region A), resulting from the covariance function of free-air gravity anomalies in this region. This could be interpreted as the ability of the m e t h o d to recover the 63% of the signal, in the case of real S G G data.
(b) Experiments for the determination of datasampling Concerning the determination of the required datasampling, experiments were carried out with terres55°
55° i
mm
mm
15°
mm
mm
mm
2o °
mm
mm
25 °
mm
mml
3o° mGal
-12-10-8 -6 -4 -2 0 2 4 6 8 10 12 14 16 Gravity Figure 3. The free-air anomaly-EGM96 gravity field in Scandinavia (simplified) Region
100 ~
~
90
(a) Experiments for the determination of the size of the area for terrestrial data collection The experiments were carried out using terrestrial gravity data with constant 10 arcmin sampling. Accuracy equal to 1 m G a l was adopted for these data. Prediction experiments were carried out collecting terrestrial data from four areas with size 5 ° x 6 ° 6 ° 8°,8 ° × 10 ° and 10 ° × 12 ° respectively. The first of them correspond to size of the control points collection area. The results of these numerical experiments in terms of the mean collocation error estimation, are shown in Fig. 5(a). It is well known that the assessment of the prediction results in collocation may be based not only on the statistics of the differences between observed (control) and predicted quantities, but also on the collocation error estimation of the prediction. Since we do not yet have any real G O C E data used the formal error estimates. With the increase of collection area from 5 ° z 6 ° to 10 ° × 12 ° a continuous i m p r o v e m e n t of the mean er-
I
I
I
I
8o
70
~2
6o
~ ~ N
50 4o 3o
•-~
20 10
o
0
I -
0
A
Simulated G O C E data used as control data in the area 58.5 ° _< q> _< 6 3 . 5 ° , - 1 1 8 ° _< X _< 112 ° , (242 values). The covariance function of gravity anomalies used (empirical and the corresponding analytical one is shown in Fig. 4(a).
I
-I
1
2
3
4
5
6
Spherical distance (Degree) 80 ~'~ 70
I
I
I
I
I
i
i
i
1
2
3
4
5
60
~2 50 = •~ o> r,.)
40 30 20 l0 0 -10
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Spherical distance (Degree) 80
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70 6o ~2 5o x ~
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(e)
40 3o
~ 2o N o
lO 0
.
L)
0
.
.
.
i
i
i
i
i
1
2
3
4
5
6
Spherical distance (Degree) Figure 4. Covariance function of free-air gravity anomalies in the regions A (a), B (b) and C (c). Solid line = empirical, Dashed line = model.
Chapter 50 • External Calibration of GOCESGG Data with Terrestrial Gravity Data: a Simulation Study trial data collected f r o m the same 6 ° x 8 ° area (58 ° _
ti k=l
We use pseudo-stochastic orbit modelling techniques (J~iggi et al., 2006) as a realization for RD-POD (Wu et al., 1991), which makes use of both the geometric strength of the GPS observations and the fact that satellite trajectories are particular solutions of an equation of motion. The attribute "pseudo" is used to distinguish our approach from methods considering the satellite motion as a stochastic process, whereas the attribute "stochastic" refers to the introduction of additional parameters to the deterministic equation of motion, which may have a priori known statistical properties. In this article we make use of two types of additional parameters, namely instantaneous velocity changes (pulses) and piecewise constant accelerations.
2.3.1 Instantaneous velocity changes Pulses are attractive for RD LEO POD mainly because a large number of pulses can be set up efficiently. This is due to the fact that a pulse-induced
(4) or, alternatively, as a linear combination with constant coefficients of the same partial derivatives and one additional partial derivative with respect to a constant acceleration pointing in the same direction and acting over the entire orbital arc. Therefore, all partial derivatives with respect to the accelerations may be constructed from a very limited set of numerically integrated partial derivatives.
2.3.3 Normal equation system We give a short overview of the structure of the resulting normal equation system for the standard leastsquares adjustment process of GPS observations. For the sake of simplicity, we consider only the six orbital elements and the pulses in three orthogonal directions at times ti, i = 1, ..., n - 1 as parameters. For a more detailed derivation, also considering different parameter types like piecewise constant accelerations and
355
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h. J~iggi • G. Beutler • H. Bock. U. Hugentobler
additional parameter types like carrier phase ambiguities, we refer to Beutler et al. (2006). The pulse-epochs divide the orbital arc into n subintervals. Let us write all no~ observation equations of the subinterval Ii = [ti, ti+l ) in a convenient matrix notation: i
Ai . AE
+ Ai . ~
B ~ . Ziv,~ - ZicPi - pi ,
the structure of Eq. (6) and found that for a large variety of applications the solution vector and the associated full variance-covariance information may be computed with sufficient efficiency. However, when striving to the kinematic limit, i.e., pseudo-stochastic parameters set up at a rate close or equal to the observation sampling rate, the procedures become inefficient due to the unavoidably large normal equation matrix, which has to be inverted.
m=l
(5) where A i is the first design matrix with no~ lines and six columns, A E the column array containing the six increments of the initial osculating elements, B ilk,j] = /~ij,k the matrix with six lines and three columns containing the coefficients of Eq. (3), a v i the column array containing the three pulses at time ti, A O i the column array containing the no~ terms "observed-computed", and Pi the column array containing the no~ residuals. Note that all pulses set up before the subinterval [i remain active and contribute to the observation equations of subinterval Ii as predicted by the linear combination of Eq. (3). Therefore, the last subinterval eventually contains the contributions due to all pulses of the orbital arc, provided that the initial conditions are still referring to the beginning of the orbital arc. To study the structure of the resulting normal equation matrix, it is instructive to use the contributions N i " A T P i A i per subinterval to the normal equation matrix of dynamic POD, i.e., POD without pulses. Obviously, these contributions form the complete normal equation matrix of dynamic POD as n--1 N " ~-~i=0 A ~ P i A i , but they are also the building blocks of the complete, symmetric normal equation matrix in the presence of pulses, which reads as /
n--1
N
~
n--1
NiB1
"'"
. Bf
Z
Bf
y~
n--1 T
.
\
Bn_
- AEi_I
NiBn-1
1 i=n-1
(8)
+ Bi" z~vi.
It is instructive to apply the transformation given in Eq. (8) each time after having processed all observations of one subinterval. The resulting normal equation matrix then reads as" / ¢
0
N
n--2
- ~ NiB1 i=0 0
....
~
'~
NiB~-I
i=0 0
• BTENiBI"'" i=0
B TZNiB~-I i=0
n--2
"
• BT-1 E
NiBn-1 J
(9) NiBn-i
i=n--1
i=1
AEi
i=0
n--1
...
where the term in parentheses denotes the column array containing the six orbital elements pertaining to epoch to, but characterizing the trajectory within this particular subinterval. This set of elements, subsequently denoted as z l E i , is simply related to the set of elements of the previous subinterval by:
k "
NiBn-1
i=n--1
NiB1
Rearranging all observation equations (Eq. (5)) of the subinterval Ii shows that the orbit may be represented within this subinterval by only six Keplerian elements"
"~
~
i=1 n--1
2.3.4 Transformation of Keplerian elements
J
(6) Equation (6) illustrates that the normal equation matrix (and also the corresponding right hand side of the normal equation system) has a simple structure, but grows monotonically after having processed all observations of one subinterval. Note in particular that it is not possible to pre-eliminate the pulses at any observation epoch which is indicated by the upper summation limit. Beutler et al. (2006) made full use of
The solution vector obtained from the transformed normal equation system contains the same pulses as the untransformed system, but the set of elements A E n _ I referring to the last subinterval (instead of A E 0 referring to the first subinterval). A comparison with the untransformed normal equation matrix (compare Eq. (6)) reveals the benefit of the applied transformation because it is now possible to preeliminate the pulses after each subinterval as the upper summation limits in Eq. (9) indicate. Beutler et al. (2006) made full use of the structure of Eq. (9) and proposed a very efficient pre-elimination and backsubstitution scheme for different types of pseudostochastic parameters, which allows it to efficiently realize the kinematic limit with RD-orbits.
Chapter52 • Kinematicand Highly-ReducedDynamicLEOOrbitDeterminationfor GravityFieldEstimation l0
,
,
+
u
,
+
,
++ +
,
IAPG MUB
+
8
x
.~~
' {x
+ 6
'~
•
5
"~
1 min. 3 rain. . -.
o -5
0
i
i
i
100
150
200
i
i
i
i
250
300
350
400
-10 0
i
i
i
i
5
10
15
20
15
20
(DOY)
th)
Fig. 1. Daily (1-dim.) RMS of differences for AIUB kinematic and IAPG kinematic orbits w.r.t, conventional RDorbits for days 071/2002 to 070/2003.
5
o
IIHIll )
IIIll
I
i1TIIII
i
-s
3 Processing
of real C H A M P
data
-10 0
5
10 (h)
The GPS final orbits and the 30s high-rate satellite clock corrections from the CODE analysis center were used together with attitude data from the star tracker on board of CHAMP provided by GFZ (GeoForschungsZentrum Potsdam) and the gravity field model EIGEN-2 (Reigber et al., 2003) to conventionally process undifferenced CHAMP GPS phase tracking data covering a one year time period from day 071/2002 to 070/2003. For a subset of tracking data, covering GPS weeks 1173-1176, 10 s GPS satellite clock corrections were generated in order to perform tests with several kinds of HRD orbit parametrization. All computations were performed with a development version of the Bernese GPS Software (Hugentobler et al., 2001). 3.1 Results of k i n e m a t i c P O D
Figure 1 shows daily (1-dim.) RMS values of orbital differences derived from our kinematic orbits (AIUB) with respect to conventional RD-orbits with pseudo-stochastic parameters set up every six minutes. As a reference, the differences emerging from the kinematic orbits (IAPG) computed by Svehla and Rothacher (2003) are displayed as well. The two curves show, on the one hand, that both sets of kinematic orbits are of similar quality, but, on the other hand, they reveal for both solutions a considerable number of poorly determined trajectories, mainly due to data quality issues. As a consequence of the very low degree of freedom per epoch, kinematic positions react very sensitively to the density and quality of GPS observations. This makes a robust preprocessing a greater challenge than for conventional RD-POD. 3.2 R D - P O D at the k i n e m a t i c limit
The estimation scheme presented in section 2.3.4 makes it possible to efficiently approach the kinematic limit with RD-orbits. For one particular day Fig. 2 (top) puts the cross-track differences for the kinematic orbit together with the differences emerg-
Fig. 2. Cross-track differences of the kinematic and different HRD-orbits w.r.t, a conventional RD-orbit (top) and differences (radial/cross-track shifted by 5 mm) between the kinematic and the MRD-orbit (bottom) for day 198/2002. ing from two HRD-orbits, i.e., orbits which are represented by six initial conditions and three unconstrained pulses set up every one and every three minutes, respectively. We see that the HRD-orbits approach the kinematic orbit when the number of pulses increases. Because the differences for the MRD-orbit would completely overlap with the differences for the kinematic orbit, Fig. 2 (bottom) displays the differences between both orbits separately and confirms their equivalence to the numerical precision provided by the SP3 orbit file format, apart from a few exceptions which are discussed in the following paragraph.
3.2.1 Properties of MRD-orbits The least-squares adjustment process for the estimation of a MRD-orbit, either based on pulses or accelerations, results in a regular normal equation system like in the case of kinematic POD. It is instructive to have a closer look at MRD-orbits in order to emphasize possible benefits of HRD-orbits. For the sake of simplicity we confine ourselves to discuss the results achieved with pulses, because MRD-orbits based on accelerations do not provide more insight in this respect. For MRD-orbits, three unconstrained pulses are set up at all hobs observation epochs, except for the very first and the very last one. Therefore, together with the six initial conditions, a total number of 3-hobs orbit parameters are estimated, which is obviously the same number of unknowns as in the case of a kinematic orbit. Provided that at least four GPS observations are available for every observation epoch, all epoch parameters can be determined for both approaches, i.e., three pulses and three kinematic coordinates, respectively, and a receiver clock correction. Both orbit ephemeris are equivalent at the observation epochs as illustrated in Fig. 3 (top).
357
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A. J~iggi • G. Beutler • H. Bock. U. Hugentobler
Table 1. Overall RMS of velocity differences w.r.t, conventional RD-orbits (GPS weeks 1173-1176). Solution o
~
,
............. ,:~ ....""-~:-
Fig. 3. MRD-orbit (solid line) based on pulses in comparison with the true orbit (dotted line) (top) and impact of an increased sampling rate (bottom). The trajectory of the MRD-orbit is defined inbetween the observation epochs, as well. The positions at the left and at the right boundary epochs provide the necessary six conditions to define a trajectory between these two points, which solves the equation of motion (Eq. (1)). Note that the positions at the observation epochs are completely independent of the force field, but the trajectory in-between and in particular the orbital velocity is given by the apriori force model. It is thus not possible to derive more independent information concerning the force field from a MRD-orbit than from a kinematic orbit. The same statement holds if accelerations are set up with the highest possible resolution, where the trajectory in-between is allowed to have large excursions ("slalom"-orbit). If there are less than four GPS observations available at a certain epoch, it is not possible to estimate all three pulses. Figure 2 (bottom) includes such epochs where the filtering due to the dynamic orbit model starts to affect the MRD-trajectory, which is responsible for a few larger differences. A closer inspection shows that this effect is limited to the neighboring epochs of intervals with few observations. For a few days, however, we also found deviations lasting longer, a case to be investigated further.
3.2.2 Properties of HRD-orbits Section 3.2.1 showed that MRD-orbits may be considered as equivalent to kinematic orbits. In order to make use of filtering effects associated with RDorbits, it is thus necessary to set up pseudo-stochastic parameters at subintervals of length Tp longer than the observation sampling interval Ts. Because Tp determines, in essence, the achievable resolution for a subsequent gravity field recovery procedure, Ts should be rather decreased than Tp increased. Figure 3 (bottom) illustrates this for pulses in comparison to the case of maximum resolution (top). The trajectory is filtered with the force model because Ts < Tp, which leads to a trajectory with a reduced scatter. As a drawback, however, the results show a dependency on the force model as illustrated in Fig. 3 (bottom), even if the orbital positions are evaluated only at the pulse epochs. A simulation study in section 4 is used
kinematic 30s pulses 30s acc. 60s pulses 60s acc.
30s GPS-sampling (mm/s) 0.19 (0.15) (9.26) 0.14 0.17
10s GPS-sampling (mm/s) 0.23 0.24 0.30 0.15 0.15
to establish the relationship between the additional force field dependency and the reduction of noise. 3.3
Results
of HRD-POD
Commonly used gravity field recovery procedures do not only use the orbital positions as pseudoobservables. The energy integral method, e.g., requires instantaneous orbital velocities to compute disturbing potential values along the orbit (FSldvfiry et al., 2004). Table 1 gives an impression of the scatter of four weeks of C H A M P orbital velocities obtained for different solutions with respect to conventional RD-orbit velocities. The solutions in parentheses (e.g., the "slalom"-orbit, see section 3.2.1) have no value for gravity field recovery and are listed just for completeness. In general, we recognize that pulse-based solutions show a smaller RMS than acceleration-based solutions for high resolutions, but would encounter an opposite behavior if the resolution was further decreased. We have to keep in mind, however, that a small noise for pulse-based solutions does not necessarily indicate a better qualification for gravity field recovery, e.g., due to dependencies on the a priori gravity field model. The results listed in Table 1 are in good agreement with the expectations from the simulation study following in section 4, with two exceptions. First, the 10 s kinematic velocities are slightly noisier than the 30s based velocities, even if the same 7-point Newton-Gregory interpolation was applied to the kinematic positions (F61dvfiry et al., 2004). This might indicate a problem with the 10 s GPS satellite clock corrections, in particular because an almost identical RMS of 0.20 mm/s results, if the velocities are evaluated every 30 s only. Secondly, simulated data predict only a slightly larger RMS for 30 s acceleration-based velocities with 10 s sampling than for the 10 s kinematic velocities. This prediction is not confirmed by the observed RMS of 0.30mm/s and needs to be investigated further.
4 Simulation study A 24 h dynamic C H A M P orbit in a true gravity field, defined by the gravity field model EIGEN-2 up to degree and order 120, served as the true orbit to sim-
Chapter52 • Kinematicand Highly-ReducedDynamic LEO OrbitDeterminationfor GravityFieldEstimation 0.3
0.3
a priori signal ( R M S = 2.5 m m ) noise signal ( R M S = 2.4 mm)
L'
0.25
I
0.25 0.2
0.2
:[
30s ki . . . .
tic ( R M S = 2.9 m m ) .......... 10s kinematic ( R M S = 2.6 m m ) noise signal ( R M S = 1.6 mm)
[i
0.15 0.1 E
0.05 0
i 50
100
150
200
250
300
50
100
150
period (s) ,~
0.3
a priori signal ( R M S = 0.6 m m ) - noise signal ( R M S = 2.4 mm)
0.25
200
250
300
period (s)
,~
0.2
,
, a priori signal (~VIS = 0.2 m m ) noise signal ( R M S = 1.6 mm)
0.25 0.2
0.15
0.15 ..
0.1 0.05
o.1 0.05
0
o
50
100
150
200
250
300
period (s)
'
50
J
.
100
.
.
.
.
.
.
.
'
-
150
' -
200
•
- '
-
250
300
period (s)
Fig. 4. Amplitude spectra due to noise and due to force model errors for HRD-orbit positions based on pulses (top) and accelerations (bottom) with T~ - 30 s and Tp - 60 s.
Fig. 5. Noise spectra (top) for 10 s and 30 s kinematic orbits and HRD-orbit based on accelerations with Ts = 10 s and Tp = 30 s and spectrum due to force model errors (bottom).
ulate undifferenced GPS phase observations, which were generated either free of noise or, alternatively, with a white noise of 1 mm RMS error. A different a priori gravity field model was then used to reconstruct the true orbit with different POD strategies. Eventually, Fourier analysis techniques were used to study the differences of the estimated orbital positions and velocities with respect to the true values in the frequency domain. A rather poor a priori gravity field, realized by the EIGEN-2 model truncated at degree and order 20, was used to reconstruct the true orbits. In addition, the low degree and order spherical harmonic coefficients (_< 20) were slightly modified according to the formal error estimates provided by the EIGEN-2 model.
range. It is in particular possible for RD-orbits based on pulses and larger values of Tp that the signal spectrum exceeds the noise spectrum also for few periods larger than 2 - T v. As expected, accelerations show a slightly better noise reduction and a greatly reduced dependency on the a priori gravity field model over the entire frequency range, because the unexplained gravity field signal is not only absorbed at certain discrete epochs. Figure 5 shows analogue spectra for the more interesting case with T8 = 10 s and Tp = 30 s based on accelerations. Figure 5 (bottom) indicates a negligible influence of the a priori gravity field signal in comparison to the noise level, at least over the considered one-day time interval (see section 4.3). Figure 5 (top) also shows the noise spectra of 10 s and 30 s kinematic orbits. Taking into account that the level of the last mentioned spectrum is only apparently higher by x/3 because of its lower sampling, we see that the HRD-orbit exhibits a better noise characteristic than both kinematic orbits in the highest frequency range and still around the region of interest at 2 • Tp = 60 s.
4.1 Fourier analysis of orbital positions Figure 4 shows amplitude spectra of orbital position differences over one day emerging from HRD-orbits based on pulses (top) and accelerations (bottom), respectively, with Ts = 30 s and Tp = 60 s. The amplitude spectra denoted as 'a priori signal' characterize the residual impact of the a priori gravity field model in a noise-free simulation. The amplitude spectra denoted as 'noise signal' characterize the impact of the 1 mm GPS phase observation noise in absence of any gravity field model errors. Fig. 4 confirms, in essence, that orbit difference signals with periods T < 2 • T v (indicated by a vertical line in all spectra) are dominated by the a priori gravity field model as the a priori gravity field induced signal amplitudes exceed the apparent amplitudes caused by the observation noise, which is strongly reduced in the highest frequency range due to the RD-filtering. The effect illustrated in Fig. 3 (bottom), however, explains that the impact of the a priori gravity field model is not only restricted to periods T < 2- Tp as one might expect from an ideal filter, but also leaks into the lower frequency
4.2 Fourier analysis of orbital velocities As mentioned in section 3.3, gravity field recovery procedures like the energy integral method also require instantaneous orbital velocities as input data. This is the motivation to perform an analogue analysis for velocities as it was done in section 4.1 for positions. Note that orbital velocities are by-products of all types of HRD-orbits, which may be obtained by taking the time derivative of Eq. (2). For kinematic orbits, however, only approximate procedures can be applied to the kinematic positions like a 7-point Newton-Gregory interpolation proposed by F61dv~ry et al. (2004). Figure 6 shows amplitude spectra of orbital velocity differences over one day emerging from HRD-
359
360
A. J~iggi • G. Beutler • H. Bock. U. Hugentobler
= ~
0.012 0.01 0.008 0.006 0.004 0.002
~.
50
~
0.012
a priori'signal(RMS = 0.044mnffs) noise signal(RMS = 0.077mm/s)
0.012 ' 0.01 0.008 0.006 0.004 0.002 0 ........~_ ' ..-~_ .?. 50
100
'
150 period (s)
200
250
f
!
I 0.0080"01
=
0.006
300
I I
:!~ i'! '": ~' ',.i ":i', '.[ I
0 0.012 0.01
a priori signal (RMS = 0.012 mln/s) noise' signal (RMS = 0.093 mm/s)
.!i 30s kinematic (,~I~IS = 0.090 mm/s) .......... :Ji !~i:[:~= ,i=[~, ,.""'"i"m =: [=~i[.n°ise : If" msignal ~. / (RMS' s ',',::i )!',!!I. = 0'"93
50
100
'
.' :.i
o.oo8 0.004 0.006 ~ ...... 100
0.002 150 period (s)
200
250
0
300
6. Amplitude spectra due to noise and due to force model errors for HRD-orbit velocities based on pulses (top) and accelerations (bottom) with T~ = 10 s and Tp = 30 s. Fig.
orbits based on pulses (top) and accelerations (bottom), respectively, with T8 = 10s and Tp = 30s, i.e., spectra corresponding to the situation shown in Fig. 5 (bottom). Figure 6 (bottom) indicates for the acceleration-based velocities a negligible influence of the a priori gravity field signal in comparison with the noise level, at least over the considered one-day time interval (see section 4.3). The pulse-based velocities (top) show a similar picture if they are computed at the pulse epochs as the mean values of the left- and fight-hand side limits of the discontinuous velocity vectors. Note that the stronger dependency on the a priori gravity field model favors accelerationbased velocities for the task of gravity field recovery. Comparing both noise spectra in Fig. 6 implies, on the other hand, that highly-resolved pulse-based solutions exhibit a more favorable noise reduction for the highest frequency range than the corresponding acceleration-based solutions, which was already observed in Table 1 for real data. Figure 7 shows the amplitude spectra from Fig. 6 (bottom) amended by spectra of kinematic velocities, which were established by the Newton-Gregory interpolation from noise-free and noisy 30s kinematic positions, respectively. Taking the apparently higher level of the last mentioned spectrum into account, we see similar noise for kinematic velocities and velocities from the acceleration based HRDorbit, except for the highest frequency range. There, the performance of the kinematic velocities is better due to a more efficient smoothing of high frequency signals by the relatively long interpolation-intervals. Application of 60 s piecewise constant accelerations would lead to a similar effect for the HRD-solution. Note that comparable results would be obtained for the kinematic velocities if the same 7-point NewtonGregory interpolation (30 s spacing between the positions used for interpolation) was applied to 10 s kine-
150 period (s)
.~
250
300
~ ~, i~ !::
I
,[ii~lll~i, ~iii~ ~ii
........... 50
200
a priori' signal (RMS = 0.012 min)s) 30s ki . . . . tic (I~M~ = 0.029 mm/s) ..........
ii i!Iii i
',
lt~!~[ ~ ~ ~ !! :i
100
150 period (s)
200
~ 250
300
Fig. 7. Noise spectrum analogue to Fig. 6 (bottom) and total spectrum for 30 s kinematic velocities (top) and corresponding model error spectra (bottom). matic positions, but much worse results if the spacing of the used interpolation points was changed to 10 s. Figure 7 (bottom) gives the corresponding noisefree spectra separately and shows that the NewtonGregory interpolation introduces comparably large interpolation errors when compared to the HRDspectrum, although the absolute errors do not exceed the level of 0.1 mm/s. The discrete spectral lines are not restricted to the period range shown in Fig. 7 (bottom), but continue to lower frequencies down to the orbital frequency. Figure 7 (top) shows that some of these lines even exceed the noise level of the total spectrum of the considered one day data set. 4.3
Impact
of data
accumulation
All gravity field recovery procedures make use of the accumulation of individual (daily) solutions in order to reduce random errors for a most reliable estimation of gravity field coefficients. Figure 8 illustrates for the most trivial error model how the HRD signal and noise spectra from Fig. 6 (bottom) would look if data had been accumulated over 400 days. It is simply assumed that all systematic errors are not reduced, which would be the (worst) case for a daily repeat orbit, whereas random errors are reduced by the square root of the number of accumulated solutions. Figure 8 indicates that for long data sets the systematic errors would become more important than the random errors under the above mentioned assumptions, because the impact of the rather poor a priori force field (truncated at degree and order 20: right vertical line) starts to exceed the noise level in the lower frequency range. It remains to be seen whether such biases towards the a priori model would actually occur in real gravity recovery experiments. It is just as well possible that small systematic errors are reduced in a combination like random errors due to a permanently changing orbit geometry.
Chapter 5 2
•
Kinematic and Highly-Reduced Dynamic LEO Orbit Determination for Gravity Field Estimation
Steady-State Ocean Circulation Mission. ESA SP-1233
0.0006
(1)
0.0005 0.0004 0.0003 0.0002
0.0001 0 0
50
100
150
200
250
300
period (s)
Fig. 8. Amplitude spectra from Fig. 6 (bottom). The noise spectrum is downscaled to simulate the worst case effect of data accumulation (see text). To get an impression of the impact of the a priori model for real data, we repeated the processing described in section 3.3 with the gravity field model EGM96 (Lemoine et al., 1997) instead of EIGEN2 and compared the corresponding orbital velocities with each other. For the most interesting solution with Ts - 10 s and Tp = 30 s we found an overall RMS of velocity differences of 0.026 mm/s for pulses and 0.016 mm/s for accelerations due to the changed force model, which is comparable to the simulated results.
5 Conclusions We presented a very efficient method to compute RDorbits based on pseudo-stochastic parameters with resolutions Tp >_ T8 and showed that MRD-orbits are, in essence, equivalent to kinematic orbits. The pre-processing of GPS data was found to be the most important aspect when generating the one-year data set of kinematic CHAMP positions. This problem is of course not removed when generating MRD-orbits. An extensive simulation study showed that not only kinematic orbits but HRD-orbits, as well, could be interesting as input data for gravity field recovery, in particular for the upcoming GOCE mission (ESA, 1999) which is expected to provide 1 s GPS data. The influence of interpolation errors on kinematic velocities was found to be larger than the influence of the a priori gravity field model on HRD velocities, even if a very poor a priori model was used. First experiences gained with four weeks of real CHAMP GPS data confirmed, in essence, the expectations from the simulation study. However, the side-issue of generating most reliable 10 s GPS satellite clock corrections and the issue of a larger velocity noise level than expected from simulations for the acceleration-based HRD solutions need to be further studied.
References Beutler G (2004) Methods of celestial mechanics. Springer, Berlin Heidelberg New York Beutler G, J~iggi A, Hugentobler U, Mervart L (2006) Efficient orbit modelling using pseudo-stochastic parameters. J Geod 80:353-372 European Space Agency ESA (1999) The Four Candidate Earth Explorer Core Missions: Gravity Field and
F61dv~iry L, Svehla D, Gerlach C, Wermuth M, Gruber T, Rummel R, Rothacher M, Frommknecht B, Peters T, Steigenberger P (2004) Gravity model TUM-2Sp based on the energy balance approach and kinematic CHAMP orbits. In: Reigber C, Liahr H, Schwintzer P, Wickert J (Eds) Earth observation with CHAMP, results from three years in orbit. Springer Verlag, Berlin Heidelberg New York, pp 13-18 Gerlach C, F61dv~iry L, Svehla D, Gruber T, Wermuth M, Sneeuw N, Frommknecht B, Oberndorfer H, Peters T, Rothacher M, Rummel R, Steigenberger P (2003) A CHAMP-only gravity field model from kinematic orbits using the energy integral. Geophys Res Lett 30(20) Hugentobler U, Schaer S, Fridez P (2001) Bernese GPS Software Version 4.2. Documentation, Astronomical Institute University of Berne J~iggi A, Beutler G, Hugentobler U (2004a) Efficient stochastic orbit modeling techniques using least squares estimators. In: Sanso F (Ed) The proceedings of the international association of geodesy: a window on the future of geodesy. Springer, Berlin Heidelberg New York, pp 175-180 J~iggi A, Hugentobler U, Beutler G (2006) Pseudostochastic orbit modelling techniques for low Earth orbiters. J Geod 80:47-60 Lemoine FG, Smith DE, Kunz L, Smith R, Pavlis EC, Pavlis NK, Klosko SM, Chinn DS, Torrence MH, Williamson RG, Cox CM, Rachlin KE, Wang YM, Kenyon SC, Salman R, Trimmer R, Rapp RH, Nerem RS (1997) The development of the NASA GSFC and NIMA Joint Geopotential Model. In: Segawa J, Fujimoto H, Okubo S (Eds) lAG Symposia: Gravity, Geoid and Marine Geodesy. Springer-Verlag, pp 461-469 Reigber C, Balmino G, Schwintzer P, Biancale R, Bode A, Lemoine JM, Koenig R, Loyer S, Neumayer H, Marty JC, Barthelmes F, Perosanz F, Zhu SY (2002) A high quality global gravity field model from CHAMP GPS tracking data and accelerometry (EIGEN-1S). Geophys Res Lett 29(14) Reigber C, Schwintzer P, Neumayer KH, Barthelmes F, K6nig R, FSrste C, Balmino G, Biancale R, Lemoine JM, Loyer S, Bruinsma S, Perosanz F, Fayard T (2003) The CHAMP-only Earth Gravity Field Model EIGEN2. Adv Space Res 31(8) 1883-1888 Svehla D, Rothacher M (2003) Kinematic and reduceddynamic precise orbit determination of CHAMP satellite over one year using zero-differences, presented at EGS-AGU-EUG Joint Assembly, Nice, France Svehla D, Rothacher M (2004) Kinematic precise orbit determination for gravity field determination. In: Sanso F (Ed) The proceedings of the international association of geodesy: a window on the future of geodesy. Springer, Berlin Heidelberg New York, pp 181-188 Touboul P, Willemenot E, Foulon B, Josselin V (1999) Accelerometers for CHAMP, GRACE and GOCE space missions: synergy and evolution. Boll Geof Teor Appl 40 321-327 Visser PNAM, Sneeuw N, Gerlach C (2003) Energy integral method for gravity field determination from satellite orbit coordinates. J Geod 77:207-216 Wu SC, Yunck TP, Thornton CL (1991) Reduced-dynamic technique for precise orbit determination of low Earth satellites. J Guid, Control Dyn 14(1): 24-30
361
Chapter 53
On the Combination of Terrestrial Gravity Data with
Satellite Gradiometry and Airborne Gravimetry Treated in Terms of Boundary-Value Problems P. Holota Research Institute of Geodesy, Topography and Cartography, 25066 Zdiby 98, Praha-vychod, Czech Republic e-mail:
[email protected], Tel." +420 323649235, Fax: +420 284890056
Abstract. The unprecedented progress in satellite, but also airborne and terrestrial measurements is associated with new possibilities for refined studies on Earth gravity field. At the same time, however, these advances open a number of new problems of theoretical nature. The purpose of this paper is to study the synthesis of satellite, airborne and terrestrial measurements and to show that it brings impulses for the formulation of boundary-value problems which together with some optimization concepts may offer a reasonable way for the combination of the mentioned data. In a sense the approach offers a generalization or modification of the problems, their importance is well-know in physical geodesy. Mathematical properties and the solution of these problems are discussed. The apparatus of spherical harmonics is applied and it illustrates the synthesis in a spectral domain. The optimization procedure enables to treat the fact that the problems are overdetermined by nature.
Keywords. Earth's gravity field, geodetic boundary-value problems, overdetermined problems, optimization
1 Introduction In gravity field modeling a challenging and also frequently discussed problem is the combination of heterogeneous gravity data. In parallel also more data become available than necessary. Hence certain kinds of overdetermined problems have to be solved. The purpose of this paper is to extend theoretical studies performed in Holota and Kern (2005) concerning the combination of terrestrial and airborne gravity data and, in view of the GOCE mission and the so-called space-wise approach, also the combination of terrestrial gravity and (satellite) gradiometry data. At the same time values of some parameters were changed. In the sequel 12 means a layer bounded by two surfaces. With some simplification we can even
suppose that 12 is bounded by two spheres of radius R i and Re, respectively, assuming that Ri < Re .
As usual we will refer our considerations to Euclidean three-dimensional Euclidean space R 3 with rectangular Cartesian coordinates xi, i = 1, 2, 3 and the origin at the center of gravity of the Earth. Then x = ( x I , x 2 , x 3 )
is a general point in R 3
In the sequel we will also use the spherical coordinates r (radius vector), q) (geocentric latitude) and 2(geocentric
longitude),
which
are related to
x 1, x 2 , x 3 by the equations x 1 = r cos (,ocos 2 ,
x 2 = r cos (p cos 2
x 3 = r sin q)
(1) (2)
We will assume that the Earth is a rigid body and that the system of coordinates rotates together with the Earth with a known constant angular velocity co around the x 3-axis.
2 Mixed boundary problem for terrestrial and airborne gravimetry We start our considerations with the problem to find iF such that in
AT-O
aT ~T c3r
--
_(2 Ag
(3) for
r
-
R i
(4)
R i
c3T --=-6g Or
for
r-R e
where A is Laplace's operator, A g gravity anomaly and 6 g
(5) is the usual
is the gravity disturbance.
Note, however, that for T as above we do not explicitly use the term "disturbing potential", since T does not necessarily coincide with this notion (which is common in geodesy) if defined by Eqs. (3)- (5), see below.
Chapter 53 • On the Combination of Terrestrial Gravity Data with Satellite Gradiometry and Airborne Gravimetry Treated in Terms of Boundary-Value Problems
In case that T really represents the disturbing potential in the sense as in geodesy, then for physical reasons T is regular at infinity, which means that asymptotically where
T(r, (,o,2) = O(r -a) for r ~ oo,
(_9(.) is the usual Landau symbol (i.e.
(17 - 1)T~/) -(17 4- 2)q"Tj e) = (, + 1)qn+'rj
')
-
, rj
(14)
R i Ag n
=
(15)
in contrast to Eqs. (7) and (8). Its determinant is D, = (17+ 1)(17+ 2) q2n+l _
n(17- 1)
(16)
IT(r, (p,2) ] < const./r for r --~ oo ). In this case one In case that D, ~ 0 one obtains
usually writes
T(,i) --[Ri17Ag~-Re(17+2)q~Ggn]/D~ T- Z ~ ,=0 \ /
T~ (~p,Z)
(6)
where T~ are the respective Laplace surface spherical harmonics. If by chance a solution of the problem above exists in the form of Eq. (6) then in view of the orthogonality of spherical harmonics (n - 1) T,, =
Ri Ag,
(7)
(17 + 1) q'+aT, = Re 6g, where
(8)
Ag, and 6gn are Laplace's surface spherical Ag and 6 g , i.e.,
harmonics in the expansions for
(17)
and T,(e) - - [ R / ( n + l ) q
n+,zag,,-Re(n
- 1)Ggn] /D~ (1 8)
However, for some 17 the determinant D n may be rather small or even zero. This illustrates the following figure.
100 -
i ~f~
50
oooo<J~o¢o
=
.999'37
I 30
40
Xk
0, ¢¢~ IL 50
oo
Ag(q~, 2) - Z Ag, (q), 2)
I "~60
(9)
n=0 oo
ag(e,
-
a g , (e,
(10)
-100
I
c~~l~l
q = 0.94099
n=0
-150
while Ri
q =-Re
(11)
Ag 1 = 0, as
a condition and from Eqs. (7) and (8) it results that 17-1 =
and selected values of R e :
R e = R i + 4 k m , i.e., q = 0.99937 ; R e = Ri + 250 k m , i.e.,
Hence from Eq. (7) we necessarily have
Ag,
Fig. 1. D n for R i = 6378km
(n+l)q
n+2
6g,
(12)
q = 0.96228 ; R e = R~ + 400 k m , i.e., q = 0.94099.
From Eq. (16) one can also deduce that D, = (2n + 1)[2- c(n + 1)(n + 2)]
(19)
Thus it is clear that the problem under consideration is an overdetermined problem, when the solution is supposed to be regular at infinity. [A similar conclusion for a non-spherical analogue of our problem can be deduced from classical potential theory as represented e.g. by Kellogg (1953, Chap. IX and XI).] On the other hand recall, that .(2 is a bounded domain and that formally in this case T = T(r, q~,2)
where c - 1 - q
is represented by
The negative root has no meaningful interpretation in our considerations, so that we can confine ourselves just to
T - ~ -°°2/ ~ ~=0 \
/ n+lT(i)((tg,~)_+ - ° ° ( ~ e l n
~
/
T(,e)(q),A)(13)
,=0
see also Holota (1995) and Grafarend and Sans6 (1984). Thus inserting into Eqs. (2) and (3), we obtain for any individual n (i.e. for the individual harmonics) the following system
and only terms linear in c where
kept. Thus Dn = 0 implies a quadratic equation
(17+1)(17+2)-2/c=0
(20)
and (taking n for a rational number) its two roots are t71, 2
--
1,5+ x/2,25 + 2 ( 1 / c - 1 )
171 - - 1,5 + x/2,25 + 2 ( 1 / ~ - 1)
(21)
(22)
In general n a is not an integer, so that rigorously, D n ~ 0 for n = 0,1,2,...,c~, but it can be very small for n close to n 1 .
363
364
P. Holota
vimetry data seems to be less complicated. We can consider the following problem
Note also that Dn - 0 means that q
2~+1 =
n(n-1)
(23)
(n + 1)(n + 2)
Hence, clearly, q - 0 for n - 0 or 1, which represents the case when R e --~ oc. On the other hand
c~T
Thus, considering our system for
2
(28)
Ag
for
r-
(29)
Ri
c~2T
Example. For q - 6378 k m / 6382 km - 0.99937 we
n - 5 5 the system of Eqs. (14) and (15) is nearly singular and in fact it can be solved only in the case that the equations are linearly dependent. Clearly, the solution is then given not uniquely.
_Q
--+--T-c3r R i
q --~ 1 as n --~ oc, see also Holota and Kern (2005).
have n 1 - 54, 86. This, however, means that e.g. for
in
AT-O
c~r 2
=G
for
r-R e
(30)
Here the input from satellite gradiometry is symbolized by oo
G(~o, 2) - ~ G~ (~o, Z)
(3 i)
n=0
and T~(e)
where again G, are the respective Laplace surface
and assuming that q - 0 , 9 9 9 3 7 and n - 55, we can
spherical harmonics. Hence, in view of the orthogonality of the spherical harmonics we can deduce for any individual n the following system
T (i)
easily verify that the ratio between the elements in the first and the second column of the matrix of the system is very close to 1, in particular n-1 (n + 1)q "+1 = 0,998
and
(n +2)q ~ n = 1,001 (24)
respectively. This means that the same ratio (close to 1 ) between the right hand sides of the system, i.e., R edg55 = R i A g 5 5
or
6g55 = q A g 5 5
(25)
actually represents the condition for the solvability of the system. In this connection recall that Eq. (25) means 2 n + l conditions between the individual harmonic components which form 6g55 and Ags5 and note that for n = 55 this actually is 111 conditions that have to be met by the respective scalar coefficients [cf. with Eq. (12) which, however, has to hold for all n ]. If the condition, i.e. Eq. (25), is met we obtain (e.g. from the first equation) that T5(~) - ( 57 / 54 ) q55 T5(~) + (1/54) R i Ag55
(26)
In consequence, using the possibility of a free choice, we naturally put Ts(~) - 0 and thus obtain T¢{) - (1 / 54) Ri Ag55
(27)
Finally, recall again that the dimension of the nullspace of the boundary problem under consideration depends on the particular value of the parameter q (on the thickness of the layer .(2 ), see also Sect. 8.
( n - 1)T(i) - (n + 2)q nT/~(e) - R i Ag~
(32)
(n + 1)(n + 2)qn+lT (i) + n ( n - 1)T(e) - R e2 G n
(33)
Its determinant is Dgn - n ( n - 1)2 + (n + 1)(n + 2)2q 2n+1
(34)
and it is clear that it always differs from zero. Thus T (i) - [Rin(n - 1)Ag~ + R e2 (n + 2)q ~ G~ ] / D g (35)
and T (e) - - [ R i (n + 1)(n + 2)q
n+l
Ag n (36)
- R e2(n
1)G,]//D g
4 Optimization in H 2 In both the cases we found a solution in the form of Eq. (13) which represents a harmonic function in the layer .(2. The problem, however, is that in general the continuation of T for r > R e is not a regular function at infinity. This can be considered a consequence of a contamination of the input data by some measurement errors and thus they do not exactly represent the same function. Indeed, the data given on the sphere of radius R i are enough to determine (under the respective solvability conditions) a harmonic function in the whole
}
3 Mixed problem for gravimetry and gradiometry
domain U2ext = { x
In comparison with the case above the combination of satellite gravity gradiometry with terrestrial gra-
nature of excess data and in general (when some measurement errors are present) give rise to the
• R 3", r > R i
and thus also in
,(2 c -Oext • Therefore, the data for r = R e have the
Chapter 53 • On the Combination of Terrestrial Gravity Data with Satellite Gradiometry and Airborne Gravimetry Treated in Terms of Boundary-Value Problems
("internal") terms (r / R e)" T~e) that are not regular
which equals zero at the point f ,
at infinity. The problem calls for a regularization that at the same time can reduce errors. In the sequel we will continue the discussion already started in Holota and Kern (2005). By nature the problem may be ranged under overdetermined problem which in general have been already treated in literature, see e.g. Sacerdote and Sans6 (1985) and Rummel et al. (1989). Here we approach it in a slightly different way, through analytical regularization. In solving the overdetermined problem as above, we will look for a harmonic function f which is
Kern (2005). In calculus of variations the identity (39) represents Euler's necessary condition for the functional @ to have a minimum at the point f
regular at infinity and meets some optimization criteria. We start with the minimization of the following functional
function. Then
@(f) - I( f - T ) 2 dx (37) 12 where dx =dxldx2dx 3 is the volume element in Cartesian coordinates. In particular, we will suppose that H 2 (X2ext) is the space of harmonic func-
The integral identity given by Eq. (39) is a natural starting point for a numerical solution. First, however, we put for the indices n = 0,1, 2,...,oc and m = - n , - n + 1,...,-1, 0,1,...,n - 1,n cosm2 form_>0 Pf/Iml(sin ('°) L sin m 2 for m < 0
t
where
vf/m
1
I f g7
dx
(38)
1 2 6"xt
and we will look for a function f • H 2 (-C2ext) that minimizes the functional @. Recall that the inner product above induces the norm ]l f l] = ( f , f)1/2 and that, roughly speaking,
H 2 (.(2e~t) is a space of harmonic functions, which are defined on .(2e~t and square integrable under the weight r -2 . Note also that the regularity at infinity of functions from H2(_C2~t) is actually implied in the definition of the space H 2 (.(2ext). The functional @ is a quadratic functional and one can show that it attains its minimum in H2(X2ext). This was already discussed in Holota and Kern (2005) in detail. The respective reasoning follows the proof based on the theory of non-linear functional, as e.g. in Necas and Hlavacek (1991). On the contrary suppose that at a point f • H 2 (.C2ext) the functional 1"2 has its minimum. In this case we can deduce that necessarily
Pnlml
(40)
is the usual (associated) Legendre
\-~/
Y~m((p, 2)
(41)
are the solid spherical harmonics and it is known that in general O0
f-
tions endowed with inner product
( f ' g) -
cf. Holota and
1Tl = f/
Z Z ff/mVf/m f/=Om=- f/
(42)
where ff/m are scalar coefficients. In consequence, using the orthogonality of spherical harmonics, Eq. (39) transforms into the following system for the coefficients ff/m :
ff/m IV2rn d x - ITvnm dX D D
(43)
Here
IVf/L d x 12
R3 ( 1 - - q 2 f / - ' ) IYn2((/9,/~)do" (44) 2n-1 o-
and do- is the surface element of the unit sphere o-. As to the integral on the right hand side of Eq. (43), we first recall that in general m--f~
T~ i) - T,(i)((,°,2) - ~
"f/m'~(i)yf/m((,O,2)
(45)
/T/=--f/ In
r(,
- r(,
----f/
- Z
.,m
Lm(e.Z)
(46)
/T/------f/
while ~"/f///n .~(i) and u.~(e) are the respective coefficients. f///n Thus we obtain
I fv dx - I Tv dx 12
(39)
12
for all v • H 2 (..C2ext ) . R e m a r k 1. To see it one has to compute the functional derivative of @ (Gfiteaux' differential of @ )
I Tvnm dx - "f/m ._,(i) I vf/m 2 dx + .(2 .(2 2 %m
(47)
2 cr
365
366
P Holota
and subsequently ~ ) + a nanm (e) fnm - a (nm
which holds for all v (2n-1)(1-q2) n-2 an = q 2n-1 ) 2(1-q
(49)
Hence, inserting into Eq . (42), we get
[T(i) + anT(e)
i
J(grad T, grad v) dx (53)
(48)
where
f -
=
J(grad f, grad v) dx
E H2 1)
(Qext ) . It represents the
respective Euler's necessary conditien in analogy to Eq . (39) . Interpreting Eq. (53) in terras of our function basis, we can again expres the function f by means of Eq . (42), but new for the coefficients fnm we obtain the following system
(50)
]
n=0~ r ~
fnm
J
grad vnm
$ (grad T, grad vnm ) dx (54)
In particular ene can show that ao = (1+ q) / 2q and that lira an = 0
n ->
as
oo .
Further values of an
which is an analogue to Eq . (43) . Here
are in Fig . 2 . J(grad T, grad v nm ) dx = an,
grad vnm 2 dx
(55)
1,2
and in consequente a nm (l)
mm
0,8
(56)
Thus we finally get
0,6
~ n+1
T(i) n
c
0,4
f
fR n=0 r
(57)
0,2
which for large lira a n 0
25
50
Fig. 2 . Values of n and a 360
a~
75
100
125
150
(q as in Fig . 1) . In addition for higher
q = 0 .99937
some further values of a n
= 0 .9919, x 720 = 0 .9672 , and
a1440
= 0
n
n
as
course, for i (e)
= 0
approaches Eq . (50), since oo , as we already know . Of (Eidl compatibility of data for
r = R . and r = Re ) both the formulas coincide .
are :
6 Optimization in
= 0 .8763 .
H21 ~ -
Traces
Considering the fact that functions belonging to
5 Optimization in Let on
H21) (sext) SZext
(f ,
Hz1 ~
Hz1) (0ext)
be the space of harmonie functions
= J
(grad f , grad g) dx
boundary of Q, we can also look for a function f u HZ> > (Qext )
which is equipped with inner product
g) 1
have precisely defined traces on the
(51)
off>=
that minimines the functional
j(f_T) 2 dS
(58)
where ( ., .) is the stalar product of two vettors in
As above ene can show, the functional 0 attains its
R 3 . We will look for a function
minimum in
f u H21) (Qext )
Hz1) (Qext)
and that new
that minimines the functional $ fv dS = $TvdS (f) =
grad ( f - T ) 2 dx
(52)
(59)
a~
As in the last sectien we can deduce that the func-
valid for all v
tional
conditien for 0 to have a minimum at the point f
attains its minimum in
H21) (Qext )
and
E H2 1) (Qext)
is the respective Euler's
that in this case the function f is defined by the
(i .e . for the function f) . Hence for the coefficients
following integral identity
fnm in Eq . (42) we new have the following system
Chapter 53
• On the
Combination of Terrestrial Gravity Data with Satellite Gradiometry and Airborne Gravimetry Treated in Terms of Boundary-Value Problems
(60)
fnm I Vn., 2 6t8--fTvnmdx
a.c2
co(f)
-
( f -- r )2
I
an
#.O
I Vnm 2 dS -- f Vnm 2 dS+ f Vnm 2 dS
- I w c ~ ( f - r ) 2 dS an ~.Q
where
~
r =R i
r =R e
(61)
: R 2 ( l + q n-l) Iy£2dcr
as -
On (69) + 2
I w grad
(f
- T)
2dx
{2
Our aim is now to find a function w such that
Aw-1
cr
and
Ow
in .(2
and
w-0
on c~.(2
(70)
(i.e. for r = R i and r = R e ). As is well-known we can split w into two parts, i.e., w = v + z , where z
f Tvnm d x _ Unrn ._,(i) f Vnm 2 6[8
(62)
+ Unm-(e)(Ri + Re)Riqn 1 8 2 doO"
is a particular solution of the Poisson equation Az = 1 and v is a harmonic function in .(-2 such that v = - z on c%O. For the particular solution we can take, e.g.,
In consequence
z = r 2/6 fn., - Unto .~(i) + r-BnCtnm _(e) ,
fin -
1 + n-1 q q n-1 l+q
(63)
and
and see that, indeed,
Hence v has to be found as a solution of the following problem
Av-O -
[ T(i) +
~--o2
tinT. {e) ]
(64)
J
One can show that /3o - 1
and that lim f i n - 0 as
n ---> oo, which qualitatively is close to what we get for the functional co.
v--R v--R
n=0 k
we will consider the following functional {P(f) - I/9 ( f - T )2 dx
in
.(2
2/6 2
e/6
(71)
for
r-R i
(72)
for
r-R e
(73)
For this aim recall that in general
7 A note on a m u t u a l tie In this section we try to throw more light upon the tie among c o ( f ) , 7-'(f) and O ( f ) . For this reason
Az = 1 in £2.
J
v, ((o,2) + Z n=0
((,o,2) (74)
Using now the orthogonality of spherical harmonics, we easily deduce that Eqs. (72) and (73) yield
v; i) + v~e) - - R 2 / 6 (65)
(75)
and
(R i/Re)v~ i) + v~e) - - R e2 /6
(76)
where /9 is a positive function (weight function). In addition we will suppose that there exists a function w such that t 9 - Aw in .(2. Thus
~ ( f ) - I ( f - T ) 2 A w dx
(66)
[2
and it follows from Greens' identity that
{~(f) - I ( f - T )2
_f w a
c?X2
(67) a(/-T)2
c~n
dS+
IwA( f
-
since f
v~i) - (R 2 + ReR i ) / 6
(77)
v~e) - - (R 2 + ReR i + R2 ) / 6
(78)
w = w ( ~ ) = v ( ~ ) + ~(~)
1
T) 2 dx
=-6[(Re+Ri)
ReRi r 2 2 r + - R e - R e R i - R[]
(79)
[2
Inspecting this function, we immediately see that w = 0 on c%O. Moreover, for r ~ (Ri,Re) it is not
Moreover, on the right hand side
A(f -T) 2 - 2 grad(f-T)
Eqs. (75) and (76) we have
and consequently
dS
On
012
-
Oral4'
while for n > 0 they give v(~i) - v(/) - O. Thus from
2
and T are harmonic functions. Hence
(68)
extremely difficult to show that w(r) is a negative function which attains its m i n i m u m Wmin for
367
368
P. Holota
r -/'min so
that
-
3%/(Re+ Ri)ReRi / 2
"
(R e + Ri) / 2
(80)
Wmi n -- W ( r m i n ) " -- ( R 2 + R ? ) / 8 .
Summing up, for w as above we have
ence of this problem if compared with the usual Stokes' problem that has a tie to the boundary condition for T , we consider on the sphere of radius R~, see Eq. (4). One can also say that for the combination of A g
~ ( f ) - c0(f) -
~ ( f - r )2 c3w dS c~n (?X2
and 6 g the solution of our auxiliary problem in the (81)
+ 2 1 w g r a d ( f - T ) 2d x In addition, c~w
c~w . Or
.
On for r
-
R i
(R e - R i)(2Ri + R e) . R e - R i . . . . 6R i 2
(82)
and
aw = a__w__w= (R e - R i ) ( R i + 2Re) _. -RRei c3n c3r 6R e 2
(83)
we are faced by a problem which is manageable more easy way, see Holota and Kern (2005). In case also the dimension of the null-space of auxiliary problem in 12 equals one for all finite
for r - R e . Hence, approximately, cO(f) - Re - R / O ( f ) 2 + 2 [ w] grad ( f - T ) 2 dx
(84)
12 where
w-w(r)
domain .(2 may be used as an aid (if combined with an optimization concept), but in this case it is better to confine the combination to higher harmonic components of the disturbing potential only. This, in fact, causes not a big problem, since a restriction like this is natural for the use of terrestrial and airborne gravity data. Nevertheless the problem needs further investigation. Note. In case that the combination concerns gravity disturbances given for r = Ri and r = R e
is given by Eq. (79). Eq. (84)
shows that there is no perfect tie between 0~, 7~ and O . Nevertheless, we can see from this equation how the values of T on the boundary c~12 and the smoothness of T in the layer 12 are represented in the functional ¢o. R e m a r k 2. From Eq. (84), if also Wmin is taken into consideration, one can deduce that
in a this the q.
As regards the problem in Section 3, we see in contrast that the use of the solution of an auxiliary boundary problem in the domain 12 offers a way for combining terrestrial gravity data and satellite gradiometry which has relatively favorable mathematical properties. It seems, that this is associated with the fact that derivatives of different orders appear in the boundary conditions on the lower and upper part of the boundary, respectively, i.e. for r = R i and r = R e . Hence for the problem studied in Section 3 the optimization in terms of the functional 0~ leads to
2
R~ - R i O ( f ) < q ) ( f ) + R~ + R2i T ( f ) 2 4
(85) n=0
( R "~"+1E
~-1Agn (86)
i.e., a special case of an important inequality which appears in a trace theorem that is well-know in functional analysis.
+ A~(e)
Re Gn (n + 1)(n + 2)
with
8 Concluding remarks In order to apply our optimization to the problems discussed in Sections 2 and 3 we still have to return to Laplace's surface spherical harmonics T~(i) and
A(O _ n 2 - 1 E n ( n - 1 ) D,g n+i
2)qn+l 1
(87)
and A(e) _ (n + 1)(n + 2)
_
T,(e) as they were derived in this two cases The problem in Section 2 (the combination of terrestrial gravity data with airborne gravity data) has somewhat more difficult nature. It is a singular problem, as we have already mentioned. The dimension of its null-space varies with the thickness of the domain (layer) 12 and may attain considerable values. This actually causes an essential differ-
-%(n+
D~
F-lL(n+2)qn+%(n_l)j
(88)
Note that for n = 1 1 A~( i ) n-1
c~1--0.33 3q
Recall also that in Eq. (86) R i A g n / ( n - l )
(89) repre-
sents the typical structure of a spherical harmonic component as it appears it the solution of Stokes'
Chapter 53 • On the Combination of Terrestrial Gravity Data with Satellite Gradiometry and Airborne Gravimetry Treated in Terms of Boundary-Value Problems
problem for the exterior of a sphere of radius R i 2 while R e G, / (n + 1)(n + 2) is a harmonic component that typically appears in the solution of the gradiometric problem for the exterior of a sphere of radius R e .
1,8 1,6 ", ~ ~
1,4 1 ,2
A (e) n
I k :
1
~
, ~
-
(i)
-
Ar~
~"'~'-/ j ,/ " ~ --~
0,8
0,6 0,4 0,2
~,q
= 0.96228
q = 0.94099
-
n
-0,2 -0,4
As regards the functional O Eqs. (86) - (89) may be used again, but in this case a , - f i n for all n . The respective figure for the coefficients A,(i) and A~(e) shows a faster attenuation of the influence of gradiometric data with increasing n again. Qualitatively, however, it does not differ too much from Fig. 3. For this reason we skip its presentation here. The illustrations also show that the optimization applied offers a natural (and analytical) concept for weighting the effect of the input data on the final solution. Possibly, the concept may also be taken for an alternative view of the balance between the terrestrial and gradiometry data which usually is solved on the basis of stochastic considerations.
Acknowledgements. The work on this paper was supported 0
25
50
75
100
125
150
Fig. 3. Coefficients A F/(i) and A /7(e) for the functional q) q = 0.96228 and q = 0.94099, i.e. for Re = R i+ 250km and Re = R i + 400kin
by the Grant Agency of the Czech Republic through Grant No. 205/04/1423 and partly by the Ministry of Education, Youth and Sports of the Czech Republic through Projects No. LC506. All this support is gratefully acknowledged. Thanks are also due to two anonymous reviewers for their constructive criticism and comments.
The values of the coefficients A~ i) and A~e) can be seen from Fig. 3. Note that the meaning of these coefficients slightly differs from their interpretation in Holota and Kern (2005). The other two optimization concepts associated with the functionals T and O yield similar results in principle, but in comparison with ¢o both of them attenuate the influence of gradiometric data with increasing n more quickly. In particular, for the functional T the optimized solution is given by Eqs. (86) - (89) again, but with a~ = 0 for all n . The respective values of the coefficients A~i) and A~(e) are depicted in Fig 4. 1,8 1,6 1,4 1,2 ~ A ,11
o,8 0,6 0,4 0,2 0
(e) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
-
A(i)
__
n
~. ~ ' " ' -
..... 0.96228
,~
" ~, ~ = l
rl
,-- q = 0 . 9 4 0
-0,2 -0,4
~
m~====~ =======,
I 0
25
50
75
100
125
150
Fig. 4. Coefficients A~i) and A~e) for q as in Fig. 3, but for the functional 7", i.e. for a n = 0 for all n.
References
Grafarend E and Sans6 F (1984) The multibody space-time geodetic boundary value problem and the Honkasalo term. Geophys. J.R. astr. Soc. 255-275 Holota P (1995) Boundary and initial value problems in airborne gravimetry. In: Proc. IAG Symp. on Airborne Gravity Field Determination (IAG Symp. G4), IUGG XXI General Assembly, Boulder, Colorado, USA, July 214, 1995, (Conveners: Schwarz K-P, Brozena JM, Hein GW). Special report No. 60010 of the Dept. of Geomatics Engineering at The University of Calgary, Calgary, 67-71 Holota P and Kern M (2005) A study on two-boundary problems in airborne gravimetry and satellite gradiometry. In: Jekeli C, Bastos L and Fernandes J (Eds.) Gravity, Geoid and Space Missions. GGSM 2004, IAG Intl. Symposium, Porto, Portugal, August 30 - September 3, 2004. Intl. Association of Geodesy Symposia, Vol. 129, Springer, Berlin-Heidelberg-New York, 2005, 173-178. Kellogg OD (1953) Foundations of potential theory. Dover Publications, Inc., New York Necas J and Hlavacek I (1981) Mathematical theory of elastic and elasto-plastic bodies: An introduction. Elsevier Sci. Publ. Company, Amterdam-Oxford-New York Rektorys K (1974) Variacni metody v inzenyrskych problemech a v problemech matematicke fyziky. SNTL Publishers of Techn. Literature, Prague 1974; also in English: Variational methods. Reidel Co., Dordrecht-Boston, 1977 Rummel R, Teunissen P and Van Gelderen M (1989) Uniquely and over-determined geodetic boundary value problem by least squares. Bull. Gdoddsique, Vol. 63, 1-33 Sacerdote F and Sans6 F (1985) Overdetermined boundary value problems in physical geodesy. Manuscripta Geodaetica, Vol. 10, No. 3, 195-207
369
Chapter 54
A Direct Method and its Numerical Interpretation in the Determination of the Earth's Gravity Field from Terrestrial Data O. Nesvadba ~'~'3, P. Holota ~ and R. Klees ~ Delft Institute of Earth Observation and Space Systems (DEOS) Delft University of Technology, 2629 HS Delft, Kluyverweg 1, The Netherlands 2 Research Institute of Geodesy, Topography and Cartography Zdiby 98, Praha vj~chod, 250 66, The Czech Republic 3 Land Survey Office. Pod sfdligt6m 9, Praha 8, 182 11, The Czech Republic
Abstract. In the determination of the gravity field of the Earth the present accuracy requiremerits represent an important driving impulse for the use of direct and numerical methods in the solution of boundary value problems for partial differential equations. The aim of this paper is to discuss the numerical solution of the linear gravimetric boundary value problem. The approach used follows the principles of Galerkin's approximations. It is formulated directly for the surface of the Earth as the boundary of the domain of harmonicity. Thus, the accuracy of the gravity field model is only limited by the accuracy of gravimetric data and the data coverage, and by the capability of the computer hardware. The approach offers a certain freedom in the choice of a function basis suitable for representing the gravity potential of the Earth. Extensive numerical simulations have been done using simulated gravity data derived from the EGM96. Solutions for spherical and more general boundaries have been computed. The solutions also show the oblique derivative effect, usually neglected in geodesy. Moreover, different function bases (such as point masses, reproducing kernels, and Poisson multi-pole wavelets) were used to represent the disturbing potential. The computed global gravity field models are compared with the EGM96 input data in terms of potential values and gravity disturbances at points on the boundary surface.
Keywords. Gravity field modelling, linear gravimetric boundary value problem, Galerkin's method, oblique derivative effect.
1
Introduction
In gravity field studies a linear gravimetric boundary value problem governs the relation between the gravity disturbance and the disturbing potential. As known, the disturbing potential meets Laplace's equation outside the Earth. (We assume that effects of extraterrestrial masses are removed by corrections.) The problem can then be solved uniquely. A uniqueness proof based on variational method can be found in Holota (1997). In addition, we will interpret the solution as a limit of a sequence of Neumann problems, see Holota (2000). This sequence can be understood as a successive rectification of an oblique derivative in the respective boundary condition, where in each iteration step the boundary data are modified. The process avoids any simplifications and leads to an exact solution of the linear gravimetric boundary value problem. The concept of the weak formulation makes it possible to solve the problem directly by means of Galerkin's method. For this purpose spherical basis functions (SBFs) are used to represent the disturbing potential. The mentioned numerical solution enables to solve the problem also for general boundary representing the surface of the Earth. Moreover, oblique derivative effects can be discussed.
2
Problem formulation
Weighted Sobolev space W~'2(fl). Following Holota (1997), in this paper we work with the weighted Sobolev's space Wl'2(t2) that is
Chapter 54 • A Direct Method and Its Numerical Interpretation in the Determination of the Earth's 6rarity Field from Terrestrial Data
equipped with the inner product given by (u, v ) s -
(gvadu, g r a d v ) d x
~-~ d x +
apply Green's identity and show that, indeed, in the classical sense Eq. (6) yields (1)
Here ft is an unbounded domain in R 3 with Lipschitz' boundary 0f~ that represents the E a r t h surface.
Neumann's problem. Loosely speaking, to solve the classical Neumann problem for Laplace's equation means to find a function u such that Au
(gradu, n)
=
0
- f
i~
(2)
on 0f~
Au
(gradu, n + a × n}
n+a×n
= f
A1 (u, v) = (f , V)L2(aa)
(4)
0
- f
in ft
(8)
on c9~2 (9)
This can be found in Holota (1997). In particular, considering the usual formulation of the L G B V P as in Koch and Pope (1972), Bjerhammar and Svensson (1983) or Grafarend (1989) and taking into account (n, n + a x n} - 1 we can show that
(3)
where n is the unit vector of the outer normal of Oft. In this paper, we will use a concept of a weak solution. A function u • Wl'2(ft) is a weak solution of the Neumann problem for Laplace's equation, if the identity
--
--
grad U on0f~ (10) {n, grad U} grad U @ on Oft (11) ( n, grad U}
where U is the normal gravity potential, 9 is the measured gravity and 5g - g grad U is the gravity disturbance on the boundary 0f~ (i.e. on the surface of the Earth). In this case the function u represents the disturbing potential as known in geodesy. In addition,
holds for all v • Wl'2(f~) and
As (u, v) - ~ (gradu, gradv} dx
(5)
A tie between classical and weak formulation may be shown by means of a reasoning based on Green's identity. Finally, note that the unique solvability of the problem in WS'2(f~) follows from the Lax-Milgram lemma. For details see Ne(:as (1967, Chap. 1, § 3.1) and Rektorys (1977, Part IV), where, however, the problem is investigated for a bounded domain. Linear
gravimetric
boundary
value
problem.
(i.e. u is regular at the infinity) which follows automatically from the properties of the weighted space Wl'2(f~). Note that O is the so-called Landau symbol.
Sequence of Neumann problems. Our aim is to solve the identity (6) by means of successive approximations. For this reason we will consider approximations u,~ E Wl'2(f~) defined by the following sequence of problems
A1 (u,~+l, v) - (f , V)L2(Oa) + A2(u,~, v)
(13)
The linear gravimetric boundary value problem (in the sequel only LGBVP) is an oblique derivative problem. In analogy to Neumann's problem above, a function u • Wl'9(f~) is a weak solution of the L G B V P if
for all v C Wl'2(ft). One can show the sequence of u,~ has a limit u C Wl'2(ft) and it yields the solution of the LGBVP, see Holota (2000). In case u,~ is sufficiently smooth, we can apply Green's identity to the bilinear form A2 and rewrite Eq. (13) as follows
A(u, v) - As (u, v) - A2(u, v) - (f , V)L~(a~) (6)
A1 (u,~+l, v) - (f,~, v)c2(On)
(14)
f,~ - f -
(15)
holds for all v • WS'e(f~). Here, in addition to As
A2(u, v)
-
./(){gradv, a × gradu} dx
+
/~ v ( c u r l a , gradu} dx
(7)
ai • L°~(~2)and x (curla)~ • L°°(ft). For sufficiently smooth functions from Wl'2(ft) we can
where
(a × n, gradu,~}
see Holota (2000). The initial iteration step of the computing process approximates the L G B V P by Neumann's problem. Next iterations then rectify the direction of the derivative in the boundary condition by modifying the right-hand side in Eq. (14).
371
372
O. Nesvadba • P. Holota. R. Klees
3
Practical realization
Numerical approximation. Galerkin's method can be directly applied to the solution of our weakly formulated problems. For any finite-dimensional subspace W~ c Wl'~(ft), l i m ~ o ~ W~ = Wl'2(ft) (n denotes the dimension of W~) we can form the respective Galerkin system of linear equations, see e.g. Rektorys (1977). This means u C W,~, i.e. /It
~(~) - } ~ ~ ( ~ )
case as vertices of some level of the sphere triangulation mentioned above. Thus we will deal with: (i) Reciprocal Distance (RD) oo
Ix -
/=0
Yil
(~9)
which is well-known in geodesy. Note that here Ibi is the angle between the position vectors x and Yi, while Yi c R (~2 U 0f~).
(16)
(ii) Reproducing Kernel (RK)
i=l
vi(x)
and for the basis functions vi, vj C V ~ we have
\
zi
oo 2 l + 1
4~
]
Z
+-----T 1 ~P~(~°~)~
(20)
/=0
i=1
where j = 1 , . . . , n . Moreover, it can be shown that Galerkin's matrix
Aij = Al(vi, vj)
i, j = 1 , . . . , n
where zi = /~2/l~llY~l, y~ ~ ~, we us~ th~ r~producing kernel of a Hilbert space of all harInonic functions froln IvVI'2(~) which is equipped with an equivalent inner product given by A~, see Holota (2004). This function can be written in a closed form
(~s)
1
(2zi
~(~) - ~ is symmetric and positive definite. Therefore, for the solution of Galerkin's system the Conjugate Gradients Method with a diagonal preconditioning can be used, see Ditmar and Klees (2002).
Integration over the boundary. In Galerkin's system, see Eq. (17), also surface integrals have to be computed. To this aim a triangulation of the faces of the icosahedron is used to generate a subdivision of the boundary surface, see Fig. 1. A Romberg integration method is used, which exploits the hierarchical subdivision of the boundary surface. Romberg's method gives a more precise approximation of the integral and also an estimate of the integration error.
Figure 1: Icosahedral refinement on the sphere. Zero, first and second subdivision levels.
Choice of the basis. In this work we use some special cases of SBFs located at points y~ of a parking grid. The parking grid points are spread over the sphere with radius lYil = const, in our
Li+zi-cos~)
- ~
~ - co~
(21)
2 Thanks to the w h e r e L ~ - V/1 2 z ~ c o s ~ + z i. reproducing property, this basis function provides very simple setup of Galerkin's matrix for a spherical boundary with radius/~.
(iii)
Poisson Wavelets (PW)
Z ( 2 l + 1)
~n(co~)
(22)
/=0
wh~r~ ~ = R/I~I. This choice gives us an advantage of spectral decomposition of the solution with respect to parameters ai (scale factor) and si (order of multipole), see Holschneider (2003).
4
I
~
Numerical example - spherical boundary
Source data. For the boundary we take a sphere of radius /~ = 6371km. Although simple, this boundary enables to show tile effect of the oblique derivative, since g r a d U does not coincide with the normal to the boundary. Note explicitly that in this work U is defined by the parameters given in Moritz (1980). The gravimetric data on the boundary are simulated by means of EGM96 published by Lemoine at al. (1997), see Fig. 2. This simulation also enables to check the quality of the solution.
Chapter 54 • A Direct Method and Its Numerical Interpretation in the Determination of the Earth's Gravity Field from Terrestrial Data
•
,,,
300mGal 4e
60.
""~
30.
---t
i
"~"'
•
./
.-
),::
O-
::-
' "i: :'~'
.t"
~~
Fi
,;,N~,,.,,. ,
';!,,
'r'
-30./
'~">-'~
!-2°°
-60.
30-
~'- 100rnGal
%!;;: ! '°°m~a'
. ~
i o-30-
reGal
| .: | ~00m~a,
....
%'0
200mGal
[,).~ [...\ .,,a~.:-,.: ~'" ......"'"i~0mGa'. .'ixk.
" ,.:.
[.,
,.~....
, '
-120
6 longitude
[deg]
6'0
120
-60
180
, 60-
,.
180 ~130 GPU
~ ~ .
;"
~-2sGPu "
.,,,,,:~,ll2oGPu
0-
For the RD basis we put [fli[ = 0.9856R (i.e. the points fli are in the depth of ca 92 kin), for the RK basis ly~l = 1.0146R (i.e. the points fl~ are in the height ca 93 kin), while the fl~ points themselves are vertices of the icosahedral refinement of the sixth level on the sphere. This produces 40962 points of the parking grid, therefore the obtained solution belongs to the W4096~, approximation subspace. The setup of Galerkin's matrix is easy for all the basis functions. Indeed, in view of the harmonicity and smoothness of the basis functions, the use of Green's identity yields
Moreover, for our basis functions and the spherical boundary cgf~ we are able to find an analytical expression for the elements of Galerkin's matrix. The matrix itself is constant during the whole computation process. Right hand side of Eq. (17) have to be computed at every iteration step by numerical surface integration described above. Starting with a trivial disturbing potential u0 (i.e. u0 = 0), the iteration process can be terminated after seven iteration steps. Actually, the obtained solution u = u7 in Fig. 3 represents the disturbing potential related to EGM96. Thus, it can be compared with the exact solution (i.e. the disturbing potential derived directly from the EGM96 and GRSS0 gravity field), for instance at the points of the boundary surface 0f~, see Fig. 3. For the IRK basis we can see the maximal difference of 67 GPU, mean value equal to 0.05 GPU and rms of 3.35 GPU, where GPU is GeoPotential Unit,
~-..~
120
30-
Solution for V~40962 approximation subspace.
(23)
60
[deg]
""~,;,:'; .:;;
Figure 2: Gravity disturbance 59 ( E G M 9 6 GRSS0) on the boundary. It simulates the input data [1 r e g a l - 10-5ms-9].
Al(vi,vj) = - ( v j , (gradvi,n))L2(O~)
0 longitude
-60-
,
........
-90
I
-180
-120
' "t
'
, " " '
-60
.
0 longitude
. . . . . . . . .
60
.
120
/=,0o 0 /11 /ml._.2SGPu ' "q~---3o
180
GPU
[deg]
Figure 3: An obtained solution u (after seven iteration steps) for RK basis (top) and differences between the obtained solution and the exact solution (bottom) depicted on the boundary [1 G P U - lm2s-2].
1 GPU - l m 2 s -2 The detected zero order harmonic term -0.0583 km 3s -2 corresponds well to 5cM - -0.0585km3s -2 for E G M 9 6 - GRSS0. The results for the RD and P W basis are similar to the RK basis. 9O
60 ~i~ll
.~ 300
_
-180
-120
-60
0 longitude
60
120
180
[deg]
Figure 4" Oblique derivative effect (as a difference between the first and the final iteration) in the case of W40962 RK basis and the spherical boundary [1 G P U - 1 m2s-2].
373
374
O. Nesvadba • P. Holota. R. Klees
Oblique derivative effect. From the iteration process it is evident that the effect of the oblique derivative cannot be neglected. When taking the oblique derivative boundary condition into account, we can see the positive impact on the LGBVP solution. This is evident from the successive iteration steps which considerably suppress low frequency error components that appear in the first iteration (Neumann's problem approximates the LGBVP) and in some cases have an amplitude of about 0.7 GPU, see Fig. 4.
..:; '.~>..~.:,.," ..-~.. ..b~
3O-
i
-30-
1.0599s7 (i.~. G753 kin).
ly,~l =
.••
0-
Solution for W2562 approximation subspace. In contrast to the spherical case treated before, the construction of the Galerkin matrix has to be done by numerical integration, see Fig. 7. Due to a considerable extent of computational work needed, only the W256~, approximation subspace has been used. For the RD basis we take ly~l- 0 . 9 4 3 5 / 7 (i.e. 6011kin), for the R K basis
l~?kl ~ I !-~ ':, •
,t :£~....,,. 160
480
640
800
960
1120 1280 1440 1600 1760 1920 2080 2 2 4 0 2 4 0 0 2 5 6 0
2400
~#*
,,
320
.
22402080= ~ _ um
_ -180
-120
-60
0 longitude [deg]
90'
. 60 . . . .
..... ~ ~ i " .::.,..~ ~ / , i t
....~,:,,~;:~ . . .
.
. • " ~"
......
.
..,
~#-
.
", . . . . . ~.~ .... , ,q,o.
0"
"-
,~ 0.
~0. +.~ ¢/3
0. 0. i
All of Argentina
i
Buenos Aires
i
Chubut
i
Mendoza
1
Neuquan
u
i
POSGAR
SantaF6
i
Tierradel Fuego
Uruguay
Ii egm96 D A R G 0 5 _ e g m 9 6 D eigen_cg01 c D A R G 0 5 _ e i g e n _ c g 0 1 c F i g u r e 5: standard deviation of the absolute differences (after fit) between the gravimetrically geoids and the GPS/levelling-derived geoid
Chapter 61 • A New High-Precision Gravimetric Geoid Model for Argentina
Mendoza and Neuqudn are GPS/levelling networks located in the rough areas in Western Argentina. In the Neuqudn area, both global models have similar standard deviation agreement with the GPS/levelling data (44 cm), but in Mendoza, the global gravity field EIGEN_CG01C is superior by 7 cm compared to EGM96. This result is reflected in the results of the corresponding gravimetric geoids. In Neuqudn, even though both global models present similar behaviour, ARG05_egm96 is better than ARG05_eigen_cg01 c by 2 cm. The Chubut GPS/levelling network shows a very different behaviour with respect to the global models. The reason could be that the ellipsoid heights are not referred to the same datum of the other GPS points The Tierra del Fuego network is located in the Southern part of Argentina; both gravimetric geoid solutions have the same level of agreement (15 cm) but these results are slightly worse than the ones obtain with the global models alone. In the Uruguay network, the differences in the standard deviation between the gravimetric solution ARG05_eigen_cg0 lc and ARG05_egm96 solution is around 4 cm.
5.2.2 Relative differences between gravimetric geoid models and the GPS/levelling geoid To evaluate the relative accuracy of the best four geoid models with respect to the GPS/levellingderived geoid, relative geoid heights differences (ANGRAV-ANGps) were formed for all the baselines
and plotted as a function of the baseline length (spherical distance in km) in parts per million (ppm) The relative differences in ppm were formed after all outliers were removed. Figure 6 and Figure 7 show the relative differences across the entire Argentina before and after fit, respectively. The two global gravity field models have the same relative accuracies up to baselines lengths of 15 km, ranging from 8.5 ppm to 1.6 ppm. For larger baseline lengths ranging from 15 to 125 km, we can see an improvement in the long wavelength structure of the EIGEN_CG01C global model compared to the EGM96. For baseline lengths larger than 125 km to near 500 km, both models show similar relative accuracies. For 500 km to 1200 km, we can observe again an improvement of the EIGEN_CG01C global model compared to the EGM96, tending to 0 ppm for lengths over 1800 km. The two new geoid models (ARG05_egm96 and ARG05_eigen_cg01 c) present for the entire country, similar behavior for all baseline lengths, except for baselines between 15 to 115 km where EIGEN_CG01C is slightly better than EGM96 and for baselines 115 km to 700 km where EGM96 performs slightly better than EIGEN_CG01C. Comparing Figure 6 and Figure 7, we can appreciate that there is a significant improvement in the relative agreement after the fit, especially for distances greater than 225 km where both gravimetric geoid models perform better than the global geopotential models. This demonstrated the importance of using local gravity data to improve the relative accuracy.
EGM96 - • - EIGEN CG01C - -, - ARG05_egm96 - • - ARG05_eigen_cg01c
0.
0.
0.
0.
0 125
250
375
500
625
750
875 1000 1125 1250 1375 1500 1625 1750 1875 2000 distance (km)
Figure 6: Relative accuracy between geoids models and GPS/levelling-derived geoid across Argentina (before fit)
421
422
C. Tocho • G. Font. M. G. 5ideris
0.8
EGM96 - •-
EIGEN CG01C
- -~ - A R G 0 5 _ e g m 9 6 - 4 - ARG05_eigen_cg01 c
0.6
0.4
0.2
0 125
250
375
500
625
750
875
1000
1125
1250
1375
1500
1625
1750
1875
2000
distance (km)
Figure 7:
R e l a t i v e a c c u r a c y b e t w e e n g e o i d s m o d e l s a n d G P S / l e v e l l i n g - d e r i v e d g e o i d a c r o s s A r g e n t i n a ( a f t e r fit)
6 Conclusions and future plans Two new gravimetric geoid models for Argentina were developed at the Department of Geomatics Engineering during the stay of the first author at the University of Calgary. The area covered by both solutions is from 20°S to 55°S in latitude and 53°W (307°E) to 76°W (284 ° E) in longitude with a grid spacing of 5'. The computation of the two new gravimetric Argentinean geoids models, namely ARG05_egm96 and ARG05_eigen_cg01c, was based on the classical remove-compute-restore technique using the most accurate current gravity database for Argentina. The comparison of both geoid solutions with the GPS/levelling data show that the absolute agreement with respect to the GPS/levelling-derived undulations (after the systematic datum differences were removed) is near 32 cm in terms of standard deviation for. ARG05_egm96 and 33 cm for ARG05_eigen_cg01 c. A regional analysis was carried out and the statistics show that the absolute agreement level of the differences between the gravimetric solutions and the GPS/levelling-derived geoid for each network are different in flat areas like the Buenos Aires province and rugged areas like the Andes. The lack of gravity data and the roughness of the topography are similar in the areas where the GPS networks are located, so it is necessary to investigate the accuracy of the GPS/levelling-derived geoid
heights especially in rough areas where the accuracy of the levelling heights is much poorer The best overall agreement with the GPS/levelling data is achieved for the whole Argentina by the ARG05_egm96 gravimetric geoid, with a standard deviation of 0.32 m. We can conclude that with the improvement of gravity data coverage, quality and density mainly in the Andes, it will be possible to improve the accuracy of the geoid to meet the requirements needed nowadays for modern geodetic, oceanographic and geophysics applications. The densification of gravity data in the Andes can be carried out with modern measurement techniques like airborne gravimetry. As digital elevations models play an important role in the removecompute-restore technique, the SRTM3 (JPL, 2004) model with a resolution of 3" x 3" will be evaluated in Argentina. Also, the spectrum of the geoid from various gravity field signals will be investigated in future work in order to be used in the optimal combination of the geopotential model and local gravity data. Finally, a numerical solution for the altimetry-gravimetry boundary value problem (AGBVP) will be evaluated in order to combine different types of gravity data along the coastline. Also the effect on geoid modeling of applying smoothing conditions along the coastline to remove data discontinuities has to be investigated as proposed by Grebenitcharsky (2004).
Chapter 61 • A New High-Precision Gravimetric Geoid Model for Argentina
Acknowledgments The visits of the first author to the Department pf Geomatics Engineering of the University of Calgary have been supported by the Natural Sciences and Engineering Research Council of Canada (NSERC). and GEOIDE NCE grants of the third author. References
Andersen OB, Knudsen P, Trimmer R (2003): Improved high-resolution altimetric gravity field mapping (KMS02 Global marine gravity field), international Association of Geodesy Symposia, vol. 128, Sans6 F (Ed.), Springer. Proceedings of the Symposium 128: A window on the future of Geodesy, Sapporo, Japan, June 30-July 11, 2003, pp. 326-331. Bajracharya S (2003): Terrain effects on geoid determination. UCGE Reports, Number 20181, The University of Calgary. Bajracharya S and Sideris MG (2005): Terrain aliasing effects on gravimetric geoid determination, Geodesy and Cartography vol. 54, no. 1, pp. 3-16. Featherstone WE and Kirby JF (2000): The reduction of aliasing in gravity anomalies and geoid heights using digital terrain data, Geophysics Journal International, no. 141, pp. 204-212. Grebenitcharsky R (2004): Numerical solutions to altimetry gravimetry Boundary Value Problem in coastal region. UCGE Reports, Number 20195, The University of Calgary. GTOPO30 (1996): http ://edc daac.us gs. gov/gtopo 30/gtopo 30.html.
Haagmans R, de Min E and van Gelderen M (1993): Fast evaluation of convolution integrals on the sphere using 1D FFT and a comparison with existing methods for Stokes's integral, Manuscripta Geodaetica, vol. 18, pp. 227-241. JPL (2004): SRTM-The mission to map the World, Jet Propulsion Laboratory, California Institute of technology. http ://www2 .jp 1.nasa. g ov./srtm/in de x.html. Lemoine FG, Kenyon SC, Factim JK, Trimmer RG, Pavlis NK, Chinn DS, Cox CM, Klosko SM, Luthcke SB, Torrence MH, Wang YM, Williamson RG, Pavlis EC, Rapp H and Olson TR (1998): The development of the joint NASA, GSFC and the National Imagery and Mapping Agency (NIMA) Geopotential Model EGM96, NASA, Technical Publication- 1998-206861, July, 1998. Sideris MG. and Li Y (1993): Gravity field convolutions without windowing and edge effects. Bulletin Gdoddsique vol. 67, no. 2. Moritz H. (2000): Geodetic Reference System 1980, Journal of Geodesy, vo|. 74, pp. 128-162. Reigber CH , Schwintzer P , Stubenvoll R, Schmidt R, Flechtner F, Meyer U , K6nig R, Neumayer H, F6rste Ch, Barthelmes F, Zhu SY, Balmino G , Biancale R, Lemoine J , Meixner H, Raimondo JC (2004): A High Resolution Global Gravity Field Model Combining CHAMP and GRACE Satellite Mission and Surface Gravity Data: EIGEN-CG01C accepted by Journal of Geodesy and abstract from Joint CHAMP/GRACE Science Meeting, GFZ, July 5-7, 2004 (page 16, no. 24 in Solid Earth. Wichiencharoen C (1982): The indirect effects on the computation of geoid undulations, Report of the Department of Geodetic Science and Surveying no. 336, The Ohio State University, Columbus, Ohio.
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Chapter 62
Local gravity field modeling using surface gravity gradient measurements Gy. T6th, L. VOlgyesi Department of Geodesy and Surveying, Budapest University of Technology and Economics; Research Group of Physical Geodesy and Geodynamics of the Hungarian Academy of Sciences, H-1521 Budapest, Hungary, Mtiegyetem rkp. 3.
Abstract. Almost 100,000 surface gravity gradient measurements exist in Hungary over an area of about 45 000 km 2. These measurements are a very useful source to study the short wavelength features of the local gravity field, especially below 30 km wavelength. Our aim is to use these existing gravity gradient data in gravity field modeling together with gravity anomalies. Therefore we predicted gravity anomalies from horizontal gravity gradients using the method of least-squares collocation. The crosscovariance function of gravity gradients and gravity anomalies was estimated over the area and a suitable covariance model was estimated for the prediction. The full covariance matrix would require about 15 GB storage, however, the storage requirement can be reduced to about 300 MB by inspecting the structure of the cross-covariance function. Using sparse linear solvers the computation proved to be manageable, and the prediction of gravity anomalies for the whole area was performed. The results were evaluated at those sites where Ag values were known from measurements in the computational area. Keywords: horizontal gradients of gravity, gravity gradient measurements, least-squares collocation, sparse matrix solvers
1 Introduction In view of the increasing accuracy demands of local gravity field determination in the GPS era, it seems advantageous to combine all available measurements to the gravity field for the purpose. Since our knowledge of the local gravity field is based mainly on gravity measurements, the combination of other kind of gravity field parameters (e.g. observations on the direction of the gravity vector or its horizontal gradient) with gravity measurements is preferable. Several authors developed methodologies to combine horizontal gravity gradient and gravity meas-
measurements for local gravity field determination. Vassiliou, for example, showed how to process and downward continue airborne gradiometer data (Vassiliou, 1986). Hein discussed many ways of dealing with gravity gradient measurements that are available in Germany in the Upper Rhine Valley (Hein, 1981). He processed altogether 21616 such measurements in view of local gravity field determination. Another method, the so-called gradient kriging with terrestrial gravity gradients was proposed and used by Menz and Knospe (2002) for local gravity field determination. The problem is particularly interesting to us since in Hungary we have almost 100,000 surface gravity gradient measurements. Our aim is therefore to combine these measurements with gravity data as well as other data in view of a new geoid solution (V61gyesi et al, 2004). This is the reason why the least-squares collocation method was chosen, since it is well known that within this method it is relatively easy to process different kind of gravity field parameters in a theoretically sound framework. The main problem of least-squares collocation is that it is computationally demanding as it requires the solution of a linear system of which the number of unknowns equal to the number of measurements. Therefore several authors proposed compactly supported covariance functions that lead to sparse matrix techniques to reduce the computational burden (Sans6 and Schuh, 1987). First, we briefly review the necessary details of the least-squares collocation method. Next within the framework of an application example (involving an area of about 45 000 km 2) the chosen method is investigated. Finally the results are discussed and several conclusions are drawn.
2 Optimum estimation of gravity anomalies The well-known method of least-squares collocation (Moritz, 1980) is proven to be suitable in grav-
Chapter62 • Local Gravity Field Modeling Using Surface Gravity Gradient Measurements ity field modeling, since it allows estimating any gravity field parameter from measurements of other gravity field parameters. The prediction s at any point is obtained through the following linear system
C~ -
I C xz,xz
Cxz'Yz 1 +
C,,,, + Cn~ - Cyz,xz
C yz, yz
Cnn ,
(4)
C,~ - [Cx~,Ag Cyz,Ag ]
(1)
the blocks each contain covariance functions (2) and (3), evaluated at the distance d(i, k) and azimuth
where ( is the measurement vector, Css and Cnn denote signal and noise covariance matrices, respec-
a(i, k) of measurement and/or prediction points Pi, Pk and Qi, Qk, respectively. The solution of Eq (1)
S -- C s ~ ( C s s --1-C n n ) - l g ,
tively, and Csc is the cross-covariance matrix of measured and predicted quantities. In our case now we would like to predict gravity anomalies, i.e. s = Ag(Qk) at points Qk, (k = 1 . . . . Kmax) from measured horizontal gravity gradients
Vx: Vy: at points Pi, (i = 1. . . . Nmax) i.e. ( = [Vxz(Pi) Vyz(Pi)]. The necessary isotropic covariances can be written as functions of distance d and azimuth a (counted anticlockwise from East towards North) between any pair of points in the local x = East, y = North
system
Cvyz, Vyz(d, a), CAg, Vyz(d, a).
as
Cvxz, Vxz(d, a), C/xg,wrz(d, a) and
is an optimum estimation in the least-squares sense to gravity anomalies Ag. An important restriction on the choice of the covariance function C(Ag,Ag), besides its positive definiteness, is the existence of its six derivatives defined in Eqs. (2) and (3). A simple analytical covariance function model, which behaves well under repeated differentiation and is physically possible, is the two-parameter Gaussian covariance function
C ( d ) - A e -Bd2 .
follows:
Cwrz, Vyz(d, a),
(5)
The covariance functions (2) and (3) can be obtained immediately as
The auto- and cross-covariance functions of horizontal
gradients Cvxz, Vxz(d, a), Cvxz, Vyz(d, a) are obtained by dif-
gravity
Cvyz, Vyz(d, a),
ferentiation from auto-covariance function of grav-
_ O__CC= 2ABd e -Bd2 cos oy Ox OC -Bd: - -- = 2ABd e
sin a'
~y
ity anomalies CAg, Ag(d) = C(Ag, Ag)
02C
---
= ABe-Bd2[1-
0x 2
2Bd 2
-(1 + 2Bd2)cos 2o(]
~2
Cvxz, Vxz ( d, ce) - - - - 7 C ( Ag , Ag ) 3x ~2
Cvxz,Vyz (d, ce) -
Ox~)yC(Ag, Ag) .
(2)
~2 Cgyz,gy z (d, ce) ----~-C(Ag,Ag) Oy Also
the
cross-covariances
CAg,
O2--~C= - A B e -Bd2 (1 + 2Bd 2)sin 2or OxOy O2C --= A B e - B d 2 [ 1 - 2 B d 2 + (1 + 2Bd2)cos 2a'] 0y 2 -
Vxz(d, a)
(6)
and
Czxg, Vyz(d, a)can be written similarly
All of the above covariance functions are azimuthdependent (non-isotropic). However, it is possible to introduce the isotropic functions
OC m__
= 2ABd e-Bd
2
Od O C Ag,Vxz (d, o~) - -~x C(Ag,Ag )
O:C = 2AB(1 (3)
c Ag, vy~ (d, cO - ~ c (Ag,/,,g) ely
The linear system (1) is composed of six covariance matrix blocks
G~,Ag, Cy~,A~"
Cxz, xz, Cyz,yz, Cxz,yz, Cyz,yz,
,
(7a,b)
2B dZ ) e -Bd2
0d 2 which are useful to estimate the parameters A and B in (5) from measured horizontal gravity gradients (Tscherning, 1976). The isotropic covariance functions (7) are illustrated for the parameters A = 6.5 mGal 2 and the parameter B, implicitly defined through the correlation length do - x/ln 2 / B - 6 km
425
426
Gy. T6th.
L. V 6 1 g y e s i
in Fig. 1. The correlation length do is by definition the distance where the covariance is half of the variance C(0) (i.e. C(do) - 0.5C(0)). 30 . . . 25 ._ %
, vd)
20
a compactly supported covariance function (Gneiting, 2002). Our second choice is to keep the covariance model (5) with infinite support, but neglect the covariances beyond a certain distance dmax. Through either of these achievements the covariance matrix will be sparse and thus efficient sparse matrix techniques can be used.
III
I
I
•
c~ 10 E
40000
.
.
I.. :
.
.
I 1
.,- ...... t'
=
.
.
.
I
-, .
.
I
.
.
.I
.
::: : I - . , - . . . . . . t'
-,
.
¢.J t"r-
0
o
• j,~.
....
o
30000. .....
_~
:"
!'_,,'", r'"'.
•
-10 -15 O
5
i
i
10
15
,
,::
"I'...~
• ,:
i
'./'~'~'": ......
,.
;:.
.':'
,
._,~
,
~
"
'
....
" .
:'
.....
!'_,,°',
:z
.-
r~-,
.~, .,.. :'.
'.,~''"':
,/7
"," r . . . ~
,.
,';
,
' ....
,
._,-t
,
"~ '
"
" .
--
20000
20
distance [km]
Fig. 1 Example of isotropic Gaussian cross-covariance function (7a) of gravity anomalies and horizontal gravity gradients C(Vd, Ag) as well as isotropic auto-covariance function (7b) of horizontal gravity gradients C(Vd, Vd). The parameters of the covariance function for this example are A = 6.5 mGal 2, do = 6 km. The two parameters necessary to define a Gaussian covariance function can be estimated from the empirical isotropic auto-covariance function (Tb) of
,oooo1
I
!;:. ..... ,~.~-!
20000
30000
,. ~
40000
I
I
I
I
I
I
I
-~ i : . : 4 / .
40000-
'4
IL
30000-
.....
iil,
........ ..
3 An
10000-
Our example application of the collocation equation (1) is the estimation of gravity anomalies from the surface gravity gradient dataset of Hungary (V61gyesi et al., 2004). The dataset contains 44 818 gravity gradients and cover an area of about 45 000 km2 (See statistical parameters in Table 1). The covariance matrix thus has slightly more than 2 billion elements and that would require 15 GB capacity to store the full matrix in double precision. To make our problem numerically tractable, we have two choices. First, we could use instead of (5)
,...
• ~ !';'!,:_,
I:~ ~:5
20000-
example
,. ;
Fig. 2 Nonzero pattern of the sparse C~ matrix (4) before preordering. The number of nonzero elements is 4 049 268 or 0.2%.
125 E 2, whereas the correlation length dg is about 0.7-1 km. Hence the parameters of the Gaussian covariance function (5) are approximately A - 0.83 mGal 2 and d o - 1-2 km. application
i;. ...... ~..'
10000
horizontal gravity gradients Vxz, Vyz. Since from (7b) the variance is 2AB, and the correlation length dg is connected to do according to the formula dg = 0.532 do, one can take these two parameters for the estimation of A and B. The actual gravity gradient data in Hungary, which were reduced to the normal and topographic effects, show an average variance
,... :-!-;..,_,
..
~. ~
..~"" • ,.;-#
,,
"'~
.~ ', .~ i
10000
,_ i
20000
i
30000
,,. i
40000
Fig. 3 Nonzero pattern of the Ce~ matrix (4) after approximate minimum degree (AMD) preordering
Our example computations were based on the second choice. All auto- and cross-covariances were truncated in the same way at dmax. If the parameters of the Gaussian covariance function are A - 0.5
Chapter 62 • Local Gravity Field Modeling Using Surface Gravity Gradient Measurements
mGal 2 and do - 1.5 km, the covariance drops below 5% of its m a x i m u m value at dmax = 4 km. Beyond this maximum distance all covariances were considered to be zero. Truncating covariances like this will tend to remove from the estimated Ag field any long-range (wavelength dmax ) systematic patterns present in the gradiometric data. This way the number of nonzero elements in the covariance matrix has been reduced to 4 049 268. In efficient compressed column format (Davies, 2005) with the necessary bookkeeping information the matrix can be stored in 46 MB. It was found that about 300 MB in-core memory was consumed during the assembly and solution stages of the problem, which is entirely acceptable even on a standard PC. Table 1 Statistical parameters of horizontal gravity gradients used in the calculations. All units are E (1 E6tv6s = 10.9 s-2) min
Vxz Vvz
max
mean
std
-82.80
98.90
-0.45
11.17
-173.90
225.40
0.92
11.79
For numerical tests we have developed Fortan 90 code and interfaced it with the C language LDL library (Davies, 2005). The preordering of the matrix for efficient factorization was performed through calls to the A M D library (Amestroy et al., 2004). It can be seen that the original covariance matrix (Fig. 2) after the approximate minimum degree (AMD) preordering step has a number of large nonzero blocks (Fig. 3), and this permutation prevents fill-in during the next step, the Cholesky factorization of the matrix.
The solution of the sparse linear system (1) provided us gravity anomalies at 22 409 points. The measurements were considered to be uncorrelated and with uniform noise variance. After several tests runs the noise standard deviation of horizontal gravity gradients was chosen to be +13.5 E. With this value the variance of predicted gravity anomalies was in agreement with the variance of the chosen covariance model. Moreover, this noise variance level is in agreement with the actual errors of _+1015 reGal found by Hein (1981) from his collocation experiments with horizontal gravity gradients in the Upper Rhine Valley in Germany. The histogram of predicted gravity anomalies can be seen on Fig. 4. Although there are several extremely big values (up to 400 reGal!), these are restricted only to a small area and more than 99% of the predictions fall within the _+25 reGal range. It was interesting to us to make comparisons of these results with gridded l ' x l . 5 ' free-air gravity anomalies. These anomalies were reduced to the effect of the EGM96 geopotential model. To get comparison also with the high frequency part of gravity anomalies, low-pass filtered anomalies with a Gaussian filter of length 15 km were removed. We found that the agreement seems better with high-pass filtered gravity anomalies (Fig. 5) than with the original ones. This was expected, since our previous experiences have shown that gravity gradients are more sensitive to local features of the gravity field than gravity anomalies. gravity anomalies
10000
, t_]~8~
760
63 1000 63
7 0
780
predicted
"'/11
o
E
790
high-pass filtered
100
e--
/ I -400
-300
-200
-100
0
100
200
300
400
gravity anomaly [mGal]
Fig. 4 Histogram of gravity anomalies predicted from horizontal gravity gradients at 22409 points with 150 bins. Notice the logarithmic scale on the vertical axis. Only less than 1% of the predictions are above _+25reGal.
760
770
780
790
760
770
780
790
x [kin]
Fig. 5 Comparison of gravity anomalies, high-pass filtered gravity anomalies and predictions over a selected 30x30 km2 nearly flat area. Contour interval is 1 reGal.
427
428
Gy. T6th. L. VSIgyesi
Fig. 5 also shows that predicted gravity anomalies in this region have less power than the actual gravity field. Truncating covariances would partially account for the loss of signal in these areas. On the other hand the chosen covariance model may be inappropriate for this almost flat area - especially the correlation length is too small. On the other hand if the area has non-flat topography, the predicted anomalies have considerably more power than reference gravity anomalies (Fig. 6). This raises the problem of non-stationarity of the gravity gradient signal, the variance of which is very strongly correlated with the topography of the area. The histogram of average point distances (Fig. 7) reflects the difference between flat and non-flat areas as well. This is a problem for least-squares collocation, since homogeneity and isotropy are essential assumptions of the method (Kearsley, 1977). Other methods like kriging may be interesting in this respect, which do not require stationarity assumption on the signal, only stationarity of signal increments, i.e. the intrinsical stationarity (Gneiting et al., 2000). We have also the possibility to smooth the gravity field by removing additional topographical effects from the gravity gradient signal or to make the predictions separately for flat and nonflat areas.
4 Conclusions and recommendations In the present study it was shown how efficient sparse matrix techniques can be used in local gravity field modeling with horizontal gravity gradients. The example computation with Hungarian gravity gradient data has suggested that non-stationarity of gravity gradients makes it difficult to achieve a uniformly good prediction in areas of different topography. On the other hand gravity anomalies predicted from horizontal gradients may show significant details at short wavelengths of the gravity field which are not necessarily present in gravity anomalies.
3000
non-flat areas
2500 t-
5
{D.
Y
2000
0
flat areas
15oo
E e--
1000 500
gravity anomalies 1
780
785
790
795
800
high-pass filtered
.1/ o i/ 780
785
790
795
contour interval: 20 mGal
800
780
x [km]
785
790
795
3 4 5 point distance (km))
6
7
Fig. 7 Histogram of average point distances of gravity gradient observations. It can be observed that fiat and non-fiat areas have different average point distances
contour int.: 5 mGal
predicted
2
800
contour interval: 5 mGal
Fig. 6 Comparison of gravity anomalies, high-pass filtered gravity anomalies and predictions over a selected 20x20 km2 non-fiat area. Contour interval is 5 mGal for the upper and right subfigures and 20 mGal for left subfigure. Notice the very high variance of predicted gravity anomalies from the horizontal gradients
Hence we propose to use gravity gradients together with topography and gravity measurements to yield a better model of the local gravity field than from gravity measurements alone. Our results have shown, however, that the topography of the area has a strong impact on the gravity gradient signal and it must be considered carefully. Further tests should also be done with other covariance models and especially compactly supported covariance functions. Non-stafionarity of gravity gradients can be a problem in combined modeling of the gravity field and it deserves further attention and research.
Ac knowledgements Our investigations are supported by the National Scientific Research Fund (OTKA T-037929 and
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Chapter 62 • Local Gravity Field Modeling Using Surface Gravity Gradient Measurements
T - 0 4 6 7 1 8 ) . T h e careful and thoughtful r e v i e w o f our p a p e r b y M. V e r m e e r is also g r e a t l y appreciated.
References Amestroy, P.R. Davis, T.A. Duff, I.S. (1996) An approximate minimum degree ordering algorithm. SlAM J. Matrix Anal. Applic., Vol. 17(4), pp. 886-905. Amestroy, P.R. Davis, T.A. Duff, I.S. (2004) Algorithm 837: AMD, an approximate minimum degree ordering algorithm. ACM Trans. Math. Softw., Vol. 30(3), pp. 381-388. Davies, T.A. (2005) Algorithm 8xx: a concise sparse Cholesky factorization package. Dept. of Computer and Information Sci. and Eng., Univ. of Florida, Gainesville, USA. (http ://w ww. cise. ufl. edu/-da vies). Gneiting, T. Sasvfiri Z. Schlather, M. (2000) Analogies and Correspondences Between Variograms and Covariance Functions. NRCSE Technical Report No. 056. Gneiting, T. (2002) Compactly Supported Correlation Functions. Journal of Multivariate Analysis, Vol. 83, pp. 493508. Hein, G (1981) Untersuchungen zur terrestrischen Schweregradiometrie. VerOffentlichungen der Deutschen Geodiitischen Kommission bei der Bayerischen Akademie der Wissenschaften Reihe C, Heft Nr. 264, Mtinchen.
Kearsley, W. (1977) Non-stationary Estimation in Gravity Prediction Problems. OSU Report No. 256, The Ohio State University, Dept. of Geod. Sci, Columbus, Ohio. Menz, J, Knospe, S (2002). Lokale Bestimmung des Geoids aus terrestrischen Gradiometermessungen unter Nutzung der geostatistischen Integration, Differentiation und Verkntipfung. Zeitschriftf~r Vermessungswesen, Vol 127. No 5. pp. 321-342. Moritz, H. (1980) Advanced Physical Geodesy. Herbert Wichmann Verlag Karlsruhe & Abacus Press, Tunbridge Wells Kent. Sans6, F. Schuh, W.-D. (1987) Finite covariance functions, Bull. Gdodesique, Vol. 61, pp. 331-347. Tscherning, C.C. (1976) Covariance expressions for second and lower order derivatives of the anomalous potential. OSU Report No. 225, The Ohio State University, Dept. of Geod. Sci, Columbus, Ohio. Vassiliou, A. A. (1986) Numerical Techniques for Processing Airborne Gradiometer Data. UCSE Report No. 20017, The University of Calgary, Calgary, Alberta, Canada. V61gyesi L, T6th Gy, Csap6 G (2004): Determination of gravity anomalies from torsion balance measurements. Gravity, Geoid and Space Missions GGSM 2004. IAG International Symposium Porto, Portugal. Jekeli C, Bastos L, Fernandes J. (Eds.) Springer Verlag Berlin, Heidelberg, New York; Series: IAG Symposia, Vol. 129. (in press)
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Part IV Earth Processes:Geodynamics, Tides, Crustal Deformation and Temporal GravityChanges Chapter 63 Chapter 64 Chapter 65 Chapter 66 Chapter 67 Chapter 68 Chapter 69 Chapter 70 Chapter 71 Chapter 72 Chapter 73 Chapter 74 Chapter 75 Chapter 76 Chapter 77 Chapter 78 Chapter 79 Chapter 80 Chapter 81 Chapter 82 Chapter 83 Chapter 84 Chapter 85
Absolute Gravity Measurements in the Southern Indian Ocean Slow Slip Events on the Hikurangi Subduction Interface, New Zealand A Geodetic Measurement of Strain Variation across the Central Southern Alps, New Zealand New Analysis of a 50 Years Tide Gauge Record at Canan~ia (SP-Brazil) with the VAV Tidal Analysis Program Atmospheric Mass Flow Reduction for Terrestrial Absolute Gravimetry in the Fennoscandian Land Uplift Network Tilt Observations around the KTB-Site/Germany: Monitoring and Modelling of Fluid Induced Deformation of the Upper Crust of the Earth Understanding Time-Variable Gravity Due to Core Dynamical Processes with Numerical Geodynamo Modelling A New Height Datum for Iran Based on the Combination of the Gravimetric and Geometric Geoid Models Monthly Mean Water Storage Variations by the Combination of GRACE and a Regional Hydrological Model: Application to the Zambezi River The Use of Smooth Piecewise Algebraic Approximation in the Determination of Vertical Crustal Movements in Eastern Canada Hydrological Signals in Gravity- Foe or Friend? Applications of the KSM03 Harmonic Development of the Tidal Potential Continental Hydrology Retrieval from GPS Time Series and GRACE Gravity Solutions Gravity Changes in Northern Europe As Observed by GRACE Investigations about Earthquake Swarm Areas and Processes Sea Level and Gravity Variations after the 2004 Sumatra Earthquake Observed at Syowa Station, Antarctica Improved Determination of the Atmospheric Attraction with 3D Air Density Data and Its Reduction on Ground Gravity Measurements Solid Earth Deformations Induced by the Sumatra Earthquakes of 2004-2005 Environmental Effects in Time-Series of Gravity Measurements at the Astrometric-Geodetic Observatorium Westerbork (The Netherlands) Numerical Models of the Rates of Change of the Geoid and Orthometric Heights over Canada Optimal Seismic Source Mechanisms to Excite the Slichter Mode Recent Dynamic Crustal Movements in the Tokai Region, Central Japan, Observed by GPS Measurements New Theory for Calculating Strains Changes Caused by Dislocations in a Spherically Symmetric Earth
Chapter 63
Absolute Gravity Measurements in the Southern Indian Ocean M. Amalvict,
[email protected] Institut de Physique du Globe de Strasbourg / Ecole et Observatoire des Sciences de la Terre (UMR 7516 CNRS-ULP), 5 rue Rend Descartes, 67000, Strasbourg, France National Institute of Polar Research, 1-9-10 Kaga, Itabashi-ku, 173 8515, Tokyo, Japan Y. Rogister, B. Luck, J. Hinderer Institut de Physique du Globe de Strasbourg / I~cole et Observatoire des Sciences de la Terre, 5 rue Rend Descartes, 67000, Strasbourg, France
Abstract. In March - April 2005, Absolute Gravity (AG) has been measured by the Strasbourg gravimetry team using gravimeter FG5#206 at 4 French scientific stations in the Indian Ocean. Crozet, Kerguelen and Amsterdam are isolated Islands located approximately 3000 km south of La Reunion Island. Four times a year, the multi-purpose ship Marion-Dufresne II connects them to La Reunion. Measurements were taken at Crozet from 8 to 10 March, Kerguelen from 14 to 18 March, and Amsterdam from 20 to 24 March. Moreover AG was measured at the Volcanic Observatory of the Piton de la Fournaise, La Reunion Island, from 31 March to 4 April. The 2005 AG measurements were the first ever made at Amsterdam Island. In contrast, AG measurements were previously done at Crozet in 2003, Kerguelen in 2001 and 2003, and finally at La Reunion in 2001 and 2003. Here, we report on the first AG measurement at Amsterdam, present the 2005 AG values at each station and compare them to previous ones. Then we pay special attention to the comparison of the gravity changes with available vertical velocities derived from precise positioning techniques (GPS, DORIS) at Kerguelen and La Reunion. The uncertainty on AG variations is too high for inferring any change and stability seems quite likely. DORIS solutions show a small uplifte (a few mm/yr) at both locations.
Keywords. Absolute Gravity, DORIS, GPS, Kerguelen, Crozet, Amsterdam, La Reunion, indian Ocean
1
Introduction
The southern Indian Ocean covers a large area with only a few emerged islands, which are mainly vol-
canic. The Kerguelen, Crozet and Amsterdam Islands host permanent scientific bases. Due to this sparseness of lands, the knowledge of AG at these few points is important. It provides data for the worldwide coverage of AG values, for the global gravity field of the Earth, and for the Mean Sea Level (MSL). A 5-year program of the Gravimetric Observatory of Strasbourg, supported by the French Polar institute, is devoted to the measurement of AG at these scientific bases. The last campaign of the program took place in March-April 2005 with measurements at each of the bases; in addition, a measurement was taken at La Reunion Island, which is the departure place of the ship.
Absolute gravity measurements in Indian Ocean in 2005 AG measurements were performed during the logistic cruise of the multi-purpose ship MarionDufresne II, from 3 to 30 March 2005. The MarionDufresne II departed from La Reunion at the end of February 2005 and came back at the beginning of April, after a counter clockwise trip, calling chronologically at Crozet, Kerguelen and Amsterdam islands. The duration of the measurements at each station was determined by the duration of the logistic stop. AG was measured with the ballistic absolute gravimeter FG5#206 of the Strasbourg Gravimetric Observatory. The instrument was unloaded, set up and loaded back on board at each station. Raw data were processed using the version 3.0806 of the "g" software from Micro-g Compaw. All the raw data were reduced in a similar way for solid Earth tides (ETGTAB), ocean loading (Schwiderski model for the Kerguelen data, Shwiderski
434
M. Amalvict. Y. Rogister • B. Luck. J. Hinderer
(1980) and FES model for the Amsterdam, Crozet and La Rdunion data), atmospheric pressure (admittance factor-0.3 gGal/hPa for the difference between observed and nominal pressures), polar motion (position of the pole from IERS website) and vertical gravity gradient measured at each station. Whenever measurements had been previously made at a given station, we used the same parameters (tide models, vertical gravity gradient, and admittance for atmospheric pressure) as previously to reduce the data. The procedure for the AG observations was the same at each station, and was the same in 2001 and 2005. Moreover, the technical operator was the same for all the measurements.
b. A G m e a s u r e m e n t s at K e r g u e l e n - P o r t - a u x Frangais
Measurements took place from March 14 to March 18 at the new reference station established in 2003 in the sacristy of the Port-aux-Frangais (PAF) church. There are 79 hourly sets of 160 drops, every 20 seconds. c.
A G m e a s u r e m e n t s at A m s t e r d a m - M a r t i n de Vivi/~s
The station, in a closed garage, has been chosen in 2001; a measurement started in 2003, which had to be interrupted because the ship had to leave for rescue mission at Kerguelen. There are 43 hourly sets of 100 drops, every 10 seconds, from March 22 to March 24. d. A G m e a s u r e m e n t s at L a R 6 u n i o n I s l a n d Volcanologic Observatory
F i g u r e 1. Geographical zone of measurements a. A G m e a s u r e m e n t s at C r o z e t - A l f r e d F a u r e
Measurements took place from March 8 to March 10, at the point in the shooting hall, which has been chosen in 2001 and established as a reference station in 2003. There are 52 hourly sets of 100 drops, every 10 seconds.
Measurements took place from March 31 to April 3 at the new reference station established in 2003 in one of the garage slots at the Volcanological Observatory of the Piton de la Fournaise volcano. There are 75 hourly sets of 100 drops, every 10 seconds. Table 1 shows the value of gravity at each station; the value is given at the ground level, after all reductions. Table 1 provides altogether the coordinates of the stations, the nominal pressure calculated according to IAGBN standards (Boedecker, 1988), the vertical gradient of gravity at the reference point, the number of measured drops and the date of measurements.
T a b l e 1 - Station parameters and gravity value (2005), at ground level. Station
Latitude Longitude Height (m)
Nominal pressure
Number of drops
Date
g value
(2005)
(~Gal)
Crozet Alfred Faure
46.430 ° S 51.861 o E 140
996.5441
-3.23
5176
8-10 March
980 964 482.69 ± 3.31
Kerguelen Port-aux-Frangais Sacristy
49.35 ° S 70.2 ° E 17
1011.21
-3.41
9224
14-18 March
981 059 354.49 ± 3.22
Amsterdam Martin de Vivi~s
37.68 ° S 77.53 ° E 35
1009.05
-3.24
3581
22-24 March
980 092 602.88 ± 3.83
R6union Volcanological Observatory
21.21 o S
840.38
-3.40
7060
31 March -3 April
978 638 097.67 ± 5.04
55.56 ° E 1550
(hPa)
Vertical gradient of gravity
(gGal/cm)
Chapter 63 • Absolute Gravity Measurements in the Southern Indian Ocean
Previous AG measurements Previous campaigns and results have been reported by Amalvict et al., 2001 and Amalvict et al., 2003.
AG measurements have been taken 3 times at both Kerguelen and La Reunion islands. Table 2 shows the value of gravity at each site and we detail below the different measurements.
T a b l e 2 - Time series of gravity values (in btGal) at Kerguelen and La Reunion, at ground level.
* Amalvict et al., 2001, ** Amalvict et al., 2003. 2001"
2003**
981 061 041.1 + 5.0
981 061 039.40 + 4.75
Kerguelen sacristy
---
981 059 349.93 + 3.76
981 059 354.49 + 3.22
La Reunion Le Port
978 919 406.7 + 5.3
978 919 424.25 + 4.36
---
---
978 638 114.34 + 2.82
978 638 097.67 + 5.04
Kerguelen B 1
2005
...........................................................................................................................................................................................................................
La Reunion Volcanological Observatory
a.
Kerguelen
AG was measured for the first time in 2001; the station was on a pillar in the shelter B1. This shelter is to be removed in the future. Therefore, in 2003, a new reference station has been established in the sacristy of the church about 100 m from B1. In 2003, AG was also measured in B 1. Taking into account the 2003 AG tie, we refer all the values to the sacristy point. The AG horizontal gradient between the 2 locations measured in 2003 is g(B1)-g(sacristy) = 1 689.47 + 6.06 btGal. We use this value to reduce to the "sacristy" the value measuremed in 2001 at the B~ site. Thus g-gr is 351.63 + 7.85 luGal in 2001, 349.93 + 3.76 gGal in 2003 and 354.49 + 3.22 gGal in 2005, where g~ is 981 059 000.00 gGal. Let us notice that the standard deviation is large in 2001, not because of the measurement itself, but because of addition of variances in the transfer of the value from B1 to the sacristy, that is to say [(4.75)2+(3.76)2+(5.0)2] 1/2= 7.85 btGal.
Thanks to the 2003 AG tie all the values are referred to the Volcanological Observatory site. The AG horizontal gradient between the 2 locations measured in 2003 is g(Meteo)-g(Obs) = 281 309.91 + 5.19 laGal. We use this value to reduce to the Observatory site the "MeteoFrance" measurement in 2001. Thus g-gr is 96.79 + 7.42 btGal in 2001, 114.34 + 2.82gGal in 2003 and 097.67 + 5.041aGal in 2005, where gr is 978 638 000.00 luGal. Let us note again that the standard deviation is large in 2001 because of addition of variances in the transfer of the value, that is to say [(4.36)2+(2.82)2+(5.3)2] ~/2 = 7.42 laGal. ¢. A G a n a l y s i s
We can now summarise the previous discussion on AG values in Table 3 and Figures 2 and 3. T a b l e 3 - Gravity values (g-gr) at Kerguelen and La Reunion, at ground level, in gGal. gr = 981 059 000.00 gGal at Kerguelen, and g~ = 978 638 000.00 gGal at La Reunion.
b. L a R 6 u n i o n
AG was measured for the first time in 2001; the station was in the building of MeteoFrance in Le Port. Unfortunately, it is built on an embankment and a new reference station had to be established in 2003 at the Volcanological Observatory, on the top of the volcano Piton de la Fournaise. AG has also been measured at the MeteoFrance station in 2003.
LaReunion Vol. Obs. Ker
Sacristy
2001
2003
2005
096.79 + 7.42
114.34 + 2.82
097.67 + 5.04
351.63 +7.85
349.93 +3.76
354.49+ 3.22
Fig. 2 shows the time series of AG values at Kerguelen (referred to the sacristy). A linear fit of
435
436
M. Amalvict. Y. Rogister • B. Luck. J. Hinderer
the data, weighted by the error bars leads to a slope of + 1.34 + 1.79 laGal/yr. Fig. 3 shows the time series of AG values at La R6union (referred to the Volcanological Observatory). A linear fit of the data, weighted by the error bars leads to a slope o f - 2.03 + 2.13 pGal/yr. This value is obviously meaningless. In other words, there is no significant linear trend at any station which is in favour of a stability of the sites. The 2003 high value of gravity at the Piton de la Fournaise could be related to volcanic activity (an eruption indeed occurred in May 2003), cf Section 4.
Kerguelen - Sacristy
365 360 355 (9
350
g 345
340 335 2000
.
. 2001
.
.
. 2003
2002
. 2004
2005
2006
Comparison of height changes and gravity variations The measurements of absolute gravity at a given station are episodic, contrary to the techniques of precise positioning such as GPS or DORIS system. It is therefore very important to operate measurements at sites equipped with different techniques. Co-located sites are now the required standard to monitor (vertical) displacements. a. DORIS analysis
DORIS (Doppler Orbitography by Radiopositioning Integrated on Satellite) observations started in 1987 at Amsterdam, Kerguelen and La R6union and at the end of December 2003 at Crozet. The present DORIS beacon is the third one at Amsterdam (AMSTB) and Kerguelen (KERB) and the second one (REUB) at La R6union. Cr6taux et al. (1998) give the first results for DORIS observations. Weekly or monthly solutions can be found on Internet at ftp ://cddis.gsfc.nasa.gov/pub/doris/products/sinexseries. DORIS values of the vertical displacement given in Table 4 come from this website.
YEAR
b. GPS analysis Figure 2 Time series of absolute gravity values at Kerguelen, at ground level and referred to the sacristy.
140
La Raunion Volc Obs
130 120 o~
The stations that we are interested in are also equipped with GPS (Global Positioning System) receivers. Kerguelen GPS receiver (KERG) is installed since November 1994 and belongs to IGS (International GPS Service). Bouin and Vigny (2000) gave the first GPS analyses in that part of the world. The GPS station at La R6union belongs to OGS for a few months. The daily JPL results are available at the website: http://wwwgpsg.mit.edu/-tah/MIT_IGS_AAC/index2.html, from where we got the solutions we present below.
(9 110 v
c. Comparison of results from different techniques
lOO
~o'0~
'
~0'0~
'
~0'0~
'
~0'0,
'
~0'0~
YEAR
Figure 3 Time series of absolute gravity values at La R6union, at ground level and referred to the Volcanologic Observatory.
The values of vertical displacement observed by DORIS and GPS are shown in Table 4, together with the AG changes. Note that this comparison is not presented for Crozet because there are only two AG measurements and no GPS data for this site. The DORIS observations are from 2001 at Kerguelen and 1999 at La R6union. The GPS solutions presented for Kerguelen are from two different
Chapter 63 • Absolute Gravity Measurements in the Southern Indian Ocean
analysis centres (jpl and cod) but for the same period of observation (from the end of 1996 to mid2001). GPS solutions are very different from one analysis centre to an other one. We have chosen the longest time series available at the website. Table 4 - . Gravity change vs height change at Kerguelen and La Reunion. The 2 GPS solutions are provided by 2 different analysis centres.
DORIS (mm/yr) GPS (mm/yr) AG (btGal/yr)
Kerguelen 4.6 + 0.3 + 2.3 + 0.2< < + 7.3 + 0.3 + 1.34 + 1.79
La Rdunion 2.1 + 0.2 not available 2.03 + 2.13
At Kerguelen, both DORIS and GPS observations suggest a few mm/yr uplift. AG measurements suggest stability. La Reunion Island is a "hot-spot" intraplate volcano. Its total height is 7500 m of which 3000 emerge; its basal diameter on the oceanic floor is 240 kin. The Observatory has been established in 1979 at 15 km from the summit of the volcano and geophysical networks are deployed since then. (http://ovp.iniv-reunion.fr). The analysis of these geodetic and geophysical observations is under investigations by several teams and not yet published. Eruptions are quite frequent, the last ones are: October 2000, March, June and November 2001, January and November 2002, May 2003, February and October 2005. Note that the 2003 eruption occurred only slightly after our AG measurements. There were no GPS solution available for this study at La Reunion island. DORIS data show a small uplift, in agreement with the decreasing gravity, but the uncertainty on the AG trend does not allow to draw any final conclusion.
5
Conclusions
AG measurements have been successfully conducted at Amsterdam for the first time and AG measurements have been successfully repeated at Crozet, Kerguelen and La Reunion, in March-April 2005. Repeated measurements at Kerguelen and La Reunion lead to the following remarks: the gravity variations at Kerguelen and at La Reunion could suggest stability of both sites, nevertheless comparison of AG changes and vertical velocity derived from positioning observations is far from being straightforward, because of the insufficient number of AG measurements. All these AG measurements
need to be remade to have a better estimate of any gravity trend that could be compared to height changes. Moreover GPS solutions are very different from one analysis centre to an other one. In addition, let us note that the knowledge of AG values is useful for both relative marine gravimetry and gravity network at the Volcanological Observatory.
Acknowledgments. This study was carried out during the stay of MA at the National institute for Polar Research (NIPR), Tokyo, Japan under a fellowship from the Japanese Society for the Promotion of Science (JSPS). Authors thank all the people from IPEV, TAAF, and Marion-Dufresne II for their support and help. The authors thank the two reviewers for their constructive comments which helped improving the manuscript.
References
Amalvict M., Hinderer J., Boy J.P. and Luck B., 2001, Gravity at Kerguelen (indian Ocean): Absolute Gravity Measurements and Tidal Analysis from a relative gravimeter data, lAG General Scientific Assembly, Budapest, September 2001, proceedings on CD-Rom Amalvict M., Bouin M-N., Hinderer J. and Luck B., 2003, Results of the first absolute gravity measurements at Crozet Island, and repetition at Kerguelen and La Reunion islands, 9 th Symposium on Antarctic Earth Sciences, Potsdam, Germany, September 2003. Boedecker, G. 1988. International Absolute Gravity Basestation Network (IAGBN). Absolute gravity observations data processing standards & station documentation. BGI Bull. Inf. 63, 5157. Bouin M.N. and Vigny C., 2000, New constraints on Antarctic plate motion deformation and deformation from GPS data, Journal of Geophysical Research Solid Earth, Vol. 105(B12), pp. 28279-28293. Cretaux J.F., L. Soudarin, A. Cazenave, F. Bouille (1998). Present-day Tectonic Plate Motions and Crustal Deformations from the DORIS Space System, Journal of Geophysical Research, Solid Earth, Vol. 103(B 12), pp. 3016730181 Schwiderski E.W., 1980, On charting global ocean tides, Rev. Geophys. Space Phys., 18, 1,243268
437
Chapter 64
Slow Slip Events on the Hikurangi Subduction Interface, New Zealand J. Beavan, L. Wallace, H. Fletcher GNS Science, P O Box 30368, Lower Hutt, New Zealand A. Douglas School of Earth Sciences, Victoria University of Wellington, P O Box 600, Wellington, New Zealand Abstract. In common with other regions where continuously-recording Global Positioning System (CGPS) networks have been established above subduction zones, several aseismic deformation episodes have been observed in New Zealand since 2002. We interpret these episodes to result from slow slip on the subduction interface, though with the current density of CGPS stations the details of most events recorded to date are not well resolved. We have observed events with accompanying surface displacements ranging from 5-30 mm magnitude, and lasting several days to more than a year. Modelling suggests that the events are occurring near the down-dip end of the locked seismogenic part of the subduction zone, in the transition zone between the interseismically coupled and creeping portions of the interface. Comparison of event sizes, inter-event deformation rates, and long-term deformation rates suggest a repeat time of 2-3 years for an October 2002 event recorded near G isborne in the northern Hikurangi margin. A similar-sized event recorded in November 2004 supports this estimate. Two longer-duration slow slip events beneath the central and southern North Island may have triggered a series of small to moderate earthquakes over the past two years. There is preliminary indication of seismic tremor associated with the Gisborne events, as has been observed in Japan and western North America, but more work is needed to confirm this. The best-documented slow-slip event in the North Island occurred beneath the Manawatu region, and lasted from early 2004 until June 2005. The event caused displacements at up to seven CGPS sites, and probably resulted from up to 300 m m of slip on the subduction interface.
Keywords. GPS, aseismic slip, slow earthquakes
1 Introduction Transient fault slip episodes, occurring over much longer time periods (days, months) than earthquakes, have been recorded with continuouslyrecording Global Positioning System (CGPS) instruments located at several subduction margins on the Pacific Rim (e.g., Dragert et al. (2001); Ozawa et al. (2001, 2003); Larson et al. (2004)). The episodes are detected at the Earth's surface as non-linear motion of CGPS sites that is often rapid compared to normal tectonic plate motions. The physics of the deformation mechanisms underlying these so-called slow slip events, or slow earthquakes, is not yet well understood. The events may trigger other types of deformation, such as actual earthquakes, and may make a significant contribution to moment release in subduction zones. Quantifying their size, location and frequency is therefore a key task in characterizing seismic hazard for subduction zones. Subduction of the Pacific Plate occurs beneath the North Island of New Zealand, and at least five distinct slow slip events have been observed at CGPS sites in the North Island over the past three years (e.g., Beavan et al. (2003); Douglas et al. (2005)). The CGPS sites have been established as part of the PositioNZ (www.linz.govt.nz/positionz) and GeoNet (www.geonet.org.nz) networks. The slow slip events have occurred in at least three different locations on the subduction interface (Figure 1). Some have lasted only a week, while others have continued for more than a year. Some events have caused small (~5 mm) displacements at the surface, while others have caused CGPS sites to move more than 30 mm. In all cases, the events seem to occur near the down-dip end of the well coupled, or seismogenic, part of the subduction interface, consistent with observations from Cascadia and Japan (e.g., Dragert et al. (2001); Ozawa et al. (2003)). We find that the recurrence intervals of some events may be two to three years,
Chapter 64 • Slow Slip Events on the Hikurangi Subduction Interface, New Zealand
and that other events may take five years or longer to recur. The diverse characteristics of slow slip events observed thus far at the Hikurangi margin highlights the importance of this location as a natural laboratory for understanding aseismic deformation events at subduction margins.
GEOLOGICAL
SC,ENCES
www.geonet.org.n7
MATW•
• MAHO
• NPLY
•
ana.a,u
HAST
,~ •
Kapiti~..."
i / ,, ** * ~ ' " ~ PAEK
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ORIPA
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:*
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KOKO .11..:
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interface appears to be well coupled
WGTN
2 Tectonic setting and interseismic coupling distribution The North Island of New Zealand lies in the b o u n d a r y zone between the obliquely converging Pacific and Australian plates. North Island active tectonics is dominated by westward subduction of the Pacific Plate beneath the eastern North Island at the Hikurangi T r o u g h (Figure 2). Clockwise rotation of the eastern North Island forearc leads to back-arc rifting in the Taupo Volcanic Z o n e (TVZ), while strike-slip faulting in the eastern North Island dextral fault belt ( N I D F B ) occurs due to the oblique convergence b e t w e e n the Pacific and Australian plates. Wallace et al. (2004) used campaign GPS data to estimate forearc rotation and interseismic coupling on the Hikurangi subduction interface. The coupled zone is wider beneath the southern North Island, and it narrows and shallows towards the north (Figures 1 and 3). Much of the North Island overlies the "transition zone" b e t w e e n the coupled and creeping portions of the subduction interface (see also Reyners (1998)).
C o n t i n u o u s G P S sites u s e d in this study
• ; .... . %, ,°~
_37 °
A r e a s w h e r e s l o w slip events have occurred
d /
o
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Fig. 1 Continuous GPS stations and regions where slow slip events have been observed. Dark shaded area approximates the well coupled part of the subduction interface (Figure 3). ,
.--.
), p-.,
.
_38 °
0 10 2 0 3 0 slip rate deficit
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y
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,fq,~, f
_42 ° 173 °
.j
Chatham . Islands
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PAClFI~ PLA TE ~
Fig. 2 Tectonic setting of New Zealand. Light coloured areas are submerged continental crust. Pacific Plate subducts beneath North Island at the Hikurangi Trough. TVZ: Taupo Volcanic Zone; NIDFB: North Island Dextral Fault Belt.
~J 174 °
....... 175 °
176 °
177 °
178 °
179 °
Fig. 3 Distribution of seismic coupling on the Hikurangi subduction interface, estimated from ~12 years of campaign GPS site velocities using an elastic crustal block model. The dots are the nodes at which slip deficit is estimated, with bilinear interpolation between nodes. Darkest areas represent slip deficit of 30 mm/yr, meaning this amount of slip is being stored as elastic energy for eventual release as slip on the subduction interface, probably in a large earthquake. White areas imply slip is occurring steadily with no elastic strain build up. The coupling distribution offshore of the eastern North Island is not well resolved. There is also strong coupling, not shown in the figure, on the strike-slip faults in the northern South Island and southern North Island.
439
440
J. Beavan • L. Wallace. H. Fletcher. A. Douglas
3 Analysis techniques and time series We process the CGPS data to give daily position estimates using Bernese version 4.2 and 5 software (Beutler et al. (2001); Hugentobler et al. (2004)). We use fixed IGS final orbits then place the daily coordinate results in a global reference frame using a least-squares fit to the ITRF2000 coordinates of a set of regional IGS stations. For the New Zealand sites, we then remove outliers and apply regional filtering to reduce remaining common-mode noise in the daily position time series, in an iterative process (Zhang et al. (1997); Beavan (2005)).
4 Gisborne events
indication that some of the Hastings events follow a few months later than Gisborne events, suggesting a possible along-strike migration with time similar to that observed by Dragert et al. (2001) in Cascadia and Ozawa et al. (2003) on the Boso Peninsula. The deformation at HAST in late 2004 and early 2005 may result largely from the Manawatu event discussed below. (a) E" 150 E E
12E~T (-5.4 _+.0.4 mm/yr)
~ 100
AC'vlV (-7.5 + 0.6 mm/yr) 1274 (-4.9 _+0.5 mm/yr) 1'gT,~ (-1.9 _+0.4 mm/yr) GISB (-12.8 _ 1.2 mm/yr)
~ 'o 5o uJ
In October 2002, a rapid (compared to normal plate motion) surface deformation event of 20-30 mm magnitude was observed over a 10 day period on two CGPS instruments near Gisborne (GIS 1 and GISB; Figure 4). The event occurred in the preliminary stages of CGPS network development above the northern Hikurangi subduction zone so was not well recorded spatially. Forward modelling by Douglas et al. (2005) indicates that the event was probably due to about 180 mm of aseismic slip on the subduction interface just offshore of the Gisborne region. The event was fairly shallow, at ~10-14 km depth, but is consistent with slip occurring near the deeper end of the well-coupled part of the plate interface in this region. By balancing the magnitude of slip and the long-term plate motion, Douglas et al. (2005) predict that events of similar magnitude could recur every 2-3 years. A similar event was, in fact, recorded in November 2004, within the predicted interval. Slow slip events in Japan and Cascadia, have been associated with a seismic noise signal, or "subduction zone tremor" (Obara (2002); Rogers and Dragert (2003); Obara et al. (2004)). Douglas (2005) has made a preliminary attempt to identify such signals in regional broadband seismic data for the 2002 and 2004 Gisborne events. There does seem to be an indication of these signals, and further work is planned.
5 Hastings events Several small displacement events have been observed at a CGPS site at Hastings (HAST) since its installation in late 2002 (Figure 5). These events are generally not observed at many CGPS sites because of poor spatial sampling. There is an
0
S
o
u
t
1996
h
1998
2000 Year , .........
,,, .........
GIS1 (-18.3 __ 1.9 mm/yr) A(\',',."~,'(~,(-8.7 _+0.7 mm/yr) AGrqP (-4.0 + 0.5 mm/yr)
~ n Sites e r 2002
2004
, .........
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Fig. $ Smoothed time series showing several deformation events detected at sites HAST and RIPA (Figure 1). Linear fits have been subtracted to make the velocities between events approximately zero. The 2003 event lags the similar Gisborne event by ~2 months. The early 2005 deformation is well modelled as part of the Manawatu slow slip event.
Chapter 64 • Slow Slip Events on the Hikurangi Subduction Interface, New Zealand
6 Kapiti Coast event A CGPS site at PAEK on the Kapiti Coast (Figure 6) moved steadily westward at 25 mm/yr (relative to the Australian Plate) from 2000 to about May 2003, when it suddenly slowed to only 15 mm/yr westward. At the same time PAEK began relative uplift at about 10 mm/yr. During the same interval, WGTN continued to show fairly steady westward motion at 30 mm/yr and no clear change in vertical motion. The changes at PAEK lasted about a year then the site resumed approximately its previous motion.
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0.369 0.420
0.106 0.002
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0.233 0.266
0.118 0.002
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CSI SN35 CSI SN36 SAGENZ SN37
CS! SN1 sites: 6718, 6720, 6732; 2 SAGENZ SN1 sites: QUAR CNCL KARA; 3 CSI SN2 sites: 6716, 6732, 6700; 4 SAGENZ SN2 sites: QUAR CNCL HORN; 5 C SI SN3 sites 6702, 6715,6718, 6719, 6720, 6732, 6733, 6735, 6736, 6706, 6737, 6714, 6734, from the Beavan et al. (1999) analysis; 6 C SI SN3 sites 6702, 6715, 6718, 6719, 6720, 6732, 6733, 6735, 6736, 6706, 6737, 6714, 6734; 7 SAGENZ SN3 sites: QUAR KARA CNCL WAKA NETT HORN VEXA PILK REDD MAKA LEOC MCKE
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Figure 4. Principal axes of the horizontal eigenvalue strain rate for the SAGENZ sub-networks. Conventions same as in Figure 3. The strain figure (grey) outside the sub-network represent the strain rate computed from 11 SAGENZ sites (QUAR, WAKA, CNCL, KARA, NETT, HORN, PILK, REDD, MAKA, LEOC,
VEXA)
(lower).
both the extension and contraction strain rates of approximately 30% and 70% respectively. Although the compression rates do not agree within the statistical uncertainty, care must be taken with any comparison as the CSI and SAGENZ regions over which the strain is estimated are not identical. The orientations of the maximum contraction do not change substantially. 4.2 Network Areal Dilation
Except for the SAGENZ SN1 and the regional CSI SN3, all the areal dilation estimates are positive. This indicates that there is an extensional regime with I~,l> Io~2l i.e. ~, + o~2 > 0, and hence an increase in area (area creation). Since the Southern Alps are caused by convergence across the plate boundary zone and as the Alpine Fault is an oblique thrust fault, it could be expected that there is an overall contraction. However, elastic dislocation theory does predict a zone of extension above the hanging wall of a thrust fault during periods of strain accumulations between earthquakes. Although the areal dilation determined by the CSI SN1 data indicates extension, the errors are considerably larger and the dilation is only marginally significant. In contrast, the precision of the areal
dilation determined from the SAGENZ SN1 data is considerably better. This is an interesting result since the Alpine fault is predominantly strike-slip with a significant contribution of reverse slip. Combined, sub-networks 2 and 3 extend across the most tectonically active part of the central Southern Alps. It is worth noting that (Walcott 1998, see Figure 22) documented the existence of normal faults east of the Alpine Fault. 4.3 SAGENZ Sub-network strain rates
Subdividing the whole SAGENZ network into smaller triangles provides further detail on strain rate variation in the region. Using the SAGENZ sites (semi-CGPS and CGPS), a network of 13 triangles was created from 10 sites (excluding NETT and MCKE). The eigenvalue strain rates are plotted in (Figure 5). For the western most triangle, the high contraction rate is most probably due to the north-south elongated triangle as the east-west spatial extent is inadequate. The contraction strain rate is likely to be poorly determined. There is evidence for an along strike variation in the geodetic strain rate. Current tectonic models of the central Southern Alps assume the Alpine fault trends at 55 ° with possibly a second parallel antithetic structure at the same strike, but opposite dip, located some hundred kilometres to the SE. The expected gradient of the strain rate is along a line
Chapter 65 • A Geodetic Measurement of Strain Variation across the Central Southern Alps, New Zealand ,
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Figure 5. Principal axes of the horizontal eigenvalue strain rates for the SAGENZ network triangles. Strain rates with negative areal dilation (contraction) are shown in grey (~1 + ~2 < 0), strain rates with positive areal dilatation (extension) are shown as black (~l + ~'2 > 0). Conventions same as in Figure 3. Note the smaller strain rate scale than in Figures 3 and 4
perpendicular to both structures (i.e., in the SE-NW direction), while position along the plate boundary zone (NE-SW direction) should not affect the strain rate (significantly). This pattern of greater extension to the north of the profile is seen in the triangle KARA-WAKA-VEXA, even though it is closer to the Alpine fault. This suggests that there is a localised area undergoing extension in the region of the network bounded by WAKA, VEXA, PILK and HORN. A shear strain pattern without any significant dilation or areal contraction is typical near an oblique reverse predominantly strike-slip fault such as the Alpine fault. The network does suggest that areal dilation (contraction or extension) is occurring and that there is regional variability (Figure 5). Comparing the strain rates for the northern regions with triangles at comparable distances from the fault on the southern side of the transect, suggests a pattern of extension along a NW-SE axis dominates along the northern side of the transect.
5 Summary This project has investigated contemporary strain variation using the SAGENZ network of semiCGPS and CGPS that crosses the Southern Alps between Karangarua and Mt Cook village. We have compared our strain rate estimates with those from a previous study and also investigated the
strain variation in the same network. Previous geodetic work, typically using data from several GPS campaigns, has provided a picture of the geodetic strain spanning a period of years at the regional scale. Using the SAGENZ data, the strain parameters were determined for several sub networks involving subsets of both the semi-CGPS and CGPS stations. In addition, the strain rates from the (localised) SAGENZ network were compared to those derived from the CSI regional campaign network that broadly covers the whole region between the east and west coasts of the South Island. Finally, a detailed investigation of the strain variability within the region covered by the SAGENZ network was considered. Consistent with the velocity estimates, a comparison of the CSI and SAGENZ strain parameters shows that they are in general agreement (within the statistical uncertainties of the data). The SAGENZ strain rate values are approximately an order of magnitude more precise. Using only the SAGENZ CGPS data, two sub networks involving 3 stations in each were compared with nearby sites from the CSI network. Although the CSI and SAGENZ strain rates and the orientation of the strain rates agree, with a denser network and the better precision of the SAGENZ data, there appears to be evidence of a zone of extension or increase in area. A wider scale regional strain rate estimate using a subset of the CSI sites and all the SAGENZ sites was also made. Again the SAGENZ strain rates broadly agree with the CSI data, but there are differences that are statistically significant. In particular the SAGENZ contraction strain rate (-0.263+0.001 ppm/yr) is nearly double the rate determined from 8 years of CSI data. This is perhaps not surprising given that the sub networks being compared are not identical (the CSI sites cover a greater region), but does demonstrate that there is regional variability at smaller scales. A shear strain pattern without any significant dilatation or areal contraction is typical around an oblique reverse predominantly strike-slip fault such as the Alpine fault. Comparing the strain rates for the northern regions with triangles at comparable distances from the fault on the southern side of the network suggests a pattern of extension along a NW-SE axis dominates along the northern side. The reason for this is still under investigation. Possible explanations are along strike variations of deformation style or subtle strain variations being caused by seasonally induced processes.
451
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P.H. Denys • M. Denham. C. F. Pearson
Acknowledgements.
This research was funded by a grant from the Earthquake Commission (New Zealand) (EQC Project 01/457). Funding for the SAGENZ project was provided by Otago University (OU) (ORG MFWB03, ORG MFWB01), Foundation for Research, Science and Technology (FRST) (contracts C05X0203, C05X0402), and the National Science Foundation (NSF) (EAR9903183). GPS data collection has been carried out by field teams including the institutions involved in the SAGENZ project (OU, GNS, MIT, UNAVCO, University of Colorado at Boulder) as well as the many different organisations involved in the collection of the CSI data. The Department of Conservation has been instrumental in granting of access and helicopter landing permits and there are several land owners from whom we have sought permission for access to their land. CGPS data has been provided by the GeoNet (funded by EQC) and the IGS data and product archives. Several figures were plotted using the public domain GMT software (Wessel and Smith 1995). Matlab Version 7.01 (R14) was used for data analysis and figures.
References Beavan, J., D. Matheson, P. Denys, M. Denham, T. Herring, B. Hager and P. Molnar (2004). A Vertical Deformation Profile across the Southern Alps, New Zealand, from 3.5 Years of Continuous GPS Data. In T. v. Dam and O. Francis, (Eds) Proceedings of the workshop: The state of GPS vertical positioning precision: Separation of earth processes by space geodesy, Luxembourg, Cahiers du Centre Europ0en de G0odynamique et de SOismologie. 23: 111-123. Beavan, J., M. Moore, C. Pearson, M. Henderson, B. Parsons, S. Bourne, P. England, D. Walcott, G. Blick, D. Darby and K. Hodgkinson (1999). Crustal Deformation During 1994-1998 Due to Oblique Continental Collision in the Central Southern Alps, New Zealand, and Implications for Seismic Potential of the Alpine Fault. Journal of Geophysical Research-Solid Earth 104(B 11): 2523325255. Bock, Y., S. Wdowinski, P. Fang, J. Zhang, S. Williams, H. Johnson, J. Behr, J. Genrich, J. Dean, M. Vandomselaar, D. Agnew, F. Wyatt, K. Stark, B. Oral, K. Hudnut, R. King, T. Herring, S. Dinardo, W. Young, D. Jackson and W. Gurtner (1997). Southern California Permanent GPS Geodetic Array - Continuous Measurements of Regional Crustal Deformation between the 1992 Landers and 1994 Northridge Earthquakes. Journal of Geophysical Research-Solid Earth 102(B8): 18013-18033. Denys, P. H., C. F. Pearson and M. Denham (2005). Strain Accumulation across the Central Southern Alps, New Zealand - a Geodetic Experiment to Characterise the Accumulation of Strain. Dunedin, New Zealand, Otago University: 58. Dong, D., P. Fang, Y. Bock, M. K. Cheng and S. Miyazaki (2002). Anatomy of Apparent Seasonal Variations from GPS-Derived Site Position Time Series - Art. No. 2075. Journal of Geophysical Research-Solid Earth 107(B4): 2075-2075. Dong, D., T. A. Herring and R. W. King (1998). Estimating Regional Deformation from a Combination of Space and
Terrestrial Geodetic Data. Journal of Geodesy 72(4): 200214. Dong, D., T. Yunck and M. Heflin (2003). Origin ofthe International Terrestrial Reference Frame. Journal of Geophysical Research-Solid Earth 108(B4). Feigl, K. L., D. C. Agnew, Y. Bock, D. Dong, A. Donnellan, B. H. Hager, T. A. Herring, D. D. Jackson, T. H. Jordan, R. W. King, S. Larsen, K. M. Larson, M. H. Murray, Z. K. Shen and F. H. Webb (1993). Space Geodetic Measurement of Crustal Deformation in Central and Southern California, 1984-1992. Journal of Geophysical ResearchSolid Earth 98(B12): 21677-21712. Feigl, K. L., R. W. King and T. H. Jordan (1990). Geodetic Measurements of Tectonic Deformation in the Santa Maria Fold and Thrust Belt, California. Journal of Geophysical Research 95(B3): 2679-2699. Hager, B. H., G. A. Lyzenga, A. Donnellan and D. Dong (1999). Reconciling Rapid Strain Accumulation with Deep Seismogenic Fault Planes in the Ventura Basin, California. Journal of Geophysical Research-Solid Earth 104(B11): 25207-25219. Hugentobler, U., S. Schaer and P. Fridez, Eds. (2001). Bernese GPS Software Version 4.2. Berne, Astronomical Institute, University of Berne. Mangiarotti, S., A. Cazenave, L. Soudarin and J.-F. Crdtaux (2001). Annual Vertical Crustal Motions Predicted from Surface Mass Redistribution and Observed by Space Geodesy. Journal of Geophysical Research 106(B3): 42774291. Shen, Z. K., M. Wang, Y. X. Li, D. D. Jackson, A. Yin, D. N. Dong and P. Fang (2001). Crustal Deformation Along the Altyn Tagh Fault System, Western China, from GPS. Journal of Geophysical Research-Solid Earth 106(B 12): 30607-30621. Shen, Z. K., C. K. Zhao, A. Yin, Y. X. Li, D. D. Jackson, P. Fang and D. N. Dong (2000). Contemporary Crustal Deformation in East Asia Constrained by Global Positioning System Measurements. Journal of Geophysical ResearchSolid Earth 105(B3): 5721-5734. Soudarin, L., J.-F. Cr6taux and A. Cazenave (1999). Vertical Crustal Motions from the Doris Space-Geodesy System. Geophysical Research Letters 26(9): 1207-1210. van Dam, T. M., J. Wahr, P. C. D. Milly, A. B. Shmakin, G. Blewitt, D. Lavallde and K. M. Larson (2001). Crustal Displacements Due to Continental Water Loading. Geophysical Research Letters 28(4): 651-654. Walcott, R. i. (1998). Modes of Oblique Compression: Late Cenozoic Tectonics of the South island of New Zealand. Reviews of Geophysics 36(1): 1-26. Wessel, P. and W. H. F. Smith (1995). New Version of the Generic Mapping Tools Released. EOS Trans. AGU 76: 329. Yetton, M. D., A. Wells and N. J. Traylen (1998). Probablility and Consequences of the Next Alpine Fault Earthquake, New Zealand. Wellington, New Zealand, New Zealand Earthquake Commission, Wellington, New Zealand. EQC Research Foundation Report 95/193: p 161. Zhang, F. P., D. Dong, Z. Y. Cheng, M. K. Cheng and C. Huang (2002). Seasonal Vertical Crustal Motions in China Detected by GPS. Chinese Science Bulletin 47(21): 1772-1779.
Chapter 66
New analysis of a 50 years tide gauge record at Canan6ia (SP-Brazil) with the VAV tidal analysis program B. Ducarme Chercheur Qualifi6 FNRS, Observatoire Royal de Belgique, Av. Circulaire 3, B- 1180, Bruxelles, Belgique. A.P. Venedikov Geophysical Institute & Central Laboratory on Geodesy, Acad. G. Bonchev Str., Block 3, Sofia 1113 A.R. de Mesquita, C.A. de Sampaio Fran~a Instituto Oceanogrfifico da Universidade de Silo Paulo, SP, Brasil. D. S. Costa, D. Blitzkow Escola. Politdcnica, Universidade de Silo Paulo, Caixa Postal 61548, 05413-001 Silo Paulo, SP, Brasil. R. Vieira Diaz Instituto de Astronomfa y Geodesia (CSIC-UCM). Facultad de Matemfiticas. Plaza de Ciencias, 3. 28040. Madrid, Spain. S.R.C. de Freitas CPGCG - Universidade Federal do Paranfi, Caixa Postal 19001, 81531-990 Curitiba, PR, Brasil.
Abstract. A homogeneous high quality tide gauge hourly record covering the period 1954-2004 was obtained at Canan6ia (SP-Brazil). A previous analysis of a 36 years data set has shown many interesting features, especially very long period signals. This re-analysis benefits from a 40% longer time series and the powerful program VAV for tidal data processing is used to determine the parameters of the tidal constituents derived from the tidal potential, including the long period tidal waves, and of the shallow water and radiation tides. Long period terms are determined from the tidal residues by a semi-automatic research algorithm. The variation of the mean sea level is estimated after subtraction of the ocean tides and estimation of the long period terms. The mean sea level rate of change is estimated to 0.5666 ___ 0.0070 cm/year. Special attention is given to the determination of the ocean pole tide as well as the 11 years term directly related to the solar activity.
Keywords. Ocean tides, mean sea level, ocean pole tide, radiation tides, Solar activity
1 Introduction The paper presents some of the results from the application of the tidal program VAV (Venedikov et al., 2001, 2003, 2005) on the ocean tide (OT) data from the Canan6ia tide gauge (SP- Brazil)
(q0 = 25 ° 01.0' S, )~ = 47 ° 55.5' W). This series of data covers 50 years in the time interval 26.02.1954 - 31.12.2004 and contains 444410 hourly ordinates. A previous analysis of a 36 years data set (Mesquita et al., 1995, 1996) has shown many interesting features, especially very long period signals. The reanalysis benefits from the longer time series and a powerful tidal analysis program. Tide gauge, station and data characteristics can be found in (Mesquita et al., 1983, 1995). The present paper is mainly focusing on the investigation of the mean sea level (MSL), including its secular variation, and the detection of various low frequency components.
2 The VAV tidal analysis program VAV is originally designed for the processing of Earth tide (ET) data. Now, it has been supplied by specific options (Ducarme et al., 2006a), corresponding to the OT characteristics and problems (Godin, 1972; Munk and Cartwright, 1966). VAV has various options for tidal data processing. The main one for the OT is the determination of the amplitudes and the phases of all 1200 tides in the development of Tamura (1987), including the long period (LP) tides. For large series of data it can also study the time variations of the tidal parameters, as shown by Figure 1 for the tidal wave M2.
454
B. Ducarme • A. P. Venedikov • A. R. de Mesquita • C. A. de Sampaio Fran(;a • D. S. Costa. D. Blitzkow • R. Vieira Diaz. S. R. C. de
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Time in days Time in years 1. Time variations of the amplitude of the main lunar tide M2 (thick line) with 95% confidence limits (grey lines) at Canan6ia Fig.
The secular increase of the amplitude of M2 was already pointed out in Mesquita et al, 1995 and Harari et al, 2004. The amplitude seems more or less stable since 1985.
7.00 6.00 ..= 5.00 (D 4.00 3.00 ".=
AIC(%,)
d(t) - L(O + ~ w ( % , t )
d(t) = L(t) + Z W ( m j ' t ) + W( w', t ) + W( w", t ) -->AI C('v~/,v()
Now w ' & w " are allowed to cover two neighboring frequency intervals. For every couple w ' & w" we get the least square solution of (7), accompanied by a corresponding AIC value. In such a way we get AIC = AIC(w',w") as a function two frequencies. The values of w ' & w" at which AIC has a minimum can be accepted as two new frequencies c0m+l and O)m+2 As shown by Table 1 and Table 2 we have found in total M = 16 frequencies. We shall demonstrate the final stage of finding the first two of them, related with the Chandler period. Final stage means that we shall use first in (5), then in (7) m = M - 2 = 14 frequencies, namely those with numbers 3 to 16, while the two first frequencies will be found through frequency variations. Figure 5 is obtained by the application of the model (5) where the variable frequency w is moving between 0.80 + 0.95 cpy. Graphics like this one can be considered as an AIC spectrum.
AIC(w) (5)
j=l In the expression (5) the new frequency w is variable within a selected frequency interval. For every value of w we get the least squares solution of (5), accompanied by a corresponding value of AIC. In such a way we obtain AIC as a function of w , i.e. AIC = A I C ( w ) . Let for a given value w = WMi. AIC(wMi o) be the minimum of the function AIC(w) in the frequency interval and, still, AIC(WMin) be lower
(7)
j=!
>
160000 159990 159980 159970 159960
159950 159940 159930
Frequency in cpy (cycles/year) Fig.5. The values of AIC = AIC(w) as function of the frequency w .
455
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B. Ducarme • A. P. Venedikov • A. R. de Mesquita • C. A. de Sampaio Fran~ja • D. S. Costa. D. Blitzkow • R. Vieira Diaz. S. R. C. de Freitas
The absolute minimum of AIC in this case is at WMin = 0.84402 cpy, i.e. at period 432.75 days. We have got also a relative minimum at frequency 0.87993 cpy, i.e. at period 415.09 days.
deeper difference is that the usual spectral analysis does not take in consideration the dependence between the frequencies found, which can be important when some of them are very close. On the contrary, our spectral analysis takes into account the effect of the frequencies already found, in this case of the m = M - 2 = 1 4 frequencies. The procedure can be repeated as many times as necessary in order to clean up completely such kind of dependences. Table 2. Summary results for very low frequency components.
%_
"
10 11 12 13 14 15 16
",7
Fig. 6. A I C = A I C ( w ' , w " ) with a minimum at w' = 0.84180 cpy and w" = 0.88102 cpy.
Due to the existence of two minima at very close frequencies we had to apply the model (7). In such a way we get the picture presented by Figure 6, through which we get the final values of the first 2 components in Table 1. The existence of two close periods at 433.6 and 414.3 day around the Chandler period of 430 day is due to the amplitude modulation of the signal. Table 1. Summary of the periodicities found by the automatic research procedure for the Chandler term, the annual one and its harmonics
Chandler Annual 1 cpy Annual at 2 cpy Annual at 3 cpy
1 2 3 4 5 6 7 8 9
Freq. cpy 0.8418 0.8810 0.9973 1.0295 1.9881 2.0154 2.9189 2.9555 3.0298
Period Amplit. Days cm 433.6 1.522 414.3 1.271 366.0 5.111 354.5 1.384 183.6 1.241 1 8 1 . 1 0.497 1 2 5 . 0 0.945 1 2 3 . 5 0.637 1 2 0 . 5 0.912
MSD cm _+0.188 _+0.188 _+0.191 _+0.192 _+0.193 _+0.193 _+0.189 _+0.189 _+0.189
The usual spectral analysis deals with the amplitude spectrum, i.e. the amplitude, represented as a function of the frequency, looking for its maximum. Here we deal with AIC as a function of the frequency, looking for the minimum of AIC. A
Freq. cpy
Period Years
Amplit. cm
MSD Cm
0.0413 0.0931 0.1479 0.1951 0.2601 0.3159 0.3795
24.22 10.74 6.76 5.13 3.84 3.17 2.64
3.39 1.48 @.97 0.87 1.39 1.61 1.36
+0.21 +0.19 +0.19 +0.19 +0.19 +0.19 +0.19
As shown in Table 2 we have identified 7 very long periods. The component with the second largest period (10.74 years) corresponds to the known solar cycle with a period close to 11 years. Harmonics of this term are also present: 5.13 year (order 2), 3.84 and 3.17 (order 3) and 2.64 (order 4). The periodicity of 6.76 year, already found by Mesquita et al. (1995), is not easy to interpret, but could be the fourth harmonic of the longest period. The total contribution to the sea level variations at Cananria of periodicities longer than two year is plotted in Figure 7. The peak to peak amplitude (double amplitude) reaches 15 cm.
8 4 -4 -8
Time in days Fig. 7. Total contribution to sea level variations of the modeled periodicities with period larger than two years.
Chapter 66 • New Analysisof a 50 YearsTide Gauge Recordat Canan~ia (SP-Brazil)with the VAVTidal AnalysisProgram 5 Evolution
of the
mean
sea
level
If M denotes the total number of the frequencies, the MSL L(t), as it was defined by (3), is found through the least square solution of the model M
d(t) - a o + ajt + Z W ( o i , t )
(8)
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'1'1'1'1'1'1'1'1'1'1'1'1'1'1'1'1'1'1
j=l
The use of the linear function a o +alt as representing L(t) requires some explanations. It appeared that when L(t) is represented by a polynomial of power higher than 1, this polynomial interferes with the low frequency components, especially with the 24.2 year wave. Namely, we start getting too high coefficients of the polynomial with too high amplitudes of the periodic constituents. Due to this, we have accepted that the non-linearity in the development of d(t) is entirely represented by the periodic constituents, while a reasonable representation of the MSL is by a polynomial L(t) of power 1. A problem in this simple model of L(t) is whether there are not some discontinuities. Namely, whether there are not one or several time points TDisc in which is an offset in L(t), i.e. some changes in the constant a 0 or changes in the general behavior, i.e. changes in the coefficient aj. Testing the data in this sense is generally necessary for Earth tides data, since the gravimeters and the clinometers are suffering from offsets due to instrumental problems. It is rather unlikely to happen with tide gauges data, except during long interruptions of the records or inside interpolated sections. Our conception is that in any case we should suppose possible discontinuities and check the data for their existence. Let us check whether in a point T is a discontinuity. Then (8) should be used with different models of L(t) before and after the point T , namely
L ( t ) - a ' o +a~t for t < T ] "
L ( t ) - a o + a~'t for t >
T~ --->AIC(T)
(9)
We can let vary the point T in a time interval. For every value of T we can apply the least squares and thus obtain the AIC = AIC(T) as a function of T. If AIC(T) at T=TMi n has a minimum and AIC(TMin) is lower than AIC for the model (8), then we have to accept that TMin is a point of discontinuity, i.e. TDisc = TMin .
Time point T in days Fig. 8. AIC as a function of a supposed point T of discontinuity with TMin - 6349 days (16.07.1971).
This is the case of Figure 8, showing a TDisc = 6 3 4 9 days. The result from the analysis with this TDisc is shown in Figure 9. A significant offset (jump) of 4.00 cm _ 0.68 cm has been found. However, the difference between the slopes of L(t) before and after the discontinuity appeared to be not significant: 0.511 +0.006 cm/year against 0.574+0.018 cm/year. Other points of discontinuities have not been found. 240 220 200 180 160 140 120
Time in days Fig. 9. The data d(t) and the MSL L(t) (white line) with a discontinuity in TDisc - 6 3 4 9 days. On the basis of this result the data d(t) have been corrected for the offset but, due to the lack of significant differences of the slopes before and after the jump, we applied later on the model (8) without discontinuity. The final result about L(t) are: a 0 - 171.279 + 0.094 at the epoch 31.01.1979 a~ - 0.5666 + 0.0070 cm/year The slope is slightly higher than the 0.405 cm/year reported by Mesquita et al, 1995 or Harari et al, 2004.
457
458
B. Ducarme • A. P. Venedikov • A. R. de Mesquita • C. A. de Sampaio Fran(ja • D. S. Costa. D. Blitzkow • R. Vieira Diaz. S. R. C. de Freitas
6 Modeling of the ocean pole tide The existence of Chandler frequencies indicates that the astro-geophysical phenomenon known as "ocean pole tide", generated by the polar motion, is effectively present in our data. The polar motion changes the position of the axis of rotation inside the Earth. At a given location on the Earth it produces a change of latitude measured by astronomical methods and a change of gravity, associated with the corresponding change of centrifugal force, measured by the superconducting gravimeters (Ducarme & al., 2006b). Moreover the associated P~ potential is able to excite a tidal deformation of the sea surface. In section 4 the study of the ocean pole tide was made in the frequency domain. It is certainly interesting to study the direct effect of the polar motion in the time domain. At a point of coordinates ((,0,)v) of the ocean surface, the equilibrium ocean pole tide can be written
Here we use only M - 2 of the frequencies, the Chandler frequencies in Table 1 being excluded. As we know already all M - 2 frequencies, the only problem to estimate 6p is the unknown At, which participates non-linearly in this model. The problem has been solved by applying again a Bayesian approach, i.e. through the least square solution of (12) for a set of different values of At. In this case, since we are mostly interested in the values of 6p we have chosen as a criterion the estimated MSD of this unknown. As shown by Figure 10 we get this MSD as a function of At. A broad minimum exists between 8 and 24 days, but we can accept, as a most reliable value of the time lag, the value of At at which we get the minimum of MSD i.e. At = 21 days. It means that the effect of the polar motion has 24 degrees of phase lag. 0.4430 -
i,
0.4425 0.4420 0.4415
.
0.4410 2/2
p(t) -- 72~'-'1
/2g
(~o)
•{x(t) COS)V+ y(t) sin )v} sin 2q~
0.4405
Time lag At in days, At > 0 : retardation where 72 - 1 + k 2 - h 2 , h 2 and k 2 being the Love numbers for radial deformation and change of the potential respectively, x(t) and y(t) are the coordinates of the pole at time t, fl is the angular velocity of the Earth, r is the radius of the Earth and g is the acceleration of gravity. The equilibrium pole tide has been computed for the daily coordinates of the pole, available from IERS since January 1, 1962, i.e. nearly 3000 days later than the start of the Canandia data. In analogy with the gravity effect of the polar motion (Ducarme & al., 2006b) we have accepted that the effect of the polar motion on our data A d(t) will be Ad(t) - 8pp(t-At)
Fig. 10. MSD of 6p as a function of the time lag At in the model (12), with a minimum at At - 21 days Finally, by applying A t = 2 1 , we have got through the solution of (12) for the ocean pole tide an amplification factor 6p - 2.19 + 0.44. 2.0 1.0 0.0 -1.0 -2.0
.
Time in days, 1.01.1962 - 31.12.2004
(11)
Fig. 11. Estimated ocean pole tide. where 6p is an unknown regression coefficient, called also amplitude factor, and At is a time lag, positive for retardation. This expression is included in our model in the following way M-2
d(t) - ao +a¢ +
~-~ff(%.,t)+apAt-m) j=l
Figure 11 displays the estimated ocean pole tide at Canandia, showing the effect of the secular drift of the pole position.. Using the model (12) the MSL variation becomes a 1 0.547 + 0.014 cm/year, with a lower precision than in the previous section. One of the reasons is that here a shorter series of data is used. -
(12)
Chapter 66 • New Analysis of a 50 Years Tide Gauge Record at Canan~ia (SP-Brazil) with the VAV Tidal Analysis Program
7 Modeling of the direct effect of the solar activity Figure 12 shows the daily series of the sunspot index, measured by the number of Wolf W(t), taken from the SIDC (Solar Index Data Center) (Vanderlinden et al, 2005). The series nearly coincides with the Canandia ocean data. The phenomenon has obviously a period, very close to the period of our component 11 in Table 2. Following the same idea as in the previous section we have attempted to study the direct effect of W(t) on our data. The model now used is
The computed effect of the solar activity on the data is shown by Figure 13. It reaches a peak to peak amplitude of 4cm. The estimated slope of L(t) obtained in this way is a 1 = 0 . 5 7 4 + 0 . 0 1 0 . Here again, as in the case of the polar data, the new value of a 1 does not differ significantly from a 1 in section 5 and, in the same time, it has a lower precision. _
-1
-
-
~-2 -3
-
M-1
d(t) - ao + alt + Z W ( m / ,t) +6wW(t-kt)
(13)
-4
'1'1'1'1'1'1'1'1'1'1'1'1'1'1'1'1'1'1
j=l
where 6 w is an amplifying coefficient, At is a time lag, positive for retardation and W(t) represents the solar activity. The number of waves is reduced by 1, because the component 11 is excluded.
Time in days, 26.02.1954 - 9.09.2003 Fig. 13. Estimated effect of the solar activity on the ocean data.
8 Conclusions 350 300 250 200 150 "~ 100
Time in days, 26.02.1954 - 9.09.2003 Fig. 12. Solar activity (sunspot index,, measured by the number of Wolf) and a least squares filtered variant (white curve): cut off 0.001cpd, half length 1,000 days. We have used W(t) in the following variants: (i) the raw data, (ii) filtered data by a least square low pass filtering with characteristics cut off frequency 0.003cpd, half length 300 and (iii) filtered with cut off frequency 0.00 l cpd, half length 1000 (the white curve in Figure 12). In all cases, using the scheme illustrated in (Figure 10), we have got a zero time lag, i.e. At - 0. With At - 0 in (13) we have got an estimate of 6 w with highest signal-to-noise ratio by using the smoothed data in variant (iii), namely 6w - -1.88 _+0.28 cm/(100 units of W(t)). It is thus in opposition of phase with the excitation.
An automatic procedure is used to determine hidden long period frequencies from the daily residues also called non-tidal components. The amplitude modulation of the annual radiation tide and its harmonic can be represented by a frequency splitting. The situation is similar for the Chandler period, associated with the ocean pole tide. Two periodicities dominate the very low frequency spectrum at 24.2 year and 10.7 year. The second one is clearly associated with the Solar cycle. The ocean pole tide was modeled using the equilibrium tide. The results show an amplification factor close to 2 with a time lag between 10 and 24 days. In a similar way we tried to associate the 10.7 year period with the daily sunspot number. We found a perfect anti-correlation, without any time lag. For what concerns the MSL variations, obtained after subtraction of the ocean tide signal and all the long period harmonic terms (pole tide and radiation tides), we found a mean linear drift rate of 0.5666 ___0.0070 cm/year. This high rate is probably due to ground subsidence. GPS observations at the same site seem to confirm this fact (Blitzkow and Costa, personal communication). The necessity of a careful determination of the very long period harmonics for the estimation of MSL has to be underlined.
459
460
B. Ducarme • A. P. Venedikov • A. R. de Mesquita • C. A. de Sampaio Franga • D. S. Costa. D. Blitzkow • R. Vieira Diaz. S. R. C. de Freitas
Acknowledgements The work of A.P. Venedikov on this paper has been supported by the Royal Observatory of Belgium and the International Center for Earth Tides in Brussels and by the project REN2001-2271/RIES of the Institute of Astronomy and Geodesy in Madrid.
References
Ducarme B., Venedikov A.P., Arnoso J., Vieira (2005a). Analysis and prediction of ocean tides by the computer program VAV. Journal of Geodynamics, 41, 119-227. Ducarme, B., Venedikov, A.P., Arnoso, J., , Chen X.D., Sun H.P., Vieira, R. (2005b). Global analysis of the GGP superconducting gravimeters network for the estimation of the pole tide gravimetric amplitude factor. Journal of Geodynamics, 41,344-344. Godin G., 1972. The analysis of tides. Liverpool University Press, 263 pp. Harari J., Fran~a C. A. S., Camargo R. (2004). Variabilidade de longo termo de componentes de mards e do nivel medio do mar na costa b ras il e ira.h ttp ://www. ma re s. io. usp. b r/aa g n/aa g n 8/ressi/ressimgf html. Mesquita A.R., Harari J. (1983). Tides and tide gauges of Canandia and Ubatuba-Brazil. Relat. Int. Inst. Oceanogr. Univ. Sao Paulo, 11, 1-14
Mesquita A.R., Harari J., Fran~a C.A.S. (1995). Interannual variability of tides and sea level at Canandia, Brazil, from 1955 to 1990. Publfao. Esp. Inst. Oceanogr., Sao Paulo, 11-20. Mesquita A.R., Harari J., Fran~a C.A.S. (1996). Changes in the South Atlantic: Decadal and Interdecadal Scales. An. Acad. Bras. Ci, 88 (Sup P1), 105-115. Munk, W. H., Cartwright, D. F. (1966). Tidal spectroscopy and prediction. Philosophical Transactions of the Royal Society of London A259 (1105) 533-581 Sakamoto, Y., Ishiguro, M., Kitagawa, G. (1986). Akaike information criterion statistics, D. Reidel Publishing Company, Tokyo, 290 pp. Tamura, Y., 1987. A harmonic development of the tide-generating potential. Bulletin informations des MarOes Terrestres, 99, 6813-6855. Vanderlinden R.A.M. and SIDC team (2005). On line catalogue of the sunspot index, http://sidc.oma.be/html/sunspot.html Venedikov, A.P., Arnoso, J., Vieira, R. (2001). Program VAV/2000 for tidal analysis of unevenly spaced data with irregular drift and colored noise. Journal of the Geodetic Society of Japan, 47 (1), 281-286. Venedikov, A.P., Arnoso, J., Vieira, R. (2003). VAV: a program for tidal data processing. Computers & Geosciences, 29, 487-502. Venedikov, A.P., Arnoso, J., Vieira, R. (2005). New version of the program VAV for tidal data processing. Computers & Geosciences, 31, 667-669
Chapter 67
Atmospheric Mass Flow Reduction for Terrestrial Absolute Gravimetry in the Fennoscandian Land Uplift Network O. Gitlein, L. Timmen Institut fiir Erdmessung, University of Hannover, Schneiderberg 50, D-30167 Hannover, Germany
Abstract. Temporal variations of the atmospheric density distribution induce changes in the gravitational air mass attraction at a specific observation site. Additionally, the load of the atmospheric masses deforms the Earth's crust and the sea surface. Variations in the local gravity acceleration and atmospheric pressure are known to be correlated with an admittance of about -3 nms -2 per hPa as an average factor, which is in accordance with the IAG Resolution No. 9, 1983. A more accurate correlation factor for a gravity station is varying with time and depends on the total global mass distribution of the atmosphere. For the absolute gravimetric observations of the Fennoscandian land uplift, the atmospheric attraction effect of the local zone has been calculated with 3D atmospheric data describing different pressure levels up to a height of 50 km. To model the regional and global attraction and deformation components with Green's functions method, 2D surface atmospheric data have been used. The improved atmospheric effects have been computed for the position-dependent absolute gravity observations in Fennoscandia performed by the Institut fiir Erdmessung (liE) in 2003. The objective is to ensure an air mass reduction within +3 nms -2 accuracy. For the 2003 campaigns, the use of atmospheric actual data has improved the reductions by about 9 rims-2 (max. 14 rims-2).
Keywords. Atmospheric attraction and deformation, Green's functions, 3D atmospheric data, air pressure reductions, absolute gravimetry
1
Introduction
Fennoscandia is a key study region for the research of glacial isostatic adjustment, and it offers an opportunity for testing the GRACE results. In the centre of the Fennoscandian land uplift area, a temporal geoid variation of 3 mm is expected (Ekman and M/ikinen, 1996) over a period of five
years (life time of GRACE). With terrestrial absolute gravimetry, the corresponding gravity change of about 100 rims -2 can be observed with an accuracy of +10 to 20 nms -2 for a 5-year period, cf. Van Camp et al. (2005). Since 2003, annual absolute gravity measurements with different FG5 absolute gravimeters are being performed. Fig. 1 shows the station distribution of the Fennoscandian land uplift network and the stations occupied by liE in 2003. The FG5 design and features are described in detail by Niebauer et al. (1995). The gravimeter is a sensor measuring gravity variations of different sources. The absolute gravity measurements have to be reduced by the effects of the solid Earth tides and ocean tides and polar motion. Also the atmospheric effects have to be modelled and removed. To ensure the high accuracy of the FG5 absolute gravimeter results, the total reduction uncertainty should be as small as possible. Especially in research applications, a total uncertainty of +10 rims -2 or even better is striven for. In this study, the effects of real global atmospheric deviations from a standard atmosphere are investigated. In general, the atmospheric effect is just considered with a correlation factor of-3 rims -2 per hPa, whereby the local air pressure measurement at the gravimetry site is applied. In the following, this common procedure is also called the "-3 nms -2 per hPa" rule. The improvements of the global atmospheric modelling compared to the commonly applied rule are pointed out. The calculation procedure of the atmospheric mass attraction and load is described in section 2. In section 2.2 the calculation of the atmospheric effects of local, regional, and global zones using 2D atmospheric surface data is presented. The data are convolved with Green's functions derived by Merriam (1992). Section 2.3 describes the calculation method of the direct Newtonian attraction effect of the local zone using 3 D atmospheric data in different pressure levels. The results are discussed in section 3.
462
O. G i t l e i n • L. T i m m e n
xl0 ~ 0
+;,oo
11.
~=-2 .G
+65°
~-3 z O
-4
-5
...............................................................
10-4 ÷60°
%
10.3
10.2
10-1
10o
101
10=
...............................................................
700
6O0 .~, 500 n
~ 400 +55
~ 300 .G
+
+ o 15
+20°
+ ° 25
+
1. Fennoscanian land uplift network. Absolute gravity stations occupied by IfE in 2003 are indicated by big dots.
Fig.
~
200
100
o -100
2 2.1
Calculation of Atmospheric Mass Attraction and Load
Fig. 2. The atmospheric load gravity Green's functions for the Newtonian attraction term GN(~) (top) and the elastic deformation term GE(~) (bottom), (Merriam, 1992).
Atmospheric Data
From the European Centre for Medium-Range Weather Forecasts (ECMWF), global 2D and 3D data are available. The data used in this study were provided by the University of Cologne in cooperation with the Deutsches Klimarechenzentrum (DKRZ, German Computing Centre for Climateand Earth System Research). They are logged every 6 hours on a 1.125°× 1.125 ° grid for 2D data and 0.75°×0.75 ° grid in 21 pressure levels (1000 to 1 hPa) for 3D data. The data sets provided for this study contain the parameters barometric pressure p, geopotential V, temperature T and relative humidity r.
2.2
~ ............................................................... 10-4 10-3 10 -2 10 -1 10 o 101 10= Sphedcel Distance ¥ in [=J
Atmospheric Attraction and Deformation from 2D Surface Data
Variations in the Earth's air pressure affect the absolute gravimetric measurements due to the direct Newtonian attraction of the air masses and due to the elastic deformation of the Earth's surface (loading effect). Farrell (1972) derived gravity Green's functions for the elastic part for a point mass load on the Earth's surface. The air mass distribution in a column of up to 60 km height follows in good approximation a mathematical approach based on surface data such as temperature and pressure. This allows to model the Newtonian contribution of the column. Merriam (1992) presented column load gravity Green's func-
tions in a tabulated form. They are plotted in Fig. 2. The Newtonian attraction of an air column of area dA and density p at a spherical distance ¢ can be computed by Zmax
g(¢) _ _ /
Gp(Z)r2sin a dAdz,
(1)
0
(Merriam, 1992; Sun, 1995; Boy et al., 2002). G is the gravitational constant, z is the vertical height in the column, r is the vector distance between the volume of the mass (dAdz) and the gravity station, and a denotes the angle between the attraction direction and local horizon at the station. The density, air pressure and temperature are related by the ideal gas law and depend on height z. Applying cosine law, rearranging and normalization of the formula (1) for attraction and deformation effects, the gravity effect is obtained using the Newtonian GN(¢) and elastic deformation GE(¢) Green's functions:
g(¢)_GN(¢)+GE(¢) 104fig[tad]
A 27T[1__COS(1O)] (P--PN) • (2)
The Green's functions are insensitive to the detailed structure of the model atmosphere. To obtain the pressure p at station height H above the sea level, the input data (global pressure at sea level P0 and
Chapter 67 • Atmospheric Mass Flow Reduction for Terrestrial Absolute Gravimetry in the Fennoscandian Land Uplift Network
z i.................. i........................ inA;m'°pressure sphericdaialevels 21 ................ ........ii:':::::':"! ........................................... "~ ii (1000 to 1 hPa) ...........
.. .=.---~.....,.. - - ~ -../] .i;~~"I--, ..~~. ' , , .-"', •
S i l l ....................................o ~ ' ~ .......... II i
:z:
i
'e
,,.
i
'o
'o
i ,,-".
,'.
......................;'-~~_o~.. =~_, lib
....-
........... .......... ........... ................................
'
dy
. .~....-"
..'""
y..'-
~Absolute ..............."'-of ~ Gravity i ............~ ~' J Station i ......... J '~
,~
Y
/7 ................................. i: . . . .
X ~"
~=1 °
/
Fig. 3. Division of the global pressure data into three grid zones (local, regional, global) around the gravity station.
Fig. 4. Atmospheric mass volume in cartesian coordinates.
the temperature T at surface) have to be transformed (Liljequist and Cehac, 1984):
putation of the local attraction effect from 3D atmospheric data considers more realisticly the real density variations within the atmosphere. Only the local zone is considered here (cf. Fig. 3). The local attraction effect induces by far the greater contribution to the total atmospheric effect. Investigation with 3D atmospheric data for the local attraction can also be found in Neumeyer et al. (2004) and Simon (2003). Atmospheric Newtonian attraction component for a small air mass volume dz dy dz in cartesian coordinates (Fig. 4) is determined by:
H g~
P-p0exp
~T,
(3)
where g,J - 9.80665 ms-2 is the mean gravity at the geoid and RL -- 287.05 J k g - ~ K - ~ is the gas constant for dry air. The normal air pressure P x at gravity station height must be substracted, to get the pressure and the corresponding gravity difference with respect to the standard atmosphere. As reference, the U.S. Standard Atmosphere, 1976 is applied, see also IGC (1988). Because the Green's functions are varying with the spherical distance ~ the globe is divided into three zones, which contribute to gravity in different proportions. The global pressure data have been subdivided into grid zones around the gravity station and interpolated as follows (see Fig. 3): • Local zone with I) _< 1° and 0.01°x0.01 °, • Regional zone with I) < 10 ° and 0.2°x0.2 ° and • Global zone with ~ _< 180 ° and 1.125°x 1.125 °.
///
9Q -- --Gpdiff
x
y
(:c2 + y2 + z2)3/2'
(4)
z
with fldi f f
--
fl -
(5)
fiN
as the difference between the density p calculated from 3D data and the density p x of the U.S. Standard Atmosphere, 1976. It is considered as the mean value of the mass volume. The density p may be derived from the input data temperature T and relative humidity r at the pressure levels (Liljequist and Cehac, 1984): p-
0.378e
Summing up all small effects g(~) of the grid points, the total gravity effect for each zone is obtained.
P
2.3
with vapour pressure e, which can be computed from the relative humidity:
Atmospheric Attraction Local Zone with 3D Data
from
the
Atmospheric pressure variations affect surface gravity by the Newtonian attraction of the atmospheric masses. Just considering the changes of the surface atmospheric data is not sufficient as the higher layers with their deviating air densities contribute significantly to the total Newtonian attraction. The corn-
t~L T[oK] '
=
r
(6)
E,
(7)
100 whereby E is the saturation vapour pressure (Bolton, 1980): ( E - - 6.112exp
17"67 T[°c] )
T[oc] + 243.5
(8) "
463
464
O. 6itlein • L. Timmen
Table 1. Contributions (atmospheric gravity effects) from the local, regional and global zones. For the local zones the results from 2D and 3D data are compared (A" attraction effect, D: deformation effect). Differences resp. improvements to the "-3 rims-2 per hPa" rule effects ("-3 rule") are shown in the right two columns.
[runs -2 ] Cop/Bud Cop/Vest Helsingor Tebstrup Onsala Borfis Metsfihovi Vaasa-AA Vaasa-AB Skelleftefi Arjeplog Kramfors 0stersund Trondheim ~lesund Trysil Honefoss mean rms + abs. max.
3D local A - 15 - 15 - 15 -2 - 17 -6 8 51 2 45 -8 -32 -28 1 12 44 9 2.0 23.9
Effects with Green's functions local regional A D A D -15 1 -2 2 -13 1 -3 0 -14 1 -2 0 0 0 -3 -2 -14 1 -1 3 -3 0 -2 1 8 -1 0 -1 51 -4 0 -9 3 0 0 -1 47 -3 -1 -9 -3 0 -3 -2 -29 2 -2 4 -24 2 -2 4 4 0 -1 -4 16
-1
-1
51 14 4.7 24.4
-4 -1 -0.3 1.7
-1 3 -1.2 1.9
47
-4
-9 1 -1.6 4.4 10
V (1 - k cos 2 ~ ) g ~
,
-4
-4 -4 -4 -1 -3 -1 -3
-2 -1 -3 -4 -2 -2 -2
-2
-2
-3 -1 -1.9 2.1 10
(9)
where k = 0.002637. The gravity attraction is determined by summing up the results for each small mass volume in z, y and z direction. The calculation method for the changes in the atmospheric mass attraction in a rectangular cartesian coordinate system is given in e.g.: Jung (1961), N a g y (1966, 2000), Sun (1995).
Improved Atmospheric Effects for Fennoscandia 2003 The atmospheric effects have been calculated for 17 absolute gravity stations in Fennoscandia occupied by IfE in June, August, and September 2003 with the absolute gravimeter FG5-220. Averagely, each station has been occupied over a measuring period of about 30 hours with about 3000 free-fall experiments. The atmospheric effects have been computed
D -1 -1 -2 -1 -1 -1
-6
1 -4 -2.9 3.3
The coordinates of the input data (interpolated to 0.1° × 0.1° grid, dz = 500 m) have to be transformed to cartesian coordinates with the origin at the gravity station. The height H at a specific latitude 6 can be calculated from the geopotential (Liljequist and Cehac, 1984): H =
from 2D global A -3 -3 -5 -3 -4 -3
Total effect with local 3D 2D -17 -17 -20 -18 -22 -22 -ll -9 -19 -16 -12 -9 -4 -3 32 32 -5 -3 25 27 -17 -12 -33 -29 -28 -24 -9 -6 2 6 28 36 8 14 -5.9
-3.2
19.7 33
19.3 36
"-3 rule" -14 -15 -13 -2 -13 1 5 36 1 36 -4 -24 -20 5 11 36 8 2.1 18.6 36
Diff. to total effect 3D 2D -3 -2 -5 -3 -10 -9 -9 -7 -5 -2 -13 -10 -9 -9 -5 -5 -6 -5 -11 -9 -14 -9 -8 -5 -8 -4 -14 -11 -9 -5 -7 0 0 5 -8.0 -5.2 8.8 6.5 14 ll
exactly for the measuring time. Interpolation is needed due to the 6 h spacing of the data. Table 1 shows the computation results of the attraction (A) and deformation (D) components for the local, regional and global zones from 2D data for the stations in chronological order. For the local zones, also the results from 3D data are given. The zones are divided as described in section 2.2. The cumulative influences from the atmospheric data (with local 3D and local 2D) and the effects computed with the " - 3 rims -2 per hPa" rule are also shown in Fig. 5. The contributions of the different zones are depicted in Fig. 6. For the local Newtonian attraction effects a max. value of 51 n m s -2, and a rms of + 2 4 n m s -2 have been obtained. The local deformation effects from the local zones are small (rms = + 1 . 7 n m s -2) compared to the attraction and have the opposite sign. The regional and global effects contribute each with up to 10 rims -2. The global zone has a significant signal and should not be neglected for precise gravity observations. The contribution of the regional zone is sitedependent and may include oceanic areas, involving an inverse barometer correction which cannot be properly performed, cf. van D a m and Wahr (1987), Merriam (1992). In this study the globe is divided
Chapter 67 • Atmospheric Mass Flow Reduction for Terrestrial Absolute Gravimetry in the Fennoscandian Land Uplift Network
CBud
-"
i- ,
i
_
3D
.....C V e s ..........i............................................... .... I Hel
t
'-~
CBud
2D -3 rule
_
C,Ves,,,,! .......~...
.......................-..-. I
....
--I
Hel
!
319 local 2 D local regional global
Teb Ons •,
r
Ons
i
B,OF......................... ~ ....................~ Mets
.,iBor .................i............................~
...................................................
==
Mets i
- V - A A ..................... ~...................................................
V-AB
;
=~D
'
....Kra ....... ,
~
~==
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i
J
El I...........................i.................................................................. ........
i
-
-
i
•T r o ..................................~...............
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--
,
i
•'Try ............................ i ................................................... ~
-40
..............................................................................................
V-AB ] .................... . ....... ]
Ost
Hen
i
....V-AA,,,, ,,,,!.................... ~
;.......... i ......
• S k e l ..................................................................................
Arje
.................................................................................................................
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,
ii
~t _ _ J
i
i
i
0
20
-20
Ale
I ......
• Try
.................... ~..................... ~
Hen
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40
-40
i -20
.............................................................................. ...........................
=:U =
i
0
20
40
60
[ n m / s 2]
[ n m / s 2]
Fig. 5 . Total effects calculated from 3D and 2D, only 2D data and with the " - 3 nms -2 per hPa" rule (columns 9 to 11, Table 1).
Fig. 6. Attraction and deformation contributions of different zones, determined from 2D and 3D atmospheric data (derived from columns 2 to 8 of Table 1).
into the three zones for all station locations in the same way, independent of the oceanic vicinity. A non-inverted barometer hypothesis for the oceanic and the Baltic Sea response has been chosen. This simplified assumption considers the still existing uncertainties in modelling the actual response of the Baltic Sea and the oceans. Atmospheric load effects at GPS sites (BIFROST stations) in Fennoscandia have been discussed and analysed in detail by Scherneck et al. (2003). Virtanen (2004) applied the non-inverse barometric response at the Baltic Sea and successfully compared the modelled sea loading with observations of the superconducting gravimeter at Metsfihovi (Finland). The inverted barometer response along Geosat altimeter tracks in the open ocean has been investigated by van Dam and Wahr (1993). Boy et al. (2002) compared loading results with non-inverted and with inverted barometer assumptions for several superconducting gravimeters of the Global Geodynamics Project (GGP) network. For the absolute gravity determinations in Fennoscandia 2003 (this paper), an uncertainty of some nms -2 in the reductions of the atmospheric mass attraction and load can not be excluded. Compared with the originally applied " - 3 nms -2 per hPa" rule for the absolute gravimetry campaign
in 2003, the largest improvement has been achieved with the 3D atmospheric data set. The differences to the " - 3 nms -2 per hPa" rule effects reach up to 14 nms -2 (column 12, Table 1) and vary with arms of +9 nms -2. The difference to the total effect from 2D data only vary with arms of +7 nms -2. Using atmospheric data distributed over different height levels (3D) changes the effects of the local zone in the range of several nms -2 compared to 2D calculations (max. 7 nms -2, rms = +3 nms-2). The greater differences have been found for the last seven stations occupied in September, when the wheater become more unstable, cf. Figs. 5 and 6. Only the 3D data imply the actual height dependent density information from real atmospheric observations. These reduction improvements should be considered for precise absolute gravity determinations. 4
Summary
In this study, the Newtonian attraction and elastic deformation effects induced by atmospheric mass flow have been computed for the absolute gravity observations performed by IfE in the Fennoscandian land uplift area in 2003. The effects have been computed globally using surface 2D data convolved with Green's functions, whereby the globe has
465
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• for high accurate absolute gravity applications, a special support with most advanced models about surface deformations due to mass loading (atmosphere, oceans and seas, continental hydrosphere) would be useful.
Ekman, M. and Mfikinen, J. (1996). Recent postglacial rebound, gravity change and mantle flow in Fennoscandia. Geophysical Journal International, 126:229-234. Farrell, W. (1972). Deformation of the Earth by Surface Loads. Reviews of Geophysics and Space Physics, 10(3):761-797. IGC (1988). International Absolute Gravity Basestation Network (IAGBN) Absolute Gravity Observations Data Processing Standards & Station Documentation Ont. Grav. Com.- WGII: World Gravity Standards). Bulletin d'Information, Bur. Grav. Int., 63:51-57. Jung, K. (1961 ). Schwerkrafiverfahren in der angewandten Geophysik. Geest & Portig K.-G., Leipzig. Liljequist, G. and Cehac, K. (1984). Allgemeine Meteorologie. Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig. Merriam, J. (1992). Atmospheric pressure and gravity. Geophysical Journal International, 109:488-500. Nagy, D. (1966). The gravitational attraction of a right rectangular prism. Geophysics, XXXI(2):362-371. Nagy, D. (2000). The gravitational potential and its derivatives for the prism. Journal of Geodesy, 74:552-560. Neumeyer, J., Hagedoorn, J., Leitloff, J., and Schmidt, T. (2004). Gravity reduction with three-dimensional atmospheric pressure data for precise ground gravity measurements. Journal of Geodynamics, 38:437-450. Niebauer, T. M., Sasagawa, G. S., Faller, J. E., Hilt, R., and Klopping, F. (1995). A new generation of absolute gravimeters. Metrologia, 32:159-180. Scherneck, H.-G., Johansson, J. M., Koivula, H., van Dam, T., and Davis, J. L. (2003). Vertical crustal motion observed in the BIFROST project. Journal of Geodynamics, 35:425-441.
Acknowledgements Within the R&D-Programme GEOTECHNOLOGIEN this study is funded by the German Ministry of Education and Research (BMBF) and the German Research Foundation (DFG), Grant Mu 1141/3-1 and 3-2. We like to thank the University of Cologne for providing ECMWF (European Centre for Medium-Range Weather Forecasts) data in cooperation with the Deutsches Klimarechenzentrum (DKRZ, German Computing Centre .for Climate- and Earth System Research). Steven S. Pietrobon is acknowledged for providing a software for the U.S. Standard Atmosphere, 1976.
Simon, D. (2003). Modelling of the gravimetric effects induced by vertical air mass shifts. Mitteilungen des Bundesamtes ftir Kartographie und Geodfisie, Frankfurt am Main. Sun, H.-P. (1995). Static Deformation and Gravity Changes at the Earth's Surface due to the Atmospherical Pressure. Dissertation, Catholic University of Louvain, Belgium. Van Camp, M., Williams, S. D. P., and Francis, O. (2005). Uncertainty of absolute gravity measurements. Journal of Geophysical Research, 110, B05406, doi: 10.1029/2004JB003497.
been devided into local, regional and global zones. Additionaly, the attraction effect of the local zone has been deduced using 3D data, including the information of the density distribution up to a height of 50 km. The total effects have been compared with the effects, computed with the common "-3 rims -2 per hPa" rule. For the absolute gravimetry purpose, the total atmospheric signal has been considered relating to a reference standard atmosphere. For the 2003 absolute gravimetry observations in Fennoscandia • the use of atmospheric data improves the reductions by 9 rims -2 in the average (rms value, max. change 14 nms -2) compared with the reductions by the "3 rims -2 per hPa" rule; • the global effect contributes with 5 rims -2 (max. 10 rims -2 ) and should not be neglected for precise gravity determinations; • local differences between 2D and 3D effects are in the range of several rims -2 (max. 7 rims -2 ) and should be considered to keep the reduction errors as small as possible; • for applications in geodynamics, the global calculation of the atmospheric reductions with Green's functions with 2D surface data should be performed at least;
References Bolton, D. (1980). The Computation of Equivalent Potential Temperature. Monthly Weather Review, 108:10461053. Boy, J.-P., Gegout, P., and Hinderer, J. (2002). Reduction of surface gravity data from global atmospheric pressure loading. Geophysical Journal International, 149:534545.
van Dam, T. M. and Wahr, J. (1993). The atmospheric load response of the ocean determined using Geosat altimeter data. Geophysical Journal International, 113:1-16. van Dam, T. M. and Wahr, J. M. (1987). Displacements of the Earth's Surface Due to Atmospheric Loading: Effects on Gravity and Baseline Measurements. Journal of Geophysical Research, 92(B2): 1281-1286. Virtanen, H. (2004). Loading effects in Mets/ihovi from the atmosphere and the Baltic Sea. Journal of Geodynamics, 38:407-422.
Chapter 68
Tilt Observations Around the KTB-Site / Germany: Monitoring and Modelling of Fluid Induced Deformation of the Upper Crust of the Earth T. Jahr, G. Jentzsch, H. Letz, A. Gebauer Institute of Geosciences, Department of Applied Geophysics, University of Jena, Burgweg 11, D-07749 Jena, Germany Abstract. At the German Continental Deep Drilling site (KTB) the pilot borehole of a depth of 4000 m was used to inject water with a medium rate of 180 litres/minute over a period of one year. To monitor the expected surface deformation five borehole tiltmeters of the ASKANIA type Gbpl0 were installed in the surrounding area of the KTB location in mid 2003. The deformation was detected at kilometre scale, together with the observations of induced seismicity in the area. We observed elastic as well as inelastic responses: changes of the rheologic properties due to pore pressure increase caused changes in the tidal parameters. First we quantified the expected additional drift for different injection scenarios at each tiltmeter site by numerical modelling. It could be demonstrated that for long term injection phases of up to four months a maximum tilt effect of about 40 nrad is modelled, which should be detectable. We expected changes of the drift curve, slow variations correlating with the injection rate as well as with changes of the rate. First results of the monitoring are presented: they reveal a slight increase of the tidal parameters (main tidal constituent O1 and M2, north-south component) and drifts associated with the long-term injection. Comprehensive numerical modelling using the Finite Element software ABAQUS is in preparation.
Keywords. Surface tilt measurements, fluid injection, KTB drill hole, ASKANIA tiltmeter, tidal parameters, upper Earth crust
1 Introduction A tiltmeter array, consisting of five high resolution borehole tiltmeters of the ASKANIA type Gbp 10 was installed in the surrounding area of the KTB location in mid 2003 (Fig. 1). There, an injection experiment of water started in June, 2004, to be completed in May 2005. The injection had an average rate of 180 litres/minute into the KTB pilot
borehole (4000 meters deep). The aim of the research project was to observe the induced deformation of the upper crust at kilometre scale. Together with the observations of tilt changes, induced seismicity in the area was monitored by a local seismograph network. Numerical modelling revealed that a constant injection of about 180 1/min would cause a deformation of about 40 nrad within 4 months.
"Fig. 1 The area around the KTB-site: The five tiltmeter stations are connected via WLAN (white lines) to transmit the data online. Data transmission from Mittelberg is routed via Stockau. Depths of boreholes are given in meters. The depths of the boreholes were chosen as such that the tiltmeters could be installed in hard bed rock well below the ground water table. Since it is necessary to eliminate locally induced interference, (e.g. arising from groundwater variations) a second borehole was drilled at all stations to observe ground water changes, especially pore pressure
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T. Jahr. G. Jentzsch • H. Letz. A. Gebauer
changes. All groundwater monitoring boreholes were fitted with tube packers (Jahr et al., 2005a). In this way the instantaneous changes of pore pressure could be monitored to avoid damping and time delay due to water exchange between the rock and the well.
Thus, the quality of the raw tiltmeter data and the remotely performed instrument calibration can be checked every day. In Fig. 3 the data are plotted. We also installed a meteorological station at KTB to monitor air pressure and precipitation.
2 I n s t r u m e n t s and Data
The locations of the tiltmeter stations are centred around KTB at distances of 1.6 to 3.2 km. The depths of the instruments range between 24.5 and 45.5 m depending on the thickness of the sediment cover, which usually is less than 20m. The tiltmeters are equipped with a 3-D geophone set in order to complement the local seismic network (Fig. 2). The groundwater level is monitored using Van Essen's DIVER sensor with a sample rate of 5 minutes. Tilt, groundwater level and seismic data are downloaded via a wireless LAN (WLAN) to KTB and further via Internet to the GeoForschungsZentrum (GFZ)-Potsdam and the Institute of Geosciences, Jena. The project is carried out in close cooperation with seismologists of the GFZ-Potsdam who operate the seismic network and the WLAN. EW - component (in msec x 10)
2000 ,ooo°
-1000 -2000 -3000
1~
Fig. 2. Top of the ASKANIA Borehole Tiltmeter GBP10 (instrument resolution < 0.2 msec) fitted with a 3-component geophone. To measure the azimuth of the tiltmeter against true North by optical methods in the borehole, two sets of Light Emitting Diodes are mounted above the geophones.
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Fig. 3. After pre-processing the time series of the tiltmeter observations the drift behaviour can be visualised. The data is displayed in milliseconds of arc times 10. The maximum tidal amplitudes are about 100 nrad (about 20 msec). Start of injection (June 04) and completion (May 05) is indicated. Data gaps at Eiglasdorf and Berg were caused by malfunctioning data loggers. In Stockau the tiltmeter failed in July 2004. Mittelberg shows the lowest drift rates, while in Ptillersreuth non-linear parts were caused by heavy precipitation.
Chapter 68 • Tilt Observations around the KTB-Site / Germany: Monitoring and Modelling of Fluid Induced Deformation of the Upper Crust of the Earth
The tilt data presented in Fig. 3 consist of long and uninterrupted sections. The comparison of the tidal amplitudes reveals the different long-term drifts at each station. There are also different reactions to precipitation and local ground water changes. Heavy rainfall usually results in strong deviations with a relaxation of several days.
3 First Results The Data are analysed for tidal parameters and for the long-term drift. The correction of the data for local meteorological effects (precipitation, air pressure) and ground water is in progress.
are quite similar, and the station Stockau as well, but with an additional strong (linear?) drift. The variations seen at Pallersreuth may be caused by local hydrological variations.
Hodograms Since the tiltmeters used are vertical pendulums with two degrees of freedom we can plot the movement of the tip of the pendulum over ground. The resulting curves contain the tidal movements, especially the 14-days beating period of M2 and $2, and the long-term drift. The stronger the drift the smaller the tidal oscillations in the figure, serving as a reference.
Tidal analyses The tidal analyses should provide 3,-values (the measured amplitudes of the tidal waves divided by the theoretical amplitudes) to be in the order of 0.7, but they vary quite a bit over the whole array, although the distances are less than 5 km (Tab. 1). There is also no correlation between the values for the tidal waves O1 and M2. We assume that local geological inhomogeneities and topography are the reason. The error bars are also quite big, which points to temporal variations of the values. On the basis of successive 2-monthly intervals (shifted by one month) this temporal behaviour is analysed: In Fig. 4 the parameter function for O1 (North-South) is given as a sample. Each point stands for the value for each interval. As can be seen, the y-values start to increase slightly at the beginning of the injection. Although the error bars are too big to take this result as significant for the individual stations, the common behaviour for all stations justifies this interpretation of an increasing value for tidal parameters (which is observed for M2 as well, but again only for the North-South-direction).
In Fig. 6 the hodograms for all five stations are given. The first figure top left shows the main geological strikes and boundaries. Two important times are also noted: The start of the injection and the time when the main borehole became artesian. It can be clearly seen that the drift directions correlate with the geological strikes; and the changes in drift correlate with the times mentioned. Also the variations of the injection rate seem to be visible, but this has still to be verified.
Table 1. Tidal analyses for O 1 and M2: calculated are the 7values for NS and EW components of the tiltmeter array at the KTB using the whole recording interval.
O1
Tiltmeter station
NS
EW
NS
EW
BER
0.696 +.138
0.653 +.055
0.758 +.028
0.790 +.018
EIK
0.595 +.085
0.707 +.166
0.656 +.ll7
0.739 +.113
STO
0.227 +.193
0.646 +.135
0.593 +.040
0.856 +.033
MIT
0.545 +.027
0.666 +.020
0.737 +.014
0.793 +.010
PUE
0.556 +.050
0.474 +.110
0.504 +.022
0.547 +.044
Long-term drifts' We can also present first results of the separation of the long-term drift. Although these plots are still preliminary because local effects have to be analysed and corrected, we can see common trends. In Fig. 5 these drifts towards the injection centre are given for all stations except Eiglasdorf, where we suspect strong local effects have masked the expected injection signal and have to be separated first. The curves for stations Berg and Mittelberg
M2
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T.Jahr. G.Jentzsch • H. Letz.A. Gebauer
Parameter Variation KTB array north-south, 01
0,8
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I
I
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I
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I
I
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Fig. 4. Parameter variation of O 1, north-south component: Since the beginning of the injection the parameter increased by about 1.2% per month as indicated by error weighted linear regression.
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~
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Nov-03 Jan-04 Mar-04 May-04 Jul-04 Sep-04 Oct-04 Dec-04 Feb-05 Fig. 5. Long-term drift of four stations: Shown is the drift in radial direction (as seen from the KTB) in msec (1 msec ~ 5 nrad). The data of the station in Eiglasdorf is superimposed by a strong additional drift and thus needs further treatment before being included here. A negative trend correlates with a movement of the bottom of the borehole in direction to the KTB and vice versa.
Chapter68 • Tilt Observationsaroundthe KTB-Site/ Germany:Monitoring and Modellingof Fluid InducedDeformationof the UpperCrustof the Earth
5~ begin of injection (9 June 2004)
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s
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Fig. 6. Hodograms: Movement of the tip of the tiltmeter at all stations; note the amplitude of the fortnightly beating period of the tides (M2 and $2 constituents) pointing to the amplitude ratios of drift and tides.
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T. Jahr. G. Jentzsch • H. Letz. A. Gebauer 4
Discussion
We found three different injection induced effects: Variations of tidal parameters, long-term drifts, and direct elastic tilts. These signals now need numerical interpretation. The tiltmeters we used at KTB are the most sensitive instruments available. The high resolution is accompanied by high sensitivity to many external sources, the first being the problem of adequate coupling to the rock. Thus, reliable measurements can only be achieved by carefully drilling the borehole and cementing the casing. In previous projects we observed tilt caused by the load of an artificial lake (Jentzsch and Kog, 1997). Our work in Finland (Weise et al., 1999) provided valuable information not only about the regional representation of the monitored tilt but also about the effects of ground water (pore pressure) changes even in granite (see KiJmpel et al., 1988). Westerhaus and Zschau (2001) reported that no earthquake precursors could be observed because of influences of the local geology and the coupling of the instrument casing. Tilt measurements are very sensitive to meteorological influences like air pressure and precipitation as well as ground water changes. Therefore, these disturbances have to be corrected prior to interpretation (Braitenberg and Zadro, 2001; Ishii et al., 2001). The injection tests at KTB were accompanied by induced seismicity (Zoback and Harjes, 1997). Here, we have a combination of surface deformation and seismicity which was already discussed by Jentzsch et al. (2001). The interpretation with regard to physical properties of the rock and fluid induced processes (Fujimori et al., 2001) will be assessed by numerical modelling. We have already modelled the expected deformation applying the program POEL (by R. Wang), but for detailed modelling of the geologic structure we plan to use the Finite Element software ABAQUS. The first model is already being developed (Jahr et al., 2005b). 5
Acknowledgements
We thank the KTB-Operational Support Group for their help. Special thanks go to G. Asch and J. K u m m e r o w (GFZ) for the organisation of the data collection and transfer. R.J. Wang provided the modelling program POEL which helped to estimate the expected deformation. The local landowners
permitted to drill the boreholes on their properties and accepted much discomfort caused by the drilling work. This is gratefully acknowledged. We thank the G e r m a n R e s e a r c h F o u n d a t i o n (DFG) for financial support, grant no. JA-542/1 11,6. For helpful corrections and comments we thank Dr. L. Wallace and an anonymous reviewer. References
Baitenberg, C., and M. Zadro, (2001). Time Series Modelling of the Hydrologic Signal in Geodetic Measurements. Proc. 14th International Symposium on Earth Tides, Special Issue of the Journal of the Geodetic Soc. of Japan, Vol.
47/1, pp. 95-100. Fujimori, K., H. Ishii, A. Mukai, S. Nakao, S. Matsumoto, and Y. Hirata, (2001). Strain and tilt changes measured during a water injection experiment at the Nojima Fault zone, Japan. The lsland Arc, 10, pp. 228-234. Ishii, H., G. Jentzsch, S. Graupner, S. Nakao, M. Ramatschi, and A. Weise, (2001). Observatory Nokogiriyama / Japan: Comparison of different tiltmeters. Proc. 14~h International Symposium on Earth Tides, Special Issue of the Journal of the Geodetic Soc. of Japan, 47/1,155-160.
Jahr, T., G. Jentzsch, H. Letz, H., and M. Sauter (2005a) Fluid injection and surface deformation at the KTB location: Modelling of expected tilt effects. Geofluids 5, pp. 20-27. Jahr, T., Letz, H. & Jentzsch, G., (2005b) The ASKANIA borehole tiltmeter array at the KTB location / Germany.Journ. of Geodynamics (accepted). Jentzsch, G., and S. KoB, (1997). Interpretation of longperiod tilt records at Blfi Sjo, Southern Norway, with respect to the variations of the lake level. Phys. Chem. Earth, 22, pp. 25-31. Jentzsch, G., P. Malischewsky, M. Zadro, C. Braitenberg, A. Latynina, E. Bojarsky, T. Verbytzkyy, A. Tikhomirov, and A. Kurskeev, (2001). Relations between different geodynamic parameters and seismicity in areas of high and low seismic hazards. Proc. 14 th International Symposium on Earth Tides, Special Issue of the Journal of the Geodetic Soc. of Japan, 47/1, 82 - 87.
Kfimpel, H.-J., J.A. Peters, and D.R. Bower, (1988). Nontidal tilt and water table variations in a seismically active region in Quebec, Canada, Tectonophysics, 152, pp. 253-265. Weise, A., G. Jentzsch, A. Kiviniemi, and J. KfifiriNnen, (1999). Comparison of long-period tilt measurements: Results from the two clinometric stations Metsfihovi and Lohja, Finland. J. of Geodynarnics, 27, pp. 237-257. Westerhaus, M., and J. Zschau, (2001). No clear evidence for temporal variations of tidal tilt prior to the 1999 Izmit and DOzce earthquakes in NW-Anatolia. Proc. 14th International Symposium on Earth Tides, Special Issue of the Journal of the Geodetic Soc. of Japan, 47/1,448- 455.
Zoback, M.D., and H.-P. Harjes, (1997). Injection-induced earthquakes and crustal stress at 9 km depth at the KTB deep drilling site, Germany. J. Geophys. Res., 102 (B8), pp. 18477-18492.
Chapter 69
Understanding Time-variable Gravity due to Core Dynamical Processes with Numerical Geodynamo Modeling W. Jiang Joint Center for Earth Systems Technology, University of Maryland at Baltimore County 1000 Hilltop Circle, Baltimore, MD 21229, USA W. Kuang, B. Chao Space Geodesy Laboratory, NASA Goddard Space Flight Center, Greenbelt, MD 20771 M. Fang Dept. Earth and Space Sciences, Massachusetts Institute of Technology, Cambridge, MA 02136 C. Cox Raytheon at Space Geodesy Laboratory, NASA Goddard Space Flight Center
Abstract. On decadal time scales, there are three major physical processes in the Earth's outer core that contribute to gravity variations: (i) the mass redistribution due to advection in the outer core, (ii) the mantle deformation in response to (i), and (iii) the (core) pressure loading on the core-mantle boundary. Except the last one, they cannot be evaluated from surface observations. In this paper we use MoSST core dynamics model and PREM model to understand the gravity anomalies from the three processes. Our numerical results show that, the gravity anomalies are comparable in magnitude, though that from the process (i) is in general the strongest. The gravity anomalies from the first two processes tend to offset each other ("mantle shielding"). Consequently the pressure loading effect contributes more to axisymmetric part of the net gravity variation, while the net effect from the first two processes is more important to non axisymmetric components.
Keywords.
Gravity anomalies, core convection, numerical model.
1 Introduction For more than two decades, space geodetic techniques have observed quite accurately the synoptic or low-degree temporal variations of the Earth's gravity field (AGU 1993; Cox and Chao 2002). These minute changes (relative to the mean gravity of the Earth) arise from various geophysical
processes on the surface and in the deep interior of the Earth that give rise to mass redistribution over space and time (e.g., Chao et al., 2000). While the major effort in studying the timevariable gravity (TVG) has been pertaining to the processes that happen in the surface "geophysical fluids" (e.g. atmosphere, hydrosphere and cryosphere) and the post-glacial rebound in the mantle, only limited attempts have been made on possible contributions from physical processes in the Earth's fluid outer core, in particular those on decadal time scales. Fang et al (1996), Dumburry and Bloxham (2004) attempted to understand the TVG from non-hydrostatic pressure acting on the core-mantle boundary (CMB). Greff et al (2004) discussed the gravity variation and surface deformation due to core flow. Kuang et al (2004) used their numerical geodynamo solutions to investigate gravity variation due to advection of core fluid density anomalies. However, those studies on the gravity variations from core are incomplete in many aspects. The incompleteness can be identified simply by examining possible geophysical processes in the core that contribute to the decadal TVG. There are at least three major geophysical processes for our attention. The first is the mass redistribution inside the Earth's fluid outer core due to convection (Kuang et al 2004). We denote by 6q~p the corresponding gravity anomaly. From surface geomagnetic observations, and likely dynamic balances in the outer core, one could estimate that 6q~p can be observed at the surface of the Earth (Kuang et a12004). The second process is the loading on the CMB by the non-hydrostatic pressure in the core. This
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W. Jiang • W. Kuang • B. Chao. M. Fang. C. Cox
loading results in mantle deformation, and then gravity changes over time and space (Fang et al 1996, Dumburry and Bloxham 2004). We denote by 601, the gravity anomaly from this CMB pressure loading. The third process is the response of the mantle to gravity changes inside the fluid outer core 6Op. This process is similar to the solid tides resulting from external gravity field changes (e.g. orbital motion of the moon). We denote by 6q)l the corresponding gravity anomaly. Thus, the results from previous studies, even if justified by themselves, are at best partial as far as a complete comprehension on the net, core-induced TVG & b - cT~p +cT~j +cT~p.
(1)
Surface geomagnetic observations can shed light only on 6q5l,, even though such information is incomplete. The observations could provide the geomagnetic field itself and its secular variation (the time derivative). With these data, the "frozenflux" and the "tangential geostrophy" assumptions, one could obtain part of the pressure p on the CMB (e.g. Jault et al 1988), and therefore 6 ~ . However, there is no surface observation on the density anomalies inside the core, and thus we are unable to determine 6~ and 6~,. Therefore, we shall use numerical model to understand the properties of the three gravity anomalies, in particular to find possible relationships among them. The model used in our study is the MoSST core dynamics model. For the details of the model, we refer the reader to Kuang and Bloxham (1999), Kuang and Chao (2002). Though it targets a (mathematically) simplified Earth's core, the numerical model is self-consistent; and all physical variables describing the outer core are determined dynamically. Therefore we hope that the dynamically consistent dynamo model solutions could shed light on the properties of the three gravity anomalies, and thus their net contribution to the TVG observable at the Earth's surface. This paper is organized as follows: the mathematical formulations are described in Section 2. In Section 3, numerical results with selected parameters are presented. Discussion on possible geophysical applications is given in Section 4.
2 Mathematical Model The numerical model used for this study is the MoSST core dynamics model in NASA Goddard Space Flight Center. For the details of the model,
we refer the readers to Kuang and Bloxham (1999), and Kuang and Chao (2003). In this paper, we simply summarize part of the model relevant to our study. In the MoSST core dynamics model, the outer core is approximated to leading order as a spherical shell. The mean radii of the inner core boundary (ICB) and the CMB are the same as those of the Earth inferred from seismic studies. The coordinate used in the model is defined relative to the "mean", spherically symmetric model Earth: it is fixed to the mantle, and centered at the geometric center (also the mass center). In this coordinate system (or reference frame), the mass center of the model Earth will change due to core convection, which results in mass redistribution in the core, and the mantle deformation discussed in this paper. However, the change of the mass center depends also on the inner core deformation due to loadings on the ICB. However, we do not discuss this phenomena in this paper. All physical quantities are scaled as follows: the length scale r ~ is the mean radius of the CMB; the time scale r is the magnetic diffusive time 2
r - rcmh lrl,
where q is the magnetic diffusivity of the core fluid; the magnetic field scale B0is given by B 0 - ~J2P0f2/ar/ , where P0 is the mean density of the core fluid, f2is the mean rotation rate of the Earth, and /1 is the magnetic permeability in the core (almost identical to that of the free space). In the model, the density anomaly kp is assumed arising from thermal effects
Ap -- --6g T (r -- T O ), /90
where c~r is the thermal expansion coefficient of the core. The temperature perturbation is scaled by r , , . h T (the ambient temperature gradient at the ICB). With these scaling rules, the momentum balance in the fluid core is given by the following nondimensional equation, Vp - J x B - 2 x v - R,h O r o
- - + ~ x V
)
(2) + EV2v,
where B is the magnetic field, a ( - V x B ) is the current density, v is the velocity, m ( - V x v) is the
Chapter 69 • Understanding Time-Variable Gravity Due to Core Dynamical Processes with Numerical Geodynamo Modelling
vorticity, ® is the density perturbation, p is the modified pressure, ~ is the unit vector along the z-axis (the mean rotation axis of the Earth), r is the position vector, R,, is the Rayleigh number, R ° is the magnetic Rossby number and E is the Ekman number. The equation (2) is not in the traditional form because we intend to use it to solve the pressure p. Taking the divergence of Equation (2) and applying the continuity condition V. v = 0, one obtains, V219 - V .
(J x B - k x v + R,hOr
(3)
-RoO~X v ) . This equation is solved together with the boundary conditions
-
f.[JxB-2xv-R,hOr
m
m
h/ '
-
(7)
where h~m are the spherical coefficients of the terms on the right hand side of (4). The pressure p in our model is non-dimensional, scaled by p0 = 2pX2~7. Therefore the dimensional pressure used in Fang et al (2005) is given by p & . Furthermore, the dimensional gravity potential is scaled by ~0-4aGP0ro,2,b (Kuang et al, 2004). Applying the relevant scales to the formulation in Fang et al (2005), we can obtain the following nondimensional gravity anomaly:
6oo
8 --pOr
Pt
-
Or
(4)
- -dP0~< _ _, 2tk;+ 1 p'~(" -~)~" (o, 0,) + c.c. (8) L
- Z~7~(o,d+c.c
- RoO~ x v + EV2v~
Ol~,~ax ,rn
with
+
_(:x&)
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--
=(:xs) =(B) _ Kl=(axv)=(B)
Clr n
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....................................................................................................
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03101101
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Time, m o n t h s
Figure 3" Monthly mean water storage variations averaged over the upper Zambezi shown in figure 2 in [mm/month] inferred from the LEW regional hydrological model (the sampling rate is 1 month).
491
492
R. Klees • E. A. Zapreeva • H. C. Winsemius • H. H. G. Savenije
Figure 4 : Delineated watersheds for the estimation of the water storage over the surrounding areas of the upper Zambezi. The total size of the watersheds is about 2.6 • 106 km2, i.e. much larger than the upper Zambezi sub-catchment.
This output of the LEW model is used to assess the contribution of the surrounding areas to the GRACE estimates of the storage variations averaged over the upper Zambezi sub-catchment.
phase is not changed when one removes these coefficients from the analysis or when they are replaced by the corresponding values from the analysis of EOP and SLR data. The estimated amplitudes of the computed storage variations strongly depend on the correlation length of the spatial filter. The larger the correlation length, the smaller the estimated amplitude of the monthly mean water storage variations (cf. figure 6). For a Gaussian filter, we find that the amplitude changes with about 15-20% if the correlation length changes with 500 km. By visual inspection, we finally choose a filter with correlation length 1000 km. If a smaller correlation length is chosen, we observe strong ground-track patterns in the estimated storage variations. There is no sound methodology of how to choose the optimal correlation length. An alternative to the approach we followed is to determine the correlation length such that on a global scale the amplitudes of GRACE estimates fits in a least-squares sense the output of a global hydrological model. This results in a slightly lower correlation length of about 800 km. Unfortunately, whether this is a better choice has to be left open.
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Figure 5 : Monthly mean water storage variations for the areas shown in figure 4 in [mm/month] inferred from the LEW regional hydrological model (the sampling rate is 1 month).
17 monthly GRACE gravity models, covering the period from April 2002 until June 2004, are used to estimate the mean water storage variations over the upper Zambezi sub-catchment. The spherical harmonic coefficients of degree 0 and 1 and the degree 2 and order 0 coefficient have always been removed from the solution. Chen et al. (2005) have shown that these coefficients do not significantly contribute to the amplitude of the monthly water storage variation averaged over the Zambezi fiver basin. They also showed that the
-~'1~Ol
o~o~o,
o~o1~o~ o~o~o,
o,~o1~ol o,~o~o~
Time, m o n t h s
Figure 6: GRACE monthly water storage variations averaged over the upper Zambezi sub-catchment in [ram/month]. A Gauss filter with various correlation lengths has been used: 600 km (dashed), 1000 km (solid), 1500 km (dashdot). AvailableGRACE solutions are shown by squares; values in between have been obtained by spline interpolation.
GRACE data are not continues (see figure 6, squares). There are about I to 3 gaps per year. Therefore the gaps are filled by applying the cubic spline interpolation to the GRACE solutions in terms of water storage. The obtained analytic function for storage is then used to compute GRACE estimates of the monthly mean water storage variation ASGRACE.
Chapter
71
• Monthly
Water Storage Variations
Mean
Combination of GRACE and a Regional Hydrological Model: Application
by the
The results are shown in figure 7.
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procedure as for the computation of the G R A C E estimates is applied using the same Gaussian filter (correlation length 1000 km).
8O
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to the
04/07/01
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::::::::::::::::::::: ] =
i" .................................
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Time, months
Figure 7: Monthly mean water storage variations in [ram/month] over the upper Zambezi sub-catchment. Dotted line: GRACE estimates if a Gauss filter with correlation length 1000 km is used. Dashed-dotted line: the unfiltered monthly mean water storage variations of the LEW hydrological model. Dashed line: the monthly mean water storage variations of LEW if a Gauss filter with correlation length 1000 km is used. The sampling rate of LEW is 1 month; available GRACE solutions are shown by squares.
The seasonal variation is clearly visible and in agreement with the output of the L E W regional hydrological model. The amplitudes, however, differ significantly. The amplitudes of the unfiltered output of the L E W model are much larger than the G R A C E amplitudes. Vice versa, if the output of the L E W model is filtered using the same filter as applied to GRACE, the L E W amplitudes are much smaller. In both cases, the differences can reach values of 3 0 - 4 0 mrrdmonth. There are several potential contributors to these differences: (i) the L E W regional hydrological model is not calibrated properly; (ii) the G R A C E estimates are not correct; (iii) significant contribution of storage variations in neighbored areas (see section 4.2); (iv) residual contributions of atmospheric and ocean mass variations (see section 4.3).
4.2 Quantification of the hydrological contribution of the surrounding areas To quantify the hydrological contribution of the surrounding areas to the G R A C E estimates of the monthly mean water storage variations over the upper Zambezi sub-catchment, the L E W regional hydrological model is used. The output of the model is expanded into spherical harmonics. Then the same
0~J(~1/01
02/07/01
03/01101
03/07101
04101101
04/07101
Time, months
Figure 8: Monthly mean water storage variations in [mm/month] over the upper Zambezi sub-catchment. Dashed line: the contribution of water storage variations over surrounding areas to GRACE monthly mean water storage variations of the upper Zambezi. Solid line: GRACE estimates corrected for the influence of the surrounding areas. Dashed-dotted line: unfiltered monthly mean water storage variations from the LEW model. Dotted line: filtered monthly mean water storage variations from the LEW model. Available GRACE data are indicated by squares. The sampling rate of the LEW model output is 1 month.
Finally, the water storage variation averaged over the upper Zambezi sub-cathment is estimated. The estimates are shown in figure 8. The influence of the surrounding areas on the monthly G R A C E estimates has amplitudes up to 20 mm/month, which is quite significant. W h e n correcting G R A C E estimates for this influence, the differences between the filtered L E W m o d e l estimates and the G R A C E estimates b e c o m e smaller. Nevertheless, the amplitudes still differ significantly: 6 - 8 mrrdmonth from L E W compared with about 20 - 30 m m from GRACE. One possible explanation for this difference is the limited size of the study area (0.5-106 k m 2) in combination with the need to filter G R A C E monthly models due to highfrequency noise. This results in a significant loss of energy for the L E W model estimates. Therefore, we compared G R A C E estimates with (i) the L E W m o d e l and (ii) the N O A A / C P C global hydrological model for the area covering the upper Zambezi subcatchment a n d the surrounding areas. The size of this area is about 3 . 106 k m 2, i.e. six times larger than the area of the upper Zambezi sub-catchment. The results are shown in figure 9. Apparently, the fit between the G R A C E estimates and the output of the regional L E W model is much better than for the upper
494
R. K l e e s
• E. A . Z a p r e e v a
• H. C. Winsemius
• H. H. G.
Savenije
Zambezi sub-catchment (figure 8). They also agree well with the global NOAA/CPC model, although the latter gives slightly higher amplitudes compared with GRACE and LEW.
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F i g u r e 10: Total contribution of atmosphere (dashed) and oceans (dashdot) to the GRACE monthly mean storage variations over the upper Zambezi when assuming that GRACE measurements are not corrected for atmosphere and oceans. GRACE monthly mean storage variations are shown for comparison (solid line). Available GRACE solutions are indicated by squares.
04107101
F i g u r e 9: Mean monthly water storage variations over the upper Zambezi sub-catchment and the surrounding area after smoothing with a Gauss filter with 1000 km correlation length. The area is about 3- 106 km 2. Black squares: available filtered GRACE solutions. Solid line: interpolated filtered GRACE estimates. Dashed line: filtered NOAA/CPC model. Dash-dotted line: filtered L E W model.
4.3 Quantification of the total contribution of atmosphere and oceans Monthly mass variations in the atmosphere and the oceans provided by GFZ are used to quantify the total contribution of the atmosphere and oceans to the GRACE monthly mean water storage variations over the upper Zambezi (i.e. assuming that no corrections for atmosphere and oceans are applied to the GRACE measurements). The same standard averaging procedure as described above is used. The results are shown in figure 10.
The contribution of the atmosphere varies between - 2 0 mm/month and +18 mm/month. The contribution of the oceans varies between - 1 0 mm/month and +18 mm/month. The effects are quite significant. Therefore, it is necessary to investigate the amplitude of the residual atmospheric and oceanic effects for the upper Zambezi subcatchment. This is an on-going activity, and the results will be reported elsewhere.
5
S u m m a r y a n d conclusions
The LEW hydrological model has been applied for the first time to the upper Zambezi sub-catchment. The storage variations consist of changes in the unsaturated zone, the saturated zone, the wetlands and the channel storage. LEW regional hydrological model estimates and GRACE estimates of the monthly water storage variations over the upper Zambezi differ significantly in terms of amplitude. Maximum differences are about 20 mm/month. The contribution of the hydrological signal over surrounding areas to the GRACE monthly mean storage variations over the upper Zambezi has amplitudes up to 20 mm/month. Therefore, when storage variations over the upper Zambezi sub-catchment are studied, it is necessary, to accurately model the contribution of the surrounding areas. Filtered LEW estimates of the monthly storage variations over the upper Zambezi sub-catchment are a factor of 10 smaller than the unfiltered estimates. This loss of energy is due to the fact that the area of the upper Zambezi sub-catchment is relatively small (about 0.5- 106 km 2) compared with the total surface of the Earth. GRACE monthly mean storage variations after correction for the contribution of the surrounding areas have maximum amplitudes of about 30 mm/month. This is much smaller than the amplitudes of the unfiltered LEW model output (maximum amplitudes of 80 mm/month, but significantly larger
Chapter 71 • Monthly Mean Water Storage Variations by the Combination of GRACEand a Regional HydrologicalModel:Application to the Zambezi River
than the amplitudes of the filtered LEW model output (maximum amplitudes of about 8 mm/month). There is no definite explanation for the discrepancies between GRACE estimates and (filtered) LEW model output. The limited size of the upper Zambezi sub-catchment and the need to smooth GRACE estimates due to high-frequency errors may be the most significant contributor. This is supported by the results shown in figure 9. The contribution of the atmosphere and oceans to the GRACE monthly mean storage variations over the upper Zambezi can reach up to 20 mm/month if GRACE measurements are not corrected for. These large amplitudes motivate further investigations into the residual atmospheric and oceanic contribution to the monthly GRACE estimates of mean water storage variations over the upper Zambezi. The contribution of the errors caused by the application of spherical harmonic expansion has also been investigated. The same methodology (see paragraph 2) is used to compute monthly mean water storage variation averaged over the upper Zambezi by using the Gauss-Legendre method of numerical integration as alternative to spherical harmonic expansions. The contribution of the Gibbs phenomenon when using spherical harmonics is found to be below 2 %. Therefore, this artefact cannot be the source of the discrepancy in the amplitudes between GRACE and LEW model. The estimated amplitudes of the monthly mean water storage variations as inferred from GRACE strongly depend on the correlation length of the filter function, which is used to reduce the influence of GRACE errors. The proper choice needs further investigations. The quantification of the uncertainties in GRACE estimates of the monthly mean water storage variations as well as in the hydrological model output is an important subject for future studies and a pre-requisite for a combined solution using hydrological data and models and GRACE data.
Acknowledgments
The project is supported by the Dutch Organization for Scientific Research (NWO) and the Water Research Center Delft (WRCD). The support is gratefully acknowledged. References
Aerts J and Bouwer L (2003) STREAM manual. Amsterdam. The Netherlands
Beven KJ and Binley AM (1992) The future of distributed models: model calibration and uncertainty prediction. Hydrol. Proc. 6:279-298 Beven KJ and Freer J (2001 ) Equifinality, data assimilation and uncertainty estimation in mechanistic modeling of complex environmental systems using the GLUE methodology. Journal of Hydrology 249:11-29 Chen JL, Rodell Matt, Wilson CR, and Famigletti JS (2005) Low degree spherical harmonic influences on Gravity Recovery and Climate Experiment (GRACE) water storage estimates. Geophys. Res. Let.32, L14405 Herman A, Kumar VB Arkin PA and Kousky JV (1997) Objectively determined 10-day African rainfall estimates created for famine early warning systems. Int. J. Remote sensing. 18(10): 21472159 Jekeli C (1981) Alternative methods to smooth the Earth's Gravity field. Report No. 327, Department of Civil and Environmental Engineering and Geodetic Science,The Ohio Sate University, Columbus, Ohio Nash JE and Sutcliffe JV (1970) River flow forecasting through conceptual models, Part 1: A discussion of principles. J. of Hydrol. 10:282-290 Seo KW and Wilson CR (2005) Simulated estimation of hydrological loads from GRACE. Journal of Geodesy 78:442-456 Swenson S and Wahr J (2002) Methods of inferring regional surface mass anomalies from Gravity Recovery and Climate Experiment (GRACE) measurements of time-variable gravity. J. of Geop. Res. 107(B9), 2193:3.1-3.13 Swenson S, Wahr J and Milly P (2003) Estimated accuracies of regional water storage variations inferred from the Gravity Recovery and Climate Experiment (GRACE). Water Resources Research 39(8), 1223:11.1-11.13 Wahr J, Molenaar M and Bryan F (1998) Time variability of the Earth's gravity field: Hydrological and oceanic effects and their possible detection using GRACE. J. ofGeop. Res. 103(30): 205-229 Winsemius HC, Savenije HHG, Gerrits AMJ, Zapreeva EA and Klees R (submitted, 2005) Comparison of two model approaches in the Zambezi river basin with regard to model reliability and identifiability. Hydrology and Earth System Sciences Discussions Xie P and Arkin PA (1997) Global Precipitation: A 17-Year Monthly Analysis Based on Gauge Observations, Satellite Estimates, and Numerical Model Outputs. Bull. Amer. Meteor. Soc. 78:2539-2558
495
Chapter 72
The use of smooth piecewise algebraic approximation in the determination of vertical crustal m o v e m e n t s in Eastern C a n a d a Azadeh Koohzare, Petr Vanfeek, Marcelo Santos Department of Geodesy and Geomatics Engineering, University of New Brunswick, P.O.Box 4400. Fredericton NB., Canada E3B 5A3
Abstract. The objective of this study is to compile
Keywords. Least Square Approximation, crustal
a physically meaningful map of vertical crustal movements (VCM) for Eastern Canada. Average vertical velocities over the past century are determined by repeated precise levelling and monthly mean sea level observations from 17 tide gauges. The spatial vertical velocities may be mathematically expressed in any number of ways.
movements, Adjustment
In this study, the uplift rate is calculated using smooth piecewise algebraic polynomial approximation. The mathematical model of the approximation for the geodetic data is given. We show how a vertical velocity surface is approximated using piecewise algebraic polynomials and what conditions should be satisfied to guarantee the smoothness of the surface. First, we divide Eastern Canada into zones. The vertical movement is represented by a different polynomial surface in each zone. The polynomials are joined together at nodal points along the border of adjacent zones in such a way that a certain degree of smoothness (differentiability) of the resulting function is guaranteed. This study shows that piecewise polynomial surfaces can represent the available data in a unified map. The pattern of a northwest to southeast gradient of crustal movements is consistent with the existing Glacial Isostatic Adjustment (GIA) models. Present-day radial displacement predictions due to postglacial rebound over North America computed using VM2 Earth model and ICE-4G adopted ice history show a zero line (hinge line) very similar to ours along the St. Lawrence River. The main advantage of the presented technique is its capability of accommodating in one model, different kinds of information when the re-levelled segments are scattered not only in time but also in space. Piecewise approximations make it easier to get the physically meaningful details of the map, without increasing the degree of polynomials.
geodynamics,
Glacial
Isostatic
1 Introduction It has been recognized for several decades that the determination of a Vertical Crustal Motion model is of importance in geosciences. In geophysics, for example, it is of primary interest in the study of the theology of the mantle and lithosphere which is crucial in understanding geodynamical processes. In geodesy, they are important in the definition of vertical datum which is in turn, required in many application areas such as navigation, mapping, and environmental studies. The first VCM model in Canada was compiled by Vanf6ek and Christodulides (1974) using scattered geodetic relevelled segments and the first study which covered the whole of Canada was carried out by Vanf6ek and Nagy (1981) using precise re-leveled segments and tide gauge records. The country was divided into regions and polynomial surfaces of order 2, 3 and 4 were calculated by the method of least squares for each region to obtain representations of the vertical movements. A considerably larger database has been gathered since then, and this, together with additional insight into the nature of the data, led to the recompilation of the map of vertical crustal movement of Canada by Carrera et al. (1994) in which a vertical polynomial was fitted to the data. In order to infer a physically meaningful VCM, it is necessary to combine the geodetic and geophysical data, theories, methodologies and techniques that are somehow linked together. Hence, finding the best approach to reconcile geodetic data with geological phenomena is required. In North America, the most significant geophysical process that has an evident effect on the shape of the viscoelastic earth is postglacial rebound
Chapter 72 • The Use of Smooth Piecewise Algebraic Approximation in the Determination of Vertical Crustal Movements in Eastern Canada
or Glacial Isostatic Adjustment (GIA). During the last major glaciation event, immense masses of ice accumulated over regions of North America, causing subsidence of the Earth's crust in these ice covered regions, and uplift in peripheral regions. When this ice has melted during the last 20,000 years, the viscoelastic rebounding of the crust in the ice covered regions started and has been ongoing since (Peltier, 1996). In this paper, some ideas are exploited in an effort to infer a more physically meaningful VCM model for Canada. Using smooth piecewise approximation method, the velocity surfaces are computed in pieces, and then they are tied together to guarantee their continuity across the zone boundaries.
2 Sea Level data and re-levelled segments The data used in this study are of two kinds: sealevel records and relevelled segments of the firstorder levelling network. A number of 17 permanent tide gauge stations with long enough records were selected in the area of interest (Figure 1). The subset of 17 sites was then selected to include all stations for which continuous records of at least 10 years duration are available. In the studies of vertical crustal motion, tide gauge records with longer time span are considered more reliable. Sea 6500W I
5000N
/
level records with duration of less than 50 years may not be taken as representative for the secular trends sought, if they are studied individually. However, when they are treated in pairs, the secular variations can be accurately estimated. There is a well documented feature of tide gauge records: their striking similarity when they are obtained at two close-by locations. (Vanf6ek and Carrera, 1993). This spatial coherence is caused by common atmospheric and oceanic noise. Clearly, a large portion of these variations disappears when the records are differenced. This behaviour offers an alternative way of treating sea level trends in closeby tide gauges. In this study, a straight--forward trend analysis was carried out on monthly mean values for all stations. Then, it was decided to use the differencing technique to treat the sea level records. The regional correlation matrices and correlation coefficient confidence interval is used to select the optimum pairing of sites, i.e., a tree diagram for optimum differencing, that gives the most precise and accurate velocity differences to be used in the modelling. Figure 1 demonstrates the optimum pairing of tide gauges in Eastern Canada. A total of 14168 relevelled segments from Maritimes and southern Quebec were chosen for this study. They were observed during the period between 1909 and 2002. The distribution of data is shown in Figure 1.
6000W I
5000N I
-
-4500N
-6000W
i',
/ .~
4500N
Tide gauge
-
~Pairing • 7000W
tide g a u g e
Levelling B M
65(~0W
Fig 1: Data distribution used in computations. The optimum tree diagram of tide-gauges for differencing is shown by red lines.
497
498
A. Koohzare • P. Vani?ek. M. Santos
3 Mathematical Model
m+z) . Here, q, represents the maximum number of the nodal points in each border.
In order to predict the spatial vertical velocities, or uplift rates, a vertical velocity surface should be fitted to the sea level linear trends and levelling height difference differences data reviewed in the previous section. Therefore, the main concern is to provide an approximation to a function V(x,y) from geodetic data. The assumptions underlying this approach are that the uplift rates are linear in time and that they vary smoothly with location. The velocity surface is first obtained in the form of
In order to piece the polynomials together, the following conditions should be satisfied:
V(x, y) - £ co x~y j
,
(1)
G
(Xm,k'
(3.a)
av,,(x, y) &
8Vm(x,y)
i,j=O
In general, if we divide the area of study into m zones and the degree of all the algebraic polynomials is n, the resulting function is a polynomial function of degree n with m zones. A given polynomial in the m-th zone looks as follows:
Vm(X,y)-- ~C~j,,.(X--X,.,~)~(y-- ym,~)j ,
m 8Vm+l (X, y) X=Xm ,k - y= ym,k (iX
_
X=Xm,k y= ym,k
ovm+,(x,y)
_
X=Xm,k - y=ym,k
where (x,y) is the location of the points in an arbitrary selected local horizontal coordinate system, n is the degree of polynomials, and c,j are the sought coefficients. Here, the algebraic functions are the simplest functions to deal with numerically and are adequate when the solution is confined to the regions where sufficient data exits; the poor behaviour appears only when the solution is used in an extrapolation mode (Vanf6ek and Nagy, 1981). The procedure of fitting a surface to the geodetic data involves the use of both the point rates and the gradients simultaneously, together with their proper weights. The point rates are determined from some of the tide gauge data which were selected to be used in the point velocity mode, and the gradients come from relevelled segments and tide gauge pairs. To get the details needed for the map to be meaningful, the order of the velocity surface would have to be too high to be numerically manageable. A practical way to avoid this is to divide the area of study into zones, and seek the velocity surface piecewise.
(2)
i,j=0
where V,,, is the algebraic least squares velocity surface for zone m, fitted to the desired data (x,y). The pair (x,,,k ,y,,,k) for k=l,2 . . . . . q represents the position of each node (P ....k) located in the predefined border between two zones (zones m and
Vk- 1,2..... q
G+I (X ....k' Y ....k )
Ym,k ) --
X=Xm ,k y=ym,~
Vk = 1,2..... q (3.b)
8~Vm(x,y)
~2Vm+l
c~x 2
X=Xm,k - y=ym,k
a 2~/~m(X, y) 2
X=Xm,k - y=ym ,/,
(x,
y)
c~x 2
X=Xm ~k y=ym,k
aaVm+l (x, y) ~17 v/, 2
X=Xm ,k y= ym ,1,
Vk = 1,2..... q (3.c)
Conditions (3.a) make sure that the piecewise polynomial fits to the nodal points (P,,,1, P ....2. . . . . Pm,k=q). These conditions imply that the function is continuous everywhere in the region. Conditions (3.b) and (3.c) ensure that the polynomials are continuous in slope and curvature respectively throughout the region spanned by the points (x,y). Assuming the velocity to be constant in time, the difference of the two levelled height differences divided by the time span between the two levellings gives the velocity difference between the two levelling segment's ends. These 'observations' are used to compute the coefficients by means of leastsquares method. The main mathematical model is equation (2) while all the conditions under (3) show the existence of constraints on the main model. To find the least square solutions, equations (2) and (3) can be simplified in a general form:
f ( ¢ , l ) - O,
(4.a)
f c (¢) - 0.
(4.b)
Chapter 72 • The Use of Smooth Piecewise Algebraic Approximation in the Determination of Vertical Crustal Movements in Eastern Canada
Here, 1 is the vector of observations and c is the vector of unknown coefficients. It will be assumed that it is possible to solve for c, using only the main model (4.a). The auxiliary modelfc consists of some constraint functions that enforce the conditions which should be guaranteed. The above models are next linearized to yield:
N - (Ar (BCrBr u-
Ar (BCrBr
)-I A)-' )-t w
(9) (10) (11)
6(1) - - N -l u .
Equation (11) represents the solution from the main A6 + Br + w-O,
D6+
(5)
w e -0.
In equations (5), r is the vector of expected residuals. Matrices A and D are the Jacobian matrices of transformation from parameter space to the two model spaces, valid for a small neighborhood of c(0,. Matrix B is the Jacobian matrix of transformation from observation space to the main model space. It is observed that equations (5) are merely the differentional form of the original non-linear mathematical model equations (4.a) and (4.b) and describe the relations of quantities in the neighborhoods of c,o,, the point of expansion in the parameter space, and w~0,, the misclosure vector, where,
d - c - c (°~,
I • + 2 k r ( A d + B r + w) + 2 k r (DO + wc ),
(7) where Cr~C/ is the covariance matrix of the observations. Here, there are two sets of Lagrange correlates: k, kc, reflecting the fact that two models are present. The minimum with respect to r is found by the Lagrange approach (Vanf6ek and Krakiwsky, 1986) as
(DN-IDr)~
(w~ + Dg(1) ),
(8) where
C~ - N - l - N - 1 D r
(DN-1Dr
)-I DN-1.
(12)
The appropriate degree of the velocity surface is determined by testing the estimated accuracy, or the 'a posteriori standard deviation'. This is computed from ~Tc[I~
d-02 = - - ,
(13)
where t; is the vector of least square residuals and v denotes the number of degree of freedom.
The variation function for finding the least-squares solution is written as,
- (j(l) _ N - I D r
The next task is to obtain the covariance matrix of the parameters. It is given by Vanf6ek and Krakiwsky (1986) as:
(6)
w(o~ - f(l~o~,c(o~).
¢ -- r T C r
model f alone, and the corrective term 6 - d (') in equation (8), arises from the enforcement of the constraints.
Due to the geophysical diversity in Eastern Canada, for example, different geological characteristics and different rate of seismicity, Eastern Canada was divided into two zones: the Maritimes zone, and the zone containing the southern part of St. Lawrence River (Figure 2). The border of these two zones is dictated by the actual data distribution and the present knowledge of the geodynamics of the area. For example, the estuary of the St. Lawrence River is an area where 50 to 100 earthquakes are detected yearly. The region, known as the Lower St. Lawrence Seismic Zone, was originally defined by spatial clustering of magnitude (M) 3000 kg/m 3 2950 bis 2990 2900 bis 2940 2850 bis 2890 2800 bis 2840 2750 bis 2790 2700 bis 2740 2550 bis 2690 2450 bis 2540
2770kg/m~
@Fichtelgebirge granite kg/m 3 kg/m 3 kg/m 3 kg/m 3 kg/m 3 kg/m 3 kg/m 3 kg/m 3
(2560 kg/m 3)
(2) sediments (2450 kg/m 3) F.L. Franconian Line M.L.F. Mari~nsk6 Lazn6 fault zone o earthquakes of the swarm 2000
Figure 4: SW-NE section of the 3-D density model.
boundaries. The most important geological units effecting the gravity field in that section are the Eibenstock granite and the Marifinzk6 Lfizn6 fault separating the Fichtelgebirge granite in the west from phyllites in the east. Regarding the periodic occurrence of earthquake swarms the existence of an upwelling mantle or a magmatic body at the crust mantle boundary is investigated. This is realised by an additional model body dealing with two different densities at the crust mantle boundary. The first case describes an upwelling mantle by a density of 3370 kg/m 3. In the second case a nephelinitic magma system with a density of 2850 kg/m 3 is assumed. The comparison of the modelled gravity of both cases with the measured one shows that obviously an upwelling mantle cannot explain the observed gravity (Hofmann et al., 2003). Therefore a magmatic system as postulated by Weinlich et al. (1999) is supported for the Vogtland/NW-Bohemia region.
4 Geodynamic modelling The geodynamic models for the Vogtland/NW-Bohemia and the Magadi/Kenya Rift area are constructed in consideration of the particular geological, tectonical and seismological circumstances (Fig. 2 & 3). According to the results of both deep seismic profiles and gravity measurements/modelling (Behr et al., 1994; Hofmann et al., 2003; Simiyu & Keller, 2001), both models consist of four layers (Fig. 5 & 6, left) reaching from the surface to the crust mantle boundary of the particular region including the upper, middle und lower crust (Kurz et al., 2003; Naujoks et al., 2004). To ensure realistic modelling, lateral variations in the material properties are likewise included. Fig. 5 (left) also shows the distribution of two different materials included in the Vogtland/NW-Bohemia model. The black boxes mark the material of granitic composition, the rest of the model has metamorphic material properties. In the Magadi/Kenya Rift area the upper 2.5 km are an interbedded stratification of sediments and volcanic rocks. Therefore, the top layer of the model was constructed with averaged material properties of that rocks. The other layers are of metamorphic composition. Tab. 1 summarises the material properties. In the Magadi/Kenya Rift area two special features of the crust, described by Ibs-von Seht et al. (2001), are included into the model: First, an upwelling of the brittle-ductile transition (dark grey in Fig. 6, left) with material properties of the underlying ductile crust. Second, a body in the earth-
Chapter 77
....
• Investigations
about Earthquake Swarm Areas and Processes
- . . . . . . ,--!-il . . . . . . . . .
0
i
:
I
:
I
Figure 5: The Vogtland/NW-Bohemia model: Construction (left) and discretisation with included geological faults (right).
V•
SSW
~"
V SSW
WNW
Figure 6: The Magadi/Kenya Rift model: Construction (left) and discretisation with included geological faults (right). quake swarm hypocenter region underneath the northern end of Lake Magadi with increased porosity and permeability (light grey in Fig. 6, left). This region in the crossing area of the passing fault zones is interpreted as strongly jointed and highly fractured (Ibs-von Seht et al., 2001). Therefore, the density of that area in comparison to the surrounding rocks is reduced by two percent. Geological faults are included in the models as weakness zones (Fig. 5 & 6, right). They differ in their physical values from the surrounding rocks. The Youngs's and bulk modulus are reduced by ten percent (Kurz et al., 2003) and the permeability in that faults is six orders of magnitude higher than that of the surrounding rocks, as found by Matthfii & Roberts (1997). In the models a porous elastic rheology as well as Mohr Coulomb plasticity in the upper crust and Newtonian creep in the middle and lower crust is realised. Therefore the brittle-ductile transition between upper and middle crust is characterised by a change in the anelastic material properties.
The question which physical processes cause the earthquake swarms in the Vogtland and Magadi areas is analysed in different steps with varying boundary conditions. At the bottom side the models are always fixed, the surface is free to move in all directions. The surrounding crust at the vertical sides of the models is simulated by elastic element foundation, which approximates the pressure of the surrounding crust to the next main geological units. The first step is to gain the long term influence of the regional stress field acting in the particular region on the model's stress and strain fields over a time period of 10,000 years. Therefore a stress field based on the results of Brudy et al. (1997), Wirth et al. (2000), Ibs-von Seht et al. (2001) and Atmaoui & Hollnack (2003) is applied to the models. This leads to values in layer 1 of about 80 MPa in the NW-SE direction and 40 MPa perpendicular for the Vogtland/NW-Bohemia and to 50 MPa in the NNE-SSW direction and 40 MPa perpendicular for the Magadi/Kenya Rift region respectively. In layer 2 363 MPa (NW-SE) and 183 MPa
531
532
M. Naujoks • T. Jahr. G. Jentzsch • J. H. Kurz. Y. Hofmann
Table 1: Parameters for the sedimentary (s), granitic (g) and metamorphic (m) rock in the Vogtland/NW-Bohemia and Magadi/Kenya Rift area after Angenheister (1982); Meissner et al. (1987); Carmichael (1989); Behr et al. (1994); Birt et al. (1997); ibs-von Seht et al. (2001); Simiyu & Keller (2001); (p density, E Young's modulus, v Poisson ratio, ~: bulk modulus, ~y cohesion yield stress (layer 1 and 2 only), 1"1viscosity (layer 3 and 4 only)). p E layer [1~r3] [GPa] Vogtland/NW-Bohemia 1m 2620 71.88 2m 2780 89.47 3m 2880 102.64 4m 3050 129.12 lg 2630 72.32 2g 2700 87.68 3g 2820 81.34 4g 3050 129.12 Magadi/Kenya Rift 1s 2550 41.90 2m 2750 91.39 3m 2850 102.09 4m 3000 130.27
~c [GPa]
•y [MPa] 11 [Pa.s]
0.252 0.228 0.234 0.229 0.255 0.239 0.233 0.229
48.35 54.87 64.37 79.41 49.14 55.94 50.89 79.41
370 370 2.5.1022 4.0.1022 510 510 2.5.1022 4.0.1022
0.264 0.247 0.240 0.231
29.53 60.30 65.54 80.67
250 370 2.5.1022 4.0.1022
(perpendicular) for the Vogtland/NW-Bohemia and 360 MPa (NNE-SSW) and 285 MPa (perpendicular) for the Magadi/Kenya Rift region are applied. For the middle and lower crust it is only known, that the differential stress (~1- (Y3 drops to zero when passing the brittle ductile zone (Brudy et al., 1997). Therefore we assume that for the middle and lower crust all three main stress components are equal and take the value of the vertical component which is the lithostatic stress. In the next steps the influence of short term periodic pore pressure variations with an amplitude of 50 MPa and a period of 10 years alone and combined with temperature changes of 0.5 K (Safanda & 0ermfik, 2000) is analysed over a time period of 100 years. The period of the pore pressure variations is an average value of the periodic occurrence of the earthquake swarms (Ibs-von Seht et al., 2001; Neunh6fer & Gtith, 1988). In the Vogtland/NW-Bohemia region the pore pressure and temperature variations are applied to the bottom of the model, because there are indications for a magmatic system (Vrfina & St6drfi, 1997). This could also explain the strong CO2degassing at the surface (Weinlich et al., 1999). Due to hot CO2 springs at the surface (Schltiter, 1997) as well as seismological investigations a magmatic system is also assumed in the upper crust in the Magadi/Kenya Rift area underneath the northern end of
lake Magadi (Ibs-von Seht et al., 2001). There the pore pressure und temperature variations are applied to the model. 5 Results
In the Vogtland/NW-Bohemia area earthquake swarms occur in depths between 7 and 15 km (Wirth et al., 2000), in the Magadi/Kenya Rift area in depths from 0 to 10 km (Ibs-von Seht et al., 2001). The aim is to analyse the influence of the local geology and the regional stress field over a time period of 10,000 years alone and in combination with periodic pore pressure changes and linear temperature changes in a time period of 100 years. The shear stresses ~512, accumulated under these boundary conditions are given in Fig. 7 for the Vogtland/NW-Bohemia region and in Fig. 8 for the Magadi/Kenya Rift region respectively. The other shear stress components (Y13 and c~23 show a similar behavier as (Y12 in the region of maximum stress as well as in the amplitude of the values. Fig. 7 (top) shows the influence of the regional stress field and the local geology in the Marifinzk6 Lfizn~ fault zone of the Vogtland after 10,000 years. Obvious is a small gradient at the intersection of the two fault zones. In the Magadi depression (Fig. 8, top) the regional stress field and the local geology lead to shear stresses which are accumulated at the intersection of the fault zones in a depth from 4 to 13 km as well. Both the amplitude and the gradients are higher compared to the Vogtland region. Fig. 7 (centre) shows an increasing shear stress in the Marifinzk6 Lfizn6 fault zone (Vogtland) under periodic pore pressure variations with an amplitude of 50 MPa and a period of 10 years over a time period of 100 years. Here the gradient in shear stress increases in the crossing region especially in a depth from 12 to 16 km which corresponds to the brittle-ductile transition zone. In the Magadi depression (Fig. 8, centre) the shear stress also increases with a maximum at a depth of about 8 km. Combining these periodic pore pressure changes with linear temperature changes of 0.5 K within 100 years the shear stress increases by the factor of 5 in the Vogtland (Fig. 7, bottom) and the factor 3 in the Magadi area (Fig. 8, bottom). Compared to the calculations, in which only the regional stress field and periodic pore pressure variations are acting, the gradients are higher and the maximum shear stresses are clustered in a smaller region. In the Vogtland as well as in the Magadi region the shear stress accumulations occur where the earthquake swarms take place in reality.
Chapter 77 • Investigations about Earthquake Swarm Areas and Processes
6 Conclusion An essential result of the modelling is that the regional stress field alone can neither explain the occurrence of earthquake swarms in the Vogtland nor in the Magadi region. Indeed, in the Magadi regionin contrast to the Vogtland region- local shear stress
accumulations appear over a long lasting time period caused by the regional stress field. But the computed stress values are too low to cause earthquake swarms. This is a clear indication that in both regions other physical processes must act additionally. As there are obvious indications from observations of fluid movement and temperature anomalies in the earth's crust,
S12 [MPa]
S12 [MPa]
- 3.3
2.8
+5.0 +3.8 +2.6 + 1.4 + 0.2
- 2.4 - 2.0
-1.6 -1.2 - o.7 - 0.3
- 1.0
-2.2
+0,1 + 0.5 + 1.0 + 1.4 + 1.8
- 3.4
-4.6 -5.8 - 7.0
3 (V) j 2
3 (V)
(SE)
~ 2 (ssw) "--. t (WNVO
1 (SW)
S12 [MPa] - 3.3
S12 [MPa]
2.8 - 2.4 - 2.0
•
+3.8
-1.6 -1.2
~" +2.6 •
+1.4
•
+_1:20
i
-2.2
- 0.7 - 0.3
+o.1 + 0.5 + 1.0 + 1,4 + 1.8
..411
-5.8
~
;
- 7.0
3 (V)
3 (V)
~
2 (SE)
-"
1 (SW)
~
> [
S12 [MPa]
•
-12.5
-15.0 •B -lO.O - 7.5
~
- 7.3 - 6.0
- 0.9
+ 0,2
+ 2.5 0.0 ~ - + 5.0 ~ - + 7.5 - +10.0
~
2 (SE) 1 (SW)
i
i 1.5 i 2.7
+4.0 i 5.2
3 (v)
"--
St2 [MPa] -9.8 -8,5 -4.7 -3,4 -2,1
~ ~ i i
- 5.0 - 2.5
~
(ssw) "-.. 1 (WNVO
i
i
Figure 7: Vogtland/NW-Bohemia: Shear stress o12 in the Mari/mzk6 Lfizn6 fault under different boundary conditions; top: Regional stress field only; center: 50 MPa pore pressure variation; bottom: 50 MPa pore pressure and 0.5 K temperature variation.
3(V) 2 (SSW) "--. t (WqVW)
Figure 8. Magadi/Kenya Rift: Shear stress (Y12 in the Magadi depression under different boundary conditions; top: Regional stress field only; center: 50 MPa pore pressure variation; bottom: 50 MPa pore pressure and 0.5 K temperature variation.
533
534
M. Naujoks • T. Jahr. G. Jentzsch • J. H. Kurz. Y. Hofmann
Table 2" Comparison and evaluation of the results of the modelling in the regions Vogtland/NW-Bohemia and Magadi/Kenya Rift as well as an estimation of the earthquake swarm areas worldwide. structures, rheologies and processes
Vogtland/NW-Bohemia
Magadi/Kenya Rift
Earthquake swarm areas worldwide
local geology and brittle-ductile transition
important for locating the shear stress gradient essential influence, particulary in crossing areas only weak shear stress accumulations in the middle crust crucial; applied to the bottom of the model crucial; intensifies geodynamic effects significantly essential; fluid movement in a southward direction
controls amongst others the local stress distribution in the focal area essential influence, likewise in crossing areas only little influence for long running model loading crucial; applied to the lower border of the upper crust crucial; intensifies geodynamic effects significantly essential; fluid circulation in the model
important for locating the geodynamic processes play obviously a very essential role in the local stress distribution little influence, but possibly important in tectonically very active regions the crucial factor for local shear stress accumulations process-controlling factor assumedly essential for all earthquake swarm areas essential; process-controlling factor in earthquake swarm areas
fault zones
regional stress field pore pressure variations e.g. by fluid movement or degassing magma temperature changes e.g. by magma systems or mobile hot fluids fluids
which can occur in the particular region in very different ways, in the next step periodic pore pressure variations and linear temperature changes are applied to the models to determine the influence of these processes on the stress and strain fields. Linear pore pressure changes do not lead to a significant change in the calculated shear stresses. But a strong effect of periodic pore pressure variations, e. g. caused by degassing of magma intrusions or fluid movements in the earth's crust, can be proved. Both in the Vogtland and in the Magadi area they lead to a concentration and local increase of shear stresses in that section of the model where the earthquake swarms occur in reality (|bs-von Seht et al., 2001; Wirth et al., 2000). The computed shear stresses are higher in the Vogtland than in the Magadi area. The consideration of temperature changes, probably caused by magma intrusions, showed that increasing temperatures are responsible for a significant rise in the occurring shear stresses. A 0.5 K temperature rise causes an amplification of the occurring shear stress by the factor of 5 in the Vogtland and 3 in the Magadi area. Probably because of the different geological settings the Vogtland model reacts more sensitively to temperature changes than the Magadi model. The influence of the regional stress field is superposed by short term and stronger processes, the pore pressure and temperature variations. They cause shear stress accumulations where the earthquake swarms occur in reality. The gravimetric modelling
in the Vogtland region shows that a magmatic system as postulated by Weinlich et al. (1999) seems to be the source of the CO2-rich mineral springs in the region, which may cause pore pressure variations and so may be the crucial factor for local stress accumulations leading to the earthquake swarms. Hence, periodic pore pressure variations may be a local trigger mechanism of the occurrence of earthquake swarms. These results are valid for both the Magadi area as well as the Vogtland area, even though they are weighed differently in each focal area. They are consistent with observations made in different earthquake swarm areas. Modelling is an progressive process. Many models were computed and the geometry, the material properties and the boundary conditions were varied. As a result an evaluation of the influence of parameters on the results of the modelling is possible. And therefore it is possible to determine which parameters in which dimension may have an effect to the occurrence of earthquake swarms in regions worldwide with different geological setting (Tab. 2).
7 Acknowledgements The authors wish to thank the German Research Foundation (DFG) for their funding of this research. Constructive reviews of the manuscript by Bert Vermeersen and an anonymous reviewer are gratefully acknowledged.
Chapter 77 • Investigations about Earthquake Swarm Areas and Processes
References Angenheister, G. (Ed.) (1982). Physical Properties of Rocks, Vol. V l a & l b of Landolt-B6rnstein, Springer, Berlin. Atmaoui, N. and D. Hollnack (2003). Neotectonics and extension direction of the southern Kenya Rift, Lake Magadi area. Tectonophysics, 364, pp. 71-83. Baker, B.H. (1958). Geology of the Magadi Area, Kenya Geol. Surv. Rept., 42. Behr, H.-J., H.-J. Dfirbaum and P. Bankwitz (1994). Crustal structure of the Saxothuringian Zone: Results of the deep seismic profile MVE-90 (East), Z geol. Wiss., 22 (6), pp. 647-769. Benoit, J. and S. McNutt (1996). Global volcanic earthquake swarm database 1979-1989, U.S. Geological Survey, Open-file Report, pp. 69-96. Birt, C. S., Maguire, P. K. H., Khan, M. A. et al. (1997). The influence of pre-existing structures on the evolution of the southern Kenya Rift Valley- evidence from seismic and gravity studies, Tectonophysics, 278, pp. 211242. Brudy, M., Zoback, M., Fuchs, K., Rummel, F. & Baumgfirtner, J., (1997). Estimation of the complete stress tensor to 8 km depth in the KTB scientific drill holes: Implications for crustal strength, J. Geophys. Res., 102 (8), pp. 18453-18475. Carmichael, R. (1989). Practical Handbook of Physical Properties of Rocks and Minerals, CRC Press, Boca Raton, USA. Grfinthal, G. (1989) About the history of the seismic activity in the focal region Vogtland/Western Bohemia. In: Monitoring and Analysis of the earthquake swarm 1985/86 in the region Vogtland/Westem Bohemia (P. Bormann, Hg.), ZIPE Ver6ffentlichung Akademie der Wissenschafien der DDR, Potsdam, 110, pp. 30-34. Hemmann, A., T. Meier, G. Jentzsch and A. Ziegert (2003). Similarity of waveforms and relative relocalisation of the earthquake swarm 1997/1998 near Werdau, J. Geodyn., 35, pp. 191-208. Hofmann, Y., T. Jahr and G. Jentzsch (2003). Threedimensional gravimetric modelling to detect the deep structure of the region Vogtland/NW-Bohemia, J. Geodyn., 35, pp. 209-220. Hofmann, Y., T. Jahr, G. Jentzsch, P. Bankwitz and K. Bram (2000). The gravity field of the Vogtland and NW Bohemia: Presentation of a new project, Studia geoph. et geod., 44 (4), pp. 608-610. Ibs-von Seht, M., S. Blumenstein, R. Wagner, D. Hollnack and J. Wohlenberg (2001). Seismicity, seismotectonics and crustal structure of the southern Kenya-Rift--new data from the Lake Magadi area, Geophys. J. Int., 146, pp. 439-453. Kurz, J.H., T. Jahr and G. Jentzsch (2003). Geodynamic modelling of the recent stress and strain field in
the Vogtland swarm earthquake area using the finiteelement-method, J. Geodyn., 35, pp. 247-258. Kurz, J.H., T. Jahr and G. Jentzsch (2004). Earthquake swarm examples and a look at the generation mechanism of the Vogtland/Western Bohemia earthquake swarms, Phys. Earth Planet. Int., 142 (1), pp. 75-88. Matthfii, S. & G. Roberts (1997) Fluid Flow and Transport in Rocks: Mechanisms and Effects, Chapter 16, pp. 263295, Chapman & Hall, London. Meissner, R., Wever, T. and E.R. Fltih, (1987). The Moho in Europe - Implications for crustal development, Anhales Geophysicae 5B (4), pp. 357-364. Mogi, K. (1963). Some discussions on aftershocks, foreshocks and earthquake swarms - The fracture of a semiinfinite body caused by an inner stress origin and its relation to the earthquake phenomena, Bull. Earthquake Res. Inst. Tokyo Univ., 41 (3), pp. 615-658. Naujoks, M., T. Jahr, G. Jentzsch and J.H. Kurz (2004). Den Schwarmerdbeben auf der Spur: Vergleichende geodynamische Modellierungen zu Kenia-Rift und Vogtland, Proc. of the 16th German-speaking ABAQUS User Conference, K6nigswinter, Germany, Sept. 20-21. Neunh6fer, H. and B. Tittel (1981). Mikrobeben in der DDR, Z. geol. Wiss., 9 (11), pp. 1285-1289. Neunh6fer, H. and D. Gtith (1988). Mikrobeben seit 1962 im Vogtland, Z. geol. Wiss., 16 (2), pp. 135-146. Prodehl, C. (1997). The KRISP 94 lithospheric investigation of southern Kenya - the experiments and their main results in Structure and Dynamic Processes in the Lithosphere of the Afro-Arabian Rift system, Tectonophysics, 278, pp. 121-147. Safanda, J. and V. Cermfik (2000). Subsurface temperature changes due to the crustal magmatic activity - numerical simulation, Studia geoph, et geod., 44, pp. 327-335. Schltiter, T. (1997). Geology of East Africa. Contributions to the regional geology of the earth, Vol. 27, Gebrfider Bomtrfiger, Berlin, Stuttgart. Simiyu, S.M. and R. Keller (2001). An integrated geophysical analysis of the upper crust of the southern Kenya rift, Geophys. J. Int., 147, pp. 543-561. Vrfina, S. and V. St6drfi (1997). Geological model of western Bohemia related to the KTB borehole in Germany, J. Geol. Sci., Vol. 47, Prague. Weinlich, F. H., K. Brfiuer, H. Kfimpf et al. (1999). An active subcontinental mantle volatile system in the eastern Eger rift, Central Europe: Gas flux, isotopic (He, C and N) and compositional fingerprints, Geochem. et Cosmochem. Acta, 63 (21), pp. 3653-3671. Wirth, W., T. Plenefisch, K. Klinge, K. Stammler and D. Seidl (2000). Focal mechanisms and stress field in the region Vogtland/Westem Bohemia, Studia geoph, et geod., 44 (2), pp. 126-141. Yamashita, T. (1998). Simulation of seismicity due to fluid migration in a fault zone, Geophys. J. Int., 132, pp. 674686.
535
Chapter 78
Sea level and gravity variations after the 2004
Sumatra Earthquake observed at Syowa Station, Antarctica Kazunari Nawa, Kenji Satake Geological Survey of Japan, AIST, AIST Tsukuba Central 7, Higashi 1-1-1, Tsukuba, Ibaraki 305-8567, Japan Naoki Suda Hiroshima University, Kagamiyama 1-3-1, Higashi-Hiroshima, Hiroshima 739-8526, Japan Koichiro Doi, Kazuo Shibuya National Institute of Polar Research, Kaga 1-9-10, Itabashi, Tokyo 173-8515, Japan Tadahiro S ato National Astronomical Observatory, Hoshigaoka 2-12, Mizusawa, Iwate 023-0861, Japan
Abstract. The Indian Ocean Tsunami reached Syowa Station, Antarctica, in approximately 12.5 hours after the 2004 Sumatra-Andaman Earthquake. We have analyzed the tsunami records of the tide gauge, including the superconducting gravimeter (SG) at the station. The synthetic tsunami and the induced gravity variations were calculated in order to compare with observations. It was found that the gravity effects of the tsunami exhibited an amplitude of microGal (10 .8 m/s2); obtained from the Syowa SG. Furthermore, the effects of the tsunami on the Earth's free oscillation records of the SG were subtracted by applying a transfer function method, using the tide gauge records as input. The improvement of S/N at frequencies of 0.3 mHz is remarkable. Keywords. Tsunami, tide gauge, superconducting gravimeter, synthetic waveform, free oscillations, noise reduction.
1 Introduction The Sumatra-Andaman earthquake on the 26 th of December 2004 generated a massive tsunami (Fig.l) in the Indian Ocean and excited the Earth's free oscillations. The tsunami propagated over the
Indian Ocean and also reached the coast of Antarctica (e.g. Dumont D'Urville, see Merrifield et al. (2005)). The tide gauge at the Syowa Station, Antarctica (69S 39.6E), detected the tsunami (the amplitude was sub-meter) approximately 12.5 hours after the occurrence of the earthquake. The Earth's free oscillations, excited by the Sumatra-Andaman earthquake, were also observed using a superconducting gravimeter (SG) at the station.
.90 °
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Fig. 1 Water height distribution of simulated Indian Ocean tsunami 12.5 hours after the 2004 Sumatra-Andaman earthquake. Red color means that the water height is higher than normal while blue means lower. Initial water height distribution is calculated by assuming a fault of 600 km in length and 150 km in width (strike 340 deg., dip 8 deg., rake 90 deg. and slip of 5 m). Syowa Station (SYO: black circle) is located at 69S, 39.6E.
Chapter 78 • Sea Level and Gravity Variations after the 2004 Sumatra Earthquake Observed at Syowa Station, Antarctica
At Syowa Station, sea level variations at frequencies between 0.2 and 2.5 mHz, in the seismic normal mode band, were detected simultaneously from on-ice GPS (Aoki et al., 2000) and SG records on calm days (Nawa et al., 2003). Since the Indian Ocean tsunami causes vibrations in the sea around the station, the gravity effects of the tsunami should be detectable using the SG. In such a case the SG could be employed as a tsunami gauge as long as the SG is installed near the coast, as at the Syowa Station. Compared to signals that originate the solid Earth, however, the signals of the gravity effects of the tsunami are on the scale of "noise". In this study we analyze sea level and gravity variations after the Sumatra-Andaman earthquake and compare observations with synthetic waves. We then try to remove the effects of the tsunami from the Earth's free oscillation spectra by applying a transfer function method.
bottom mount (CT-type), replaced the old SG in 2003, which was of a regular size and employed a top mount (TT-type). In figure 3 the original sea level and gravity variation data are presented. The sampling rates of the tide gauge and SG are 30 sec and 1 sec, respectively. One can clearly see in Figure 3 how the tsunami and seismic waves modify the tide waves. The induced Tsunami arrived 12 hours and 40 minutes after the start of the Sumatra-Andaman earthquake. In Fig. 3(b), although the amplitude becomes small, gravity variations caused by the Earth's free oscillations persist while the variation in the sea level continues, Fig. 3(a). Hence, in the time domain, it is difficult to resolve the tsunami effects from the SG records.
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2 Observation The Syowa Station is on East Ongul Island in the coastal zone of the southern Indian Ocean (Fig.2). Tide gauge (ocean bottom pressure gauge) observations, maintained by the Japan Coast Guard, are on-going at the Nishi-no-ura Cove. The SG observations have continued since 1993. However, a new type of SG, compact in size and employing a
min.
Fig. 3 Time series of (a) sea level and (b) gravity variations, caused by the 2004 Sumatra-Andaman earthquake. The data is presented as a function of time, where t=0 is the beginning of the earthquake.
The spectrum of sea level variations after the arrival of the tsunami, obtained from tide gauge, is presented in figure 4 where the data exhibits a maximum power at approximately 0.3 mHz. The tsunami observed at the Syowa Station exhibits a large power in the low frequency seismic mode band (see Fig.7). Although the power is large, the frequencies of these peaks (including around 0.3 mHz) in this frequency band coincide with that of signals on calm days detected simultaneously from sea level variation records by on-ice GPS and gravity variation records by the SG (Nawa et a1.,2003).
537
538
K.Nawa.K.Satake• N.Suda.K.Doi. K.Shibuya.T.Sato the synthetic sea level variation. The high frequency component (> about 1 mHz) of the observation reflects influences of local topography that our modeled bathymetry cannot represent. We have only confirmed that the tsunami waveform of the ocean bottom pressure gauge installed at Lutzow-Holm Bay (Doi et al., this meeting) is comparable to the synthetic tsunami of that site.
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Fig. 4 A spectrum of the Indian Ocean tsunami after the 2004 Sumatra-Andaman earthquake, obtained from 48 hours of tide gauge data. Vertical broken lines indicate frequencies of sea level variations observed by on-ice GPS and SG, with high coherency at the Syowa Station on calm days in 1998 (Nawa et al., 2003). Gravity effects of the tsunami are overlapped in the Earth's free oscillations frequency band. However, the tsunami exhibits a power at frequencies lower than that of the lowest frequency modes, 0S2 (0.3 mHz). In order to extract the tsunami effects and reduce the free oscillation signals, gravity variations are bandpass filtered at frequencies between 0.1 and 0.2 mHz (Fig.6a). This will be compared with a synthetic gravity variation in the next section.
3 Modeling of tsunami and comparison with observations In order to calculate the gravity effects of the tsunami we first need to calculate the tsunami waveforms of the global ocean. Initial water height distribution based on seafloor deformation is computed using Okada's (1985) formulas from an assumed fault model (see caption of Fig. 1). The tsunami source was estimated from the analysis of tide gauge records (Lay et al., 2005). Tsunami waveforms are computed by assuming linear long-waves using the finite-difference method. The grid size is defined as 10 min of the arc. The bathymetry grid was made from a global digital topography dataset (ETOPO2). The details of the tsunami numerical computation are described in Satake (2002). Synthetic sea level variations at the Syowa Station are presented in figure 5b. The high-cut filtered tide gauge record (Fig. 5a) is very similar to
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Fig. 5 A comparison of (a) observed and (b) synthetic tsunami sea level variations. Black and gray lines in (a) show bandpass (0.1 - 1 mHz) filtered data and residual data that only removed tidal components, respectively. The time axis starts at the origin time of the earthquake.
Second, we compute the gravity variation at the Syowa Station induced by the tsunami. We compute the gravity effects, attraction and loading (elastic) effects, assuming the 1066A earth model with a modified version of GOTIC (a program for Global Oceanic Tidal Correction by Sato and Hanada, (1984)) for synthetic global water height distributions (Fig.l) every 5 minutes. The computed gravity variation induced by the tsunami is very similar to the filtered SG record (Fig.6) with peak to peak maximum amplitudes at approximately 1400 rain. 1.5 microGal for observation and 1.7 microGal for synthetic, respectively. Furthermore, the Root-mean-square amplitides from 1200 to 2600 rain. are 0.40 microGal from the observation and 0.45 microGal from the synthetic, respectively. The "noise" level of the SG in this frequency band is found to be less than 0.1 microGal.
Chapter 78 • Sea Level and Gravity Variations after the 2004 Sumatra Earthquake Observed at Syowa Station, Antarctica
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Fig. 6 A comparison of (a) observed gravity variation and (b) synthetic gravity effects, of the tsunami. Black and gray lines in (a) are the bandpass (0.1 - 0.2 mHz) filtered data and residual data (with tidal components and atmospheric effects removed), respectively. Right scale is for bandpass filtered data (black) and left scale is for residual data (gray). Black and gray lines in (b) are bandpass filtered synthetic data at the same frequencies as observation and original synthetic data, respectively. The data is presented as a function of time, where t=0 is the beginning of the earthquake.
4 R e d u c t i o n of s e a level v a r i a t i o n effects for seismic normal mode observation We apply the reduction of the sea level variation to the SG Sumatra-Andaman earthquake record using the transfer function presented by Nawa et al. (2003), with the tide gauge records as input. The dominant frequencies of the tsunami spectrum are the same as those of the spectrum of a calm day, as described previously. We also confmned that the response fac-tor (1.6 microGal/m) between the synthetic sea level and gravity variations computed in section 3 is comparable to the findings of Nawa et al. (2003). As a result, the sea level variation effects from low-frequency free oscillations were reduced (Fig. 7). From the data given in figure 7, it is clear that a significant improvement in the S/N, at frequencies of 0.3 mHz near the 0S2 mode, has been gained (Fig.7). Two peaks (0.291, 0.330 mHz), observed on both sides of the 0S2 m o d e (Fig.7), are clearly
Fig. 7 An example of the reduction sea level variation, in order to resolve the Earth's free oscillations excited by the 2004 Sumatra-Andaman earthquake. Black and gray lines show spectrum after reduction and that before reduction of the sea level variation by the tsunami, respectively. The power spectral densities (PSDs) are averages of three PSDs. Each PSD is calculated from 3 day long and 1 day overlapped data. Vertical dashed lines indicate the frequencies of the fundamental spheroidal modes. The vertical line of the lowest frequency indicates 0S2 mode.
reduced in amplitude by applying the correction for the effect of sea level variations. The S/N of the 0S2 mode increases from 1.4 to 2.9. It is important to note that the frequencies of the two peaks are similar to the observed splitting of the 0S2 mode (e.g. Park et al., 2005). The difference, however, between the frequencies is larger than the difference between the lowest and the largest singlet frequencies of 0S2 (Rosat et al., 2005). Therefore, we recognize that the two peaks are not due to the Earth's normal mode. However, it may be advantageous to estimate the tsunami effect at the mid latitude stations, especially close to the sea coast, in order to increase the analysis accuracy at the sites discussed thus far, and to characterize the effect of the tsunami on the Earth's normal modes. In 1998, we employed a transfer function obtained from sea level variations observed using GPS and gravity variations observed using the TT-type (before 2003) SG. In order to confirm the stability of the transfer function and to increase the accuracy of the reduction the transfer function should be recalculated using recent data of sea level variations of the tide gauge and gravity variations of the new CT-type SG, installed in 2003.
539
540
K. Nawa. K. Satake • N. Suda. K. Doi. K. Shibuya. T. Sato
5 Concluding remarks The tide gauge at the S y o w a Station, Antarctica, collected data of the sub-meter Indian Ocean tsunami triggered by the 2004 S u m a t r a - A n d a m a n earthquake. A superconducting gravimeter at the station also detected the tsunami, which exhibited an amplitude of less than 2 m i c r o G a l (10 .8 m/s2), peak to peak, at frequencies of 0.1 - 0.2 mHz. The results give clear indication that the SG could be e m p l o y e d as a tsunami gauge. With respect to the solid Earth observations, the tsunami and/or seiche are on the scale of "noise". Continuous observation of sea level variations around the S y o w a Station is important for the "noise" reduction of the S G record.
A c k n o w l e d g m e n t s . W e would like to thank the Japan Coast Guard for obtaining and distributing tide gauge data at S y o w a Station, Antarctica.
References Aoki, S., T. Ozawa, K. Doi and K. Shibuya (2000). GPS observation of the sea level variation in Lutzow-Holm Bay, Antarctica. Geophys. Res. Lett., 27, 2285-2288. Doi, K. et al. (2005). Presented at Dynamic Planet 2005, session GP03, August 22-26, Cairns, Australia. Lay, T., H. Kanamori, C. J. Ammon, M. Nettles, S. N. Ward, R. C. Aster, S. L. Beck, S. L. Bilek, M. R. Brudzinski, R.
Butler, H. R. DeShon, G. Ekstorom, K. Satake, and S. Sipkin (2005). The Great Sumatra-Andaman Earthquake of 26 December 2004, Science, 308, 1127-1133. Merrifield, M. A., Y. L. Firing, T. Aarup, W. Agricole, G. Brundrit, D. Chang-Seng, R. Farre, B. Kilonsley, W. Knight, L. Kong, C. Magori, R Manurung, C. McCreery, W. Mitchell, S. Pillay, F. Schindele, E Shillington, L. Testut, E. M. S. Wijeratne, E Caldwell, J. Jardin, S. Nakahara, E -Y. Porter, and N. Turetsky (2005). Tide gauge observations of the Indian Ocean tsunami, December 26, 2004. Gephys. Res. Lett., 32, L09603. Nawa, K., N. Suda, S. Aoki, K. Shibuya, T. Sato and Y. Fukao (2003). Sea level variation in seismic normal mode band observed with on-ice GPS and on-land SG at Syowa Station, Antarctica. Geophys. Res. Lett., 30, 1402. Okada, Y. (1985). Surface deformation due to shear and tensile faults in a half-space. Bull. Seism. Soc. Am., 75, 1135-1154. Park, J., T. A. Song, J. Tromp, E. Okal, S. Stein, G. Roult, E. Clevede, G. Laske, H. Kanamori, R Davis, J. Berger, C. Braitenberg, M. V. Camp, X. Lei, H. Sun, H. Xu and S. Rosat (2005). Earth's free oscillations excited by the 26 December 2004 Sumatra-Andaman earthquake. Science, 308, 1139-1144. Rosat, S., T. Sato, Y. Imanishi, J. Hinderer, Y. Tamura, H. McQueen and M. Ohashi (2005). High-resolution analysis of the gravest seismic nomal modes after the 2004 Mw = 9 Sumatra earthquake using superconducting gravimeter data. Geophys. Res. Lett., 32, L13304. Satake, K. (2002). Tsunamis. In: International Handbook of Earthquake and Engineering Seismology, 81A, edited by W. H. K. Lee, H. Kanamori, E C. Jennings and C. Kisslinger, 437-451. Sato, T. and Hanada, H. (1984). A program for the computation of oceanic tidal loading effects 'GOTIC'. Publ. Int. Latitu. Mizusawa, 18, 63-82.
Chapter 79
Improved determination of the atmospheric attraction with 3D air density data and its reduction on ground gravity measurements J. Neumeyer, T. Schmidt GeoForschungZentrum Potsdam, Dept. Geodesy and Remote Sensing, Telegrafenberg, 14473 Potsdam, Germany C. Stoeber Institute for Geodesy, Technical University Berlin, Germany Abstract. Ground gravity measurements based on a test mass (relative and absolute gravimeters) are influenced by mass redistribution within the atmosphere which induces gravity variations (atmospheric pressure effect) in ~gal range (about 15 ~gal for the Sutherland superconducting gravimeter (SG) station). These variations are disturbing signals in gravity data and they must be reduced very carefully for detecting weak gravity signals. The atmospheric pressure effect consists of a deformation and a Newtonian attraction term. The deformation term can be modelled well with twodimensional (2D) surface atmospheric pressure data e.g. with the Green's function method. For precise modelling of the Newtonian attraction term threedimensional (3D) data are required. From European Centre For Middle Weather Forecasts (ECMWF) 3D atmospheric pressure, humidity and temperature data are available. These data are used for modelling of the Newtonian attraction term. Two 3D models for the attraction term have been tested based on mass point attraction and gravity potential of the air masses. The results show a surface pressure independent (SPI) part of gravity variations induced by mass redistributions within the atmosphere in the order of some ~gal. In the past, different methods have been developed for modelling of the atmospheric pressure effect. These methods use 2D atmospheric pressure data measured at the Earth's surface and a standard model for the height dependency of the atmospheric pressure. With these models the SPI part couldn't be detected. For different SG sites and one absolute gravimeter location the 3D attraction models have been applied and the SPI part was calculated. Its influence is shown on tidal parameter computation for long periodic tidal waves, SG measured polar motion, comparison of SG with GRACE and hydrology model derived gravity variations. Furthermore it is shown how absolute gravity measurements are
affected by the SPI part. The omission of the SPI part correction in the gravity data leads to a misinterpretation of about 2 pgal. It can be shown that the application of the SPI part increases the precision of the atmospheric pressure reduction on gravity data. Keywords. Atmospheric pressure correction, gravimetry, 3D attraction models, superconducting gravimeter, absolute gravimeter, surface pressure independent gravity variations (SPI).
1 Introduction The redistribution of the air masses induces temporal Earth gravity field variations (atmospheric pressure effect) up to about 20 ~gal. These variations are disturbing signals in the gravity recordings and they must be removed as far as possible for detection of weak gravity effects. In the past different methods have been developed for modelling of the atmospheric pressure effect which generally fall into two categories: empirical and physical approaches. These methods use local or two-dimensional (2D) atmospheric pressure data measured at the Earth's surface and a standard height-dependent air density distribution. The empirical methods, Warburton and Goodkind (1977), Crossley et al. (1995), Neumeyer (1995) use the local atmospheric pressure for determining the single and complex admittance based on regression and cross-spectral analysis. The physical approaches (Merriam 1992, Sun 1995, Kroner 1997, Boy et al. 1998, Kroner and Jentzsch 1998, Neumeyer et al. 1998, Vauterin 1998) based on atmospheric models determine the attraction and deformation terms according to Green' s function (Farrell 1972). The atmospheric pressure effect is composed of the attraction and elastic deformation terms. The deformation term can be well modelled with 2D surface atmospheric pressure data, for instance with
542
Neumeyer J.. T. Schmidt • C. Stoeber
the Green's function method. For modelling of the attraction term, three-dimensional (3D) data are required in order to consider the real density distribution within the atmosphere. Preliminary studies have been done by Hagedoorn et al, (2000). A first approach with data of radio sounding launchings was developed by Simon (2002). From European Centre for Medium-Range Weather Forecasts (ECMWF) 3D atmospheric pressure, humidity and temperature data are available. The ECMWF data are characterised by a spacing of AdO= 0.5 ° and A)~ = 0.5 °, 60 height levels up to about 60 km and an interval of 6 hours. These data have been used for modelling of the atmospheric attraction term by Neumeyer et al. (2004). Two attraction models have been developed based on point mass attraction of the air segments and gravity potential of the air masses. According to Etling (2002), the air density is calculated as a function of height from 6 hourly ECMWF data at coordinates ~b= 32.5 ° and A = 21 ° for a time span of 31 days (July 2003). Fig. 1 shows the vertical changes in density at different heights. The density changes at the same height can reach up to about 0.1 kg/m 3.
mass redistribution within the atmosphere. For calculation the atmosphere is divided into spherical air segments sV. These air segments are assumed as point masses at segment centre positions. The air density p of the air segments is calculated from ECMWF atmospheric pressure, humidity and temperature data. Equation 1 describes the vertical component of the gravitational acceleration gaz~ which acts on the test mass of the gravimeter caused by mass redistributions within the atmosphere.
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7"= Gravitational constant, p - density of the air segment, s V - spherical air volume segment, R~s = Earth radius at the gravimeter station, r - R~s + height of the spherical air segment sV, ORs - colatitude of the gravimeter site, )~Rs- longitude of the gravimeter site, 0 and 2 - coordinates of s V .
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2 3D-models for the attraction term 2.1 Point mass model
The model (Neumeyer et al. 2004), based on the law of gravitation, calculates the gravitational acceleration at the gravimeter test mass caused by
The second attraction model has been derived from the gravitational potential of the air masses. For mathematical reasons a coordinate system is introduced in such a way that the north pole of this system coincides with the gravimeter site (GS). The gravitational potential of the air masses is calculated with respect to the gravimeter site By partial derivation of the gravitational potential of air masses and summation over all spherical air segments we obtain the vertical component of the gravitational acceleration ga;r(Rcs, 0, 0) (Eq. 2) caused by mass redistributions within the atmosphere. More details can be found in Neumeyer et al. (2004). For calculating of g a i r ( R a s , O , O ) a coordinate transformation of the air density data into the gravimeter global coordinate system (X, Y, Z) is necessary for the reason of the virtual gravimeter site location at the Z-axis (pole) of this system. The coordinate transformation is performed by two rotations according to Torge (2003). Because of the wide spacing (0.5 °) the ECMWF data must be interpolated, which is accomplished by a two-dimensional (0 and 2) bi-linear interpolation. The amount of interpolated points of 10 x 10 can be regarded as adequate.
Chapter 79 • Improved Determination of the Atmospheric Attraction with 3D Air Density Data and Its Reduction on Ground Gravity Measurements
gair(RGs, O, O) -- - - y Z t91 .... (/~1 --/El-l) 1 Bmn l .... 6R2s '
(2)
-4r2_1 + R~s - 2rm_,RGscos,9- , •(2r2_,- R~s + 2R~srm_lCOS~9,,_1+ 3R~s cos 2,9,,_,
-4r2m + R~s-ZrmRGsC°SO. 1 "(2rZ-R~s + 2Rcsrm c°sO. 1 + 3R~s c°s2'9,, 1)
-4r2_, + R~s-Zrm_,RGsCOSO. • (2r2_, - P~s +
2RGsrm_lCOSO n +
3R~s cos20.)
+4rm2 + R~s - 2rmRvscos#.-(2rm2 - R~s + 2Rvsrmcos ~9,,+ 3R~s cos 2~9.)
Bm.n= -lrl[rml-RGscosO n l+4r21+R~s-2rmlRGsCOSOn II6RGsCOSOn 1sin2tgn 1 +ln[r m - RGSCOS~
1-t- 4 4 -t- R~s-
2rmI~s cos,9.116P~s cos,9., sin2 tgn 1
+ln[rm 1- R~s cos0. + 4 4 1 + R2s- 2r iR~s cos,9 16R~s cos,9 sin2 ~9
-l.[r m- RGsc o s a + 4 4 + RZs - 2rmRvscosO. ]6R3s cos a sin2 ~9.
7gravitational constant, p air density as function of the spherical coordinates (r, O, 2 ) , Rcs = Earth radius at gravimeter's site.
Green's function model using 3D-ECMWF atmospheric pressure, humidity and temperature data.
4.1 Attraction term The attraction term was calculated with the GFZ program 3DAP based on point or potential model, Stoeber (2005). Both models deliver the same results within small error bars of about +0.1%. The application of the 3D attraction model shows a seasonal surface atmospheric pressure independent part (SPI) of the attraction term that is caused by the movement of the air masses without changes of the surface atmospheric pressure. In the summer season the air masses move up and the atmospheric attraction term decreases whereas in winter the air masses move down, causing a larger attraction term. Fig. 2 illustrates this fact.
3. Model for the deformation term 2
This model is based on the calculation of gravity changes caused by a point load on the Earth's surface using the appropriate Green's functions, The Green's functions for atmospheric loading have been calculated and tabulated by Merriam (1992) and Sun (1995). They used a column load of the COSPAR standard atmosphere for mid-latitudes and calculated the atmospheric pressure admittance coefficients as functions of the angular distance between the footprint centre of the column load and the gravimeter site. For calculation of the deformation term the Green's function delivers appropriate results by using 2D surface atmospheric pressure data according to equation 2, Sun (1995).
la
a horizontal plane Earth s u r f a c e
Fig. 2 Mass relocation within the atmosphere
g (~/)._ GE(w)r dr2 105~ 2~c[l_cos(lO)] ~gal/hPa
(3)
With G E ( q / ) r - tabulated temperature corrected Green's function for the elastic deformation term, = angular distance from gravimeter to the air column and d.(2 - footprint element of the air column.
Calculation of the atmospheric pressure induced gravity variations For different superconducting gravimeter sites atmospheric attraction term was calculated with point mass and potential model whereas deformation term has been determined with
the the the the
The gravity effect caused by mass relocation is different above and to the side of the gravimeter. When we assume a mass relocation from point 1 to point 2 above the gravimeter, the attraction term becomes smaller (gair_2 < gait_l). When we assume a mass relocation from point l a to point 2a to the side of the gravimeter the vertical component of the attraction term, which acts on the gravimeter test mass, becomes larger (g~r_2a > g~r_la). This effect is height dependent. With local or 2D data we can not detect this effect. For demonstration of this effect the attraction term (Sg~r_attr) is calculated for different air columns above and beside the gravimeter with ECMWF data from January to December 2001 for station Moxa. The results are displayed in Fig. 3.
543
544
Neumeyer J.. T. Schmidt • C. S t o e b e r
,..,
6
4 ~ Jt~ ~1~ I
A.q,~0.;°,'O o.~.oi~,
1
II
t. 1.5 °, the effect is small and not considered. Therefore, the following calculations were carried out with ~ = 1.5°. For comparison of the different seasonal trends in the northern and southern hemisphere an annual wave is fitted to the attraction term calculated with E C M W F data form January 2001 to December 2003 (Fig. 5).
".l., •
'
o
o5-
o-
•
. e
.
"" ~'.-,J
~'*'.
,.
•
J • ..t
"::
iil
, :
~"
.,
•
.
" .
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. .:
.
"
..
• 1
'
-0.5 -
~.~ ,~.~ ,~.~ ...~ ^.~q. ,~.~q.,~.~m...~q. _~ ,~.~ ,\~ ~..~ ^~ Fig. 6 Surface pressure independent part (SPI) of the attraction term at surface pressure of (1001 +0.25) hPa (Metsahovi/Finland)
This signal is the surface pressure independent part SPI of the attraction term for the selected surface pressure. It includes both the seasonal behaviour and gravity changes at shorter periods.
Chapter 79 • Improved Determination of the Atmospheric Attraction with 3D Air Density Data and Its Reduction on Ground Gravity Measurements 1-
0.5-
I
i
I
J
,J
-0.5--
-1-
1t'[1
'"~
'
' '
-1.5 -
Fig. 7 Surface pressure independent part (Metsahovi/Finland) as the difference of sadm and 3Dattr atmospheric pressure corrections.
The total SPI part was separated by subtraction of the single admittance attraction term (single admittance coefficient -4.31agal/hPa) from the 3D attraction term calculated with 3 D - E C M W F data. The result is shown in Fig. 7 for the SG station Metsahovi. The total effect is about 2 lagal.
gravity variations of different sources, the atmospheric pressure induced gravity variations were removed. Two reduction methods were applied: a) 3Dmodel for the attraction term using 3 D - E C M W F data in combination with the Green's function model for the deformation term using 2 D - E C M W F data. The results are summarised to 6g apsD; b) single admittance method (sadm) using local atmospheric pressure data. It calculates the total atmospheric pressure effect 6g_ap~adm.
Table 1. Tidal analysis of long periodic waves Tidal wave ME Ampl.
4.3 Deformation term With the GFZ program 2DAP based on equation (3) the deformation term has been calculated using 2D E C M W F surface atmospheric pressure data applied to a data grid of about 10 ° around the gravimeter. The influence of larger angular distances can be neglected.
5 Applications The importance of the SPI part correction on ground gravity measurements has been investigated by analysing different gravity effects. With reduction of the SPI part the gravity signal becomes smaller in summer and larger in winter compared to the reduction based on local or 2D atmospheric pressure data. The total effect can reach up to 2 ggal.
5.1 Long-periodic tidal waves In a pre-processing procedure, spikes larger than 0.2 ggal and steps that do not have their origin in atmosphere or groundwater level induced gravity variations like instrumental (liquid helium transfers or lightening strikes) and other perturbations such as earthquakes are carefully removed from the raw superconducting gravimeter recordings at stations Moxa/Germany (~) = 50.645 °, )~ = 11.616 °, h = 455 m), Metsahovi/Finland and Sutherland/South Africa. Then, the data are low-pass filtered with a zero phase shift filter (corner period 300 sec) and reduced to 1 hour sampling rate. From these preprocessed gravity data (6graw), which include
MO
SU
theor 8WD 8osadm 8o3D ~:osadm ~:°3D Ampl. theor 8oWD 8°sadm 8°3D ~csadm ~:o3D Ampl. theor 8°WD 8osadm 8°3D ~:osadm Ko3D
SA 33.91
SSA 37.62
Mm 42.72
Mf 80.87
1.161 1.408 1.544 -9.04 -9.05 21.31
1.161 1.276 1.286 0.89 -0.32 23.64
1.161 1.177 1.186 0.47 -0.55 26.84
1.161 1.159 1.161 -0.10 -0.05 50.81
1.165 1.074 1.398 -9.19 -9.91 3.89
1.165 1.143 1.162 -0.22 0.41 4.31
1.165 1.174 1.174 -0.09 0.70 5.45
1.165 1.161 1.160 -0.35 -0.15 9.27
1.115 0.752 1.6439 -12.434 -12.80
1.115 0.995 0.894 4.06 14.92
1.073 1.063 1.126 2.94 2.96
1.115 1.142 1.134 0.66 0.06
The attraction term was calculated with the program 3DAP and the deformation term with 2DAP. The single admittance coefficient between the local atmospheric pressure and the gravity variations was determined by regression analysis. Both reductions (6g_aps D and 6g_aP,adm) were compared for analysing the longperiodic tidal waves using the E T E R N A analysis program, Wenzel 1996. Table 1 shows the theoretical amplitude, the WahrDehant-model amplitude factor 8, Dehant (1987), and the analysed tidal parameters amplitude factor 8 and phase v: for the longperiodic tidal waves SA, SSA, M m and Mf. For the annual tidal wave SA the amplitude factor 8 is larger with 3D atmospheric pressure correction compared to single admittance (local atmospheric pressure) correction, whereas the semi-annual waves SSA and M m or Mf show nearly no differences.
545
546
NeumeyerJ.. T.Schmidt.C.Stoeber 5.2 Polar motion Polar motion causes changes in centrifugal acceleration, which can be measured with the SG. In order to separate polar motion from the preprocessed gravity data dgr~~ the following quantities were subtracted (Eq. 4): -
-
-
-
Analysed Earth tides (ET) calculated with the ETERNA program based on Wahr-Dehantmodel Ocean loading calculated with the GFZ program OCLO, Dierks (2004), based on Francies and Mazzega (1990). OCLO calculates the gravity variations induced by the ocean loading (dgo) in the time domain using data from the ocean tidal model FES2002 (Lefevre et al. 2002, Le Provost et al. 2002). Local groundwater level effect ( 6 g ~ ) based on a single admittance coefficient between local ground water level changes and gravity. Instrumental drift (dr) based on polar motion measured by SG and calculated from IERS data (PT). It is simulated by a first-order polynomial d r ( t ) = a o + alt and the drift parameters a 0 and
For comparison, the gravity effect of the polar motion was calculated for the Sutherland station with IERS (International Earth Rotation Service) polar motion data xp(t) and yp(t) (Bulletin B) (black curve) according to Torge (1989). The 3D attraction reduction 6g_ap3 D fits better to the IERS polar motion than the single admittance reduction 6g_ap ~d~.
5.3 Comparison of hydrology models variations
SG, G R A C E and derived gravity
A further consideration of the SPI part is shown by comparison of SG measurements with GRACE satellite (GFZ Potsdam solutions) and the global
"0
c
0 SG._3D = SG._sadm ~ . - ~ - - ~ GRACE (GF-Z)Imax=10 ~ VVGHMmodel Imax=10
6--
a~ are determined by a linear fit of d gsopo~ -
and PT. Atmospheric pressure 6g_ap~,~ respectively 6gso eo,(t)
=
effect
dg_ap~
6gr~(t)- ET(t)-6g_ap(t)
_
and
(4)
- 6 go~(t) - 6 gg~z(t) - dr(t)
The polar motion due to equation (4) measured at Sutherland station is shown in Fig. 8. The two different atmospheric pressure reductions have been applied, in dark grey 6g_ap~D and in grey 6g_aps,~ ~.
6-
-4--
Fig. 9 Comparison of superconducting gravimeter (SG_3D and SG_sadm), GRACE and hydrological model (WGHM) derived gravity variations hydrological model WGHM, D611 et al. (2003), derived gravity variations (Fig. 9). Details about GRACE and hydrology model data processing can be found in Neumeyer et al (2005). The SG gravity variations are calculated due to equation (5). 6gso(t ) = 6graw(t ) -- E T ( t ) - P T ( t ) - 6 g _ a p ( t )
(5)
-6go, U) - 6 gg~,(t) - dr(t)
-6-
Fig. 8 Polar motion (Sutherland/South Africa) with 6g apsn (dark grey) and 6g apsadm (grey) atmospheric pressure reduction in comparison with IERS polar motion (black)
The two different atmospheric pressure reductions 6g_ap3 ~ (SG_3D) and 6g_apsad m (SG_sadm) have been applied (Fig. 9). The SG gravity variations (SG_3D) with 6g_ap3 ~ atmospheric pressure reduction are closer to the hydrological model and also closer to GRACE derived gravity variations at Metsahovi station compared to the single admittance reduction with cYg_apsad~ (SG_sadm). We clearly see the difference of both gravity signals of about +1 lagal between winter and summer season.
Chapter79 • Improved Determination of the AtmosphericAttractionwith 3D Air Density Data and Its Reductionon GroundGravityMeasurements 5.4 A b s o l u t e gravity m e a s u r e m e n t s
Another example shows the influence of the SPI part on absolute gravity measurements. At station Pecny/Czechia gravity measurements have been carried out frequently with the absolute gravimeter FG5 from August 2001 to August 2003. Both atmospheric pressure correction methods (6g_ap~ D and 6g aP~adm)have been applied on these gravity measurements. Fig. 10 shows the results, single admittance correction with local atmospheric pressure (FG5_sadm) and 3D reduction with ECMWF data (FG5_3D). In winter season the li ~ ~ FG5_sadm 7 o FG5._3D I 6--
FG5_3D_Hy~
-4-
N1-
11"
~ 0 ~
~.
'~
-1-
Fig. 10 3D (FG5_3D) and sadm (FG5_sadm) atmospheric pressure reduction on absolute gravity measurements (Pecny/Czechia) and subtraction of WGHM model derived gravity variations (FG5_3D_Hy)
single admittance method subtracts to less compared to the 3D reduction. The difference is in total about 2Mgal. For interpretation of the absolute gravity measurements this fact must be considered. When we apply the gravity correction according to the hydrological model WGHM (Neumeyer et al. 2005), the oscillation of the signal is reduced to about _+2 Mgal. This is close to the accuracy of the absolute gravimeter.
6 Conclusions
The mass redistribution within the atmosphere induces a gravitational attraction on the test mass of the gravimeter which is independent from surface atmospheric pressure. The gravity recordings contain this attraction (SPI) that has a magnitude of about 2 Mgal. It can only be detected with 3D atmospheric pressure, humidity and temperature data. These data are globally available at European Centre for Middle Weather Forecasts (ECMWF).
The SPI part influences the metering precision of longperiodic gravity variations (Earth tides, polar motion, hydrology signal measured with e.g. SG's) and the absolute gravity measurements. Disregarding the SPI part can introduce a reduction error of up to a few Mgal. The quality of the attraction term computation depends first of all on the quality of the 3D atmospheric pressure data. Unfortunately the spacing (0.5 °) and the interval (6 hours) of the present 3D-ECMWF data is inadequate for precise calculation. A spacing of about 0.1 o and an interval of 1 hour or shorter is required. Consequently, higher precision can be achieved at the moment only by interpolation. For determination of the attraction term a data grid of 5 ° around the gravimeter with 60 height levels (up to 60 km) is sufficient. The deformation term should be determined with a data grid of about 10 ° around the gravimeter. The deformation effect of the air masses redistribution can be determined adequately with 2D surface pressure data using the Green's function method With the present quality of the 3D atmospheric pressure data seasonal gravity atmospheric pressure corrections on gravity data can be performed more precisely compared to corrections with local or 2D data. For the future, a further improvement is anticipated using pressure data with higher resolution in space and at shorter time intervals. For improving the attraction program 3DAP the topography around the station should be considered in more detail.
Acknowledgments
We like to thank Jan Kostelecky Department of Advanced Geodesy, Faculty of Civil Engineering CTU Prague, Czechia; Corinna Kroner Institute of Geosciences Friedrich Schiller University of Jena; Germany; Bruno Meurers Institute of Meteorology and Geophysics University of Vienna, Austria; Heiki Virtanen Finnish Geodetic Institute Masala; Finland; Herbert Wilmes Federal Agency for Cartography and Geodesy (BKG), Germany for providing the gravity data.
References
Boy J.-P., Hinderer J., Gegout P. (1998) The effect of atmospheric loading on gravity. Proc. of the 13t" Int. Symp. on Earth Tides, Brussels, 1997, Eds.: B. Ducarme and P. Paquet, 439-440.
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Neumeyer J.. T. Schmidt. C. Stoeber
Bulletin B, http://www.iers.org/iers/products/eop/ monthly.html Crossley D. J., Jensen O. G., Hinderer J. (1995) Effective barometric admittance and gravity residuals. Phys. Earth Planet. Int., 90, 221-241. Dierks, O., 2004. Ocean loading program OCLO. Internal report GFZ Potsdam. D611 P, Kaspar F, Lehner B (2003) A global hydrological model for deriving water availability indicators: model tuning and validation. J. Hydrol. 270:105-134. Etling D. (2002) Theoretische Meteorologie: Eine Einftihrung, Springer Verlag Farrell W. E. (1972) Deformation of the Earth by surface loads. Rev. Geophys. Space Phys., 10, 761-797. Francis, O., Mazzega, P., (1990). Global charts of ocean tide loading effects. J. Geophys. Res., 95, 11411-11424 Hagedoom J., Wolf D., Neumeyer J. (2000) Modellierung von atmosph~irischen Einfltissen auf hochgenaue Schweremessungen mit Hilfe elastischer Erdmodelle. Scient. Technical Report GFZ Potsdam STR 00/15. Kroner C. (1997) Reduktion von Luftdruckeffekten in zeitabh~ingigen Schwerebeobachtungen, Dissertation, Technische Universit~it Clausthal Kroner C., Jentzsch G. (1998) Methods of air pressure reduction tested on Potsdam station. Marees Terrestres Bulletin d'Informationes, Bruxelles, 127, 9834-9842. Lefevre F., Lyard F. H., Le Provost C., Schrama E. J. O., 2002. FES99: a global tide finite element solution assimilating tide gauge and altimetric information. J. Atmos. Oceanic Technol., Vol. 19, 1345-1356. Le Provost C, Lyard F, Lefevre F, Roblou L (2002) FES 2002 - A new version of the FES tidal solution series. Abstract Volume Jason-1 Science Working Team Meeting, Biarritz, France Merriam J. B. (1992) Atmospheric pressure and gravity. Geophys. J. Int., 109, 488-500. Neumeyer J. (1995) Frequency dependent atmospheric pressure correction on gravity variations by means of cross spectral analysis. Marees Ten'estres Bulletin d'Informationes, Bruxelles, 122, 9212-9220.
Neumeyer J., Barthelmes F., Wolf D. (1998) Atmospheric Pressure Correction for Gravity Data Using Different Methods. Proc. of the 13 th Int. Symp. on Earth Tides, Brussels, 1997, Eds.: B. Ducarme and P. Paquet, 431-438. Neumeyer J.,. Barthelmes F., Dierks O., Flechtner F., Harnisch M., Harnisch G.,. Hinderer J.,. Imanishi Y., Kroner C.,. Meurers B., Petrovic S., Reigber Ch., Schmidt R.,. Schwintzer P., Sun H.-P., Virtanen H. (2005) Combination of temporal gravity variations resulting from Superconducting Gravimeter recordings, GRACE satellite observations and global hydrology models. J. Geodes., (accepted). Neumeyer, J., Hagedom, J., Leitloff, J., Schmidt, T.,(2004) Gravity reduction with three-dimensional atmospheric pressure data for precise ground gravity measurements. J.Geodyn. 38,437-450.. Sun H.-P. (1995) Static deformation and gravity changes at the Earth's surface due to the atmospheric pressure. Observatoire Royal des Belgique, Serie Geophysique Hors-Serie, Bruxelles. Simon D. (2002) Modelling of the field of gravity variations induced by seasonal air mass warming 1990-2000. Marees Ten'estres Bulletin d'Informationes, Bruxelles, 136, 1082110836. Stoeber C. (2005) Modellierung und Analyse des Einflusses der 3D Luftdruckkorrektur in Supraleitgravimeter Registrierungen auf die langperiodischen Gezeitenparameter. Diplomarbeit, Institut for Geod~isie und Geoinformationstechnik Technische Universit~it Berlin (unpublished) Torge W. (2003) Geodesy, Walter de Gmyter, Berlin New York Vauterin P. (1998) The correction of the pressure effect for the Superconducting Gravimeter in Membach (Belgium). Proc. of the 13 th Int. Symp. on Earth Tides, Brussels, 1997, Eds.: B. Ducarme and P. Paquet, 447-454 Warburton R. J., Goodkind J. M. (1977) The influence of barometric - pressure variations on gravity. Geophys. J. R. Astr. Soc., 48, 281-292. Wenzel H.-G. (1995) Tidal data processing on a PC. In Hsu H.T. (ed), Proceedings of the 12~hInternational Symposium on Earth Tides, August 4 - 7, 1993, Science Press Beijing China, 235-244.
Chapter 80
Solid Earth Deformations Induced by the Sumatra Earthquakes of 2004-2005: GPS Detection of Co-Seismic Displacements and Tsunami-Induced Loading H.-P. Plag, G. Blewitt, C. Kreemer, W.C. Hammond Nevada Bureau of Mines and Geology and Seismological Laboratory, University of Nevada, Reno, Mailstop 178, Reno, NV 89557, USA.
Abstract The two great Sumatra earthquakes of December 26, 2004, and March 28, 2005, are associated with a number of geodynamic processes affecting Earth's shape, gravity field and rotation on a wide range of spatial and temporal scales, including co-seismic strong motion, static co-seismic displacements, surface loading due to the tsunami, normal modes, and transient postseismic deformations. The December 26, 2004, earthquake is the first one with Mw > 9.0 observed with space-geodetic techniques. Thus, new insight into the earthquake processes and associated phenomena can be expected from studies of space-geodetic, and particularly GPS, observations. Here we focus on the static co-seismic offsets and the tsunami loading signal. The global patterns of the static co-seismic offsets induced by the earthquakes are determined from time series of daily displacements resulting from an analysis of the GPS data from selected stations of the global IGS network. Our estimates are compared to similar estimates computed in other studies. The differences between the independent estimates are mostly within the uncertainties of the offsets (2 to 4 mm). The global pattern of the computed offsets is in agreement with the spatial fingerprint predicted by a reasonable rupture model. Initial model predictions of the tsunami-induced loading signal show that the peak amplitudes are of the order of 10 to 20 mm. If detectable in GPS time series with low latency, the loading signal could be utilized in a tsunami early warning system. Keywords: earthquakes, coseismic displacements, tsunami loading.
1
Introduction
Great earthquakes like the Sumatra events of 26 December 2004 (Mw = 9.2, denoted as "event A") and 28 March 2005 (Mw = 8.7, denoted as "event B") are associated with a number of geodynamic phenomena on a wide range of spatial and temporal scales including co-seismic strong motion and static displacements, free oscillations of the solid Earth, and, if located in an oceanic region, tsunamis. The redistri-
bution of water mass in the ocean associated with the tsunami induces transient perturbations of the Earth's surface and gravity field. Moreover, post-seismic deformations can continue for months and years after such great earthquakes. Event A was the first earthquake of Mw > 9.0 to be observed with space-geodetic techniques, in particular the global network of tracking stations for the Global Positioning System (GPS), that is coordinated by the International GNSS Service (IGS). The expected new insight from studies of the co- and post-seismic displacements into the rupture process of the earthquake and the associated phenomena has stimulated a number of GPS-based studies (e.g. Khan & Gudmundson, 2005; Banerjee et al., 2005; Vigny et al., 2005; Kreemer et al., 2005). GPS observations potentially provide information on the co-seismic displacements, including the static co-seismic offsets, post-seismic non-linear displacements, and tsunami induced loading signals. In order to determine the static offset with high accuracy, time series of daily or sub-daily coordinate estimates are required, and these time series also contain the postseismic signal. For co-seismic motion and tsunamiinduced loading, time series with high temporal resolution down to 30 sec or better are needed. The static co-seismic offsets are valuable constraints for models of the rupture processes. Moreover, the magnitudes of earthquakes determined from initial broadband estimates tend to be too low for large earthquakes (Menke, 2005), compromising early warning efforts (Kerr, 2005). Therefore, if available in near-real time, GPS estimates of the static offsets could help to improve the initial magnitude estimates for large earthquakes. Tsunamis travel the ocean as barotropic waves and thus induce a loading signal. If detectable by GPS with low latency, these signals could be integrated in an early warning system. Here, we first consider the static co-seismic offsets determined from GPS observations and, by comparing our own estimates to those computed by others, assess the accuracy to which these offsets are determined. Then we will briefly comment on the post-
550
H.-P. Plag. G. Blewitt • C. Kreemer. W. C. Hammond
Table 1° Selected GPS stations used in this study and their nominal distances (D) from the rupture zone. Not all farfield stations are shown. No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
Stat. samp ntus coco bako iisc mald hyde kunm dgar lhas pimo karr wuhn tnml seyl darw pert kit3 reun alic bahr guam lael tskb mali tow2 mbar hrao
D km
Long. o
Lat. o
498 1064 1722 1757 2142 2304 2340 2605 2688 2973 3208 3654 3711 3744 4394 4440 4502 4870 5003 5220 5313 5677 5953 5956 6047 6281 7072 7841
98.72 103.68 96.83 106.85 77.57 73.53 78.55 102.80 72.37 91.10 121.08 117.10 114.36 120.99 55.48 131.13 115.89 66.89 55.57 133.89 50.61 144.87 146.99 140.09 40.19 147.06 30.74 27.69
3.61 1.35 -12.19 -6.49 13.02 4.19 17.42 25.03 -7.27 29.66 14.64 -20.98 30.53 24.80 -4.67 - 12.84 -31.80 39.14 -21.21 -23.67 26.21 13.59 -6.67 36.11 -3.00 -19.27 -0.60 -25.89
seismic displacements observed at regional stations after the two events. The surface loading caused by the tsunami is studied on the basis of a model prediction of the sea surface heights variations caused by the tsunami. We use the well validated Green's function approach to estimate the solid Earth deformations induced by the tsunami on the basis of modeled sea surface displacements. Finally, we will consider the necessary improvements to the observing system and the analysis strategy in order to fully utilize the potential of GPS and other Global Navigation Satellite Systems for studies of large earthquakes and early warning systems.
2
Observed co-seismic displacements
surface
In order to determine the static co-seismic offset for the two events, GPS data from a total of 39 stations within 7,600 km of the rupture zone (Table 1) were processed for the interval 1 January 2000 to 21 May 2005 using the GIPSY-OASIS II software package from the Jet Propulsion Laboratory (JPL). Daily station coordinates were estimated using the precise point posi-
tioning method (Zumberge et al., 1997) with ambiguity resolution applied successfully across the entire network by automatic selection of the ionospheric- or pseudorange-widelane method (Blewitt, 1989). Satellite orbit and clock parameters, and daily coordinate transformation parameters into ITRF2000 were obtained from JPL. Ionosphere-free combinations of carrier phase and pseudorange were processed every 5 minutes. Estimated parameters included a tropospheric zenith bias and two gradient parameters estimated as random-walk processes, and station clocks estimated as a white-noise process. For all stations, secular velocities where estimated for the interval 1 January 2000 to 25 December 2004. These velocities are assumed to represent the interseismic tectonic motions prior to event A, and they are used to detrend the data. Moreover, using the stations more than 4000 km away from the epicenter as reference stations, all daily solutions were then transformed by a 7-parameter Helmert transformation onto the constant velocity solution (denoted as "spatial filtering"). In Fig. 1, the detrended time series for four stations close to the rupture zones are shown for the interval 1 January 2004 to 21 May 2005. Particularly the sites SAMP and NTUS show large offsets at the time of the two events and significant post-seismic deformations in the months following each of the events. For sites further away, the offsets appear to be hidden in the long-period variations present in the time series. As illustrated by these examples, the determination of the static co-seismic offsets is hampered by the presence of noise and other signals in the time series. In particular, it is difficult to estimate reliable errors. An early analysis of the time series of SAMP and NTUS for event A was presented by Khan & Gudmundson (2005). They used 5 day averages of the north and east displacements to estimate simultaneously a linear trend and the offsets at the time of event A. Their static offsets are (given in mm for east, north) ( - 1 4 5 . 2 + 3.2, - 1 2 . 1 + 1.8) and ( - 2 2 . 0 + 2.0, 6.1 -+- 1.6) for SAMP and NTUS, respectively. Banerjee et al. (2005) analysed the data of 41 farfield and regional stations using the GAMIT/GLOBK software. Static offsets for event A were then determined by differencing the mean positions in five days before and after the event A. They claim to see a coherent surface motion for distances up to 4500 km from the epicenter. Vigny et al. (2005) computed static co-seismic offsets from time series of 79 regional and global CGPS sites. These time series were determined using the GIPSY-OASIS ii software in precise point positioning mode. The daily solutions for 14 days before and after
Chapter80 • Solid Earth DeformationsInducedby the SumatraEarthquakesof 2004-2005 80
80
RA
RA
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2005.60
IlSC
Fig. 1: Detrended displacement time series for permanent GPS sites close to the rupture zone. For each station, the vertical (RA), north (LA) and east (LO) component of the displacement vector are shown. Lower left diagrams are for SAME upper left for NTUS, lower right for IISC, and upper right for HYDE. Vertical lines indicate the times of the two events.
event A were combined into campaign-like solutions, which were then projected onto ITRF. The offsets were computed as the differences of the 14-day averages after and prior to event A. Kreemer et al. (2005) using the same time series as in the present study (see above), computed the coseismic displacements by differencing the average coordinates from 28 days after and before the event. While increasing the span of days before and after the event reduces the noise in the computed average, there is a trade-off with bias from post-seismic displacement, at least for the two stations in the near field of the event (SAMP and NTUS). Therefore, for these two stations, Kreemer et al. (2005) estimated a logarithmic function describing the post-seismic displacements simultaneously with the static offsets. The alternative approach chosen here is to model the time series x by the function NH
Nc
x(t) - a + bt + E ~iH(ti) + E Aj sin(wit + Cj) i=1
j=l
(1) where t is time, a is a constant, b a constant rate, and l l is a Heaviside function, with ai giving the displacements associate with event i. We have chosen NH = 2 with the times tl and t2 coinciding with the events A and B, respectively. The harmonic constituents are used to represent a seasonal cycle in the GPS time
series, and we have chosen to include an annual and semi-annual constituent (i.e. N c = 2). In the fit, we solve for a, b, Oil, OL2 and the amplitudes of the cosine and sine terms of the harmonic constituents. The data prior to approximately December 2002 was found to be of lower quality. Therefore, the data interval for the fit ofEq. 1 to the displacement time series was constrained to 1 January 2003 to 21 May 2005. Examples of the resulting models and the residuals for the stations included in Fig. 1 illustrate that the model function is appropriate for most stations not having significant post-seismic motion (Fig. 2). Particularly for SAMP, the large post-seismic displacement after event B biases the estimated offset for this event if the postseismic displacement is not modeled properly. For stations with significant post-seismic deformations, the approach by Kreemer et al. (2005) is appropriate. The resulting offset parameters are listed in Table 1. The co-seismic offsets estimated by Banerjee et al. (2005), Vigny et al. (2005) and Kreemer et al. (2005) are also given for comparison. For most stations, the estimates differ by several mm and the error bars do not overlap for all stations. This indicates that the determination of the offsets depends on the GPS processing strategy and the model used to approximate the time series prior and after the offsets. However, there is no clear systematic difference between the four sets (Fig. 3).
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Fig. 2" Determination of co-seismic static offsets using eq. (1). Diagrams marked with O show the observations and the model function, while those marked/ig show the residuals, that is the difference between the observations and the fitted model function.
0
0
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Station
Fig. 3: Comparison of the computed offsets. For each station, the deviation from the mean of all estimates are shown for the east (lower) and north offsets. Triangles: Banerjee et al. (2005), inverted triangles: Vigny et al. (2005), squares: Kreemer et al. (2005), diamonds: this study. Kreemer et al. (2005) computed predicted static offsets using a layered elastic spherical Earth model (Pollitz, 1996) and a rupture model that best fits their static offsets computed from the GPS series. The predicted displacement field is a dipole with the main motion on both sides of the fracture being directed towards the rupture zone (Fig. 4). Surprisingly large displacements of the order of 5 mm are predicted for areas
as far away as South Africa and South America. The offsets computed in the various studies are in general agreement with the predictions of the model derived by Kreemer et al. (2005) (Fig. 5). Despite the fact that the model is a best fit to the Kreemer et al. (2005) offsets, no systematic differences are detected between the predictions and any of the four sets of estimates. Deviations are generally of the order of a few mm. The estimates for SAMP and NTUS computed by Khan & Gudmundson (2005) fall well into the range covered by the other four estimates. The largest differences are found for SAME where the east offset derived here is biased by the unaccounted post-seismic deformations. For the stations being further than 4000 km away from the rupture zone, the offsets derived here are generally smaller than those determined by Kreemer et al. (2005). Since these offsets should be close to zero, this may be taken as an indication that using a model function provides slightly better estimates. However, in order to prove this, a more detailed comparison of the observed spatial fingerprint to the predicted fingerprint would be required. It is emphasized here that Kreemer et al. (2005) used identical time series for the determination of the static offsets as the present study but a different ap-
Chapter 80 • Solid Earth Deformations Induced by the Sumatra Earthquakes of 2004-2005
T a b l e 2" Comparison o f static co-seismic offsets determined b y four independent analyses.
Stat. samp ntus COCO
bako iisc mald hyde kunm dgar lhas pimo karr wuhn tnml seyl darw pert kit3 reun alic bahr guam lael tskb mali tow2 mbar hrao
Banerjee et al., 2005 (~E (~N OE ON mm mm mm mm -135.0 -14.8 6.0 2.2 -13.8 2.4 3.0 1.6 3.7 1.1 3.4 1.8 0.9 -3.7 3.6 1.8 14.9 -1.4 2.7 1.5 9.9 -2.8 11.5 1.0 -10.8 -1.8 -3.8 -9.2 -1.6 -2.5 1.8 1.6
-2.7 -8.5 2.2 -4.2 -4.2 1.5 -4.5 -2.1 0.3 -0.6 1.0 -1.5
-1.5
-6.0
2.6 3.5 3.8 5.2 4.0 3.8 3.5 3.4 7.7 4.5 1.4 1.8
6.6
1.5 1.8 2.1 2.6 2.1 2.1 2.0 2.0 4.3 2.3 1.3 1.1
Vigny et al., (~E (~N mm mm -142.8 -12.7 -19.4 6.4 2.2 5.0 -1.7 1.0 11.7 -0.1
2005 O'E mm 3.6 2.6 2.5 4.1 2.7
O'N mm 2.0 1.3 2.4 1.9 2.4
7.0 -6.4 5.7 1.7 -7.1 0.9 -2.0
-0.8 -6.9 5.3 -1.3 -1.3 1.1 -2.9
2.5 3.9 3.0 2.5 5.4 4.4 3.5
3.2 5.1 1.7 6.6 2.8 3.3 4.0
0.1 1.3 -0.4
1.8 0.3 -1.3
4.9 3.8 3.3
2.3 4.1 4.6
1.5 -1.0
1.7 1.6
8.6 4.9
6.7 2.4
0.3
-2.3
3.8
3.7
3.3
Kreemer et al., 2005 (~E (~N O'E mm mm mm -139.0 -9.0 1.1 -22.0 8.0 0.6 1.3 4.9 0.6 2.3 3.7 1.3 11.7 -1.1 0.6 9.9 4.0 1.1 5.8 -1.4 0.6 -6.2 -5.0 1.1 5.4 6.0 0.6 -1.1 -1.3 0.8 -7.9 0.7 0.6 -1.9 2.3 0.6 -3.9 -1.9 0.6 -5.5 -1.8 0.6 -3.5 -3.1 0.8 -3.6 0.8 0.6 1.0 1.6 -1.9 0.1 -4.2 1.9 -2.6 2.0 1.6 1.4 1.5
1.1 3.7 1.1 2.5 2.3 -2.0 -1.8 0.7 -0.7 0.5 -0.4
O'N mm 0.6 0.4 0.4 0.6 0.4 0.6 0.4 0.6 0.4 1.1 0.6 0.4 0.6 0.4 0.4 0.4
0.6 1.3 0.4 0.4 0.6 0.8 0.4 0.8 0.6 0.6 0.6
0.6 0.6 0.4 0.4 0.4 0.6 0.6 0.6 0.4 0.4 0.6
~E mm -160.0 -24.6 -1.5 -0.7 11.7 9.0 6.8 -8.4 6.7 -1.6 1.0 -1.3 -6.4 -10.2 -8.8 -3.4 -0.8 -1.4 -1.4 -0.6 -0.9 -1.9 -0.5 -4.2 0.1 0.6 0.1 0.5
This study (~N O'E mm mm -15.0 0.6 7.7 0.6 1.6 0.6 2.7 0.6 -2.7 0.6 1.9 1.0 -1.4 0.6 -5.5 0.6 6.1 1.1 0.2 0.8 3.9 0.4 -1.2 0.6 -0.9 0.6 -0.1 0.6 -3.8 0.6 2.0 1.0 -1.4 0.6 -1.1 0.7 0.5 0.9 -1.6 0.6 1.3 0.6 5.1 0.7 1.0 1.1 2.7 0.6 -0.7 0.8 0.3 0.6 -0.1 1.2 1.9 0.7
O'N mm 0.6 0.6 0.6 0.6 0.6 1.0 0.6 0.6 1.1 0.6 0.4 0.6 0.6 0.6 0.6 1.0 0.6 0.7 0.9 0.6 0.6 0.7 1.1 0.6 0.8 0.6 1.2 0.7
0
-60 300 330
0
30
60
90 120 150 180 210 240 270
Fig. 4- Predicted co-seismic displacement field. The predictions are for a layered elastic spherical Earth model and a rupture model that best fits the offsets computed b y K r e e m e r et al. (2005). p r o a c h , w h i l e V i g n y et al. ( 2 0 0 5 ) u s e d t h e s a m e s o f t ware package for the GPS analysis but a different ap-
and the other estimates. T h e c o m p a r i s o n o f t h e f o u r sets o f e s t i m a t e s r e v e a l s
p r o a c h to r e a l i z e t h e r e f e r e n c e f r a m e a n d to c o m p u t e
t h a t (1) t h e e r r o r e s t i m a t e s o f K r e e m e r
t h e o f f s e t s . B a n e r j e e et al. ( 2 0 0 5 ) , o n t h e o t h e r h a n d ,
o u r l e a s t s q u a r e s e r r o r s a p p e a r to b e o v e r - o p t i m i s t i c ,
et al. ( 2 0 0 5 ) a n d
used time series obtained independently with a differ-
w h i l e t h e e r r o r e s t i m a t e s o f V i g n y et al. ( 2 0 0 5 ) a n d
ent s o f t w a r e p a c k a g e . N e v e r t h e l e s s , no s y s t e m a t i c dif-
a l s o B a n e r j e e et al. ( 2 0 0 5 ) m i g h t b e p e s s i m i s t i c ; (2)
f e r e n c e s a r e f o u n d b e t w e e n t h e B a n e r j e e et al. ( 2 0 0 5 )
the actual u n c e r t a i n t i e s o f the steps are m o r e o f the
553
554
H.-P. Plag. G. Blewitt. C. Kreemer. W. C. Hammond
number of observed time series without distorting the predicted spatial pattern. Eq. (2), in fact, can be used directly in the search for the best-fit model. Static offsets detected with statistical significance for event B are constrained to relatively few stations (Fig. 6). It is interesting to note that the offset for SAMP is too large to be explained by a dislocation model.
A
14
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6
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Fig. 5: Comparison of the observed and predicted offsets for
event A. Symbols are the same as in Fig. 3. ,0J
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7;
;
order of 2 to 4 mm; and (3) only the offsets for stations experiencing displacements larger than a few mm, i.e. if they are in the areas on both sides of the rupture zone, can be determined with statistical significance. We expect that using the spatial finger-print as predicted by models (see Fig. 4 above) in a multi-station regression for the static offsets, will allow a much better determination of the observed displacement field. For that, we suggest using the modified model function Np
Nc
~(t) - a + bt + E 7i~H(ti) + E Aj sin(wit + Cj) i=l
j=l
(2) to represent the vector of the horizontal components, w h e r e / ~ is the horizontal displacement vector predicted by a rupture model. While the c~i in eq. (1) are the displacements determined independently for each time series, the regression coefficients ~/i are global quantities scaling the model predictions to fit a large
Tsunami
loading
signal
A tsunami travels barotropically through the ocean and causes movements of the ocean water mass comparable to ocean tides. Tsunami similarly load and deform the solid Earth. The displacements of the Earth's surface and the changes in the Earth's gravity field induced by surface loads can be computed using the theory of Farrell (1972). Predictions of the perturbations are computed through a convolution of the surface load expressed through the surface pressure and mass density field with the appropriate Green's function. The initial computations carried out here to estimate the order of magnitude of the induced displacements uses a static Green's function, i.e. the load Love numbers required to compute the Green's functions (see Farrell, 1972, for the details) are calculated neglecting the acceleration term in the field equations for the displacements. Considering that the longest elastic eigenmodes for a non-rotating Earth are of the order of 53 minutes (e.g. Lapwood & Usami, 1981), this static approximation is appropriate for loading with periods of several hours or longer. Tsunamis waves, however, have periods of 30 minutes and less, and the static solution can only give a first order estimate of the amplitude, while arrival times of the loading signal will be strongly biased. Using dynamic Green's function requires not only a convolution in space but also a convolution over the complete history of the tsunami. Thus, for a more accurate modeling of the amplitudes and particularly the temporal variation of the loading signal, a far more complex computation needs to be implemented. For the tsunami caused by Event A, the ocean bottom pressure variations are computed from the sea surface height anomalies predicted by the MOST model of the NOAA Tsunami Research Center (see Titov et al., 2005, and the reference therein). The sea surface heights are given with a mean spatial resolution of 0.3°in longitude and ~ 0.1°in latitude and a temporal resolution of 5 minutes. For the initial loading cornputation, a spatial resolution of 2.5 °was used, with the resolution being increased to 0.25 °in coastal areas and
Chapter80
3O
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Earth Deformations Induced by the Sumatra Earthquakesof 2 0 0 4 - 2 0 0 5
120 .
30
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240
300
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270
600 240
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close to the observer point. Shortly after the earthquake the peak deviation in ocean bottom pressure reaches values of over 200 hPa in the vicinity of the earthquake region (Fig. 7). After two hours, the peak values are still close to 200 hPa with all land areas around the Indian ocean experiencing vertical displacements of 5 to 12 mm (Fig. 7). After nearly a day, the energy of the tsunami has been distributed through the global ocean, with peak bottom pressure values nearly reaching 10 hPa (Fig. 8). The resulting vertical displacements of the Earth's surface are of the order of +1 ram. Peak signals of the order of 20 mm were found in vertical displacements. These peak signals lend to the prospect of detecting tsunami loading using GPS. In particular, the deformational signal with spatial wavelength of the order of 102 km should be detectable. However, the processing of GPS observations with high temporal resolution of better than five minutes will have to be improved in order to be able to detect such signals with low latency. In particular, sidereal filtering (Choi et al., 2004) and appropriate regional filtering (Wdowinski et al., 1997, e.g.) will have to be utilized. Peak signals of the order of 20 mm were found in vertical displacements. These peak signals lend to the prospect of detecting tsunami loading using GPS. In particular, the deformational signal with spatial wavelength of the order of 100 km might be detectable if improvements can be made to the quality of current high rate GPS estimates of the vertical. To as-
o
-
.
-30
-30
-60
7ooo 14ooo21;oo
Fig. 7: Ocean bottom pressure and induced vertical displacements computed from predicted sea surface heights. Upper diagrams: 30 minutes after the earthquake, lower diagrams: 2 hours after the earthquake. Left: ocean bottom pressure in Pa; right: vertical displacement of the solid Earth's surface in mm.
30
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-0.9
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-0.3
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0.3
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1.2
Fig. 8: Same as Figure 7 but for 23 hours after the earthquake.
sess the potential of GPS to detect tsunami loading, we estimated station positions for a 10-station network as white noise every 30 seconds while simultaneously estimating satellite and station clocks as white noise, for the 7 days before and after the earthquake. We then applied a sidereal filter to calibrate for multipath (Nikolaidis et al., 2001), and then averaged the 30second vertical estimates into 5-minute normal points. The resulting time series show a scatter of the order of 10 ram, however systematic variations can be seen that sometimes exceed 20 mm during a 24-hour period. Moreover, a tsunami might happen to occur on a day of poor weather, when passing weather fronts can cause significant systematic error (Gregorius & Blewitt, 1999). Clearly, much research and development is needed to improve high rate GPS vertical positioning, for example, by the application of more effective methods of sidereal filtering (Choi et al., 2004) and spatial filtering (Wdowinski et al., 1997), which might be adapted for high rate network solutions.
4
Conclusions
Event A (26 December 2005) caused the entire Earth surface to deform at a geodetically significant level (> 0.1 ram), with important implications for terrestrial reference frame maintenance. Co-seismic displacements > 2 mm were clearly observed at GPS stations as far away as 4000 km from the rupture zone. The comparison of the co-seismic offsets determined in four different studies shows that the determination of these offsets depends on the GPS analysis as well as the assumptions for the subsequent analysis of the
555
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H.-P. Plag. G. Blewitt. C. Kreemer. W. C. Hammond
displacement time series. Most of the differences are attributed to noise in the displacements time series. Based on the comparison, the uncertainties in the offsets are estimated to be 2 to 4 mm. Using the predicted spatial fingerprint of the displacements in the estimation of the observed offsets is expected to improve the estimates of the displacement field. However, a main limitation for the determination of the displacement fields of large earthquakes arises from the number of available permanent GPS sites. At a minimum, attempts should be made to densify the global tracking network in regions with potentially large or great earthquakes to a spatial resolution of 500 km to 1000 km. Combined with models, the stations could be used to determine the effect of these great earthquakes on the global reference frame and to account for it. The tsunami caused by event A led to transient redistributions of oceanic water mass comparable to those associated with ocean tides. Similar to ocean tidal loading, the tsunami induced displacements of the Earth's surface and gravity changes, with the vertical displacements reaching up to 20 mm. With a sensitive observing system based on e.g. kinematic GPS, these deformations in principle could be sensed in real-time and integrated into a warning system. However, in order to extract this signal in near-real time, other effects such as the apparent displacements caused by variations in multi-path associated with orbital repletions will have to be reduced sufficiently.
Acknowledgment The authors are grateful to the IGS, IERS and JPL for providing the GPS data, the reference frame, and GIPSY OASIS II software as well as precise estimates of satellite orbits, clocks, and reference frame transformations, respectively. The sea level height data for the tsunami was made available by Vasily Titov of the NOAA Tsunami Research Center. Paul Denys and an anonymous reviewer provided valuable comments. The work at UNR was funded by grants from NASA Solid Earth and NASA Interdisciplinary Science.
References Banerjee, R, Pollitz, F. F., & Bfirgmann, R., 2005. The size and duration of the Sumatra-Andaman earthquake from far-field static offsets, Science, 308, 1769-1772. Blewitt, G., 1989. Carrier phase ambiguity resolution for the Global Positioning System applied to geodetic baselines up to 2000 km, d. Geophys. Res., 94(B8), 10187-10283.
Choi, K., Bilich, A., Larson, K. M., & Axelrad, R, 2004. Modified sidereal filtering: Implications for high-rate GPS positioning, Geophys. Res. Lett., 31, L22608, doi:l 0.1029/2004GL021621. Farrell, W. E., 1972. Deformation of the Earth by surface loads., Rev. Geophys. Space Phys., 10, 761-797. Gregorius, T. L. H. & Blewitt, G., 1999. Modeling weather fronts to improve GPS heights: A new tool for GPS meteorology?, J. Geophys. Res., 104, 15,261-15,279. Kerr, R., 2005. Failure to gauge the quake crippled the warning effort, Science, 307, 201. Khan, S. A. & Gudmundson, O., 2005. GPS analysis of the Sumatra-Andaman earthquake, EOS, Trans. Am. Geophys. Union, 86, 89-94. Kreemer, C., Blewitt, G., Hammond, W. C., & Plag, H.-R, 2005. Global deformations from the great 2004 Sumatra-Andaman earthquake observed by GPS: implications for rupture process and global reference frame, Earth Planets Space, In press. Lapwood, E. & Usami, T., 1981. Free Oscillations of the Earth, Cambridge University Press, Cambridge. Menke, W., 2005. A strategy to rapidly determine the magnitude of great earthquakes, EOS, Trans. Am. Geophys. Union, 86, 185,189. Nikolaidis, R. M., Bock, Y., de Jonge, R J., Agnew, D. C., & Van Domselaar, M., 2001. Seismic wave observations with the Global Positioning System, J. Geophys. Res., 106, 21,897-21,916. Pollitz, F. F., 1996. Coseismic deformation from earthquake faulting on a layered spherical Earth, Geophys. J. Int., 125, 1-14. Titov, V., Rabinovich, A. B., Mofjeld, H. O., Thomson, R. E., & Gonzfilez, F. I., 2005. The global reach of the 26 December 2004 Sumatra tsunami, Science, 309, 2045-2048. Vigny, C., Simons, W. J. F., Abu, S., Bamphenyu, R., Satirapod, C., Choosakul, N., Subarya, C., Socquet, A., Omar, K., Abidin, H. Z., & Ambrosius, B. A. C., 2005. Insight into the 2004 SumatraAndaman earthquake from GPS measurements in southeast Asia, Nature, 436, 201-206. Wdowinski, S., Bock, Y., Zhang, J., & Fang, R, 1997. Southern California Permanent GPS geodetic array: spatial filtering of daily positions for estimating cosesimic and postseismic displacements induced by the 1992 Landers earthquake, J. Geophys. Res., 102, 18,057-18,070. Zumberge, J. F., Heflin, M. B., Jefferson, D. C., & Watkins, M. M., 1997. Precise point positioning for the efficient and robust analysis of GPS data from large networks, J. Geophys. Res., 102, 5005-5017.
Chapter 81
Environmental effects in time-series of gravity measurements at the Astrometric-Geodetic Observatorium YYest-
erbork (The Netherlands)
I. Prutkin and R. Klees Delft Institute of Earth Observation and Space Systems (DEOS) Delft University of Technology, 2629 HS Delft, Kluyverweg 1, The Netherlands
Abstract. The effect of air pressure variations, soil moisture variations and groundwater level variations on time-series of gravity measurements acquired with the tidal gravimeter ET-15 at the Astrometric-Geodetic Observatorium Westerbork (WAGO), The Netherlands, has been investigated. The gravity measurements have been corrected for gravimeter drift, Earth body tide, and ocean loading effects. The residual signal has been smoothed using a variational smoothing algorithm. An agro-hydrological model provided the change in soil moisture in the vicinity of the gravity bunker over the measurement period. A precise finite element model of the gravity bunker and the surrounding layers was developed to compute the impact of changes in soil moisture content and ground water level variations on gravity. The combined effect of soil moisture variations and groundwater level variations varies between-7 #Gal a n d - 3 #Gal. It is dominated by soil moisture variations in the layers above the gravimeter between-7 #Gal a n d - 5 . 5 #Gal. Soil moisture variations below the gravimeter and groundwater level variations contribute between 0 #Gal and 2.5 #Gal. The analysis of two years of gravity and local air pressure variations show high correlation factors above 80% for periods of one day and shorter, whereas periods between one day and half a week are correlated between 50% and 80%. Over periods shorter than half a week, an admittance factor of-0.37 # G a l / m b a r has been obtained. The correlation for periods longer than half a week is very low, i.e. these periods should not be used to determine an admittance factor from local air pressure data. Rainfall events have a significant influence on gravity measurements at the WAGO site. A proper modeling requires measurements of precipitation, evaporation and run-off. This is the subject of flmlre studies.
Keywords.
Environmental effects on gravity,
admittance factor, soil moisture, groundwater level variations.
1
Introduction
In 1998, the Westerbork Astrometric-Geodetic Observatory (WAGO) became operational. WAGO is the successor of the Observatory for Space Geodesy at Kootwijk (KOSG). It has facilities for satellite laser ranging, GPS tracking, VLBI, and high-accurate relative and absolute gravity measurements corresponding to international standards. The gravimetry platform is a concrete cube of 3 x 3 x 3 m 3, in order to obtain the highest stability. Moreover, it is protected against the nuisance of moving VLBI telescopes in the vicinity of the station by the construction of surrounding barrages. Over the platform, a climatecontrolled bunker is built wherein absolute and relative gravimeters can operate. A weather station controls pressure, humidity, temperature, speed and direction of wind. Since 2002 permanent gravity measurements are performed with the LaCoste-Romberg ET15 tidal gravimeter and several Scintrex CG3M gravimeters. Moreover, absolute gravity is measured yearly with an FG-5 gravimeter. Since 2004 hydrological information (soil moisture, ground water level, and precipitation) is being gathered. The fluctuation of the groundwater level in the aquifers due to precipitation was measured using several piezometers with divers. In the shaft of the bunker, the height of the free water level was measured, too. At the Royal Dutch Meteorological Institute (KNMI) stations Hoogeveen and Eelde, which are located within a radius of 20 km around the bunker, evaporation, radiation and temperature measurements were acquired. These data are used as input to an agro-hydrological model, which quantifies soil
558
I. P r u t k i n • R. K l e e s
moisture variations at the WAGO site. Soil toolsture variations, groundwater level variations and local atmospheric pressure variations are used to analyze their effect on time series of gravity measurements at the WAGO site. In this study, we will discuss the results of these activities. The main focus is on the quantification of the hydrological constituents soil moisture and groundwater level variations on gravity observed with the ET-15 gravimeter. Moreover, the relation between local air pressure data and gravity data at the Westerbork station is investigated. Finally, the effect of ocean-tide loading on the Westerbork station is quantified using different ocean-tide loading models. The outline of the paper is the following: in section 2 we review the data pre-processing strategy applied. The main focus is on the effect of gravimeter drift and the amplitude and quality of ocean tide loading corrections. The latter is important because the WAGO site is only 50 km away from the coastline. After subtraction of the contributions of tides and drift, we applied some smoothing to reduce residual effects caused by the imperfectness of the models. The smoothing is done using a new variational smoothing algorithm, see section 2. The effect of soil moisture variations, groundwater level variations, and atmospheric pressure variations on gravity is addressed in sections 3 and 4. In section 3, the results of the hydrological modeling are presented. In particular, we quantify the contribution of soil moisture and ground water level variations to the time series of gravity measurements. Some other examples of the estimation of hydrological effects on gravity data can be found in e.g. [5] and [7]. Section 4 is devoted to the admittance between local atmospheric pressure variations and gravity variations. The frequency dependent admittance is well known and investigated by different authors (see, e.g. [2] and {9]). It will be shown that significant correlations between the two time series can only be obtained for certain frequency bands. Almost no correlation between air pressure and gravity variations is found for periods above half a week, whereas correlations for shorter periods vary between 50% and 90%. Periods of anti-correlation can be attributed to strong rainfall events. Section 5 contains the main conclusions of this study.
2
Pre-processing of gravity data
The body Earth tides of the Westerbork station have been calculated by means of the package ETERNA, version 3.30 ([11]). To estimate the effect of ocean tide loading, three programs have been compared: LOAD89 [4], O L F G / O L M P P [10] and a new program CARGA [1]. CARGA and O L F G / O L M P P use the ocean tide model FES99, whereas LOAD89 uses the model FES95.2. These ocean tide roodels are quite similar. CARGA uses the most sophisticated approach and its accuracy for the Westerbork station is better than 1% [1]. Therefore, this model is used as a reference. The differences between O L F G / O L M P P and LOAD89 w.r.t, the CARGA model are 11% (0.24 #Gal) and 16% (0.34 #Gal), respectively. Ocean tide
loading (CARGA)
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Figure 1: Gravity ocean loading at the WAGO site calculated with various programs. Top panel: output of CARGA [ 1 ] . Bottom panel: difference between CARGA and O L F G / O L M P P [10].
Among several approaches to model the gravimeter drift, the best results have been obtained using Chebyshev polynomials. For the standard time-series of approximately 40 days, we found a Chebyshev polynomial of the 5th order to give the best results. The order of the polynomial was changed successively until the best fit in terms of amplitude and frequency character of residual gravity and expected gravity signal of local atmospheric pressure variations has been found. Of course, usage of such a drift model means the elimination of the lowest frequencies in the time series of gravity measurements. This is not critical for the analysis of at-
Chapter 81
• Environmental Effects
in Time-Series of Gravity Measurements at the Astrometric-Geodetic Observatorium Westerbork (The Netherlands)
mospheric pressure, soil moisture and groundwater level variations, because they contain almost no power at these frequencies. The same holds for the very high frequencies, which mostly represent noise and un-modeled signal. Therefore, these frequencies have been removed using a variational smoothing algorithm. Assume that u0 is a function to be smoothed (e.g. the residual gravity data). The main idea is to find the smoothest possible function u among all functions, equally distant from the given function u0. The mathematical formulation leads to two conditions, which have to be met simultaneously: I /z In
2 ~zO L L2
3
Evaluation of the hydrological signal
The time-series of gravity measurements have been acquired at the Westerbork station with the tidal gravimeter ET-15. The gravimeter platform in the bunker (Fig. 3) is located 75 cm below ground level, which has significant consequences for the effect of soil moisture variations on gravity. The overall contribution of soil moisture
__ (~2
i rain
(1)
Eq. (1) are discretized, which gives the following variational problem: S
-
u °
_
~//Gravitimete Centre line of gravltl Z'mete r 75 crn b e l o w surface
Problem (2) is solved iteratively by means of Lagrangian multipliers. The effect of this algorithm on the time series of gravity measurements and on the signal power spectral density are shown in Fig. 2. Effect o f
i 200
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i 1000
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Figure 3: Cross section of the gravity bunker at the WAGO site. Notice the location of the gravimeter, which is 75 cm below ground level.
variational smoothing algorithm
,
0
Groundwaterlevel Grou [
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Frequency[1/day]
Figure 2" The effect of variational smoothing in the time domain (top panel) and in the frequency domain (bottom panel).
variations to the gravity signal is a combination of three factors: (i) soil moisture variations in the layers above the gravimeter level; an increasing soil moisture in this layer reduces the observed gravity; (ii) soil moisture variations in the layers below the gravimeter level; an increasing soil moisture in these layers will increase the observed gravity; (iii) the direct gravitational effect of groundwater level variations. Note that there is a relation between (ii) and (iii) in the sense that an increased groundwater level reduces the volume of the layers below the gravimeter level. Special emphasis has been made on a proper modeling of the constituents (i)-(iii). To model soil moisture variations in the layers above the gravimeter level, a precise finite element model of the bunker and the surrounding subsurface layers has been developed. The whole volume was divided into a number of small prisms (see
559
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I. Prutkin • R.
Klees
Fig. 5). The gravitational effect of soil moisture variations over each prism was computed using Gauss-Legendre cubature formulas. The total gravitational effect of soil moisture variations in these layers was obtained by summing up the contribution of all prims. To calculate the Newton integral with 4 correct digits, the area or 350 m around the gravimeter has been taken into account. The information about the variation of groundwater level and soil moisture between the driest and the wettest days of the summer 2004 has been provided by Wageningen University [3]. The information is taken as the output of an agro-hydrological model driven by measurements of precipitation, evaporation, radiation, temperature, and groundwater levels in the aquifers around the Westerbork station. The layer between the ground surface and the centre line of gravimeter is the most influenced by precipitation (see Fig. 4, from [3]). Due to increase of soil moisture, which amounts to 0.3 cma/cm a, this layer of thickness 75 cm generates quite considerable additional gravitational signal. Because the layer is located above the gravimeter (see Fig. 3), an increased soil moisture content gives a negative gravitational signal. For summer 2004, this signal varies between - 5 . 5 and - 7 #Gal. Maximum variation of soil moisture (%) 0 0
5
10
15
20
,
,
,
,
25
30
-100
~-~-200
-300
-400
Figure 4: Distribution of soil moisture variation with depth at the WAGO site during summer 2004 [3]. The contribution of soil moisture variations in the layer below the gravimeter level and of groundwater level variations to the gravity signal has been approximated conventionally using the Bouguer plate approximation. This is sufficiently accurate for modeling the soil moisture
variation in this layer, because the amplitude is very small. The Bouguer plate approximation is also sufficiently accurate to model the groundwater level variations at the WAGO site. The maximum groundwater level at WAGO is 70 cm below the gravimeter level; the lowest groundwater level is 290 cm below the gravimeter level (Fig. 4). The positive effect of groundwater level variations and soil moisture variations in the subsurface layer between gravimeter level and groundwater level is less than 2.5 #Gal. Thus, the combined seasonal effect of soil moisture and groundwater level variation at the WAGO site is in the range [-7,-3] #Gal. This is significantly above the noise level of the ET-15 gravimeter, which is about 0.5 #Gal.
3 2.5 2 1.5 1 0.5 0
0
~ ~
14
Figure 5: Finite-element model of the bunker to compute the gravity signal of soil moisture variations above the gravimeter level.
4
Correlation between air pressure and gravity variations
We used a time series of about 2 years to analyze the relation between local air pressure variations and gravity variations. Our study has confirmed, that the correlation between air pressure and gravity variations strongly depends on the frequency band. For periods of one day and less, we observe a high correlation of 8 0 - 9 5 % . This is in agreement with known results (e.g. [8]). One typical record is shown in Fig. 6. The correlation between gravity and air pressure for periods longer than one day is more difficult to access. To compute an admittance factor, we analyzed the whole two-year time-series of gravity and air pressure observation. For all frequencies below 1 cycle per day, we found an averaged value of-0.37 #Gal/mbar, which is not too far from the value-0.356 #Gal/mbar, suggested by
Chapter 81 • Environmental Effects in Time-Series of Gravity Measurements at the Astrometric-Geodetic Observatorium Westerbork (The Netherlands) lO
r
"r
5
0.5
/ ~;,t~
A .,:
/i IV" ~.
i
o
i
:
~\
i
-5
o
31-12-03 15 o
t
'
10-1-04
20-1.04
80-1-04
'
'
'
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,
,
-0.5
-5
-15 31-12-03 0
5
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35,
Time [hours]
Figure 6: Gravity variations and gravitational effect of air pressure variations for periods less than one day. The correlation is 81°-/0. The gravity variations are corrected for Earth body tides, ocean loading tides and gravimeter drift prior to correlation analysis.
Merriam [6] as a local admittance on the base of theoretical considerations. However, the correlation between gravity variations and air pressure variations is very low for frequencies below 1 cycle per day. For this reason, we split the frequency band in two parts: a high-frequency part covering periods from one day till half a week, and a low-frequency part, covering periods longer than half a week. For the high-frequency part we found reasonable correlation coefficients between gravity and air pressure variations; typical correlation coefficients vary between 54% and 71%. For the low frequency part, however, we could not find any significant correlation. Therefore, we only determined an admittance over the frequencies between 1 cycle per day and 0.3 cycles per day. Again, we found a value of-0.37 # G a l / m b a r for this frequency band. Un-modeled rainfall events have a strong influence on the correlation between gravity and atmospheric pressure variations. This situation is displayed in Fig. 7. After a long relatively dry period the hydrological effect on gravity is minimal. During a strong rainfall event, air pressure drops down, and gravity goes up. In the high frequency part of gravity variations, the same maximum could be easily observed. But the low frequency part of the gravity signal has at this moment not a maximum, but a minimum. The moment of 'anti-correlation' can be clearly related to a strong rainfall event. It is quite in agreement
"f 15
J
Figure 7: Gravity and air pressure variations for different frequency bands during a strong rainfall event. Top panel: periods between one day and half a week; mid panel: periods longer than half a week; bottom panel: rainfall events.
with the considerations in the previous section, that after a strong rainfall event, the total gravirational effect of the increased soil moisture and of the uplift of groundwater level should be negative. This result has been noticed many times when comparing time series of gravity and atmospheric pressure variations at the WAGO site.
5
Summary and conclusions
A first analysis of the gravity signal of atmospheric pressure variations, groundwater level variations and soil moisture variations at the Westerbork Astrometric-Geodetic Observatory (WAGO) has been done. The following conclusions are drawn: i. The ocean tide loading effect at the WAGO site is ±2 #Gal. The differences between various ocean tide loading algorithms can reach values op to 0.5 #Gal, which is comparable with the noise level of the ET-15 gravimeter. 2. The drift of the ET-15 gravimeter has to be modeled properly prior to studying environmental effects on gravity. Best results have been obtained using Chebyshev polynomials. The degree of the polynomial has to be
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I. Prutkin • R. Klees
chosen carefully and is mainly a function of the length of the time series. A methodology for the automatic selection of the optimal polynomial degree is still missing. 3. The total effect of soil moisture and groundwater level variation is negative at the WAGO site, due to the particular location of the platform below ground level. The effect varies between [-7,-3] #Gal, and is significantly above the noise level of the ET-15 gravimeter. 4. Strong rainfall events show-up as a negative minimum in the low-frequency part of the spectrum of gravity variations. This is in agreement with the modeling of the total gravitational effect of soil moisture and ground water variations at the WAGO site. So far, only two hydrological constituents have been analyzed at the WAGO site. Recently, the station has been equipped with several rainfall meters and a lysimeter. They will provide more detailed information about precipitation, evaporation, and run-off. Moreover, the network of piezometers will be extended and the reading will be automized. The results of the analysis of these data will be the subject of a forthcoming paper.
Acknowledl~ments Prof. Trevor Baker from P r o u d m a n Oceanographic Laboratory in Liverpool, United Kingdora, has provided us with the ET-15 gravimeter. This support is gratefully acknowledged.
References [1] Bos, M.S. and Baker, T.F. (2005). An estimate of the errors in gravity ocean tide loading computations. Y. Geod., 79:50-63.
[2] Crossley, D. J., Jensen, O. G., Hinderer, J.
[3]
[4]
[5]
[6]
(1995). Effective barometric admittance and gravity residuals. Phys. Earth Planet. Int., 90:221-241. de Jong, B. and Ros, G. (2004). The effect of water storage changes on gravity near Westerbork. MSc thesis, Hydrology and Quantitative Water Management Group, Wageningen University, The Netherlands. Francis, O. and Mazzega, P. (1990). Global Charts of Ocean Tide Loading Effects. J. Geophys. I~es., 95(C7):11,411-11,424. Kroner, C. (2001). Hydrological effects on gravity data of the Geodynamic Observatory Moxa. J. Geod. Soc. Japan, 47(1):353-358. Merriam, J.B. (1992). Atmospheric pressure and gravity. Geophys. J. Int., 109:488-500.
[7] Meurers, B., Van Camp, M., Petermans, T., Verbeeck, K., Vanneste, K. (2005). Investigation of local atmospheric and hydrological gravity signals in Superconducting Gravimeter time series. Geophysical Research Abstracts, 7:07463. [8] Mukai, A., Higashi, T., Takemoto, S., Naito, I. and Nakagawa, I. (1995). Atmospheric effects on gravity observations within the diurnal band. J. Geod. Soc. Japan, 41:365-378. [9] Neumeyer, J. (1995). Frequency dependent atinospheric pressure correction on gravity variations by means of cross spectral analysis. Bulletin d'Information Mardes Terrestres, 122:92129220. [10] Scherneck, H.-G. (1991). A parametrized solid Earth tide mode and ocean loading effects for global geodetic base-line measurements. Geophys. J. Int., 106(3):677-694. [11] Wenzel, H.-G. (1996). The Nanogal Software: Earth tide data processing package ETERNA 3.30. Bulletin d'Information Mattes Terrestrcs, 124:9425-9439.
Chapter 82
Numerical models of the rates of change of the geoid and orthometric heights over Canada E. Rangelova, W. van der Wal, M.G. Sideris, Department of Geomatics Engineering, University of Calgary, 2500 University Drive NW, Calgary, Alberta, Canada, T2N 1N4 P. Wu, Department of Geology and Geophysics, University of Calgary, 2500 University Drive NW, Calgary, Alberta, Canada, T2N 1N4
Abstract. For the purpose of modernization of the vertical datum in Canada, rates of change of geoid and orthometric height are computed independently of postglacial rebound simulations. The designed procedure is based on the general least-squares collocation approach and uses as input rates of change of absolute gravity and absolute vertical motion velocities. The procedure is applied in a stepwise manner in order to account for new available data. The predicted geoid rate has a peak value of 1.3-1.4 mm/yr over Hudson Bay and agrees within the computational accuracy, i.e. _+0.1 mm/yr, with the postglacial rebound simulated rate. The predicted rate of change of orthometric height has the general pattern of the vertical motion of the rebounding crust, but also shows some local effects mainly inherited from the rates of gravity data, e.g., the uplift predicted over Southern Alberta as a result of the strong negative gravity rate signal and the subsidence in the Mackenzie River delta. It is found that the achieved prediction accuracy of _+2.4 mm/yr can be increased by using more accurate GPS velocities and extending their coverage to the north as well as incorporating precise relevelled data. Also, improvement is expected from using vertical velocities from a combination of tide gauge records and satellite altimetry sea surface heights. Keywords. Postglacial rebound, rate of change of gravity, vertical crustal motion, rate of change of geoid, vertical datum, inverse multiquadrics.
1 Introduction Traditionally, the definition of the vertical datum is based on the concept of the geoid as a particular equipotential surface of the gravity field that coincides with the mean sea level (MSL). MSL is computed by averaging records of sea level variations in order to define a conventional "zero" (origin at the geoid) for orthometric heights determined and propagated via precise levelling. In
this way, a vertical control network, through which the vertical datum is accessible, is established. The vertical datum is subject to large systematic errors and distortions. They are one of the main factors that contribute to discrepancies between orthometric, geoid, and GPS (geodetic) heights (H, N, and h, respectively) at benchmarks of the vertical control network (see, e.g., Fotopoulos 2003). The discrepancies are computed via a simple relationship given by Heiskanen and Moritz (1967, Eq.4.58, p. 176) On the other hand, with the new geopotential models (provided by CHAMP and GRACE) of increased accuracy at long wavelengths, and expected considerable improvement in accuracy of medium and short wavelengths by GOCE, the computation of a cm-level geoid will become realizable in the very near future (Tscherning et al. 2000). Thus, the geoid becomes an attractive alternative to the traditional vertical datum in those countries, such as Canada, where large territories with harsh environmental conditions are not covered by vertical control networks (V6ronneau 2001). Large scale and magnitude secular vertical crustal motion and temporal changes in the geopotential can also contribute to the discrepancies between orthometric, geoid, and GPS heights, especially if they refer to different epochs. In Canada, the temporal effects are mainly due to the prominent postglacial rebound (PGR), a viscoelastic response of the Earth to the melting of the Laurentide ice-sheet about 18,000-20,000 years ago. Although not as large as the distortions in the levelling network, these temporal effects must be quantified if the vertical datum is to be updated or re-established using the geoid as a reference surface for orthometric heights. In the context of the temporal variations, the absolute vertical crustal motion measured by GPS, 1~, and the rate of change of orthometric height, Iit, differ by the rate of change of geoid, lq, i.e., 1~ - Iq + I2I.
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E. Rangelova • W. van der Wal. M. G. Sideris • P. Wu
In Canada, the availability of absolute rates of change of gravity, g, and measured GPS vertical velocity, t~, allows studies of the spatial variations of the temporal changes of gravity field and vertical crustal motion over the whole country to be conducted independently of postglacial rebound simulations. Compiled maps of vertical crustal motion in Canada based on relevelled data are documented in a series of papers starting with Vanf~ek and Christodulidis (1974), Vanf~ek and Nagy (1981), and later Carrera et al. (1991). Relevelled segments of the first order network contain information about vertical motion, but they are not considered at this stage of the study. The objective of this paper is to investigate a procedure based on the general least-squares collocation (LSC) approach for modelling rates of change of geoid and orthometric heights using solely the available gravity rate data and GPS vertical velocities. However, the procedure can also accommodate repeated levelling data as well as rates of vertical motion from tide gauges and radar satellite altimetry data. In addition to PGR, assumed to be the only factor that contributes to the general trend in data, the procedure is designed to also account for gravity and height rates of change over Canada due to subsidence and erosion described by Pagiatakis and Salib (2003). In the next section, the computational algorithm is explained. That is followed by a description of the numerical tests and a comparison with postglacial rebound simulations. Finally, future improvements in the proposed methodology are suggested.
2 Methodology 2.1
Stepwise Least-Squares Collocation Procedure
The two-dimensional space representation of velocity surfaces derived from scattered relevelling data, traditionally uses a bivariate polynomial series (see, e.g., Vanfeek and Christodulidis 1974). This pure functional approach, though offering a simple mathematical representation of the velocity surfaces, has disadvantages such as numerical instabilities due to the failure of the polynomials to reproduce accurately a surface in areas with lack of data and a risk to over-parameterize the surface in the presence of a limited number of data. In this paper, a general least-squares collocation approach is adopted. It was used by Hein and Kistermann (1981) to separate the local motion of a non-tectonic origin (which takes on a stochastic
description) from the regional trend due to tectonics (described analytically). In the case of isostatic adjustment of the crust, the LSC approach has been followed, e.g. by Danielsen (2001), to model the rates of the vertical uplift in Fennoscandia. A polynomial surface of order 4 has been used to describe the trend in vertical crustal motion data. In this paper, in contrast, inverse multiquadrics (MQs) of Hardy (1990) are used. The MQ analysis is based on a linear combination of (different shape) hyperboloids (basis functions). The basis functions can be located at data points or at arbitrary nodes. In this respect, MQs are able to interpolate more accurately a velocity surface from scattered data than polynomials are (see Holdahl and Hardy 1979, Hardy 1990). Another important difference from the studies of PGR using LSC approach is that the output of the procedure consists of the rates of change of geoid in addition to the rates of change of orthometric heights. This requires considering an additional step in the procedure, i.e., converting the estimated rate in the data into the rate of the geoid change. For example, after the trend of the rate of vertical motion is estimated in the LSC procedure, it can be transformed into the rate of geoid change via a mass flow model (see Eq.7). The observation equation of the general leastsquares collocation model is given by:
1= A X + t + n
(1)
where 1 is the data vector, t is the signal vector component of the data, n is the observation noise, and A is the coefficient matrix of the unknown trend parameters, X. With the a priori covariance matrices of the signal and noise, Ctt = coy(t, t) and Cnn = cov(n,n), respectively, the least-squares collocation solution is given by the well known formulas (Moritz, 1980, p.144) as follows: - (ATC-'A) -jATC-jl,
~ - CstC-' ( 1 - AX) (2a)
where C =Ctt +Cnn ; and s is the signal to be predicted. The error covariance matrices are given by: E x x - ( A T C - 1 A ) -1,
E s s - C s s - C s t C - 1 C t s (2b)
In case a new data set is available and the covariance matrices between the old and new data sets are known, the solution can be derived in a stepwise manner. The estimated parameters, X , and the predicted signal, ~, are corrected as follows"
Chapter 82 • Numerical Models of the Rates of Change of the Geoid and Orthometric Heights over Canada
corr -- ~ nt- 8 X ,
Scorr -- ~ + 8 S .
(3)
Eqs.(3) correspond to the formulas (19-20) and (1927) in Moritz (1980, pp. 144-156). The corrections 8X and 8s stand for the improvement of the initial estimates due to the new data included. Analogously, the error covariances at the first step are corrected using the formulas (20-23) and (2024) given in Moritz (1980, pp. 144-156).
2.1.1
Modelling the Trend Components
The inverse MQs are used to model the trend component of the data. The inverse MQ basis function is defined as follows:
lp = O ( d p j ) [ O ( d ~ j
) f cD(dij )]-10(d ij )f 1.
(6)
O(dij ) will correspond to the coefficient matrix A in Eqs.2a and b if the least-squares collocation approach is followed. The trend in the data, attributed entirely to PGR, is converted into a trend surface of the rate of geoid change using a mass flow model. It is based on the assumption that the crustal uplift is accompanied by an inflow of masses (with density 9) from the upper mantle (see Sj6berg 1982). According to this model, the geoid rate at a point i depends on the differential m a s s pfljd(yj
at a point j located on the internal
sphere with radius r as follows: l~li = 1 Gpla jdoj
(I)(dj) - (dj2 + A2 ) -1/2 ,
where
(4)
d j is the Euclidean distance between the
points (x, y) and (x j, y j ), and A is a parameter that controls the shape of the basis function. A can be varied, so that the basis functions range from a cone (A=0) to flat sheet-like surfaces in order to interpolate accurately the particular data. In the interpolation mode, the basis functions are centered at nodes coinciding with the data points. The MQ equation is then given (in matrix notation) by: ocO(dij ) = 1
(5a)
where dij is the distance between the locations of the i th data point a n d jth basis function; 0c is a vector of
the unknown coefficients to be determined from the known components 1i of the data vector in the linear system of Eq.5a. Values at the grid points p are interpolated as follows:
Y /
where Lij -- ~R2 -k- r 2 - 2 r R c o s l l / i j
is
the
spatial
distance between points i and j, R is the mean Earth radius, and y is the mean gravity.
2.1.2
Modelling the Stochastic Components
The absolute vertical motion (measured by GPS) itself does not contain information about changes in the geopotential due to the mass redistribution below the Earth's crust. Only combined with measurements of gravity changes does it suffice to derive the changes in the geopotential. Since the rate of change of geopotential, w, is given by the following integral relation (see, e.g., Heck 1984): 2y @ - S t ( g +--~I2I),
R St(. ) - - ~ g ~ ( . )S(gt)d•,
(8)
(y
the two components of w can be evaluated as: I - St(g),
lp =O(dpj )O(dij )-11.
(7)
Lij
x~2 - --~ St(tit) •
(9)
(5b) Following Heck (1984),
In least-squares approximation the number of nodes and their locations can be varied and the selection is usually based on accuracy tolerance. A forward selection algorithm based on orthogonal least-squares (see, e.g., Chen et al. 1991) is designed to select the nodes that can explain certain amount of data power. The unexplained power defines the accuracy tolerance. The least-squares prediction of the v e c t o r lp is as follows:
( K i j ) k m = COV{(XV i ) k , ( X ~ q j ) m },
k,m = 1,2
(10)
defines the covariance kernel as covariance between @k and v~m , k , m = 1,2 at any two points i and j. Note that the points i and j are both located on the sphere with radius R that represents the geoid, while, in Eq.7, j was used to designate the differential mass on the interior sphere.
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The covariance kernels can be described by the Gaussian model as follows: 2 Kk m _ Ak m ~km
d 2
exp(-
),
k , m = 1,2
-q [ Rate of geopot, change / via modified kernel Input ~g,G~ -gravity / ~I~ Stokes integral 1= {l~,Oh _vertical vel./ wl = St(g) W2 : (23'/R)St(fl)
rate|
!
(11) 0.06 < cy~ < 0.88 ~tGal/yr
2%km
c~fi > 1 m m / y r
Here d denotes the distance, A k m ' C 2 m / 2 k , m = 1,2 stand for variances, and the parameter "c is related to the correlation length. The auto- and cross-covariances between the rates of change of gravity, orthometric, and geoid heights can be formed using the inverse of the relationships in Eq.9:
I2I - R--~-St-l(0e2).
Stepwise LSC : - estimate the trend - predict the signal in data
(12)
A flowchart of the applied computational algorithm is shown in Figure 1. The input data consist of the absolute rates of change of gravity and GPS vertical velocities, and the output is in terms of rates of change of geoid and orthometric height. Since the rate of geopotential depends on the latter (see Eq.9), the computations are iterated after the absolute vertical velocities, 1~, are replaced by estimated I2I. The evaluation of the Stokes integral (Eqs.8 and 9), via fast Fourier transform (FFT) requires gridding of noisy data by an appropriate method. Among all investigated methods (continuous curvature splines in tension, (inverse)MQs and thinplate splines), only inverse MQs have been able to produce at the same time an accurately interpolated and smooth surface from the irregularly distributed noisy gravity rates and GPS vertical velocities. In order to derive the stochastic information, the long wavelengths below a certain degree are filtered out via applying a two-dimensional (2D) FFT with a spheroidal Stokes kernel (see, e.g., Vanfeek and Featherstone 1998, Eq.23, p. 687) in the evaluation of Eq.9. The components of the rate of geopotential, Wl and w z, are interpolated at the scattered data point locations, and the empirical covariances are computed by averaging products of any two values in the predefined bins with respect to the distance. Then, the parameters Akm and 1;km of the covariance models in E q . l l are fitted to the empirical covariance functions by least-squares adjustment with constraints on the variances
variance factor: accept the model Yes I Compute the trend and signal components of the rates of change of geoid and orthometric heights
The reader can consult Heck (1984) for a list of analytical auto- and cross-covariance functions.
Iterative Computational Algorithm
Corrections for a new
data set
Z 2 - test on a posteriori
23'
2.1.3
Fitting analytical kernels - Gaussian model for the rates of geopotential change due to g and h
!
Correct the model
- st-l(wl),
1~is replaced by ~I
Output N, I:I,o s, on
Figure 1 The iterative computational algorithm based on stepwise least-squares collocation.
2 Akm'l;km
[ 2 k , m = 1,2 to be equal to the empirically estimated variances. A Z2 test on the a posteriori variance factor at a 95% confidence level is set as a criterion for accepting or rejecting tested models. The algorithm can test automatically a number of models defined by combinations of adopted functional descriptions for the data trend component and the covariance matrices computed from the stochastic information derived with different degree (e.g., 10, 15, or 20) spheroidal kernels.
2.2
Postglacial Rebound Modeling
The postglacial rebound simulation consists of two parts. The first is the response of the Earth to a surface load; the Peltier-Wu normal mode theory is used to simulate this (see Wu and Peltier 1982). The input is a spherical, radially symmetric Earth with 6 layers that represent the major discontinuities found in the Earth. The stress-strain behaviour is modeled as Maxwell rheology, which assumes a linear relation between stress and strain. Density and rigidity for each of the six layers are obtained by volume averaging values from the widely used Preliminary Reference Earth Model (PREM) (Dziewonski and Andersen 1981). Viscosity values are selected close to the VM2 model of Peltier
Chapter 82 • Numerical Models of the Rates of Change of the Geoid and Orthometric Heights over Canada
(2004), see Table 1. The output of this part of the simulation are visco-elastic Love numbers, which represent the response of the Earth to a unit load in the spectral domain. The second part of the simulation describes the ice-ocean-earth interaction during growth and melt of ice sheets in an ice age cycle. The process is described by the sea-level equation, which in its basic form is (Farrell and Clark 1976): S(% ~, t ) = C(% ~, t){G(q~, ~, t ) - R(q), ~,, t)}. (13) ~'bU
G and R are the perturbations of the geoid and solid surface, respectively, induced by postglacial rebound. Both have a convolution form of the surface load with the respective Greens functions that contain the visco-eastic Love numbers. The function C, by definition, takes the values 1 for ocean and 0 for land. It is time dependent to account for ice which occupies ocean area (e. g., Hudson Bay) during part of the glaciation. Apart from this so-called marine-based ice, the coastline is taken to be equal to the present-day coastline. Also, the rotational feedback mechanism is not considered in the simulation because it leads to a degree 2 order 1 signal with geoid rate in North America less than 0.1 mm/year (Peltier 1999, Figure 13). Since Eq.13 contains the sea level both on the left side and on the right side (as part of the load that causes change in G and R), the equation is an integral equation solved by the method of Mitrovica and Peltier (1991). With sea level computed and ice level known, the present day geoid perturbation can be computed. The ice model used is the smoothed ICE-3G of Tushingham and Peltier (1991), which provides a good fit to observed sea level data and is independent of the data used in this study. The growth of the ice sheets is simulated by reversing the ICE-3G history, taking 7,000 years instead of 1,000 years for each step, as in Milne et al. (1999). The simulated rate of the geoid change is plotted in Figure 2. The maximum rate of 1.3 - 1.4 mm/yr is located over Hudson Bay; it gradually decreases to 0.6 - 0.8 mm/yr in the Great Lakes area.
Table 1. Stratification for the Earth model used in the postglacial rebound simulation, with viscosities selected to
be close to the VM2 model (Peltier 2004). Layer Depth, [km] Visc., [Pa.s]
Lith. UM1 UM2 LM1 0 115 400 670 ~
4.10 20 4"10 20 2"10 21
LM2
Core
1171 4"10 21
2891 0
. 0.1
.
. 0.3
.
ZIU
. 0.5
0.7
0.9
I 1.1
I 1.3
I mm/year 1.5
Rate of Geoid C h a n g e Figure 2 Rate of change of geoid from a PGR simulation (ICE-3G), in mm/yr.
;ites
, 0.1
, 0.3
, 0.5
LDU , 0.7
Z/U , 0.9
I I .I
I 1.3
I mm/yr 1.5
Rate of Geoid G h a n g e (trend) Figure 3 Trend of the geoid rate from a mass flow model, in mm/yr.
3. Numerical Tests and Discussion of Results The first data set used in the numerical tests consists of the historical time rates of change of absolute gravity from the readjustment of the primary Canadian Gravity Standardization Network (Pagiatakis and Salib 2003). The second data set is the GPS vertical velocities provided by GSD/Natural Resources Canada. Although data from two networks with a different time span of measurements are combined, it is assumed that both data represent the same temporal changes. This would be true for the secular changes associated with postglacial rebound; for local effects this might be disputable. The LSC procedure can be designed such that corrections for the trend and covariance functions are computed from gravity rates and GPS velocities at collocated absolute gravity/GPS sites.
567
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E. R a n g e l o v a • W. v a n d e r W a l . M. G. Sideris • P. W u
Model b
Model a
;ites
;ites
, 0.1
, 0.3
LbU , 0.7
, 0.5
/bU
V_./U I 0.9
i I .I
, 1.3
I mm/yr 1.5
. 0.1
.
. 0.3
Z/U I 0.9
. 0.5
0.7
I 1.1
I 1.3
I mm/yr 1.5
Rate of Geoid Change
Rate of Geoid Change
ites
ites
. -8
.
. -6
. -4
-'' . . -2
260" . 0
2
4
270 ° I 6
I 8
~,~v I 10
I 12
I 14
I mm/yr 16
, -8
, -6
, -4
-,-,u , -2
, 0
260" , 2
, 4
270" I 6
I 8
~,~v I 10
I 12
I 14
I mm/yr 16
Rate of Change of Orthometric Height
Rate of Change of Orthometric Height
ites
ites
LDU Z/U . . . . . . . . . I I I I l i I I mm/yr 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 4.4 4.8 5.2 5.6 6.0 6.4 6.8 7.2
LbU Z/u . . . . . . . . . I I I I I I I mm/yr 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 4.4 4.8 5.2 5.6 6.0 6.4 6.8 7.2
St. Dev. of Rate of Change of Orthometric Height
St. Dev. of Rate of Change of Orthometric Height
Figure 4 Rate of change of geoid and orthometric height predicted by LSC and accuracy of rate of change of orthometric height, in mrrdyr.
Chapter 82 • Numerical Models of the Rates of Change of the Geoid and Orthometric Heights over Canada
While the rates of change of gravity are well distributed over the territory of Canada, almost all GPS sites are located south of the 60 degree parallel. Since long wavelengths of postglacial rebound can be modelled by the gravity rates alone, the procedure is modified so that, at the first step, only gravity information is considered. To derive the vertical velocities, a value of-0.16 gGal/mm for the gravity-to-height ratio has been computed using Eq.18 in Fang and Hager (2001, p. 541). This value is characteristic for PGR, so the vertical motion and changes in geopotential will be entirely interpreted in terms of the latter. The rates of change of geoid and orthometric height and the standard deviation are plotted in Figure 4, Model (a); see also Tables 2 and 3. To predict the signal, the wavelengths below degree 15, which contain 62 percent of the total data power, have been filtered out (see section 2.1.3). To model the data trend component, 13 nodes that can explain the filtered data power have been selected as significant. At the second step, the Model (a) estimates are corrected by adding the GPS velocities to obtain the combined solution, i.e. Model (b) in Figure 4. The comparison between the postglacial rebound geoid rate (Figure 2) and the trend component of the LSCestimated geoid rate of Model (b) in Figure 3 reveals a similar pattern with a peak value over Hudson Bay, an identical surface gradient in most areas, and agreement at the level of the standard error of the predicted rate, i.e. _0.1 mm/yr. In Western Canada, where other processes govern gravity and height changes as well as in the areas with lack of data, the difference increases to 0.2-0.3 mm/yr. It is worth repeating that the PGR model is completely independent of the data used in this study. This result verifies the ability of the inverse MQs to approximate correctly the general changes in the velocity surface using the absolute rates of change of gravity and GPS velocities and also the correctness of the assumed mass inflow model. The use of accurate vertical velocities is critical for prediction of the rate of change of orthometric height. It can be seen from the comparison between Models (a) and (b) and Table 2 that GPS velocities only slightly decrease the magnitude of the geoid rate and the prediction error. In contrast, the peak value for the rates of orthometric height in the eastern part of Hudson Bay decreases by 2-3 mm/yr, while accuracy is improved by 1.3 mm/yr. Note that the extreme prediction errors are encountered in the areas with lack of data (see Figure 4). The minimum error, associated with the sites where both gravity and GPS data are available, is below 1 mm/yr.
Table 2. Standard deviation of the predicted rate of change of geoid, in mrrdyr. mean _+0.13 _+0.10
Model (a) Model (b)
min _+0.11 _+0.08
max _+0.14 _+0.12
Table 3. Standard deviation of the predicted rate of change of orthometric height, in mm/yr. mean ___3.7 ___2.4
Model (a) Model (b)
min ___2.1 _+0.8
max __.7.2 _+5.2
14 12 o10 c-~
. . . j~ jl' o,~}, ii~,,,&~.
.
~-
LSC geoid rate ' LSC orth. heigth rate ..... PGR geoid rate _ - . - PGR v ert. displ, r_ate_
--,
~8 ~6 o
~4 .c_ 2 0
5
10
Figure 5 Power
15
20
25
30
35
angular degree
40
45
50
spectra of rates of change of geoid and orthometric height.
Finally, the power spectra of the LSC-estimated and PGR-simulated rates are compared in Figure 5. Both spectra of the PGR and LSC-predicted rates of change of geoid have a peak at degree 5. The slight difference (less than 1% of the total power) in the band from degrees 13 to 25 can be due to mismodelling effects, but it also reflects the signal component in the geoid rate (being between-0.2 to 0.2 mm/yr across the studied territory) which may include the contribution from local processes. Significant decrease in the low degrees and a shift of the power towards higher degrees is observed for LSC-estimated rate of orthometric height. The major contribution comes from the dome-like structure of rates that is due to still insufficient data density mainly in north, but some local patterns, e.g. in Western Canada contribute as well.
4 Conclusions The rates of change of geoid and orthometric heights can be predicted in a stepwise least-squares collocation procedure at the level of accuracy of 0.1 and 2.4 mm/yr, respectively. The good agreement
569
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E. Rangelova • W. van der Wal. tvl. G. Sideris • P. Wu
of the postglacial r e b o u n d simulation and L S C p r e d i c t e d geoid rates (within the given data density and the c h o s e n Earth and ice m o d e l p a r a m e t e rs) confirms that the d e s i g n e d p r o c e d u r e is able to predict the m a i n g e o d y n a m i c signatures over C a n a d a using solely absolute rates of c h a n g e of gravity and GPS vertical velocities. T h e s e results are subject to further i m p r o v e m e n t s before being useful for the m o d e r n i z a t i o n of the vertical d a t u m in Canada. For instance, the accuracy of the p r e d i c t ed rate of c h a n g e of o r t h o m e t r i c height, m o s t l y in Southern Canada, will increase after precise relevelling data are included. U s i n g additional information in terms of vertical velocities derived f r o m water tide gauges and radar satellite altimetry data in the Great L a k e area as well as using c o l l o c a t e d absolute g r a v i t y / G P S sites to i m p r o v e the trend c o m p o n e n t and signal c o v a r i a n c e functions are next possible steps.
Acknowledgements The authors gratefully acknowledge Dr. L.L.A. Vermeersen for kindly providing normal mode codes and GSD, Natural Resources, Canada for providing GPS data, as well as the funding received by the GEOIDE NCE. The authors are also very grateful to reviewers for their constructive criticism and suggestions. Most of the figures are potted by The Generic Mapping Tools (Wessel and Smith 1998).
References Carrera, G., P. Vanf6ek and M.R. Craymer (1991). The compilation of a map of recent vertical crustal movements in Canada, Open file 50SS.23,244-7-4257, Dep. of Energy, Mines and Resour., Ottawa, Canada. Chert S., C.F.N. Cowan and P.M. Grant (1991). Orthogonal Least Squares Learning Algorithm for Radial Basis Function Networks. IEEE Transactions on Neural Networks, Vol. 2, No. 2, pp. 3 0 2 - 309. Danielsen, J.S. (2001). A land uplift map of Fennoscandia. Survey Review, Vol.36, pp. 282-291. Dziewonski, A.M. and D.L. Anderson (1981). Preliminary reference Earth model. Phys. Earth Planet. Inter., Vol. 25, pp. 279-356. Fang, M. and B.H. Hager (2001). Vertical deformation and absolute gravity. Geophys. J. Int., Vo1.146, pp. 539-548. Farrell, W.E. and J.A. Clark (1976). On postglacial sea level. Geophys. J. R. Astr. Soc., Vol. 46, pp. 647-667. Fotopoulos, G. (2003). An Analysis on the Optimal Combination of Geoid, Orthometric and Ellipsoidal Height Data. Doctoral thesis. URL: http://www, geomatic s. uc al gary. c a/links/GradThes es. html. Hardy, R.L. (1990). Theory and applications of the multiquadric-biharmonic method: 20 years of discovery 1968-1988. Computers Math. Applic., Vol.19 (8/9), pp. 163-208. Heck, B. (1984). Zur Bestimmung vertikaler rezenter Erdkrustenbewegungen und zeitlicher )~nderungen des Schwerefeldes aus wiederholten Schweremessungen und
Nivellements. Deutsche geodi:itische commission, Reihe C, Nr.302, Mtinchen 1984. Hein, G.W. and R. Kistermann (1981). Mathematical foundation of non-tectonic effects in geodetic recent crustal movement models. Tectonophysics, Vol. 71, pp. 315-334. Heiskanen, H. and H. Moritz (1967). Physical Geodesy. Freeman, San Francisco. Holdahl, S. and R.L. Hardy (1979). Solvability and multiquadric analysis as applied to investigations of vertical crustal movements. Tectonophysics, Vol. 52, pp. 139-155. Milne, G.A., J.X. Mitrovica, J.L. Davis (1999). Near-field hydro-isostasy: the implementation of a revised sea-level equation. Geophysical Journal International, Vol. 139, pp. 464-482. Mitrovica, J.X. and W.R. Peltier (1991). On Postglacial Geoid Subsidence over the Equatorial Oceans. J. Geophys. Res., Vol. 96, No B 12, pp. 20053-20071. Moritz, H., (1980). Advanced Physical Geodesy. H.Wichmann, Karlsruhe, Germany. Pagiatakis, S.D. and P. Salib (2003). Historical relative gravity observations and the time rate of change of gravity due to postglacial rebound and other tectonic movements in Canada. J. Geophys. Res., Vol. 108, No B9, 2406, doi:10.1029/2001JB001676, 2003. Peltier, W.R. (1999). Global sea level rise and glacial isostatic adjustment. Global and Planetary Change, Vol. 20, pp. 93-123. Peltier, W.R. (2004). Global Glacial Isostasy and the Surface of the Ice-Age Earth: The ICE-5G (VM2) Model and GRACE. Annu Rev. Earth Palnet. Sci., Vol. 32, pp. 111. Sj6berg, L.E. (1982). Studies on the land uplift and its implications on the geoid in Fennoscandia. University of Uppsala, Institute of Geophysics, Department of Geodesy, Report No 14. Tscherning, C.C., D. Arabelos and G. Strykowski (2000). The 1-cm geoid after GOCE. In: International Association of Geodesy Symposia, Vol. 123 Sideris (ed.): Gravity, Geoid, and Geodynamics 2000, Springer- Verlag Berlin Heidelberg 2001, pp. 267-270. Thushingam, A.M. and W.R. Peltier (1991). ICE-3G: a new global model of late Pleistocene deglaciation based upon geophysical predictions of post-glacial relative sea-level change. J. Geophys. Res., B 96, pp. 4497-4523. Vanf6ek, P. and D. Christodulidis (1974). A method for the evaluation of the vertical crustal movement from scattered geodetic relevelling. Canadian Journal of Earth Sciences, Vol. 11, No 5, pp. 605-610. Vanf6ek, P. and D. Nagy (1981). On the compilation of the map of contemporary vertical crustal movements in Canada. Tectonophysics, 71, pp. 75- 86. Vanf6ek, P, and W.E. Featherstone (1998). Performance of three types of Stokes's kernel in the combined solution for the geoid. Journal of Geodesy, 72, pp. 684-697. V6ronneau, M. (2001). The Canadian vertical datum: A new perspective for the year 2005 and beyond, Presented at lAG Symposium 124 Vertical Reference Systems, Cartagena, Columbia, Feb. 20-23,2001. Wessel, P., and W.H.F. Smith (1998). New, improved version of Generic Mapping Tools released, EOS Trans. Amer. Geophys. U., Vol. 79 (47), pp. 579. Wu, P. and W.R. Peltier (1982). Viscous gravitational relaxation. Geophys. J. R. Astron. Soc., Vol. 70, pp. 435486.
Chapter 83
Optimal Seismic Source Mechanisms to Excite the Slichter Mode S. Rosat National Astrogeodynamics Observatory, Mizusawa, Iwate, 023-0861 Japan
[email protected] Abstract. The recent M w = 9.3 SumatraAndaman earthquake on December 2 6 th 2004 has strongly excited the low-frequency seismic modes of the Earth. In particular the large amplitude of the fundamental radial mode 0S0, which consists of a roughly uniform compression and dilatation of the whole Earth, and of the seismic core modes, like 3S2, means that much energy was radiated toward the core. Also the degree one mode 2S~, which corresponds to a translation of the core in the mantle, was clearly observed for the first time without any stacking process on seismometers and on gravimeters. 2S, is the first harmonic of the degree one mode ,S,, the so-called Slichter triplet. In fact, the Slichter mode is not an elastic spheroidal mode like 2S, since its main feedback force is the Archimedean force. The Slichter mode corresponds to a translation of the solid inner core inside the fluid outer core. The theory concerning this mode is still poorly constrained and no convincing detection has been suggested yet. This paper presents theoretical predictions of the amplitude of the Slichter mode after the 2004 Sumatra-Andaman earthquake as well as after some largest events in the past decades. The computation is based on the usual normal mode theory for a spherically symmetric non-rotating Earth model. The source parameters are finally investigated to find the optimal seismic mechanism to excite the translational motion of the inner core. Keywords.
Normal modes, gravimeter, Slichter modes
superconducting
1 Introduction The development of a global Superconducting Gravimeters network under the framework of the Global Geodynamics Project (Crossley et al., 1999)
has given new opportunities of studies. These instruments are very stable and the recent generations are less noisy than the spring gravimeters and even less noisy than the longperiod seismometers for frequencies below 1 mHz (Van Camp, 1999; Widmer-Schnidrig, 2003; Rosat et al., 2003-2004). The large Sumatra-Andaman earthquake that occurred on December 26, 2004 with a moment magnitude larger than 9 (M w = 9.3 by Stein and Okal, 2005; M w = 9.15 by Park et al., 2005), has strongly excited the low-frequency seismic modes; in particular the fundamental radial mode 0S0, associated with changes in the Earth's circumference, has been largely excited. Although the source was shallow, the earthquake was strong enough to excite the core sensitive seismic modes. Therefore much energy was radiated toward the core. The harmonic degree one seismic mode 2S, that can be seen as a translational oscillation of the core has been observed for the first time directly on individual records of seismometers and gravimeters (e.g. Rosat et al., 2005a). In the normal mode theory, the seismic mode 2S1 is the first harmonic of the sub-seismic mode ,S,, the so-called Slichter mode (Slichter, 1961). The Slichter mode corresponds to a translation of the solid inner core inside the fluid outer core. The predicted period of this mode is 5.42 h for the seismological reference Earth's model PREM (Dziewonski and Anderson, 1981). The latest theoretical developments predict that the period of the Slichter mode is between 4 h (Rieutord, 2002) and 6 h (Rogister, 2003). Since the first claim by Smylie (1992) of a possible detection of the Slichter mode, many unfruitful attempts have been performed to detect this mode in superconducting gravimeter records (Hinderer et al., 1995; Jensen et al., 1995; Rosat et al., 2003; 2005b). The observation of this free oscillation of the inner core is fundamental: since the main restoring
572
S. Rosat
forces are the Archimedean forces, the period of the Slichter mode is directly linked to the density jump at the inner core boundary (ICB). The PREM model predicts a density jump at the ICB of 600 kg/m 3. However two recent papers have reached diverging conclusions whether the actual density jump is larger or smaller than for PREM model. The first paper, by Masters and Gubbins (2003), is based on the analysis of the free oscillations of the Earth and proposes that the density jump at the inner core boundary could be as large as 800 kg/m 3. While the second paper by Koper and Pyle (2004) suggests, from the analysis of amplitude ratios of PKiKP/PcP reflected waves that the density jump at the ICB could be as small as 300 kg/m 3. It is in fact difficult to constrain the density jump from these two techniques as, in the case of the seismic modes, the resolution is weak and in the case of the seismic waves, the phases used have very low signal-tonoise ratios. Rosat et al. (2005b) have developed a detection tool based on the splitting, by rotation and ellipticity of the Earth, of the Slichter mode into a triplet of frequencies in their search of the surface gravity effect of the Slichter mode in superconducting gravimeter (SG) data. They computed the splitting for different PREM-like Earth's models having density jumps at the ICB ranging from 300 to 900 kg/m 3. However no convincing evidence for a probable Slichter triplet has been found. The objective of this paper is to investigate a possible excitation of the Slichter mode by the 2004 Sumatra earthquake and by the past largest events like the 1960 Chile and to estimate the best seismic source mechanism to excite ,S,. In the next section we present the formulations to compute synthetic seismograms in a spherically symmetric nonrotating Earth. In section 3, a comparison of our results to previous computations is presented and then, in a last part, we discuss the seismic source mechanism to excite the Slichter mode.
2 Amplitude excitation computation in a spherically symmetric Earth 2.1 Moment tensor response In the following, we use the notation U, V and P for the radial eigenfunctions which respectively correspond to the radial displacement, tangential displacement and the Eulerian potential perturbation. The response of the Earth to a moment
tensor source has been explained in details by Dahlen and Tromp (1998). We only recall here the main equations we have used to compute the amplitude excitation of the harmonic degree one Slichter mode to a moment tensor source M situated at x = (r, t3, qb) in the spherical coordinate convention used here. The receiver is located at x = (r, 0, qb). The angular epicentral distance ® between the source and the receiver is given by cos ® = cos 0 cos 0~ + sin 0 sin 0~ cos (qb- qb) and we note • the azimuth to the receiver measured counter clockwise from due south at the source. The excitation amplitude A x at the receiver x is expressed by: A×(x) =
(2/+llD(r,O, OO)a(o,o0),(l) 4zc
where D is the displacement operator that in the case of the response of a gravimeter in a nonrotating Earth reduces to the radial displacement eigenfunction U at the receiver position x. The real scalar function A is defined for degree harmonic 1=1 modes by: A ( O , ~ ) = A 0 c o s O + A 1 sinOcos~+BlsinOsin • (2) where A o, A, and B, have the following expressions:
A0 -
/
+ (Moo +
' v,-
' kv,
,
The equations (3) result from the contraction of the moment tensor with the deformation tensor. The components M 0 are the elements of the moment tensor M given a s :
M-- MrO MOO MOO The dot on the radial eigenfunctions U and V denotes the derivative with respect to the radius vector r and the subscript s means that it is evaluated at the source x S. k (1(1+1)) 1/2 = x/2. The radial eigenfunctions U, V and P have been computed as in Crossley (1975) and are plotted in Figure 1. These radial functions are close to zero near the Earth's surface, yet they contribute directly (and to the square) to the excitation amplitude through the displacement operator D = U in =
Chapter 83 • Optimal Seismic Source Mechanisms to Excite the Slichter Mode
equation (1) and through the deformation tensor (derivative of the radial functions) implicitly present in equations (3). Therefore, near the Earth's surface, the excitation amplitude is the square of a very small quantity. Dis ~lacements
Ufree -
2 co-2g a-' U and
Upo t -
(1+ 1) co-2a-' P,
where a is the mean Earth's radius, co the pulsation of the angular order 1 mode (in our case/=1) and g is the acceleration of gravity. The relative magnitudes of the inertial, free-air and potentialperturbation for 1S1 are compared in Table 1 with the values obtained by Dahlen and Tromp (1998). The values differ only in the case of the potentialperturbation contribution because of differences in the radial eigenfunction P.
CMB
CMB
We can now compute the excitation amplitude of the Slichter mode after some major earthquakes.
.
.
.
.
ICB
Table 1. Relative magnitudes of the inertial, free-air and potential-perturbation contributions to the accelerometer response for the Slichter mode 1S1. The tabulated values are for the PREM model.
Fig. 1 Displacement and potential eigenfunctions of the Slichter mode.
U
and V
are respectively the radial
displacement and tangential displacement and P (right hand side plot) is the perturbation of the gravitational potential.
Dahlen and
Tromp (1998) This paper
U/U*
UereJU*
Upo/U*
0.032
0.960
0.008
0.032
0.967
2.2e-5
ICB refers to the inner core boundary and CMB to the coremantle boundary. From equation (2), the excitation amplitude of the Slichter mode can be estimated. This computation takes into account the self-gravitation but not the gravitational effects on the instrument itself, in addition to the gravitational effect on the housing (Dahlen and Tromp, 1998, pp. 143-144), that are quite significant for some of the low-frequency spheroidal oscillations, like ,S,.
2.2 Instrumental gravitational effect A gravimeter as well as a seismometer on the Earth's surface responds to changes in the Earth's gravitational field in addition to the acceleration of the instrument housing. In order to account for the free-air change in gravity due to the radial displacement of the instrument Ufree and for the perturbation P in the gravitational potential due to the redistribution of the Earth's mass Upot, the radial displacement eigenfunction U must be replaced by U :~ -- U "/r" Ufree "~- Upo t. The free-air effect Ufree dominates the vertical response of the Slichter mode 1S, so we must incorporate these self-gravitational corrections in the excitation amplitude A x. Ufree and Upot are defined by:
3 Computation results after past major earthquakes We consider the 1960 Chilean, the 1964 Alaska, the 1994 Bolivian, the 2001 Peruvian and the 2004 Sumatra earthquakes. The source mechanisms used and the predicted excitation amplitudes of ,S, for the spherically symmetric PREM model at 10 SG sites well-distributed on the globe (see figure 2) are summarized in Table 2. The last row shows the results on the whole Earth's surface by Crossley (1992) based on the method described in Crossley (1988). To estimate his results at each SG site, the value in Table 2 must be multiplied by the degree one spherical harmonic function at the station coordinates, namely by the sine of the latitude in a spherical model. His results predict an amplitude excitation of the Slichter mode of the same order than our computation. SGs are presently the most sensitive instruments in the Slichter mode frequency range. However their detection threshold is of the order of 1 nGal (10 -12 g) and the noise level of the best sites is a few nGal in the sub-seismic frequency band (Rosat et al., 2004). So in order to be able to detect the Slichter mode in SG records, its amplitude should reach at least 1 nGal at the Earth's surface. From
573
574
S. Rosat
T a b l e 2, o n l y the 1960 C h i l e a n e a r t h q u a k e c o u l d
Chilean
h a v e s u f f i c i e n t l y e x c i t e d 1S1 for it to b e d e t e c t e d b y
Kanamori and Cipar,
S G on the E a r t h ' s surface, if w e c o n s i d e r the t w o
source.
I
l:~¢'cc'~,on l,lr,,w..l of ~
~
Ny-Alesund
l
L ± ~ I
Cant ley
_ 7 i ~ ~ I
[:~t¢ctI¢io~1 ofn~ l % Boulder
iil°,
lil
~
I~,.,:
.......
(foreshock
..
y
Iil
Sutherlnnd
~--
I
~ ~
~..,~.. ! 1
~ I
=2
°
shock,
l:~l~le£tlle~l on of
~,~o,~o o" ~. ~i'~ro~ ~80"
t-
main
Strasb°ur9
•
•
i
.._t,
. . :_o ~
and
1974) in o n e m a i n s e i s m i c
-~'-~i i I Karnioki3
,
iii ,
events
N
0o,u, ;--
Sy0~0 , I H~ o_ I :
~
ii
i
= II
~
~
~
:~
Fig. 2 Excitation amplitude of the Slichter mode at 10 SG sites for the seven seismic events considered in Table 2. From left to right, at each site, the events considered are the 1960 Chile 1, Chile 2, Chile 1+2, 1964 Alaska, 1994 Bolivia, 2001 Peru and 2004 Sumatra earthquakes. The horizontal line corresponds to the 1 nanoGal detection threshold. Table 2. Excitation amplitude of the Slichter mode at various SG sites after the major past earthquakes for PREM model. Event Date Moment (N.m) Mw Depth (km) Dip (o) Strike (°) Slip (o) Reference Site (latitude in degrees N)
Chile l 1960 2.7 1023
Chile2 1960 3.5 1023
9.5 25 10 170 80 Kanamori
Chile 1+2 1960 6.2 1023
Alaska 1964 7.5 1 0 22
Bolivia 1994 2.6 1021
9.2 8.2 8.4 9.6 9.8 50 38 640 30 50 10 10 20 10 18 170 170 114 302 310 80 80 270 -60 63 Harvard CMT* and Cipar (1974) Kanamori (1970) Surface gravity effect in nGal (= 10-2 nm/s2) for PREM model
Sumatra 2004 1.1 1023 9.31 28 8 329 110
Boulder (40.13)
0.263
0.342
0.605
0.117
0.005
0.004
0.253
Canberra (-35.32) Cantley (45.58)
0.399 0.006
0.519 0.008
0.918 0.014
0.175 0.160
0.004 0.005
0.006 0.0007
0.049 0.198
Kamioka (36.43) Ny-Alesund (78.93)
0.503 0.019
0.654 0.024
1.157 0.043
0.047 0.196
0.005 0.007
0.002 0.007
0.177 0.179
Strasbourg (48.62) Sutherland (-32.38)
0.353 0.656
0.459 0.853
0.811 1.509
0.193 0.037
0.006 0.004
0.010 0.006
0.042 0.286
Syowa (-69.01) Tigo-Concepcion (-36.84)
0.350 0.243
0.455 0.315
0.804 0.558
0.153 0.074
0.007
0.001
0.277
Bandung (-6.90)
0.173
0.225
0.398
0.068
0.005 0.0005
0.007 0.004
0.093 0.072
Crossley (1992) (whole surface) 0.724 0.835 1.52 0.336 I 0"022 Stein and Okal (2005) 2 personal communication * The Harvard Centroid Moment Tensor is available at: http://www.seismology.harvard.edu/CMTsearch.html 1
Peru 2001 4.7 1021
_
Chapter 83
•
Optimal Seismic Source Mechanismsto
Excite the Slichter Mode
Table 3. Moment magnitude needed to excite the Slichter mode at the nanoGal level in surface gravity effect for different ideal source mechanisms. The source is situated at a depth of 500 km on the equator at the longitude 90 ° and the receiver is located at the longitude 40°E also on the equator. Type
Moment Tensor
/°iJ /i°)
Explosion
1
Vertical dip-slip
0
1
Pure compensated linear vector dipole
"eyeball" or "fried-egg"
Crossley (1988) came to the conclusion that even under very favourable conditions (i.e. an M w - 9.3 earthquake), it is not possible to exceed a 1 nGal signal at the Earth's surface. We have confirmed it by our computation after the M w - 9.3 Sumatra earthquake. The maximum excitation amplitude expected is 0.3 nGal at the ideal receiver position from the source, which is 47 ° N and 210 ° E.
4 Discussion Intuitively we can think that to excite the free oscillation of the inner core, we need a very deep source with a fault mechanism that enables to radiate as much energy as possible toward the core. This type of focal mechanism would be a vertical dip-slip source. We have computed the moment magnitude required to excite the S lichter mode at the nanoGal level in surface gravity effect, for different focal mechanisms. The results are summarized in table 3. A vertical strike-slip mechanism should not happen and can not excite the S lichter mode. So it is not considered here. The best natural focal mechanism to excite ~S~ is therefore a vertical dip-
dip 10 ° slip 80 °
dip 8° slip 110 °
Fig. 3 Focal mechanisms of the 1960 Chilean earthquake (left) and of the 2004 Sumatra event (right).
/i0 / 0
~/~
0
Mw 9.6
0 ~
45 °-dip thrust
Focal mechanism
I
)
)
9.7
9.8
1
/ ° i/
9.8
r-2 0 0" 0 1 0 q6 0 0 1
9.6
1
~ -2
'/6 , 0 1
/'7"_
slip source as we need a smaller energetic earthquake than other mechanisms to excite 1S1 at the nanoGal level. However compared to a 45°-dip thrust for instance, the difference in magnitude is only of the order of one tenth of a magnitude unit. A pure compensated linear vector dipole, indicative of simultaneous vertical extension and horizontal compression, is not a usual focal mechanism. It has been observed in shallow earthquakes of moderate size (M>5) beneath volcanoes (Nettles and Ekstr6m, 1998). An "eyeball" mechanism would give the largest amplitude excitation of 1S1. However such non-double-couple source mechanisms are unusual and usually associated with shallow earthquakes induced by volcanism or geothermic, so they can unlikely induce an Mw = 9.6 earthquake. Both 1960 Chile and 2004 Sumatra earthquakes were dip-slip sources. If we consider their focal mechanisms represented in figure 3, we can notice that the Chile fault plane was closer to a pure vertical dip-slip focal mechanism than the Sumatra earthquake. If we consider the 1960 Chile source parameters with different moment magnitudes, we obtain the upper graph of figure 4. It represents the estimated excitation amplitude of ~$1 as a function of the magnitude Mw. With a mechanism similar to the one of 1960 Chile event, the nanoGal level is reached with a moment magnitude of about 9.7. The combination of Chile 1 and 2 events had a moment magnitude of 9.8.
575
576
S. Rosat
Now, if we consider the source parameters of the Chile event but we modify the source depth, we obtain the lower graph of figure 4. We have also computed the excitation amplitude of 1S1 as a function of the source depth with the source parameters of the 2004 Sumatra event (dot-dashed line of lower graph in figure 4). With source parameters similar to the Chile 1+2 event and with a moment magnitude of 9.8, the nanoGal level in surface gravity is reached even for shallow sources. In the case of the Sumatra earthquake, even with a source in the lower mantle, the excitation amplitude of the Slichter mode does not reach the nanoGal signal amplitude at the surface. Some "core-quakes" would be needed, which is impossible.
10 -1
i 10_2 ......
12.G._al.d._e
10_3
__=1o-4 E
F
0- 5
10 .6
10 .7 6
f
/ ~ 6.5
~ 7
~ 7.5
~ 8 i
~ 8.5
~ 9
i
10
w
ICB
CMB
1¢
7
~ 9.5
i
..... 2004 M = 9.3 Sumatra w
102
~
--
1960Mw - 9.8 Chile 1+2
10°
© x
< 10-4 mantle
10-6 0
inner core
outer core
i
i
i
i
i
i
1000
2000
3000
4000
5000
6000
From figure 4 we can clearly see that the excitation amplitude depends much more on the magnitude of the earthquake than of its source depth. The effect of the source depth is represented by 1/rs in equations (3), while the seismic moment directly scales the excitation amplitude. The weak influence of the source depth is mainly due to the radial eigenfunctions, which are almost constant with radius in the mantle. In fact they are even close to zero (figure 1).
5 Conclusion The excitation of the translational motion of the inner core by a seismic source is difficult to detect via its surface gravity effect even with the superconducting gravimeters, which are presently the most appropriate instruments. A 1960 Chile-like earthquake with a vertical dip-slip mechanism and huge energy release (Mw > 9.7) is the only way to excite the Slichter mode with amplitude that can reach and even overpass the nanoGal level in surface gravity effect, if we consider the usual normal mode theory in a spherically symmetric PREM Earth's model. The introduction of a more realistic Earth's model that accounts for the rotation, ellipticity and lateral heterogeneities will be necessary to complete this study; however we can a priori think that it will not increase the amplitude excitation of the Slichter mode. Smith (1976) has demonstrated that the surface gravity effect of the Slichter triplet is strongly dependent on the stability or not of the density stratification of the fluid outer core. This point will also have to be further studied. A seismic excitation of the Slichter mode is not the best way to have it detectable at the Earth's surface. Therefore we must consider other possible sources of excitation, like turbulent flows in the liquid outer core at the core-mantle boundary or an atmospheric and oceanic (individually or coupled) excitation. No studies have been done yet about a possible excitation of the Slichter mode by one of these phenomena, so the debate is still open.
Source depth (kin)
Acknowledgements Fig. 4 Influence on the Slichter m o d e ' s surface excitation amplitude of (upper plot) the magnitude of an earthquake with focal mechanism similar to the one of the 1960 Chilean (Chile 1+2) event and of (lower plot) the source depth of an earthquake with focal mechanism respectively similar to the 2004 Sumatra event (dot-dashed line) and to the Chilean (Chile 1+2) event (solid line).
1960
The author is grateful to David Crossley for his code to compute the eigenfunctions of the Slichter mode and to Luis Rivera and Jacques Hinderer for their useful comments on this paper. This study is granted by the Japan Society for the Promotion of Science.
Chapter 83 • Optimal Seismic Source Mechanisms to Excite the Slichter Mode
References Crossley, D.J. (1975). The free-oscillation equations at the centre of the Earth. Geophys. J. R. Astron. Soc., 41, pp. 153-163. Crossley, D.J. (1988). The excitation of core modes by earthquakes. In: D.E. Smylie and R. Hide (Editors), Structure and Dynamics of Earth's Deep Interior. Geophys. Monogr. Am. Geophys. Union, 46(1), pp. 41-50. Crossley, D.J. (1992). Eigensolutions and seismic excitation of the Slichter mode triplet for a fully rotating Earth model, EOS, 73, p. 60. Crossley, D., Hinderer, J., Casula, G., Francis, O., Hsu, H.T., Imanishi, Y., Jentzsch, G., K~i~iri~iinen, J., Merriam, J., Meurers, B., Neumeyer, J., Richter, B., Shibuya, K., Sato, T. and T. Van Dam. (1999). Network of superconducting gravimeters benefits a number of disciplines, EOS, 80, 11, pp. 121/125-126. Dahlen, F.A. and J. Tromp (1998). Theoretical Global Seismology, Princeton: Princeton Univ. Press., Princeton, NJ, 1025 pp. Dziewonski, A. M. and D. L. Anderson (1981). Preliminary reference Earth model (PREM), Phys. Earth Planet. Int., 25, pp. 297-356. Hinderer, J., Crossley, D. and O. Jensen. (1995). A search for the Slichter triplet in superconducting gravimeter data, Phys. Earth Planet. Int., 90, pp. 183-195. Jensen, O.G., Hinderer, J. and D.J. Crossley. (1995). Noise limitations in the core-mode band of superconducting gravimeter data, Phys. Earth Planet. Int., 90, pp. 169-181. Kanamori, H. (1970). The Alaska earthquake of 1964: Radiation of long-period surface waves and source mechanism. J. Geophys. Res., 75, pp. 5029-5040. Kanamori, H. and J. J. Cipar (1974). Focal process of the great Chilean earthquake May 22, 1960. Phys. Earth Planet. Int., 9, pp. 128-136. Koper, Keith D. and L. Moira Pyle (2004). Observations of PKiKP/PcP amplitude ratios and implications for Earth structure at the boundaries of the liquid core. J. Geophys. Res., 109(B3), B03301 10.1029/2003JB002750. Masters, G. and D. Gubbins (2003). On the resolution of the density within the Earth, Phys. Eart Planet. Int., 140, pp. 159-167. Nettles, M. and G. Ekstr6m (1998). Faulting mechanism of anomalous earthquakes near Bfirdarbunga Volcano, Iceland, J. Geophys. Res., 103 (B8), pp. 17,973-17,984.
Park, J., Song, T-R A., Tromp, J., Okal, E., Stein, S., Roult, G., Clevede, E., Laske, G., Kanamori, H., Davis, P., Berger, J., Braitenberg, C., Van Camp, M., Lei, X., Sun, H. and S. Rosat (2005). Earth's Free Oscillations Excited by the 26 December 2004 Sumatra-Andaman Earthquake. Science, 308, pp. 1139-1144. Rieutord, M. (2002). Slichter modes of the Earth revisited. Phys. Earth Planet. Int., 131, pp. 269-278. Rogister Y. (2003). Splitting of seismic free oscillations and of the Slichter triplet using the normal mode theory of a rotating, ellipsoidal Earth. Phys. Earth Planet. Int., 140, pp. 169-182. Rosat, S., Hinderer, J., Crossley, D. and L. Rivera (2003). The search for the Slichter mode: Comparison of noise levels of superconducting gravimeters and investigation of a stacking method. Phys. Earth Planet. Int., 140, pp. 183-202. Rosat, S., Hinderer, J., Crossley, D. and J.-P. Boy (2004). Performance of superconducting gravimeters from longperiod seismology to tides. J. of Geodynamics, 38, 3-5, pp. 461-476. Rosat, S., Sato, T., Imanishi, Y., Hinderer, J., Tamura, Y., McQueen, H. and M. Ohashi (2005a). High resolution analysis of the gravest seismic normal modes after the 2004 Mw=9 Sumatra earthquake using superconducting gravimeter data, Geophys. Res. Lett., 32, L13304, doi: 10.1029/2005GL023128. Rosat, S., Rogister, Y., Crossley, D. and J. Hinderer, J. (2005b). A search for the Slichter Triplet with Superconducting Gravimeters: Impact of the Density Jump at the Inner Core Boundary. J. of Geodynamics, accepted August 30, 2005. Slichter, L. B. (1961). The fundamental free mode of the Earth's inner core. Proc. Nat. Acad. Sci., 47 (2), pp. 186-190. Smith, M.L. (1976). Translational Inner Core Oscillations of a Rotating, Slightly Elliptical Earth. J. Geophys. Res., 81 (17), pp. 3055-3065. Smylie, D.E. (1992). The Inner Core Translational Triplet and the Density Near Earth's Center, Science, 255, pp. 1678-1682. Stein S. and E. Okal (2005). Speed and size of the Sumatra earthquake. Nature, 434, p. 581. Van Camp, M. (1999). Measuring seismic normal modes with the GWR C021 superconducting gravimeter. Phys. Earth Planet. Int., 116, pp. 81-92. Widmer-Schnidrig, R. (2003). What can superconducting gravimeters contribute to normal mode seismology?. Bull. Seismol. Soc. Am., 93 (3), pp. 1370-1380.
577
Chapter 84
Recent dynamic crustal movements in the Tokai Region, Central Japan, observed by GPS Measurements S. Shimada and T. Kazakami Solid Earth Research Group, National Research Institute for Earth Science and Disaster Prevention (NIED), 3-1 Tennodai, Tsukuba-Shi, Ibaraki-Ken 305-0006 Japan e-mail:
[email protected]; Tel.: +81-29-863-7622; Fax: +81-29-854-0629
Abstract. We present the recent dynamic crustal movements in the Tokai Region, Central Japan, observed by Global Positioning System (GPS) measurements. Since October 2000 the abnormal crustal movements began to move southeastward in the area inland Tokai Region, the reverse motion to the secular crustal movements, suggesting the slow slip between the subducting Philippine Sea slab and the subducted southwest Japan (SWJ). The abnormal crustal movements were fast and wide during October 2000 and December 2001, and slow and narrow during January and December 2002, and again fast and wide during January 2003 and April 2004. Since around June 2004, the abnormal crustal movements became again slow and narrow, and then in September 2004 a M7.4 earthquake occurred at around 200 km southern southwest of the Tokai Region near the Nankai Trough. Co-seismic motions are southward 10 - 30 mm displacement in the Tokai Region. Since the occurrence of the earthquake the southward motion began in the southeast end of SWJ including the Tokai Region. On the other hand the abnormal crustal movements are still ongoing to be slow and narrow.
network (Fig. 1.2) routinely, until midst of 2000, secular crustal movements consistent with the tectonic setting in the region progressed. GEONET is the Japanese nation-wide GPS permanent network established and managed by Geographical Survey Institute (GSI) (Miyazaki et al. (1998)). Abnormal crustal movement began around October 2000, after the wide-area crustal motion occurred associated with the earthquake swarm and dyke intrusion near Kozu Island about 150 km southeast of the Tokai Region during the period of July and August 2000. The slow event indicates the reverse motion in the Tokai Region and caused by the aseismic slip of
i
36 ° •
~ /. / /
'-~%---~ ~"
I
'
I,," .{-...,l
"_ I
Keywords: Global Positioning System., Crustal movement, Slow slip, Plate subducting zone, Tokai Region
1. I n t r o d u c t i o n
In the Tokai Region, Central Japan, the Philippine Sea Plate (PHS) is subducting along the Suruga Trough beneath SWJ (Fig. 1.1), and usually subducted Tokai Region is compressed and deformed north-westward. Since 1996 when the National Research Institute for Earth Science and Disaster Prevention (NIED) began the analysis of the part of the Tokai Region of the GEONET
~' •
~~,/~
[I J 1,36"
PHS 138"
_22
Fig. 1.1. Tectonic Setting around the Tokai Region. The Izu Peninsula, the northern tip of the Philippine Sea Plate (PHS) is colliding to the main part of Honshu. PHS is subducting along the Suruga trough and Nankai Trough beneath the southwest Japan (SWJ), and along the Sagami trough beneath the northeast Japan (NEJ). The 2004 South coast of Honshu earthquake occurred near the Nankai Trough in the PHS slab.
Chapter 84
•
Recent DynamicCrustal Movements in the Tokai Region,Central Japan,Observed by GPS Measurements
I ÷ USUD
36"
i....
* USUD IGS SITE . N IED SITi;
I
. GEONET SITE
35-5" I
i
=, ,,, " ,,, ,, •
II i I
,"
. . . . ,
•
,,.I
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.
.,
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,-:
.
.
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.
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~
-', ,
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,
POINT
__..', PAc'~tc oc~N 137,5"
138"
• •
138.5"
I SEP 1996 - JUN 2000
~USUD
.s
~." ~.
-, 139"
.,
,.,
\
13c)..5"
Fig. 1.2. GPS sites monitored in NIED with the tectonic setting of the area. The dashed line is the focal region of the hypothesized Tokai earthquake, which is thought to occur between SWJ and subducting PHS slab. The Usuda (USUD) IGS site indicates solid diamond, the NIED GPS sites solid squares, and the GEONET sites solid circles.
SWJ against the Philippine Sea slab (Ozawa et al., 2002). On this paper we will introduce the time evolution of the abnormal crustal movements, which seems to be still ongoing. From the observation of the borehole tiltmeter, the slow event also occurred around 1988 although the period is only about two years and the magnitude of the slow event is smaller than the ongoing event (Yamamoto et al., 2005). On September 2004, a M7.4 large earthquake occurred at about 200 km southern southwest of the Tokai Region near the Nankai trough where PHS is subducting to SWJ. The earthquake affects considerably on the crustal movements in the Tokai Region. This paper also mentions on the co-seismic displacement and after slip of the earthquake. 2. A n a l y z i n g
r e f e r e n c e site
".~ -~ /
" /
,'_/._
I
• "/-',,;.*-'~/ "'~ • / /.
~J._W, ~ ,~7> ~g/~
L--...... ,ok~~ -~--~..~ , ./ / :HAMANA OMAEZAKt • !
Q
GLOBK GPS analyzing software (King and Bock, 2004; Herring 2004). Some of the GEONET sites, mainly western part of Tokai Region, are excluded during September 1996 and December 1999, some of the GEONET sites are abandoned during the period, and some of the GEONET and NIED sites established in the midst of the period.
data
Fig. 1.2 shows the analyzing network in Tokai Region, the Izu Peninsula, and the north-west of the Izu Peninsula, where GEONET network sites and NIED network sites are introduced, with the Usuda IGS site (USUD). The north-west of the Izu Peninsula belongs to northeast Japan (NEJ). About 55 GEONET sites and 5 NIED GPS permanent sites exist in the analyzing network by NIED. We analyzed the network data with twenty IGS sites in and around Japan as fiducial site during September 1996 to July 2005. We use GAMIT/
y
0
m.r
Fig. 3.1. The velocity field during September 1996 and June
2000 as background secular motion. Circles indicates 90% confidential ellipses.
3.
Background
ongoing
slow
secular
motion
and
event
Fig. 3.1 shows usual secular motion as the velocity field during September 1996 and June 2000. In the Tokai Region, the velocity is largest at the Omaezaki point locating the southeast end of SWJ, and the large velocity along the coast facing the Suruga Trough and along the area facing the Pacific Ocean, decreasing inland Tokai Region. The mean velocity field of the ongoing abnormal crustal movements is shown in Fig. 3.2 for the period of October 2000 and July 2005 in the inland area of SWJ. Excluding the area in and around Izu Peninsula, the fastest velocity can be seen in the area northeast of Lake Hamana, just covering some part of the boundary of the focal area of the Tokai earthquake. The area of abnormal crustal movements extended to the southern coast region facing the Pacific Ocean and the west coast of the Suruga Bay. The western extension was not obvious in the term of observed velocity. According to Ozawa et al. (2002), the abnormal crustal movements are extend-
579
580
S. Shimada• T. Kazakami
O C T 2000 - JUL 2005
• U S U D reference site
JAN 2002 - DEC 2002
,
• U S U D reference site
,
L_ -.--=--,'Y I
:'
'20 mm/yr
Fig. 3.2. The velocity field during October 2000 and July 2005, showing the velocity field of the whole period of the slow event ongoing. Circles indicate 90% confidential ellipses.
ed 50 km more to the west in SWJ.
4. T i m e e v o l u t i o n
of the slow event
By estimating the velocity fields using shorter time span, it is possible to study the episodic feature of the slow event. The time evolution of the velocity field is seen in Figures 4.1 to 4.3. During October 2000 and December 2001, the area affected by the slow event was very wide, extending to the areas
O C T 2000 - DEC 2001
• U S U D reference site
Fig. 4.2. The velocity field during January and December 2002. Circles indicate 90% confidential ellipses.
facing the Pacific Ocean in the south and facing the coast of the Suruga Bay to the east (Fig. 4.1). The affected area must continue to the ocean bottom of the Pacific Ocean in the south. The velocity was also very fast in general. During January and December 2002 the area affected by the slow event shrank only to the inland Tokai Region, and the velocity became slower (Fig. 4.2). During the period of February and May 2003, GEONET system was upgraded by replacing the antenna, the antenna radome, and the receiver at each site to minimize the multi-path effects and to improve the phase center variation correction. HowJUN 2 0 0 3 - A U G 2004
~ U S U D reference site
t I
"~
Fig. 4.1. The velocity field during October 2000 and December 2001. Circles indicate 90% confidential ellipses.
i,.,.~
'''~er
.
'~
Fig. 4.3. The velocity field during June 2003 and August 2004. Circles indicate 90% confidential ellipses.
Chapter84
• Recent Dynamic
CrustalMovementsin the Tokai Region, Central Japan, Observed by GPS Measurements
ever, the upgrade caused offsets in the site coordinate solutions in our GPS analysis. GSI also has changed the elevation cut-off angle of the GEONET receiver from 15 ° to 5 ° to improve the vertical repeatability. This change caused significant offset both in horizontal and vertical components of the site coordinate solutions. It is very difficult to obtain the accurate offset estimates due to such a major changes in the observation setup. Therefore we omitted the data during January and May 2003, Fig. 4.3 shows the annual velocity field during June 2003 and August 2004, revealing that the areas affected by the slow event again became very wide and the velocity was also very fast, almost same as those in the period of Fig. 4.1.
5. Co-seismic and after slip displacements of South coast of Honshu earthquake On September 5, 2004, a M7.4 earthquake occurred at about 200 km southern southwest of the Tokai Region. The earthquake occurred in the PHS slab near the Nankai Trough where PHS is subducting under SWJ. The main shock mechanism is almost pure strike slip approximately perpendicular to the trough axis. The foreshock and the largest after shock are nearly parallel to the trough axis and the
• =_.::-.-
i [
"jd,.~ •,
~ .... :~:I' ,~,
, ..
.... .
•
; ..
,~.~,'~.. . :
...~,~
....- .~-...,:..'
j
l
]20ram Fig. 5.2. Co-seismic displacement at the September 4, 2004, South coast of Honshu earthquake.
mechanisms of the earthquakes are almost pure dip slip in the PHS slab. Fig. 5.1 shows the epicenter of large shocks, rupture displacement contour of large shocks, distribution of the epicenter of the earthquakes during September 5 and 8, and the asperity between SWJ and the PHS slab. Mostly caused by the main shock of the above earthquakes, large co-seismic step arose in the Tokai Region. The amount of step is 1 0 - 30 mm. Fig, 5.2 shows the co-seismic displacement at each site in the network. Fig. 5.3 shows the annual velocity field during
•
i:~?: :~: SEP 12 2004 - JUL 2005
.~.j~
.
.
.
.
~ -.~.:,• •
135
~ U S U D reference site
==:. ~.
•
s
i ~ ~
,~;:
t36
...
[37
138
t39
Fig. 5.1. September 5, 2004, South coast of Honshu earthquake. Stars are the epicenters of large earthquakes. Dots are the epicenters of the earthquakes occurred during September 5 and 8. Black contours are the rupture displacements of large earthquakes. The grey contours are the rupture displacements of the M8.3 Tonankai earthquake in 1944 and the white shades are the asperity between the PHS slab and SWJ. Grey shade and grey contours are the bathymetry (after Yamanaka, 2004).
:
'20 m m / y r
Fig. 5.3. The velocity field during September 12, 2004 and July 2005. Circles indicate 90% confidential ellipses.
581
582
S. Shimada. T Kazakami
TAT-USUD B A S E L I N E V E C T O R
USUD-TSKB BASELINE VECTOR N-S C O M P O N E N T
4
0
COMPONENT 40 .E-W ...........
~
N-S C O M P O N E N T
40..
,
.
,
. . . . .
,
.
.
•
-40 ~. 2000
U-D C O M P O N E N T 60
,
,
,
,
40 2 0
,
.
,
•
,
.
,
.
,
60
,
' 2002
.
'
.
•
,
,
i
. . . . . . . 2000 2002
:
f
~
%00
2006
.
,
.
,
'
.
t
• 2006
-.40/ 2000
,
.
•
I 2002
2004
2006
BASELINE LENGTH
,
•
40
,
,
,
,
,
.....
¢
" 0 ' "
.40f
' ' ' -40 . . . . . . . . . 2006 2000 2002 2004
.
. .,..i,~,~..
2 0 1
2004
. . . .
,,~i. ~oI
._..20~
20
' 2004
. . . . .
~ot
20 ~
~u--
.
,
20
U-D C O M P O N E N T
BASELINE LENGTH 40
'
E-W C O M P O N E N T
40..
t
°
-2_420!0
26o2 2oo4
2oo6
2002
6
2oo4 2006
Fig. 5.4. Time series of the USUD-TSKB baseline during 2000 and 2005. The data are the weekly values estimated from the daily values using Kalman filtering.
Fig. 6.1. Time series of the TAT-USUD baseline during 2001 and 2005. The data are the weekly values estimated from the daily values using Kalman filtering.
September 12, 2004, and July 2005. Although the error ellipses are large, the slow event in the Tokai Region became slow, especially the area around the site 3097, the central area of the former slow slip, and the center of the largest velocity moves to northeast of the former center, to the area around the 0819 site. The fast velocity sites are also seen some sites northeast inland of the Tokai Region, one of them the 3078 site. The sites on the Izu Peninsula and NEJ seem to move northward. Actually the USUD reference site moved southward, affected by the after slip of the South coast of Honshu earthquake. To confirm the southward motion of the USUD sites, we plot the time series of the Tsukuba (TSKB) - USUD baseline vector. The TSKB site is the IGS site in NEJ. Fig. 5.4 shows the time series, eliminating the co-seismic step at the South coast of Honshu earthquake. From the occurrence of the South coast of Honshu earthquake, showing at the epoch of the arrow in the figure, southward motion of the USUD site began, and terminated in December 2004. Actually for the period of June 2003 and August 2004, the N-S component velocity of the USUD-TSKB baseline is 0.07 +0.01 (+ means one sigma), although the velocity is -13.94 + 0.61 for the period of September 12 and the end of December 2004.
background secular motion in the Tokai Region (Fig. 3.1), and those two points still indicates westward motions even in the slow slip period (Figs. 4.1, 4.2, 4.3, and 5.3), which suggests that the slow slip event does not reach to the upper-most end of the PHS slab along the Suruga Trough near the Omaezaki Point. To examine the slow slip phenomena during January and May 2003 when GEONET sites were not available because of the system revision, we examine the time series of the baseline between the TAT NIED site indicated in Fig. 5.3 and the USUD IGS sites, both are free from the noise of the GEONET maintenance. Fig. 6.1 shows the time series of the TAT- USUD
3097-USUD BASELINE VECTOR N-S C O M P O N E N T 40
..401 2000
,of
.
,
i
,
.
,
i 2002
.
.
.
~.2o~r 01
.
,
.
,
i
,
_~(ll
.
.
.
"~
.
"
~ ~ ~ "
~-2o 2000
The area near the Omaezaki Point, especially two GEONET sites in the southeast end of the Tokai Region, indicates the largest westward motion in the
,
,
i 2004
k
E-W C O M P O N E N T
,
.
i
,
40
.401 2000
2006
,
U-D C O M P O N E N T 60
E
6. Discussion
.
,
I
I
2002
,
I
40
"~ ,
I
2004
.
i
,
,
i 2002
.
,
,
i
,
,
, 2004
I
,
.
,
, / 2006
BASELINE LENGTH
.
,
,
.
,
,
,
~22,
/
,
,
I
I
|
.401
,
o I
I
I
,
I
I
I
I
0
Fig. 6.2. Time series of the 3097-USUD baseline during 2000 and 2005. The data are the weekly values estimated from the daily values using Kalman filtering.
Chapter
84
• Recent
Dynamic
Crustal
Movements
0819-USUD BASELINE V E C T O R N-S
C O M P O N E N T
E-W
C O M P O N E N T
,!ooo 60
U-D C O M P O N E N T . , . , . . . . .
+o
.
40
,+o!, +ooo BASELINE . , . ,
.
LENGTH . ,
,
,
.
,.
-6£)
~
2000
,
~ , 2002
1 '
V
~
I
,
2004
2006
-40 2000
........ 1 2002
2004
2006
Fig. 6.3. Time series of the 0819-USUD baseline during 2000 and 2005. The data are the weekly values estimated from the daily values using Kalman filtering.
baseline, eliminating the co-seismic step at the South coast of Honshu earthquake. For the period of January and December 2002, the E-W component velocity of the baseline is 1.81 + 0.26 (+ means one sigma), although the velocity is 8.36 + 0.04 for the period of January 2003 and June 2004. Thus the E-W component indicates that the slow event became faster significantly at the beginning of 2003. We re-evaluate the start time of the slow-down of the abnormal crustal movements at midst of 2004 concerning the relation of the occurrence of the South coast of Honshu earthquake and the change of the velocity field in the SWJ. Fig. 5.3 indicates the sites whose time series were examined. Fig. 6.2 shows the time series of the 3097 - USUD baseline,
3078-USUD BASELINE V E C T O R 40
.
N-S ,
.
C O M P O N E N T , . , . ,
,
.
40
20
~
-20 ,
2000
-201"
C O M P O N E N T , . , . , ,
I
Eo g
•
.
L
-20
•..40
g+ o
E-W , .
20
"fro g
60
•
.
2002
2004
U-D C O M P O N E N T , . , . , . ,
-:;L' ' "
,
.
-40 21000
40
2002
•
BASELINE , • , .
"
" "
"
"~
, 2006
2004
.
.
LENGTH . .
,,
,
,?. •
2006
2006
2000
2002
2004
Tokai
Region,
Central
Japan,
Observed
by
GPS
Measurements
elimiating the co-seismic step at the South coast of Honshu earthquake. The arrow indicates the epoch of the change of the trend. The E-W component velocity of the baseline for the period of April 2003 and March 2004 is 11.17 +0.21 (+means one sigma), although the velocity is 2.29 + 0.23 for the period of April 2004 and March 2005. It seems that the change of the trend occurred around April 2004, obviously before the occurrence of the South coast of Honshu Earthquake, judging from the data of the E-W component. Fig. 6.3 shows the time series of the 0819-USUD baseline, eliminating the co-seismic step at the South coast of Honshu earthquake. The arrow indicates the epoch of the change of the trend, judging mainly from the E-W component. The E-W component velocity of the baseline for the period of July 2003 and June 2004 is 12.00 + 0.26 (+ means one sigma), although the velocity is 1.77 + 0.37 for the period of July 2004 and June 2005. The E-W component seems to indicate some gap in the midst of the year and then the velocity changes around July 2004, also before the occurrence of the South coast of Honshu earthquake, although the change delayed three or fore months after that of 3097 site. Fig. 6.4 shows the time series of the 3078-USUD baseline, eliminating the co-seismic step at the South coast of Honshu earthquake. The arrow indicates the epoch of the change of velocity judging from the N-S and E-W components. The N-S and E-W components velocities of the baseline for the period of August 24 2003 and September 4 2004 are 0.11 + 0.22 and-1.67 + 0.22 (+ means one sigma), although the velocities are -0.51 + 0.35 and 3.02 + 0.34 for the period of September 12 2004 and September 17 2005. The epoch seems to coincide with occurrence of the South coast of Honshu earthquake. Thus the epochs of the velocity changes in 2004 are earlier at the sites near Hamana Lake and later at the sites NE Tokai Region, suggesting the migration of the place of the slow slip at the plate boundary.
7. Conclusion
•
t
"
+o4' '
in the
,, t 2006
Fig. 6.4. Time series of the 3078-USUD baseline during 2000 and 2005. The data are the weekly values estimated from the daily values using Kalman filtering.
The recent dynamic crustal movements in the Tokai Region, Central Japan, are examined, observed by GPS measurements. The abnormal crustal movements began to move southeastward, the reverse motion to the secular crustal movement in the Tokai Region, since October 2000, indicating the slow slip between the subducting PHS slab and the subducted SWJ.
583
584
S. Shimada • T. Kazakami
The abnormal crustal movements were fast and wide during October 2000 and December 2001, and slow and narrow during January and December 2002, and again fast and wide during January 2003 and May 2004. Since around April 2004, the abnormal crustal movements became again slow and narrow in the area northeast of Lake Hamana, and then in September 2004 a M7.4 earthquake occurred around 200km southern southwest of the Tokai Region near the Nankai Trough. Co-seismic motions are southward 1 0 - 30 mm displacement in the Tokai Region. During the occurrence of the earthquake and December 2004 the after slip southward motion occurred in the southeast end of SWJ including the Tokai Region. On the other hand the abnormal crustal movements are still ongoing to be slow and narrow.
References
Herring, T. A., (2004). GLOBK global Kalman filter VLBI and GPS analysis program. Mass. Inst. of Technol., Cambridge, Massachesetts. King, R. W., and Bock, Y., (2004). Documentation for the GAMIT GPS analysis software. Mass. Inst. of Technol., Cambridge, Massachusetts. Miyazaki, S., Hatanaka, Y., Sagiya, T., and Tada, T. (1998). The nationwide GPS array as an Earth observation system, Bull. Geogr. Surv. Inst., 44, pp. 11-22. Ozawa, S., Murakami, M., Kaidzu, M., Tada, T., Sagiya, T., Hatanaka, Y., Yarai, H., and Nishimura, T., (2002). Detection and monitoring of ongoing aseislnic slip in the Tokai Region, Central Japan. Science, 298, pp. 1009-1012. Yamamoto, E., Matsumura, S., and Ohkubo, T., (2005). Repeatedly occurring slow rebound of the continental plate at the Kanto-Tokai area of Japan detected by continuous tilt observation. Earth Planetary Space, 57, pp. 917-923. Yamanaka, Y., (2004). http://www.eri.u-tokyo.ac.jp/sanchu/ Seismo Note/2004/EIC153e.html.
Chapter 85
New Theory for Calculating Strains Changes Caused by Dislocations in a Spherically Symmetric Earth Wenke SUN, Shuhei OKUBO and Guangyu FU Earthquake Research Institute, University of Tokyo, Tokyo, Japan sunw@ eri.u-tokyo, ac.j p
Abstract. A new theory is presented for calculating co-seismic strain caused by four independent types of seismic source in a spherically symmetric, non-rotating, perfectly elastic, and isotropic (SNREI) Earth model. Expressions are derived by introducing strain Green's functions. A proper combination of these expressions is useful to calculate co-seismic strain components resulting from an arbitrary seismic source at any position in the Earth. Numerical computations are performed for four independent sources at a depth of 32 km inside the 1066A Earth model. Results in the near field agree well with that calculated for a half-space Earth model. A case study is performed and Earth model effects are investigated. Furthermore, the effects of spherical curvature and the stratified structure of the Earth in computing co-seismic strain changes are also investigated using the present dislocation theory and Okada's (1985) formulation. Curvature effects are small for shallow seismic events, but they are larger for greater source depths. Stratified effects are very large for any depth and epicentral distance, reaching a discrepancy greater than 30% almost everywhere. Key Words: Co-seismic Deformation, change, Dislocation, Earthquake.
1.
Strain
Introduction
Advances in modern geodetic techniques such as GPS and InSAR allow better detection of co-seismic deformations such as displacement, gravity change, and strain. Such geophysical geodetic information is useful for studying seismic mechanisms, Earth structure, and even earthquake forecasting. A quasi-static dislocation theory is necessary to properly apply the observed geophysical phenomena to interpret or invert the seismic parameters. To study co-seismic deformation in a half-space Earth model, numerous studies have been undertaken by many scientists; among them are Steketee (1958), Maruyama (1964), and Okada (1985). They presented analytical expressions for
calculating surface displacement, tilt, and strain resulting from various dislocations buried in a semi-infinite (half-space) medium. However, the validity of these theories is strictly limited to a near field because Earth's curvature and radial heterogeneity are ignored. As modern geodetic techniques can detect and observe far field crustal deformation, even a global co-seismic deformation, a dislocation theory for a more realistic Earth model is demanded to interpret far-field deformation. Efforts to develop formations for such an Earth model were advanced through numerous studies (e.g., Ben-Menahem and Singh, 1968; Ben-Menahem and Israel, 1970; Smylie and Mansinha, 1971). Such studies have revealed that Earth's curvature effects are negligible for shallow events, whereas vertical layering may have considerable effects on deformation fields. However, Sun and Okubo's (2002) recent study comparing discrepancies between a half-space and a homogeneous sphere (accounting for self-gravitation) and between a homogeneous sphere and a stratified sphere indicates that both curvature and vertical layering have marked effects on co-seismic deformation. Stratified sphere models, such as the 1066A model (Gilbert and Dziewonski, 1975) or the PREM model (Dziewonski and Anderson, 1981), are the most realistic: they reflect both sphericity and stratified structure of the Earth. For such an Earth model, Pollitz (1992) solved the problem of regional displacement and strain fields induced by dislocation in a viscoelastic, non-gravitational model. Sun and Okubo (1993, 1998) and Sun et al. (1996) presented methods to calculate co-seismic displacements and gravity changes in spherically symmetric Earth models introducing dislocation Love numbers and Green's functions. Okubo (1993) proposed a reciprocity theorem for connecting solutions of dislocation and tidal, shear, and load deformations. That study found that deformation on the Earth's surface caused by dislocations at source radius r = ~ .
are expressible simply by a linear
combination of tide, shear, and load deformations at
586
W.
Sun. S. Okubo. G. Fu
Theoretically, a co-seismic strain tensor can be obtained numerically by taking spatial derivatives of the co-seismic displacement results from first principles. However, it would be a very tedious work for repeat computations. On the other hand, since co-seismic deformations very rapidly as distance increases, the numerical methods may cause a considerable error. Therefore, it calls a convenient solution in a straightforward manner. This paper presents a new theory for calculating the co-seismic strain for a realistic (e.g., SNREI: Spherically Symmetric, Non-rotating, purely Elastic and Isotropic) Earth model. The theory is given by a set of expressions by strain Green's functions multiplied by an epicentral distance related factor for four independent seismic sources. These formulations can be used to calculate any strain components caused by any kind of dislocation at an arbitrary position in the Earth. 2.
Basic E q u a t i o n s of Elastic R e s p o n s e s of a Sphere to a Point Dislocation
We start with a dislocation model (Fig. 1), which is defined at radial distance r s on an infinitesimal fault dS by slip vector v, normal n, slip angle ~, and dip angle 6 in the coordinate system ( e l , e 2 , e 3); unit vectors e 1 and e 2 are taken in the equatorial plane in the directions of longitude ~ = 0 and 7r/2, respectively, and e 3 along polar axis r. movement (dislocation) of the two fault defined as (U/2)-(-U/2) =U. Note that for opening, the slip vector and the normal equal: v=n.
Relative sides is a tensile become
form of three components along with spherical coordinates ( e r, e o, e e ) as (Sun et al., 1996): u(a,O,(p)= ~
y
m
n~ij
m
I(~R. ( O, (19) + Y 3,m S n (0,(19)
i,j
.~,t,n,ijTm (0, (/9)] vin j ++l,m
UdS a
(1)
2
where
R~(o, ~)- ey2(o,~) I 0 S~(O, q~) - e o -0--~+ e~
1 0] sin 0 Oq~
Y~(O, (p)
e~
r2(o, ~)
E '
T~ (0,~o)- e o sin0 c~(p
(2)
Y~ (0, ~o) = P~ (cos 0)e i~e P [ (cos 0) is the associated Legendre function and a is the Earth radius; the superscript s represents spheroidal deformation and t stands for toroidal deformation. The y-variables ,,n,is :~,m(a) and y t,n,ij
~,m (a)are obtainable by solving the linearized
first order equations of equilibrium, stress-strain relation, and Poisson's equation for excited deformation (Saito, 1967; Takeuchi and Saito, 1972). Because i=1, 2, 3 and j = l , 2, 3, the combination of i and j is nine, i.e., total solutions of all y~ should be nine. However, because of the symmetry and intrinsic symmetry within fault geometry of source functions S ' " , the number of independent solutions of :~,m(a)""'u and :~,~,,""'iJ(a) is four. Consequently, if any four independent solutions are obtained, other solutions among the nine are also obtained easily. In this study, we choose ( Y~im n 12 _ n,32 n 22 n 33 ' Y~,m , Yk;m , Y~im ) (and the
Fig. 1. The top row shows three Earth models: l e f t homogeneous half-space; m i d d l e - homogeneous sphere; r i g h t - heterogeneous sphere. The bottom represents four seismic sources. If dislocation occurs in a spherical Earth, such as in a homogenous sphere or a SNREI Earth (spherically symmetric, non-rotating, perfectly elastic and isotropic, Dahlen 1968), excited displacement u (a, 0, ~o) (radial distance, co-latitude and longitude) are describable in the
corresponding toroidal solution) for four independent solutions. They are excited respectively by a vertical strike-slip, a vertical dip-slip, a horizontal opening along a vertical fault, and a vertical opening along a horizontal fault. Once the displacement components ui(a,O,~o ) in (1) are obtained (Sun et al., 1996), the co-seismic strain tensor can be derived and numerically calculated, i.e., the work of this study. 0
E x p r e s s i o n s of Co-seismic Strain for the F o u r I n d e p e n d e n t Sources
According to conventional theory of elasticity, the components of the co-seismic strain are expressible
Chapter 85 • New Theory for Calculating Strains Changes Caused by Dislocations in a Spherically Symmetric Earth
in spherical coordinates as (Takeuchi and Saito, 1972) the following.
e~, = -C~U - r
(3)
8r
1 8u o
1
eoo = - - +-u r r c~O r 1
eoe
rsin0
e oo = e~
e"°
c~o
r
r
1
1
--+-uo
r 80
cot 0 + - u ~ r
u o cot 0 + - 8u o + - -
c~u o
(6)
8r
8u o 1 . . . uo c~r r
18u,. - - r c30
+
(22)
c o s (/5~32O (a,
0)
e0022'0(1:7/,O, 0/9) -- eoo~22'°(a, O) - 0
(23)
%" (a, o, ~o) - %~"(a, o) - o
(24)
ek z'`ij (a, 0) are the strain Green's functions from"
err-
Yl,2 (6/)
a(2 + 2//) n=2
(25)
- n(n + 1)y3~'~2 (a))/~,2 (cos O)
1
---ue
r sin 0 c~(p .
1
(5)
r sin 0 8 ( , o
8u,.
=
CO32o (a, 0, (,o) -
Therein, the epicentral distance-related variations 1
1
(21)
(4)
c~u o
1 c~u~o
12 (a, 0, (,o) - cos 2(,~;2o (a, 0) CO o
(7)
r
,,32 2,~ £(2yn,32 = ~ ,,, err a(,,~ + 2/./) n=l
(8)
(a)
(26)
1\ n,32
- n(n + UY3,1 (a))P2(cosO) = a(A + 2/a) ,=0
err
Yl,0 (a)
(27)
ix n,22
+ rt(n + l)Y3, 0 ( a ) > , (cos0) Using the displacement components defined in the section above, the expressions of the strain components are derived in the following. The last two components e ro and e,.o vanish on the free Earth surface. Only the remaining four components are considered hereafter. Inserting the displacement components in (1) into (3)-(8), taking into account the responses to the four independent sources, the above four strain components are reducible into the following 16 components, expressed as an epicentral distance-related variation (called Green's function) multiplied by a (,o-related factor, yields:
,2 (a, 0, O)) - sin 2(,~I. 2 (a 0) C/./,
(9)
32(a, O, (,o) - sin Oer~ ,,32(a, O) Crr
(10)
"22,0 (a O) err22,0 (a, O, (/9) -- err ,
(11)
~" (a, o) err" (~ o, ~o) - err
(12)
12 (a, O, (,o) - sin 20~; 2 (a, O) eoo
(13)
32 (a , 0, O) - sin ~eoo "`32(a, O) eoo
(14)
22,0 (a, O, 0/9) -- "22,0 (a O) Coo Coo ,
(15)
"(a Coo
0, e) - Coo ~" (a, o)
(16)
e~012(a, O, (p) - sin 2q~% 0"`12(a, O)
(17)
e~32 (a,
(18)
,
,
O, (,0) - sin
"`32(a, 0) ee~
eoe22'0(a, O, 0/9) -- C00"22'0(a, O)
(19)
eoo33(a, 0, q)) - eoo~33(a, 0)
(20)
"`33 -
/~
err
£(
n33 (a) y,,~
a(/~ + 2/a)~0
(28)
. . . . " ( a ) ~ (cos O) +n(n+UY3,o d 2 P,) (COS 0) Coo
--
--
--
Y3,'2
(a)
dO 2
a n=2
-- yl~)12P f (cos {9) - ~zyl,t,n,12 2 (a)
I 1 dPf(c°sO) dO
(29)
c°s-O 2
• sin 0
I]
sin2 0 P~ (cos O)
"32 - - 2~_.[_yn,32 eoo 3,, (a) d2 C 1(cosO)
a
1
dO 2
-- yln] 32 ( a ) P 1 (cos 0) - yl]l '32 (a)
dP~(cosO) dO COO =
cosO 1 sin 2 0 P~ (cos O)
.y3n '22 (15/) a n=o L
sin 0
(30)
J
/Dn (COS 0)
dO2
(31)
+ yln~22 (a)P n (cos 0)] "33 _ 1~-]~
COO
~,33
d2p~(c°sO)
-~__o~Y~,o (~) + yl~O"(a)P~ (co~ 0)]
dO ~
(32)
587
588
W. Sun. S. Okubo. 6. Fu
"`12 __ Z eel, a ~:2 -
{ yn,12 ( 3,2 (a) 4P] (cosO) sin 0 sin 0
m=2. The above formulas give solutions for the case of m=0; solutions of the m=2 case can be derived from e,l,2 (a, 0, (p) because the following relation holds.
cos 0 dp2 (cos 0) 1 "'~ dO - Y~,2 (a)
(33)
t,n,12
0:1,2 (a)
• p ] ( c o s o) +
t,n,22
Vj - 1,2"s.1:,+2
sin 0
c°s°, OPn2 1 t
•
-- COS 0
cot
-
dO
"32 __ Z eel, a ,--1
{ yn,32
3,1 (a) sin 0
( | Pn (COS0) sin 0
dpl (COS 0) )
n 32
dO
- y~'] (a)
(34) 0
• C' (cos o) + -
[
- cot Of" (cos O)
dO
"`22,0 1 £ I c o t 0 y n , 2 2 ~,0 ( a ) e~°~° = -a n=0
]}
dP. (cos 0) dO
(35)
+ y/,~)22(a)p n(COS 0)] "`33 -_1 ~0Icot0yn,333,0(a) dPn(cosO)
e~
a :
dO
a n,12(a) I "`12 2 ~ ~Y3,2 e°e--- ~ a :2 sin0
(36)
dP n2 (COS0)
dO
+ cot Op2 (cosO)]
-'ky;:2'12Icot O(a) dP2 (cos O) dO I 4P] (cos O) sin 20
"`32 2 eoe-~ a ,, 1
(37)
d2p2(cosO)]} dO ~
3,1 (a) sin 0
Numerical Earth
sin 0
d n (COS 0)
dO
+ cot OP2(cos O)) + y[:i''32 (a)/cot 0 alP1(cos 0)
(38)
dO P2(coS0)sin20 _ d2p2(cosO))}_dO 5
22 (a , 0, (/9) is The case of e~l
--
(39)
__ • t n 12
-t-lSj',+2
Expressions (9)-(38) illustrate the main results of this study. They are useful to calculate strain components excited by four types of sources buried in a spherically symmetric Earth model. In combination, these components allow calculation of a strain field that is excited by an arbitrary seismic source.
(cos O)
y~:i'32(a) •
n,122 V j - 1 , - . . 6 " s ~ ( 22 - -T",sji_+
special because
the source function contains two parts with m=0 and
Results
for
a Homogeneous
Model
To bolster the validity of the theory described above, a numerical test is made by considering two Earth models: a half-space model and a homogeneous sphere. Both Earth models have identical media parameters, which are equivalent to those of the top layer of the 1066A Earth model (Gilbert and Dziewonski, 1975). The numerical calculations are performed for the two models and the results are compared. The above co-seismic strain theory can be considered valid if the results agree well in the near field. For this purpose, Okada's theory (1986) is used for the half-space Earth model; the current theory is used for the homogeneous model. The Green's functions in (25)-(38) show that the important computations are the y-solutions and Legendre functions and their derivatives. The y-solutions
,, "'/j e~,m(a)
and
,, t,,,/j (a) :~,,,~
have already
been discussed in previous papers (e.g., Sun and Okubo (1993) and Sun et al. (1996). The Legendre functions and their derivatives can be calculated using recurrence formulas. Green's functions are then obtainable through series summations. The computation for the near field is relatively difficult in comparison to that for the far field because of the slow convergence of the series of the former. Therefore, some skills are required to accelerate computation (see Sun and Okubo, 1993). On the other hand, because co-seismic deformations in near field dominate over those of the far field, they are useful for comparison with the result of a half-space model. Therefore, the following discussions are limited in the near field. Assuming four independent point sources at a depth of 32 km in the homogenous Earth model, the
Chapter 85 • New Theory for Calculating Strains Changes Caused by Dislocations in a Spherically Symmetric Earth
y2;~(a)
y-solutions
and
y~',"2y(a)
are
~,,mm 11
,.
Computations are performed under the condition of UdS/a2=l, and the Earth radius a=6371 km is taken for all the Green's functions. This assumption requires that, for a practical computation of a strain component, the parameters UdS and a should use the same kilometer unit so that the final strain components become dimensionless. Once these Green's functions ekl"O(a,O) are calculated, the co-seismic strain components in Eqs. (9)-(24) are easily computed. Figure 2 depicts the co-seismic strain components err12 (a, 0, (p) caused by the vertical strike-slip source at a depth of 32 km in the Earth model. The horizontal axis indicates co-latitude and longitude to 2°; the vertical axis represents the strain magnitude with the unit of km -1. As expected, the strain component err12 ( a , 0 , (/9) for the vertical strike-slip source appears as a quadratic distribution pattern. For comparison, the corresponding strain components caused by the four sources in a half-space Earth model are also calculated using Okada's formulation (Okada, 1985), and plotted in Fig. 3. Comparison of Fig. 2 and Fig. 3 illustrates that the two results agree well in both distribution pattern and magnitude. However, discrepancies in magnitude exist for some components (details to be seen in Figs. 5 and 6). These discrepancies are inferred to result from sphericity and vertical inhomogeneous effects.
) s,
W
lllm •
m
m •
m
L ;; .i
.ram
"
l[
ii
l
m" m
|m "i . m m m~
Fig. 3. Co-seismic strain caused by the vertical strike-slip source at a depth of 32 km in the half-space Earth model, as calculated by Okada' formulation (Okada, 1985).
0
Numerical Model
Results
for a SNREI
Earth
Next, we test the above co-seismic strain theory by performing a numerical calculation. Assuming the four independent point sources at a depth of 32 km in the 1066A Earth model (Gilbert and Dziewonski, 1975), we first calculate the strain Green's functions
by the vertical strike-slip source at a depth of 32 km in the 1066A Earth model. Results for other strain ij (a, O, q~) , e~i; (a, O, q~) , and components eoo
,i
\
nun "
Once these Green's functions e~1 ~/; (a,0) are calculated, the co-seismic strain components in Eqs. (9)-(24) are easily computable. Figure 4 depicts the co-seismic strain components err12(a, 0, ~p) caused
i¢
4,,
I¢
IN
m
^/; (a, 0) in Eqs. (25)-(38). Again, the computations are performed under the condition of UdS/a2=l; Earth radius a=6371 km is assumed for all the Green's functions.
Ai
12
)Ni
first
computed. Then the strain Green's functions e~1~i;(a,0) are obtainable using Eqs. (25)-(38).
m $ i-,¢.
m
Fig. 2. Co-seismic strain components err,2 (a, 0, ~) caused by the vertical strike-slip source at a depth of 32 km in a homogeneous Earth model.
589
e~o(a,O,~o ) are also computed, but the plots are omitted here. They show a similar distribution pattern to the component e~r(a,O,~o ) , but with different magnitude.
590
W. Sun. S. Okubo • G. Fu
,mm
m(l
Ill m, _-m
! I .
11
k II
•
..i in m
.l."
l m
•
I I
%
•m
,m_',
",m, •
m m n ~
Fig. 4. Co-seismic strain components elr~(a, 0, q~) caused by the vertical strike-slip source at a depth of 32 km in the 1066A Earth model. 0
Case S t u d y - Application to 1994 Far Off Sanriku Earthquake (M7.5)
In this section, the above co-seismic strain theory is applied to compute the strain changes caused by the 1994 Far Off Sanriku earthquake, which occurred on 28 December 1994, about 180 km east of Hachido, Aomori prefecture, Japan. The fault plane of the earthquake comprises two rectangles with parameters listed in Table 1. This earthquake caused crustal deformations in a large area: displacement detected by GPS, and strain changes observed by extensometers installed at observation points HSK and FDA. The following computations consider only those strain changes to which the above theory applies. Table 1. Source parameters for the main shock of the 1994 Far Off Sanriku earthquake. (Faculty of Science, Tohoku University, 1994) Parameter Fault # 1 Fault #2 3.4 x 1027 1.7 x 1 0 27 M0 dyne x cm dyne x cm Mw 78 Strike angle N 184°E N184°E Dip angle 8° 35 ° Slip angle 70 ° 90 ° LengthxWidth 60x 100 km 50x60 km Dislocation 1.57 m 1.57 m Latitude 40.24°N 40.55°N Longitude 144.04°N 142.85 °N Depth 10 km 24 km The point source theory cannot be used directly because a large error is expected to occur as a result of the large geometrical fault size relative to
the epicentral distance. To surmount this obstacle, a numerical integration of the point source over the fault plane could be done, or a segment-summation scheme could be used, as pointed out by Fu and Sun (2004). The latter is adopted in our practical computation. First, a homogenous half-space model and a homogenous sphere model are used to calculate the strain changes caused by this earthquake. The former is calculated using Okada's (1985) formulation, whereas the latter is computed using the present theory. Results are given in Table 2 (second and third rows). Both results are nearly identical and basically coincide with the observed results. This fact indicates that the current theory is valid and correct. Table 2. Observed and calculated strain changes at the points HSK and FDA Dilatation N87°E N177°E at FDA at HSK at HSK 0.77x 10-6 0.77x 10-6 -0.66 x 10-6 Observed 0.20xlO -6 0.10xl04 -0.78x10 -6 Okada 0.20x 10-6 0.11xl04 -0.80x 10-6 HomoSph O.16xlO -6 0.11xl04 -0.82 x 10-6 1066A Then, we perform a computation for a spherically symmetric Earth model, 1066A (Gilbert and Dziewonsk, 1975). Results in Table 2 (fourth row) show that the volume change is 20% different from that of the homogenous models. Although the results calculated by the above theory fundamentally agree well with observed strain changes, some discrepancies remain for two reasons: 1) the fault model is inaccurate, and 2) the topography effect. The second reason is considered as dominant because this earthquake occurred in the deep Pacific trench and this theory ignores complicated terrain effects. This effect will be addressed in a future study.
7.
Effects of the Earth's Spherical Curvature and Radial Heterogeneity
Two Earth models are used to study effects of sphericity: a homogeneous half-space and a homogeneous sphere. Numerical calculations are made for the four independent seismic sources mentioned previously: two shear strikes and two tensile openings. Media parameters used in both models are equal to those of the top layer of the 1066A Earth model. A comparison of the near field co-seismic 12(a, 0, (,o) strain changes of component err calculated for the 1066A Earth model and the
Chapter 85 • Hew Theory for Calculating
homogeneous half-space caused by a vertical strike-slip source at a 32 km depth show that the two Earth models are almost identical: it is difficult to identify their differences. This similarity indicates that the effect of sphericity is very small for a shallow seismic source. However, as depth increases, their discrepancy becomes large. This fact is illustrated in Fig. 5, which shows a comparison of the co-seismic strain changes of the component err12(a , 0,(,o) for source depths of 3, 10, 100 and 300 km, respectively. We conclude that the effect of spherical curvature (proportional) increases as source depth increases. Because the homogeneous half-space model is non-gravitating, whereas the homogeneous sphere is self-gravitating, it is notable that the difference obtained by comparison in this section includes both the sphericity effect and the self-gravitation effect. Independent comparisons should be made to distinguish the two effects: a comparison between a homogeneous half-space and a homogeneous sphere with non-gravitation, and a comparison between two homogeneous spheres with and without self-gravitation. However, to assume that the self-gravitating effect is less than the spherical effect (to be confirmed) and to simplify discussion, we do not discuss them separately, and refer to them as effect of sphericity for convenience.
Strains
Changes Caused by Dislocations in a Spherically Symmetric Earth
Numerical calculations are made using the same parameters as those in the previous section. 32(a, 0, (,o) Figure 6 shows the strain components ekt (from top to bottom: k l = r r , 0 0 , ~o~o, 0~o ) calculated for the dip-slip source at 32-km depth. The blue curve shows results calculated for the 1066A Earth model; the red line shows homogeneous sphere results. The figure shows clearly that discrepancies between the two models are greater than 30% almost everywhere, including the epicenter. Discrepancies caused by the stratified structure are much larger than those of sphericity. Hence, we infer that the half-space dislocation theory might create an error of 30%. To investigate the effect of stratified structure, we further consider the 1066A Earth model with a new top layer of 11 km with parameters equivalent to those at 30 km. The corresponding calculated numerical results are plotted in Fig. 6 using a green color line. A difference appears for t h e er32 component (and also for all strain components of other sources) from comparison of the homogeneous and 1066A Earth models. It again confirms the importance of the Earth structure in computing co-seismic strain changes. "ll~l .-gO, • I~.l.
• I.'l"
..,. L"
J.
m £ I
"iI
£ •
'11
£ ".
'11
£ ."
I1
£ I
I
L I
.i.I
L ."
.l'l
i. ~.
.Iq
i. "
.i I
i. ~
I
='F
r ~
I
.1. L. ."gO, •
[ I ~'.IL • I r~z HiT •
i..
•
~,
• L
~.
• ~:,:
I, ..~
I I'1", ;4"C,.
g
I:'.Cl
L'"
I: I "
L-."
I a'J
L J
I ."
L II
3-1:1
L :l
f.
:~,.
.
.
.
.
.
.
.
i',.
~
~ . .
I.
I1~. I" I ' -
r,
r
"
."El', • ,:-'.,.
, .,.
;
, .,.
.I "lgl',
~I
r.
C'l
1".~
c".
1"~
c ,.
Li-.¢-
."Hi L - - . I r
l"d
r -
..n
(34.
r ~1. 1...£
... I
"l
]
i Hi
L..
/
I
II 'd
i
I
l
II
L'rIJLr
Fig. 5. Comparison of co-seismic strain components 12
err (a, 0, (,o) calculated for the 1066A Earth model and the homogeneous half-space caused by a vertical strike-slip source at depths of 3, 10, 100 and 300 km, respectively. The blue curve shows results calculated for the 1066A Earth model; the red line shows homogeneous sphere results. To study the effect of stratified structure, we consider the homogeneous sphere and the heterogeneous sphere (1066A model) and compare results calculated from the two models.
Fig. 6. Comparison of co-seismic strain changes 32 ( a , O , ~o) calculated for dip-slip source at 32-kin ekt depth in the homogeneous sphere, 1066A Earth model and with its new top layer. Top to bottom panels show respective results for k l = r r , OO , fafa , O~o . The blue curve shows results calculated for the 1066A Earth model; the red line shows homogeneous sphere results. 8.
Summary
This research presents a new theory for calculating co-seismic strain change for a SNREI Earth model. Four independent point sources are assumed to be
591
592
W.Sun. S. Okubo. G. Fu
located at the polar axis. Corresponding expressions for calculating co-seismic strain components are derived by introducing strain Green's functions. A proper combination of these expressions is useful to calculate co-seismic strain components caused by an arbitrary seismic source at any geographical position in the Earth. Numerical computations are performed for calculating surface strain components in the near field caused by four independent sources at a depth of 32 km inside the 1066A Earth model. Results agree well with those calculated for a half-space Earth model. Results confirm the validity of the theory presented herein. This theory is also applicable to compute co-seismic strain effects caused by the 1994 Far Off Sanriku earthquake. Results indicate that the new theory can approximately predict the observed results. Effects attributable to the Earth models themselves were also investigated. Furthermore, the effects of spherical curvature and the stratified structure of the Earth in computing co-seismic strain changes are also investigated using the present dislocation theory and Okada's (1985) formulation. Curvature effects are small for shallow seismic events, but they are (fractionally) larger for greater source depths. Stratified effects are very large for any depth and epicentral distance, reaching a discrepancy greater than 30% almost everywhere Acknowledgements. This research was supported
financially by JSPS Grant-in-Aid for Scientific Research (C16540377) and "Applications of Precise Satellite Positioning for Monitoring the Earth's Environment". Helpful comments by J. Beavan and G. Blewitt are highly appreciated. References
Ben-Menahem, A., and M. Israel (1970), Effects of major seismic events on the rotation of the Earth, Geophys. J. R. Astron. Soc., 19, 367-393. Ben-Menahem, A., and S. J. Singh. (1968), Eigenvector expansions of Green's dyads with applications to geophysical theory, Geophys. J. R. Astron. Soc., 16, 417.452. Dahlen, F. A. (1968) The normal modes of a rotating, elliptical Earth, Geophys. J. R. Astron. Soc., 16, 329-367. Dziewonski, A. M., and D. L. Anderson (1981), Preliminary Reference Earth Model, Phys. Earth Planet. Inter., 25, 297-356. Faculty of Science, Tohoku University (1994), Faulting Process of the 1994 Far East Off Sanriku Earthquake inferred from GPS observation, The Report of CCER The
coordinating Committee for Earthquake Prediction, Japan, Vol. 54, 97-98. Fu, G., and W. Sun (2004), Effects of Spatial Distribution of Fault Slip on Calculating Co-seismic Displacements- Case Studies of the Chi-Chi Earthquake (m=7.6) and the Kunlun Earthquake (m=7.8), Geophys. Res. Lett., Vol. 31, L21601, doi: 10.1029/2004GL020841. Gilbert F., and A. M. Dziewonski (1975), An application of normal mode theory to the retrieval of structural parameters and source mechanisms from seismic spectra, Phil. Trans. R. Soc. London A, 278, 187-269. Maruyama, T. (1964), Statical elastic dislocations in an infinite and semi-infinite medium, Bull. Earthquake Res. Inst. Univ. Tokyo, 42, 289-368. Okada, Y. (1985), Surface deformation due to shear and tensile faults in a half-space, Bull. Seism. Soc. Am., 75, 1135-1154. Okubo, S. (1993), Reciprocity theorem to compute the static deformation due to a point dislocation buried in a spherically symmetric Earth, Geophys. J. Int., 115, 921-928. Pollitz, F. F. (1992), Postseismic relaxation theory on the spherical Earth, Bull. Seismol. Soc. Am., 82, 422-453. Saito, M. (1967), Excitation of free oscillations and surface waves by a point source in a vertically heterogeneous Earth, J. Geophys. Res., 72, 3689-3699. Smylie, D. S., and Mansinha L. (1971), The elasticity theory of dislocation in real Earth models and changes in the rotation of the Earth, Geophys. J. R. Astron. Soc., 23, 329-354. Steketee, J. A. (1958), On Volterra's dislocations in a semi-infinite elastic medium, Can. J. Phys., 36, 192-205. Sun, W., and S. Okubo (1993), Surface potential and gravity changes due to internal dislocations in a spherical E a r t h - I. Theory for a point dislocation, Geophys. J. Int., 114, 569-592. Sun, W., and S. Okubo (1998), Surface potential and gravity changes due to internal dislocations in a spherical E a r t h - II. Application to a finite fault, Geophys. J. Int., 132, 79-88. Sun, W., Okubo S., and P. Vanicek (1996), Global displacement caused by dislocations in a realistic Earth model, J. Geophys. Res., 101, 8561-8577. Sun, W., and S. Okubo (2002), Effects of the Earth's Spherical Curvature and Radial Heterogeneity in Dislocation Studies - For a Point Dislocation, Geophys. R.L., V. 29, No. 12, 46 (1-4). Takeuchi, H., and M. Saito (1972), Seismic surface waves, Methods Comput. Phys., 11, 217-295.
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Part V Advances in the Realization of Global and Regional Reference Frames Chapter 86
Advances in Terrestrial Reference Frame Computations
Chapter 87
The Status and Future of the International Celestial Reference Frame
Chapter 88
Is Scintillation the Key to a Better Celestial Reference Frame?
Chapter 89
Improvement and Extension of the International Celestial Reference Frame in the Southern Hemisphere
Chapter 90
Limitations in the NZGD2000 Deformation Model
Chapter 91
Implementing Localised Deformation Models into a Semi-Dynamic Datum
Chapter 92
Definition and Realisation of the SIRGAS Vertical Reference System within a Globally Unified Height System
Chapter 93
Tests on Integrating Gravity and Leveling to Realize SIRGAS Vertical Reference System in Brazil
Chapter 94
Accessing the New SIRGAS2000 Reference Frame through a Modernized Brazilian Active Control Network
Chapter 95
Deformations Control for the Chilean Part of the SIRGAS 2000 Frame
Chapter 96
Estimation of Horizontal Movement Function for Geodeticor Mapping-Oriented Maintenance in the Taiwan Area
Chapter 97
Activities Related to the Materialization of a New Vertical System for Argentina
Chapter 98
An Analysis of Errors Introduced by the Use of Transformation Grids
Chapter 99
Preliminary Analysis in View of the ITRF2005
Chapter 100
Long Term Consistency of Multi-Technique Terrestrial Reference Frames, a Spectral Approach
Chapter 86
Advances in terrestrial reference frame computations Detlef Angerrnann, Hermann Drewes, Manuela Kriigel, Barbara Meisel Deutsches Geod5tisches Forschungsinstitut (DGFI), Marstallplatz 8, D-80539 Munich, Germany e-mail:
[email protected] Abstract. In its function as an ITRS Combination Center, DGFI has developed refined methods for the terrestrial reference frame computation. The advanced approach is based on the combination of epoch normal equations (weekly/ daily data sets) containing station positions and Earth orientation parameters (EOP) obtained from different geodetic space techniques such as VLBI, SLR, GPS and DORIS. This refined approach allows to account for nonlinear effects (e.g., periodic signals and discontinuities) in station positions and to ensure consistency between T R F and EOP. The ITRF2004 submissions were used as input for a refined realization of the terrestrial reference frame. This paper presents the combination methodology and the current status of the ITRF2004 computations at DGFI. Key words: ITRF computation, EOP, VLBI, SLR, GPS, DORIS
1
Introduction
Today, space geodetic techniques allow to determine geodetic parameters (e.g., station positions, Earth rotation) with an accuracy up to the millimeter level. However, this high accuracy is not fully reflected in current realizations of the terrestrial reference frame. The reasons are manyfold and reach from remaining biases between different observation techniques to deficiencies in the combination methodology. With the high accuracy of the space geodetic techniques time-variable effects of station positions and datum parameters (e.g., T R F origin) become detectable, that are not considered in recent T R F realizations based on multi-year solutions with station positions and velocities. The most recent realization of the International Terrestrial Reference Frame, the ITRF2000, has been performed in 2000 based on solutions of the space geodetic techniques available at that time (Altamimi et al., 2002; Boucher et al., 2004). Since then almost five years of addi-
tional data have become available, new sites have joined the global network, the processing strategies and models have been improved and some station positions and velocities are no longer valid because of events (e.g., earthquakes, equipmerit changes, etc.). Furthermore, until now, the major products of the International Earth Rotation and Reference Systems Service (IERS), namely the ITRF, the International Celestial Reference Frame (ICRF), and the EOP are computed (combined) separately by different IERS Product Centers. Consequently, the IERS products are not consistent. The results of the IERS Analysis Campaign to align EOP to I T R F 2 0 0 0 / I C R F reveal that significant biases between EOP series exist (e.g., Dill and Rothacher, 2003). This means that there are clear deficiencies in the present IERS product generation, which have to be overcome by a rigorous combination of station positions, EOP (and quasar positions). Towards this aim, the IERS Combination Pilot Project (CPP) has been initiated in 2004 (as a follow-on project of the SINEX Combination Campaign) to develop suitable methods for a rigorous combination of the IERS products, and to prepare the product generation on a weekly basis. The general scope and the objectives of the CPP are presented in Rothacher et al. (2005). Taking into account the deficiencies of current realizations of the terrestrial reference frame a Call for long time series of "weekly" SINEX files for ITRF2004 and a supplementation of the CPP was released in December 2004. The ITRF2004 is based on the combination of time series of station positions and EOP. Weekly or (daily VLBI) contributions will allow better monitoring of nonlinear motions and other kinds of discontinuities in the time series. The ITRS Combination Centers, namely DGFI, IGN (Institut Geographique National, Paris), and NRCan (National Resources Canada, Ottawa), coordinated by the ITRS Product Center (IGN), are in charge for the generation of the ITRF2004 solution.
596
D. A n g e r m a n n • H. Drewes • M. KriJgel • B. Meisel
2
ITRS Combination Center at DGFI
DGFI serves as an ITRS Combination Center (ITRS-CC) within the IERS. Various software programmes of the DGFI Orbit and Geodetic Parameter Estimation Software (DOGS) are implemented. The methodology for the terrestrial reference frame computations at DGFI is based on the combination of unconstrained normal equations. The processing flow and the major comportents of the ITRS-CC are shown in Fig. 1. A detailed description of the combination methodology is provided in Angermann et al. (2004). DGFI has computed a terrestrial reference frame realization 2003 based on multi-year solutions with station positions and velocities (see Angermann et al., 2004; Angermann et al., 2005; Drewes et al., 2005). The contributing space geodetic techniques are Very Long Baseline Interferometry (VLBI), Satellite Laser Ranging (SLR), the Global Positioning System (GPS) and the Doppler Orbitography and Radiopositioning Integrated by Satellite (DORIS). The T R F computations provided valuable results to assess the current accuracy of the terrestrial reference frame, to identify remaining deficiencies and to enhance the combination methodology. The refined approach is based on the com-
Auxilliary Data
I
• StationInformation • Localties
IndividualSolutions/ NEQ's (VLBI, SLR, GPS, DORIS) .
.
.
.
.
.
.
.
.
bIN~-Z,t-orma[
SNXCHK CheckSINEX Format
IndividualFormats (e.g. DOGS,OCCAM)
/
DOGS-AS Analysis of solutions
SNX2DOGS ConvertionSINEX--~--DOGSFormat Generationof free NEQ's
Individual NEQ's (unconstrained) DOGS-OV OutputVisualisation I P,otso~are I
DOGS-C8 Combination& Solutionof NEQ's IPrep ...... ing of individualNEO's I
[ Analysisof ti. . . . . ies ]
I
Intra-technique I Comparison/ Combination I Inter-technique I Comparison/ Combination
I Helmert-T.... formation I
l
Final
CombinedSolution
)
Figure 1" T R F combination procedure.
1
bination of epoch (weekly/daily) SINEX files of the different space geodetic techniques containing station positions and EOP. The advantages are that nonlinear effects in site motions (e.g., caused by earthquakes or instrumentation changes) are detectable and that secondly consistency between the terrestrial reference frame and the EOP can be achieved. The results of an exemplary T R F realization obtained from an accumulated five years solution (1999-2004) are encouraging (Meisel et al., 2005). The ongoing activities at DGFI focus on the processing for the official IERS product ITRF2004.
3
ITRF2004 input data
According to the specifications of the IERS Call for long time series of epoch SINEX files for ITRF2004 and a supplementation for the IERS Combination Pilot Project the input data have to follow the SINEX Version 2.0 format standard and should comply with one of the following constraints categories: (1) Free normal equations; (2) Loosely constrained solutions; (3) Solutions with removable constraints; (4) Solutions with T R F minimum/inner constraints. The temporal resolution of the SINEX files is one week (Sunday to Saturday) for DORIS, GPS and SLR, and one 24 hour session (17 hr to 17 hr) in the case of VLBI. The parameters to be included should comprise site positions, a set of EOP for each day (pole offsets and rates, LOD as well as UT1 for VLBI). If SINEX files contain the variancecovariance matrix, all constraints applied to the solution should be given in the a priori variancecovariance matrix. Solicited solutions are of three types: (A) Official single-technique combined solutions from the Technique Centers. (B) If solutions of type A are not available or for specific reasons, individual Analysis Center solutions are solicited under the coordination of the corresponding Technique Center. (C) Solutions that result from a combination of various techniques at the observation level may be submitted as well. Table 1 summarizes the major characteristics of the ITRF2004 submissions as available by August 2005. In the case of GPS, SLR and VLBI official single-technique combined solutions were submitted by the Techniques' Combination Centers, namely NRCan for the International GNSS
Chapter86
• Advances in Terrestrial Reference
FrameComputations
Table 1: Summary of ITRF2004 submissions. Techn.
Service AC
Data
Time period
Parameters
Constraints
GPS
IGS NRCan
weekly solutions
1996 2004 from June 1999 from March 1999
Station positions EOP (pole rates, LOD) geocenter
NNT: 0.1 mm NNR: 0.3 mm NNS: 0.02 ppb
VLBI
IVS GIUB
daily sessions free NEQs
1984
Station positions EOP (pole, UT1 + rates)
none
SLR
ILRS ASI
weekly solutions
1993 - 2004
Station positions EOP (pole + LOD)
1m 1m
DORIS
IGN
weekly sol.
1993 2004
loose
INA
weekly sol.
10/92
LCA
weekly sol.
1993
Station positons EOP (pole, UT1 + rates) Station positions EOP (pole, UT1 + rates) Station positions EOP (pole)
2004
06/2004 2004
Service (IGS), the Geodetic Institute of the University Bonn (GIUB) for the International VLBI Service for Geodesy and Astrometry (IVS), and the Agenzia Spaziale Italiana (ASI) for the International Laser Ranging Service (ILRS). Until now, no combined DORIS solution is available by the IDS. Three solutions of individual DORIS Analysis Centers (IGN; INA: Institute of Astrononly, Russian Academy of Sciences; LCA: LEGOS/CLS, France) were submitted as ITRF2004 contributions. In addition to the SINEX solutions the Technique Centers also provided a list with information about discontinuities (e.g., equipment changes, earthquakes) in station positions, which are used as input by the ITRS Combination Centers. The ITRF2004 submissions were analysed by the ITRS Combination Centers and intratechnique combinations have been performed to validate their combination procedures and to give feedback to the Technique Centers and contributing Analysis Centers. The current status of the ITRF2004 submissions is that the ILRS and IVS will provide updated SINEX files until Oct./Nov. 2005, which will then serve as official input for the ITRF2004 computation. In the case of GPS the input data are almost fihal, however an updated version will be submitted as well. Hopefully, also the IDS will provide combined DORIS solutions as official ITRF2004 submission.
4
none
loose loose
Combination methodology
The combination methodology for the ITRF2004 computation applied at DGFI comprises the fob lowing major steps: • Analysis of ITRF2004 submissions as input data and generation of normal equations • Analysis of time series and combination pertechnique (intra-technique combination) • Comparison and combination of different techniques (inter-technique combination) • Final combined solution
Analysis of input data and generation of normal equations: In a first step the ITRF2004 submissions were analysed concerning various aspects, such as the SINEX format compatibility, the suitability for a rigorous combination of station positions and EOP, and the a priori constraints that were applied by the Technique Centers or individual Analysis Centers. If possible, the reported constraints were removed and free normal equations were generated. The resulting SLR normal equations contain information to realize the scale and the T R F origin; they have a rank defect of three w.r.t, the rotations. The VLBI submissions in the form of normal equations could be used directly (without inversion and reduction of constraints). They
597
598
D. Angermann • H. Drewes • M. Kffigel • B. Meisel
have a rank defect of six (three translations and three rotations) and contain information to realize the scale. Since the weekly GPS and DORIS solutions are defined in an '~arbitrary '~ reference frame~ it was necessary to reduce the datum information by setting up respective Hehnerttransformation parameters. I n t r a - t e c h n i q u e c o m b i n a t i o n : The epoch normal equations were accumulated for each technique separately to compute multi-year solutions with station positions, velocities and EOP. In a first step we include all stations available except those without sufficient data to estimate velocities (e.g.~ less than 2.5 years} and we use the information on discontinuities that is provided by the Technique Centers. We compute epoch solutions by applying datum constraints and inverting the matrices and align them to the multi-year reference solution by a seven parameter similarity transformation to investigate the resulting time series (station positions and transformation parameters} w.r.t, discontinuities or other nonlinear effects (e.g.~ periodic signals and postseismic deformations). To incorporate these effects we change the parameterization for the accumulation of the epoch data (e.g.~ by setting up new positions and velocities where necessary). Postseismic deformation may be parameterized by piecewice linear functions. To account for annual signals it is possible to introduce amplitudes and phases as additional parameters. Finally the intra-technique solutions are obtained by applying minimum datum constraints. Inter-technique combination: Input for it are the accumulated intra-technique normal equations for VLBI~ SLR~ GPS and DORIS. The parameters comprise station positions, velocities and daily EOP. Concerning the combination of EOP of the different space techniques it has to be considered~ that the VLBI estimates are referred to the midpoint of a daily VLBI session (from 17 hr to 17 hr)~ whereas the EOP values of the other techniques are referred to 12 h. Thus the VLBI EOP estimates have to be transformed to the reference epochs of the other techniques. A key issue within the inter-technique combination is the implementation of local tie information. For this purpose the EOP are essential to validate the local tie selection and to stabilize the inter-technique combination as additional "global ties". Other issues include the equating of station velocities of co-located instruments and the weighting between different tech-
niques. The intra-technique normal equations are added by applying the weighting factors. Final c o m b i n e d solution: The combined inter-technique normal equations are completed by pseudo-observations for the selected local ties and for equating station velocities at co-location sites. To generate the final combined solution~ we add datum conditions and invert the resulting normal equation system. The geodetic datum is realized by no-net-rotation (NNR) conditions~ minimizing the common rotation of the ITRF2004 solution w.r.t, its approximate values for the orientation at the reference epoch 2000.0~ and minimizing the horizontal velocity field over the whole Earth for the time evolution of orientation. For the realization of the kinematic reference frame it is necessary to compute a present-day model representing the entire motions of the Earth surface~ such as the Actual Plate Klnematik and Deformation Mode]~ APKIM2002 (Drewes and Meise], 2003). The origin (translation and their rates) is realized by SLR, and the scale and its rate by SLR and VLBI. The final ITRF2004 solution will comprise station position~ velocities and daily EOP estimates as primary results. In addition epoch position residuals and geocenter coordinates will be obtained from the time series combination. Status
of
ITRF2004
computations
at
DGFI
The ITRF2004 submissions were analysed at DGFI and first intra-technique solutions were computed on the basis of the weekly input data. As the ITRF2004 submissions are not final at the moment~ the major focus of the computations is (1) to identify remaining deficiencies; (2) to provide feedback to the Technique Centers and to the contributing Analysis Centers; (3) to validate and enhance the combination procedure; (4) to perform first comparisons among the ITRS Combination Centers. The current status is that the intra-technique combination for GPS can be considered as almost final (see sect. 5.1)~ and comparisons among the three ITRS Combination Centers were performed (see sect. 5.2). The intra-technique combination for SLR, VLBI and DORIS are preliminary as new ITRF2004 input will be submitted by the corresponding Technique Centers. Concerning the inter-technique combination specific investigations were performed and re-
Chapter86 • Advancesin TerrestrialReferenceFrameComputations
fined methods were developed (see chapter 6), which can directly by implemented for the computation of the final ITRF2004 solution. 5.1
G P S intra-technique combination
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In a first step the weekly SINEX solution files of the IGS were analysed and normal equations in DOGS-format were generated. As the resulting GPS normal equations are not singular, we reduced the a priori datum information by setting up seven Helmert-transformation parameters for each week within the accumulation to avoid possible network deformations caused by the a priori datum. We used the information about discontinuities that is provided by the IGS for the accumulation of the weekly normal equations. Specific information related to the GPS intratechnique combination is provided below:
82.6 82.58 1000
• So far the velocities are not set equal yet for different solution numbers at the same station; this procedure will be done within the inter-technique combination. The accumulated GPS normal equation has 14 degrees of freedom as it should be since the datum information was reduced by setting up Helmert-transformation parameters.
• Finally, minimum constraints were applied to generate the final combined GPS solution. The estimated parameters comprise station positions, velocities and daily EOP. As an example for the GPS intra-technique computation Fig. 2 shows the time series of weekly station positions along with the position and velocity estimations of the accumulated multi-year solution for the GPS station HOFN, Iceland. A discontinuity in the station heights of about 5 cm was caused by an antenna and receiver change, which led to two different solutions on this station.
0
Time lid2000]
Figure 2: Position time series and velocity estimates for the height component of GPS station HOFN, Iceland.
Table 2: Vertical station velocity estimates of different solutions at GPS station HOFN, Iceland.
• A few additional discontinuities were introduced, which were obtained from the time series analysis. • All sites with a time span with less than 2.5 years were excluded (since no reliable station velocities can be estimated). In addition solutions of a specific solution number for a station with less than 26 weeks were not used.
500
Velocities
Epochs JD2000
Estimates [mm/yr]
Solution 1 Solution2
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As shown in Table 2 the velocity estimates of these two solutions before and after the event differ by 4.2 mm/yr. An important issue (also for many other GPS stations) is the question, whether the estimated velocities for different solutions on a station should be equated or not. For the application of statistical tests it is important that the standard deviations are realistic, which requires sophisticated weigthing methods. 5.2
Comparison of G P S solutions I T R S C o m b i n a t i o n Centers
among
The new IERS structure with three ITRS Combination Centers (IGN, NRCan and DGFI) provides an optimal basis for the accuracy evaluation of the terrestrial reference frame computations. For first comparisons among the ITRS Combination Centers results of the GPS intratechnique combination were used. It has to be considered that different strategies were applied (e.g., at DGFI and NRCan velocities of different solutions for a station were not equated, IGN equated most of the velocities). Thus we used for the comparisons a subset of about 65 IGS reference frame stations without discontinuities to estimate RMS differences for station positions and velocities between the different GPS solutions. As shown in Table 3 the results of the
599
600
D. Angermann • H. Drewes • M. Kriigel • B. Meisel
Table 3: RMS differences for station positions and velocities. RMS position differences [mm]
ITRS CC
DGFI IGN DGFI NRCan IGN NRCan
RMS velocity differences [mm/yr]
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Imaging Results
Imaging observations of all existing Southern Hemisphere ICRF sources have been made at least once. This imaging program will continue to image both the new ICRF sources (see Section 5) and to obtain second epoch images of all sources concentrating on those that show extended structure and thus most need their suitability as ICRF sources to be monitored and evaluated. Final images of 110 of these sources (some with images at multiple epochs) are presented in [16] and [18]. Reduction and imaging of the remaining sources is in progress. A preliminary morphological classification of the imaged sources was made by inspection of the images and of Gaussian models fitted to the self-calibrated visibility data using the Caltech Difmap package. Barely 40% of these sources exhibit a compact structure with a single fitted component. A more robust way to quantify the effect of intrinsic source structure on calculations of bandwidth synthesis delay is to define a source "structure index", following [6], according to the median value of the structure delay corrections calculated for all projected VLBI baselines that could possibly be observed with Earth-based VLBI. About 35% of the sources in our imaged sample have a structure index of I or 2, indicative of compact or very compact structures as exemplified by Figure 4 and Figure 5. The remaining sources have a structure index of 3 or 4 indicating the presence of more extended emission structures (see Figure 6 and Figure 7). Pointing out that such structures are more likely to affect the observed VLBI synthesis delays, [6] recommended that they be avoided in astrometric and geodetic VLBI experiments requiring the highest accuracy. This stipulation applies particularly to sources with a structure index of 4 which form approximately 35% of this sample. This high percentage of sources with extended structure underscores the importance of this imaging program in maintaining and improving the accuracy of the ICRF. Apart from its stated goal of providing morphological information necessary to evaluate the continuing suitability of ICRF sources, this imaging program forms the most extensive VLBI imaging program of Southern Hemisphere extragalactic radio sources undertaken. As such, it provides an unique opportunity for two types
Chapter 89 • Improvement and Extension of the International Celestial Reference Frame in the Southern Hemisphere
of study. First, it furnishes a large (over 200 sources including the newly added ones) sample of sources for study of a wide range of astrophysical phenomena associated with extragalactic sources in a manner analogous to the PearsonReadhead [19] and follow up surveys in the northern hemisphere. The sample includes subgroups of great physical interest e.g. objects with energy spectra extending up to gamma-ray energies (the so called E G R E T sources) and sources with X-ray jets (radio jets which also emit in the X-ray) detected by the Chandra space observatory. It also contains morphologically defined sub-groups such as BLLacertae objects (blazars with weak emission lines) and quasars which are suitable for study of unification schemes that invoke geometric orientation effects and the presence of an obscuring torus to account for the variety of morphologies exhibited by extragalactic radio sources. Second, the survey includes a number of sources that are individually very interesting and have remained relatively unstudied only because they are too far south for northern hemisphere telescopes to observe. Just one example is the gigahertz peaked spectrum (GPS) galaxy PKS 1934-638 (Figure 7) where we may have detected proper motion over the longest time baseline for any such object [17].
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At least one epoch of VLBI imaging observations of all existing Southern Hemisphere sources have been concluded and imaging of new sources as well as second epoch observations, necessary to develop a time-dependent model of source
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An ATNF and USNO VLBI observing program to increase the density of Southern Hemisphere ICRF sources and to image their structure is yielding rich dividends. To date, milliarcsecond accurate astrometric positions for 74 new Southern Hemisphere sources have been determined and made available for inclusion in the next realization of the ICRF. This is the largest group of new Southern Hemisphere positions made available since the initial definition of the ICRF. Astrometric observations of more new sources are in progress. Not only are these new sources vital for strengthening the ICRF, they also densify a grid of "phase reference" calibrator sources in the Southern Hemisphere that are essential for the study of weaker objects of astrophysical interest such as radio stars, pulsars and weak quasars.
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Figure g" Example of an ICRF source with structure index of 2. Almost all of the flux is still contained in the core component with only a small fraction in a nearby weak component.
621
622
R. Ojha. A. Fey. P. Chariot. K. Johnston. D. Jauncey. J. Reynolds. A. Tzioumis • J. Lovell • J. Quick. G. Nicolson • S. Ellingsen • P. McCulloch • Y. Koyama Clean R R map. 0925-203 ,
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Figure 6: Example of an ICRF source with structure index of 3. A significant fraction of the total flux is located in a component some distance away from the core. Such structures are more likely to affect observed VLBI synthesis delays. Clean
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structure, are in progress. Images and "structure index" calculations for 110 of these sources are complete. About 35% of these sources have "structure index" indicative of compact or very compact structure. Thus ahnost twothirds of these sources may only be suitable for high-precision astrometry or geodesy after making correction for structure. This imaging program furnishes the largest set of VLBI images of extragalactic radio sources in the Southern Hemisphere and makes available unique data for the study of both individually interesting sources and large samples of interesting types of sources that have remained unstudied due to their southerly location. Fundamentally, this program of imaging and addition of new sources strengthens the ICRF, which is the fundamental celestial reference frame, by allowing the choice of better and additional sources as defining sources in the next realization of the ICRF.
References [1] Cannon, W. H., Baer, D., Feil, G., Feir, B., Newby, P., Novikov, A., Dewdney, P. E., Carlson, B. R., Petrachenko, W. T., Popelar, J., Mathieu, P., Wietfeldt, R. D. (1997). The $2 VLBI system. Vistas in Astronomy 41, pp. 297-302 [2] Charlot, P. (1990). Radio-source structure in astrometric and geodetic very long baseline interferometry. The Astronomical Journal 99, pp. 1309-1326
9 f o o
E
[3] Clark, T. A. et al. (1985). Precision Geodesy Using the Mark-III Very Long Baseline Interferometer System. IEEE Trans. Geosci. Remote Sens. 23, pp. 438-449
g o c
o
cl
6 o o rY
o cN I
[4] Dehant, V., Feissel-Vernier, M., de Viron, O., Ma, C., Yseboodt, M., Bizouard, C. (2003). Journal of Geophysical Research (Solid Earth) 108, pp. ETG 13-1
a o
210
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,
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-60
Right Ascension (mas) RA: 19 39 25.027, Dec: - 6 3 42 45.625 (2000.0)
Map center: Map peak: 0 . 3 7 8 d y / b e a m Contours: 0 . 0 1 3 5 J y / b e a m x (-1 1 2 4 8 16 ) Beam FWHM: 3.78 x 2.9 (mas) at 70.5 °
Figure 7: Example of an ICRF source with structure index of 4. Such double structures are best avoided in those astrometric and geodetic experiments requiring high accuracy.
[5] Feissel-Vernier, M., Ma, C., Gontier, A.-M., Barache, C. (2005). Sidereal orientation of the Earth and stability of the VLBI celestial reference frame. Astronomy and Astrophysics 438, pp.1141-1148 [6] Fey, A. L. & Charlot, P. (1997). VLBA Observations of Radio Reference Frame Sources. II. Astrometric Suitability Based on Observed
Chapter 89 • Improvement and Extension of the International Celestial Reference Frame in the Southern Hemisphere
Structure. The Astrophysical Journal Supplement Series, 111, pp. 95-142
ometry. The Astronomical Journal, 116, pp. 516-546
[7] Fey, A. L., & Charlot, P. (2000). VLBA Observations of Radio Reference Frame Sources. III. Astrometric Suitability of an Additional 225 Sources. The Astrophysical Journal Supplement Series 128, pp. 17-83
[15] MacMillan, D. S. & Ma, C. (1997). Atmospheric gradients and the VLBI terrestrial and celestial reference frames. Geophysical Research Letters 24, pp. 453-456
[8] Fey, A. L., Ojha, R., Jauncey, D. L., Johnston, K. J., Reynolds, J. E., Lovell, J. E. J., Tzioumis, A. K., Quick, J. F. H, Nicolson, G. D., Ellingsen, S. P., McCulloch, P. M., & Koyama, Y. (2004a). Accurate Astrometry of 22 Southern Hemisphere Radio Sources. The Astronomical Journal 127, pp. 1791-1795 [9] Fey, A. L., Ojha, R., Reynolds, J. E., Ellingsen, S. P., McCulloch, P. M., Jauncey, D. L., & Johnston, K. J. (2004b). Astrotnetry of 25 Southern Hemisphere Radio Sources from a VLBI Short-Baseline Survey. The Astronomical Journal 128, pp. 2593-2598 [10] Fey et al. (2006). In Preparation [11] Johnston, K. J. et al. (1995). A Radio Reference Frame. The Astronomical Journal 110, pp. 880-915 [12] Johnston, K. J. & de Vegt, C. (1999). Reference Frames in Astronomy. Annual Reviews of Astronomy and Astrophysics 37, pp. 97-125 [13] Kingham, K. A. (2003). Washington Correlator. In: International VLBI Service for Geodesy and Astrometry 2002 Annual Report N. R. Vandenberg & K. D. Bayer (eds) (NASA Tech. Pap. 211619)(Greenbelt, MD:GSFC), pp. 201-202 [14] Ma, C., Arias, E. F., Eubanks, T. M., Fey, A. L., Gontier, A . - M . , Jacobs, C. S., Sovers, O. J., Archinal, B. A., & Charlot, P. (1998). The International Celestial Reference Frame as Realized by Very Long Baseline Interfer-
[16] Ojha et al. (2004a). VLBI Observations of Southern Hemisphere ICRF Sources. I. The Astronomical Journal 127, pp. 3609-3621 [17] Ojha, R., Fey, A. L., Johnston, K. J., Jauncey, D. L., Tzioumis, A. K., Reynolds, J. E. (2004b). VLBI Observations of the Gigahertz-Peaked Spectrum Galaxy PKS 1934-638. The Astronomical Journal 127, pp. 1977-1981 [18] Ojha et al. (2004). VLBI Observations of Southern Hemisphere ICRF Sources. II. Astrometric Suitability Based on Intrinsic Structure. The Astronomical Journal 130, pp. 25292540 [19] Pearson, T. J., & Readhead, A. C. S. (1988). The milliarcsecond structure of a complete sample of radio sources. II - First-epoch maps at 5 GHz. The Astrophysical Journal 328, pp. 114-142 [20] Reynolds, J. E., Jauncey, D. L., Russell, J. L., King, E. A., McCulloch, P. M., Fey, A. L., Johnston, K. J. (1994). A radio optical reference frame. 7: Additional source positions from a Southern hemisphere short baseline survey. The Astronomical Journal 108, pp. 725-730 [21] Russell, J. L. et al. (1994). A radio/optical reference frame. 5: Additional source positions in the inid-latitude southern hemisphere. The Astronomical Journal 107, pp. 379-384 [22] Wilson, W. E., Roberts, P. P., Davis, E. R. (1995). The ATNF Long Baseline Array Correlator. In: Proceedings of ~th A P T Workshop Sydney, King, E. A. (ed) pp. 16-20
623
Chapter 90
Limitations in the NZGD2000 Deformation Model J. Beavan GNS Science, PO Box 30368, Lower Hutt, New Zealand G. Blick Land Information New Zealand, Private Box 5501, Wellington, New Zealand
Abstract. The New Zealand Geodetic Datum 2000 (NZGD2000) is a semi-dynamic datum, in that coordinates are fixed to their values at 1 January 2000 and velocities from a horizontal deformation model are used to transform the coordinates of data collected before or after that date. The deformation model was calculated from GPS campaign data collected between 1991 and 1998, and was aligned with ITRF96. We examine the performance of this model in 2005 from two points of view: (1) how different are the ITRF2000 velocities from ITRF96, and (2) for new stations, and older stations where additional data have been collected, how well do the newly estimated velocities match those in the deformation model (after the ITRF96-ITRF2000 transformation)? We have calculated ITRF2000 velocities at points throughout New Zealand. The ITRF96 and ITRF2000 velocities differ by 4.8 mm/yr at azimuth -101 ° in the southwest of the country, and by 5.4 mm/yr at azimuth -111 ° in the northeast. The differences between newly-calculated ITRF2000 site velocities and velocities from the deformation model transformed to ITRF2000 range between zero and about 4 mm/yr. Velocities of some continuous GPS stations installed in the past few years differ by more than this (in two cases by >7 mm/yr), in part because the new velocities cannot be estimated reliably from relatively short spans of data, and in part because the velocities at some sites are not linear. Significant vertical velocities up to a few mm/yr are estimated for continuous GPS stations that have been established for at least four years. These comparisons suggest that an upgrade to the deformation model should be considered. Alignment of the deformation model with ITRF2000 (or its successors) will have benefits in combining newly collected data with existing data, as the ITRF96 to ITRF2000 transformation step will no longer be needed.
Keywords. Geodetic datum, deformation model, crustal deformation, New Zealand
1 Introduction In 1998 Land Information New Zealand (LINZ) implemented a new geocentric datum for New Zealand, New Zealand Geodetic Datum 2000 (NZGD2000) with a reference epoch of 1 January 2000 (2000.0). NZGD2000 is realised in terms of ITRF96 and uses the GRS80 ellipsoid; (see Grant et al., 1999; Blick et al., 2003; Office of the SurveyorGeneral, 2003a). A major conceptual departure from the definition of the previous national datum (New Zealand Geodetic Datum 1949) and other international datums is that NZGD2000 accommodates the effects of crustal deformation. This is achieved by applying a deformation model when generating new coordinates, enabling them to be transformed from one epoch to another following a method similar to that described by Snay (1999). For most users, it has the appearance of a static datum. The accuracy criteria aimed at for NZGD2000 are that a mark's coordinate accuracy relative to adjacent marks of the next highest order shall not exceed 0.05 m horizontally and 0.15 m vertically (Office of the Surveyor-General, 2003b). The deformation model must be of sufficient accuracy to enable these accuracy requirements to continue to be achieved over time. Where the computed positions of marks using the deformation model differ from the surveyed positions by greater than these limits, consideration will need to be given to refining the deformation model. The deformation model must be able to reflect the true deformation field with adequate accuracy and resolution. A model should include both the long term deformation trends and, potentially, discrete events such as earthquakes, where the model definition could include surface fault ruptures. This would depend on the extent to which fault movement should be reflected by the deformation field, and the extent to which it should be represented by changing the coordinates of survey marks. The deformation model used in NZGD2000 (Figure 1) is now seven years old and it is time to
Chapter90 • Limitationsin the NZGD2000DeformationModel
consider if the accuracy requirements of the datum are still being met. This paper examines the performance of the current model in 2005 from two points of view: (1) how different are the ITRF2000 velocities from ITRF96, and (2) for new stations, and older stations where additional data have been collected, how well do the newly estimated velocities match those in the deformation model (after the ITRF96-ITRF2000 transformation)?
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coordinates and velocities for all other stations in the solution. A 3-parameter (3 orthogonal rotations) transformation was derived by comparing the horizontal velocities of these 29 points in the deformation model with their estimated ITRF96 values. This transformation was used to convert the Australia-fixed deformation model into ITRF96, and this is the model used by LINZ for the NZGD2000 datum. The surveys used to determine the deformation model are now on average nearly 10 years old. As time passes, errors in the determination of the velocities used in the deformation model lead to increasing errors in the calculated position of marks in terms of the reference epoch of 2000.0. In effect, the spatial accuracy of the datum is steadily degrading. Also, the datum and the effectiveness of the deformation model may be degraded by localised and temporally non-linear deformation, for example earthquakes and recently observed "slow earthquakes" (e.g., Douglas et al., 2005; Beavan et al., 2006). In addition, the current deformation model is aligned with ITRF96, and NZGD2000 will therefore drift from future and more accurate realisations of the ITRF. These limitations are shown schematically in Figure 2.
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Frame
Improvements
We can easily convert the deformation model used in NZGD2000 from ITRF96 to ITRF2000 (or another realisation of the ITRF). We simply need to repeat the global or regional GPS analysis of the 29 1st order stations using exactly the same data that went into the original analysis, but with the positions and velocities of the global or regional
625
626
J.Beavan.G.Blick reference stations constrained to ITRF2000 rather than ITRF96. Figure 3 shows the differences between the ITRF96 and ITRF2000 positions of the 29 1st order stations between epochs 2000.0 and 2010.0. The differences have grown by -50 mm over this 10 year period, because of the -5 mm/yr velocity difference between ITRF96 and ITRF2000 in the New Zealand region. Our estimate of this velocity difference is 4.8 mm/yr at azimuth -101 ° in the southwest of the country, rising to 5.4 mm/yr at azimuth -111 ° in the northeast (westward velocities are faster in ITRF2000 than in ITRF96). Because the differences are quite uniform across the country, the absolute position accuracy of the datum is compromised; however, the relative accuracy between marks is maintained.
*~~
-35 °
-
CoordinateDifferences ~ ~ , ~ 0 , , 2000 to 2010 .~,~~3 ~400 OX
11.S~
BSC
1
....
-45°
~
....
.I~{ ''" -~ ~ •
165 °
170 °
50 mm
175 °
180 °
Fig 3. Horizontal coordinate differences between ITRF96 and ITRF2000 from epoch 2000.0 to epoch 2010.0.
While the change from ITRF96 to ITRF2000 velocities can be managed by a standard 7parameter transformation, the difference o f - 5 mm/yr is quite significant; after 10 years it exceeds the NZGD2000 horizontal accuracy criterion cited earlier. We expect that the changes from ITRF2000 to any future realisation of ITRF will be much smaller than this, so that it will be worthwhile aligning any update of NZGD2000 to ITRF2000 (or whatever ITRF realisation is current at the time of the update), rather than retaining ITRF96.
4 Deformation Model Distortions There are two types of potential distortion in the velocity model itself, even assuming that all points move linearly in time (i.e., with constant velocity). The first is that the velocity estimated at an individual site from 1992-1998 data is likely to be improved by the inclusion of additional (later) data collected from the same site (i.e., more time). The second is that the interpolation of the velocity field between those points where the velocity was actually measured is likely to be improved by the inclusion of velocities from additional spatially distributed points (i.e., more sites). At the time of computation of the NZGD2000 deformation model 371 points were used. There are significant regions where the data were geographically relatively sparse, and the interpolation relied heavily on the minimum strain-rate constraint and the plateboundary velocity conditions applied at the margins of the model. More than 800 points are now available where velocity estimates can be made and which could be included in a recalculation of the velocity model. These new data have largely been collected in GPS surveys carried out for scientific purposes associated with plate tectonic and earthquake hazard research. We can test the two types of distortion by comparing the NZGD2000 velocity model with (1) observed velocities at points used in the velocity model where additional data have been collected since 1998, and (2) observed velocities at points not used in the velocity model where at least two wellseparated epochs of data have been collected since 1998. In these comparisons, we use a version of the NZGD2000 velocity model that has been transformed from ITRF96 to ITRF2000 as explained in Section 3. We compare this model with observed velocities that we estimate in ITRF2000 by including a set of regional IGS stations in the GPS data analysis and aligning the daily solutions with the ITRF2000 coordinates of these stations. This means we are testing the velocity model itself, with minimal effect from differences in the datum on which it is based. There are a large number of points that fulfill one or other of the above criteria, but for this study we consider a limited number of points with rather stronger criteria. In the first category (points updated from the NZGD2000 deformation model), we use the 20 1st order points where additional data have been collected since 1998 (Table 1). These were the points that had the best history of occupation in NZGD2000, with at least four epochs
Chapter 90 • Limitations in the NZGD2000 Deformation Model
o f observation between 1993 and 1998. In general either one or two additional epochs o f data have been obtained since 1998. In the second category (new points since 1998) we take velocities from 24 continuous GPS stations with at least 2 years o f available data. W e use these, rather than campaign stations, because the extra data from continuous stations are likely to provide more accurate velocity estimates than we could obtain from the two (or at most three) available campaign occupations.
5 Comparison
at u p d a t e d p o i n t s
Figure 4 and Table 1 show the velocity differences and estimated uncertainties for the first category: points w h o s e velocities have been updated since 1998 by the collection o f additional data. Since the new velocities use the 1992-1998 data as well as the new data, the uncertainties between the two estimates are not independent. The uncertainties we plot are from the deformation model (which are larger than those from individual site velocity estimates). Only a few o f the differences (3 out o f 40; shown in bold type in Table 1) fall significantly outside 3 standard deviations. This suggests that the individual site velocites used in the construction o f N Z G D 2 0 0 0 were quite reliable, at least for the frequentlymeasured 1 st order sites.
Table 1. Tabulated velocity differences and 1a uncertainties shown in Figure 4. Site
(~(Ve)
~(Vn)
(J(Ve)
a(Vn)
Yrs a
Nb
1004 1103 1153 1181 1231 1259 1273 1305 1314 1367 1501 5508 5509 6731 A31C A33D AUCK B03W OUSD WGTN
-0.50 -0.08 0.18 0.41 3.59 -0.07 -0.24 -1.94 0.64 1.53 1.14 -0.67 -0.26 0.82 -0.64 0.34 2.27 0.09 1.40 2.30
-1.16 -1.41 -0.87 0.46 1.74 -0.66 0.98 2.54 2.06 0.48 0.88 -2.69 0.41 0.38 -2.47 -0.17 -0.02 -1.12 0.72 1.56
1.03 0.81 0.91 0.90 0.91 0.83 0.95 0.97 0.87 0.86 0.92 0.60 1.12 0.90 1.12 0.76 0.74 1.17 0.60 0.79
0.88 0.71 0.72 0.74 0.75 0.65 0.76 0.78 0.69 0.69 0.71 0.52 0.93 0.75 0.94 0.61 0.54 0.99 0.52 0.66
7.9 11.0 10.9 10.0 11.0 10.0 11.0 11.0 11.0 11.0 11.0 10.9 11.0 7.9 5.9 10.0 12.3 7.9 10.3 8.3
1 2 1 2 3 2 2 1 2 2 2 2 2 1 1 2
l~J uncertainties are from the deformation model. East-north correlations are -0. Velocities and uncertainties are in mm/yr. Differences greater than 3c~ are shown in bold type. aTotal duration of data series in years. bNumber of additional epochs of data at campaign sites.
6 Comparison
-350,
A~!~~11305 ~i
1273
. 4 0 o,
,,
-45 ° ,
B 0 3 W ~ D
~ 5509
95%confidenceregions
lSs o
17oo
5 mm/yr
175o
,
leo o
Fig 4. Velocity differences and 95% confidence regions for points used in NZGD2000 whose velocities have been updated with 1 to 3 epochs of new data since 1998.
2
at n e w p o i n t s
Figure 5 and Table 2 show the velocity differences and estimated uncertainties for the second category o f points, those newly observed since the NZGD2000 calculations. In this case the uncertainties in the model and those in the site velocity estimates are independent, so we combine them by summing variances. A substantially larger fraction o f the differences (7 out o f 48; shown in bold type in Table 2) fall significantly outside 3 standard deviations in this case. Also, the absolute values o f the differences are considerably larger, e x c e e d i n g - 5 m m / y r at five stations and 7 m m / y r at two o f these. This indicates either that the velocity model is not performing particularly well, or that the new velocity estimates are inaccurate in some way. Differences o f this size indicate that distortions in the current velocity model may, in some regions, exceed the N Z G D 2 0 0 0 50 m m horizontal accuracy criterion in less than 10 years.
627
628
J. Beavan. G. Blick
~,
"35°
W
7 Vertical motion :e
~ CORM J TAUP7'~" GISB
-400 J
WAN~ ~'7"~/
QUAR~ -45 °
DUNT
Vertical velocities were also computed for continuous sites with a >4 year data span. The computed velocities all fall b e t w e e n - 1 . 6 m m / y r and +4.3 mm/yr. The largest rates are within the Southern Alps where a more rigorous analysis by Beavan et al. (2004) has shown sites to be rising at 3-5 mm/yr relative to sites on the east coast of the South Island. The definition of vertical rates is subject to some fundamental questions (e.g., Blewitt, 2003), and there are probably regional slopes within the ITRF vertical motion field which affect our estimated velocities. The vertical rates are everywhere small enough that they do not need to be considered in N Z G D 2 0 0 0 , given the vertical accuracy criterion detailed above.
5 mm~/r
165 °
17o0
li'50
1so*
Fig 5. Differences between velocities predicted by the deformation model and velocities observed at continuous sites established since the model was computed, and having > 2 years data. Uncertainties not shown (see Table 2).
Table 2. Tabulated velocity differences shown in Figure 5. Site 6(vo) 6(Vn) o(v°) O(Vn) Years a CNCL -0.15 2.14 0.99 0.86 5.1 CORM -0.38 0.20 1.49 1.81 2.2 DNVK 2.12 3.21 1.93 1.85 2.7 DUNT 1.35 0.72 0.54 0.51 5.6 GISB -1.68 0.45 1.14 1.01 3.0 GRAC 2.54 0.31 0.82 0.72 6.5 HAMT -0.73 -0.78 1.20 1.11 2.2 HAST 4.70 0.80 1.40 1.50 2.8 HIKB -7.24 2.63 2.12 2.44 2.1 HOKI 0.95 1.27 1.05 0.92 6.7 KARA -0.96 1.90 0.95 0.81 5.1 MAST 3.77 -0.17 1.23 1.21 2.5 MQZG -0.32 -0.26 0.85 0.76 5.2 MTJO -0.17 -1.08 0.75 0.74 4.6 NETT 4.33 -0.21 1.05 0.95 5.1 NPLY -0.14 -2.06 0.98 0.93 2.3 PAEK 5.00 -0.39 0.91 0.79 5.2 QUAIl 0.80 1.54 0.96 0.84 5.1 TAKL 2.16 0.08 0.85 0.78 3.7 TAUP 5.27 2.29 1.08 1.01 3.3 TRNG 2.78 1.59 1.36 1.15 2.4 WANG 7.09 -1.05 1.81 1.92 2.2 WGTT 2.63 1.50 0.79 0.70 5.1 WHNG 1.66 -0.87 1.02 0.83 2.2 Uncertainties are combined from deformation model and site velocity estimates. Other details as in Table 1.
8 Discussion It is interesting to explore the reasons for the large horizontal velocity differences between the deformation model and several of the new sites (Table 2). A majority of the large differences are at sites where we have observed non-linear site velocities since the establishment of continuous GPS stations (e.g., HAST, W A N G , D N V K and PAEK). We believe this non-linear deformation is caused by slip episodes lasting from days to months on the deeper part of the subduction interface where the Pacific Plate descends beneath the North Island (Douglas et al., 2005; Beavan et al., 2006). These events are often known as slow slip events or slow earthquakes; the adjective "slow" refers to their rate of slip compared to the several km per second slip rate in normal earthquakes. Another site with a large velocity discrepancy is HIKB. Though this has shown a fairly steady velocity since the continuous station was installed, it is in a region where slow earthquakes appear to be common. It is possible that the data from this region used in the construction of the N Z G D 2 0 0 0 deformation model had been affected by slow earthquakes that were unrecognised in the campaign GPS data available at the time. The widespread occurrence of slow slip events, at least in the North Island, has implications for the N Z G D 2 0 0 0 deformation model. So far, the largest event we have observed (in 3 years) has had a magnitude of - 3 0 m m at the Earth's surface. This is within the N Z G D 2 0 0 0 horizontal accuracy specification, so it is possible that such events can be ignored at the N Z G D 2 0 0 0 accuracy level. However, to achieve
Chapter 90 • Limitations in the NZGD2000 Deformation Model
this it is important that the velocities in the N Z G D 2 0 0 0 deformation model are estimated using long enough spans of data that an average velocity is obtained. In the Gisborne region we have evidence that such events recur as often as twoyearly, implying that it should be easy to obtain an average velocity here; the small residual at GISB in Figure 5 indicates we may have achieved this. Though we have noted that steady vertical velocities may be neglected in respect of the N Z G D 2 0 0 0 deformation model, it is still important to estimate these velocities accurately when constructing the datum. This is particularly the case when positions need to be extrapolated to the reference time of the datum. An example is the 2000.0 reference epoch of N Z G D 2 0 0 0 , for which the data defining the datum were collected between 1992 and 1998. We know of at least one case where the vertical velocity estimated from 1992-98 data was significantly in error. This meant that the height coordinate for that point extrapolated to 2000.0 was in error (though not by more than the N Z G D 2 0 0 0 vertical criterion). To mitigate this problem it is desirable to estimate velocities using long data spans, and to set the reference epoch within the available data span.
9 Conclusions Our tests indicate that the ITRF96 datum drifts at about 5 mm/yr relative to ITRF2000 in the N e w Zealand region and the N Z G D 2 0 0 0 deformation model probably has errors >5 mm/yr in some regions, implying that the N Z G D 2 0 0 0 horizontal accuracy criterion will be exceeded within the next few years. Some N Z G D 2 0 0 0 stations have larger than desirable vertical position errors because of poor vertical velocity estimates and extrapolation to the 2000.0 reference epoch. There are now >800 sites in N e w Zealand with GPS velocity estimates, whereas there were 120) of determining Wo, because their anomalies refer to the level surfaces passing through the reference tide gauges rather than to a unique global surface; i.e. they are affected by the influences of vertical datum inconsistencies. Figure 3 shows the variation of Wo, if the MSS block size is changed. The largest deviation (0,1 mZs-2)
occurs between block sizes of 30' and 45', while Wo is very similar when using the remaining cell sizes. In this way, one can say that 1° x 1° is a representative cell size, and it will be taken as the standard in the following computations. The Wo estimates presented above are based on the models EGM96 and CLS01. In order to verify the reliability of these values, 141ois also computed using other GGMs and MSS models. In these computations GM corresponds to the value of each GGM and co to 7 292 115 x 10 -11 tad s -1 (IAG SC3 Rep. 1995). The GGMs computed up to n = 360 (EGM96, EIGENCG03C) are also truncated by n = 200 and n = 120 to compare the Wo values with those derived from lower degree models (TEG4, GGM02S). Table 1 summarizes the results. The largest Wo variations (1,26 m2s-2) are due to the extension of the computation area ((,o = 60 ° N/S or ~o = 80 ° N/S). However, over the same latitudinal range the Wo values are consistent. The Wo variation from one GGM to another (at the same degree and latitude coverage) is less than 0,02 m2s-2. The discrepancies between the Wo values derived from different MSS models are greater by including data from high latitudes O-=80°N/S than by middle latitudes
Chapter 92 • Definition and Realisation of the SIRGAS Vertical Reference System within a Globally Unified Height System
(p=60°N/S. It is assumed these differences are a consequence of the diverse models applied to analyse the altimetric data in each MSS model, and also the
inter-annual ocean variability averaged over distinct periods of time, which does not permit to define a specified reference epoch for the MSS heights.
Table 1.14Io values derived from different GGMs and MSS models [in m2s-2].
I
MSS
n
EIGEN-CG03C
EGM96
TEG4
GGM02S
120
62 636 853,35
62 636 853,37
62 636 853,38
62 6368 53,36
CLS01
200
53,35
53,37
53,37
q~
[N/S]
60/60 60/60
360
53,35
53,36
60/60
KMS04
360
53,24
53,26
60/60
GSFC00.1
360
53,58
53,59
60/60
120
62 636 854,61
62 636 854,62
62 636 854,65
200
54,61
54,62
54,64
360
54,61
54,61
82/80
KMS04
360
54,46
54,45
82/82
GSFC00.1
360
54,93
54,93
80/80
CLS01
62 636 854,61
82/80 82/80
WO [mzs 'z]
"--....
53.50
"
"
.
" " " " "
53.35
" I
"~
if' ~
m ~I
i
~
4
i
h
53.30
~ . . . .
53_25
1993
EGM96 [rb=3601 m 1994
.
. T E G 4 In=200] m 1995
-
- GGM02S [n=120] 1.996
CG03[n=1201 1.997
'
year 1.998
1999
2000
2001
Fig. 4 Annual Wo values derived from different GGMs and yearly MSS models from T/P 1-365 cycles (1 ° x 1°, (p = 60°N/S), the value 62 636 800 m2s -2 should be added.
To identify the variation of Wo as a function of time, annual MSS models are computed using the T/P altimetric data (cycles 1-365) available in the MGDR's Version C AVISO altimetry project (AVISO 1996). The FES2004 global tidal solution (Lettelier et al. 2004) and the sea state bias models of Chambers et al. (2003) were applied to the MGDR-C data. The yearly Wo values are obtained combining each annual MSS model (latitude limits (p = 60 ° N/S) with the GGMs reduced to the same epoch, i. e. MSS heights at 1997.0 are combined with GGM coefficients at 1997.0 to estimate Wo at 1997.0. Although the Wo changes as a function of the reference epoch of the GGM coefficients is insignificant (~0,002 mZs-2 from 1986 to 2000), they are reduced to the corresponding annual epochs (1993...2001) to be consistent with the MSS models. The results are shown in figure 4.
The variation of 14/o is highly correlated with the variation of MSS. The largest annual changes happened in 1996/1997 (-0,06 m 2s-2), 1997/1998 (-0,05 mZs-2) and 2000/2001 (-0,05 mZs-2). The total Wo variation between 1993 and 2001 is-0,20 mZs-2. The absolute level differences between the used GGMs are assumed insignificant, since they are lower than 5 mm. According to the above presented Wo values, one may conclude: Wo is almost independent of the GGM, slightly dependent on the MSS model and strongly dependent on the latitudinal extension. Its variation with time (until now) is almost negligible, but the sea surface is constantly changing and after some years this dependence could reach significant values. Therefore, it is necessary to define a reference epoch of 141oas a convention. This convention should also include, among other adoptions, the 14/o annual velocity, the latitudinal extension of the computation
641
642
L.Sanchez
area and the spatial resolution of the MSS model to be used. From the computations presented in this study, the value Wo = 62 636 853,4 m2s-2 is recommended. It is derived using the following specifications: _
_
[3]
being W~the actual gravity potential of a point referred to the height system i.
_
Extension:
q) = 60 ° N ... q0= 60 ° S
Resolution:
1° x 1°
MSS model:
T/P derived MSS heights at the epoch 2000.0
GGM:
EIGEN-CG03C, n - 120, reference epoch 2000.0
Constants:
o~, - w 0 - ~
P
W,
,~WJ-,,,,
W~
GM = 398 600,4415 x 10 9 m3s -2
dWj,,,,
co = 7 292 115 x 10-11 rad s-1 Procedure:
Eqs. 1 and 2, averaging Wi values with weight equal to cosq)
The proposed Wo value differs from previous computations by ~3 m2s-2 (Bursa et al. 1997, Bursa et al. 1998, Bursa et al. 1999, Bursa et al. 2002, Bursa et al. 2004). The reasons for this disagreement are not yet clear. It is necessary to make a detailed comparison between the different methodologies applied and the used GGMs and MSS models. In fact, the first computations of the presented work were carried out using the models EGM96 and CLS01 only. Since the obtained Wo value varies from those proposed by Bursa et al., it was decided to evaluate the reliability of our results by using two additional independent software suites (Smith 1998, Rapp 1982, Pavlis 1996), other MSS models (KMS04, GSFC00.1) and different GGMs (TEG4, GGM02S, EIGEN-CG03C). All the combinations yield to the here proposed Wo value. In consequence, it will be used as reference level in the next section.
3. Unification of the South American height datums into a global vertical system The relationship between the classical height datums and the proposed global vertical system is represented in figure 5. The existing physical heights (HN, I4°) and the local (quasi)geoid models (N, Q refer to the corresponding surfaces passing through the vertical datum WA or We, respectively. They disagree among each other by dWAB and with respect to the global reference level Wo by dWA or dWB, respectively. (e.g. Rummel and Teunissen 1988, Pavlis 1991, Rapp and Balasubramania 1992, Rapp 1994, Heck 2004). In general, dW~ can be written as:
B ~'T. ,.
;
.
.
,..
Fig.
5 Relationship between classical height datums and a global vertical reference system
In units of meters, Eq. 3 corresponds to: [4]
l~Hi = ~ Yi
where ~ is the normal gravity of the evaluation point on the Earth's surface. Having already introduced a normal gravity field (associated, of course, to a reference ellipsoid), Eq. 4 can be expressed, in terms of normal heights H N, as:
~Hi - h - H i - (
[5]
where the ellipsoidal heights h and the height anomalies ( m u s t be associated to the same reference ellipsoid. Similarly, the discrepancies between two vertical datums/,j would be: 5/-/u ~ = Y
5Wj N N =Hj -H i Y
[6]
Eqs. 4, 5, and 6 indicate that the unification of classical height datums into a global system is feasible by combining adequately high resolution gravity field models, precise geometrical coordinates, and physical heights derived from levelling and terrestrial gravity data. According to this, the proposed methodology is: a)
Determination of 6W~rc (dH/c) at the main tide gauges (including reference gauge site) in each country (or datum zone). It requires GNSS positioning, spirit levelling and a high resolution quasigeoid at the tide gauge marks.
Chapter92 • Definition and Realisation of the SIRGASVertical Reference System within a Globally Unified Height System
b)
Determination of ~W~ssJ~p (6H ss~°p) at the marine areas surrounding the tide gauge. It requires the estimation of the SSTop at each tide gauge included in a).
c)
Determination of c~WiR~ ( 6 H F ) at the vertical reference frame stations, i. e. a set of reference points (well-defined and reproducible benchmarks) should be provided with accurate geometrical coordinates (referred to the ITRS), physical heights and a high resolution quasigeoid model.
d)
Determination
of
6~; (6H!j) by
connecting
precisely neighbouring levelling networks. e)
Combination of the different solutions given by a), b), c), and at) by least squares adjustment.
This process should be iterated until the required reliability (1 ram-level) is achieved. It implies also the adoption of a reference epoch (to) and the determination of changes to h,/4 N, (, and SSTop over time, i.e.:
074 ssr'') (t/, ) - h ss~,,, (t k ) _ ((t/, ) oT-Ii
- h
- I4i
(~Hij(t~) = H j ( t ~ ) -
[71 -
)
H i (tk)
with: h(t
) - h(to) +
8h
- to)
81-1
n(t~) - H(t0) + - ~ ( t ~ -to) a( ((t~) - ((to)
+ -~(t,,
-
[8]
to)
In the particular case of South America, the new vertical reference flame was established in 2000 through a 10 day GPS campaign (Luz et al 2002). It includes 180 stations, among them the main tide gauges of each country, levelling points at the international borders, and the SIRGAS reference flame determined in 1995. The geometrical component corresponds to SIRGAS2000, i.e. ITRS realized by ITRF2000, epoch 2000.4. The physical component is being improved by controlling the national first order levelling networks, by checking the existing terrestrial gravity data and by calculating geopotential numbers as input data for the normal height determination at the reference stations. Once each country had readjusted its levelling networks, 3"Wir~, 3"WiR~ and cyW° can be determined and the continental adjustment of the geopotential numbers
will be feasible. To complement these results, the refined terrestrial gravity data shall be combined with global gravity models derived from CHAMP, GRACE, and GOCE to obtain a unified high resolution quasigeoid model within a global vertical level. The time dependence of heights (h, H N) and SSTop is also being estimated by analysing tide gauge records, satellite altimetry data and continuous GNSS positioning. As a first approximation, the discrepancies 6 H ~ from the existing South American height datums with respect to the proposed vertical system Wo are estimated by means of Eq. 4 or Eq. 5 (the results are identical); the actual gravity potential W~ at each tide gauge is computed using Eq. 2 at [(p, 2, h-/-F] and the EIGEN-CG03C model, n = 360. Since the geometrical coordinates refer to the SIRGAS datum (i.e. GRS80), ~ (in Eq. 4) is derived from the GRS80 ellipsoid, and ((in Eq. 5) takes into account the degree zero terms arising from the different G M values and the difference between Uo and Wo (Eq. 2-182, Heiskanen and Moritz 1967). All height coordinates are in a tide-free system. Figure 6 illustrates the corresponding 6/4/C values at the reference (main) tide gauges. Chile has three different height datums, because the tide gauges located in the south (Chile I and Chile II) cannot be connected by spirit levelling with each other or with the central gauge (Chile III). Those countries that included more than one tide gauge in the reference frame have been denoted with (A). For instance, if Colombia takes into account only the reference tide gauge, c~H~~ would be +52 cm, while by including the other two, it would be -6 cm. The same occurs with Chile III and Venezuela. These large variations of ~H/c as a function of the included tide gauges show the low reliability of linking a height system to another by just one point or few points. It is necessary to include a well-distributed network with very high accuracy on h, H x and (.. Table 2 compares the cyHire" and (~H~fsr°p derived using the EIGEN-CG03C model and the high resolution quasigeoid GeoCol2004 (GGM + high quality terrestrial gravity data, Sanchez 2003) at the Colombian tide gauges. The 6//rc values derived from the high resolution local quasigeoid are (of course) more consistent with each other than those obtained from the lower resolution GGM. One identifies a bias of--22 cm, which can be assumed as an absolute level difference between the local height system and the global one. ;HiSSr°P shows a very similar behaviour but, in this
643
644
L.Sanchez
vertical level 14/o and combined with those of the neighbouring countries. The results presented here should be understood as preliminary. They shall be refined by determining 6H Re which are not computable at this current stage, "-'J Col '
case, the systematic difference reaches -26 cm. The inconsistency (4 cm) between 6H i~C and 6H iss>,' should be solved by including 6~;/-/ie~ and performing the corresponding adjustment (Eq. 8). Once a unique 6Hco l (6Wow) for Colombia is obtained, the existing
since the normal heights still require processing.
geopotential numbers can be linked to the global Ecuador 82 cm
Chile I 19 cm
Colombia (A) 6 cm "1"
i±
IIl -5 cm
Chile
Chile II - 17
cm
Venezuela (A) -13 cm
Chil -
II
Venezuela -19 cm
(A)B
cm
Argentina 6 cm T
H0
i Brazil -20 cm
la
-35 cm
Colombia -52 cm Fig. 6 Discrepancies between the individual classical height datums in South America and the proposed global vertical reference system.
Table 2. Level differences between the Colombian height datum and the global vertical system derived from a high resolution quasigeoid model and the model EIGEN-CG03C. GeoCol2004 Name
EIGEN-CG03 C
( ~ H iTG
~ iSST°P
av£
( ~ H SSTop i
[cm]
[cm]
[cm]
[cm]
TG !
- 18
-22
56
43
Ref. TG
-25
-30
-52
-41
TG ii
-23
-20
15
29
-22_+3
-26_+6
6_+53
10+49
Average II
4 Final c o m m e n t s The empirical determination of I47obecomes feasible since accurate derived satellite altimetry MSS models and precise GGMs are available. Nevertheless, as in any reference system, 147oshould be based on some adopted conventions, which guarantee its reliability and repeatability. It is needed, among others, to define a reference epoch (common to both, GGM and MSS model), the spatial resolution and the latitudinal extension of the MSS model, and to use a GGM derived from satellite gravity data only. This study concludes with a value of Wo = 62 636 853,4 m2s-2 recommended as a global reference level. It differs b y - 3 mZs-2 from other computations, but the empirical evaluation of the gravity potential using
different combinations of four GGMs (EGM96, TEG4, GGM02S and EIGEN-CG03C) and four MSS models (CLS01, KMS01, GSFC00.1 and one derived from T/P with yearly representations) proves the reliability of our results. The unification of the classical height systems should be done in a global flame. If they refer to a selected tide gauge mark or to the average of several, its related heights would be confined to be valid only in a determined region and they may not be combined with the geometrical reference system (ellipsoidal heights). The transformation of the height datums into the new global vertical reference system should be done through a set of accurate reference flame stations with precise geometrical coordinates (GNSS positioning and satellite altimetry in sea regions), high resolution (quasi)geoid models (GGM + refined terrestrial gravity data), and physical heights derived from spirit levelling and gravity data.
References Andersen, O. B., A. L. Vest, P. Knudsen, (2004). KMS04 mean sea surface model and inter-annual sea level variability. Poster presented at EGU Gen. Ass. 2005, Vienna, Austria, 24-29, April. AVISO (1996). A VISO user handbook. Merged Topex/Poseidon products (GDR-Ms). CLS/ CNES, AVI-NT-02-101-CN. 3 d Ed., July. Bursa, M., K. Radej, Z. Sima, S. True, V. Vatrt, (1997). Determination of the geopotential scale factor from
Chapter 92 • Definition and Realisation of the SIRGASVertical Reference System within a Globally Unified Height System
Topex/Poseidon satellite altimetry. Studia geoph, et geod. 41: 203-215. Bursa, M., J. Kouba, K. Radej, S. True, V. Vatrt, M. Vojtiskova, (1998). Mean Earth's equipotential surface from Topex/Poseidon altimetry. Studia geoph, et geod. 42: 456-466. Bursa, M., J. Kouba, M. Kulnar, A. Miiller, K. Radej, S. True, V. Vatrt, M. Vojtiskova, (1999). Geoidal geopotential and world height system. Studia geoph, et geod. 43: 327-337. Bursa, M., S. Kenyon, J. Kouba, K. Radej, V. Vatrt, M. Vojtiskova, J. Simek, (2002). World height system specified by geopotential at tide gauge stations. IAG Symposia, 124:291-296. Springer. Bursa, M., S. Kenyon, J. Kouba, Z. Sima, V. Vatrt, M. Vojtiskova, (2004). A global vertical reference frame based on four regional vertical datums. Studia geoph, et geod. 48: 493-502. Chambers, D., S. A. Hayes, J. C. Pies, T. J. Urban, (2003). New Topex sea state bias models and their effect on global mean sea level. J. Geophys. Res. 108 (C10), 3305,10.1029/2003JC001839. Drewes, H., L. Sanchez, D. Blitzkow, S. de Freitas, (2002): Scientific foundations of the SIRGAS vertical reference system. IAG Symposia 124:297-301. Springer. FGrste, C., F. Flechtner, R. Schmidt, U. Meyer, R. Stubenvoll, F. Barthelmes, R. KGnig, K.H. Neuanayer, M. Rothacher, Ch. Reigber, R. Biancale, S. Bruinsrna, J.-M. Lemoine, J.C. Raimondo, (2005) A New High Resolution Global Gravity Field Model Derived From Combination of GRACE and CHAMP Mission and Altimetry/Gravimetry Surface Gravity Data. Poster presented at EGU General Assembly 2005, Vienna, Austria, 24-29, April. Heck, B. (2004). Problems in the definition of vertical reference frames. IAG Symposia 127:164-174. Heiskanen W. And H. Mofitz (1967). Physical Geodesy. W. H. Freeman and company. San Francisco. Hernandez, F., Ph. Schaeffer (200 l a). MSS CLSO1 http:// www'cls'fr/html/°cean°/pr°jects/mss/cls 01_en.html Hernandez, F., Ph. Schaeffer (2001b). The CLSO1 mean sea su~ace." a validation with the GFSCO0.1 surface. Available at http://www.cls.fr/html/oceano/projects/ mss/cls 01 en.html IAG SC3 Pep. (1995). IA G SC3 final report, Travaux de L 'Association Internationale de Gdod&ie, 30:370 - 384 Koblinsky et al. (1999). NASA Ocean Altimeter Pathfinder Project, Report 1." Data processing handbook, NASA/TM- 1998 -208605, April. Lemoine, F., S. Kenyon, J. Factor, R. Trimmer, N. Pavlis, D. Chinn C. Cox, S. Kloslo, S. Luthcke, M. Torrence, Y. Wang, R. Williamson, E. Pavlis, R. Rapp, T. Olson. (1998). The Development of the Joint NASA GSFC and the National Imagery and Mapping Agency (NIMA)
Geopotential Model EGM96, NASA, Goddard Space Flight Center, Greenbelt. Lettelier, T., F. Lyard, F. Lefebre, (2004). The new global tidal solution." FES2004. Presented at: Ocean Surface Topography Science Team Meeting. St. Petersburg, Florida. Nov. 4-6. Luz, R. T., L. P. S. Fortes, M. Hoyer, H. Drewes, (2002): The vertical reference frame for the Americas - the SIRGAS 2000 GPS campaign, lAG Symposia 124: 301-305, Springer. Mather, R. S. (1978). The role of the geoid in fourdimensional geodesy. Marine Geodesy, 1:217-252. Pavlis, N. (1991). Estimation of geopotential differences over intercontinental locations' using satellite and terrestrial measurements. The Ohio State University, Department of Geodetic Science and Surveying. Report No. 409. Pavlis, N. (1996). Modification of program f477 (Rapp 1982) Rapp, R. (1982). A FORTRAN program for the computation of gravimetn'c quantities from high degree spherical harmonic expansions. Rep. No. 344. Dept of Geodetic Science and Surveying, The Ohio State University, Columbus Ohio. Rapp, P.; N. Balasubramania. (1992). A conceptual formulation of a world height system. The Ohio State University, Department of Geodetic Science and Surveying. Report No. 421. Rapp, R., (1994). Separation between reference surfaces of selected vertical datums. Bull. GGod. 69:26-31. Rummel, R.; P. Teunissen. (1988). Height datum definition, height datum connection and the role of the geodetic boundary value problem. Bull. GGod. 62: 477498. S/mchez, L. (2003): Bestimmung der HShenreferenzfliiche fiir Kolumbien. Diplomarbeit. TU Dresden. SIRGAS (1997): Final Report Working Groups I and IISIRGAS RelatGrio Final Grupos de Trabalho 1 e 1L Insfituto Brasileiro de Geografia e Estatistica, Rio de Janeiro. Smith (1998). Program geopot97, v. 0.4c. http://www. ngs.noaa.gov/GEOID/RESEARCH_SOFTWARE/rese arch soflware.html. Tapley M. Kiln, S. Poole, M. Cheng, D. Chambers, J. Ries, (2001). The TEG-4 Gravity field model. AGU Fall 2001. Abstract G51A-0236 Tapley J., Ries, S. Bettadpur, D. Chambers, M. Cheng, F. Condi, B. Gunter, Z. Kang, P.Nagel, R. Pastor, T. Pekker, S.Poole, F. Wang,. (2005). GGM02." An improved Earth gravity field model from GRACE. Journalof Geodesy, doi 10.1007/s00190-005-0480-z. Torge (2001). Geodesy. 3rd. Edition. De Gruyter. Berlin, New York.
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Chapter 93
Tests on Integrating Gravity and Leveling to Realize SIRGAS Vertical Reference System in Brazil R. T. Luz 1'2, S. R. C. de Freitas 1, R. Dalazoana ~, J. C. Baez ~'3, A. S. Palmeiro ~ Geodetic Sciences Graduation Course, Federal University of Paranfi (UFPR), Curitiba, Brazil 2 Coordination of Geodesy, Brazilian Institute of Geography and Statistics (IBGE), Rio de Janeiro, Brazil 3 Department of Geomatics, University of Concepci6n, Chile ( robtluz, sfreitas, regiane ) @ ufpr.br, jbaez @ udec.cl, ale_palmeiro @ yahoo.com.br Abstract. The integration of gravity data within the Brazilian leveling network is very difficult due to the historical dissociation between leveling and gravity surveys. This research was started, in the context of SIRGAS Project, to resolve this problem by evaluating different strategies and procedures with the aim of establishing some kind of gravity coverage over vertical reference stations. Data employed in this article comes from one of the few areas where the points of the Brazilian fundamental leveling network are entirely covered with gravity surveys and, in addition, connect three permanent GPS stations, two of which belong to the SIRGAS 2000 Reference Network. This will allow for several analyses regarding the Brazilian realization of SIRGAS Vertical System, including: an evaluation of the effects of adopting different types of heights; and the investigation of strategies and procedures to solve the absence of gravity values over benchmarks. Values of geopotential differences were computed and adjusted, for a 2300 km network consisting of six loops with perimeters ranging from 136 km to 690 km. Dynamic, Helmert, Normal and Normal-Orthometric reductions were generated and compared. The status of the development of a GIS designed for the tasks involved in geopotential numbers computation to be applied in the studies related to the SIRGAS Vertical Reference System is also presented.
mation to leveling lines is an underlying cause for this problem. Primary vertical surveys conducted by IBGE from the 1940's to the 1980's spread to cover almost the entire country (except for the Amazon Region) (Luz et al., 2002b). IBGE also conducted gravity surveys, which were initially concentrated on horizontal datum region and, since the 1980's, on filling in the gravity gaps. Gravity surveys over new leveling lines were only recently established as a routine procedure at IBGE (Blitzkow et al., 2002). Other institutions also carried out gravity densification, but these efforts were not sufficient to configure an adequate coverage of the leveling lines. An example of this situation is presented in Figure 1.
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Keywords. Heights, Geopotential numbers, SIRGAS.
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1 Introduction The present status of the Brazilian Geodetic System (SGB) does not allow for obtaining precise values of physical heights using modem space geodesy techniques. The inability to integrate gravity infor-
0
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Fig. 1. Map (upper) and profile (lowed of the distribution of vertical stations (RRNN, black crosses) of the leveling loop including Brazilian Vertical "Imbituba" Datum, and gravity stations (EEGG, gray circles) in the same area (see Figure 2).
Chapter 93 • Tests on Integrating Gravity and Leveling to Realize SIRGASVertical Reference System in Brazil
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Consequently, heights of the SGB Vertical Network have a certain conceptual weakness due to the absence of gravity information and, therefore, cannot be called orthometric. The normal-orthometric reduction is used since 1993, but it deals only with the effects of theoretical variations of gravity derived from differences in latitude. There are similar difficulties with regard to the height systems of other South American countries (Drewes et al., 2002). The effects of this situation include, e. g., inconsistencies in the geopotential models covering the continent, difficulties for the integration of results from other space techniques, like satellite altimetry, and problems in connecting vertical datums
around the world (e. g., Hemfindez et al., 2002; Ihde and Augath, 2002). To deal with these kinds of problems in the South American context, the Working Group on Vertical Datum (WG-III) of the SIRGAS Project ("Geocentric Reference System for the Americas") was created in 1997. The main goal of WG-III is to improve and unify the vertical reference for the entire continent through the establishment of a network of stations with geometric and physical heights (Drewes et al., 2002). Ellipsoidal heights were achieved after the SIRGAS 2000 GPS Campaign (Luz et al., 2002a), whose stations form the SIRGAS Reference Network (Figure 2). For the
647
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R.T. Luz. S. R. C. de Freitas • R. Dalazoana. J. C. B~ez. A. S. Palmeiro
physical component of the SIRGAS Vertical Reference System, WG-III recommended the use of normal heights based upon geopotential numbers obtained from leveling lines with gravity, connecting those stations. Detailed analysis of the joint distribution of Brazilian leveling and gravity stations led to the conclusion that almost all leveling paths have at least some gravity points, even thought all leveling stations were not occupied by gravity surveys (Figure 1). Despite being far from ideal, this configuration allows for the interpolation of gravity values, as recommended by the SIRGAS WG-IlI. The following sections describe the procedures and preliminary results from the integration of gravity data to the leveling lines of a test area, whose observed height differences were reduced aiming the computation of dynamic, normal and Helmert heights.
The tests described in next section used the following concepts (e. g., Freitas and Blitzkow, 1999, Drewes et al., 2002). Geopotential (number) difference (AC) between successive leveled points are computed with the respective observed height difference (AH°b') and mean observed gravity (gOb,) : g oh, AHOb,
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3 Integrating Gravity to Leveling
2 Types of Heights
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(4)
where garbitris a constant, arbitrarily chosen value. For the test area, the value 978 170 mGal was used, and additional analyses were performed using the value 978 270 mGal.
As previously stated, the leveling lines of the SGB do not commonly have adequate gravity coverage. Consequently, the computation of geopotential numbers is difficult, except for those few areas where recent leveling stations were occupied by homogeneous gravity surveys. One of these subnetworks, shown in Figure 2 ("Nortesul"), was chosen as a test area for verifying conditions, procedures and impacts regarding the implementation of the WG-III resolutions and recommendations. Besides providing good gravity integration, these leveling lines connect three continuous monitoring GPS stations, two of them (Braz and Bomj) belonging to the SIRGAS Vertical Reference Network. This initial effort considered only gravity surveys performed by IBGE, to avoid problems regarding quality heterogeneity, data access and documentation. Parallel activities are under development to collect the large amounts of data from other institutions. A Fortran program was developed to analyze leveling networks, helping to criticize connections of lines observed in different dates and to integrate gravity data. The program identifies network nodes, filters out the open lines, computes geopotential differences and respective gravity reductions (dynamic, Helmert, and normal) to the observed height differences. Future improvements include the analysis and implementation of some kind of gravity interpolation. The test network will also be im-
Chapter 93 • Tests on Integrating Gravity and Leveling to Realize SIRGASVertical Reference System in Brazil
portant as a standard to be used in the validation of procedures and in the definition of corresponding minimum gravity spacing values. Initial runnings of the program helped to identify some problems regarding the stability of 20 of the stations and the lack of gravity for 7 other stations, which were removed from the tests. Figure 3 and Table 1 represent the resulting network used in the tests, with 861 stations, forming six loops with a total length of about 2300 km.
With the network defined, the computation o f reductions was performed in two steps. First, geopotential and respective dynamic height differences were computed, considering a reference gravity value of 978 170 mGal. Figure 4 presents the results for loop 3. To investigate the effects of using different reference gravity v a l u e s , the reduction was computed using also the value 978 2 7 0 m G a l , which is approximately symmetrical to the other value in terms o f observed gravity in the region.
Table 1. Loop closures (mm) for the test network Loop 1 2 3 4 5 6
Perimeter (km) 231.46 198.94 592.22 670.34 690.35 136.59
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Fig. 4. Leveled heights, observed gravity, gravity anomalies and dynamic reductions (also in Fig. 6) for loop 3. Position of the leveling stations identified in the "Height" graph can be observed in Fig. 3.
649
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Dynamic height differences (garbitr = 978 170 reGal) were adjusted according the common parametric model, referring to an approximate height value for one of the central nodes of the network. Figure 5 shows the standard deviations of the adjusted dynamic heights. After the adjustment of dynamic height differences, normal and Helmert reductions were computed. Figure 6 shows these results. Interesting direct and inverse correlations to topogr aphy can be observed. The direct correlation of the first set of dynamic reductions is clearly dependant upon the gravity reference value used. Using the second value reverses the correlation. So, for a larger network, it can be expected that the reductions will also become larger, as stated by many authors (e. g., Torge, 2001, p. 251). Helmert and normal reduction values show larger variations than those for dynamic reductions. This is probably caused by the geographical configuration of loop 3, i. e., incorporating the effects of latitude variations. This can be observed in the normalorthometric reduction.
-10 -20 -30 -40
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Fig. 6. Leveled heights, observed gravity and dynamic, Helmert, normal and normal-orthometricreductions for loop 3.
Chapter 93 • Tests on Integrating Gravity and Leveling to Realize SIRGASVertical Reference System in Brazil
4 Next Activities A Geographical Information System (GIS) dedicated to the preparation of leveling lines for adjustment including integration of gravity data is being developed. This GIS will also perform the interpolation of gravity, where needed, and the analysis of the effects of the future adoption of SIRGAS heights. The system should also aid in the study and integration of results from the continuous monitoring GPS stations with regard to possible vertical crustal movements. Considering the existence of three such GPS stations in the test area and the existence of older leveling lines in the same area, some interesting studies could be carried out employing the proposed GIS. The linking of the Santana Vertical Datum (Figure 7) to the main Imbituba Datum is another important study associated to the realization of the SIRGAS Vertical System in Brazil. These studies can provide interesting aspects for the integration of various types of geodetic and oceanographic tools, such as the data from the Geodetic Tide Gauge Network (RMPG), observations from satellite altimetry and results from a local hydrodynamic circulation model ( e. g., Heck and Rummel, 1990; Sans6 and Usai, 1995).
n
RMPG data are also being used to link satellite altimetry results in the Imbituba region to the data used in the definition of the Imbituba Datum, stored at PSMSL (Dalazoana et al., 2005). The integration of present RMPG results (starting in 2001) to the old PSMSL data (1949-1969) is difficult, but the recent recovering of records for 1985 to 1988 promises to be an important way to aid in that integration.
5 Final Remarks The identification of a region where leveling and gravity surveys are already integrated to the SIRGAS Vertical Reference Network allowed for beginning implementation and analysis of procedures for computing geopotential numbers and physical heights. Data corresponding to 2300 km of leveling were selected, forming a network with 861 stations and six loops, whose perimeters and closures range from 136 km to 690 km and from 0.1 mm(km) 1/2 to 3.1 mm(km)l/2, respectively. Values of geopotential differences were computed and adjusted in a preliminary way due to the arbitrary reference value chosen, as there is yet no connection of this test network to any reference surface. A Fortran program that performs not only the integration of gravity to leveling, but also identifies and builds the network, was developed. Dynamic, Helmert, normal and normalorthometric reductions, based on the adjusted geopotential differences, were computed and compared. This comparison could not highlight any pattern that would be suitable to guide the choice for any kind of reduction. An enlargement of the test network would be needed to verify and implement procedures for gravity interpolation. A first analysis, without any interpolation, based only on the results already achieved, will be performed by the increase of gravity spacing. Quantifying differences from respective solutions will allow a preliminary evaluation of the required gravity distribution over leveling stations.
Acknowledgements
Fig. 7. Marajo Island area, where the connection of the local datum for the small state network (Santana Datum) to the main Imbituba Datum will be studied.
The authors wish to acknowledge the Brazilian agencies for research and education (CNPq and CAPES) for financial support ; IBGE, for the data made available and for the study license for R. T. Luz ; B. Heck (GIK/Uni-Karlsruhe, Germany), for
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R.T. Luz. S. R. C. de Freitas • R. Dalazoana. J. C. B~ez. A. S. Palmeiro
kind s u g g e s t i o n s and c o m m e n t s ; H. Drewes. W. B o s c h an d W. Seemfiller ( D G F I , Mfinchen, G e r m a n y ) , for fruitful d i s c u s s i o n s ; and I A G , for the grant s u p p o r t i n g R. T. L u z a t t e n d a n c e at the D y n a m i c P l a n e t 2005 Joint A s s e m b l y .
References Blitzkow D, Cancoro de Matos AC, Lobianco MCB (2002) Data Collecting and Processing for Quasi-Geoid Determination in Brazil. In: Drewes H et al. (eds) Vertical Reference Systems (lAG Symposia, vol. 124). Springer, Berlin, pp. 148-151. Dalazoana R, Baez JC, Luz RT, Freitas SRC (2005) Brazilian Vertical Datum Monitoring - Vertical Land Movements and Sea Level Variations. In: Program & Abstract Book, Dynamic Planet 2005, IAG-IAPSO-IABO Joint Assembly, Cairns, p. 257. Drewes H, Sfinchez L, Blitzkow D, Freitas S (2002) Scientific Foundations of the SIRGAS Vertical Reference System. In: Drewes H et al. (eds) Vertical Reference Systems (IAG Symposia, vol. 124). Springer, Berlin, pp. 297-301. Freitas SRC, Blitzkow D (1999) Altitudes e Geopotencial. In: Sans6 F et al. (eds) Bulletin No. 9, International Geoid Service (Special Issue for South America). IgeS, Milano, pp. 47-61. Freitas SRC, Medina AS, Lima SRS (2002a) Associated Problems to Link South American Vertical Networks and Possible Approaches to Face Them. In: Drewes H et al. (eds) Vertical Reference Systems (IAG Symposia, vol. 124). Springer, Berlin, pp. 318-323. Freitas SRC, Schwab SHS, Marone E, Pires AO, Dalazoana R (2002b) Local Effects in the Brazilian Vertical Datum. In: Adam J, Schwarz KP (eds) Vistas for Geodesy in the New Millennium (IAG Symposia, vol. 125). Springer, Berlin, pp. 102-107. Heck B (2004) Problems in the Definition of Vertical Reference Frames. In: Sans6 F (ed) V Hotine-Marussi Symposium on Mathematical Geodesy (IAG Symposia, vol. 127). Springer, Berlin, pp. 164-173.
Heck B, Rummel R (1990) Strategies for solving the vertical datum problem using terrestrial and satellite geodetic data. In: Sfinkel H, Baker T (eds) Sea Surface Topography and the Geoid (IAG Symposia, v. 104). Springer, New York, pp. 116-128. Heiskanen WA, Moritz H (1967) Physical Geodesy. Freeman, San Francisco, 364 pp. Hernfindez JN, B1 itzkow D, Luz R, Sfinchez L, Sandoval P, Drewes H (2002) Connection of the Vertical Control Networks of Venezuela, Brazil and Colombia. In: Drewes H et al. (eds) Vertical Reference Systems (IAG Symposia, vol. 124). Springer, Berlin, pp. 324-327. Hwang C, Hs iao YS (2003) Orthometric corrections from leveling, gravity, density and elevation data: a case study in Taiwan. Journal of Geodesy 77:279-291. Ihde J, Augath W (2002) The European Vertical Reference System (EVRS), Its Relation to a World Height System and to the |TRS. In: Adam J, Schwarz KP (eds) Vistas for Geodesy in the New Millennium (lAG Symposia, vol. 125). Springer, Berlin, pp. 78-83. Luz RT, Fortes LPS, Hoyer M, Drewes H (2002a) The Vertical Reference Frame for the Americas - the SIRGAS 2000 GPS Campaign. In: Drewes H et al. (eds) Vertical Reference Systems (|AG Symposia, vol. 124). Springer, Berlin, pp. 302-305. Luz RT, Guimarfies VM, Rodrigues AC, Correia JD (2002b) Brazilian First Order Levelling Network. In: Drewes H et al. (eds) Vertical Reference Systems (IAG Symposia, vol. 124). Springer, Berlin, pp. 20-22. Sans6 F , Usai S (1995) Height datum and local geodetic datums in the theory of geodetic boundary value problems. Allgemeine Vermessungs-Nachrichten 102:343 355. Tenzer R, Vanicek P, Santo s M, Featherstone WE, Kuhn M (2005) The rigorous determination of orthometric heights. Journal of Geodesy 79: 82-92. Torge W (2001) Geodesy. 3rd edn., Walter de Gmyter, Berlin, 416 pp. Vanicek P, Krakiwsky EJ (1986) Geodesy." the Concepts. 2nd edn., Elsevier, Amsterdam, 697 pp.
Chapter 94
Accessing the New SIRGAS2000 Reference Frame through a Modernized Brazilian Active Control Network L.P.S. Fortes, S.M.A. Costa, M.A.A. Lima, J.A. Fazan Directorate of Geosciences Brazilian Institute of Geography and Statistics, Av. Brasil 15671, Rio de Janeiro, RJ, Brazil, 21241-051 M.C. Santos University of New Brunswick, Fredericton, Canada
Abstract. Since the beginning of its establishment, in December of 1996, the Brazilian Network for Continuous Monitoring of GPS - RBMC has been playing the role as the fundamental geodetic frame in the country, providing users with a direct connection to the Brazilian Geodetic System- SGB. This role has become more relevant with the adoption of the new geodetic system, SIRGAS2000, as of February 25, 2005. In this paper, the current RBMC status is presented, as well as the expansion and modernization plans for its structure, functionality and services to be provided to users. RBMC currently works in postmission mode, where users are able to freely download from the |nternet data collected by each of its 19 stations 24 hours after the observations are collected. The modernization plans specify, in a first step, the network expansion with six additional stations in the Amazon region, including the reactivation of Manaus station, and the connection of all stations to the lnternet, to support real time transfer of 1 Hz data to the control center, in Rio de Janeiro. When available at the control center, the data will support WADGPS (Wide Area Differential GPS) corrections to be transmitted, in real time, to users in Brazil and surrounding areas. This new service is under development based on a cooperation signed at the end of 2004 with the University of New Brunswick, supported by the Canadian International Development Agency and the Brazilian Cooperation Agency. It is estimated that users will be able to achieve a horizontal accuracy around 0.5 m (1-~) in static and kinematic positioning. The expected accuracy for dual frequency receiver users is even better. The availability of the WADGPS service- at no costwill allow users to tie to the new SIRGAS2000 system in a more rapid and transparent way in positioning and navigation applications, it should be emphasized that support to post-mission static
positioning will continue to be provided to users interested in higher accuracy levels.
Keywords. RBMC, real time, SIRGAS2000
1 Introduction The Brazilian Network for Continuous Monitoring of GPS - RBMC (Fortes et al., 1998; IBGE, 2005a) is an active geodetic network which constitutes the main geodetic framework of the country, providing users with the possibility of precise linking to the Brazilian Geodetic System- SGB. This role is even more relevant at the current moment, when the new geodetic system SIRGAS2000 (Drewes et al., 2005) has been officially adopted in Brazil as of February 25, 2005 (IBGE, 2005b), considering that this new system is mainly realized in the country throughout the RBMC stations. In this paper, the current RBMC status is presented, as well as the expansion and modernization plans for its structure, functionality and services provided to users.
2 Current RBMC Status The Brazilian Institute of Geography and Statistics IBGE started to establish RBMC at the end of 1996, when the Curitiba/PR and Presidente Prudente/SP stations were installed with the support of the National Fund for the Environment- FNMA and of the Politechnic School of the University of Silo Paulo -EPUSP. Nowadays, the network is composed of 19 continuous operating GPS stations (Fig. 1), distributed across the national territory, being automatically monitored and remotely controlled by the control center localized in Rio de Janeiro. Among these 19 stations, the ones in Brasilia and Fortaleza are part of the International GNSS Service - IGS global network (IGS, 2005a), whereas the remaining ones compose the IGS
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densification network in South America and surroundings, whose data is processed on a weekly basis by the IGS Regional Network Associate Analysis Center for the continent- IGS RNAAC SIR (Seemueller and Drewes, 2004). These characteristics include the Brazilian geodetic reference framework in the global structures in a consistent way, which guarantees its continuous monitoring and update. Each RBMC station is equipped with a dual frequency GPS receiver and a chokering antenna. At the end of each 24 hour observing session, the collected data are automatically transferred to a local computer. Few minutes later, the computer server at the control center downloads data from the local computer to Rio de Janeiro, through a dial-up connection or using the Internet. After being transferred, data is checked and made freely
available at the site http://www.ibge.gov.br/home/ geociencias/geodesia/rbmc/rbmc.shtm in general within 24 hours after the observation date. During its almost ten years of operation, the network has been largely used by the national and international communities, as demonstrated by many projects and published papers carried out based on the RBMC. Currently around 3500 daily observation files are downloaded each month (Fig. 2). This high demand is related to the increasing use of GPS for positioning applications in general. Among these applications, the following can be listed: • •
Support to GPS relative positioning in general; Topographic and cadastral systematic mapping;
Chapter 94 •
Accessingthe New SIRGAS2000 Reference Frame through a Modernized Brazilian Active Control Network DOWNLOADS RBMC
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• • • • • • • •
Area delimitation (political-administrative, environmental, etc.); Land use (land reform, precise agriculture, etc.); Navigation (currently in post-mission mode); Ionosphere modeling; Support to climatology and meteorology; Fleet control and management; Integration into global geodetic networks; Regional and global Geodynamics.
3 RBMC Structure Expansion and Modernization Plans As it can be seen in Fig. 1, the RBMC interstation distances vary from over 200 km in the Southeast, to more than 1000 km, in the Amazon region, where due to regional characteristics the network is sparser. Increasing the density of network points becomes a necessity. In its current configuration, RBMC supports GPS positioning applications using long baselines, requiring longer observation sessions to generate adequate results, due to the spatial decorrelation of the positioning residual errors, especially those caused by the ionosphere. A denser RBMC will provide users with stations closer to their area of interest,
allowing them to achieve their desired accuracy faster. Five stations are expected to be installed in the Amazon, including the reactivation of the Manaus station (Fig. 1). The installation of these stations, as well as of the Beldm and Macapfi stations, is result of cooperation with SIVAM project (System for the Vigilance of the Amazon), in order to improve the coverage of the network in the region. It must be emphasized the understandings being established with the National Institute for Land R e f o r m - 1NCRA, towards integrating their Network of Community GPS Base S t a t i o n s RIBaC to RBMC. RIBaC was established to support land surveys directly or indirectly carried out by INCRA. It is currently composed of 31 continuously operating GPS stations distributed in the Brazilian territory. RIBaC stations are equipped with single frequency receivers, which significantly restricts the coverage area of each station, especially in Brazil, where error gradients caused by the ionosphere on GPS signals can easily reach values around tens of parts per million, (Fortes, 2002). The RIBaC stations must also have their coordinates tied to SGB through RBMC. Based on the afore mentioned understandings, around 30 to 40 last generation
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dual frequency GPS receivers are expected to be purchased in order to replace the majority of receivers currently used in both RBMC and RIBaC. This receiver replacement has the following objectives:
Fig. 3 presents the intended configuration for the resulting network, with the stations to be modernized being shown. The selection of these stations satisfied the following priorities: •
• •
•
To equip RIBaC stations with dual frequency receivers; To equip RBMC and RIBaC stations with receivers with good GPS signals tracking performance, especially with respect to L2, as Brazil is located under the Equatorial Anomaly where occurrence of scintillations is very common (Fortes, 2002); To equip both networks with receivers capable of real time operation, at 1 Hz, directly connected to the Internet, without the need of a local computer, in order to support real time applications described in next section.
In order to achieve the above objectives, the new receivers have to satisfy the following specifications" •
•
• • •
•
•
•
• • •
At least 12 L1 and 12 L2 channels to track carrier phase and C/A (L1) e P (El e L2) codes; Low carrier phase and code noises (few decimeters for codes and < 0.01 cycle for carrier phase); Chokering or equivalent antenna for multipath mitigation; Observation rates up to 1 Hz; L2-tracking technique with good performance under high ionospheric activities (e.g., semicodeless or equivalent); IP network port for connecting the receiver to LAN/Internet with no local computer interface; Possibility of remote controlling the receiver and real time transferring of observations through the Internet; Possibility of storing observations on the receiver memory at the same time as transferring them through the Intemet to the network Control Center; Enough memory to store 30 days of observations at 1 Hz; External oscillator port; Possibility of L2C signals tracking or upgradeability to that.
• •
Availability of local connection to the Internet, with good stability and quality (i.e., broad band); Existence of long time dual frequency data series collected at the station; Existence of stable monuments.
4 Plans for Modernization of RBMC's Functionality and Services to Users Until today, RBMC has provided support to applications that rely on post-processing data, mostly in relative mode. A modernization of RBMC's functionality is being proposed in order to increase the range of applications, most notably those which require real-time information. Those applications include navigation, either air, maritime or terrestrial. The new functionality planned for a modernized RBMC involves:
•
• • •
Real-time transmission of the data collected by each station to the Control Center in Rio de Janeiro; Reduction of the current 15-second observation interval to 1 second. Real-time computation of WADGPS-type corrections at the Control Center. Real-time availability of the corrections to users, at no via the Internet or satellite link.
The WADGPS corrections are those for the satellite and clock orbits and for the delay provoked by signal propagation through the ionosphere and troposphere. Since the RBMC stations have highly accurate coordinates the data collected can be used to quantify the actual errors and to predict the corrections to be transmitted to users.
This new service is being developed in cooperation with the University of New Brunswick under the National Geospatial Framework Project (PIGN), a technology transfer project sponsored by the Canadian International Development Agency (CIDA) with the support of the Brazilian Cooperation Agency (ABC). The PIGN (PIGN, 2005) has a main object to
Chapter 94 • Accessing the New SIRGAS2000 Reference Frame through a Modernized Brazilian Active Control Network
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collaborate and assist in Brazilian efforts towards the adoption of a geocentric coordinate system (SIRGAS2000) compatible with modern satellite positioning technology. Project activities include technical issues, study on the impacts resulting from the adoption of the new system and communication with user community. The modernization of the RBMC corresponds to PIGN Demonstration Project#7. This Demo Project aims at providing the background for the implementation of a modern reference structure that facilitates the connection to the Brazilian geodetic system by users. Since the corrections will be implicitly attached to SIRGAS2000, their application by the users will result in SIRGAS2000 coordinates. Users will be directly attached to SIRGAS2000 in their positioning and
navigation applications. The participation of the Geodetic Survey Division of Natural Resources Canada is being discussed considering the expertise this institution holds from the development of the Canada-Wide DGPS ServiceCDGPS (CDGPS, 2005a). It is expected that users will be capable of performing (real-time) static and kinematic positioning at the 1 m 95% confidence level (0.5 m DRMS). For dual-frequency users, these figures drop to 0.3 m at 95% confidence level (less than 0.2 DRMS) (CDGPS, 2005b; Rho et al., 2005). The real-time functionality of RBMC will allow an even closer collaboration with the IGS
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within the IGS Real Time Working Group (IGS, 2005b). 5
Conclusions
The development of global navigation satellite system (GNSS) positioning technology has set up new standards to national geodetic infrastructure via active networks such as RBMC. RBMC has been the fundamental geodetic infrastructure in Brazil since its inception in 1996, providing accurate connection to the Brazilian Geodetic System, in post-processing mode. In an attempt to follow the technological evolution, IBGE is proposing the expansion and modernization of the geodetic infrastructure, functionality and services provided to users. With these purposes in mind, the densification of the network in the Amazon region is being planned with the establishment of five new stations. In addition, an on-going understanding with INCRA may result in the purchase of 30 to 40 new generation dualfrequency GPS receivers, to be used in RBMC and also in the network maintained by INCRA in support of land reform activities. The resulting infrastructure will provide capabilities for realtime services, by means of computation of WADGPS-type corrections and consequent transmission of these corrections to users in Brazil, and possibly in neighboring regions. The modernization of the RBMC is being carried out under the National Geospatial Framework Project, supported by CIDA.
The application of the WADGPS corrections will allow users to be attached to SIRGAS2000 in a direct and clear way in positioning and navigation applications. It is believed that users will be capable of real-time static and kinematic positioning with a 2D accuracy of 0.5 m (DRMS) or better, depending on the type of receiver used. The current post-processing service will still be offered, allowing users to reach the highest accuracy possible.
CONDERCompanhia de Desenvolvimento Urbano do Estado da Bahia EPUSP - Escola Polit6cnica da Universidade de Silo Paulo DSG - Diretoria do Servigo Geogrfifico do Ex6rcito, 4 a DE, Manaus F N M A - Fundo Nacional do Meio Ambiente I M E - Instituto Militar de Engenharia I N P E - instituto Nacional de Pesquisas Espaciai, Cuiabfi e Euzdbio Marinha do Brasil - Capitania dos Portos, Bom Jesus da Lapa Pr6 G u a i b a - Fundo Prd-Guaiba, Governo do Estado do Rio Grande do Sul SIVAM/SIPAM- Sistema de Vigilfincia/Proteg~o da Amaz6nia UFPE- Universidade Federal de Pernambuco UFPR- Universidade Federal do Paranfi UFRGS - Universidade Federal do Rio Grande do Sul UFSM- Universidade Federal de Santa Maria U F V - Universidade Federal de Vigosa UNESP - Universidade Estadual Paulista, Campus de Presidente Prudente U R C A - Fundag~o Universidade Regional do Cariri
7
In M e m o r i a m
This paper is dedicated to the memory of Eng. Kfitia Duarte Pereira. Kfitia was responsible for the operation of the RBMC since its establishment. Kfitia passed away, prematurely, in April 2005.
References
CDGPS (2005a). The Real-Time Canada-Wide DGPS Service. http ://www.cdgps.com/. CDGPS (2005b). CDGPS Features. http://www.cdgps. com/e/features.htm.
The RBMC network is a reality with the support of the following institutions:
Drewes, H.; K. Kaniuth; C. V61ksen; S.M.A. Costa; L.P.S. Fortes (2005). Results of the SIRGAS Campaign 2000 and Coordinates Variations with Respect to the 1995 South American Geocentric Reference Frame. In A Window on the Future of Geodesy (ed) F. Sans6, International Association of Geodesy Symposia, Vol. 128, pp. 32-37.
CEFET/UNEDI - Centro Federal de Educagfio Tecnoldgica, imperatriz C E M I G - Companhia Energdtica de Minas Gerais
Fortes, L.P.S. (2002). Optimising the Use of GPS MultiReference Stations for Kinematic Positioning. PhD Thesis, UCGE Report Number 20158, The University of Calgary, Calgary, 323 p.
6 Acknowledgments
Chapter 94 • Accessing the New SIRGAS2000 Reference Frame through a Modernized Brazilian Active Control Network
Fortes, L.P.S.; R.T. Luz; K.D. Pereira; S.M.A. Costa; D. Blitzkow (1998). The Brazilian Network for Continuous Monitoring of GPS (RBMC): Operation and Products. In Advances in Positioning and Reference Frames (ed) F.K. Brunner, International Association of Geodesy Symposia, Vol. 118, pp. 73-78. IBGE (2005a). Rede Brasileira de Monitoramento Continuo. http ://www.ibge.gov.br/home/geociencias/ geodesia/rbmc/rbmc, shtm. IBGE (2005b). Projeto Mudanga do Referencial Geod6sico. http ://www.ibge.gov.br/home/geociencias/ noticia_sirgas.shtm. IGS (2005a). Network. html.
International GNSS Service Tracking http ://igscb.jpl.nasa.gov/network/netindex.
IGS (2005b). IGS Real Time Working Group. http ://igscb .jpl.nas a. gov/proj ects/rtwg/index.htm 1. INCRA (2005). Rede INCRA de Bases Comunitfirias do GPS. http ://www.incra.gov.br/_htm/serveinf/_htm/_asp/ estacoes_dcn/default.asp. PIGN (2005). National Geospatial Framework Project. http ://www.pign.org/. Rho, H.; R. Langley; A. Kassam (2003). The Canada-Wide Differential GPS Service: Initial Performance, in Proceedings of the 16th International Technical Meeting of the Satellite Division of the Institute of Navigation ION GPS/GNSS 2003. Portland, Oregon, pp. 425-436. Seemueller, W.; H. Drewes (2004). Annual Report 2002 of IGS RNAAC SIR. In IGS 2001-2002 Technical Reports (eds) K. Gowey, R. Neilan e A. Moore, IGS Central Bureau, Pasadena, California, pp. 129-131.
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Chapter 95
Deformations Control for the Chilean Part of the S IRGAS 2000 Frame J. C.
B f i e z 1'2 ,
S. R. C. de Freitas ~, H. Drewes 3, R. Dalazoana ~R. T.
L u z 1'4
1 Geodetic Sciences Graduation Course (CPGCG), Federal University of Paranfi (UFPR), Curitiba, Brazil. 2 Department of Surveying, Universidad de Concepcidn (UdeC), Chile. 3 Deutsches Geod~itisches Forschungsinstitut, Manchen, Germany. 4 Brazilian Institute of Geography and Statistics (IBGE), Coordination of Geodesy, Brazil. With the adoption of the Geocentric Reference System for the Americas (SIRGAS), the determination of temporal variations and deformations in its frame are a necessary task. These tasks are fundamental in order to maintain the consistency of the frame with respect to the definition of SIRGAS, especially in the Chilean deformation area, where the variations are larger compared with the rest of the plate due to the convergence between the South American and Nazca plates. 21 permanent GPS stations was used.14 of them are regional stations of the International GNSS Service (IGS), and 7 permanent GPS stations are from different regional networks. 10 of these 21 stations are located in the Chilean Andes were used to study the kinematic effects in this area. Some of them are also SIRGAS stations. Five years of GPS observations from 2000 to 2004 were selected in order to have a sufficient time period with enough coverage for the considered stations. Daily solutions were generated with the Bernese GPS Software 5.0 using IGS precise ephemeris, clocks and ERP. The normal equations were then accumulated and solved in a final combination where the coordinates and velocities of selected stations are weighted for the datum definition compatible with ITRF2000. The estimated velocities of the north Chilean Andes regions are -2630 + 2mm/yr, decreasing up to -16-25 + 2mm/yr and for the south Chilean Andes the estimated velocities are-9-12 + 2ram/yr. For the stable part of the continent the estimated velocities are -8-12 + 2ram/yr. The final results are discussed and compared with the NUVEL-1A and SIRGAS velocity models (SIRGAS-VM).
ties to observe and measure the traditional triangulation network for the entire continent, and to define a unique Geodetic Reference System (GRS). Space geodetic techniques like GNSS are suitable to develop a GRS and its realization. The SIRGAS project is the most important effort to develop an unique GRS, and it was realized in two extensive campaigns during 1995 and 2000, the later one performing the SIRGAS 2000 realization (IBGE, 2002). One of the necessary tasks is the observation of kinematic effects in the frame due to the continental drift which is affecting the position of the stations. One special case is the Andes region which is affected by the subduction area of the Nazca under the South American plate, and produces large movements and deformations of the continental crust. Several plate models have been derived based on evidences from geophysical studies; one of them is the NUVEL-1A (DeMets etal., 1994), which explains the kinematic effects for the past million of years, and is mostly sufficient for the stable part of the crust. Geodetic models were developed to explain the recent kinematic effects, like the Actual Plate Kinematic Model (APKIM 2002) (Drewes, 2003). Nevertheless, the geophysical and geodetic models are not sufficient to completely explain the kinematic effects in the deformation areas. Continuous GPS observations from 21 stations, 14 of them of the International GNSS Service (IGS), are used in selected periods to estimate velocities for the Andes region, and to study the kinematic effects and its correlation with the deformations of the Chilean part of SIRGAS 2000 frame.
Keywords. effects.
2 Chilean part of SIRGAS 2000 frame
Abstract.
GRS, velocities estimation, kinematic
1 Introduction Almost one century after the first realizations of geodetic reference networks in South America, it was necessary to improve them, due to the difficul-
After the SIRGAS 2000 realization, the lnstituto Geogrfifico Militar (IGM) of Chile, responsible for the realization and maintenance of the Chilean Geodetic Reference Frame (GRF), decided to increase the number of 20 original stations included in the SIRGAS 2000 realization (IBGE, 2002), by more than 300 stations, including 10 permanent
Chapter 95 • Deformations Control for the Chilean Part of the SIRGAS 2000 Frame
GPS stations. With this frame it was possible to adopt the SIRGAS2000 as the official Geodetic Reference System (GRS), and its frame was realized for all stations. Figure 1 shows the stations in the Chilean SIRGAS realization (small circles) and the 10 permanent GPS stations (triangles). The IGS and permanently observing GPS stations are serving as fiducials for the final coordinate and velocity estimation of the whole Chilean Network. Considering the deformations of the Andes regions, the IGM of Chile also decided to adopt the SIRGAS 2000 using a mean epoch of 2002.0 which differs from the 2000.4 adopted for the SIRGAS 2000 final solution. The network densification is based on the original 20 SIRGAS stations. In that way, the error propagation is small, because some of the stations used during the campaign are monitored, and its velocities are well known. But, the remaining station, velocities must be estimated from a model with uncertainties, because in deformation areas simple geophysical models fail and geodetic models do not represent the true velocities.
can Geodynamic Activities Project (SAGA). To stabilize the geometry, some other stations were included: 14 IGS stations, 4 Argentinean Geodetic Position stations (POSGAR), and 2 Brazilian Network of Continuous Monitoring (RBMC) stations. Data of ten days were selected from the epochs 2000.4, 2000.9, 2001.4, 2001.8, 2002.4, 2002.7, 2003.3, 2003.8 and 2004.2. Data files contain 24 hours of continuous observations with a sampling interval of 30 s. The stations used are shown in Figure 2. IGS products (precise ephemeris, EOPs, and clocks) were used to process the daily solutions. Carrier phase double differences were formed using the ionospheric-delay free observable. The troposphere was modeled using a combination of Saastamoinen zenith path delay and Niell mapping function. A tropospheric parameter was estimated every two hours. Daily ionospheric maps were used during the ambiguities fixing solutions.
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Fig. 1. SIRGAS Chilean reference frame (small circles) and permanent GPS network (triangles).
3 Data collection and processing The network used in this experiment was originally formed by 8 stations from the permanent GPS network of Chile and 2 stations from the South Ameri-
Fig. 2. Permanent GPS stations included in the processing. The GPS data processing was carried out with Bernese GPS Software (Hugentobler et al., 2004). Baselines were created using the maximum observations strategy. During the data processing the observations were referred to the antenna phase centre using the National Geodetic Survey (NGS) GPS antenna calibrations and variations. Daily solutions were generated in a free network and saved in normal equation files.
661
662
J.C. B&ez. S. R. C. de Freitas • H. Drewes • R. Dalazoana • R. T. Luz
For the final solution, all the normal equations were combined in a least squares adjustment, including the ITRF2000 coordinates of five IGS stations (SANT, LPGS, RIOG, PARA, CORD) and velocities from the Regional Network Associate Analysis Centre for SIRGAS (RNAAC-SIR), and its precisions to estimate coordinates and velocities compatible with the IRTF2000 frame. Table 1 summarizes the stations velocities and its standard deviations.
stations coordinates of the individual daily solutions (Xi), and the modeled coordinates (X0) epoch (to) and velocities V At=(ti-t0) (1)
4 Velocity comparison
As shown in Figure 3, the estimated velocities (grey vectors), are consistent with those of the NNR-NUVEL-1A model (black vectors) for the stable part of the South American plate (stations: RIOG, PARC, PARA, UEPP, LPGS, VBCA, IGM0, RWSN).
Considering all lithospheric plate motions, a kinematic reference frame must be introduced for its temporal variations (e.g., Larson et al., 1997). The geophysical plate models like NNR-NUVEL-1A provide quit good velocities for the stable part of the continent, but inconsistent, however, in the deformation areas. The estimated solutions (gray vectors) and NNR-NUVEL-1A (black vectors) are compared in Figure 3.
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Table 1. Estimated Velocities. Station
North
East
(mm/yr)+o
(mm/yr) +o
ANTC
7.7+0.5
15.9+0.5
ANTO
15.9+0.9
30.0+ 1.1
CFAG
8.7+1.3
6.4+1.6
CONZ
15.8+1.0
25.7+1.0
COPO
16.6+ 1.1
20.1+0.9
CORD
9.1+0.7
1.0+0.9
COYQ
8.7+0.5
-3.9+0.6
IGM0
8.2+ 1.7
-1.2+0.7
IQQE
10.1+1.3
25.4+1.6
LPGS
9.6+0.6
-1.8+0.5
PARA
8.5+0.6
-4.6+0.7
PARC
11.2+0.5
3.6+0.7
PMON
11.7+0.8
0.0+0.6
RIOG
11.0+0.3
4.0+0.3
RWSN
9.5+0.9
-2.4+0.5 21.1 +0.8
SANT
13.8+0.5
TUCU
6.6+0.5
1.6+0.6
UEPP
10.1+0.5
-3.4+0.4
UNSA
10.9+1.2
5.7+1.6
VALP
20.6+ 1.1
24.3+ 1.2
VBCA
8.9+0.6
-0.8+0.8
Table 1 present the estimated velocities and their standard deviations (~). The latter ones were computed from the differences between the estimated
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Fig. 3. Comparison between estimated and NNR-NUVEL1A velocities.
The SIRGAS-VM model (Drewes and Heidbach, 2005) was developed using data from SIRGAS campaign (1995 and 2000), and point velocities from different geodynamic projects. The SIRGAS executive recommended using this model for the South American plate. The estimated solutions for the stable part of the plate (UEPP, PARA, IGM0, LPGS, VBCA, RWSN) are in agreement with the SIRGAS-VM
Chapter95 • DeformationsControl for the Chilean Part of the SIRGAS2000 Frame
model with RMS value of N=2.8mm and E=I.0 mm. For the Andes regions the differences are larger with RMS N=3.8mm and E=3.9 mm, and the variations in three stations located in the northern part (IQQE, ANTO, COPO) are rather large, due to the seismic activities. For the central part (SANT, VALP), the velocities are in complete agreement with the SIRGAS model. For the southern part (PARC, RIOG), the comparison with the SIRGASVM model was not possible due to the limitation of the model, which is developed to -43 ° of latitude only. Figure 4 shows the SIRGAS-VM model and Table 2 summarizes the velocity differences between the two models, (SIRGAS-VM and NNRNUVEL-1A), and the estimated velocities in this research paper. The station AREQ was not used in this investigation, because two earthquakes happened in 2001 affecting the position of the station (Kaniuth et al., 2002). More data are necessary to estimate reliable velocity of this station. The same is valid for the station IQQE (earthquake June 2005), therefore data from this period were not included in this research. The comparison of estimated solution and the SIRGAS-VM model is shown in Figure 5.
280"
30{)"
Table 2. Tabulated differences between velocities of NNRNUVEL-1A and SIRGAS-VM model, and that from this study.
Station
Differences Esti.-NUVEL-1A North East (mm/yr) (mm/yr)
Differences Esti.-SIRGAS-VM North East /mm/yr) /mm/yr)
ANTC
-1.6
16.0
-4.4
ANTO
6.4
32.5
-1.3
-2.7 8.7
CFAG
-1.0
7.9
-3.9
-2.2
CONZ
6.7
25.7
-5.2
-5.3
COPO
7.1
22.1
-0.6
-1.2
CORD
-1.0
2.9
-2.5
0.2
COYQ
-0.6
-5.6
-4.2
-5.5
IGM0
-2.5
1.0
-3.4
-0.3
IQQE
0.6
28.4
-6.5
4.9
LPGS
-1.1
0.5
-2.0
-0.3
PARA
-2.8
-0.5
-3.8
-1.2
PARC
1.8
-0.2
PMON
2.5
-0.9
-3.7
-2.3 -1.1
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1.2
0.7
RWSN
-0.6
-2.5
-2.6
SANT
4.4
22.1
-1.1
1.2
TUCU
-3.5
4.1
-5.3
-1.8
UEPP
-1.1
0.7
-2.5
-0.3
UNSA
0.9
8.4
-1.7
-0.6
VALP
12.3
25.2
1.0
-0.0
VBCA
-1.4
0.3
-2.2
-2.1
4.1
14.8
3.5
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5 Final remarks
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3.1 Control points In our investigation a set of 200 control points from the State of Paranfi, in Brazil, was used. The geographic location of these points can be seen in
Figure 2 Normalized shifts at control points.
2 For a list of free programs, please contact the first author; for commercial ones see ICSM (2005) and GSD (2005).
3 From now on, whenever we show the shifts in latitude and in longitude together, we will be showing normalized shifts.
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F.G. Nievinski
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3,2 Reference transformation
3.3 Generation of the transformation grid
To analyze the errors introduced by the grid we need a reference transformation to be used as a benchmark for the comparisons to follow. After obtaining that reference transformation we assume it as "true" or, conversely, that it does a "perfect" job modelling the observed shifts, which are then abandoned. This is valid because we are not interested in the errors introduced by the reference transformation itself, only in the errors introduced by the grid alone. To allow us to answer the third question posed in this paper (section 4.3), we needed a transformation that would yield results with high spatial variability. Therefore we discarded simple well-behaved models such as global polynomial surfaces of low degree. Other than that, the choice was arbitrary. The chosen reference transformation is as follows. We interpreted the shifts in latitude and longitude as two separate two-dimensional scalar fields, varying over the horizontal space. Then, we used the trianglebased bi-cubic interpolation (Fortune, 1997) to interpolate the shifts at the desired points. A sample of the reference transformation's result is shown in Figure 3. Figure 4 shows the so-called Delaunay triangulation of the control points, an intermediate result required by the chosen transformation.
We generated an array of regularly spaced nodes (spaced 13 ° 30' in latitude and longitude) enclosed by the convex hull of the control points, as shown in Figure 5. Then, we evaluated the reference transformation at each grid node. The results of this evaluation are shown in Figure 6.
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3.4 Test points
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To evaluate the performance of the grid we overlaid on it 10,000 points at random positions, as shown in Figure 7. The number of random points was chosen arbitrarily. We transformed the coordinates of each random point using (i) the reference transformation, and (ii) the grid. The difference between (i) and (ii) represents the error introduced by the grid.
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• An
Analysisof ErrorsIntroducedbythe UseofTransformationGrids
As we sample the grid error with more and more test points, we find that there is an upper bound in the error curves. In our case, we noticed that after 5,000 test points the error stops increasing, meaning that (i) that sample is representative of the grid error, and (ii) the error is no greater than 3.5 x 10 .6 degrees or 0.0126". The upper bound on the error curves depends on a balance between grid spacing and spatial variability of the reference transformation results. The next two questions below address the problem of "tuning" a grid so that it introduces only negligible errors, in an efficient manner. 1/] 6
Error in longitude for each random point
3.5 Use of the transformation grid We have used a program conforming to the transformation grids specification (the Canadian NTv2) to verify that the grids we were generating followed the specified text format. The grid is to be used for interpolating bi-linearly (Press et al, 1992) the shifts given at the grid nodes. We did so with a Matlab implementation (function interp2) of that interpolator.
4 Results and discussion 4
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Figure 8 Absolute error at each point (thin dots), maximum 4.1 Is there an upper bound in the error introduced by the transformation grid?
cumulative error (thin, stepped curve), and mean cumulative error (thick, continuous line) versus number of test points.
Obviously the grid may introduce an error with respect to the reference transformation. That is so because the grid transformation model (which is made of the grid itself a n d the bi-linear interpolator to be applied on it) m a y not be able to capture all the variability described by the reference transformation. We would like to assess how bad this error can get. To answer this question we analyzed the cumulative mean and m a x i m u m errors of the r a n d o m test points as we increased the n u m b e r of points (see Figure 8). In that figure, each dot represents an individual test point; the thin, stepped curve is the m a x i m u m error; the thick continuous line is the mean error. These error curves are cumulative, meaning that each value along them is calculated from the test points to the left of it.
4.2 What is the coarsest spacing between nodes for a transformation grid to introduce only negligible errors?
4 Due to lack of space, in the following we have figures only for longitude. The corresponding figures for latitude show curves with similar behaviour and values 10 times larger.
First of all it is required that the ones generating the grid define what error would be negligible in their application. This value might be based, e.g., on the error already introduced by the reference transformation itself. W e will describe an example later in this section. The question will help us to tune the grid so that the inevitable errors introduced by it do not affect the application we have in mind. Here we will assume that there is only one uniformly spaced grid covering the area of interest. In the next question we will be interested in the case of reducing the grid spacing locally (instead of globally, as we do here). To answer the question we have obtained, for each test point, the "distance ''5 to its nearest grid
5 The "distance" is actually the Euclidean norm of the difference in geodetic coordinates between a given point and
681
682
F.G. Nievinski • M. C. Santos
node. We sorted the points based on that distance. Then we analyzed how the error increases as that distance increases (see Figure 9). Intuitively, the closer a point is to a grid node (i.e., the smaller that distance), the better the grid model represents the actual shift at that point or, conversely, the smaller the error is at that point. lO .0
Error in longitude versus distance to nearest grid node
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I
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one grid, it should satisfy both latitude and longitude maximum errors. Therefore we use the most stringent maximum distance, which is 0.025 ° in this case. 0.035 ° is the spacing between grid nodes that yields 0.025 ° as maximum distance (i.e., 0.035 ° = 2 x 0.025°). Figures 11 show the nodes of this new, denser, grid, and Figure 12 shows its corresponding error curve. Now the maximum distance is 0.025 ° (see horizontal axis), and the maximum error is close to 2x 10-5 degrees (see vertical axis), as specified.
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nearest grid node. Figure
Figure 9 can be used to find the coarsest spacing between nodes for a transformation grid to introduce only negligible errors. To do so, first, we specify the maximum acceptable error; second, we find the corresponding maximum distance to a grid node, using the error curve in Figure 9; third, we regenerate the grid using that distance times 2 (see Figure 10) as the spacing between its nodes.
11 Nodes of the denser grid.
x 10e
Error in longitude versus distancs to nsarsst grid nods
2.5
Maximum ] } N°des A
distance
A
spacing
.
" 0.0~
Figure 10 Relationship between maximum distance and nodes spacing (equal spacing in both directions). As an example, the maximum acceptable error was chosen arbitrarily as 2 x 10 .5 degrees, which equals 0.072". The corresponding maximum distance found in the maximum latitude and longitude error curves is 0.04 ° (Figure 9) and 0.025 ° (figure not shown), respectively. As there is only
its nearest node. It is not, e.g., the ellipsoidal distance between the two.
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12 Error at each point (thin dots), maximum
cumulative error (thin, stepped curve), and mean cumulative
error (thick, continuous line) versus distance to nearest grid node - denser grid.
4.3 How does the error introduced by a transformation grid vary spatially? In the previous question we assumed that we would be using only one, uniformly spaced grid. But transformation grids allow the use of sub-grids, i.e., denser grids covering a subset of the main grid. This property is important when a national grid is
Chapter 98 • An Analysis of Errors Introduced by the Use of Transformation Grids
densified by a state or provincial grid. It is useful to be able to predict the maximum error spatially because the areas with high maximum errors would be strong candidates for sub-gridding. To help us tackle the posed question we have used a denser 6 set of regularly spaced test data. We did so to improve visualization. The denser point set depicts the patterns in more detail, and the fact that it is regularly spaced allows us to plot the set as an image, which is a lot faster than to plot each individual point. We expected to find a spatial portrayal of the behaviour shown in Figure 9, i.e., error increasing as a function of distance to nearest node. But what we found was the intriguing pattern shown in Figure 13. Maxima seem to concentrate near edges and vertices of the Delaunay triangulation.
case, the piece-wise maximum error depends strongly on the absolute value of the second derivative of the reference function. We could not find or develop an expression describing a similar dependence in the 2dimensional case. Then we went to investigate empirically whether a similar behaviour is observed. To do so, we calculated numerically the second gradient of shift in each coordinate, by means of the central difference numerical derivative (Press et al., 1992) in each direction. The norm of that gradient is shown in Figure 14.
Figure 14 Norm of the second gradient of shift in longitude (red- large values, blue- small values), versus latitude and longitude, with edges of triangulation overlaid.
Figure 13 Error (red- large values, blue- small values) in longitude versus latitude and longitude, with edges of triangulation overlaid. In the search for an explanation for that behavior, it was brought to our attention the existence of the following formal error bound for the 1-dimensional linear interpolation (Wikipedia, 2005):
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max f"(x),
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where e is the error, f i s the reference function being approximated by the interpolation procedure, and x o and x 1 define the limits of a piece of the domain on which the interpolation is taking place. The expression above shows that, in the 1-dimensional
6 The test points were 10 times denser than the transformation grid nodes.
We see roughly the main peaks at the same positions (see, e.g., SE-, NE-, and NW-corners). Despite that, the overall matching is weak, as attested by the correlation coefficients: 0.4 and 0.3 for shifts in latitude and in longitude, respectively. The main difference between the error field and the second gradient norm field is that the latter is better defined, with sharper variations, while the former seems like a locally-averaged version of the latter. To verify the interpretation above, we defined grid cells, which are rectangular areas delimited by four nodes of the original grid (the grid on which the errors are b a s e d - see Figures 5 and 10). Then, we computed the maximum values of error and norm of second gradient p e r g r i d cell. At this time the correlation is stronger: 0.76 and 0.75 for shifts in latitude and longitude, respectively (see scatter plot in Figure 15). Therefore, in this case, the norm of the second gradient of shifts predicts partially the error per grid cell.
683
684
F.G. Nievinski• M. C. Santos x 10 "~ 6
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As stated at the beginning of this section, in the process of developing and tuning transformation grids, the areas with high errors would be strong candidates for sub-gridding. We should recognize that for this purpose it might be more practical to simply obtain and analyze the error field directly (Figure 13) instead of predicting it only partially. 5
Conclusions
In this paper we analyzed the error introduced by the use of transformation grids. We have shown that: (i) there is an upper bound in the error introduced by the grid; (ii) the coarsest spacing can be found by plotting the error versus distance to nearest grid node; and, (iii) the m a x i m u m error vary spatially partially in proportion to the norm of the second gradient of the shifts. We believe these conclusions and, more importantly, the analyses presented in this paper, might be useful to individuals and agencies considering, developing, or tuning transformation grids to support the transition from a classic to a modern reference frame.
Acknowledgements
Funds for this study have been provided by the Canadian International Development Agency (CIDA) in support of the Brazilian National Geospatial Framework Project ().
References
Australian Intergovernmental Committee on Surveying and Mapping (ICSM). Geocentric Datum of A u s t r a l i a Software. Available at: . Last access on 08/May/2005. Brazilian Institute of Geography and Statistics (IBGE) (1996). Adjustment of the planimetric network, Brazilian Geodetic System (in Portuguese). Canada Geodetic Survey Division (GSD). Commercial Software with NTv2 feature. Available at: . Last access on 08/May/2005. Collier P.A., Argeseanu V., Leahy F. (1998). "Distortion Modelling and the transition to GDA94", The Australian Surveyor, Vol. 43, No. 1, March 1998. Collier, P.A. (2002). "Development of Australia's National GDA94 Transformation Grids". Consultant' s Report to the Intergovernmental Committee Surveying and Mapping. February, 2002. Costa, S. M. A., M. C. Santos and C. Gemael (1999). "The integration of Brazilian geodetic system into terrestrial reference systems." Abstracts, XXII General Assembly, International Union of Geodesy and Geophysics, Birmingham, England, 19-24 July, p. A.415. Fortune, S. (1997). Voronoi diagrams and Delaunay triangulations, Handbook of Discrete and Computational Geometry, J.E. Goodman, J. O'Rourke, eds., pp. 377-388, CRC Press, New York. Junkins, D.R. and Erickson, C. Version 2 of the National Transformation Between NAD-27 and NAD-83 and Its Importance for GPS Positioning in Canada. Geodetic Survey Division Geomatics. Canada, 1996. Oliveira, L.C., J.F.G. Monico, M.C. Santos and D. Blitzkow (1998). "Some considerations related to the new realization of the SAD-69 in Brazil." Advances in Positioning and Reference Frames, F. K. Brunner (Ed.), International Association of Geodesy Symposia, Vol. 118, Springer, Berlin, pp. 205-210. McCarthy, D.D. and G. Petit. (2004). IERS Conventions 2003 (IERS Technical Note 32) Frankfurt am Main: Verlag des Bundesamts fiir Kartographie und Geoddsie. 127 pp., paperback, in print. Mitchell, D.J., Collier, P.A. (2000). GDAit Software Documentation Version 2.0. Available at: . Last access on: 07/Sep/2004. Press W.H., S.A. Teukolsky, W.T. Vetterling and B.P. Flannery (1992). Numerical Recipes in C - The Art of Scientific Computing. Second Edition, Cambridge University Press. Vaniek, P., R.R Steeves (1996) "Transformation of coordinates between two horizontal geodetic datums". Journal of Geodesy, Vol, 70, No, 11, pp. 740-745. "Linear interpolation." Wikipedia: The Free Encyclopedia. 2005. Available at . Last access on 01/Aug/2005. Wolberg, G. (1990) Digital Image Warping, IEEE Computer Society Press, Los Alamitos, CA.
Chapter 99
Preliminary Analysis in view of the ITRF2005 Z. Altamimi, X. Collilieux Institut Geographique National, ENSG/LAREG, 6-8 avenue Blaise Pascal, 77455 Marne-la-Vallee, France C. Boucher Conseil gdndral des ponts et chaussdes, tour Pascal B, 92055 La Ddfense, France
Abstract. Unlike the previous versions of the International Terrestrial Reference Frame (ITRF), the ITRF2005 will be constructed with input data under the form of time series of station positions and Earth Orientation Parameters (EOP' s). This paper presents some preliminary results of the analysis of the time series submitted to the ITRF2005, focusing on the frame parameters and in particular the origin and the scale, as well as the EOP alignment to the combined frame. Quality assessment of the preliminary combinations is addressed in view of the ITRF2005 official solution.
Keywords. Reference System, Reference Frame, Earth Rotation, Combination, Space Geodesy, ITRF
site instability, seasonal loading effects, etc. rigorously and consistently including EOPs in the combination and ensuring their alignment to the combined frame. The results presented in this paper are intended to be preliminary and should not be taken as final or official products of the ITRF2005. Most importantly, some individual solutions analyzed here will not enter the official ITRF2005 combination. They are analyzed in order to assess the current accuracy of some frame parameter, in particular the scale and origin. Moreover, the ITRF2005 analysis will be performed by the 3 ITRF combination centers Natural Resources Canada (NRCan), and Deutsches Geod~idtisches Forschungsinstitut (DGFI) and IGN. The official final ITRF2005 will be delivered by the ITRF Product Center hosted by IGN.
1 Introduction 2 ITRF2005 Input Data Contrary to previous ITRF versions, the ITRF2005 will integrate time series of station positions and daily Earth Orientation Parameters (EOP's). The time series solutions are now provided in a weekly basis by the Services of the International Association of Geodesy (lAG) of satellite techniques (IGS: International GNSS Service, ILRS: International Laser Ranging Service, IDS: International DORIS Service) and in a daily (VLBI session-wise) basis by the International VLBI Service (IVS). Reasons for which it was decided to use time series of station positions and EOP as input to ITRF2005 include: monitoring of non-linear station motions and all kinds of discontinuities in the time series: Earthquake related ruptures,
As input data to the ITRF2005, it was decided to consider official combined time series of solutions provided by the IAG services, known as Technique Centers (TC) by the IERS. These official TC's solutions result obviously from a combination of the corresponding individual analysis centers solutions. Official time series of solutions were submitted to the ITRF2005 by the IGS, IVS and ILRS. At the time of writing, the IVS and ILRS solutions still need some refinement as the quality of their solutions needs to be improved. For DORIS, official weekly combined solutions do not exist yet so that individual solutions are provided by 3 analysis centers. Table 1 summarizes the submitted solution to ITRF2005 and individual additional solutions analyzed in this paper.
686
Z. Altamimi. X. Collilieux • C. Boucher
Table 1. ITRF2005 Submitted and analyzed solutions TC
AC*
Time span
Type of constraints/solution
Comment
IVS
1984-2005
Normal Equation
Official submission but still need refinement at the time of writing
IVS - GSFC
1984-2005
Normal Equation & var-covar
Analyzed in this paper
IVS - GEOS
1984-2005
Loose; var-covar
Analyzed in this paper
IVS - DGFI
1984-2005
Normal Equation
Analyzed in this paper
ILRS
1992-2005
Loose; var-covar
Official submission but still need refinement at the time of writing
ILRS - ASI
1992-2005
Loose; var-covar
Analyzed in this paper
ILRS - DGFI
1992-2005
Loose; var-covar
Analyzed in this paper
ILRS - GEOS
1992-2005
Loose; var-covar
Analyzed in this paper
ILRS - GFZ
1992-2005
Loose; var-covar
Analyzed in this paper
ILRS - JCET
1992-2005
Loose; var-covar
Analyzed in this paper
ILRS - NSGF
1992-2005
Loose; var-covar
Analyzed in this paper
IGS
1996-2005
Minimal/Inner; var-covar
Official submission to ITRF2005
IDS - IGN/JPL
1993-2005
Loose; var-covar
Official submission to ITRF2005
IDS
INA
1993-2005
Loose ; var-covar
Official submission to ITRF2005
IDS
LCA
1993-2005
Loose ; var-covar
Official submission to ITRF2005
* TC: Technique Center, AC: Analysis Center, GSFC: Goddard Space Flight Center, NASA, USA, GEOS: Geosciences Australia, DGFI: Deutsches Geod~idtisches Forschungsinstitut, Germany, ASI: Agencia Spaziale Italian, GFZ: GeoForschungsZentrum Potsdam, JCET: Joint Center for Earth System Technology, at GSFC, NSGF: NERC Space Geodesy Facility (NSGF), formely RGO Satellite Laser Ranging Group, U K , IGN: Institut Gdographique National, France, JPL: Jet Propulsion Laboratory, INA: INstitute of AStronomy Russian Academy of Sciences, LCA: Laboratoire d'Etudes en Geophysique et Oceanographie Spatiale (LEGOS) in cooperation with Collecte Localisation par Satellite (CLS), France. I I R F 2 0 0 5 Derivation
3 Analysis Strategy "W1
Y~'2
...
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I ' I R F (X, V)+ EOP (SINEX)
The strategy adopted for the ITRF2005 generation consists of the following steps, illustrated in Figure 1.:
~LRI I
I
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•
• • •
• •
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I
I LocalIies
I
I
v
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Fig l. I T R F 2 0 0 5 Derivation
3.1 Stacking of Times Series CATREF Software is used to combine (rigorously stacking) the per technique time series of station positions and EOP using the following two sets of equations, (Altamimi and Boucher, 2003):
Chapter99 • PreliminaryAnalysisin Viewof the ITRF2005 -
+ (t i - to)2
. +
+
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=
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where X R and X c are the reference and the combined solutions, respectively. A is the design matrix of partial derivatives of the 14 TRF parameters. The reference solution used in the analyses presented in this paper is extracted from the ITRF2000 solution (Altamimi et al, 2002) over a reference set of station for each one of the 4 techniques.
+
i
i
(1)
.) R3 k
4 Discussion results
(2) Jc~
=
~cP+ R2 k
y ps
_
j,v + R1 k
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-
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time
derivatives
2f,
f = 1.002737909350795
~9~
and
LOD.
is the conversion factor
unit, as the EOPs considered here are provided at daily basis. Note that the link between EOP and the TRF is ensured by the 3 rotation angles and their time derivatives.
3.2 D a t u m D e f i n i t i o n When stacking the daily/weekly solutions using equations (1) and (2) the constructed normal equation system is singular and has 14 degrees of freedom. This rank deficiency corresponds to the number of parameters which are necessary for the datum definition of the combined Terrestrial Reference Frame (TRF). The latter is defined through minimum constraint approach using the equation: -
0
(3)
some
preliminary
In addition to the combinations (rigorously stacking) of the individual time series (per technique and or per analysis centers), a first tentative global combination was also performed for the purpose of this paper. We present here some results, focusing on the frame parameters, namely the origin and the scale. In addition to the analysis of the time series, we provide some results on apparent seasonal variations and a quality evaluation of the individual solutions. Preliminary results of Earth Rotation Parameters included in our combinations are also included here.
'
from universal to sidereal time. Considering LOD--A 0 --7-, dc;r A 0 is equal to one day in time
a
of
4.1 Origin results
and
Scale
preliminary
It is expected that the origin of the ITRF2005 will be defined by the ILRS solution and the scale by the average of IVS and ILRS solutions. In order to illustrate the origin and the scale behavior over time, Figure 2 displays the three translation components as well as the scale of the 6 AC's composing the ILRS combined time series. Figure 3 shows the scale behavior of the IVS preliminary time series as well as 3 other IVS AC solutions. All results show here are with respect to ITRF2000. From these results we see, in average, a good origin and scale consistency between the individual analyzed series and the ITRF2000. Meanwhile the IVS scale variation exhibits significant seasonal variation which might be attributed, at least partly, to thermal deformations of the VLBI radio telescopes. For completeness, Figure 4 presents the time variations of the origin and the scale of the two analyzed DORIS solutions.
687
688
Z. A l t a m i m i .
X. Collilieux • C. B o u c h e r
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origin and
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4.2 Seasonal Variations GS F(_', GEOS
IVS
The authors plan to make available to users the residual time series of the station positions as results from the individual combination/stacking. In order to illustrate the type of seasonal variations as seen for some individual stations, Figure 5 displays the residual time series of some GPS/IGS stations as indicated. We note that this type of seasonal variations is more pronounced and clearly identified in the IGS GPS station time series, compared to the other techniques and that this effect is most significant in the vertical component. In order to illustrate that effect, Figure 6 shows the annual amplitude and phase of the vertical components of most pertinent IGS sites.
(mm)
Scale
Fig. 3. Time variation of IVS AC's scale in mm.
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BAI-IR Annual Amplitude and Phase
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Chapter 99 • Preliminary Analysis in View of the ITRF2005
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4.3 E a r t h O r i e n t a t i o n
1995
2000
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1995
2000
2005
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2000
2005
YpO GP~/IGS , 1.o
, 1995
, 2000
,-1.0 2005
(mas) as result from the global combination test.
Parameters
As results from the preliminary global combination, similar to what will be the ITRF2005, Figure 7 displays a zoom of _+ 1 milli-arcseconds (mas) of polar motion residuals. This combination test includes the following solutions: VLBI/GSFC, SLR/ILRS preliminary solution, GPS/IGS official solution as well as IGN DORIS solution.
The approximate average of the Weighted Root Mean Square Error (WRMS) on polar motion is around (in gas ) 50 for GPS, between 150 and 200 for VLBI and SLR and around 1000 for DORIS. Moreover, a comparison of the resulting polar motion series with the IERS C04 series was performed and the differences are plotted in Figure 8. The mean of these differences indicates a significant bias of about 100 gas in the Y component between our EOP series (expressed in ITRF2000) and the IERS C04.
689
690
Z. A l t a m i m i . X. Collilieux • C. Boucher
4.4 Quality evaluation
conclusions from this preliminary analysis are the following:
As an internal quality indicator of the analyzed times series, we utilise the WRMS of the individual series. Figure 9 displays the weekly (daily for VLBI) WRMS per solution for the horizontal and vertical component. The rough average of these values is summarized in Table 2.
•
• Table 2. Rough average of internal precision per solution, based on computed weekly (daily for VLBI) WRMS resulting from the stacking of the individual time series. Solution
2-D WRMS (mm) 7 6 10 20
VLBI GSFC GPS IGS SLR ILRS DORIS IGN
UP-WRMS (mm) 3 3 10 20
•
•
5 Conclusions • Preliminary analyses are presented in this paper as preparation for the ITRF2005. The major
the use of time series as input to the ITRF2005 combination definitely allows monitoring the stations behavior and consequently improve their estimated linear velocities. the origins of the SLR solutions analyzed are, in average, consistent with the ITRF2000. Their scales as well as the scale of VLBI solutions are also consistent with the ITRF2000. However some significant VLBI scale seasonal variation is apparent. significant seasonal variations are detected for GPS/IGS station time series and most significant in the vertical component. the quality of the analyzed time series is encouraging, promising an improved ITRF2005 solutions compared to previous ITRF versions more work and analysis are still to be done before the official release of the ITRF2005.
X pole '
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1997
1998
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mas) with IERS C04.
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99 • Preliminary Analysis in View of the ITRF2005
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400
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Fig. 9. Weekly (daily VLBI) WRMS
Acknowledgments The ITRF activities are funded by the Institut G6ographique National, France and partly by the Group de Recherche de G6od6sie Spatiale. The IERS ITRF Product Center is indebted to the Analysis Centers and Technique Services of the 4 techniques for their contribution to the ITRF2005.
References Altamimi, Z., P. Sillard, C. Boucher (2002). ITRF2000, A new release of the International
Terrestrial Reference Frame for earth science applications. Journal of Geophysical Research, Solid Earth, vol. 107(B 10), 2214. Altamimi Z, Boucher C (2003) Multi-technique combination of time series of station positions and Earth orientation parameters, in Proceedings of the IERS Workshop on Combination Research and Global Geophysical Fluids, IERS Technical Note No. 30, Richter B, Schwegmann W, Dick W (eds.), Verlag des Bundesamts ffir Kartographie und Geodfisie, Frankfurt am Main, Germany, 102-106
691
Chapter 100
Long term consistency of multi-technique terrestrial reference frames, a spectral approach K. Le Bail Institut Gdographique National/LAREG and Observatoire de la C6te d'Azur/GEMINI (UMR 6203) 8 Av. Blaise Pascal, Champs sur Marne, 77455 Marne la Vallde Cedex 2, France M. Feissel-Vernier Observatoire de Paris/SYRTE and institut Gdographique National/LAREG 8 Av. Blaise Pascal, Champs sur Marne, 77455 Marne la Vallde Cedex 2, France J.-J. Valette Collecte Localisation Satellites (CLS) 8-10 rue Hennas, 31526 Ramonville-Saint-Agne Cedex, France
W. Zerhouni Centre National des Techniques Spatiales (CNTS) BP 13, Arzew - 31200, Oran, Algeria Abstract. Analysing time series of space-geodetic station coordinates, we show that VLBI, SLR and DORIS station motions have a white noise error spectrum, while the majority of GPS station motion have a flicker noise error spectrum. In the latter case, discontinuities in the series do not account for this spectrum. Atmospheric loading has a white noise spectrum, at a much lower level than geodetic measurement errors in the long term. The series of transformation parameters derived for the GPS colocation sub-networks with VLBI, SLR and DORIS have very close spectral characteristics, reaching the 1 mm stability level at one-year interval in a white noise context.
Keywords. Space Station stability.
geodesy.
Reference
frames.
1 Introduction With the advent of the combination of time series of station coordinates to construct ITRF2004 (Altamimi et al. 2005), a key element in the long term stability of a multi-technique Terrestrial Reference Frame (TRF) is the control of the time consistency of the motion measured by the various techniques for colocated stations. Time series of station coordinates are available from global space geodetic programs operated with GPS, SLR, VLBI and DORIS. Good quality series go back to the early 1990's. The major signature in time series of station coordinates is usually modelled as a tri-dimensional linear drift. The horizontal component is mostly related to tectonic plate motion, while the vertical
component is assumed to reflect local uplift or subsidence. The remaining component may be interpreted as noise related to local geophysical phenomena, instrumentation, or to the analysis strategies and modelling. The hypothesis of linear motion is also a key one in most uses of space geodetic positioning. For this reason, our study is focusing on the non-linear content of the time series of coordinates; i.e. residuals relative to a linear motion model for the station. The derivation of spectral characteristics of the time behaviour of the station coordinates from the four ITRF techniques is described in Section 2. In Sections 3 and 4 the possible influence of two types of perturbing phenomena on station stability is investigated. These are 1) confirmed or suspected coordinate discontinuities in position or velocity, and 2) atmospheric loading. The long term consistency of the combined reference frame depends directly on the stability of the tie between the VLBI, SLR, GPS and DORIS networks. In Section 5, based on the stability analysis and associated quality criteria, a detailed analysis of the time behaviour of the GPS-colocated sub-networks is proposed.
2 Spectral characteristics of non linear signals in time series of station coordinates The data analysed are series of station coordinates derived from the VLBI, SLR, DORIS and GPS techniques. They are typically provided as series of geocentric Cartesian coordinates or, equivalently, series of offsets in local directions to the East, North
Chapter 100 • Long Term Consistency of Multi-Technique Terrestrial Reference Frames, a Spectral Approach
and Up (ENU), in some defined terrestrial reference frame. GPS and DORIS series are available at oneweek intervals. The average time distribution of the SLR and VLBI series is comparable but somewhat irregular, due to observing conditions
The time series analysed are listed in Table 1. The VLBI series was derived by Ma (2005) using the CALC-SOLVE software package; the SLR series was derived by Coulot and Bdrio (2005) using GINS-DYNAMO; the GPS series was derived by R. Ferland (IGS 2005a) as the combination of the IGS analysis centres series; the IGN DORIS series was derived by Willis (2005) using GIPSY-OASIS; the LCA DORIS series was derived by Soudarin and Crdtaux (2005) using GINS-DYNAMO. The IGN series was used as the DORIS series for figures 4, 5, 7, and table 3; the LCA series was used as the DORIS series in figures 1, 4, 6, and table 3.
Table 1. The sets of station coordinates analysed. The numbers of stations that (a) passed the time density threshold (see text), and (b) that are sparsely observed are given. Techn. VLBI SLR DORIS DORIS GPS
Solution name gsfc_2004b slr oca05 ignwd05 lcawdl 2 igs_2004
Data span 90.0-04.3 93.0-05.0 93.0-05.3 93.0-05.1 95.0-05.4
Sites # 34 20 59 59 175
Stations (a) (b) . 24 13 18 3 44 38 47 36 137 52
The data analysis is based on the time series of residuals with respect to linear motion for each station individually. Station time series are selected on the basis of time span (at least four years), missing weeks (less than 30%), and data gaps (shorter than 200 days). Data with postfit residuals larger than three times the standard deviation are edited. Table 1 gives the numbers of sites and stations available from each technique, considering separately the stations whose time series of coordinates did or did not pass the above threshold. The analysis approach is described in detail by Le Bail (2004) and Le Bail and Feissel-Vernier (2005). The three-dimensional geodetic station position signal is submitted to Principal Component Analysis (PCA) in the time domain. Then the analysis is made in both the local reference frame and the 3D Principal Components (PCTs) derived from PCA.
The PCTs are the projections of the initial series on the space generated by a set of eigenvectors which are defined so that the leading component describe the temporally coherent pattern that maximises its variance. The spectral behaviour of the series is characterised by the change of the Allan variance of the time series of coordinates as a function of the sampling time (Allan 1987). This statistics allows one to identify white noise (spectral density S independent of frequency j), flicker noise (S proportional to f-l), and random walk (S proportional to f-2). In a white noise context, the variance of residual motion is lessened as the data span is extended. In a flicker noise context, the errors are not diminished with the expansion of the data span. A convenient and rigorous way to relate the Allan variance of a signal to its error spectrum is the interpretation of the Allan graph, which gives the changes of the Allan variance for increasing values of the sampling time "c. In logarithmic scales, slopes -1, 0 and + 1 correspond to white noise, flicker noise and random walk, respectively. Note that the signature of a linear slope in the signal in the Allan graph is a +2 slope. Here, the upstream correction for a linear motion makes it little probable to find a random walk spectrum in the residuals. The presence of a cyclic variation is recognised by the superimposition of a dip when "c is equal to the cycle period and a bump at about 1/2 the cycle period, with a size that depends on the signal/noise ratio of the amplitude of the cyclic component. To avoid biasing due to this effect, the Allan graph slopes are computed on time series corrected for their annual term. The following statistical parameters are derived. - Stability for a one-year sampling time of the first Principal Component in the Time domain, measured by the Allan standard deviation, and its projection in the local ENU frame. - Spectral law followed by the same parameters, measured by the slope of the Allan graph.
Figure 1 shows the values of these two indicators obtained for the coordinate time series (a) of Table 1 (dense series). The two-dimensional graphs show for each station the one-year Allan standard deviation as a function of the slope of its Allan graph. The quantity considered is the first principal component in the time domain (PCT), which explains in general
693
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K. Le Bail. M. Feissel-Vernier. J.-J. Valette • W. Zerhouni
over 80% of the non-linear, non-seasonal signal. To relate this component to the stations local conditions, the projections of the PCT characteristics in the local ENU frame are also shown. A remarkable, and not unexpected, feature is the occurrence of largest values of the Allan standard deviation in the Up direction for GPS and VLBI, and for a part of the SLR series. The case of DORIS is different, with relatively homogeneous levels in all three directions and some noisier values in the East direction.
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I Iv 9.0) since continuous earthquake monitoring. It will be described how the different sensors at Fundamental Station Wettzell record the signals induced by the seismic waves. While deformation due to an earthquake in terms of elastic rebound could only be detected in local or regional distances to the centre of the earthquake by means of GPS, the horizontal deformation due to seismic surface waves was large enough to be analysed with highrate GPS data even 9000 km apart from the centre. The rotational component of the seismic waves is clearly recorded by the ring laser "G" which is the only type of instrument being able to measure this quantity. The results of the different sensors will be compared, the benefit of each sensor referred to seismic waves will be summarized. Conclusions for further investigations with respect to the Global
Geodetic Observing System (GGOS) will be discussed. Keywords. Seismic waves, highrate GPS, superconducting gravimeter, ring laser, Sumatra earthquake
1 Introduction The Fundamental Station Wettzell (Fig. 1) of Bundesamt ftir Kartographie und Geodfisie (BKG) is jointly operated with Forschungseinrichtung Satellitengeodfisie (FESG) of Technical University Munich. It is located in the Bavarian Forest. The Fundamental Station Wettzell is one of a few stations worldwide possessing main geodetic observing systems which are complemented by a number of other sensors and instrumentation: • Very Long Baseline Interferometry (VLBI): a 2 0 m radio telescope specially designed for VLBI • Laser Ranging: Wettzell Laser Ranging System (WLRS) designed for Satellite Laser Ranging (SLR) as well as Lunar Laser Ranging (LLR) • Global Navigation Satellite System (GNSS): Different GPS and GPS+GLONASS receiver/antenna pairs mounted on a concrete survey tower, about 7.5 m over ground, observing permanently. The stations are included in the Network of the International GNSS Service (IGS), the EUREF Permanent Network (EPN) and the German Reference Network (GREF) • Superconducting gravimeter: a GWR SG-29 with a relative resolution of 10-11
Chapter 109 • Combination of Different Geodetic Techniques for Signal Detection - a Case Study at Fundamental Station Wettzell
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• Ring Laser: a 4 m x 4 m ring laser for monitoring Earth rotation, installed in a special subterranean laboratory • Seismometer: standard b r o a d b a n d s e is m o m e te r STS-2 installed in a seismic vault; additionally Lennartz seismometers installed in the ring laser laboratory • Tiltmeter: 6 tilt meters installed in the ring laser laboratory • Time and frequency: three cesium frequency standards, three h y d r o g e n masers and two GPS time receivers
• Meteorological sensors: sensors for air temperature, pressure, humidity, precipitation, wi n d speed and direction, g r o u n d moisture; g r o u n d water level; water v a p o u r radiometer. The location o f the main c o m p o n e n t s is s h o w n in Figure 2.
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Fig. 3 Principle scheme of surface wave propagation caused by the Sumatra-Andaman earthquake to Germany The white circles indicate the wave front, the ray path is shown in black.
761
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J. Ihde. W. S6hne. W. Schwahn • H. Wilmes • H. Wziontek • T. KKigel. W. SchKiter
2 Seismic waves The ground deformation following the SumatraAndaman earthquake of Dec 26, 2004, 0:58:50 UTC (second-of-day 3530) was large enough to be detected at far distances to the epicentre. The main types of seismic waves are P, S, Love and Rayleigh waves. Since P and S waves are body waves they will arrive first at far distance locations. Surface waves like the Love waves caused by the SumatraAndaman earthquake are expected to arrive in Germany from the East direction (Fig. 3) with a travel time of approx. 2000 seconds. Therefore the Love wave will reach Germany approx. 5530 seconds after 00:00:00 UTC, Dec 26. The sensitivity of the various sensors to the main seismic waves induced by the Sumatra-Andaman earthquake and detectable at Fundamental Station Wettzell is expected to be different. Due to their spatial design the seismometers should be able to detect all four main seismic waves. GPS with its higher noise level mutually
will only detect the horizontal displacements caused by the surface waves while the ring laser only measures rotations around its sensitive axis with high precision. This unique property led to the development of the Geosensor, a new instrument for seismology, Schreiber et al. (2003). 3 GPS With GREF and the network of the Satellite Positioning Service (SAPOS ®) operated by the Surveying Authorities of the German Lfinder a dense network of GNSS permanent stations is available in Germany. While for most applications a data sampling rate of 30 seconds is sufficient the data are internally stored with a sampling rate of 1 Hz. The derivation of positioning time series was carried out using the Bernese GNSS analysis software version 5.0. The software has the option to output
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4 Time series of the North-South component of GREF site WTZT (12.9°/49.1 °) with S A P O S ® site Kaiserslautern (7.8 ° / 49.4°) as reference station (baseline length 373 km) for three consecutive days, differences against the individual mean coordinate. The curves for days-of-year 359 and 360 were shifted by 472 and 236 seconds, resp., to take the sidereal day into account. Beside the good overall agreement the signatures as a result of the Love wave can clearly be recognized.
Fig.
Chapter 109 • Combination of Different Geodetic Techniques for Signal Detection - a Case Study at Fundamental Station Wettzell
kinematic coordinates at every observation epoch after fixing the ambiguities for the whole session. From the unfiltered kinematic positioning results for a West-East orientated baseline (Fig. 4) it can be seen that there are common signatures of the corresponding time series reflecting the common behaviour as a result of, e.g. satellite constellation or multi-path. The East-West component (not shown) in general has the smallest noise level (rms + 5-7 mm) whereas the rms values for the North-South component are in the range of + 7-10 mm. For the vertical component the noise level is higher with + 13-20 mm. A distinct signal can be detected in the North-South component for day-of-year 361 starting approximately at second 5650 followed by two extrema at seconds 5720 and 5780 with an amplitude of 20-30 mm. Results for other baselines in Germany can be found in Soehne et al. (2005).
4 Superconducting Gravimeter The superconducting gravimeter GWR SG-29 at Fundamental Station Wettzell records the gravitational and inertial acceleration in the local vertical direction with two identical sensors (sensor 1 and 2). Main use of the instrument is the observation and investigation of Earth gravity tides and other long-period and non-periodic signals (Richter et al. (2004)). For this purpose the instrument is equipped with an analogue low-pass filter (approx. 44 sec phase lag). The recorded gravity data (sampling rate 5 s) allow the interpretation of observations in the frequency band 0 to 8 mHz.
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The gravimeter includes an active verticality control with two cryogenic tilt meters separated at an angle of 120 deg and operated in a feedback loop using thermally controlled levellers. During ~ 85 minutes after arrival of the first seismic signals of the Sumatra-Andaman earthquake, the gravity record shows a systematic deviation which is obviously correlated with a perturbation of the tilt compensation system. Whereas an interpretation of the gravity records during the first two hours after the earthquake does not seem reasonable due to the signal deviation and the applied low-pass filtering, an analysis of the following data series clearly reveals spheroidal modes of the excited Earth free oscillations and their decay. The spheroidal modes can be detected very clearly and e.g. the 0S0 mode can be observed until June 2005. Figure 5 shows the amplitude spectrum in the frequency range up to 2 mHz for the time period immediately after the earthquake (Dec 28 - Dec 31, 2004). Figure 6 demonstrates the decay of the radial mode 0S0 until June 2005 which was excited again by the seismic event on March 28, 2005 (Northern Sumatra, magnitude 8.4).
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Ring lasers are rotation sensors. The 4 m x 4 m ring laser at Fundamental Station Wettzell (Fig. 7) is designed for measuring variations in Earth rotation with high precision in near real time. Rotational components of passing seismic waves are recorded as well. The horizontal installation makes the instrument sensitive for rotations around the vertical axis, e.g. Love waves. Regarding a plane shear wave, the rotation rate is in phase with acceleration
763
764
J. Ihde. W. S6hne. W. Schwahn • H. Wilmes • H. Wziontek • T. KliJgel. W. SchliJter
having an amplitude of a/2c (a: acceleration, c: phase velocity), Igel et al. (2005). A number of tilt meters, measuring horizontal accelerations, are located on top of the ring laser body to monitor the orientation of the instrument in different directions. A Lennartz seismometer type LE3D/20s is co-located at the same place. Beside this a STS-2 broadband seismometer of the German Regional Seismic Network is operated in a seismic vault in a distance of 250 m. The time series are numerically differentiated to obtain accelerations and integrated to obtain displacements (Figs. 9 and
and 5780 s. This signal is present in records of other sensors, too (Fig. 4, Fig. 9). It is not visible in the East-West component (not presented here).
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Chapter 109 • Combination of Different Geodetic Techniques for Signal Detection - a Case Study at Fundamental Station Wettzell
6 Comparisons
6.2 Vertical accelerations
6.1 Horizontal displacements
The monitored accelerations from the superconducting gravimeter and the seismometer show almost no correlation (Fig. 10). The low pass filter implemented in the SG-29 damps the high frequency parts and causes a frequency dependent phase lag. Therefore, the upper curve for SG-29 is mainly a result of this filter. For the seismometer the values are calculated by numerical derivation of the original vertical velocity output. Although both signals represent vertical accelerations, discrepancies appear caused by the low sampling rate and the applied filtering of the superconducting gravimeter. It is recommended for future investigations to use a low pass filter with significantly smaller time constant and an increased sampling rate of 1 second.
Time series of horizontal displacements are available from GPS and from the seismometer. For the seismometer the values were derived by integration from the velocity output. Since the GPS solution is derived by differential positioning the seismometer results are also differentiated (Fig. 9). The station BFO is used for this, nearly in the same direction and distance to Wettzell (see map in Fig. 4). The second peak shows a small time offset. This is the result of the arrival of the surface waves at the reference stations which are not on the same longitude.
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Fundamental Stations play an important role with respect to a global observing system like GGOS since they provide the instrumentation and the observations for the determination of the fundamental geodetic observables. Beyond this, the co-location of geodetic and geophysical sensors at the Fundamental Station Wettzell gives the unique possibility to study the sensitivity of each sensor to small signals induced by the Sumatra-Andaman earthquake. All of the five different sensor types considered within this paper were able to detect those signals. With respect to the principle of construction of the sensor and/or the noise level of the results the sensors were differently sensitive to the individual signals: - GPS is able to directly give a quantification of the displacement. As a pure geometric technique GPS may help verifying or even calibrating displacements derived by seismometer data. For detecting small seismic signals with GPS improved solutions with additional processing, e.g. sidereal filtering and spatial stacking, are necessary, Bock et al. (2004). - The superconducting gravimeter primarily reveals spheroidal modes of the excited free oscillations of the Earth and their decay. - The ring laser measures rotations around its sensitive axis (the vertical in our case). This new quantity in seismology has never been recorded before with such a precision and gives access to the complete description of the deformation field.
765
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J. Ihde. W. S6hne. W. Schwahn • H. Wilmes • H. Wziontek • T. KKigel. W. SchKiter
- Seismometers measure inertial ground velocities in each direction. Low noise and high resolution allow the derivation of accelerations and displacements by differentiation and integration, resp. Tilt meters record tilts with respect to the plumb line as well as horizontal accelerations and are thus sensitive to horizontal and vertical components of seismic waves. -
References
Bock, Y., L. Prawirodirdjo, T.I. Melbourne (2004). Detection of arbitrarily large dynamic ground motions with a dense high-rate GPS network. Geophys. Res. Let., 31, L06604. Igel, H., U. Schreiber, A. Flaws, B. Schuberth, A. Velikosetsev, A. Cochard (2005). Rotational motions induced by the M8.1 Tokachi-Oki earth-
quake, September 25, 2003. Geophys. Res. Let., 32, L08309. Richter, B., S. Zerbini, F. Matoni, D. Simon (2004). Long-term crustal deformation monitored by gravity and space techniques at Medicina, Italy and Wettzell, Germany. Journal of Geodynamics, 38 (2004) p. 281-292. Schreiber, U., A. Velikosetsev, H. Igel, A. Cochard, A. Flaws, W. Drewitz, F. Miiller (2003). The GEOSENSOR: a new instrument for seismology. Geotechnologien science report, no. 3, 148-151. Soehne, W., W. Schwahn, J. Ihde (2005). Earth surface deformation in Germany following the Sumatra Dec 26, 2004 earthquake using 1 Hz GPS data. Report on the Symposium of the IAG Subcommission for Europe (EUREF), Vienna, 01-03 June 2005, Mitteilungen des Bundesamtes fiir Kartographie und Geodiisie, (in press)
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Part VII Systems and Methods for Airborne Mapping, Geophysicsand Hazardsand DisasterMonitoring Chapter 110
A Multi-Scale Monitoring Concept for Landslide Disaster Mitigation
Chapter 111
High-Speed Laser Scanning for near Real-Time Monitoring of Structural Deformations
Chapter 112
A Method for Modelling the Non-Stationary Behaviour of Structures in Deformation Analysis
Chapter 113
Volcano Deformation Monitoring in Indonesia: Status, Limitations and Prospects
Chapter 114
Vehicle Classification and Traffic Flow Estimation from Airborne Lidar/CCD Data
Chapter 115
Fine Analysis of Lever Arm Effects in Moving Gravimetry
Chapter 116
Improving LiDAR-Based Surface Reconstruction Using Ground Control
Chapter 117
The Use of GPS for Disaster Monitoring of Suspension Bridges
Chapter 110
A Multi-Scale Monitoring Concept for Landslide Disaster Mitigation H. Kahmen, A. Eichhorn, M. Haberler-Weber Institute of Geodesy and Geophysics, Vienna University of Technology, Gusshausstrasse 27-29/E1283, 1040 Vienna, Austria Abstract. There has already been much research on how the kinematic and geodynamic behaviour of landslides can be predicted. However, until now, there has been no decisive breakthrough for a monitoring and evaluation system combined with an alert system. Therefore an interdisciplinary international project OASYS (Integrated Optimization of Landslide Alert Systems) was commenced to progress research in this area. The members of the project, supported by the European Union, believe a multidisciplinary integration of different methods has potential for substantial progress in natural hazards management. This project proposes a new method consisting of three different steps:
1. Detection of boundary lines of potential landslides based on large scale information. 2. Detection of "taking-off-domains" and permanent local scale monitoring of these regions with high sensitivity geotechnical measurement methods. 3. Knowledge-based derivation of real time information regarding actual risks to support alert systems. Following the methodology of the OASYS project, a mathematical method for the detection of takingoff-domains and the concept of a knowledge-based alert system is described in this paper. Keywords. Landslides, Risk Assessment, Alert System, Fuzzy Techniques, Kalman Filtering Techniques. 1 Existing Technologies introduction
- an
There has already been a wide range of research work undertaken on landslides (see, e.g., AlmeidaTeixeira et al., 1991). Most of this work had a bias towards one discipline, such as remote sensing or geology.
Currently the investigation of landslides and unstable slopes is based on two groups of information sources: 1.1 Information sources on a regional scale
- Data on historical landslides: The relevant data should include information of when and where landslides occurred in the critical areas. - General conditions of the areas: Map data of the general conditions, including digital elevation models, geology, tectonics, geomorphology, vegetation, climate, and land use, may be indicators for active landslides (Cui, 1999). - Remote sensing data: Further sources for potential landslide risk can be differential satellite image analysis, time series of airborne sensor data (photogrammetry, laser scanners, radar systems, geophysical measuring devices) and terrestrial mapping. 1.2 Information sources on a local scale
- Geodetic: GPS, precise levelling, tacheometers (measurement robots), multisensor remote sensing techniques (using optical- and radarsensors). - Geophysical: geoelectric and electromagnetic field measurements. - Geodynamical: borehole tiltmeters, extensometers, hydrostatic tiltmeters. - Seismological: sensors for microseismic activity, seismic reflexion measurements. -Hydrological: sensors for groundwater level variations, water level variations, groundwater stream variations. - Meterological: sensors for temperature, air pressure, precipitation. The information of the first source enables detection of the site of the landslide, while those of the second source are used to describe the mechanism of the process(es). Normally, however, only some of the parameters (e.g. deformation vectors, water pore pressure) are used for the investigations and the measurement points are widely spaced across the
770
H. Kahmen. A. Eichhorn• M. Haberler-Weber
landslide domain. The deformation measurements are frequently based only on one measurement method (e.g. GPS, tacheometry). As these measurement methods are relatively costly, usually only a limited number of observation points are observed and the measurement systems are not operated continuously. This methodology has made it possible to follow the evolution of the landslides precisely but yet it is difficult to predict the evolution. By a better integration of these information sources a more reliable prediction should be possible in the future. 2
Integrated
results will be fields of vectors describing the displacements and velocities (Theilen-Willige, 1998). An advanced and generalised deformation analysis algorithm in addition based on geometrical as well as topographical, geological, hydrological and meteorological information has to be developed in order to improve the detection of taking-offdomains, see e.g. Haberler, 2003; Zhang et al., 2001.
Optimization
An advanced model is now under investigation based on large scale monitoring as a first step, regional monitoring as a second step, culminating in a multi-component knowledge-based alert system. 1 st step: Detection of potential landslides (large scale monitoring). To get information about the long-term geodynamical processes a large scale evaluation has to be performed. This includes the historical data, and any information describing the general conditions of the area, as outlined in section 1.1, and remote sensing data, such as aerial photographs, optical and radar images from satellites (Fig. 1).
Fig. 1 Large scale monitoring with airborne and satellite remote sensing.
More specific remote sensing techniques (e.g. InSAR), differential GPS (using phase observations) and tacheometric measurements shall be used to obtain additive information about the deformation process (Fig. 2). The measurements will be performed only three or four times a year, and the
Fig. 2 Geodetic landslide monitoring using GPS and total stations.
2 "0 step: High precision permanent measurements in the taking-off-domains. High precision borehole tiltmeters, extensometers, hydrostatic levelling and further relative measurement systems shall be used in the area of the taking-off-domains to obtain online information about the geodynamical process, see Savvaidis et al., 2001. This multi-sensor system will be running continuously and can therefore support a real time alert system (Fig. 3).
Fig. 3 Continuous local scle measurements and the information transfer.
3 ra step: Impact and risk assessment; development of strategies for knowledge-based alert systems. Risk assessment comprises the analysis of the empirical data and the development of an alert system. The analysis of empirical data includes the
Chapter 110
detection and interpretation of velocity fields in order to define zones of increased deformation. The final analysis will be supported by expert systems, using methods of cluster analysis, neural networks, fuzzy techniques and others (Haberler, 2003). The process of risk analysis can be divided into hazard analysis and vulnerability analysis. Hazard analysis is the review of the potential hazardous processes. Scenarios for the evolution of a landslide area of interest have to be set up, including an estimation of the probability of these scenarios. Different scenarios will affect different areas and therefore have a different impact on people and property. The assessment of the impact of different hazard processes on the affected population and its property is called vulnerability analysis. The integration of hazard and vulnerability analysis will lead to an estimation of the actual risk situation of the affected population. The risk management measures will depend heavily on the specific conditions and will include landuse planning, technical measures (e.g. build drainage systems), biological measures (e.g. afforestation) and temporary measures (e. g. evacuation). The integrated workflow for landslide hazard management is depicted in Fig. 4. Two research areas concerning the integrated work flow are described below. A
I
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_
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• A Multi-Scale
Monitoring Concept for Landslide Disaster Mitigation
following step geotechnical sensors can be installed across these boundaries to obtain high precision monitoring data of the critical areas as an input for the knowledge-based system in the risk assessment. In the last few years modern techniques like fuzzy systems, neural networks and knowledge-based systems have started to be used also in geodesy (see, e.g., Heine, 1999; Wieser, 2002). One advantage of these methods is that they can reproduce the human way of thinking, so that problem solving is done in a rather intuitive way. Here, fuzzy techniques are used for finding a modern method for the automated detection of consistent block deformation. The classical deformation analysis results in a set of displacement vectors for the observed points in the landslide area. The task is now to find groups of points with a similar pattern of movement so that the boundaries between these blocks can be identified. Two different types of parameters are used here to do the block separation. Firstly, geodetic influence factors are determined to assess the state of deformation within this block. But for an automated block detection these indicators are not sufficient. Hence a second group of parameters is used. The human way of assessing a graph of displacement vectors is imitated by finding displacement vectors with similar direction and length, which are called "visual" influence factors here. Fuzzy systems, which are the suitable tools for imitating the human way of thinking, are used for the analysis of both groups of indicators mentioned above. A deeper insight into these parameters is given in the next sections.
classification of landslides
,,,
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I
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I
o _1
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planning
V
of monitoring concepts for alert systems
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Fig. 4 Integrated workflow for integrated landslide hazard management.
An over-determined affine coordinate transformation is used to assess the movement of the observed points between two subsequent epochs of measurements. This means that the coordinates x - (x, y ) r of the points of epoch n are mapped
3 Detection of taking-off-domains based on a Fuzzy System
onto the coordinates points of epoch n+ 1"
x'-(x',y') v
of the same
x'-F.x+t One task within OASYS is that from classical geodetic monitoring measurements, the boundaries between the stable and the unstable, or between unstable areas, moving with different velocities in different directions have to be found, so that in the
(1)
where: x - (x, y)~ ... coordinates of epoch n
oordin.tes of epoch deformation)
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H. Kahmen. A. Eichhorn • M. Haberler-Weber
t
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(t,, ty )r ...translational parameters c3x' c3x' -~x @,
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ax
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The indicators used as input parameters in the fuzzy system can be determined by the results of the sequence of affine transformations. A group of points moving in the same direction (assuming that they are lying on one common block) is characterized by, e.g., a small standard deviation of unit weight So within an over-determined affine coordinate transformation. On the other hand, if points of different blocks are considered simultaneously the indicating parameters are significantly larger. In the next step, the following strain parameters are derived from the affine transformation parameters: the infinitesimal strain components exx, eyy (rate of change of length per unit length in the direction of the x-axis and y-axis respectively), exy (= eyx, rate of shear strain) and the derived rotation angle co (see Haberler, 2005). Since these strain parameters are dependent on the coordinate system it is better to transform them into the principal strain axes system, represented by the strain ellipse (Tissot indicatrix). The elements of the strain ellipse (the semi-axes el, e2 and the orientation 0 of the maximum strain rate), which fully describe the state of deformation, are calculated from the strain parameters according to the geodetic point error ellipse (for further explanations see, e.g., Welsch et al., 2000). The strain ellipse parameters are the basic geodetic indicators for the block detection.
vectors with similar directions can belong to one common block. As a second indicator, two or more vectors are said to be similar if the lengths of the displacement vectors are similar. The combined analysis of the direction and the length of the vectors gives a clear distinction if points show the same pattern of movement. In addition, the property of "neighbourhood" is determined from a Delaunay triangulation. The example in Fig. 5 shows the modelling of the input variable "direction" in the fuzzy system. If the directions of several vectors under investigation are within a range of approximately 20 gon, they are assessed as "similar" by the fuzzy system. The greater the difference in azimuth, the smaller the property of "similarity" will be according to this human thinking.
not similar, neg
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3.3 Structure of the algorithm 3.2 Visual indicators Another group of indicators for the block detection is based on the human way of thinking, represented by fuzzy systems. The human mind is able to determine blocks of a similar pattern of movement simply by looking at the graph of displacement vectors. Reproducing this intuitive pattern recognition in the fuzzy systems, the similarities of the length and of the direction of the vectors and the property of neighbourhood are basic indicators. According to human thinking it seems to be clear that only
The block detection algorithm was implemented in Matlab ®. With the displacement vectors given by the deformation analysis, the algorithm starts by finding all possible blocks consisting of four neighbouring points (see Fig. 6). A minimum of four points per block is necessary due to the over-determined calculation of the strain analysis. The fuzzy system selects the 'best' set and in an iterative process the best fitting neighbouring points are added to the block until the following fuzzy systems 3 and 4 determine that the block is complete, i.e. that no neighbouring points with a similar pattern of
Chapter 110 • A Multi-Scale Monitoring Concept for Landslide Disaster Mitigation
movement exist. Then the algorithm starts again finding four neighbouring points out of the remaining points.
of stabilizing forces (= shear strength) to destabilizing forces (= shear stress) (Wittke and Erichsen, 2002):
FS= Find all combinations of 4 neighbouring points
_.,
Soil Shear Strength Equilibrium Shear Stress
FS >> 1 (stable)
(1)
ys ~1 (failure)
Fuzzy System 1' choose best block /
Fuzzy System 2 choose I . next best point FFuzzy System 3 assess block properties
yes
oq
Remove last point, terminate block
Fig. 6 Structure of the developed algorithm.
Using this method the task of block detection as a subsequent step following the geodetic deformation analysis can be automated for the first time. Several examples have shown the applicability of this method for local and regional scale landslide areas.
and must be calculated in different parts of the slope. It is very sensitive to external influences (i.e. rain, and other loads) and internal structural changes (Crozier, 1986). The reliability of the allocation can be enhanced by combination of additional quantitative and qualitative expert knowledge. The basic idea for the development of the alert system is the combination of a classical data-based system analysis with a knowledge-based system analysis. The data-based part of this knowledgebased alert system is responsible for prediction and statistical evaluation of the slope's motion using calibrated deformation transfer functions (i.e. finite element models, VOLTERRA-Series or neural networks). The integration of additional sources of hybrid (expert) knowledge and the establishment of automated decision processes is realized by the knowledge-based part.
4.1 Architecture of the alert system 4 Concept of a knowledge-based alert system Impact and risk assessment in the (possible) landslide area primarily requires the definition and reliable separation of different kinematic / geomechanical conditions of the slope. In OASYS five decision levels are defined to evaluate the current stability status, and to take adequate measures for instrumentation, monitoring and alerting (TU Braunschweig et al., 2004): Normal operation ~ Low Margin Operation Warning ~ Emergency ~ Post Mortem To provide suitable indicators for allocation of the different levels is one major goal of the analysis of the landslide process and the task of an alert system. A typical numerical value is represented by the factor of safety FS which is defined as the ratio
In Fig. 7 the architecture of the alert system is shown. In the configuration phase the knowledgebased decision is made, to determine which deformation models and observation designs are suitable for the quantification of the present state of the landslide. The knowledge base must include, e.g., available instrumentation and measurement results from preliminary investigations, possible measuring rates, economic restrictions and accessible additional expert knowledge (geophysics, soil mechanics, etc.). The selected deformation models can be descriptive or causative, static or dynamic, respectively parametric or non-parametric. The identification of the deformation process is realized using geodetic and geotechnical measurements. This calibration step is a precondition for the close-to-reality prediction of the progress of the landslide.
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Fig. 7 Architecture of the knowledge based alert system.
The calculated predictions of the slide's motion are used as one main input for the knowledge acquisition component of the expert system. The system has to decide whether the predicted progress will conform with reality or not (verification step). Considering additional hybrid knowledge, the verification is not only based on incoming measurements (classical procedure) but also on expert prognoses. It enables the evaluation of the predictions at an early stage (preverification). If the predictions are identified as not conforming with reality a feedback to the system's configuration is induced for modification / new adaptation of the deformation models (e.g. static to dynamic). In addition this may be an efficient indicator for a significant change in the slide's motion. The predictions are used as input for the first dec&ion step in order to decide whether to change the alert level (decision level) or not. Additional quantitative or qualitative expert knowledge (excluding the predictions) is used to define the
second dec&ion step. The final dec&ion is made combining the results from the first and the second decision steps. In the case of a change of the alert level the verification component of the alert system will indicate if an additional modification (and identification) of the deformation models is required. In this concept the integration of suitable deformation models into the data-based system analysis, and its calibration to deformation predictors, must be the first principal task.
4.2 Main task" parametric identification of structural landslide models Structural models of the landslide, i.e. FLAC-2D/3D (Kampfer, 2005), SLOPE/W (Geoslope International, 2005) or PFC2D/3D (HcItaska Company, 2005), offer the most comprehensive possibilities for the analysis and prediction of critical states of the slope. Representing its inner structure, the stress distribution indicators such as the FS (1) can be calculated. Nevertheless one main problem is the model adaptation to reality. A suitable tool for the adaptation of structural models (= parametric identification) adaptive KALMAY-filtering techniques (see Fig. 8, Gelb, 1974) are suggested. In contrast to common trial and error-methods KALMAY-filtering offers the optimal combination of theoretical model and empirical measurements. In Fig. 8 the basic principle for the identification of a structural model is shown. Observed influence quantities u (i.e. changes in water regime, mechanical loads, etc.) at time t~ are used as input to calculate the prediction x of the geometrical and structural state of the slope at time tk+l. Together with geodetic and geotechnical measurements L and the linearized model (represented by the transition matrix T, coefficient matrix of correcting B and disturbing variables S) the least-squares estimation x is calculated at tk+l for model update. The recursive adaptation algorithm is finished when predictions and measurements finally show no more significant deviations. This is statistically proven by testing the filter innovations d.
Chapter 110 • A Multi-Scale Monitoring Conceptfor Landslide Disaster Mitigation
Structural model of landslide
Xk+1
Structural parameters
Kinematic State
Xp,~
x,
Prediction
Geometrical and geotechnical observations
Observed influences
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11
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Fig. 8 Parametric identification of a landslide model with adaptive KALMAN-filtering.
The statistical evaluation of the KALMAN-filter's innovation is an efficient tool for detecting contradictions between the predictions and observations. The innovations are sensitive to changes in the slope's structure and motion, and can be used as an additional indicator for the change of alert levels in the data-based component of the alert system.
References Almeida-Teixeira M.E. et al. (1991). Natural hazards and engineering geology: Prevention and control of landslides and other mass movements. Proceedings of the European School of Climatology and Natural Hazards course held in Lisbon from 28 March to 5 April 1990. Crozier M.J. (1986). Landslides- Causes, Consequences and Environment. Croom Helm, London. Cui Z.Q. (1999). Geology and geomechanics of the Three Gorges Projects. Bureau of Investigations and survey, Changjiang Water Resources Commission, MWR. Gelb A. (1974). Applied Optimal Estimation. The M.I.T. Press, Cambridge London. Geoslope International (2005). www.geo-slope.com, last access 07/2005. Haberler M. (2003). A fuzzy system for the assessment of landslide monitoring data. Osterreichische Zeitschrifl ftir Vermessung und Geoinformation, Heft 1/2003, 91. Jahrgang, 92-98. Haberler M. (2005). Einsatz von Fuzzy Methoden zur Detektion konsistenter Punktbewegungen. Geowissenschaflliche Mitteilungen, Schriftenreihe der Studienrichtung Vermessungswesen und Geoinformation der Technischen Universitfit Wien, Nr. 73. HcItaska Company (2005). www.itascacg.com/pfc.html, last access 07/2005.
Heine K. (1999). Beschreibung von Deformationsprozessen durch Volterra- und Fuzzy-Modelle sowie Neuronale Netze, Deutsche Geod~itische Kommission, Reihe C, Heft Nr. 516, Verlag der Bayerischen Akademie der Wissenschaften, Mtinchen. KampferG. (2005). Zusammenfassung der Ergebnisse der geotechnischen Modellierung der RWE Testb6schung. Report of the project OASYS. Savvaidis P., K. Lakakis, A. Zeimpekis (2001). Monitoring ground displacements at a national highway project: The case of "Egnatia Odes" in Greece. Proceedings of the IAG Workshop on Monitoring of Constructions and Local Geodynamic Process, Wuhan, China, 2001. Theilen-Willige B. (1998). Seismic hazard localization based on lineament analysis of ERS- and SIR-C- radar-data of the Lake Constance Area and on field check. Proceedings of the European Conference on Synthetic Aperture Radar. 25-27 May 1998 in Friedrichshafen, VDE-Verlag, Berlin, 447-550. TU Braunschweig, TU Vienna, Geodata GmbH (2004). Instrumentation, Data Analysis and Decision Support for Landslide Alarm Systems. Report of the project OASYS. Welsch W., O. Heuneke, H. Kuhlmann (2000). Auswertung geod~itischer lJberwachungsmessungen. Wichmann Verlag, Heidelberg. Wieser A. (2002). Robust and fuzzy techniques for parameter estimation and quality assessment in GPS, Shaker Verlag, Aachen. Wittke W., C. Erichsen (2002). Stability of Rock Slopes. Geotechnical Engineering Handbook, Ernst & Sohn. Zhang Z.L., Huang Q.Y., Chmelina K. (2001). Research on geological landslide problems related to the Three Gorges Project. Proceedings of the IAG Workshop on Monitoring of Constructions and Local Geodynamic Process, Wuhan, China, 2001.
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Chapter 111
High Speed Laser Scanning for Near Real-Time Monitoring of Structural Deformations Hansj6rg Kutterer, Christian Hesse Geodfitisches Institut, Universit/it Hannover, Nienburger Strasse 1, D-30167 Hannover, Germany E-Mail: [kutterer,hesse] @gih.uni-hannover.de
Abstract. Several tasks in structural deformation
monitoring require real-time or near real-time data acquisition methods. In one case, a lock gate is being deformed within a time period of 14 minutes due to the change of the water level inside the lock. In order to accomplish a comprehensive monitoring of the gate's deformations, the high speed laser scanner Z÷F Imager 5003 has been used. The main advantage of this procedure is that object deformations and the corresponding variances can be derived without reflectors and with data acquisition rates of up to 500,000 points per second. The gate was scanned completely at different stages as well as in horizontal profiles at high speed. The analysis of the observed deformations was carried out by using methods from statistics such as Principle Component Analysis, which is well suited for the analysis of the interrelations of multiple data series. For this purpose, the object data was segmented into equally spaced blocks, which were treated as realizations of a time series. In combination with derived statistical measures, it was possible to determine the regions of the object which were mainly affected by deformations. Keywords. Laser scanning, deformation analysis, structural deformation monitoring, lock gate, near real-time, principle component analysis, statistics
1 Introduction Terrestrial laser scanners are new and promising tools for monitoring deformations of artificial and natural objects. At present, there are several types of scanners which can meet different requirements. For example time-of-flight-based laser scanners offer long ranges and can acquire data from widespread objects. Phase-shift scanners offer much higher data rates and are thus the preferred scanner for kinematic tasks or the monitoring of objects in motion. Note that the particular measurement task,
the environmental conditions and other factors such as measurement speed, maximum range, object dimensions and accuracy have to be considered when choosing the most appropriate scanner (Schulz and Ingensand, 2004; B6hler et al., 2003) In any case, three-dimensional (3d) Cartesian coordinates defined in a scanner system are derived from spherical coordinates. The number of scanned points typically numbers up to several million or more (3d point clouds). Hence, the objects are observed quasi-continuously in space and depending on the speed of scanning - with a (very) high temporal resolution. The goal of this study was the derivation of twoand three-dimensional deformation patterns due to varying external forces, such as water pressure, with high temporal and spatial resolution. Here, the changing deformations of a lock gate are of interest when the water level falls or rises. Consequently, the potential of kinematic terrestrial laser scanning for geodetic object monitoring will be assessed. For this study the Leica HDS 4500 scanner was used as it is very convenient for kinematic close range applications. The original equipment manufacturer (OEM) of this scanner is Zoller + Fr6hlich (Z+F), hence it is also known as the Z+F Imager 5003. At present it is one of the fastest scanners on the market with data acquisition rate reaching 500,000 points per second. It has a field of view of 360 ° horizontal and 310 ° vertical with a maximum unique measurement range of 53.5 m. The accuracy of the distance measurement is denoted by the manufacturer as being >3 mm at 25 m distance. A more detailed overview of the technical data of the scanner can be obtained from Table 1. Besides its use in 3d mode, it is possible to observe in 2d mode (single horizontal or vertical profiles) by disabling the low speed motor that is responsible for a rotation around the vertical axis. Then a
Chapter 111
• High-Speed
frequency of 33 Hz can be reached with the noise reduction parameter set to default noise, or a reduced frequency of 12 Hz for the low noise mode. In addition to this it is possible to stop both rotating mirrors in order to realize a one-dimensional distance measurement with up to 500 kHz. This option was not used for this study. Table 1. Technical specifications ofLeica HDS 4500 Parameter Field of view Min/Max range Data acqusition rate
Range accuracy @ 25 m Spot size @ 25 m Min spot spacing @ 25 m Weight
Performance 360 ° (Hz); 310 ° (V) 0.1 m / 5 3 . 5 m Points/s: 500.000; Profiles/s: 33 (default)or 12 (low noise) 9 mm (20% reflectivity); 3 mm (100% reflectivity) 8.5 mm 4.4 m m (Hz); 7.8 mm (V) 13 kg
The paper is organized as follows. In the subsequent section time series from 3d and 2d kinematic terrestrial laser scans observed in order to describe the deformations of a lock gate are presented and discussed. For the analysis, methods from multivariate statistics were used: the derivation of an empirical variance-covariance matrix and its Principal Component Analysis.
2. Lock gate Uelzen I 2.1 Object description
Laser Scanning for near Real-Time Monitoring of Structural Deformations
The lock Uelzen I is located between Hamburg and Hannover in northern Germany. The lock (Fig. 1) was built at the Elbe side channel which connects two major domestic shipping routes in Germany: the river Elbe and the Midland channel. The lock was built to transcend a difference in water level of about 23 m. It has an overall length of 185 m and therefore can accommodate one Euro Class II pusher unit, which has a defined length of 183.6 m. In filled state the lock keeps a water volume of 5 4 , 0 0 0 m 3. When bringing ships downwards, 60% of this water can be reused for filling because of three basins beside the lock. The Geodetic Institute of the University of Hannover carried out five major deformation measurement campaigns at this site within the last 20 years to monitor the deformations induced by the periodic change of the water level inside the lock. Since the completion of the lock its opposing side walls moved continuously away from each other with a magnitude of nearly 15 cm. Because of these deformations two additional campaigns took place in April 2004 (Hesse and Stramm, 2005) and July 2005 at which the deformations of the steel lift gate (Fig. 2) were to be determined. The vertically moving gate has a width of 12 m, a height of 11 m and a thickness of approximately 1.2 m. The water level inside the lock changes from 42 m (low level) to 65 m (high level) to bring ships upwards or downwards. Hence, a water column of 23 m resides behind the gate. The water level has a sink rate of 2.15 m/min and a climb rate of 1.94 m/min. These parameters suggest that a very high temporal resolution of the scanner is needed for measuring deformations at discrete epochs.
Fig. 2" Lock gate Uelzen (width: 12 m, height: 7m above outside water level) from an outside point of view.
2.2 3D scans
Fig. 1; Lock gate Uelzen (width:
12 m, height: 7m above outside water level)from an outside point of view.
In order to obtain a temporal sequence of the deformation patterns of the lock gate between the two extreme water levels (maximum and minimum) the gate was scanned repeatedly with the Leica HDS 4500 in July 2005. The fastest scanning mode
777
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("preview") was used so a reduced point spacing has to be taken into account. This procedure yielded 32 single scans in total with each one consisting of about 8500 points. Each scan took about 21 s. Fig. 3 shows an example of a 3d point cloud of such a scan.
In the next step the matrix elements (i.e. the associated time series of median values) were mapped row by row into new row vectors r (k) starting with the lowest vector N}k). Formally this reads:
r(k) =[l~}k)~k)~k)~k)l
(3)
These vectors were then rearranged as rows of the matrix:
Fr O)
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/r °t
Fig. 3: 3D point cloud of the lock gate
where n denotes the number of epochs and which comprises the time series of all grid elements. The vectors Ni,j contain the time series of median
For all further analysis the point clouds were referred to a regular vertical grid which covers the lock gate more or less completely. Therefore, grid elements (classes) with the size 1 m x 1 m were chosen, which are defined column by column starting at the lower left comer (1,1) and ending at the upper right comer (4,10). Each point of the cloud was uniquely assigned to a grid element. The coordinate components orthogonal to the lock gate plane were considered as deformation values. For each grid element and each scan epoch the median value was calculated. This reduced the data noise significantly since each grid element contained more than 200 single points. The noise level of the original data before the analysis was about + 2cm. For further references about laser scanning of a lock gate see Kopacik and Wunderlich (2004).
values of each grid element. The result of this procedure is shown in Fig. 4. In addition, the respective median values are given in Fig. 5 for a 50 cm × 50 cm grid in the lock gate plane for four selected epochs. Actually, the temporal evolution along the axis 'Epochs' shows the release of the deformation since the initial state reflects the maximum deformation. Some properties are of interest. First, there is a clear linear deformation of about 14 to 16 mm combined with a slight trend (translation) for each grid element from Epoch 1 to Epoch 29. It is caused by the reduction of water load on the surface of the lock gate. Second, beginning with Epochs 27 and 28 but mainly between Epoch 29 and Epoch 30, there is a sudden translation of about 16 to 18 mm which is caused by a movement of the whole gate away from the sealings into its regular position. Finally, the actual noise is on average below 1 mm and the number of irregularities (which might be potential outliers) is zero.
r~
Uelzen from outside," grey value coded by intensity. The light vertical lines are caused by water on the surface.
By this way a temporal sequence (time series) of 32 matrices of the size 4 x 10 such as"
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m41(k im4,1 m4,2 ... m4,10 (k) n (k) = m3 = / m3'1 m3'2 m3'1° m2 /m2,1 m2,2 m2,10 ml [_ml,1 ml,2 ml,lO with
the respective mi,j, i = 1,..., 4, j = 1,..., 10
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median values as elements was
obtained; the index k denotes the respective epoch. Afterwards the matrix elements were referred to the initial deformation state M (1) (maximum water level) by subtracting the values of the first epoch. This yielded: ~(k) = M(k) _ M(1). (2)
Epochs [ ]
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Time series of the deformation of the grid elements representing the lock gate plane aligned row by row from the left to the right (matrix R according to Eq. (4)).
Chapter 111 Epoch 10
•
High-Speed Laser Scanning for near Real-Time Monitoring of Structural Deformations
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Std.-Deviation vs. Principle Component Analysis
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Fig. 5: Temporal slices of the deformation of the grid elements represented the lock gate plane.
For a more refined interpretation a Principal Component Analysis based on the empirical variance-covariance matrix (vcm) of the column vectors ~ i j of the grid elements was computed. The components of this vcm are the empirical covariances of each pair of the time series vectors ~ i j . Hence, the empirical variances are its main diagonal elements.
Fig. 6 shows the eigenvalue spectrum of this vcm. There is one dominant eigenvalue which represents 94 % of the sum of all eigenvalues (trace of the vcm). Looking at the associated principal component (Fig. 6), is it obvious that the principal component (dominant eigenvector) explains most of the actual variations. There is also a clear null space of the matrix which has not been understood yet. It will be analyzed in detail in a further study.
The angle on the right edge of Fig. 7 is of some interest; it is also visible in Fig. 4. This right hand part is connected with the uppermost row of the grid. Obviously the deformations in the centre element and at the edges differ in their magnitude. This could be due to the steel bar construction on the inside of the lock gate.
2.3 2D scans
In a second scenario a large number of horizontal profiles were scanned in May 2004 with the Z+F Imager 5003. Observation of profiles only allows a significantly higher temporal resolution than of complete surfaces. Nearly 7600 epochs were scanned, with each consisting of about 7300 single points. Hence, the corresponding frequency was about 12 profiles/s. The profile points were associated with 22 equidistant classes with a width of 50 cm. As in the 3d case the noise level of the original data was reduced by calculating median values for each class. A further smoothing was gained by a moving median filter in the time domain, i.e. medians for each class using a window length of 151 epochs.
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The following figures show the resulting time series. In Fig. 8 all epochs are represented. As in Fig. 4 there are clear translation and deformation patterns until a sudden change during the final quarter of the epochs. The first three quarters are shown in Fig. 9. The properties are here similar to the ones in the 3d case. The final peak at the centre of the profiles has not been explained yet. Note the similar effect in Fig. 4 which was already referred to in Section 2.2.
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Eigenvalue [ ]
~o
Fig. 10: Eigenvalue spectrum obtained for the empirical Fig. 8: Time series of the deformation of the lock gate profile for
a number of nearly 7600 epochs and equidistant classes along the profile (width." 50 cm)
variance-covariance matrix of the time series of the profile classes in semilogarithmic scaling. Std.-Deviation vs. Principle Component Analysis
x 10-s
A [,nl
K I~.,
s
;/a 2
% .,, ,, 4
6
8
10
12 Class [ ]
14
16
18
20
22
Fig. 11: Empirical standard the principal component (eigenvec-
tor associated to the maximum eigenvalue of the variance-covariance matrix multiplied by the square root of the maximum eigenvalue)
4. Discussion and conclusions
Fig. 9: Same time series as shown in Fig. 8 but without the
values of the sudden translation
Another possibility for smoothing the denoised point data, and obtaining deformation values at constantly spaced grid points, is using spline interpolation (Schfifer et al., 2004). As in the 3d case, an empirical covariance matrix was derived for the time series of the 22 profile classes. Again an eigenvalue decomposition was computed. The obtained eigenvalue spectrum is shown in Fig. 10. One eigenvalue is dominant. It represents about 89 % of the sum of all eigenvalues (trace of the matrix). In contrast to the 3d case all eigenvalues are positive, so there is no null space. The deformation between the centre of the profile and its edges is given in Fig. 11. The empirical standard deviations reflect the variability which was already visible in Fig. 9 and Fig. 10. The amount of deformation on the left side of the lock gate is significantly smaller than for the right side. The
Kinematic terrestrial laser scanning is a new and promising method for the fast observation of surfaces and profiles of artificical and natural objects of local scale. Thus, it is a basis for a refined deformation analysis. The experiences gained and the results derived for the lock gate Uelzen I show clearly that today 3d scans are repeatable within time spans which are significantly below one minute. The quality of the results is high. With standard statistical methods it is already possible to obtain a sub-cm precision. In the case of 2d scans (profiles) even higher frequencies for the repetition of scans are possible. In the example described here a frequency of 12 profiles/s was achieved. The statistical processing of the data showed that the noise could be reduced significantly through averaging of observations. Hence, there is a good possibility to control the noise level by the respective spatial and temporal resolution. Without doubt it is possible to apply more sophisticated statistical methods such as estimation, prediction and filtering, Fourier or wavelet transforms, etc. These topics were not the subject of this study but will be investigated in future.
Chapter 111 • High-Speed Laser Scanning for near Real-Time Monitoring of Structural Deformations
Successful tests with other structures, such as the pylon of a wind energy plant (Hesse et al., 2005), indicate further the great potential of terrestrial laser scanning for geodetic monitoring and deformation analysis - in particular in the kinematic mode. Future work has to consider aspects of the instrumentation such as systematic effects and calibration, of the observation configuration such as the merging of scans, the integration with other (complementary) observation techniques, and efficient and effective statistical analysis.
References B6hler, W., Bordas, V., Marbs, A. (2003). Investigating Laser Scanner Accuracy. The International Archives of Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXIV, Part 5/C15, 696-701. Hesse, C., Stramm, i4. (2004). Deformation Measurements with Laser Scanners- Possibilities and Challenges. International Symposium on Modem Technologies, Education and
Professional Practice in Geodesy and Related Fields, Sofia, Nov. 04-05 2004, pp. 228-240. Kopacik, A., Wunderlich, T. (2004). Usage of Laser Scanning Systems at Hydro-technical Structures. In: Proceedings of the FIG Working Week- The Olympic Spirit in Surveying, Athens, May 22-27 2004, http://www.fig.net/pub/athens/papers/ts23/TS23 4 Kopacik Wunderlich.pdf. Schfifer, T., Weber, T., Kyrinovic, P., Zamecnikova, M. (2004). Deformation Mesaurement using Terrestrial Laser Scanning at the Hydropower Station of Gabcikovo. INGEO 2004 and FIG Regional Central and Eastern European Conference on Engineering Surveying, Bratislava, Slovakia, Nov. 11-13 2004,
http ://www.fig.net/pub/bratislava/papers/ts_O2/ts_O2_schaef er etal.pdf. Schulz, T., Ingensand, H. (2004). Influencing Variables, Precision and Accuracy of Terrestrial Laser Scanners. INGEO 2004 and FIG Regional Central and Eastem European Conference on Engineering Surveying, Bratislava, Slovakia, Nov. 11-13 2004,
http://www.fig.net/pub/bratislava/papers/ts_O2/ts_O2_schulz _ingensand.pdf.
781
Chapter 112
A method for modelling the non-stationary behaviour of structures in deformation analysis H. Neuner Geodetic Institute, University of Hanover, Nienburger Str. 1, 30167 Hanover, Germany
Abstract. The standard description and modelling of the normal behaviour of structures under influence of factors like temperature or tides is done by means of correlation functions and power spectra that assume the condition of stationarity of the observed process. As a consequence of influencing parameters with highly irregular patterns, aging or damage to the structure the measured reaction of the structure may have non-stationary characteristics. Hence these functions give only a coarse description of the object's reaction and more detailed analysis techniques are needed. For these cases a method of data analysis based on the discrete wavelet decomposition and variance homogeneity testing of the wavelet coefficients is applied in this paper. After identifying the frequencies with high energy in a spectral analysis, the signal is decomposed on a scale base in order to analyse the different stationary and non-stationary components separately. The wavelet coefficients of each scale are separated into intervals with homogeneous variance by performing a test based on the cumulative sum of squares. The amplitude and phase shift of the detected frequencies are calculated for each interval in an adjustment step. Finally, using the modelled data the signal is reconstructed by wavelet synthesis. Because the wavelet transform is done without loss of information, an objective evaluation of the modelled data is possible. Continuously recorded tilt measurements at the tower of a wind energy plant were analysed both in the classical manner and with the proposed method, in order to test the latter's performance. It is shown that the new method performs well and lowers the standard deviation of the residuals by 30%, while the standard procedure does not remove the high energy components. Keywords.
Discrete Wavelet Variance homogeneity test.
Transformation,
1 Introduction One of the main goals of structural deformation analysis is to characterise the behaviour of the monitored object by means of a dynamic deformation model. In the standard approach, the causality between influencing factors acting on the object and the deformation signals they are generating is expressed by a linear filtering model. The gain and phase of the transfer function expresses the amplification and delay of the input and thus reflects the object's physical properties. To estimate these two components, correlation or spectral functions are used. This method provides good results, as shown in previous engineering projects (Kuhlmann, 1996; Neuner et al., 2004), if the analysed deformations are slow in relation to the recording rates, and the time series are at least stationary up to order two. That means, all time points must have the same mean and variance, and the covariance must depend only on the interval between the time points (Priestley, 1981). This property is not always fulfilled by the recorded time series because of blunders or effects like non-stationary influences on the structure (e.g. wind, traffic), measuring of various direction-dependent components of the deformation signal, changing of calibration parameters due to automatic calibration, or high-speed recordings with modem sensors. These effects may cause various types of non-stationarity like jumps, spikes or change of variability. To properly account for nonstationarity, other data analysis methods are required. The present paper deals with a method of analysing time series that are periodic, but encounter changes of variance in time. For this kind of signal, the dominant frequencies can be detected by a Fourier Transformation, and the corresponding amplitude and phase can be estimated separately in an adjustment model of the kind: m
X k -X
qt_ Z A i
i .COS(2rcfitk)+B i .sin(2rcfitk)
(1.1)
Chapter112 • A Methodfor Modellingthe Non-StationaryBehaviourof Structuresin DeformationAnalysis
For a correct functional model, one needs to detect the time points at which the statistical properties change, such that the coefficients A~ and B~ are estimated only for data with homogeneous statistical moments. This can be done by using a statistical goodness-of-fit test. The sensitivity of such tests is low if signals with different power and statistical properties overlap. A way to improve the sensitivity is the a priori filtering of the recorded signal in order to separate its different periodic components, test these separately for variance homogeneity, and estimate the corresponding amplitude and phase individually. Once this step is completed, one needs to find an inverting filter operation that should put the signal components together, to regain a unique modelled signal. The Wavelet Transform is most appropriate for accomplishing this task, because of its orthogonality property. The remainder of the paper is organised as follows. The second and third section contain the theoretical background of the proposed analysis procedure. The results obtained by applying this procedure for modelling the oscillations of the tower of a wind energy plant are presented in the fourth section and compared to the ones obtained by the "overall" estimation. Concluding remarks and possible future work are discussed in the last section. 2 The Wavelet Transform time series analysis
method
for
The Wavelet Transform offers the possibility to extract and study local characteristics of the signal at subsequent resolution levels. This can be done in different ways according to the purpose of the data decomposition (B/ini, 2002). The Discrete Wavelet Transform (DWT) is appropriate if further processing of the transformed data is needed, and therefore this paper is focused on this particular analysis method. The DWT is formulated in terms of an orthogonal filter bank and consists of passing the n-valued low frequency signal component u, separated in a previous step j, through a quadrature mirror filter pair, and decimating the components for reasons of energy conservation by retaining every other value:
Uj+I, n -- Z hk-2n " U J, k k Vj+l,n -- Z gk-2n " Uj,k" k
The high-pass filter g is referred to as the wavelet filter and the output signal components v, form the wavelet coefficients. Similarly, the low-pass filter h is referred to as the scaling filter and the output signal components u, form the scaling coefficients. In terms of the wavelet theory, every decomposition level j is referred to by a scale sj=2j-1. To interpret the physical meaning of signal components in the scale sj, the transfer function of the applied cascade of scaling and wavelet filters, denoted by capital letters of the corresponding filters, can be derived from the relations (Percival et al., 2002): j-2
Gj (f) - G(2 j-'. f). ]--[ H(2 k. f) k=0 j-1
Hi(f) - UH{21' •f).
(2.2)
k=0
The first relation is the transfer function of a band-pass filter with nominal passband 1 / 2 j + l - - 1/2j, while the second one represents the transfer function of a low-pass filter with nominal passband 0 - 1/2j+l. Fig. 1 illustrates the gain functions of the 4 th order Daubechies scaling and wavelet filters at the third decomposition level. Because these filters are not ideal, signal components with frequencies in the transition band, as well as those introduced by leakage effects, can occur in the spectrum of the wavelet coefficients. Beside these effects, the wavelet coefficients, generated at each level of the DWT, are mainly signal components with frequencies located in the band-pass of the equivalent filter sequence. Due to downsampling, the respective frequencies will appear in the frequency spectrum of the wavelet coefficients as aliases. Accounting for these facts, it is
31....
2.5
,i
=2[
-g
li
'~//
/ii, 02
nominal passband of thewavelet-filter
0.2
0.3
0.4
o15
frequency(Hz) (2.1)
Fig. 1 Gain function of the third level Daubechies scaling (dashed line) and wavelet-filter (continuous line)
783
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possible to relate dominant frequencies in the spectrum of the wavelet coefficients to frequencies contained in the original series and to derive their corresponding magnitudes. In order to perform the DWT it is necessary to know the coefficients of the filters to be used. Daubechies extended the mathematical constraints imposed on filters of orthogonal perfect reconstruction filter banks with regularity conditions, and obtained a filter family appropriate for decomposing and analysing the signal components according to the goals of the wavelet transform (BriM, 2002). The main properties of the filters are determined by the order of the root f = 0.5 of the equation H(f)=0. By increasing the order of this root the transition bands of the filters represented in Fig. 1 become steeper and thus the frequency localisation improves. On the other hand, the filter length also increases and more coefficients become affected by boundary conditions in the convolution operation. These opposite effects make it necessary to select the filter's structure according to the goals of the data analysis. The DWT performed in this analysis aims to separate the signal components of different frequencies. To do so, good localisation properties in the frequency domain are required. Due to the sufficient spacing between the peaks of dominant frequencies, a Daubechies filter of the fourth order seemed to be a good choice. Because of its orthogonality, the DWT is an energy preserving transform. Thus, without concern about the stationarity properties, the original time series and the resulting wavelet and scaling coefficients have the same information content. This can be expressed by the relation:
j-1
where V and U stand for the series of wavelet and scaling coefficients at scale 2j-1 and J for the maximal decomposition level. Regarding the energy of a stationary process in terms of variability, it can be claimed that the DWT decomposes its variance on a scale-by-scale basis (Percival et al., 2002). This is a useful property that permits the separate analysis of the contributions of each scale to the sample variance. For the kind of non-stationary processes treated in this paper, the property (2.3) makes it possible to follow the buildup of the variance in each scale, and hence to study the change of variance on a scale-by-scale basis.
The time points of variance change in each scale can be identified by using a goodness-of-fit statistical test, like the one presented in the following section.
3 Testing for variance homogeneity To test the variance of time series for homogeneity, several procedures are available, but some of them aim to detect a single variance change, or refer to particular models. A more general approach comes from Inclan and Tiao (1994), who investigated the detection of multiple variance changes of a sequence of uncorrelated normally-distributed random variables ak, with mean 0 and variance c~, k = 1..N, based on their cumulative sum of squares Ck -- ~ i k l a i2 • Because Ck is a positive, monotonically increasing function with a slowly varying slope, a sharp detection of a variance change point is difficult. Hence the test value is based on the normalised and centred cumulative sum of squares, defined as: (3 1)
Dk = C k _ k
CN
N'
In the case of variance homogeneity, Dk oscillates around the horizontal zero axis and changes its slope drastically (even in sign) at the change points of the variance. The search for variance change points is done sequentially, in points k* where ]Dk*] attains its maximum. To evaluate the significance of Dk, a distribution relation of the measure defined in (3.1) has to be found. If Dk, exceeds a certain percentage quantile, than a change of variance occurs and the identified point k* is a first evaluation of the change point. For a certain k, the F-statistic could be used to test the equality between the variances c~02and c~ 2 of the samples ai i=l..k and aj j=k+l..N respectively. Expressing the ratio of the two variances in terms of the measure Dk, the testing of rio: C~o2 = c~12leads to:
k.(N-k) (1--FNk.k., c~) / P Dk-~--N k+(N ,~-~----z--~--~--,,1 k)-FNk ~ =l-c~ (3.2) To correctly estimate the quantile and the degrees of freedom, the exact location of k must be known, while looking for max IDkl requires the determination of the location of the change point. The asymptotic distribution of Dk can be determined by using convergence properties of probability measures presented in Billingsley (1999). Under the assumption that the sequence of random variables ak have ho-
Chapter 112 • A Method for Modelling the Non-Stationary Behaviour of Structures in Deformation Analysis
mogeneous variances c~, the random variables ~k build as ~k -- ak2 -- ~a2 with E(~k) -- 0 and Var (~k) -2~S~, can be used to form a function XN(t) of the type:
Table 1. Quantiles of the probability distribution of the Brownian bridge function W°
1 LNtJ ( N . t - k N . t J ) (3.3) XN(t)- ,]-2(s~x[-N ~ ~ + x/~rs~x/-N '~N-tJ+l According to Donsker's theorem (Billingsley, 1999, p. 90) this function converges weakly to the Wiener probability measure W, used to describe the path of a particle in Brownian motion. Based on the Wiener measure, a Brownian bridge function may be defined as W( = W t - t W l . Its sample version is obtained by introducing (3.3): 1 (k k.~i;+ W°(t) = XN(t)-t'X'(t) = x/-2~x/~ i ~ i --N- i l
@
(N.t-~N.tJ) ~V/~ 2~-N " ~N.t~+l
(3.4)
Reversing the prior change of variables, one gets the normalised and centred cumulative sum of squares in the first term:
N2/ / W°(t)=,4/2~77/~ .~a k
~ a 2k k ~a,2, N i1 i=l
-t-
(N.t-LN.tJ)
N i~a2k (N.t-[N.tJ) %/-20"~ %/-N-" Dk @ -~/-2-~a~--~N/r--N - " ~N.tJ+l
(3.5)
For large sample numbers the second summand vanishes, and a concise relation between the Brownian bridge and the cumulative sum of squares follows:
W'~ (t) - ~-~ • Dk
(3.6)
The distribution relation of the Brownian bridge function is introduced in Billingsley (1999): P(maxlW([ < b) - 1 + 2 ~ (_1) k -2k2b2 k=l e
(3.7)
The quantiles were determined numerically with a precision of 10 .6 and are presented for the most common confidence levels in Table 1. For the data analysis presented in the next section, a confidence level of 95% was chosen. The method presented so far offers the possibility to examine a sample of zero-mean uncorrelated
normally-distributed random variables for variance homogeneity. The test value based on their cumulative sum of squares fulfills under the null hypothesis of variance equality, the distributional relation (3.7). If the test value exceeds the quantile corresponding to a prior fixed confidence level, then k* corresponding to max(Dk) is marked as a possible change point. To check for further variance change points the samples are divided into subseries [al..ak,] and [ak*+l..aN], and the procedure is applied again for each of them. This process is iterated until the test value falls below the quantile for every subset of variables [atl..at2]. Thus, a set of variance change point candidates is obtained. To avoid overestimation, the algorithm includes a further step, in which the number and location of the change points is refined. Therefore the candidates are included in ascending order in a vector c, whose first and last elements are 0 and N respectively. The change point detection method is applied successively on each interval c[i- 1]+l..c[i+ 1], aiming to detect c[i]. If a new change point location occurs, it will be added to the vector c and considered in a following iteration step. Similarly if the change point is not confirmed it will be removed from c. This step is iterated until the number of points remains unchanged and their location varies below a certain level. For the data analysis in the following section a maximum modification of two positions was allowed. Because the convergence of this step has not been proven mathematically, it is advisable to set a maximum number of iterations. Before using this method for testing the homogeneity of the variance of the wavelet coefficients in each scale, one needs to make sure that the wavelet coefficients match the stochastic properties imposed on the sample variables ak. While the expectation value of the wavelet coefficients is always zero due to the structure of the high-pass filter, the condition on stochastic independency requires further attention. For long- and short-memory processes the decorrelating effect of the wavelet transform was pointed out by Whitcher (1998). As presented earlier, this effect does not hold in the case of periodic signals. A way to account for existing correlations is
785
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H. Neuner
to replace in the test value (3.6) the sample number N by the equivalent degrees of freedom Nedf. If the samples of the analysed time series are normallydistributed the squares of the wavelet coefficients will follow a )(,2 distribution and the equivalent degrees of freedom can be calculated by (Priestley, ]98]):
Nedf =
o¢
(3.8)
2 c2 (k) k=-oe vv
where Cvv denotes the autocovariance function and (~v2 the variance of the wavelet coefficients unaffected by the boundary conditions. With this modification of the test value (3.6), the test procedure based on the cumulative sum of squares can be used to analyse in each scale the build-up of the variance and hence to set the intervals on which separate estimation of the parameters in the model (1.1) is necessary.
4 Modelling the non-stationary behaviour of the pylon of a wind energy turbine The wind energy sector owes its continuous development to wind turbine plants with ever larger dimensions, and to the improved technique of acquiring and transforming the mechanic energy into electric power. The increased financial investment necessary for the use of modem technology, coupled with the placement of this structures in relatively inaccessible locations, makes the monitoring of their stability particularly relevant. The static properties of the wind energy turbine distinguish themselves from those of other structures by their ability to resist very high short-term loads, such as those generated by strong wind squalls. This is why the monitoring activity aims more at the detection of material fatigue than at describing the effect of an unusual event. This can be done by continuously measuring the oscillations of the turbine's components at established epochs, and by assessing the change of dominant frequencies and their corresponding amplitude. Hence, the precise estimation of the oscillation's amplitudes is one of the most important tasks of monitoring wind energy turbines. A project performed at the Geodetic Institute Hanover had the main goal to point out possible contributions of engineering geodesy to the monitoring of a wind energy turbine. The studied object
was a wind power plant of the type Tacke 1.5s with variable rotation speed and nominal output of 1.5 MW. To achieve an optimal power level at each wind condition the nacelle rotates following the wind's direction and the blade's pitch can be varied. At high wind speeds the variable pitch prevents the build-up of extreme loadings and throttles, maintaining the rotation speed at a maximal constant velocity of 21 rpm. The project was focused on the determination of the behaviour of the tower, which consists of a 5 m high concrete base and a steel pylon of 77 m. The pylon comprises 3 segments with heights of 25.9 m, 25.9 m and 25.0 m, joined by circular flange connections. A platform is situated inside the pylon at each connection level between the segments. The data were collected over a period of ten days. Due to the variation of the wind speed it was possible to analyse the oscillating behaviour for rotation speeds between 0 and 20.1 rpm. The monitoring was performed with GPS, which was mounted on the top of the nacelle, and with inclinometers, set up on the three platforms and on the base. Only the inclinometer measurements will be treated in this paper because they have the highest sampling rate. However effects similar to those presented in this paper have been detected also from the GPS data. The tilts were measured in one direction only, in order to achieve maximal sampling rates. The directions of the various sensors are roughly, but not exactly, the same. On the first and third platform Rotlevel inclinometers were installed with a maximal recording rate of 1 Hz. At the second level, a Schaevitz inclinometer was used, with a recording rate that was restricted by the attached A/D converter at 6.1 Hz. Because these data have the highest sampling rate, the presentation of the modelling results refers mainly to them.
The analysed time series contains 14671 samples, which correspond to a period of 40.08 minutes. During this period the turbine had an almost constant power output of 110 kW and a rotor velocity of 12 rpm. The spectral analysis was performed only on measurements recorded over periods with nearly constant wind and functioning conditions in order to give a description of the structure's behaviour for distinct operation states. This analysed period is one of the longest in the project which meets these requirements. The dominant frequencies in the spectrum are visible in Fig. 2.
Chapter 112 • A Method for Modelling the Non-Stationary Behaviour of Structures in Deformation Analysis
0.01
0.01
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0.009
0.008
/
0.007 g
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'
,
i
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'.
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ii
.... "~.?,
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_
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Fig. 2 Amplitude spectra of the time series.
Fig. 3 Amplitude spectra of the residuals of the overall model.
The leftmost peak labelled l p corresponds to the rotation velocity of 0.2 Hz. f0 and fl denote the first and second eigenfrequency of the pylon respectively. The frequency 3p is attributable to the blade frequency and hence is three times l p. The other higher dominant frequencies are harmonics of the blade frequency. Some aspects can be noticed from the spectrum. The first one is the high amplitude resulting from the overlapping of the second eigenfrequency of the pylon with the fourth harmonic of the blade frequency. More important for the estimation of the amplitudes is the energy spreading over neighbouring frequencies in the region corresponding to the second harmonics of the blade frequency. For these only the frequency with the maximal amplitude in the group was introduced in the adjustment model. In addition to these seven frequencies, further peaks occur at other frequencies (e.g. 0.42 Hz, 1.52 Hz or 1.92 Hz) for which no physical interpretation could be given, nor repeatability established. Hence, they were treated as local effects and were not further considered in the adjustment model. If the amplitudes and phases of the seven dominant frequencies are estimated in the adjustment model (1.1) using the data of the entire time series, one expects white noise in the residual series. The residuals spectra are shown in Fig. 3. For comparison purposes the ordinate axes are scaled equally. The diminishing of the magnitude of the peaks is clearly observable, but the leftover energy visible primarily in the eigenfrequencies indicates that further improvement of the model can still be attained. The insufficient model accuracy could come either from small variations of the frequencies, or from changes of the corresponding amplitudes in time.
To account for variation of the frequencies one can introduce further neighbouring frequencies in the model. This is justified by the small variations of the wind speed only in the case of the eigenfrequencies. For rotation-induced frequencies this action could cause an overfitting of the model, because during the analysed time period the rotation velocity and pitch angle remain unchanged and thus new introduced frequencies cannot be justified from a physical point of view. Variations of the amplitudes could be caused by the rotation of the nacelle and by the changing of the wind velocity. They express themselves in a change of the variance. A useful method to detect variance change points is the iterative cumulative sum of squares algorithm presented in the third section. If the test is applied directly on the data of the original time series, the test statistic does not indicate any variance change. But this procedure is rather insensitive because the larger variance on some frequencies might cover effects occurring on frequencies with lesser variance. Additionally, the nonstationary effect cannot be attributed to a certain frequency, which makes it hard to interpret. To overcome these disadvantages the signal was decomposed by a Discrete Wavelet Transform using a Daubechies filter of the fourth order. After the transformation the signal components corresponding to the scale's pass-band were obtained. The distribution of the seven dominant frequencies over the scales is presented, together with the nominal passbands in Table 2. To project all dominant frequencies onto the corresponding wavelet scales, four decomposition levels were necessary. Some of the frequencies, like f0 and 9p, can appear in more than one scale due to the FIR nature of the filters.
787
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H. Neuner
2. Distribution of the dominant frequency components over scales Table
Scale Nominal frequency band (Hz) Contained frequencies
20 2] 1.530,763.05 1,53 15p, 9p, 6p, 9p 12p + fl
22 0,380,76 3p, f0
0.015
alias of fO at the fourth level
23 0,190,38 f0 +lp
/
0.01
fO at the third level 0.005
The amplitudes corresponding to these frequencies were estimated in both scales on the intervals of homogeneous variance. Their separate processing improves the reliability of the estimates, though it is an increased computational burden. In the spectra of the wavelet coefficients the peaks do not appear at the same frequencies as in the original time series, but on aliases with respect to the lower end of the nominal pass-band. This effect occurs due to the downsampling made at each level of the decomposition. In the adjustment model (1.1) based on the wavelet coefficients the alias frequencies were used instead of the original ones. But not all frequencies occurring in a scale's spectra must be interpreted as aliases. Frequencies in the transition band of the filters below the nominal lower end appear as real ones. This situation occurred for f0 and is illustrated in Fig. 4. In order to follow and understand the characteristics exposed by the spectra of the wavelet coefficients of a certain scale, not only the ideal pass-band but also the gain of the applied filter sequence must be derived and analysed carefully. The variance homogeneity test was applied to each of the four series of wavelet coefficients. At each scale the test indicated several points of variance change. The maximum number of change points was 14 and occurred in the second scale. This can be explained by irregularities on the 6p frequency that already showed up in Figures 2 and 3. Because this frequency contributes little to the total energy budget, the following presentations focuses on effects related to the first eigenfrequency. In each scale containing f0, seven variance change points were identified. Thus, the idea that effects on this frequency cause non-stationarity seems justified. Fig. 5 illustrates the test value (3.6) based on the centred cumulative sum of squares of the wavelet coefficients of the third and fourth level. Vertical dashed lines indicate the variance change points. To compare visually the respective homogeneity intervals, the graphic of the fourth de-
0.4 0 Frequency (Hz)
0.2
0.1
0.2
Fig. 4 Peaks corresponding to f0 at the third (left) and at the fourth (right) decomposition level.
composition level is scaled by two. As can be noticed, only the last three intervals are equal, clearly indicating non-stationary effects on f0. At the other intervals the additional effects on l p and 3p lead to different variance homogeneity domains. This result confirms the justification to decompose the signal into frequency components. It also shows, however, that an even finer segmentation should be aimed at in order to gain good insight on the effects associated with each frequency. One way to accomplish this task is to use the Wavelet Packet Transform, which further decomposes each series of wavelet coefficients contained in a scale. The amplitudes of the f0 oscillation estimated on intervals of homogeneous variance are presented in Table 3. The amplitudes in brackets are obtained on the corresponding intervals of the third level. The good agreement between them confirms the non-
4
,
,
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1000
4
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I
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looo
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2000
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Fig. 5 Test value (3.6) for the wavelets coefficients of the third (top) and fourth (bottom) scale.
Chapter
112 • A
Method for Modelling the Non-Stationary Behaviour of Structures in Deformation Analysis 0.01
Table 3. Amplitudes of the f0 oscillation estimated from wavelet coefficients of the fourth (third) level.
0.009 0.008
Interval 1 Amplitude 40 (cc)
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3 54
4 5 192 49
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Fig, 4 The effects of relative tropos~te tic delay on the point displacements. These results suggested that for GPS volcano deformation monitoring surveys requiring a relatively high accuracy, the relative tropospheric delay has to be estimated.
3.7 Multipath in a Volcanic Environment GPS observations in volcanic environment are also prone to multipath, as illustrated in Figure 5. The GPS signals can be reflected either by cliffs, caldera rim, surface of crater lake, volcano flank, or other reflective objects in the area.
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-6
.4
similar in other years. In other words, Figures 3a and 3b show the errors that have been introduced into the estimates of GPS or VLBI station heights that have been obtained previously when using the NMF based on a common assumption for the global weather.
-2
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Fig. 3a. GPS station height changes (in mm) simulated from ERA40 data for January 2001 when using VMF1 instead of NMF. From these simulations, large positive station height changes (>10 mm) can be expected for Antarctica and around Japan and negative height changes can be expected for the northern part of Europe. Fig. 3b. GPS station height changes (in mm) simulated from ERA40 data for July 2001 when using VMF1 instead of NMF. In July 2001, there are only relatively small height changes which can be expected in other years, too. Fig. 3e. Station height differences from GPS analysis using either NMF or VMF1 for January 2005. Black bars indicate positive height changes, grey bars negative height changes. It is evident that these analysis data confirm the simulations from ERA40 data for January 2001 (Figure 3a), i.e. positive station height changes can be found in the southern hemisphere and around Japan, and negative station height changes occur at stations in northern Europe, the western part of Canada, and Alaska. Fig. 3d. Station height differences from GPS analysis using either NMF or VMF1 for July 2005. Clearly, these analyses confirm the simulations from ERA40 data for July 2001 (Figure 3b), i.e. the estimated station height changes are moderate compared to January (see Figure 3c).
3 Vienna Mapping Functions in the GAMIT software
A global network of more than 100 GPS stations was analysed with the software package GAMIT software Version 10.21 (King and Bock, 2005) applying first the NMF and then the VMF1 mapping functions. We analysed a full year of global observations from July 2004 until July
2005, producing a fiducial-free global network for each day analysed. The elevation cutoff angle was set to 7 ° and no downweighting of low observations was applied. Atmospheric pressure loading (tidal and non-tidal) (Tregoning and van Dam, 2005) was applied along with ocean tide loading and the IERS2003 solid Earth tide model (IERS Conventions 2003). We estimated satellite orbital parameters, station coordinates, zenith tropospheric delay parameters every 2 hours, and
839
840
J. B o e h m • P. J. M e n d e s Cerveira • H. Schuh • P. T r e g o n i n g
and Figures 3b/3d). This confirms that the NMF has temporal deficiencies, with a maximum around January, especially at high southern latitudes, for Japan, the northern part of Europe, the western part of Canada, and Alaska.
resolved ambiguities where possible. We used-60 sites to transform the fiducial-free networks into the ITRF2000 by estimating 6-parameter transformations (3 rotations, 3 translations) (Herring 2005). For the investigations described below the time series of estimated station heights at 133 sites were used. Each of these sites has more than 300 daily height estimates. The site distribution is shown in Figure 3c. Amplitudes A and phases doyo (in terms of day of year) of annual periodic signals were estimated by the method of least-squares for all station heights from the NMF and VMF1 time series (see Figure 2). Then these sinusoidal functions (Equation 2) were used to calculate the station height differences on 1 January 2005 and 1 July 2005, respectively (Figures 3c and 3d). h - A . sin l, doy - doy o . 2 rc)
Figure 4 shows the amplitudes and phases for all 133 GPS stations. (Figure 5 shows amplitudes and phases in Europe for clarity.) Generally, the VMF 1 reduced the amplitudes of annual variations on around 50% of sites. However, there is a systematic improvement (reduction of amplitude larger than 5 mm, see Figure 2) at sites situated below 45 ° S, indicating deficiencies of the NMF at higher southern latitudes. The agreement between the amplitudes and phases when changing from NMF to VMF1 is rather good; however, at some stations, especially in the southern hemisphere and in northern Europe, large discrepancies occur. This may be due to the short time series that has been used in this analysis (only one year). If significant, these changes in amplitudes of annual signals might influence the determination of normal modes of the Earth according to Blewitt (2003).
(2)
365
A comparison of the estimated height differences from GPS with those predicted from the NWMs shows a very high correlation (cf. Figures 3a/3c
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Chapter 1 1 8
The Impact of Mapping Functions for t h e
•
Neutral
Based on Numerical Weather Models in GPS Data Analysis
Atmosphere
deviation of the daily station heights with regard to the annual signal is smaller for 117 of the 133 stations, and the average relative improvement is about 6%. Thus, using the VMF1 not only changes the terrestrial reference frame but it also improves considerably the precision of the GPS analysis. The progression from the old N M F to the new mapping functions based on N W M s influences the terrestrial reference frame by changing the heights of some stations - in particular, in Japan and in some regions of the northern hemisphere (Figure 7). Thus, there will be a distortion of the whole frame and rather likely a general shift along the zaxis. As radio-wave techniques play an important role in the realization of the International Terrestrial Reference Frame (ITRF), a significant influence on the next ITRF can be expected if weather-based mapping functions are used in the analysis of the GPS and VLBI observations.
Fig. 5. Amplitudes A and phases doyo (see Equation 2) of annual variation in the station height time series determined from GPS analyses using NMF or VMF1 in Europe. The grey bars correspond to NMF, the black bars to VMF1. The phase angles doyo are counted from north (January) to east (April). The standard respect to decreases for compared to
The results presented here are derived from analyses where no elevation-dependent weighting of the observations has been performed. Very similar results are obtained when such weighting is used, although the influence of the more accurate tropospheric mapping functions is reduced.
deviation of the station heights with the sinusoidal functions clearly almost all stations using the VMF1 N M F (Figure 6). The standard
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:30
Chapter 119 • Validation of Improved Atmospheric Refraction Models for Satellite Laser Ranging (SLR)
5 Conclusions The current accuracy of the new zenith delay model, FCULzd has sub-millimeter differences when compared with ray-tracing of radiosonde data at 180 stations for a full year [Mendes and Pavlis, 2004]. In our results (Figure 2) we see that all three methods, ART, ERT and NRT have sub-millimeter mean biases in the zenith with RMS values not exceeding 1 mm. We have found similar values at other stations and during different times of the year, using a wide range of wavelengths from 0.355 1.064 /~m. At lower elevations, the largest biases occur due to errors in the mapping function, FCULa which has a two-year average RMS of 7 mm (model minus ray-tracing through radiosonde data) at 10°elevation [Mendes et al., 2002]. Our ART results show sub-centimeter biases with the model at 10 ° elevation at all stations, but with a large scatter about the mean. ERT results also show subcentimeter biases, but only at European stations such as Herstmonceux, Graz and Zimmerwald. We see differences above the centimeter level at other stations in Australia and North America. NRT on the other hand has sub-centimeter biases at all North American sites, but has biases at the 12 mm level at all other stations.
from l0 ° to 20 ° elevation, and are at the submillimeter level in the zenith direction. By raytracing at 12 different azimuth angles around stations with varying climates and in mountainous regions, we found noticeable gradient delay differences in each azimuth direction. This was expected since horizontal gradients are largely inhomogeneous and the delay should be dependent on azimuth angle as a result. In conclusion, our study addresses the validation of the latest atmospheric delay model as well as the contribution of horizontal refractivity gradients to the computation of the total atmospheric delay. In order to accomplish this we have developed a robust 3-d ray-tracing program with input atmospheric fields from AIRS, ECMWF and NCEP that will calculate the total atmospheric delay including the contribution from horizontal gradients at selected azimuths, elevation angles and at any given location on the globe.
Acknowledgments.
We
gratefully
acknowledge the support from NASA's Cooperative Agreement with JCET NCC5-339 and NGA's Grant NURI NMA201-01-BAA-2002. ECMWF
for
providing
us
We also thank
with
data
during
February and August 2004 in order to make
Although we only have made comparisons at 12 globally distributed stations and for two months during 2004, we have found that FCULzd does perform well when compared with our ray-tracing methods and for a wide range of wavelengths. However, the use of a mapping function at low elevations still produces errors at sub-centimeter levels.
References
We have found that the NS and EW refractivity gradients are affected significantly by changes in climate from summer to winter seasons at the majority of SLR stations. Sites situated in mountainous regions had larger horizontal pressure gradients while a station's proximity to a large body of water resulted in larger horizontal temperature gradients. No significant non-hydrostatic gradients were found, with maximum wet delays only reaching a few tenths of a millimeter at stations such as Greenbelt. We have found that NS horizontal gradient delays can have month-long average values of up to 6 mm at 10 ° elevation. Gradient delays decrease by approximately half
Ciddor, P.E., (1996), Refractive index of air: New equations for the visible and near infrared, Applied @tics, 35, No. 9, pp. 1566-1573. Ciddor, P.E. and R. J. Hill, (1999), Refractive index of air. 2. Group index, Applied @tics, 38, pp. 1663-1667. Chen, G.E. and T.A. Herring, (1997), Effects of atmospheric azimuthal asymmetry on the analysis of space geodetic data, J. Geophys. Res., 102, No. B9, pp. 20,489-20,502. MacMillan, D.S. and C. Ma, (1997), Atmospheric gradients and the VLBI terrestrial and celestial reference frames, J. Geophys. Res.,24, No.4, pp. 453-456.
comparisons with AIRS results, and also Scott Hannon
from
UMBC
for
processing
and
distributing the data to us.
851
852
G. Hulley. E. C. Pavlis. V. B. Mendes
MacMillan, D.S., (1995), Atmospheric gradients from very long baseline interferometry observations, Geophys. Res. Lett., 22, 1041-1044. Marini, J.W., and C.W. Murray, (1973), Correction of laser range tracking data for atmospheric refraction at elevations above 10 degrees, NASA Rep. X-591-73-351, Goddard Space Flight Cert., Greenbelt, MD.
Mendes, V. B., G. Prates, E. C. Pavlis, D. E. Pavlis, and R. B. Langley, (2002), Improved Mapping Functions for Atmospheric Refraction Correction in SLR, Geophys. Res. Lett., 29(10), 1414, doi: 10.1029/2001GL014394. Mendes, V.B., and E. C. Pavlis, (2004), HighAccuracy Zenith Delay Prediction at Optical Wavelengths, Geophys. Res. Lett., 31, L14602, doi: 10.1029/2004GL020308.
Chapter 120
Correlation Analyses of Horizontal Gradients of Atmospheric Wet Delay versus Wind Direction and Velocity Torao TANAKA Department of Environmental Science and Technology Meijo University, 1-501 Shiogamaguchi, Tenpaku-ku, Nagoya 468-8502, Japan
Abstract.
Total water vapor contents were
phenomenon found by Mousa was real and not
monitored along lines of sight at such low
due to instrumental origin, setting of instruments
elevation angles as 10 and 15 degrees with two
nor
water vapor radiometers, WVR1100 TM, WVR05
coefficients used for data transformation from
and 06. They were installed in two directions of
observed brightness temperatures with Water
improper
determination
of
retrieval
N-S and E-W in Uji city, southwest Japan in the
Vapor Radiometers, WVR1100 TM WVR05 and 06
period from 1997 to 99. Results show that
produced by Radiometrics Corp., to the excess
differences of wet delays between N and S
path delay of micro waves, namely wet delay
directions, which correspond to the gradient of
(hereafter WD) (Tanaka, 2003).
wet delays of microwaves, sometimes reach to
Since the amount of water vapor is the largest
3cm or more and continue to exist stably for a
in the lowest part of the troposphere
few days or longer.
It is ascertained that the
decreases exponentially with height, WD gives
and
horizontal gradient of water vapor distribution in
larger directional differences at lower elevation
the N-S direction is caused by atmospheric
angles of GPS satellites. The most probable
conditions, especially by wind direction and
sources of such spatial inhomogeneous WD
velocity, and also probably by sunlight. Similar
distribution, namely the horizontal gradient, are
correlations are apparent between E-W gradients
the topography around an observation site and the
of wet delay and wind velocity.
However, the
meteorological condition such as air temperature,
data is not enough to draw definite conclusions on
wind direction and velocity, air pressure, humidity.
the E-W component.
and sunlight. Aonashi et al. (2004) estimated the horizontal
Keywords.
wet
delay,
wind
velocity,
positioning, water vapor radiometer
scales of the gradient of precipitable water vapor content using three water vapor radiometers. They concluded that the horizontal scale was less than
1 Introduction
10km. Although their aim was to have statistical
Inhomogeneous distribution of water vapor in
information on temporal variations of the gradient.
the atmosphere causes errors of the precise
the temporal variations themselves are very
positioning
important
and
surveying
techniques such as GPS.
with
the
space
information
to
correct positioning
Mousa(1997) found a
errors in observations of stepwise or transient
stable inhomogeneous water vapor distribution
ground movements with GPS due to spatially
continuing for one day or longer at Shionomisaki
inhomogeneous
Promontory in southwest Japan. Tanaka et al.
GPS-derived water vapor and horizontal wind
(2004) carried out similar observations in Uji city
was obtained from Doppler radar-derived radial
located
such
wind data above Kanto Plain in Japan by Seko et
another
al. (2004). However, the spatial scale is too large
at
inhomogeneous
inland
to
investigate
distributions
geographical setting.
at
They ascertained that the
WD
distributions.
Mesoscale
to apply their results to correct positioning errors
854
T.
Tanaka
Pl~isht
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0.6) barely occurred at our monitor stations, whereas moderate to strong phase scintillation occurrence (%~>0.5) was significantly higher, with values reaching that level for 5% of the time on many days at our northern-most stations, especially during the night time sectors.
Also, long term comparisons between the Kp geomagnetic index (from the USA NOAA SEC) and scintillation activity were carried out, statistically corroborating the existence of significant correlation between the two. GPS phase scintillation at high latitudes has a daily pattern mainly controlled by the receiver location moving into the aurora oval (Aarons, 1997). During magnetic quiet periods the aurora oval has the largest latitudinal extension at magnetic midnight, as a consequence of its natural expansion in the anti-sunward direction, and phase scintillation activity is greatest around that time. Occurrence of phase scintillation during daytime is mostly due to the oval's expansion during geomagnetic storms. Establishing the correlation between scintillation occurrence and geomagnetic events is of particular relevance to the development of scintillation prediction and warning mechanisms. We analysed the correlation between the Kp and scintillation activity recorded at our monitoring stations, in particular Bronnoysund (-65°N) and Hammerfest (-71°N), these being the most likely stations to be affected by the oval expansion/contraction. For
instance, at Bronnoysund we observed strong correlation of the maximum daily occurrence of ~,p>0.5 for 3-hour time bins with the maximum daily value of Kp (which is computed for each 3-hours) for every day of 2002 and 2003. Correlation was particularly significant during 2003, when a greater number of Kp values over 6 occurred (this value of Kp corresponds to a major geomagnetic storm according to NOAA's classification). It was also seen, through similar analysis, that geomagnetic, more than solar activity, was predominant in controlling scintillation occurrence at Hammerfest in 2003 (the reader is referred to Aquino et al, 2005a for more details of these analyses).
However, to assess how detrimental the effects of scintillation may be to a GPS user, it is necessary to investigate how many satellites are likely to be simultaneously affected by strong scintillation, in particular phase scintillation. Whether a certain level of scintillation (defined as moderate or strong) will have a direct influence on the quality of the measurement made at the receiver, or lead to loss of lock on the satellite, an estimate of how many satellites may be affected at the same time is of relevance, as this will impact on the user positioning accuracy or the user ability to calculate position at all. For that analysis, the probability of a number of GPS satellites simultaneously observing different levels of phase scintillation was calculated. It was shown that at our station in Hammerfest there was a probability of about 0.1% of 2 satellites being simultaneously affected by c~>0.5 in 2002. This probability increased to 0.25% in 2003, amounting to about 22 hours within that one year. This may pose a concern if lock on satellite is lost due to this level of scintillation, especially for applications requiring high availability, such as in civil aviation. A similar analysis for station Bronnoysund indicated probabilities of 0.06% and 0.1%, respectively for the years of 2002 and 2003. More importantly, when analysing the severe geomagnetic storm that occurred during October 29th and 30th, 2003, probabilities of about 3% and 2.5% of 2 satellites observing ~ > 0 . 5 simultaneously, respectively at Hammerfest and Bronnoysund were found. It was also seen that under such severe storm conditions 3 satellites observed c~>0.5 simultaneously with probabilities of 1% and 0.75%, respectively at Hammerfest and Bronnoysund.
Loss of lock on satellites possibly correlated with high levels of scintillation is thus a primary concern
Chapter 121 • Ionospheric Scintillation Effects on GPS Carrier Phase Positioning Accuracy at Auroral and Sub-Auroral Latitudes
for GNSS users in Northern Europe, in particular due to the greater susceptibility of the L2 to loss of lock in the presence of high phase scintillation. A codeless receiver exhibited up to 50% of L2 data loss during a severe geomagnetic storm and L1 or L2 data from up to 7 satellites was not present in the corresponding RINEX file on some occasions. On several occasions loss of lock was observed only on the L2 signal, with a direct implication for SBAS reference stations, as they must track both GPS signals in order to compute and disseminate ionospheric delay corrections. Further analyses on loss of lock involved studying its dependence on elevation angle, and loss was found to be more evident for elevations lower than 20 °. Losses of satellite lock correlated with phase/amplitude scintillation even occurred with our GPS scintillation monitors, highlighting the potential impact of ionospheric scintillation in the tracking performance of conventional GPS receivers. In this paper, however, we mainly concentrate on experiments aimed to investigating possible accuracy degradation on carrier phase positioning related to scintillation effects. In the next section we review some of our overall positioning accuracy results when analysing GPS positioning errors concurrent with scintillation occurrence. The following sections describe and discuss results of experiments performed in particular with carrier phase positioning. In section 3 we deal with static carrier phase positioning, whereas in section 4 we discuss results of using a pseudo-kinematic carrier phase technique. Section 5 contains our conclusions.
series of normalised horizontal positioning errors (the dark triangles), with the observed values of the phase scintillation index for all satellites shown in the background. We use the normalised horizontal errors, which are given by the horizontal errors divided by the HDOP (Horizontal Dilution of Precision) of each epoch, in order to enable a meaningful comparison between epochs as the effect of the geometry is thereby neutralised. Although the errors are greater during the period of high scintillation levels, an epoch by epoch comparison does not show a direct correlation between the two. The bottom plot shows that in fact the errors correlate better with the background TEC, which is shown in the plot by the (non-calibrated) slant TEC values for all satellites being observed.
B r o n n o y s u n d , 30 O c t o b e r 2 0 0 3
•
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.
Ill
•~
10
2O
•
18
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V
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2 Summary of Previous Work 2 , 0
Our investigations of GPS C/A code point positioning errors have shown there to be no significant accuracy degradation which can be strongly correlated with increased levels of scintillation, but rather any degradation in standalone code positioning occurred as a result of increased levels in the background TEC. A representative example is given in figure 1, which refers to C/A code stand-alone positioning errors computed for every 1 minute at our station in Bronnoysund (--65°N) during a period of increasing scintillation levels (given by the 60 seconds cy~ index), between 14:00UT and 24:00UT on 30 October 2003. In the top plot we show the time
16
18
20
22
0
Universaltime
Fig.
1 Scintillation
Effects
on
C/A
code
stand-alone
positioning
Due to the nature of the algorithms used to compute pseudorange corrections in a conventional DGPS, where clock, orbit and atmospheric errors are combined together, it is expected that varying ionospheric conditions will lead to spatial decorrelation between reference station and user.
861
862
M. Aquino • A. Dodson. J. Souter • T. Moore
Varying ionospheric conditions between reference and user may take place due to north-south TEC gradients that occur at sub-auroral and auroral latitudes during the solar high, in particular under active geomagnetic conditions. This was also investigated and when comparing the 2drms errors from a (approximately) north-south baseline with those from an (approximately) east-west oriented baseline in the UK, during a severely disturbed period (5 th to 8th November 2001), a degradation of about 30% in the horizontal positioning accuracy was observed, confirming the potential influence of TEC gradients on DGPS positioning (Moore et al, 2002). A way of improving stand-alone or DGPS user accuracy is to access and apply location specific ionospheric corrections disseminated by Augmentation Systems such as the European Geostationary Navigation Overlay Service (EGNOS). Such corrections would in principle resolve the problem of spatial decorrelation due to strong TEC gradients. However, the computation of the ionospheric correction by these systems relies on dual frequency GPS data from their reference stations, that may also be affected by scintillation during periods of unfavourable geomagnetic conditions. In particular the acquisition of the less robust L2 observations may be compromised. We have observed missing corrections in the EGNOS ionospheric grid during periods of occurrence of high values of phase scintillation which are likely to relate to the inability of the EGNOS reference stations to track one or both of the GPS signals of some satellites. Users at mid-latitudes opting to avoid satellites not monitored by the EGNOS ionospheric grid were shown to, thereby, indirectly suffer the effects of scintillation due to ionospheric irregularities occurring further northwards. On occasion such users were left with only 4 satellites to compute position, which has been shown to cause problems in the least squares solution due to lack of redundant measurements (Aquino et al, 2005b). This could be an issue if users do not have a choice to include (non-monitored) satellites in their solution.
Experiments were carried out to analyse possible position and measurement accuracy degradation when using carrier phase observations during periods of observed high levels of scintillation. These are described in the next two sections.
3 Static Carrier Phase Experiments Initial experiments with static carrier phase positioning have been reported in Aquino et al (2005b). Carrier phase data from the 30 th of October 2003 was analysed. On that day significant enhancement in phase scintillation (a~) values was observed even at our mid-latitude station in Nottingham (~53°N). The increased levels began quite markedly during the night time hours, with values changing from low during the day to moderate and tending to high (figure 2 top plot). For our analyses, the 24 hours RINEX file of our scintillation monitor was split in 2 hours sessions for processing.
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Each session was processed using IGS (International GPS Service) precise orbits and three IGS permanent stations with data available in the region, to form a network and solve for the coordinates of our scintillation monitor. The observable used was the ionosphere corrected L1 double difference carrier phase. Estimated coordinates were compared with accurate ground truth coordinates, with results for each 2 hours session shown in figure 2 (bottom plot). Temporal correlation of the RMS of the recorded phase scintillation values with, respectively, the 3D errors
Chapter 121 • IonosphericScintillationEffectson GPSCarrier Phase Positioning Accuracyat Auroral and Sub-Auroral Latitudes
and the RMS of the measurement residuals was then analysed for every 2 hours session. Results are shown in the first row of table 1. NSF06 indicates our scintillation monitor located in Nottingham. An entirely similar analysis was conducted involving data from a permanent Ashtech ZXII semicodeless GPS receiver (station IESG), whose antenna is located just a few meters from our scintillation monitor's, with accurately known coordinates. Resulting correlation coefficients are seen in the second row of table 1. A comparison in the performance of the two receivers, not shown on the table, revealed that the scintillation monitor overall provided better positioning accuracy than the Ashtech semicodeless receiver, especially when scintillation levels were low. However when scintillation was at its peak on that day, the latter seemed to have been less affected. Influence of scintillation on the RMS residuals did not show any appreciable difference between the two receivers (notice correlation coefficients of 0.96 and 0.98 respectively), however both receivers indicate a strong correlation between phase scintillation and the quality of their measurements. These results clearly deserve further investigation, which is however outside the scope of this paper. We also analysed the performance of our monitor in Bronnoysund using a similar approach, during the same day. Resulting correlation coefficients are shown in the third row of table 1, where NSF03 indicates our monitor located in Bronnoysund.
shortening the time span of the processing session to 15 minutes. Also, a baseline was compared with a network approach in order to assess whether the latter could improve the solution in the presence of strong scintillation. The baseline was formed by our Bronnoysund receiver (NSF03) and EUREF station VILO (Vilhelmina), approximately 240km to the east, which was held fixed while solutions were obtained for NSF03. The network solutions included another 2 permanent IGS stations with data available in the region. Figure 3 shows the results. The top plot represents the baseline solutions, with the RMS of the 3D errors of 0.90m, mainly due to the poor performance (spikes in the plot) during the sessions where high scintillation occurred. Average values of the ~ index for all satellites are shown in the bottom plot. The middle plot shows the network solutions, with the RMS of the 3D errors of 0.43m. In both cases the correlation between the errors and the phase scintillation values is clearly visible, and the results also indicate that the network solution leads to a significant improvement in accuracy, as expected due to the greater redundancy and geometrical strength given by the network solution. It is noted however that the poorer solutions during the periods of stronger scintillation remain.
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Fig. 3 Comparison of baseline and network static carrier phase solutions and correlation with phase scintillation values
The correlation of errors with the o~ index suggests the possibility of de-weighting carrier phase measurements with the use of a scintillation related parameter. These findings require further investigations, but are nevertheless encouraging and give scope for the potential development of
863
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M. Aquino • A. Dodson. J. Souter • T. Moore
warning/mitigation mechanisms that could be based on phase scintillation indices.
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The correlation between phase scintillation levels and position accuracy degradation observed in section 3 prompted more detailed investigations. Although correlation was seen when 2 hour and 15 minute sessions were used, ideally a temporal resolution compatible with the time span which the scintillation indices referred to (1 minute) should be used for comparisons. We used the Bernese software, of the Astronomical Institute of the University of Berne, to shorten the processing interval to 1 minute, through their 'pseudokinematic' solution. This is a kinematic solution whereby epoch-wise corrections to the a priori coordinates of the rover receiver are computed, under the assumption that its displacement is small, so that it 'remains within the linear regime of the partial derivatives' (Hugentobler et al, 2001). In our case this solution was indeed suitable, as in all of our experiments the 'rover' station was actually static. That therefore allowed us to assess more directly the relationship between positioning accuracy and its degradation under the influence of increased levels of phase scintillation, in our case estimated by the 60 seconds cy~ index. The result may be assessed by examining the plots of figure 4. We processed the same baseline as above (i.e. between our monitor in Bronnoysund and EUREF station VILO) using Bernese's epochwise (1 minute) pseudo-kinematic solution. Again station VILO was held fixed and solutions were obtained for NSF03. The time span was again 29 th to 31 st October 2003. Besides the ionosphere corrected L1 double difference carrier phase solution (bottom plot) another two different observables were used in an attempt to assess and compare the effects respectively on the L1 C/A code (middle plot) and on the phase smoothed L1 C/A code (top plot) solutions. For each plot the dark line represents the epochwise 3D positioning errors, sorted in ascending order. The small dots are the corresponding average values of the 60 seconds phase scintillation index for all satellites tracked within the corresponding epochs, as measured at Bronnoysund. It can be seen that correlation is visible on the top and bottom plots, where scintillation values begin to increase as the errors are greater, towards the right of the plots.
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A less clear correlation is seen on the middle plot, confirming that the C/A code solution does not seem strongly affected by phase scintillation (as noted for instance by Conker et al, 2003, that 'phase scintillations have little effect on code tracking errors'). However, simply introducing the carrier phase data to smooth the code measurements causes the positioning accuracy to degrade, as seen in the top plot. Also, in 9.6% of the epochs an ionospheric free solution was not possible, due to missing L2 data (extreme right of the bottom plot). Analyses at the lower latitude of Nottingham (-53°N) were also undertaken, using the same pseudo-kinematic approach, in order to further the investigations of section 3. The same time span (29 th to 31 st October 2003) was analysed with two different baselines. The first baseline was between our scintillation monitor (NSF06) and station IESG of section 3, whose antenna is separated by about 3m from our scintillation monitor's antenna. The second baseline was between NSF06 and the permanent Ordnance Survey of Great Britain (OSGB) station NOTT, located about 2km to the north, where an Ashtech ZXII receiver is also
Chapter 121 • Ionospheric Scintillation Effects on GPS Carrier Phase Positioning Accuracy at Auroral and Sub-Auroral Latitudes
installed. In both cases the baselines were processed to obtain a solution for NSF06 coordinates, with the opposite end of the baseline fixed. The aim was to carry out the analysis when all stations were, in all likelihood, subjected to similar, if not identical, levels of scintillation. In the previous analyses involving our Nottingham monitor (section 3) a regional network had been formed, with other stations probably not sensing the same scintillation effects. Results are seen in figures 5 and 6, where the dark dots show the deviation of the 3D positioning errors from the mean error and the light crosses are the average phase scintillation values for all satellites within each 1 minute epoch. Firstly, we focus on figure 5. Despite the slight departure from the mean near epoch 2800 (late night on the 3 0 th October and early morning of 31 st October), in correspondence with the peak in scintillation levels, no significant increase in the errors is observed that could be clearly correlated with the higher scintillation values. Analysing the pseudo-kinematic solution file it was possible to verify that, although over the 3 days there were 1.8% of epochs when no solution was possible, these epochs did not concentrate particularly around the period of high scintillation.
separated by only about 3m, implying that even in the presence of small scale irregularities (tens of meters), it is much likely that the link from each satellite to the receivers traverse the same electron density, causing the effect to cancel out. When analysing figure 6, it is possible to see that for the 2km baseline a significant effect takes place in line with the higher scintillation levels. A visible degradation in accuracy correlates with the peak in the 60 seconds c~ values. In contrast with the shorter baseline now the percentage of epochs without solution nearly doubles, to 3.2%. The pseudo-kinematic solution, clearly, was computed in post-processing. When no solution is possible Bernese interpolates the values of the coordinates using the surrounding valid solutions. It can be seen that next to epoch 2881 (beginning of 31 st October) the errors increase quite remarkably in line with the increase in the phase scintillation levels. During that period of time many of the solutions seen in the plot are the result of interpolations (sparse dark dots), however these (intermediate) results can only be obtained because for some of the epochs a solution was actually computed.
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If the quality of the carrier phase measurements was somehow degraded during that period, both receivers were affected to a similar degree, as only very marginal positioning accuracy degradation could be seen during that period. Given the short separation of about 3m between the two antennas, the corresponding ionospheric pierce points are also
The conclusion is that for the epochs where the solution was possible (e.g. the top-most dark dot) the phase measurements used in the computation seem to have been corrupted to the point of notably degrading the positioning accuracy. In this particular case the maximum error observed in the chosen period was over 7m. Clearly the two receivers involved are not the same and differences in the robustness to scintillation by individual receivers is
865
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M. Aquino • A. Dodson. J. Souter • T. Moore
well documented (e.g. Skone et al, 2001), so this may be partly responsible for the effect not cancelling out. For this baseline length, the separation of the pierce points for the two receivers is just under 2km, implying that in the presence of small scale irregularities, which could be smaller than 2km, the effect may not be expected to cancel out. This may have implications for applications that rely on the spatial correlation of errors over short baselines, such as RTK GPS, with potential adverse effects on carrier phase ambiguity resolution. 5
Conclusions
Studies carried out based on long term ionospheric scintillation data recorded in Northern Europe revealed that the worst case scenario for a GPS/GNSS user is the loss of lock on satellites due to strong scintillation. Statistical analyses on the data showed that during periods of severe geomagnetic activity the probability of 2 satellites being simultaneously affected by strong scintillation may be as much as 3% at auroral latitudes, with implications in particular for safety critical applications such as civil aviation. In this paper we concentrated on effects that occur when degradation of measurement quality and positioning accuracy take place as a result of scintillation occurrence, rather than on the implications of loss of satellite lock. Clearly these effects will be the more harmful the greater the number of satellites that are subjected to strong scintillation. Effects on C/A code standalone positioning were seen not to show a direct correlation with scintillation occurrence and conventional DGPS positioning was seen to be affected by TEC gradients that develop under adverse geomagnetic conditions. It was also shown that EGNOS ionospheric corrections may falter when they are most needed, potentially due to scintillation occurrence affecting its reference stations. We analysed in particular carrier phase data and resulting positioning performance during periods of severe geomagnetic activity, with which strong scintillation is normally associated at sub-auroral and auroral regions. Correlation was seen between phase scintillation and static and pseudo-kinematic carrier phase data processing results, both in terms of 3D errors and RMS of residuals. A baseline of approximately 240km was analysed using 15 minute sessions on a carrier phase static solution, where errors were seen to correlate with
scintillation occurrence. Also in this case a network approach was shown to improve positioning accuracy. The adverse influence of phase scintillation was also seen when this baseline was processed using a (1 minute) epochwise pseudokinematic solution. More importantly it was verified that effects may be significant at the sub-auroral latitude of Nottingham (-53°N). Analyses carried out on a 2km baseline in that area showed the occurrence of errors not observed on a significantly shorter baseline, of approximately 3m in length. In the 2km baseline, errors reached values over 7 meters on a (1 minute) epochwise pseudo-kinematic solution, suggesting that scintillation may cause problems for RTK GPS, with possible adverse effects on carrier phase ambiguity resolution.
References
Aarons, J. (1997). Global Positioning System Phase Fluctuations at Auroral Latitudes. Journal of Geophysical Research, 102, A8, 17219-17231. Aquino, M., F. S. Rodrigues, J. Souter, T. Moore, A. Dodson and S. Waugh (2005a). Ionospheric Scintillation and Impact on GNSS Users in Northern Europe: Results of a 3 Years Study. Accepted .for publication on the Space Communications Journal, 10S Press. Aquino, M, T. Moore, A. Dodson, S. Waugh, J Souter and F. S. Rodrigues (2005b). Implications of Ionospheric Scintillation for GNSS Users in Northern Europe (2005b). The Journal of Navigation, 58, pp 241-256. Conker, R. S., M. B. E1-Arini, C. Hegarty and T. Hsiao (2003). Modelling the Effects of Ionospheric Scintillation on GPS/SBAS Availability, Radio Science, 38, 1. Hugentobler, U., S. Schaer and P. Fridez (2001). Bernese GPS Software Version 4.2. Astronomical Institute, University of Berne. Mitchell, C., L. Alfonsi, G. De Franceschi, M. Lester, V. Romano and A. Wernik (2005). GPS TEC and Scintillation Measurements from the Polar Ionosphere during the October 2003 Storm. Geophysical Research Letters, 32, L 12S03, doi."10.. 1029/2004GL021644 Moore, T., M. Aquino, A. Dodson, S. Waugh and C. Hill (2002). Evaluation of the EGNOS Ionospheric Correction Model under Scintillation in Northern Europe, In: Proc. of ION GPS 2002 Conference, Copenhagen, Denmark, pp1297-1306. Rodrigues, F. S., M. Aquino, A. Dodson, T. Moore and S. Waugh (2004). Navigation, Journal of the Institute of Navigation, 51, 1, pp 59-76. Skone, S., K. Nudsen and M. de Jong (2001). Limitations in GPS Receiver Tracking Performance under Ionospheric Scintillation Conditions, Physics and Chemistry of the Earth (A), 26, 6-8, pp 613-621.
Chapter 122
The impact of severe ionospheric conditions on the GPS hardware in the Southern Polar Region D. A. Grejner-Brzezinska, C-K. Hong, P. Wielgosz
Satellite Positioning and Inertial Navigation (SPIN) Lab Department of Civil and Environmental Engineering and Geodetic Science The Ohio State University, 470 Hitchcock Hall, Columbus, OH 43210-1275, USA L. Hothem United States Geological Survey 521 National Center, 12201 Sunrise Valley Dr., Reston, VA 20192 USA
Abstract. The primary objective of this paper is to present results of an experiment to determine the effects of moderate and severe ionospheric conditions on the GPS signal tracking by different L1/L2 receivers operating in the Southern Polar region. In this study, data collected by the Ohio State University (OSU) and the U.S. Geological Survey (USGS) joint team within the TAMDEF (Transantarctic Mountains Deformation) network were used together with the IGS and UNAVCO Antarctic stations. Seventeen Antarctic stations equipped with different dual-frequency GPS hardware were selected, and data were evaluated for two 24-hour periods of severe ionospheric storm (October 29 th, 2003) and active ionospheric conditions (moderate storm of November 11th, 2003). The UNAVCO QC software was used to carry out the analyses. Depending on the data sampling rate and the elevation mask angle, the expected number of observations per receiver/satellite was compared to the actual number of measurements collected during the ionospheric storms, with a special emphasis on L2 data. Depending on the receiver model, the number of lost measurements during the severe ionospheric conditions ranged from 0.5% to 30.0%. In addition, the number of cycle slips (CS) per number of observations as a function of receiver model was computed; it shows great variation for different hardware. The possible variability of the ionospheric conditions at some of these sites (due to their separation) is considered in the conclusions. The results indicate that depending on the severity of ionospheric conditions, there is a significant difference in the impact on the operations of different hardware models. Thus, careful hardware selection is needed to assure data quality/continuity when observations may be affected by severe ionospheric disturbances.
Keywords. Ionospheric effects on GPS, GPS hardware, GPS signal tracking conditions
1 Introduction The ionosphere is one of the primary error sources that affect the GPS observables. The carrier phase is subject to phase advance while the pseudorange is subject to a code delay. The group delay/phase advance effect is due to the propagation through an ionized medium where the refractive index changes throughout the medium, while the increased signal dynamics accounts for total electron content (TEC) fluctuations. Under disturbed geomagnetic conditions, the error due to ionosphere may reach tens or even hundreds of meters, and still under relatively benign conditions this effect amounts to meters or tens of meters, depending on the geographic location of the GPS station, the time of day (solar angle) and the 11year Sun-spot cycle. Another ionospheric effect is the signal loss of lock, due to radio wave scintillations that are amplitude and phase fluctuations originated by small-scale electron density irregularities. The regions where ionospheric scintillations are more present are low, auroral and polar latitudes in a geomagnetic sense. Rapid ionospheric disturbances may cause cycle slips (CS) and losses of signal lock, since a Doppler shift of the signal caused by ionospheric disturbances may exceed the bandwidth of the phase tracking loops (note that tracking loops are tunable to account for Doppler shift within certain limits). A loss of code lock can occur as a result of an increase in fading amplitude and a drop in the signal-to-noise (SNR) ratio. Since the ionosphere is a dispersive medium, the L2 signal with a wavelength of 24.45 cm is subject to larger ionospheric code delays than the L1 signal with a wavelength of 19.04 cm. By design, the L2 signal
868
D.A. Grejner-Brzezinska • C-K. Hong. P.Wielgosz • L. Hothem
is weaker compared to L1; the transmitted power levels are 23.8 dBW and 19.7 dBW for the encrypted P-code signals on L1 and L2, respectively, and 28.8 dBW for the L1 C/A-code signal (Langley, 1998). Moreover, due to L2 signal encryption, it is tracked using codeless and semicodeless techniques, which are more likely to give rise to loss of lock due to the lack of despreading gain. Most of the receivers provide a hybrid tracking technique, where the L1 carrier phase is reconstructed by code-correlation using the C/A code, while a codeless or semi-codeless technique is used to acquire the L2 carrier phase. The codeless technique assumes no knowledge of the Y-code, while the semi-codeless method relies on the knowledge of the known P-code, and the fact that the Y-code is a modulo-2 sum of the P-code and the W-code, making it possible to remove the P-code component of the modulation using a locally generated replica of the P-code (see, Woo, 1999 for more details on semi-codeless and codeless tracking). The codeless techniques, such as signal squaring, and cross correlation, result in SNR changes by -30 dB and-27 dB, respectively, while the corresponding numbers for the semicodeless techniques (code correlation plus squaring, and Z-tracking) are -17 dB and-14 dB, respectively (Ashjaee and Lorenz, 1992; Woo, 1999; Hofman-Wellenhof et al., 2001). In addition, the L2 phase tracking loops apply a narrower bandwidth, compared to L1. As a result, L2 is often lost, while the L 1 signal may still be tracked. Thus, the tracking performance of a GPS receiver depends not only on the state of the ionosphere (and thus, the amount of the ionospheric disturbances), but it is also a function of the receiver tracking capabilities. Therefore, the purpose of the study presented here is to determine the level of differences in the receiver tracking performance under severe ionospheric conditions for different types of geodetic-grade receivers. The motivation behind this study is to determine the most suitable GPS hardware for operational deployments in Antarctica, where ionospheric conditions are much more severe, as compared to mid-latitudes, and thus, the hardware tracking continuity and signal quality are of major concern. It should be noted that the comparison was performed for receivers significantly spread in latitude and longitude, under the assumption that the ionospheric conditions were roughly the same on average for the analyzed periods. Due to general ionospheric structure dynamics, it is a
reasonable assumption; however, more appropriate would be to test the receivers connected to the same antenna, to assure no time and space dependency as a function of the storm evolution.
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Seventeen GPS stations in Antarctica (Figure 1), where nine different types of geodetic-grade receivers are deployed, were analyzed. Several stations of the TAMDEF (Transantarctic Mountains Deformation) network were used together with the IGS and UNAVCO Antarctic stations; TAMDEF is a joint project between The Ohio State University and the US Geological Survey, to measure crustal motion in the Transantarctic Mountains of the Southern Victoria Land (http ://www.geology.ohiostate.edu/TAMDEF/). Figure 2 illustrates examples of the antenna location at some of the analyzed sites. The test datasets include the ionospheric superstorm of October 29 th, 2003 and a moderate ionospheric storm of November 11 th, 2003", the Kp index for both days is plotted in Figure 3. The UNAVCO QC (hereinafter QC) version 3.0 software was used for the analyses; the ionospheric effects based on L2 data are presented, with a special emphasis on the number of cycle slips and amount of data loss under increased ionospheric activity; an elevation cut-off of 10 ° was used. The series of geomagnetic and radiation storms, including the superstorm of October 29 th, 2003 represents a very interesting phenomenon. Solar Cycle 23 began its 11-year cycle in May 1996, and reached its peak in April 2000. While the level of ionospheric activity was low at the beginning of October 2003, by the end of the month, massive sunspot groups developed on the solar surface, leading to the most severe solar activity observed in recent years. For example, 17 major solar flares occurred between October 19 th and November 5 th, 2003; geomagnetic storms observed at that time reached extreme (G5) levels, and solar radiation storms reached severe ($4) levels, according to NOAA's Space Weather Scales (http ://www. sec.noaa.gov/NOAAscales/), and the 6th most intense geomagnetic storm since 1932, and the 4th most intense radiation storm since 1976 occurred on October 29 th, 2003. The largest solar flare recorded occurred on November 4 th, 2003 (U.S. Department of Commerce, 2004).
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-.,:,~:,.:,
l
........ ........
2006
Fig. 2 History of the POD quality fits, estimated via the variations of the empirical drag (FD) and solar pressure (Fs) coefficients, and r.m.s, of the Doppler measurement residuals here given in hertz (multiply by 35 and 17.5 for one and two-way Doppler respectively to obtain residuals in mm/sec). The mean and r.m.s, values of F D are respectively 0.10, 0.30 for period 1 and 0.46, 0.79 for period 2; Fs has more stable mean and r.m.s, values of 0.99, 0.15 for period 1 and 0.99, 0.11 for period 2.
Three main types of solutions, derived at ten day intervals (coverage permitting) for 6C20 and 6C30, have been obtained: the first type uses continuity constraints in the form of a priori o's on the difference between two consecutive values of these parameters; handling of large gaps between some arcs is delicate and these solutions are not presently satisfactory; - in the second type a priori o's on the parameter values themselves are introduced; after several attempts we took a unique value of 3 . 1 0 -9 which corresponds to the magnitude of the variations predicted by G C M simulations (Smith et al. (1999)); these solutions will be refered to as simply
regularized; - the last type introduces an a priori annual (period of 687 terrestrial days) and semi-annual behaviour of the gravity coefficients with given o's of 3.10 -9 o n the sine and cosine factors of the periodic terms; these solutions will be refered to as forced periodic. In all solutions the Love number k 2 w a s solved for, without any constraint. Figure 3 shows the time variations of 6C20 and 6C30 for different periods and different types, with the fitted annual and semiannual terms (adjusted a posteriori). Also shown are the variations predicted from the NASA-AMES GCM and those derived from the HEND instrument on Mars Odyssey (see section 4).
,-:-.:,.:.
....
2,:,,:,,:,
=.,:,t:,,
2,:,':,--,
...,..,_,~.. - ' - -'
_:,.. . . . . ......
. . . .
Fig. 3 Time variations of 6C20 and 6C30 for different solution classes and types. (al) , (bl): forced periodic, period 1;(a2) , (b2): forced periodic, period 2; (c) and (d): forced periodic, full period. The error bars correspond to formal errors (non calibrated). Also shown are results from GCM and HEND (periodized). Table 1 lists the amplitudes or. and phases c9. of the periodic terms and the k 2 value in each case. For sake of comparison with Yoder et al. (ibid.) the periodic terms are here written as: 0% sin(n MMar~ + Cgn) with MMars being the mean anomaly of Mars and with n=l and 2 for the annual and semi-annual variations respectively. We did not solve for terannual or higher order terms for we do not believe (from the graphs of the 6C20 and 6C30 variations and from the magnitude of the formal errors) that they could be reliably estimated. We do not show the simply regularized solutions for periods 1 and 2 separately for they look very close to the corresponding parts in the full period case. For all solutions the phase values come with large formal errors, which precludes any physical interpretation. We believe that the scattering of the Table 1 amplitude values is in line with the true errors (which we cannot estimate and which are probably much larger than the formal ones) and is indeed
899
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G. Balmino. J. C. Marty. J. Duron • O. Karatekin
related to the geometry and to the quality of the POD which result in more or less observability of some of these p a r a m e t e r s and/or significant correlations. For instance the clear increase of the formal error bars on 6C20 between mid-2000 and the first quarter of 2001 (see fig. 3) corresponds to an edge-on geometry. In an attempt at mitigating some of these effects we derived monthly solutions under similar conditions, but these do not show significant differences with the ten day solutions apart from some (expected) reduction of the amplitudes of some periodic terms. We found that the simply regularized solutions for ~C20 and ~C30 have smaller annual amplitudes and that ~C20 has a weaker semi-annual signal compared to the HEND and the GCM estimates; h o w e v e r the phases of 6C30 are in fairly good agreement despite the large formal errors as pointed out above. The forced periodic solution over period 1 (Fig. 3- a l and b l) looks comparable to previous results such as those of Yoder et al. (ibid.) which is no surprise since these authors solved explicitly for periodic terms; the fact that an apparently fiat signal for 6C20 in the simply regularized solutions t r a n s f o r m s into an o r g a n i s e d signal can be explained by the strong decorrelation brought by the forcing equation. This is also clear from the solution over the full period (Fig. 3-c). It leads us to favor the forced periodic solutions. Our amplitudes of the periodic terms fitted to ~C30 are significantly smaller than those found by Yoder et al. even in the forced periodic solution over period 1 (close to the period analyzed by these
authors); the 6C20 variations are closer, and the semi-annual term is also in reasonable agreement with the value given by Smith et al. (2001). But our strategy is different: Yoder evaluates the 6 C 2 0 , 6C30 and k 2 coefficients separatly whereas we estimate all parameters simultaneously. The forced periodic solutions for 6C30 (Fig 3- b2 and d) show similar phases and smaller amplitudes than those of HEND and GCM, as in the case of the simply regularized solution, but for period 1 (Fig.3b 1) the annual amplitude is in good agreement with the HEND and GCM d a t a - but not the phase. The agreement between forced 6C20 variations is also best over this period (Fig. 3-a 1). The 6C20 forced periodic signal obtained from period 2 (Fig 3-a2) has larger annual and semi-annual components than those of the signal from the full period (Fig. 3-c) and there is a large phase difference compared to the period 1 results, as well as to the HEND and GCM solutions (which look almost anti-correlated). Values of k2 are also scattered. The smallest one (0.099) is obtained with the period 1 forced periodic solution (with a better 6C20 match up), where Yoder et al. (ibid.) find a high value of 0.153 (_+ 0.017). Our largest value (0.154), close to the value found by these authors, comes with the period 2 solution. The value of 0.130 found in between with a small formal error of 0.006 uses the full period data set, comes with forced periodic ~C20, 6 C 3 0 and should be the most reliable from the dynamical viewpoint; it is compatible with the hypothesis of a fluid Martian core (Yoder et al. (ibid.).
Table 1. Annual (A) and semi-annual (SA) amplitudes (in units of 10 -9) and phases (in degrees) of ~C20 and 6C30, and values of the k 2 Love number for the most significant solutions, and comparisons with Smith et al. (2001) and Yoder et al. (2003) solutions, s.regul, are simply regularized solutions, forced per. are forced periodic solutions as explained in the text.
Solution Full period s. reguL Full period forcedper. Period 1 forcedper. Period 2 forcedper. Smith et aL,
2001, forced per. Yoder et aL, 2003, forced per.
eoeff. 6C20 ~C30 ~C20 6C30 ~C20 ~C30 6C20 6C30
A
A
SA
SA
amplitude
phase
amplitude
phase
0.3 + 0.3 0.4 +0.3 1.8+0.4 0.6+0.5 1.9 + 1.4 1.3+ 1.1 2.9 + 1.0 0.4 + 0.3
13+61 43 ±40 4+63 40+ 42 09 + 28 65 + 28 7 + 80 13 + 68
0.1 +0.2 0.05 ± 0.3 0.7 +0.3 0.1 +0.3 1.3+ 1.6 0.4 + 0.8 1.4 + 1.0 0.07 + 0.1
67+34 -11 +45 63+33 -9 + 48 -87 + 25 -94 + 28 21 + 55 10 + 67
6C20 ~C20 ~C30
1.96 + 0.69 1.81 + 1.02 6.59 + 0.28
12 -7
2.32 + 0.94 1.34 + 0.26
Love number kz 0.113 + 0.004 0.130 + 0.006 0.099 + 0.007 0.154 + 0.021 0.055 + 0.008
-3 -15
0.153 + 0.017
Chapter 125 • Mars Long Wavelength Gravity Field Time Variations:a New Solution from MGS Tracking Data 6 Conclusions
References
Seasonal variations of the Martian gravity field at very large scale have been confirmed from the analysis of five years of MGS tracking data. The behaviour of the lumped zonal harmonics of degrees two and three shows annual and semiannual patterns, which are more or less clear depending on the analyzed period and orbital geometry. The amplitudes of these terms, considering their formal errors, are compatible with those found earlier by other authors. The solutions depend on the period of the data set, which was processed in two sub-sets, then in one set. Regularized and forced periodic solutions give similar results for 6C30 but not for dC20. Comparisons with external data (GCM and HEND) are best generally for the forced periodic solutions and especially over the first sub-set period, and the match up is satisfactory for 6C30. The degree two Love number value is in the range 0.10-0.15 depending on the solutions. The intermediate value of 0.130 _+ 0.006, obtained with the full data set and a forced periodic solution for the zonal terms, may be considered the most probable; it supports the hypothesis of a fluid core of Mars. However the forced period fits for 6C20 show some dependence o n k 2 and disagree significantly with the models, casting some question on the k2 recovery.
Bruinsma, S., and F. G. Lemoine (2002), A preliminary semiempirical thermosphere model of Mars: DTM-Mars, J. Geophys. Res., 107(E 10), doi: 10.1029/2001JE001508. Cazenave, A., and G. Balmino (1981), Meteorological effects on the seasonal variations of the rotation of Mars, Geophys. Res. Lett., 8, 245-248. Chapront-Touz6, M. (1990), Orbits of the martian satellites from ESAPHO and ESADE theories, Astro. Astrophys., 240, 159-172. Davies, M. E., et al. (1992), Report of the I A U / I A G / C O S P A R working group on cartographic coordinates and rotational elements of the planets and satellites, Celestial Mech. Dyn. Astron., 53, 377-397. Haberle, R. M., et al. (1999), General circulation model simulations of the Mars Pathfinder atmospheric structure investigation/meteorology data, J. Geophys. Res., 104(E4), 8957-8974. Karatekin, (), J. Duron, P. Rosenblatt, T. Van Hoolst, V. Dehant, and J. P. Barriot (2005), Mars'time-variable gravity and its determination: Simulated geodesy experiments, J. Geophys. Res., l l O(E06001), doi: 10.1029/2004JE002378. Kaula, W. M. (1966), Theory of Satellite Geodesy, Blaisdell, Waltham, Mass. Lemoine, F. G., D. E. Smith, D. D. Rowlands, M. T. Zuber, G. A. Neumann, D. S. Chinn, and D. E. Pavlis (2001), An improved solution of the gravity field of Mars (GMM-2B) from Mars Global Surveyor, J. Geophys. Res., 106(ElO), 23,359-23,376. Litvak, M. L., et al. (2004), Seasonal carbon dioxide depositions on the Martian surface as revealed from neutron measurements by the HEND instrument onboard the 2001 Mars Odyssey spacecraft, Sol. Syst. Res., 38, 167-177, doi: 10.1023/B: SOLS.0000030856.83622.17. McCarthy, D. D. (Ed.), (1996), IERS Technical Note 21, U.S. Naval Obs., Washington D. C. Moyer, T. D. (2000), Formulation for observed and computed values of Deep Space Network data types for navigation, Monograph 2, Deep Space Communications and Navigation series. Reasenberg, R. D. , and R. W. King (1979), The Rotation of Mars, J. Geophys. Res., 84(B11), 6231-6240. Smith, D. E., M. T. Zuber, R. M. Haberle, D. D. Rowlands, and J. R. Murphy (1999), The Mars seasonal CO2 cycle and the time variation of the gravity field: A general circulation model
Acknowledgments
This study used the GINS software and DYNAMO libraries developped by the Satellite Geodesy team at CNES. We acknowledge F.G. Lemoine and D.D. Rowlands (GSFC) for their help in making comparisons between their own software (GEODYN) and GINS, for providing assistance and software for handling some delicate aspects of the spacecraft attitude. MGS Radio Science team members and especially Dick Simpson (Stanford University) are thanked for their help in providing data, advises and encouragements. M. Litvak from SRI and R. Haberle from NASA Ames are acknowledged for providing the HEND and GCM data. This research was supported by the CNES Programme Directorate and the Toulouse Space Center on one hand, and by the Observatoire Royal de Belgique on the other hand. It benefited from PRODEX and Action-2 grants of the Belgian Science Federal Policy and from the support of the E u r o p e a n ' s Community Improving Human Potential Programme, under contract RTN-200100414, MAGE.
901
902
G. Balmino. J. C. Marty. J. Duron • O. Karatekin
simulation, J. Geophys. Res., 1 04(El), doi: 10.1029/1998JE900024. Smith, D. E., M. T. Zuber, and G. A. Neumann (2001), Seasonal variations of snow depth on Mars, S c i e n c e , 294, 2141-2146, doi:10.1126/science. 1066556. Smith, D. E., and M. T. Zuber (2003), Seasonal changes in the masses of the polar ice caps of Mars as derived from Mars Global Surveyor gravity, Mars Atmosphere Modelling and Observations Workshop, Cent. Natl. d'Etudes
Spatiales, Granada, Spain, 13-15 Jan. Standish, M. E., X X Newhall, J. G. Williams, and W. M. Folkner (1995), JPL planetary and lunar ephemerides DE403/LE403, JPL Interoff. Memo. 314.10-127, Jet Propul. Lab., Pasadena, Calif., May 22. Yoder, C. F., A. S. Konopliv, D. N. Yuan, E. M. Standish, and W. M. Folkner (2003), Fluid core size of Mars from detection of the solar tide, Science, 300, 299-303, doi: 10.1126/science. 1079645.
Chapter 126
Potential Capabilities of Lunar Laser Ranging for
Geodesy and Relativity Jfirgen Mtiller Institut ffir Erdmessung, University of Hannover, Schneiderberg 50, 30167 Hannover, Germany
[email protected] James G. Williams, Slava G. Turyshev Jet Propulsion Laboratory, 4800 Oak Grove Drive, Pasadena, CA 91109, USA Peter J. Shelus University of Texas at Austin, Center for Space Research, 3925 W. Braker Lane, Austin, TX 78759, USA
Abstract. Lunar Laser Ranging (LLR), which has been carried out for more than 35 years, is used to determine many parameters within the Earth-Moon system. This includes coordinates of terrestrial ranging stations and that of lunar retro-reflectors, as well as lunar orbit, gravity field, and its tidal acceleration. LLR data analysis also performs a number of gravitational physics experiments such as test of the equivalence principle, search for time variation of the gravitational constant, and determines value of several metric gravity parameters. These gravitational physics parameters cause both secular and periodic effects on the lunar orbit that are detectable with LLR. Furthermore, LLR contributes to the determination of Earth orientation parameters (EOP) such as nutation, precession (including relativistic precession), polar motion, and UT1. The corresponding LLR EOP series is three decades long. LLR can be used for the realization of both the terrestrial and selenocentric reference frames. The realization of a dynamically defined inertial reference frame, in contrast to the kinematically realized frame of VLBI, offers new possibilities for mutual cross-checking and confirmation. Finally, LLR also investigates the processes related to the Moon's interior dynamics. Here, we review the LLR technique focusing on its impact on Geodesy and Relativity. We discuss the modern observational accuracy and the level of existing LLR modeling. We present the near-term objectives and emphasize improvements needed to fully utilize the scientific potential of LLR. Keywords. Lunar Laser Ranging, Relativity, EarthMoon dynamics
1
Motivation
Being one of the first space geodetic techniques, lunar laser ranging (LLR) has routinely provided ob-
servations for more than 35 years. The LLR data are collected as normal points, i.e. the combination of round trip light times of lunar returns obtained over a short time span of 10 to 20 minutes. Out of ~ 1019 photons sent per pulse by the transmitter, less than 1 is statistically detected at the receiver Williams et al. (1996); this is because of the combination of several factors, namely energy loss (i.e. 1 / R 4 law), atmospherical extinction and geometric reasons (rather small telescope apertures and reflector areas). Moreover, the detection of real lunar returns is rather difficult as dedicated data filtering (spatially, temporally and spectrally) is required. These conditions are the main reason, why only a few observatories worldwide are capable of laser ranging to the Moon. Observations began shortly after the first Apollo 11 manned mission to the Moon in 1969 which deployed a passive retro-reflector on its surface. Two American and two French-built reflector arrays (transported by Soviet spacecraft) followed until 1973. Since then over 16,000 LLR measurements have by now been made of the distance between Earth observatories and lunar reflectors. Most LLR data have been collected by a site operated by the Observatoire de la Cgte dAzur (OCA), France. The transmitter/receiver used by OCA is a 1.5m altaz Ritchey-Chr&ien reflecting telescope. The mount and control electronics insure blind tracking on a lunar feature at the 1 arcsec level for 10 minutes. The OCA station uses a neodymium-YAG laser, emitting a train of pulses, each with a width of several tens of picoseconds). LLR station at the McDonald Observatory in Texas, USA is another major provider of the LLR data. The McDonald Laser Ranging Station (MLRS) is built around a computer-controlled 0.76m x-y mounted Cassegrain/Coudd reflecting telescope and a short pulse, frequency doubled, 532nm, neodymium-YAG laser with appropriate computer, electronic, meteorological, and timing interfaces. Until 1990, Haleakala laser ranging station
904
J. MLiller. J. G. Williams. S. G. Turyshev • P. J. Shelus
on the island of Maui (Hawaii, USA) contributed to LLR activities with its 0.40 m telescope. Single lunar returns are available from Orroral laser ranging station in Australia (closed 1 November 1998) and the Wettzell Laser Ranging System (0.75 m) in Germany. Other modem stations have demonstrated lunar capability, e.g. the Matera Laser Ranging Station (0.50 m) in Italy and Hartebeesthook Observatory (0.762 m) in South Africa. A new site with lunar capability is currently being built at the Apache Point Observatory (New Mexico, USA) around a 3.5 m telescope. This station, called APOLLO, is designed for mm accuracy ranging (Williams et al. 2004b). Today MLRS and OCA are the only currently operational LLR sites achieving a typical range precision of 18-25 mm. Fig. 1 shows the number of LLR normal points per year since 1970. As shown in Fig. 2, the range data have not been accumulated uniformly; substantial variations in data density exist as a function of synodic angle D, these phase angles are represented by 36 bins of 10 degree width. In Fig. 2, data gaps are seen near new Moon (0 and 360 degrees) and full Moon (180 degrees) phases. The properties of this data distribution are a consequence of operational restrictions, such as difficulties to operate near the bright sun in daylight (i.e. new Moon) or of high background solar illumination noise (i.e. full Moon).
600
._.q500 rn L Q Q.
t 400 O
300 e~
o ? 200 e~
E z 100
30
60
90
120 150 180 210 240 270 300 330 360 Synodic Angle D [degrees]
Fig. 2. Data distribution as a function of the synodic angle D.
range measurements. 25
.........................................................................................................................................................................................
20
..i . . . . . . . . . . . . . . . . . . . . . . . .
~. . . . . . . . . . . . . . . . . . . . . . . . .
~. . . . . . . . . . . . . . . . . . . . . . . .
: ........................
~. . . . . . . . . . . . . . . . . . . . . . . . .
~. . . . . . . . . . . . . . . . . . . . . . . . .
- ........................
, ....
.....
15 1 ....................... ~......................... i................................................. i................................................ i........................ i
lo
i ........................ i .............
5[',
........................
........... ~........................
; ........................ ',........................ i ........................ i .....
i ........................
i ........................
i
i ........................
~...................
i ........................
.....
i ........................ ~
1200
0
1000
I
1970
1975
1980
1985
1990
1995
2000
2005
years 800
Fig. 3. Weighted residuals (observed-computed Earth-Moon distance) annually averaged.
6 0 0
400
200
OLm.m 1970
1975
1980
1985
years
1 ggo
1995
2000
2005
Fig. 1. Lunar observations per year, 1970 - 2005.
While measurement precision for all model parameters benefit from the ever-increasing improvement in precision of individual range measurements (which now is at the few cm level, see also Fig. 3), some parameters of scientific interest, such as time variation of Newton's coupling parameter ( ~ / G or precession rate of lunar perigee, particularly benefit from the long time period (35 years and growing) of
In the 1970s LLR was an early space technique for determining Earth orientation parameters (EOP). EOP data from LLR are computed for those nights where sufficient data were available (approximately 2000 nights over 35 years). Although, the other space geodetic techniques (i.e., SLR, VLBI) dominate since the 80s, today LLR still delivers very competitive results, and because of large improvements in ranging precision (30 cm in 1969 to 2 cm today), it now serves as one of the strongest tools in the solar system for testing general relativity. Moreover, parameters such as the station coordinates and velocities contributed to the International Terrestrial Reference Frame ITRF2000, EOP quantities were used in combined solutions of the International Earth Rotation and Reference Systems Service IERS (~ = 0.5 mas).
Chapter 126 • Potential Capabilities of Lunar Laser Ranging for Geodesy and Relativity
2
LLR Model
The existing LLR model has been developed to compute the LLR observables the round trip travel times of laser pulses between stations on the Earth and passive reflectors on the Moon (see e.g. Mfiller et al. 1996, Mfiller and Nordtvedt 1998 or MLiller 2000, 2001, Mfiller and Tesmer 2002, Williams et al. 2005b and the references therein). The model is fully relativistic and is complete up to first post-Newtonian (1/c 2) level; it uses the Einstein's general theory of relativity- the standard theory of gravity. The modeling of the relativistic parts is much more challenging than, e.g., in SLR, because the relativistic corrections increase the farther the distance becomes. The modeling of the 'classical' parts has been set up according to IERS Conventions (IERS 2003), but it is restricted to the 1 cm level. Based upon this model, two groups of parameters (170 in total) are determined by a weighted least-squares fit of the observations. The first group comprised from the so-called 1Newtonian' parameters such as - g e o c e n t r i c coordinates of three Earth-based LLR stations and their velocities; a set of EOPs (luni-solar precession constant, nutation coefficients of the 18.6 years period, Earth's rotation UT0 and variation of latitude by polar motion);
-
- selenocentric reflectors;
coordinates
of
four
retro-
- rotation of the Moon at one initial epoch (physical librations); orbit (position and velocity) of the Moon at this epoch;
-
The second group of parameters used to perform LLR tests of plausible modifications of general theory of relativity (these parameter values for general relativity are given in parentheses): - geodetic de Sitter precession ~dS of the lunar orbit (_~ 1.92"/cy); space-curvature parameter 7 (= 1) and nonlinearity parameter ~ (= 1); time variation of the gravitational coupling parameter GIG (= 0 yr -1) which is important for the unification of the fundamental interactions; -
s
t
equivalence principle (EP) parameter, which for metric theories is r/ - 4~ - 3 - 7 r
o
n
g
(=0); -
-
EP-violating coupling of normal matter to 'dark matter' at the galactic center; coupling constant c~ (= 0) of Yukawa potential for the Earth-Moon distance which corresponds to a test of Newton's inverse square law;
- combination of parameters ~1 - ~0 - 1 (= 0) derived in the Mansouri and Sexl (1977) formalism indicating a violation of special relativity (there: Lorentz contraction parameter ~1 = 1/2, time dilation parameter ~0 = - 1 / 2 ) ; -- OL1 (= 0) and c~2 (= 0) which parameterize 'pre-
ferred frame' effects in metric gravity. Most relativistic effects produce periodic perturbations of the Earth-Moon range n
ArEM -- E Ai cos(w/At + ~i).
(1)
i=1
orbit of the Earth-Moon system about the Sun at one epoch;
-
- mass of the Earth-Moon system times the gravitational constant; -
t
h
lowest mass multipole moments of the Moon; e
- lunar Love number and a rotational energy dissipation parameter; -
lag angle indicating the lunar tidal acceleration responsible for the increase of the Earth-Moon distance (about 3.8 cm/yr), the increase in the lunar orbit period and the slowdown of Earth's angular velocity.
Ai, czi, and ¢i are the amplitudes, frequencies, and phases, respectively, of the various perturbations. Some example periods of perturbations important for the measurement of various parameters are given in Table 1.Note: the designations should not be used as formulae for the computation of the corresponding periods, e.g. the period 'sidereal-2-annual' has to be calculated as 1/(1/27.32 d - 2/365.25 d) ~ 32.13 d. 'secular + emerging periodic' means the changing orbital frequencies induced by GIG are starting to become better signals than the secular rate of change of the Earth-Moon range in LLR. Fig. 4 represents the sensitivity of the Earth-Moon distance with respect to a possible temporal variation of the gravitational constant in the order of 8 . 1 0 -13,
905
906
J. Mailer. J. G. Williams. S. G. Turyshev. P. J. Shelus
10 e
,
,
,
10 r 10 4
o c
, E -~
+ ~, ., C~l
em
II
¢N
in
~
in lii
m
10 2
~
•"o
*'-
~
E
u~
"o
© ~
,~
o
, E E,-®
u~ ~ ,
= ILl ~.lll
,,
Q~ ~IN10 °
10 .2 n
lO 4
10 "s
0
2000
4000 6000 8000 10000 Days since December 2 7 , 1 9 6 9
12000
10-el 10 -4
Fig. 4. Sensitivity of LLR with respect to
AG/G -
14000
........
,
. . . . . . . .
10 .3
i
. . . . . . . .
10 .2 F r e q u e n c y [1 / d]
i
10 "1
. . . . . . . .
10 °
GIG assuming
8.10 -13 yr -1.
Fig. 6. Power spectrum of a possible equivalence principle violation assuming A(raa/m±) ~ 10-13
10 s
with different amplitudes, so that they can be distinguished and separated from the effects investigated•
,~,E } 10 4
U)
~
o~ "=
E o
= ~
"
10 2
~
II)
==
"10
= I/ j[
O4 o
~o
Table 1. Typical periods of some relativistic quantities, taken
from Mfiller et al. (1999). 10 °
?
Parameter
~'10-2 n
~h-~:o-1
10 -4
6ggalactic
al
10 "s
10 .8 10 .4
........
i 10 -3
........
i 10 .2
Frequency
........
i 10 -1
a2 O/G
........ 100
Typical Periods synodic (29 d 12h44m2.9s) annual (365.25d) sidereal (27d7h43m11.5s) sidereal, annual, sidereal-2.annual, anomal. (27d 13h 18m 33.2s)+annual, synodic 2.sidereal,2-sidereal-anomal., nodal (6798a) secular + emerging periodic
[1 / d]
Fig. 5. Power spectrum of the effect of GIG in the EarthMoon distance assuming AG/G - 8.10 -13 yr -1.
3
the present accuracy of that parameter. It seems as if perturbations of up to 9 meters are still caused, but this range (compared to the ranging accuracy at the cm level) can not fully be exploited, because the lunar tidal acceleration perturbation is similar. The largest periods for dl/G are shown in Fig. 5 and for the EP-parameter in Fig. 6. Obviously many periods are affected simultaneously, because the perturbations, even if caused by a single beat period only (e.g. the synodic month for ~7), change the whole lunar orbit (and rotation) and therefore excite further frequencies. Nevertheless these properties can be used to identify and separate the different effects and to determine corresponding parameters (note that relativistic phenomena show up with typical periods). Other effects, like the asteroids cause similar orbit perturbations (i.e. with the same frequencies) but
The global adjustment of the model by least-squaresfit procedures gives improved values for the estimated parameters and their formal standard errors, while consideration of parameter correlations obtained from the covariance analysis and of model limitations lead to more 'realistic' errors. Incompletely modeled solid Earth tides, ocean loading or geocenter motion, and uncertainties in values of fixed model parameters have to be considered in those estimations. For the temporal variation of the gravitational constant, GIG (6 + 8 ) - 1 0 -13 has been obtained, where the formal standard deviation has been scaled by a factor 3 to yield the given value. This parameter benefits most from the long time span of L L R data and has experienced the biggest improvement over the past years (cf. Mfiller et al. 1999). In contrast, the EP-parameter r/ (-- (6 + 7). 10 -4) benefits most from highest accuracy over a sufficient
Results
-
Chapter 126
long time span (e.g. one year) and a good data coverage over the synodic month, as far as possible. Its improvement was not so big, as the LLR RMS residuals increased a little bit in the past years, compare Fig. 3. The reason for that increase is not completely understood and has to be investigated further. In combination with the recent value of the space-curvature parameter ")/Cassini (")/ -- 1 - (2.1 + 2.3). 10 -5) derived from Doppler measurements to the Cassini spacecraft (Bertotti et al. 2003), the non-linearity parameter ~ can be determined by applying the relationship r/ = 4 / 3 - 3 - ")/Cassini- One obtains / 3 - 1 - (1.5 4-1.8)- 10 -4 (note that using the EP test to determine parameters r/and/3 assumes that there is no composition-induced EP violation). Final results for all relativistic parameters obtained from the IfE (Institute ftir Erdmessung) analysis are shown in Table 2. The realistic errors are comparable with those obtained in other recent investigations, e.g. at JPL (see Williams et al. 1996, 2004a, 2004b, 2005b). Table 2. Determined values for the relativistic quantities and their realistic errors.
Parameter
Results
diff. geod. prec. f~GP - f~deSit ["/cy]
Yuk. coupl, const. O~A=4.105 km spec. relativi. ¢1 ~0 1 intl. of dark matter ~9galactic [cm/s 2 ] 'preferred frame' effect Ctl
(6±io).io -3 (4 4- 5). 10 - 3 (--2 ± 4). 10 -3 (1.5 ± 1.8). 10 -4 (6-q- 7)" 10 -4 (6 4- 8). 10 -13 (3 4 - 2 ) . 10 -11 ( - 5 ± 12). 10 - 5 (4 ± 4) • 10 -14 ( 7+9).I0 -5
'preferred frame' effect c~2
(1.8 4- 2.5) • 10-5
metric par. 7 1 metric par. ~ 1 (direct) and from r/= 4/~ 3 ")'Cassini equiv, principle par. r/
time var. grav. const. G/G [yr-1]
4
•
Potential Capabilities of Lunar Laser Ranging for
Geodesy
and Relativity
timated simultaneously in the standard solution, contribute to the realization of the international terrestrial reference frame, e.g. to the last one, the ITRF2000. 3. Earth rotation: LLR contributes, among others, to the determination of long-term nutation parameters, where again the stable, highly accurate orbit and the lack of non-conservative forces from atmosphere (which affect satellite orbits substantially) is very convenient. Additionally UT0 and VOL (variation of latitude) values are computed (e.g. Dickey et al. 1985), which stabilize the combined EOP series, especially in the 1970s when no good data from other space geodetic techniques were available. The precession rate is another example in this respect. The present accuracy of the longterm nutation coefficients and precession rate fits well with the VLBI solutions (within the present error bars), see (e.g. Williams et al. 2005a,c) 4. Relativity: In addition to the use of LLR in the more 'classical' geodetic areas, the dedicated investigation of Einstein's theory of relativity is of special interest. With an improved accuracy the investigation of further effects (e.g. the Lense-Thirring precession) or those of alternative theories become possible. 5. Lunar physics: By the determination of the libration angles of the Moon, LLR gives access to underlying processes affecting lunar rotation (e.g. Moon's core, dissipation), cf. Williams et al. (2005a). A better distribution of the retroreflectors on the Moon (see Fig. 7) would be very helpful.
Further Applications
In addition to the relativistic phenomena mentioned above, more effects related to lunar physics, geosciences, and geodesy can be investigated. The following items are of special interest: 1. Celestial reference frame: A dynamical realization of the International Celestial Reference System (ICRS) by the lunar orbit is obtained (~ = 0.001") from LLR data. This can be compared and analyzed with respect to the kinematical ICRS from VLBI. Here, the very good longterm stability of the orbit is of great advantage. 2. Terrestrial reference frame: The results for the station coordinates and velocities, which are es-
6. Selenocentric reference frame: The determination of a selenocentric reference frame, the combination with high-resolution images and the establishment of a better geodetic network on the Moon is a further big item, which then allows accurate lunar mapping. The LLR frame is used as reference in many application, e.g. to derive gravity models of the Moon (e.g. Konopliv et al. 2001). 7. Earth-Moon dynamics: The mass of the EarthMoon system, the lunar tidal acceleration, possible geocenter variations and related processes as well as further effects can be investigated in detail.
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8. Time scales: The lunar orbit can also be considered as a long-term stable clock so that LLR can be used for the independent realization of time scales. Those features shall be addressed in the future.
Fig. 7. Distribution of retro-reflectors on the Moon surface. To use the full potential of Lunar Laser Ranging, the theoretical models as well as the measurements require optimization. Using the 3.5 m telescope at the new Apollo site in New Mexico, USA, millimeter ranging becomes possible. To allow the determination of the various quantities of the LLR solution with a total gain of resolution of one order of magnitude, the models have to be up-dated according to the IERS conventions 2003, and made compatible with the IAU 2000 resolutions. This requires, e.g., to better model • higher degrees of the gravity fields of Earth and Moon and their couplings; • the effect of the asteroids (up to 1000); • relativistically consistent torques in the rotational equations of the Moon; • relativistic spin-orbit couplings; • torques caused by other planets like Jupiter; • the lunar tidal acceleration with more periods (diurnal and semi-diurnal); • ocean and atmospheric loading by updating the corresponding subroutines; • nutation using the recommended IAU model;
• the tidal deformation of Earth and Moon; • Moon's interior (e.g. solid inner core) and its coupling to the Earth-Moon dynamics. Besides modeling, the overall LLR processing shall be optimized. The best strategy for the data fitting procedure needs to be explored for (highly) correlated parameters. Finally LLR should be prepared for a renaissance of lunar missions where transponders (e.g. Degnan 2002) or new retro-reflectors may be deployed on the surface of the Moon which would enable many pure SLR stations to observe the Moon. NASA is planning to return to the Moon by 2008 with Lunar Reconnaissance Orbiter (LRO, 2005), and later with robotic landers, and then with astronauts in the middle of the next decade. The primary focus of these planned missions will be lunar exploration and preparation for trips to Mars, but they will also provide opportunities for science, particularly if new reflectors are placed at more widely separated locations than the present configuration (see Fig. 7). New installations on the Moon would give stronger determinations of lunar rotation and tides. New reflectors on the Moon would provide additional accurate surface positions for cartographic control (Williams et al. 2005b), would also aid navigation of surface vehicles or spacecraft at the Moon, and they also would contribute significantly to research in fundamental and gravitational physics, LLR-derived ephemeris and lunar rotation. Moreover in the case of co-location of microwave transponders, the connection to the VLBI system may become possible which will open a wide range of further activities such as frame ties. 5
Conclusions
For the IERS, LLR has contributed to the realization of the International Terrestrial Reference Frame ITRF2000 and to combined solutions of Earth Orientation Parameters. Additionally, LLR has become a technique for measuring a variety of relativistic gravity parameters with unsurpassed precision. No definitive violation of the predictions from general relativity are found. Both the weak and strong forms of the EP are verified, while strong empirical limitations on any inverse square law violation, time variation of G, and preferred frame effects are also obtained. LLR continues as an active program, and it can remain as one of the most important tools for testing Einstein's general relativity theory of gravitation if appropriate observations strategies are adopted and
Chapter 126 • Potential Capabilities of Lunar Laser Ranging for Geodesy and Relativity
if the basic L L R m o d e l is further e x t e n d e d and imp r o v e d d o w n to the m i l l i m e t e r level o f accuracy. Additional r a n g i n g devices on the M o o n w o u l d have benefits for lunar science, f u n d a m e n t a l physics, control n e t w o r k s for surface m a p p i n g , and navigation. D e m o n s t r a t i o n o f active devices w o u l d prepare the w a y for very accurate r a n g i n g to Mars and other solar s y s t e m bodies. Acknowledgments. Current LLR data is collected, archived and distributed under the auspices of the international Laser Ranging Service (ILRS). All former and current LLR data is electronically accessible through the European Data Center (EDC) in Munich, Germany and the Crustal Dynamics Data Information Service (CDDIS) in Greenbelt, Maryland. The following web-site can be queried for further information: http://ilrs.gsfc.nasa.gov. We also acknowledge with thanks, that the more than 35 years of LLR data, used in these analyses, have been obtained under the efforts of personnel at the Observatoire de la Cote dAzur, in France, the LURE Observatory in Maui, Hawaii, and the McDonald Observatory in Texas. A portion of the research described in this paper was carried out at the Jet Propulsion Laboratory of the California Institute of Technology, under a contract with the National Aeronautics and Space Administration. References
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(Part i); No. 7, 515-524 (Part ii); No. 10, 809-814 (Part iii), 1977. Mtiller, J.: FESG/TUM, Report about the LLR Activities. ILRS Annual Report 1999, M. Pearlman, L. Taggart (eds.), R 204-208, 2000. Mfiller, J.: FESG/TUM, Report about the LLR Activities. ILRS Annual Report 2000, M. Pearlman, M. Torrence, L. Taggart (eds.), R 7-35 - 7-36, 2001.
Mtiller, J., and Nordtvedt, K.: Lunar laser ranging and the equivalence principle signal. Physical Review D, 58, 062001, 1998. Mfiller, J., Nordtvedt, K., Schneider, M., and Vokrouhlickg, D.: improved Determination of Relativistic Quantities from LLR. In: Proceedings of the l lth International Workshop on Laser Ranging Instrumentation, held in Deggendorf, Germany, Sept. 21-25, 1998, BKG Vol 10, R 216-222, 1999.
Mfiller, J., Nordtvedt, K., and Vokrouhlick), D.: Improved constraint on the c~1 PPN parameter from lunar motion. Physical Review D, 54, R5927-R5930, 1996.
Mfiller, J., and Tesmer, V.: Investigation of Tidal Effects in Lunar Laser Ranging. Journal of Geodesy, Vol. 76, R 232237, 2002. Williams, J.G., Newhall, X.X. and Dickey, J.O.: Relativity parameters determined from lunar laser ranging. Physical Review D, 53, 6730, 1996. Williams, J.G., Turyshev, S.G., and Boggs, D.H.: Progress in lunar laser ranging tests of relativistic gravity. Phys. Rev. Lett., 93, 261101, 2004a, gr-qc/0411113. Williams, J.G., Turyshev, S.G., and Murphy, T.W., Jr.: Improving LLR Tests of Gravitational Theory. (Fundamental Physics meeting, Oxnard, CA, April 2003), International Journal of Modem Physics D, V13 (No. 3), 567-582, 2004b, gr-qc/0311021. Williams, J.G., Boggs, D.H., and Ratcliff, J.T.: Lunar Fluid Core and Solid-Body Tides. Abstract No. 1503 of the Lunar and Planetary Science Conference XXXVI, March 14-18, 2005a. Williams, J.G., Turyshev, S.G., and Boggs, D.H.: Lunar Laser Ranging Tests of the Equivalence Principle with the Earth and Moon. In proceedings of 'Testing the Equivalence Principle on Ground and in Space', Pescara, Italy, September 20-23, 2004, C. Laemmerzahl, C.W.F. Everitt and R. Ruffini (eds.), to be published by Springer Verlag, Lect. Notes Phys., 2005b, gr-qc/0507083. Williams, J.G., Turyshev, S.G., Boggs, D.H., and Ratcliff, J.T.: Lunar Laser Ranging Science: Gravitational Physics and Lunar Interior and Geodesy, In proceedings of "35th COSPAR Scientific Assembly," July 18-24, 2004, Paris, France. Accepted, to be published in Advances of Space Research, 2005c, gr-qc/0412049.
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