VI Hotine-Marussi Symposium on Theoretical and Computational Geodesy: Wuhan, China 29 May - 2 June, 2006 (International Association of Geodesy Symposia) (International Association of Geodesy Symposia)

International Association of Geodesy Symposia Fernando Sansò, Series Editor International Association of Geodesy Symp...

Author: Peiliang Xu | Jingnan Liu | Athanasios Dermanis

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International Association of Geodesy Symposia Fernando Sansò, Series Editor

International Association of Geodesy Symposia Fernando Sansò, 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 110: 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 Its 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 … Symposium 131: Geodetic Deformation Monitoring: From Geophysical to Engineering Roles Symposium 132: VI Hotine-Marussi Symposium on Theoretical and Computational Geodesy

VI Hotine-Marussi Symposium on Theoretical and Computational Geodesy IAG Symposium Wuhan, China 29 May - 2 June, 2006

Edited by Peiliang Xu Jingnan Liu Athanasios Dermanis

Volume Editors

Series Editor

Dr. Peiliang Xu Kyoto University Disaster Prevention Research Institute Uji, Kyoto 611-0011 Japan

Professor Fernando Sansó Polytechnic of Milan D.I.I.A.R. – Surveying Section Piazza Leonardo da Vinci, 32 20133 Milan Italy

Professor Jingnan Liu Wuhan University GNSS Engineering Res. Center 430079 Wuhan China Professor Athanasios Dermanis Aristotle University of Thessaloniki Department of Geodesy & Surveying University Box 503 54124 Thessaloniki Greece

ISBN: 978-3-540-74583-9

e-ISBN: 978-3-540-74584-6

International Association of Geodesy Symposia ISSN: 0939-9585 Library of Congress Control Number: 2007933501 c 2008 Springer-Verlag Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover Design: WMXDesign GmbH, Heidelberg Printed on acid-free paper 9 8 7 6 5 4 3 2 1 springer.com

Preface

The famous Hotine-Marussi Symposium series is held once every four years and has been traditionally focused on mathematical geodesy. The VI HotineMarussi Symposium was organized by the Intercommission Committee on Theory (ICCT) and successfully held from 29 May to 2 June, 2006, at Wuhan University, PR China, with 162 registered scientists and students from 20 countries and regions, in addition to many more unregistered attendees. It was kindly sponsored by the International Association of Geodesy and Wuhan University. The VI Hotine-Marussi Symposium was unique in the senses that: (i) this is the first Hotine-Marussi symposium to go beyond mathematical geodesy; (ii) this is the first time for a Hotine-Marussi symposium to be held outside Europe; and (iii) this is the first time that a Hotine-Marussi symposium was organized by an IAG entity instead of by Prof. F. Sanso and his group, as was traditionally the case. An attentive reader might soon notice the change of the title for the VI Hotine-Marussi Symposium. Indeed, this should be one of the most important aspects of the Symposium and was carefully designed as a result of many hours of discussion among Prof. A. Dermanis (ICCT Vice President), Prof. F. Sanso (IAG Past President and past organizer of the Hotine-Marussi symposia), Prof. J.N. Liu (President of Wuhan University) and P.L. Xu (ICCT President), in particular, also among the Scientific Committee members Prof. J.Y. Chen, Prof. B. Chao, Prof. H. Drewes, Prof. H.Z. Hsu, Prof. C. Jekeli, Dr. N.E. Neilan, Prof. C. Rizos and Prof. S.H. Ye. In fact, as part of the IAG restructuring, the ICCT was formally approved and established after the IUGG XXIII Assembly in Sapporo, to succeed the former IAG Section IV on General Theory and Methodology, and more importantly, to actively and directly interact with other IAG Entities. The most important goals and/or targets of the ICCT are: (1) to strongly encourage frontier mathematical and physical research, directly motivated

by geodetic need/practice, as a contribution to science/engineering in general and the foundations for Geodesy in particular; (2) to provide the channel of communication amongst the different IAG entities of commissions/services/projects, on the ground of theory and methodology, and directly cooperate with and support these entities in the topics-oriented work; (3) to help the IAG in articulating mathematical and physical challenges of geodesy as a subject of science and in attracting young talents to geodesy; and (4) to encourage closer research ties with and directly gets involved with relevant areas of the Earth Sciences, bearing in mind that geodesy has been playing an important role in understanding the physics of the Earth. In order to partly materialize the ICCT missions, we decided to use the VI Hotine-Marussi Symposium as a platform for promoting what we believe would be of most importance in the near future and for strengthening the interaction with commissions. This should clearly explain why we further decided to modify the traditional title of Hotine-Marussi symposia from “Mathematical Geodesy” to “Theoretical and Computational Geodesy”, with a subtitle to emphasize challenge, opportunity and role of modern geodesy, and why you could see from our symposium programs that the IAG President Prof. G. Beutler, the IAG Secrectary General Prof. C.C. Tscherning and IAG commission Presidents Prof. H. Drewes, Prof. C. Rizos were invited to deliver invited talks at the Symposium, with our great honour, pleasure and gratitude. Scientifically, recognizing that geodetic observing systems have advanced to such an extent that geodetic measurements:

(i) are now of unprecedented high accuracy and quality, can readily cover a region of any scale up to tens of thousands of kilometers, consist v

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of non-conventional data types, and can be provided continuously; (ii) consequently, demand new mathematical modeling in order to obtain best possible benefit of such technological advance; and (iii) are finding applications that were either not possible due to accuracy limit or were not thought of as part of geodesy such as space weather and/or earth-environmental monitoring, we designed and selected for the symposium the following five topics: (i) Satellite gravity missions: open theoretical problems and their future application; (ii) Earth-environmental, disaster monitoring and prevention by Geodetic methods; (iii) GNSS: Mathematical theory, engineering applications, reference system definition and monitoring; (iv) Deterministic and random fields analysis with application to Boundary Value Problems, approximation theory and inverse problems; and (v) Statistical estimation and prediction theory, quality improvement and data fusion.

Preface

Some of these are either of urgent importance to geodesy or are of potentially fundamental importance to geodesy, but not necessarily limited to geodesy, at the very least, from our point of view. To name a few examples, let us say that: (i) satellite gravity missions are of current importance in and far beyond geodesy, environmental monitoring, for example; (ii) seafloor geodesy will become essential in the next one or two decades in Earth Sciences, even though the invited speakers could not find time to contribute their papers on the topic; and (iii) mixed integer linear models should be a subject that geodesists can make greatest possible contributions to mathematics and statistics. Finally, we thank the International Association of Geodesy and Wuhan University for financial support. We thank all the conveners: B. Chao, D. Wolf, N. Sneeuw, J.T. Freymueller, K. Heki, C.K. Shum, Y. Fukuda, D.-N. Yuan, P. Teunissen, A. Dermanis, H. Drewes, Z. Altamimi, B. Heck, Karlsruhe, P. Holota, J. Kusche, B. Schaffrin, Y.Q. Chen, H. Kutterer and Y. Yang, for their hard work to convene and to take care of the review process of the Proceedings papers, which are essential to guarantee the success of the Symposium and the quality of the Proceedings. We also thank the LOC team, in particular, Dr. X. Zhang and Ms Y. Hu, for all their hard work. Peiliang Xu Jingnan Liu Athanasios Dermanis

Contents

Part I: Satellite Gravity and Geodynamics Do We Need New Gravity Field Recovery Techniques for the New Gravity Field Satellites? . . . . . . . . . . . . . . K.H. Ilk, A. Löcher, T. Mayer-Gürr

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A Localizing Basis Functions Representation for Low–Low Mode SST and Gravity Gradients Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 W. Keller Gravity Field Modeling on the Basis of GRACE Range-Rate Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 P. Ditmar, X. Liu The Torus Approach in Spaceborne Gravimetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 C. Xu, M.G. Sideris, N. Sneeuw Gravity Recovery from Formation Flight Missions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 N. Sneeuw, M.A. Sharifi, W. Keller GRACE Gravity Model Derived by Energy Integral Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Z.T. Wang, J.C. Li, D.B. Chao, W.P. Jiang Robust Estimation and Robust Re-Weighting in Satellite Gravity Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 J.P. van Loon Topographic and Isostatic Reductions for Use in Satellite Gravity Gradiometry . . . . . . . . . . . . . . . . . . . . . . . . . 49 F. Wild, B. Heck Gravity Change After the First Water Impoundment in the Three-Gorges Reservoir, China . . . . . . . . . . . . . . . 56 S. Sun, A. Xiang, C. Shen, P. Zhu, B.F. Chao Continental Water Storage Changes from GRACE Line-of-Sight Range Acceleration Measurements . . . . . . 62 Y. Chen, B. Schaffrin, C.K. Shum Atmospheric De-Aliasing Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 T. Peters First Results of the 2005 Seismology – Geodesy Monitoring Campaign for Volcanic Crustal Deformation in the Reykjanes Peninsula, Iceland . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 J. Nicolas, S. Durand, S. Cravoisier, L. Geoffroy, C. Dorbath The Statistical Analysis of the Eigenspace Components of the Strain Rate Tensor Derived from FinnRef GPS Measurements (1997–2004) in Fennoscandia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 J. Cai, E.W. Grafarend, H. Koivula, M. Poutanen GPS Research for Earthquake Studies in India . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 M. N. Kulkarni Preliminary Results of Subsidence Measurements in Xi’an by Differential SAR Interferometry . . . . . . . . . . . 94 C. Zhao, Q. Zhang, X. Ding, Z. Li vii

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Contents

Part II: Reference Frame, GPS Theory, Algorithms and Applications Accuracy Assessment of the ITRF Datum Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Z. Altamimi, X. Collilieux, C. Boucher The ITRF Beyond the “Linear” Model. Choices and Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 A. Dermanis Approach for the Establishment of a Global Vertical Reference Level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 L. Sánchez The Research Challenges of IAG Commission 4 “Positioning & Applications” . . . . . . . . . . . . . . . . . . . . . . . . . . 126 C. Rizos Integrated Adjustment of LEO and GPS in Precision Orbit Determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 J.H. Geng, C. Shi, Q.L. Zhao, M.R. Ge, J.N. Liu Reduced-Dynamic Precise Orbit Determination Based on Helmert Transformation . . . . . . . . . . . . . . . . . . . . . . 138 J. Chen, J. Wang GNSS Ambiguity Resolution: When and How to Fix or not to Fix? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 P.J.G. Teunissen, S. Verhagen Probabilistic Evaluation of the Integer Least-Squares and Integer Aperture Estimators . . . . . . . . . . . . . . . . . . . 149 S. Verhagen, P.J.G. Teunissen The Evaluation of the Baseline’s Quality Based on the Probabilistic Characteristics of the Integer Ambiguity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 R. Xu, D. Huang, C. Li, L. Zhou, L. Yuan Kinematic GPS Batch Processing, a Source for Large Sparse Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 M. Roggero Optimal Recursive Least-Squares Filtering of GPS Pseudorange Measurements . . . . . . . . . . . . . . . . . . . . . . . . . 166 A.Q. Le, P.J.G. Teunissen A Comparison of Particle Filters for Personal Positioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 D. Petrovich, R. Piché An Effective Wavelet Method to Detect and Mitigate Low-Frequency Multipath Effects . . . . . . . . . . . . . . . . . 179 E.M. Souza, J.F.G. Monico, W.G.C. Polezel, A. Pagamisse Regional Tropospheric Delay Modeling Based on GPS Reference Station Network . . . . . . . . . . . . . . . . . . . . . . 185 H. Yin, D. Huang, Y. Xiong Research on GPS Receiver Antenna Gain and Signal Carrier-to-Noise Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 J. Liu, J. Huang, H. Tian, C. Liu Optimal Combination of Galileo Inter-Frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 B. Li, Y. Shen Closed-Form ADOP Expressions for Single-Frequency GNSS-Based Attitude Determination . . . . . . . . . . . . 200 D. Odijk, P.J.G. Teunissen, A.R. Amiri-Simkooei

Contents

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Safety Monitoring for Dam Construction Crane System with Single Frequency GPS Receiver . . . . . . . . . . . . 207 W. Wang, J. Guo, B. Chao, N. Luo PPP for Long-Range Airborne GPS Kinematic Positioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 X. Zhang, J. Liu, R. Forsberg The Uniform Tykhonov-Phillips Regularization (α-weighted S-homBLE) and its Application in GPS Rapid Static Positioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 J. Cai, E.W. Grafarend, C. Hu, J. Wang

Part III: Statistical Estimation: Methods and Applications Collocation with Integer Trend . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 P.J.G. Teunissen Multidimensional Statistical Tests for Imprecise Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 H. Kutterer, I. Neumann Multivariate Total Least – Squares Adjustment for Empirical Affine Transformations . . . . . . . . . . . . . . . . . . . . 238 B. Schaffrin, Y.A. Felus Robust Double-k-Type Ridge Estimation and Its Applications in GPS Rapid Positioning . . . . . . . . . . . . . . . . . 243 S. Han, Q. Gui, C. Ma Adaptive Robust Sequential Adjustment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 L. Sui, Y.Y. Liu, W. Wang, P. Fan Application of Unscented Kalman Filter in Nonlinear Geodetic Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 D. Zhao, Z. Cai, C. Zhang Order Statistics Filtering for Detecting Outliers in Depth Data along a Sounding Line . . . . . . . . . . . . . . . . . . . 258 M. Li, Y.C. Liu, Z. Lv, J. Bao Stepwise Solutions to Random Field Prediction Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 M. Reguzzoni, N. Tselfes, G. Venuti Maximum Possibility Estimation Method with Application in GPS Ambiguity Resolution . . . . . . . . . . . . . . . 269 X. Wang, C. Xu Variance Component Estimation by the Method of Least-Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 P.J.G. Teunissen, A.R. Amiri-Simkooei Noise Characteristics in High Precision GPS Positioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 A.R. Amiri-Simkooei, C.C.J.M. Tiberius, P.J.G. Teunissen Helmert Variance Component Estimation-based Vondrak Filter and its Application in GPS Multipath Error Mitigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 X.W. Zhou, W.J. Dai, J.J. Zhu, Z.W. Li, Z.R. Zou Statistical Analysis of Negative Variance Components in the Estimation of Variance Components . . . . . . . . . 293 B. Gao, S. Li, W. Li, X. Wang A Method to Adjust the Systematic Error along a Sounding Line in an Irregular Net . . . . . . . . . . . . . . . . . . . . . 297 M. Li, Y.C. Liu, Z. Lv, J. Bao Research on Precise Monitoring Method of Riverbed Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 J.H. Zhao, H.M. Zhang

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Contents

Part IV: Geodetic Boundary Value Problems and Inverse Problem Theory On the Universal Solvability of Classical Boundary-Value Problems of Potential Theory: A Contribution from Geodesy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 F. Sansò, F. Sacerdote Model Refinements and Numerical Solutions of Weakly Formulated Boundary-Value Problems in Physical Geodesy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 P. Holota, O. Nesvadba On an Ellipsoidal Approach to the Singularity-Free Gravity Space Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 G. Austen, W. Keller Local Geoid Modelling From Vertical Deflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 W. Freeden, S. Gramsch, M. Schreiner Monte Carlo Integration for Quasi–linear Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 B. Gundlich, J. Kusche Wavelet Evaluation of Inverse Geodetic Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 M. El-Habiby, M.G. Sideris, C. Xu Correcting the Smoothing Effect of Least-Squares Collocation with a Covariance-Adaptive Optimal Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 C. Kotsakis Analytical Downward and Upward Continuation Based on the Method of Domain Decomposition and Local Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356 Y.M. Wang, D.R. Roman, J. Saleh Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361

Contributors

Z. Altamimi Institut Géographique National, LAREG, 6-8 Avenue Blaise Pascal, 77455 Marne-la-Vallée, France

B.F. Chao College of Earth Sciences, National Central University, Taiwan; also NASA Goddard Space Flight Center, USA

A.R. Amiri-Simkooei Delft Institute of Earth Observation and Space Systems (DEOS), Delft University of Technology, Kluyverweg 1, 2629 HS Delft, The Netherlands, e-mail: [email protected]

D.B. Chao The Key Laboratory of Geospace Environment and Geodesy, Ministry of Education, School of Geodesy and Geomatics, Wuhan University, Wuhan 129 Luoyu Road, Wuhan 430079, P.R. China

G. Austen Stuttgart University, Geodetic Institute, GeschwisterScholl-Str. 24/D, 70174 Stuttgart, Germany

J. Chen Department of Surveying and Geo-informatics, Tongji University, Siping Road 1239, 200092 Shanghai, P.R. China; GeoForschungsZentrum Potsdam, Telegrafenberg A17, 14473 Potsdam, Germany

J. Bao Department of Hydrography and Cartography, Dalian Naval Academy, Dalian 116018, P.R. China J. Bolte GeoForschungsZentrum Potsdam, Telegrafenberg A17, 14473 Potsdam, Germany C. Boucher Conseil Général des Ponts et Chaussées, tour Pascal B, 92055 La Défense, France J. Cai Department of Geodesy and GeoInformatics, University of Stuttgart Geschwister-Scholl-Str. 24, D-70174 Stuttgart, Germany, e-mail: [email protected] Z. Cai Global Information Application and Development Center of Beijing, 100094 Beijing, P.R. China B. Chao The School of Geodesy and Geomatics, Wuhan University, Wuhan, P.R. China

Y. Chen Geodetic Science, School of Earth Sciences, The Ohio State University, 125 S. Oval Mall, 275 Mendenhall Lab., Columbus, Ohio 43210, USA, e-mail: [email protected] X. Collilieux Institut Géographique National, LAREG, 6–8 Avenue Blaise Pascal, 77455 Marne-la-Vallée, France S. Cravoisier Laboratoire de Géodynamique des Rifts et des Marges Passives (LGRMP), Université du Maine, UFR Sciences et Techniques, Bât. Géologie, Avenue O. Messiaen, F-72085 Le Mans Cedex 09, France W.J. Dai Department of Survey Engineering and Geomatics, Central South University, Changsha, Hunan Province 410083, P.R. China xi

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A. Dermanis Department of Geodesy and Surveying, Aristotle University of Thessaloniki, University Box 503, 54124 Thessaloniki, Greece X. Ding Department of Land Surveying and Geo-Informatics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, P.R. China P. Ditmar Delft Institute of Earth Observation and Space Systems (DEOS), Delft University of Technology, Kluyverweg 1, P.O. Box 5058, 2600 GB Delft, The Netherlands C. Dorbath Institut Physique du Globe de Strasbourg (IPGS), 5 Rue René Descartes, F-67084 Strasbourg Cedex, France S. Durand Laboratoire de Géodésie et Géomatique (L2G), Ecole Supérieure des Géomètres et Topographes (ESGT/CNAM), 1 Boulevard Pythagore, F-72000 Le Mans, France M. El-Habiby Department of Geomatics Engineering, The University of Calgary, 2500 University Drive N.W., Calgary, Alberta, Canada T2N 1N4 P. Fan Zhengzhou Institute of Surveying and Mapping, 66th Longhai Middle Road, Zhengzhou 450052, P.R. China Y.A. Felus Surveying Engineering Department, Ferris State University, Big Rapids, Michigan, USA R. Forsberg Geodynamic department, Danish National Space Center, Juliane Maries Vej 30, 2100 Copenhagen, Denmark W. Freeden University of Kaiserslautern, Geomathematics Group, 67653 Kaiserslautern, P.O. Box 3049, Germany, e-mail: [email protected]

Contributors

B. Gao Civil Engineering Department, Shijiazhuang Railway Institute, Shijiazhuang 050043, P.R. China M.R. Ge GNSS Research Center, Wuhan University, Wuhan, Hubei, P.R. China J.H. Geng GNSS Research Center, Wuhan University, Wuhan, Hubei, P.R. China L. Geoffroy Laboratoire de Géodynamique des Rifts et des Marges Passives (LGRMP), Université du Maine, UFR Sciences et Techniques, Bât. Géologie, Avenue O. Messiaen, F-72085 Le Mans Cedex 09, France E.W. Grafarend Department of Geodesy and GeoInformatics, University of Stuttgart Geschwister-Scholl-Str. 24, D-70174 Stuttgart, Germany S. Gramsch University of Kaiserslautern, Geomathematics Group, 67653 Kaiserslautern, P.O. Box 3049, Germany, e-mail: [email protected] V. Grund GeoForschungsZentrum Potsdam, Telegrafenberg A17, 14473 Potsdam, Germany Q. Gui Institute of Science, Information Engineering University, No. 62, Kexue Road, Zhengzhou 450001, Henan, P.R. China B. Gundlich Central Institute for Electronics Forschungszentrum Jülich GmbH, 52425 Jülich, Germany; GFZ Potsdam, Telegrafenberg, 14473 Postdam, Germany J. Guo The School of Geodesy and Geomatics, Wuhan University, Wuhan, P.R. China B. Han School of Architectural Engineering, Shandong University of Technology, Zibo, P.R. China

Contributors

S. Han Institute of Science, Information Engineering University, No. 62, Kexue Road, Zhengzhou 450001, Henan, P.R. China B. Heck Geodetic Institute, University of Karlsruhe, Englerstr 7, D-76128 Karlsruhe, Germany P. Holota Research Institute of Geodesy, Topography and Cartography, 25066 Zdiby 98, Praha-východ, Czech Republic, e-mail: [email protected] C. Hu Department of Surveying and Geo-informatics, Tongji University, Siping Road 1239, 200092 Shanghai, P.R. China D. Huang Department of Geomatic Engineering, Southwest Jiaotong University, Chendu 610031, P.R. China J. Huang School of Geodesy and Geomatics, Wuhan University, Wuhan, China K.H. Ilk Institute of Theoretical Geodesy, University of Bonn, Nussallee 17, D-53115 Bonn, Germany W.P. Jiang School of Geodesy and Geomatics, Wuhan University, 129 Luoyu Road, Wuhan 430079, P.R. China W. Keller Stuttgart University, Geodetic Institute, GeschwisterScholl-Str. 24/D, 70174 Stuttgart, Germany J. Klotz GeoForschungsZentrum Potsdam, Telegrafenberg A17, 14473 Potsdam, Germany H. Koivula Department of Geodesy and Geodynamics, Finnish Geodetic Institute, Geodeetinrinne 2, FI-02430 Masala, Finland C. Kotsakis Department of Geodesy and Surveying, Aristotle University of Thessaloniki, University Box 440, GR-54124, Thessaloniki, Greece, e-mail: [email protected]

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M.N. Kulkarni Department of Civil Engineering, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India, e-mail: [email protected] J. Kusche Delft Institute of Earth Observation and Space Systems (DEOS), TU Delft, Kluyverweg 1, P.O. Box 5058, 2600 GB Delft, The Netherlands; GFZ Potsdam, Telegrafenberg, 14473 Potsdam, Germany H. Kutterer Geodetic Institute, Leibniz University of Hannover, Nienburger Straße 1, D-30167 Hannover, Germany A.Q. Le Delft Institute of Earth Observation and Space Systems (DEOS), Delft University of Technology, Delft, The Netherlands B. Li Department of Surveying and Geo-informatics, Tongji University, 1239 Siping Road, Shanghai 200092, P.R. China C. Li Center for Geomation Engineering, Southwest Jiaotong University, Chengdu, P.R. China J.C. Li The Key Laboratory of Geospace Environment and Geodesy, Ministry of Education, School of Geodesy and Geomatics, Wuhan University, 129 Luoyu Road, Wuhan 430079, P.R. China M. Li Department of Hydrography and Cartography, Dalian Naval Academy, Dalian 116018, P.R. China; Geomatics and Applications Laboratory, Liaoning Technical University, Fuxin 123000, P.R. China; Institute of Surveying and Mapping, Information Engineering University, Zhenzhou 450052, P.R. China S. Li Civil Engineering Department, Shijiazhuang Railway Institute, Shijiazhuang 050043, P.R. China; College of Geodesy and Geomatics, Wuhan University, Wuhan 430079, P.R. China W. Li Civil Engineering Department, Shijiazhuang Railway Institute, Shijiazhuang 050043, P.R. China

xiv

Contributors

Z. Li Department of Land Surveying and Geo-Informatics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, P.R. China

C. Ma Institute of Science, Information Engineering University, No. 62, Kexue Road, Zhengzhou, 450001 Henan, P.R. China

Z.W. Li Department of Survey Engineering and Geomatics, Central South University, Changsha, Hunan Province 410083, P.R. China

T. Mayer-Gurr ¨ Institute of Theoretical Geodesy, University of Bonn, Nussallee 17, D-53115 Bonn, Germany

C. Liu School of Geodesy and Geomatics, Wuhan University, Wuhan, China

J.F.G. Monico Department of Cartography, São Paulo State University, UNESP, Roberto Simonsen, 305, Pres. Prudente, SP, Brazil

J. Liu School of Geodesy and Geomatics, Wuhan University, 129, Luoyu Road, Wuhan, China

M. Moreno GeoForschungsZentrum Potsdam, Telegrafenberg A17, 14473 Potsdam, Germany

J.N. Liu GNSS Research Center, Wuhan University, Wuhan, Hubei, P.R. China

O. Nesvadba Land Survey Office, Pod S´ıdliˇstˇem 9, 182 11 Praha 8, Czech Republic, e-mail: [email protected]

X. Liu Delft Institute of Earth Observation and Space Systems (DEOS), Delft University of Technology, Kluyverweg 1, P.O. Box 5058, 2600 GB Delft, The Netherlands, e-mail: [email protected]

I. Neumann Geodetic Institute, Leibniz University of Hannover, Nienburger Straße 1, D-30167 Hannover, Germany

Y.C. Liu Department of Hydrography and Cartography, Dalian Naval Academy, Dalian 116018, P.R. China; Geomatics and Applications Laboratory, Liaoning Technical University, Fuxin 123000, P.R. China; Institute of Surveying and Mapping, Information Engineering University, Zhenzhou 450052, P.R. China Y.Y. Liu Zhengzhou Institute of Surveying and Mapping, 66th Longhai Middle Road, Zhengzhou 450052, P.R. China A. Löcher Institute of Theoretical Geodesy, University of Bonn, Nussallee 17, D-53115 Bonn, Germany N. Luo The School of Geodesy and Geomatics, Wuhan University, Wuhan, P.R. China Z. Lv Institute of Surveying and Mapping, Information Engineering University, Zhenzhou 450052, P.R. China

J. Nicolas Laboratoire de Géodésie et Géomatique (L2G), Ecole Supérieure des Géomètres et Topographes (ESGT/CNAM), 1 Boulevard Pythagore, F-72000 Le Mans, France D. Odijk Delft Institute of Earth Observation and Space Systems (DEOS), Delft University of Technology, Kluyverweg 1, 2629 HS Delft, The Netherlands, e-mail: [email protected] A. Pagamisse Department of Mathematics, São Paulo State University, UNESP, Roberto Simonsen, 305, Pres., Prudente, SP, Brazil T. Peters Institute of Astronomical and Physical Geodesy, Technische Universität München, Arcisstr. 21, D-80290 München, Germany, e-mail: [email protected] D. Petrovich Institute of Mathematics, Tampere University of Technology, P.O. Box 553, 33101 Tampere, Finland

Contributors

R. Piché Institute of Mathematics, Tampere University of Technology, P.O. Box 553, 33101 Tampere, Finland W.G.C. Polezel Department of Cartography, São Paulo State University, UNESP, Roberto Simonsen, 305, Pres., Prudente, SP, Brazil M. Poutanenr Department of Geodesy and Geodynamics, Finnish Geodetic Institute, Geodeetinrinne 2, FI-02430 Masala, Finland M. Reguzzoni Italian National Institute of Oceanography and Applied Geophysics (OGS), c/o Politecnico di Milano, Polo Regionale di Como, Via Valleggio, 11, 22100 Como, Italy

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B. Schaffrin Geodetic Science Program, School of Earth Sciences, The Ohio State University, 125 S. Oval Mall, 275 Mendenhall Lab., Columbus, Ohio 43210, USA M. Schreiner University of Buchs, Laboratory for Industrial Mathematics, Werdenbergstrasse 4, CH-9471 Buchs, Switzerland, e-mail: [email protected] M.A. Sharifi Institute of Geodesy, Universität Stuttgart, Geschwister-Scholl-Str. 24D, D-70174 Stuttgart, Germany C. Shen Institute of Seismology, CEA, Wuhan 430071, P.R. China

C. Rizos School of Surveying and Spatial Information Systems, University of New South Wales, Sydney, NSW 2052, Australia

Y. Shen Department of Surveying and Geo-informatics, Tongji University, 1239 Siping Road, Shanghai 200092, P.R. China

M. Roggero Politecnico di Torino, Department of Land Engineering, Environment and Geotechnologies, Corso Duca degli Abruzzi 24, 10129, Torino, Italy, e-mail: [email protected]

C. Shi GNSS Research Center, Wuhan University, Wuhan, Hubei, P.R. China

D.R. Roman National Geodetic Survey, Silver Spring, MD 20910, USA F. Sacerdote Dipartimento di Ingegneria Civile, Università di Firenze, Via S. Marta 3, 50139 Firenze, Italy J. Saleh National Geodetic Survey, Silver Spring, MD 20910, USA L. Sánchez Deutsches Geodätisches Forschungsinstitut, DGFI, Alfons-Goppel-Str. 11, D-80539 Munich, Germany F. Sansò DIIAR, Politecnico di Milano, Polo Regionale di Como, Via Valleggio, 22100 Como, Italy

C.K. Shum Geodetic Science, School of Earth Sciences, The Ohio State University, 125 S. Oval Mall, 275 Mendenhall Lab., Columbus, Ohio 43210, USA M.G. Sideris Department of Geomatics Engineering, University of Calgary, 2500 University Drive N.W., Calgary, Alberta, Canada T2N 1N4, e-mail: [email protected] N. Sneeuw Geodätisches Institut, Stuttgart Universität, Stuttgart D-70147, Germany, e-mail: [email protected] E.M. Souza Department of Cartography, São Paulo State University, UNESP, Roberto Simonsen, 305, Pres., Prudente, SP, Brazil

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L. Sui Zhengzhou Institute of Surveying and Mapping, 66th Longhai Middle Road, Zhengzhou 450052, P.R. China, e-mail: [email protected] S. Sun Institute of Seismology, CEA, Wuhan 430071, P.R. China P.J.G. Teunissen Delft Institute of Earth Observation and Space systems (DEOS), Delft University of Technology, Kluyverweg 1, 2629 HS Delft, The Netherlands, e-mail: [email protected] H. Tian School of Geodesy and Geomatics, Wuhan University, Wuhan, China C.C.J.M. Tiberius Delft institute of Earth Observation and Space systems (DEOS), Delft University of Technology, Kluyverweg 1, 2629 HS Delft, The Netherlands N. Tselfes DIIAR, Politecnico di Milano, Polo Regionale di Como, Via Valleggio, 11, 22100 Como, Italy J.P. van Loon Delft Institute of Earth Observation and Space Systems (DEOS), TU Delft, Kluyverweg 1, P.O. Box 5058, 2600 GB Delft, The Netherlands G. Venuti DIIAR, Politecnico di Milano, Polo Regionale di Como, Via Valleggio, 11, 22100 Como, Italy S. Verhagen Delft Institute of Earth Observation and Space systems, Delft University of Technology, Kluyverweg 1, 2629 HS Delft, The Netherlands, e-mail: [email protected] J. Wang Department of Surveying and Geo-informatics, Tongji University, Siping Road 1239, 200092 Shanghai, P.R. China W. Wang The School of Geodesy and Geomatics, Wuhan University, P.R. China, e-mail: wangwei [email protected] W. Wang Zhengzhou Institute of Surveying and Mapping, 66th Longhai Middle Road, Zhengzhou 450052, P.R. China

Contributors

X. Wang School of Geodesy & Geomatics, Wuhan University, 129 Luoyu Road, Wuhan 430079, P.R. China; Key Laboratory of Geomatics and Digital Technology of Shandong Province, Shandong University of Science and Technology, 579 Qianwangang, Qingdao 266510, P.R. China; Research Center for Hazard Monitoring and Prevention, Wuhan University, 129 Luoyu Road, Wuhan 430079, P.R. China Y.M. Wang National Geodetic Survey, Silver Spring, MD 20910, USA Z.T. Wang The Institute of Geodesy and Geodynamics, Chinese Academy of Surveying and Mapping, 16 Bei Tai Ping Lu, 100039, Beijing, P.R. China F. Wild Geodetic Institute, University of Karlsruhe, Englerstr 7, D-76128 Karlsruhe, Germany, e-mail: [email protected] A. Xiang Institute of Seismology, CEA, Wuhan 430071, P.R. China Y. Xiong Department of Geomatic Engineering, Southwest Jiaotong University, Chendu 610031, P.R. China C. Xu Department of Geomatics Engineering, The University of Calgary, 2500 University Drive N.W., Calgary, Alberta, Canada T2N 1N4, e-mail: [email protected] C.Q. Xu School of Geodesy & Geomatics, Wuhan University, 129 Luoyu Road, Wuhan 430079, P.R. China; Key Laboratory of Geomatics and Digital Technology of Shandong Province, Shandong University of Science and Technology, 579 Qianwangang, Qingdao 266510, P.R. China; Research Center for Hazard Monitoring and Prevention, Wuhan University, 129 Luoyu Road, Wuhan 430079, P.R. China R. Xu Center for Geomation Engineering, Southwest Jiaotong University, Chengdu, P.R. China H. Yin Earthquake Administration of Shandong Province, Jinan 250014, P.R. China; Department of Geomatic

Contributors

Engineering, Southwest Jiaotong Chengdu 610031, P.R. China

xvii

University,

Road, No. 66#, Zhengzhou, Henan Province, P.R. China

L. Yuan Center for Geomation Engineering, Southwest Jiaotong University, Chengdu, P.R. China

J.H. Zhao School of Geodesy and Geomatics, Wuhan University, Wuhan, Hubei 430079, P.R. China

C. Zhang Department of Geodesy and Navigation Engineering, Zhengzhou Institute of Surveying and Mapping, Longhai Middle Road, No. 66#, Zhengzhou, Henan Province, P.R. China

Q.L. Zhao GNSS Research Center, Wuhan University, Wuhan, Hubei, P.R. China

H.M. Zhang School of Power and Mechanical Engineering, Wuhan University, Hubei 430072, P.R. China Q. Zhang College of Geological Engineering and Geomatics, Chang’an University, No. 126 Yanta Road, Xi’an, Shaanxi, P.R. China X. Zhang School of Geodesy and Geomatics, Wuhan University, 129, Luoyu Road, Wuhan 430079, P.R. China C. Zhao College of Geological Engineering and Geomatics, Chang’an University, No. 126 Yanta Road, Xi’an, Shaanxi, P.R. China D. Zhao School of Geodesy and Geomatics, Wuhan University, Hubei Province, P.R. China; Department of Geodesy and Navigation Engineering, Zhengzhou Institute of Surveying and Mapping, Longhai Middle

L. Zhou Center for Geomation Engineering, Southwest Jiaotong University, Chengdu, P.R. China X.W. Zhou Department of Survey Engineering and Geomatics, Central South University, Changsha, Hunan Province 410083, P.R. China, e-mail: [email protected] J.J. Zhu Department of Survey Engineering and Geomatics, Central South University, Changsha, Hunan Province 410083, P.R. China P. Zhu Royal Observatory of Belgium, Belgium X. Zhu School of Computer Science and Technology, Shandong University of Technology, Zibo, P.R. China Z.R. Zou Department of Survey Engineering and Geomatics, Central South University, Changsha, Hunan Province 410083, P.R. China

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Part I

Satellite Gravity and Geodynamics

Do We Need New Gravity Field Recovery Techniques for the New Gravity Field Satellites? K.H. Ilk, A. Löcher, T. Mayer-Gürr Institute of Theoretical Geodesy, University of Bonn, Nussallee 17, D-53115 Bonn, Germany

Abstract. The classical approach of satellite geodesy consists in deriving the spherical harmonic coefficients representing the gravitational potential from an analysis of accumulated orbit perturbations of artificial satellites with different altitudes and orbit inclinations. This so-called differential orbit improvement technique required the analysis of rather long arcs of days to weeks; it was the adequate technique for satellite arcs poorly covered with observations, mainly precise laser ranging to satellites. The situation changed dramatically with the new generation of dedicated gravity satellites such as CHAMP, GRACE and – in a couple of months – GOCE. These satellites are equipped with very precise sensors to measure the gravity field and the orbits. The sensors provide a very dense coverage with observations independent from Earth based observation stations. The measurement concepts can be characterized by an in-situ measurement principle of the gravitational field of the Earth. In the last years various recovery techniques have been developed which exploit these specific characteristics of the in-situ observation strategy. This paper gives an overview of the various gravity field recovery principles and tries to systemize these new techniques. Alternative in-situ modelling strategies are presented based on the translational and rotational integrals of motion. These alternative techniques are tailored to the in-situ measurement characteristics of the innovative type of satellite missions. They complement the scheme of in-situ gravity field analysis techniques. Keywords. CHAMP, GRACE, GOCE, differential orbit improvement, in-situ measurement principle, integrals of motion, energy integral, balance equations, gravity field recovery

1 Introduction The success of the Global Navigation Satellite Systems (GNSS), the development of microcomputer

technology and the availability of highly sophisticated sensors enabled space borne concepts of gravity field missions such as CHAMP and GRACE and – to be realized in a couple of months – GOCE. The innovative character of these missions is based on the continuous and precise observations of the orbits of the low flying satellites and the extremely precise range and rangerate K-band measurements between the satellites in case of GRACE. In addition, the surface forces acting on these satellites are measured and can be considered properly during the recovery procedure. In case of GOCE components of the gravity gradient are measured by a gravity gradiometer. The orbit decay of GOCE is compensated by a feedback system coupled with the measurement of the surface forces acting on the satellite so that the kinematically computed orbit is purely gravity field determined. For the analysis of the observations frequently the classical approach of satellite geodesy has been applied. It consists basically in deriving the spherical harmonic coefficients representing the gravitational potential from an analysis of accumulated orbit perturbations of artificial satellites with different altitudes and orbit inclinations and of sufficient arc lengths. This was an indispensable requirement in case of the satellites available during the last three decades with its poor coverage with observations. On the other hand, the results based on the data from satellite missions such as CHAMP and GRACE demonstrated that a variety of satellites with varying inclinations and altitudes is not necessary for the new generation of dedicated gravity satellites. The measurement concept of these missions can be characterized by an in-situ principle and the analysis of accumulated orbit perturbations caused by the inhomogeneous structure of the gravity field seems to be not necessary. The question arises whether the gravity field recovery techniques which were tailored to the classical observation configurations are still the proper tools for these new observation scenarios?

3

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K.H. Ilk et al.

In the following section we will shortly characterize the classical techniques of satellite geodesy and point out the characteristical features of these techniques. Then a scheme of alternative techniques is sketched, which tries to take into account the characteristical features of the innovative type of satellite missions. Gravity field recovery results have demonstrated that very precise competitive models can be achieved with these new in-situ techniques.

2 The Classical Techniques of Satellite Geodesy The classical techniques of satellite geodesy are based on the use of satellites as high targets, as test bodies following the force function acting on the satellites and as platforms carrying sensors to detect various features of the Earth system by remote sensing principles. The determination of the gravitational field and selected position coordinates of terrestrial observation stations by using the satellites as test masses can be performed by a differential orbit determination procedure which is based on the classical (in most cases non-relativistic) Newton–Euler formalism 1 d p(t) = K(r, r˙ ; t) → r¨ = a, dt M

(1)

with the force function K(r, r˙ ; t) or the specific force function a, the position, velocity and the acceleration vectors r, r˙ and r¨ , as well as the linear momentum p. A numerical as well as an analytical perturbation strategy has been applied, frequently in a complementary way. The numerical perturbation concept can be characterized by the definitive orbit determination process where differential corrections to the various observed or unknown parameters are determined numerically. It is based on the basic geometric relation ri (t) = Rli (t) + Rl (t),

(2)

with the geocentric position vector ri (t) to the satellite i , the respective topocentric position vector Rli (t), referred to the observation station l and the station vector Rl (t). This equation constitutes the observation model which reads for a specific observation time tk after inserting the observations b¯ i (ranges, direction elements, etc.) and the approximate values for the (unknown) station coordinates x0S and the respective residuals dbi and corrections to the station coordinates dx S

ri (tk ) = Rli tk ; b¯ i + dbi + Rl tk ; x0S + dx S . (3) The orbit model is based on Newton–Euler’s equation of motion r¨ i (t) = a F (t; x F ) + a D (t; xi ),

(4)

where the specific force function is composed of the Earth-related specific force function a F (t; x F ) with the parameters x F and the orbit-related specific disturbance forces a D (t; xi ) with the corresponding model parameters xi . This equation has to be integrated twice based on the initial values ␣0i for the orbit i , so that the non-linear model results in ri (tk ; ␣0i + d␣i , x0i + dxi , x0F + dx F ) = = ri (tk , b¯ i + dbi , x0S + dx S ).

(5)

A linearization leads to the so-called mixed adjustment model. The partial differentials are determined numerically by integrating the variational equations or by approximating the partial differentials by partial differences. Obviously, this model requires satellite arcs of sufficient lengths because of two reasons. On the one hand, the coverage of the satellite arcs with observations was very poor in the past compared to the situation nowadays. Therefore, to achieve a sufficient redundancy it was necessary to use medium or long arcs. On the other hand, to cover the characteristic periodic and secular disturbances caused by the small corrections to the approximate force function parameters it was necessary – at least useful – to use medium or long satellite arcs as well. This fact becomes even more visible by having a closer look at the analytical perturbation strategy. The explicit Lagrange’s perturbation equations expressed by classical Keplerian elements a, i, e, Ω, ω, σ and the disturbing potential R read, e.g. for the orbit inclination i and the right ascension of the ascending node Ω 1 ⭸R ⭸R di = − √ cos i , dt ⭸ω ⭸Ω na 2 1 − e2 sin i (6) 1 dΩ ⭸R = . √ dt na 2 1 − e2 sin i ⭸i Inserting Kaula’s expansions of the disturbing function in terms of the Keplerian elements leads to the famous Kaula’s perturbation equations, with the inclination function Fnmp , the excentricity function G npq , etc. (refer to Kaula, 2000, for the explanation of additional quantities):

Do We Need New Gravity Field Recovery Techniques for the New Gravity Field Satellites? Fnmp G npq Snmpq

di n = · GM⊗ a⊗ dt n,m, p,q GM⊗ a 1 − e2 a n+1 sin i · ((n − 2 p) cos i − m) , ⭸Fnmp /⭸i G npq Snmpq dΩ n = . GM⊗ a⊗ dt n,m, p,q GM⊗ a 1 − e2 a n+1 sin i

the gravity field of the Earth is observed. The only difference to low-low-SST is the fact that the specific force function is dominated mainly by the gravitational acceleration of the Earth, g⊗, and not by the tidal force field G(21)⊗ as in case of low-low-SST or SGG: r¨ = g⊗ .

(7) These equations demonstrate after a careful analysis that the secular effects and the various periodicities can be detected only with arcs of sufficient length which are able to cover these typical disturbance patterns of the Keplerian elements. As typical effects we only want to mention the dependency of the rotation of the nodal line of the orbit plane and the line of apsides by the zonal spherical harmonics. The situation is similar also in case of the numerical perturbation techniques. The practical experiences underline these numerical characteristics of the perturbation strategies.

3 What Is New with the New Gravity Field Satellite Missions? A common feature of the new gravity field measurement techniques is the fact that the differences of the free-fall motion of test masses is used to derive more or less in-situ the field strength of the gravity field. This is obvious in case of Satellite Gravity Gradiometry (SGG); here the relative acceleration of two test masses M1 and M2 in the sensitivity axis r12 is measured. The main part of the acceleration is represented by the tidal force field G(21)⊗ of the Earth which can be approximated by the gravitational tensor ∇g⊗ : r¨ 12 = r12 · ∇g⊗ .

(8)

There is no basic difference to the measurement principle in case of Satellite-to-Satellite Tracking (SST) in the low–low mode where the Earth’s gravity field is measured also in form of the tidal field acting on the relative motion of two satellites. It reads with the line-of-sight unit vector e12 , the reduced mass μ12 and the mutual gravitational attraction of both satellites K21 : e12 · r¨ 12 =

1 K21 + G(21)⊗ · e12 . μ12

(9)

In this case the tidal force G(21)⊗ cannot be approximated with sufficient accuracy by the gravitational tensor. The same principle holds also in case of the free-fall absolute gravimetry or by the use of precisely determined kinematical orbits for gravity field recovery; here the free fall of a test mass with respect to

5

(10)

Obviously, the in-situ character of these measurement principles does not require the analysis of long arcs with respect to accumulated gravity field effects, because the gravity field is detected more or less directly. It should be pointed out that in all these different measurement scenarios the in-situ observations contain the complete spectral band of the gravity field. Therefore, the frequently expressed argument that long wavelength features of the gravity field cannot be detected in such an in-situ way is certainly not true. The restrictions with respect to the signal content in certain observables are caused by the spectral limitations of the measurement apparatus, such as in case of a satellite gravity gradiometer, as envisaged for the GOCE mission.

4 A Systematic of In-Situ Gravity Field Recovery Techniques We define in-situ gravity field recovery concepts as those which are based in principle on the precisely observed free-fall motion of a test mass within the Earth’s gravity field. This group of gravity measurement techniques covers not only the absolute gravity measurement concepts based on the free-fall principle, but also SGG and SST in the high–low or low–low mode or by analyzing short precisely determined kinematic arcs (POD) with respect to the Earth (Figure 1). In the following, we will refer without loss of generality on the motion of a single satellite or test mass with respect to the Earth, but formulated in an Inertial Reference System. The gravity field recovery techniques can be divided in three analysis levels (Figure 2). The analysis level 1 is based directly on the observed precisely determined kinematic positions, derived from GNSS observations. It is related directly to the specific force function via an integral equation of Fredholm type (with the integral kernel K t, t ):

t r(t) = r¯ (t) −

K t, t g(r; t )dt .

(11)

t0

This equation has been applied by Mayer-Gürr et al. (2005) for the determination of the precise

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K.H. Ilk et al.

test masses of a satellite gravity gradiometry ex-periment

POD satellite-to-satellite tracking

absolute gravimetry: free fall

point wise positions: single or twofold differentiation

Fig. 1. In-situ free fall gravity field measurement techniques.

CHAMP gravity field models CHAMP-ITG 01E, 01K and 01S. The solution of this equation can be formulated as well in the spectral domain: r(t) = r¯ (t) +

∞

rν sin (νπτ ) ,

(12)

various authors to determine the gravity field with a sort of generalized Jacobi or energy integral (see e.g. Jekeli (1999) or Gerlach et al. (2003) for the derivation of the CHAMP gravity field model TUM-1s). The use of energy balance relations for the validation of gravity field models and orbit determination results has been treated by Ilk and Löcher (2003) and Löcher and Ilk (2005). In Löcher and Ilk (2006) new balance equations have been formulated for validation and gravity field recovery. These various integrals of motion can be derived from Newton’s equation of motion starting with an operation which transforms the acceleration term into a function f and the force into a function h. If f has the primitive F, the transformed equation ˙ R ¨ − h M, R, R, ˙ K =0 f M, R, R,

results by integration over the time interval [t0 , t] in

˙ − F M, R, R

ν=1

1

sin νπτ g(r; τ )dτ .

(13)

τ =0

The analysis level 2 requires the numerical differentiation of the time series of precise kinematically determined positions at the (left) observation model side and an integration of the force function at the (right) orbit model side. Up to now this possibility has been applied frequently in the last years by

analysis level 1

integral equation of Fredholm type

r simple differentiation

r

analysis level 2

twofold differentiation

analysis level 3

r

t

˙ K dt = C. h M, R, R,

(15)

t0

with the normalized time τ (t). The sinus coefficients are related to the specific force function by the relation (see, e.g., Ilk et al., 2003) 2T 2 rν = − 2 2 π ν

(14)

twofold integration

integrals of motion simple integration

equation of motion

Fig. 2. The three analysis levels of the in-situ gravity field recovery techniques.

The first term represents the “kinetic” term of the observation model, the second term the force function integral of the orbit model. Figure 3 gives an overview of all possible integrals of translational motion and its functional dependencies and Figure 4 shows a similar flow chart for the integrals of rotational motion (Löcher, 2006). Despite their dependencies the various balance equations show specific characteristics if they are applied for validation and gravity field determination tasks. Investigations demonstrated that these alternative balance equations show partly much better properties for validation and gravity field improvements than the frequently used Jacobi integral. The analysis level 3 requires a twofold numerical differentiation at the observation model side and the direct use of the orbit model. This approach is based directly on Newton’s equation of motion, which balances the acceleration vector and the gradient of the gravitational potential. By a twofold numerical differentiation of a moving interpolation polynomial in powers of the (normed) time τ , with a proper degree N, r(τ ) =

N n=0

τn

N

wn j r(tk + τ j ),

(16)

j =0

the parameters of the orbit model can be determined directly based on the discretized Newton–Euler

Do We Need New Gravity Field Recovery Techniques for the New Gravity Field Satellites?

7

− K = 0 MR considered as a vector equation

considered as a triple of scalar equations

i − K i = 0 MR

i ∈{x, y, z}

multiplying two equations crosswise by R i and R j

multiplying eache quation by the other components of the velocity

adding the pairs of equations

adding the triple of equations

time integration scalar multiplication by R

t

− K dt = P MR 0 ∫ t0

b.e. of thelinear momentum kg m / s

multiplying each equation by R i

time integration

t

time integration

1 2 dt = E MR − ∫ K ⋅ R 2 t0 kg m 2 / s 2 total energy

1 2 t MRi − ∫ K i Ri dt = Eii 2 t0

∂E =P ∂R

energyin the coordinates

time integration

kg m 2 / s 2 t

E xx + E yy + E zz = E

time integration

MR i R j − ∫ ( K i R j + K j R i ) dt = Eij t0

2

energy in the coordinate surfaces kg m / s

2

t

MR x R y R z − ∫ ( K x R y R z + K y R x R z + K z R x R y ) dt = E xyz

∂Eii = Pi ∂Ri

t0

b.e. of the momentum volume

kg m 3 / s 3

∂E xyz = E yz ∂R

∂Eij = Pi ∂R

x

j

Fig. 3. Integrals of translational motion and its functional dependencies.

equation of motion. This technique has been successfully applied for the gravity field recovery based on kinematical orbits of CHAMP (Reubelt et al., 2003). A similar technique based on weighted averages of three successive positions in the form of 1 K(r, r˙ ; t) M r(t − ⌬t) − 2r(t) + r(t − ⌬t) = (⌬t)2

r¨ (t) =

(17)

has been applied successfully by Ditmar et al. (2006). Obviously the latter analysis level requires in principle only a subsequent set of precise positions which represents again a short arc and the procedure can again be characterized by the in-situ measurement principle as defined before.

5 Conclusions In this paper alternative in-situ gravity field recovery procedures, applied in the last couple of years,

have been reviewed and additional ones have been proposed. These recent techniques are tailored to the specific characteristics of the new gravity field missions. In the past, only few observations, mostly laser ranging data to the satellites were available. This fact required the use of long arcs and the analysis of accumulated gravity field effects in the observations to cover the periodicities of specific gravity field disturbances. Numerical or analytical differential orbit improvement techniques have been applied to solve for the unknown parameters. Especially the analytical techniques required the modelling of the gravity field by series of spherical harmonics. A disadvantage of these techniques is the accumulation of improperly modelled disturbing forces. The requirement of comparably long arcs causes problems also in case of gaps in the series of observations. The recent gravity field missions such as CHAMP, GRACE and – in a couple of months – GOCE are characterized by the fact that the orbits show a very dense coverage of precise GNSS observations

8

K.H. Ilk et al. vector multiplication by R

MR − K = 0

MR × R − M = 0 considered as a triple of scalar equations

considered as a vector equation

M(R×R)i −Mi =0

i∈{x, y, z}

time integration scalar multiplication by R × R

t

L − ∫ M dt = L 0

multiplying lt each equation by ( R × R )ii

time integration

t0

multiplying each equation by the other components of

Multiplying two equations crosswise by ( R × R ) i and ( R × R )j

R ×R

2

b.e. of the angularr m momentum m kg m / s

1 2 1 t L − ∫ M ⋅ L dt = E 2M M t0

e ttriple adding the of equations

adding the pairs of equations

time integration

total rotational energy kg m 4 / s 2

1 1 t 2 Li − ∫ Mi Li dt = Eii 2M M t0

∂E =L ∂(R × R )

time integration

4 2 rotational energy in the coordinates kg m / s

1 1 t Li L j − ∫ ( M i L j + M j Li ) dt = Eij M M t0

E xx + E yy + E zz = E

time integration 4

rotational energy in the coordinate surfaces kg m / s

2

1 1 t Lx Ly Lz − 2 ∫ ( M x L y Lz + M y Lx Lz + M z Lx Ly ) dt = E xyz 2 M M t0

∂Eii = Li ∂(R × R )i

b.e. of the angular momentum volume

∂Eij ∂(R × R ) j

∂E xyz

= Li

∂(R × R ) x

kg m 6 / s 3

= E yz

Fig. 4. Integrals of rotational motion and its functional dependencies.

and – as a result – very precise kinematical orbits. In addition, highly precise range and range-rate measurements between the GRACE satellites are available and very precise gravity gradient components in case of GOCE will be available soon. Instead of analyzing accumulated orbit disturbances, the gravity field can be determined in a more direct way by in-situ measurement and analysis techniques by using short arcs. This has some advantages: the accumulation of improperly modelled disturbing forces can be avoided. Observation gaps are not critical and it is possible to perform regional gravity field refinements by space localizing base functions. Various investigations have shown that there are additional gravity field signals in the observations over rough gravity field regions. In a forthcoming paper under preparation gravity field recovery tests will be performed based on these different in-situ analysis techniques.

Acknowledgement The support of BMBF (Bundesministerium für Bildung und Forschung) and DFG (Deutsche Forschungs Gemeinschaft) of the GEOTECHNOLOGIEN programme is gratefully acknowledged.

References Ditmar P, Kuznetsov V, van Eck van der Sluijs AA, Schrama E, Klees R (2006) “DEOS CHAMP-01C 70”: a model of the Earth’s gravity field computed from accelerations of the CHAMP satellite, Journal of Geodesy (2006) 79: 586–601. ˇ Gerlach C, Földvary 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, Geophysical Research Letters (2003) 30(20), 2037. Ilk KH, Feuchtinger M, Mayer-Gürr T (2003) Gravity Field Recovery and Validation by Analysis of Short Arcs of a Satellite-to-Satellite Tracking Experiment as CHAMP and GRACE, In: F. Sansò (ed.) A Window on the Future of Geodesy, IUGG General Assembly 2003, Sapporo, Japan, International Association of Geodesy Symposia, Vol. 128, pp. 189–194, Springer. Ilk KH, Löcher A (2003) The Use of Energy Balance Relations for Validation of Gravity Field Models and Orbit Determination Results, In: F. Sansò (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. Jekeli Ch (1999) The determination of gravitational potential differences from satellite-to-satellite tracking, Celestial

Do We Need New Gravity Field Recovery Techniques for the New Gravity Field Satellites? Mechanics and Dynamical Astronomy (1999) 7582, pp. 85–100. Kaula WM (2000) Theory of Satellite Geodesy, Applications of Satellites to Geodesy, Dover Publications, INC, Mineola, New York. Löcher A (2006) A scheme of integrals of motion for gravity field determination based on precisely observed low Earth satellites, poster presented at the General Assembly 2006 of the EGU, April 02–07, 2006, Vienna, Austria. Löcher 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. Lühr, P. Schwintzer, J. Wickert (Eds.) Earth Observation with CHAMP, Results from Three Years in Orbit, pp. 53–58, Springer.

9

Löcher A, Ilk KH (2006) A Validation Procedure for Satellite Orbits and Force Function Models Based on a New Balance Equation Approach, In: Proceedings of the Internat. Symposium Dynamic Planet 2005, Monitoring and Understanding a Dynamic Planet with Geodetic and Oceanographic Tools, August 22–26, 2005, Cairns, Australia. Mayer-Gürr T, Ilk KH, Eicker A, Feuchtinger M (2005) ITGCHAMP01: a CHAMP gravity field model from short kinematical arcs of a one-year observation period, Journal of Geodesy (2005) 78:462–480. Reubelt T, Austen G, Grafarend EW (2003) Harmonic analysis of the Earth’s gravitational field by means of semicontinuous ephemerides of a low Earth orbiting GPStracked satellite. Case study: CHAMP, Journal of Geodesy (2003) 77:257–278.

A Localizing Basis Functions Representation for Low–Low Mode SST and Gravity Gradients Observations W. Keller Institute of Geodesy, Universität Stuttgart, Geschwister-Scholl-Str. 24D, D-70174 Stuttgart, Germany

Abstract. For geophysical/ oceanographic/ hydrological applications of dedicated gravity field missions regional gravity field solutions are of higher interest than the usual global solutions. In order to derive regional solutions, so-called in-situ observations like line-of-sight accelerations or satellite gradiometry data are optimal, since they do not change, if the potential outside a infinitesimal neighborhood of the observation point changes. Therefore, in-situ observations do not introduce influences from outside the region under consideration. The localization on the observation-side has to be balaced by a localization on the model-side. The usual spherical harmonics representation is not appropriate for the desired regional solution, because spherical harmonics have a global support. In order to model local phenomena by base functions with a global support, the superposition of a large number of those global base functions is necessary. For this reason the paper aims at an establishment of a direct relationship between several types of in-situ observations and the unknown coefficients of a localizing basis functions representation of the regional gravity field. Keywords. Satellite-to-satellite tracking, localizing base functions, representation of rotation group, Wigner functions

1 Introduction The temporal data-sampling of the Earth’s gravity field by an orbiting satellite is transformed via the orbital movement of the satellite and the rotation of the Earth into a spatial sampling on the surface of a sphere. In general the resulting data-spacing on the Earth is non-uniform and coarser than the theoretical resolution limit, stemming from the temporal data sampling. The usual technique for the analysis of dedicated gravity field satellite missions is the the representation of the resulting gravity field solution as a 10

series expansion in spherical harmonics. Due to the fact that the related surface spherical harmonics have a global support on the unit sphere and the data sampling is non-uniform, the theoretical resolution limit, deduced from the temporal data sampling rate, cannot be reached and the spherical harmonics solution includes a certain smoothing of details in the gravity field. This becomes obvious when the original observations are compared with synthetic observation, computed from an existing gravity field solution. In Figure 1 the difference between the original GRACE range-rates and the synthetic range-rates computed from the GRACE gravity field solution GGSM02 is plotted. It is clearly visible that the difference is not white noise but contains a residual signal. This residual signal is caused by the fact that, due to their global support and due to the given data-distribution, spherical harmonics are not able to capture all signal details. In order to capture also the residual signal components, two measures have to be taken 1. Representation of the residual (so far not captured gravity field) by localizing basis functions in the region under consideration. 2. Usage of so-called in-situ observation, as e.g. line-of-sight accelerations or satellite gradiometry data, for sensing of the residual field, to make sure that no influences from outside the region under consideration enter the observations. In Keller and Sharifi (2005) it was shown that with proper reductions low–low mode SST observations can be treated as along-track gravity gradients. Therefore, the results to be presented here for gravity gradients do implicitly also hold for low–low mode SST observations. So far the only in-situ observation with a clear relationship to the unknown parameters of a localizing basis function representation are the radial gravity tensor components observations (cf. Freeden et al. (1999)). To the author’s knowledge no

A Localizing Basis Functions Representation for Low–Low Mode SST and Gravity Gradients Observations

Spectral Modeling

residual signal

range-rate differences [m/s]

0,0001

11

measured - GGSM02 derived range-rates

Spectral modeling can be applied in those cases, where the gravity field-related observation can be represented by a so called invariant pseudodifferential operator (PDO) p on C ∞ (σr ), the space of all infinite often differentiable functions on a sphere of radius r . A PDO is called invariant, if it is invariant against rotations g out of S O(3)

5e-05

0

[ pu](g −1x) = p[u(g −1 x)].

-5e-05

52869,63

52869,632 52869,634 time [JD]

52869,636

Fig. 1. Difference between original range-rates and synthetic range-rates of GRACE along a 10 min arc.

other gravity-field related observations have been expressed in a simple analytic form as functionals on localizing base functions in geodetic literature so far. The paper aims at an establishment of simple relationships also for the along-track and the out-ofplane gravity tensor components.

This leads to the consequence, that all surface spherical harmonics Yn,m of the same degree n are eigenfunctions belonging to the same eigenvalue p ∧ (n) ω ω pYn,m ( ) = p ∧ (n) · Yn,m ( ). r r

The eigenvalues p ∧ (n) are called the spherical symbols of the PDO p. Examples for invariant PDOs are the radial derivatives and the Poisson operator PRr for harmonic upward continuation:

2 State-of-the-Art Gravity field modeling by localizing base functions means to approximate the unknown potential V by a linear combination of special base functions: V (x) =

ci ψi (x).

(1)

i

Here the base functions ψi are localizing base functions having the following structure ψi (x) := ψ(gi−1 x), and ψ(x) =

n∈N

gi ∈ S O(3)

(2)

x ) x

(3)

σn2 Pn (e3 ·

p PRr

p ∧ (n) R n+1 r

⭸u/⭸r

− n+1 r

⭸2 u/⭸r 2

(n+1)(n+2) r2

From the addition theorem n 2n + 1 ∗ Pn (ζ · η) = Yn,m (ζ )Yn,m (η), ζ, η ∈ σ1 2 m=−n (5) for each invariant PDO p immediately follows p

where Pn are the Legendre-polynomials and e3 is a unit-vector pointing in the direction of 3rd axis of the underlying cartesian coordinate system. The sequence {σn } controls the decay of the base function ψ. The generic base function ψ is located at the north-pole of the sphere and the actual base functions are the rotated copies of this generic function. So far the only well-established method to relate in-situ observations to a localizing base function representation of the field is an approach which could be called spectral modeling.

(4)

pψi (x) = ψi (x) x p ψ (x) := σn2 · p ∧ (n) Pn (e3 · ) x

(6) (7)

n∈N

Hence the application of an invariant PDO on a base function results in a change of its decay. The spectral modeling assumes that a certain quantity ⌫ ∈ C ∞ (σr ) is given on the sphere σr , which is the image of a unknown function u ∈ C ∞ (σ R ) under the invariant PDO p ⌫ = pu. (8)

12

W. Keller

Both the given data ⌫ and the unknown function u can be represented as linear combinations of systems p of localizing base functions ψi and ψi , respectively. ⌫=

p

ci ψi ,

u=

i

di ψi ,

(9)

i

with the known coefficients di and the unknown coefficients ci . Which leads via

p

ci ψ1 = ⌫ = pu

i

=

di pψi

3 Representation Theory of S O(3) Both the definition of a system of localizing radial basis functions and the establishment of a relationship between such a representation and in-situ observations make use of the representation theory of S O(3). For this purpose the necessary results from representation theory are to be compiled here. The group of rotations of R3 around the origin is denoted by S O(3). It consists of real 3-by-3 orthogonal matrices of determinant +1. To each g = u(γ )a(β)u(α) ∈ S O(3) an operator ⌳(g) acting on L 2 (σ ) can be associated

(10)

(⌳(g) f )(ω) := f (g −1 ω),

i

=

(11)

p

di ψi .

with the matrices a, u given by

i

to a comparison of coefficients di = ci and from there to the desired solution u. The spectral combination is an inversion-free and stable method, but restricted to first and second order radial derivatives as observations. There is an extended literature about spectral modeling. Without attempting to be close to completeness the following newer references are to be mentioned: Freeden et al. (1999), Freeden and Hesse (2002), Freeden and Maier (2003), and Schmidt et al. (2005, 2006). Unfortunately, the spectral modeling is not directly applicable for alongtrack and out-of-plane gravity gradients. The idea to relate those observations to a localizing base function representation of the unknown potential is similar to the classical Lagrangian disturbing theory. There the observed orbital disturbances are expressed as linear combination of multi-periodic functions, weighted by the unknown coefficients of the spherical harmonics expansion of the potential. There are two differences between the classical Lagrangian disturbing theory and the development the paper is aiming at:

1. Instead of spherical harmonics here localizing base functions are to be used. 2. Instead of orbital disturbances gravity gradients in three orthogonal directions are used as observations. The way this goal is to be achieved is similar to the classical Lagrangian disturbing theory: Transformation of the potential representation to a coordinate system, which follows the movement of the satellite cf. Sneeuw (1992).

⎡

⎤ 0 − sin α 1 0 ⎦ 0 cos α

cos α 0 a(α) := ⎣ sin α and

⎡

cos β u(β) := ⎣ − sin β 0

sin β cos β 0

⎤ 0 0 ⎦. 1

(12)

(13)

Every rotated version ⌳(g)Y¯nm of a surface spherical harmonic is the following linear combination of the non-rotated surface spherical harmonics of the same degree: ⌳(g)Y¯nm (ϑ, λ) =

n

¯ λ), ¯ D(g)nkm Y¯nk (ϑ,

(14)

k=−n

with l l (g) = eıkα dkm (β)eımγ , Dkm

(15)

¯ ϑ and λ¯ , λ are co-latitude and longitude in where ϑ, the non-rotated and the rotated system, respectively. l (β) are called Wigner-d funcThe functions dkm tions and are defined as follows (l + m)!(l − m)! l dkm (β) = (−1)m−k (l + k)!(l − k)! β β ×(sin )m−k (cos )k+m (16) 2 2 (m−k,m+k) ×Pl−m (cos β), with Pl(m,n) being the Jacobi Polynomials (Vilenkin 1968).


4 Transformation to Orbital System Local gravity field representation means an approximation of the residual field by rotated versions of the radial base functions δV (ω) =

N

ci ψi (ω).

(17)

to the free parameters ci of the field representation is known. In what follows S O(3) representation theory will be used to establish a similar relationship for the remaining two tensor components δVyy , δVzz . The relationship between the body-fixed and the space fixed system is approximatively given by the following rotation

i=1

In order to determine the unknown coefficients ci and the unknown placements g1−1 e3 in the radial base function representation of the residual field, the residual field has to be related to residual SST or gradiometry observations. If a body-fixed coordinate system x, y, z is attached to the satellite in such a way that x points in radial, y points in along track and z points in out-of-plane direction (see Figure 2), only for the radial tensor component δVx x a simple relationship

z

(⌳(g)ψi )(x) = ⌳(g)

n∈N

n

2 σn 2n + 1

σn

n∈N n

M

2 2n + 1

=

x1

σn

n∈N n

i

R x

R x

n+1

x ) x

n+1

∗ Yn,m (gi e3 ) · ⌳(g)Yn,m (

m=−n

x2

∗ Yn,m (gi e3 )Yn,m (

m=−n

=

ω

(18)

where ω, ⍀, i, M are the mean elements of the orbital arc under consideration. The representation of a radial base function in the rotating system is given by

y

Ω

π π )a(i )u( + ω + M) 2 2

g = u(⍀ − ⌰ −

x x3

13

2 2n + 1

R x

n+1

x ) x

∗ Yn,m (gi e3 ) · Yn,m (g −1 ω). ¯

m=−n

Here, ω¯ is the position of the satellite in the rotating system. Since for an exact circular orbit ω¯ = e1 holds also for weakly eccentric orbits approximatively holds: (⌳(g)ψi )(x) =

2 σn 2n + 1

n∈N n

R x

n+1

∗ Yn,m (gi e3 ) · Yn,m (g −1 e1 )

m=−n

=

σn

n∈N

R x

n+1

Pn ((gi e3 · (g −1 e1 )).

Besides this an equivalent representation of (⌳(g)ψi )(x) is useful: Fig. 2. Body-fixed coordinate system (bottom) and its relationship to the space-fixed system (top).

(⌳(g)ψi )(x) =

n∈N

σn

2 2n + 1

R x

n+1

14

W. Keller n

∗ Yn,m (gi e3 ) · ⌳(g)Yn,m (

m=−n

=

2 σn 2n + 1

n∈N n

σn

n∈N n m=−n n

R x

n+1

∗ Yn,m (gi e3 ) · ⌳(g)Yn,m (

m=−n

=

2 2n + 1

R x

n+1

x ) x

x ) x

×Pn (gi e3 )

R n+1 2 − σn (2n + 1)a 2 x

π

Yn,m (gi e3 )eım( 2 +ω+M) · π

n eık(⍀−⌰− 2 ) dm,k (i )Yn,k (ω). ¯

With the introduction of the abbreviations n

π

n eı[k(⍀−⌰− 2 )] dk,m (i )Yn,k (ω) ¯

⭸2 ⌳(g)ψi ⭸z 2

=

and G n,m (gi , ω, M) := Yn,m (gi e3 )e

ım( π2

+ω+M)

(20)

(⌳(g)ψi )(x) =

2 σn 2n + 1

n∈N n

R x

m=−n ⭸2 ⌳(g)δV (ω)

n∈N

+

×Pn (gi e3 ) 1 a 2 sin2 (M + ω)

2 R n+1 σn 2n + 1 x

n∈N n

this leads to the final result

n+1 (21)

m 2 Yn,m (gi e3 )Yn,m (ge1 )

a 2 sin2 (M + ω)⭸i 2 1 ⭸⌳(g)δV (ω) + a ⭸r (n + 1) R n+2 = σn Ra x

k=−n

(19)

×

m=−n

G n,m (gi , ω, M) ·

G n,m (gi , ω, M) ×

m=−n

×Fn,m (i, ⍀, ⌰)

5 Observation Equations

(22)

n∈N

n∈N n

k=−n

Fn,m (i, ⍀, ⌰) :=

1 ⭸2 ⌳(g)δV (ω) ⭸2 ⌳(g)ψi = ⭸y 2 a 2 ⭸(M + ω)2 1 ⭸⌳(g)δV (ω) + a ⭸r (n + 1) R n+2 σn = Ra x

⭸2 Fn,m (i, ⍀, ⌰) ⭸i 2

Relations (22) establish the analytic relationships between the localizing base functions representation and gravity gradient observations in three orthogonal directions.

The second order derivatives in – x -radial direction – y -along-track direction – z -across-track direction are given by (see Koop 1993): ⭸2 ⌳(g)δV (ω) ⭸2 ⌳(g)ψi = 2 ⭸x ⭸r 2 R n+3 = σn x n∈N

(n + 1)(n + 2) R2 ×Pn (gi e3 · (g −1 e1 )) ×

6 Numerical Example In order to verify the derivations above, a simple forward computation was carried out. For a single GOCE arc the along track gravity-gradient tensor component δVyy was computed twice: Once by numerical orbit computation and once using the relations (22). As gravity field a three-basis functions regional model δV on top of GGSM02 was used. In order to relate the arc to the residual potential, a projection of the satellite ground track onto the residual potential is displayed in Figure 3. Along this track the quantities δVyy were computed both numerically and analytically. In Figure 4 the difference between the true gradiometry signal (i.e. the


15

Fig. 5. Gradient signal (top) and difference between gradient signal and model (botton).

Fig. 3. Track over residual potential.

result of the numerical computation) and its analytical computation is shown. It is clearly visible that the numerical and the analytical solution match almost perfectly. This is an indication of the correctness of the obtained relationship between along-track in-situ observations and regional basis functions representation. For the test of the inverse computation, i.e. the recovery of the regional disturbing potential δV from the observed along-track gravity gradients the same arc was used. Along the arc equidistantly 12 base functions were placed and the amplitudes of these base functions were estimated using the gravity gradient data along the arc. Figure 5 shows in the upper panel the difference between the true potential and the estimated potential. Since the gravity gradient is basically a second order derivative it is blind

for bias and tilt. Therefore, the recovered potential differs from the true potential by a bias and a drift. After bias and drift removal the recovered potential almost perfectly matches the true potential.

7 Conclusions A simple analytic relationship between in-situ observations of the Earth’s gravity field and its representation by localizing base functions has been known only for radial gravity gradients observations. In the paper a similarly simple relationship also for two other gravity gradient components has been found: The along-track and the out-of-plane component. For the derivation of these relations the same techniques was used as in Lagrangian disturbing theory for the conversion of the spherical harmonics representation of the gravity field to orbital elements: The representation theorie of the rotation group S O(3). In a forward computation the derived analytic formulea were compared to gravity gradients obtained by numerical differentiation. Both results match almost perfectly. For the inverse computation gravity gradients along a single arc were resolved by 12 equally spaced base functions along the ground-track of the satellite arc. The resolved potential along the arc differs from the true potential by a bias and a tilt. After bias and tild removal both data agree very well.

References

Fig. 4. Gradient signal (top) and difference between gradient signal and model (bottom).

Freeden, W., Glockner, O. and M. Thalhammer (1999) Multiscale Gravitational Field Recovery from GPS-Satelliteto-Satellite Trackin. Studia Geophysica et Geodaetica, 43: 229–264. Freeden W. and K. Hesse (2002) On the Multiscale Solution of Satellite Problems by Use of Locally Supported Kernel

16 Functions Corresponding to Equidistributed Data on Spherical Orbits. Studia Scientiarum Mathematicarum Hungarica, 39:37–74. Freeden W. and T. Maier (2003) SST-Regularisierung durch Multiresolutionstechniken auf der Basis von CHAMPDaten. Allgemeine Vermessungs-Nachrichten (AVN) 110 :162–175. Schmidt M., Fabert O. and C.K. Shum (2005) On the estimation of a multi resolution representation of the gravity field based on spherical harmonics and wavelets. Journal of Geodynamics 39: 512–526. Schmidt M., Han S.-C., Kusche J., Sanchez L. and C.K. Shum (2006) Regional high-resolution spatio-temporal

W. Keller gravity modeling from GRACE data using spherical wavelets. Geophysical Research Letters, 33 L08403, doi:10.1029/2005GL025509. Sneeuw N. (1992) Representation Coefficients and their Use in Satellite Geodesy, Manuscr. Geod. 17:117–123. Vilenkin N.J. (1968) Special functions and the theory of group representations, American Mathematical Society, Translations of Mathematical Monographs, Vol. 22. Koop R. (1993) Global gravity modelling using satellite gravity gradiometry, Netherlands Geodetic Commission, Publications on Geodesy No 38, Delft 1993 Keller W. and M. Sharifi (2005) Satellite gradiometry using a satellite pair. Journal of Geodesy 78: 544–557.

Gravity Field Modeling on the Basis of GRACE Range-Rate Combinations P. Ditmar, X. Liu Delft Institute of Earth Observation and Space System (DEOS), Delft University of Technology, Kluyverweg 1, P.O. Box 5058, 2600 GB Delft, The Netherlands, e-mail: [email protected]

Abstract. A new functional model is proposed for gravity field modeling on the basis of KBR data from the GRACE satellite mission. This functional model explicitly connects a linear combination of gravitational potential gradients with a linear combination of range-rate measurements at several successive epochs. The system of observation equations is solved in the least-squares sense by means of the pre-conditioned conjugate gradient method. Noise in range-rate combinations is strongly dependent on frequency, so that a proper frequency-dependent data weighting is a must. The new approach allows a high numerical efficiency to be reached. Both simulated and real GRACE data have been considered. In particular, we found that the resulting gravity field model is rather sensitive to errors in the satellite orbits. A preliminary gravity field model we obtained from a 101 day set of GRACE data has a comparable accuracy with the GGM01S model derived by CSR. Keywords. GRACE, range-rate combinations, frequency-dependent weighting, Earth’s gravity field

satellites, (ii) accelerometers, which measure nongravitational satellite accelerations, and (iii) star cameras needed to determine the satellite attitudes. A number of functional models for processing GRACE KBR data have been already proposed and applied, e.g. variational equation approach (Tapley et al., 2004; Reigber et al., 2005), energy balance approach (Jekeli, 1999; Han et al., 2005b), acceleration approach (Rummel, 1979), the approach based on integration of short arcs (Ilk et al., 2003) and the gradiometry approach (Keller and Sharifi, 2005). In the paper, we propose a new approach for gravity field modeling, which is based on so-called range-rate combination. The structure of the paper is as follows. In Sect. 2, we present the theoretical basis of the proposed approach. In Sect. 3, we tackle some implementation issues. To verify the approach, one-month GRACE data set is simulated and processed (Sect. 4). Next, we process 101-day set of real GRACE data (Sect. 5). Finally, conclusions are given and the future outlook is discussed.

2 Functional Model 1 Introduction The GRACE (Gravity Recovery And Climate Experiment) satellite mission was launched in March 2002 mainly for the purpose of high-precision mapping of the Earth’s gravity field (Tapley et al., 2004). The mission consists of two satellites co-orbiting at about 480 km altitide with a 220 ± 50 km alongtrack seperation. The satellites are equipped with a K-band ranging (KBR) measurement system, thanks to which the inter-satellite range-rates can be continuously determined with an accuracy of better than 0.5 ␮m/s (Biancale et al., 2005). Other important on-board sensors are: (i) GPS receivers needed to determine the satellite orbits and to synchronize time tags of KBR measurements of the two

The functional model makes use of a local frame at each particular epoch (see Figure 1(a)). In the frame, the X-axis is defined as the Line-Of-Sight; the Z -axis is orthogonal to the X-axis in the plane formed by two satellites and the center of the Earth (i.e. this axis is approximately radial) and the Y -axis is orthogonal to the X- and Z -axes forming a right-hand frame (i.e. the Y -axis is cross-track). In order to build up one observation equation, three succesive epochs are considered (say, i − 1, i and i + 1). Let us introduce inter-satellite average accelerations between the epochs i − 1 and i (i.e. g¯ i− ) and between the epochs i and i + 1 (i.e. g¯ i+ ) as: 0 g¯ i− :=

−⌬t

g(ti + s) ds ⌬t

⌬t , g¯ i+ :=

0

g(ti + s) ds , ⌬t 17

18

P. Ditmar, X. Liu

where g(t) is point-wise inter-satellite acceleration as a function of time, and ⌬t is sampling rate. Obviously, the following equalities hold: ⌬t · g¯ i− = vi − vi−1 ,

(1)

⌬t · g¯ i+ = vi+1 − vi ,

(2)

where vi−1 , vi and vi+1 are inter-satellite velocities at three successive epochs. The accelerations in the left-hand side of Equations (1) and (2) can be related to the gravitational potential gradients, while the inter-satellite velocities in the right-hand side can be related to the range-rates. As a result, a linear combination of three successive range-rates di−1 , di , di+1 can be directly related to the average inter-satellite accelerations g¯ i− and g¯ i+ (Ditmar and Liu, 2006): z z x x + g¯ i+ ) − (τi− )g¯ i− − (τi+ )g¯ i+ νi (g¯ i−

= (i− )di−1 + i di + (i+ )di+1 ,

(3)

x and g¯ z are the X- and Z-component of where g¯ i± i± the vector g¯ i± at the epoch of i , respectively; νi , τi− , τi+ , εi− , εi , and εi+ , the so-called navigation parameters, are functions of the unit vectors ei−1 , ei , and ei+1 that define the line-of-sight directions at the three successive epochs (see Figure 1(b)). Strictly speaking, equation (3) is only valid in the 2-D case, i.e. if all 3 line-of-sight unit vectors coincide with the orbital planes of the satellites. However, real data can be reduced to the (locally) 2-D case by applying small corrections to di−1 , di and di+1 , respectively. The corrections are calculated from the Y-components of velocity differences at epochs i − 1 and i + 1 projected onto the X- and Z -axes of the epoch i . As the orbit radius approaches infinity, equation (3) turns into a double-differentiation formula: x − gx gi+ i−

⌬t

=

di−1 − 2di + di+1 . (⌬t)2

(4)

x A series of average inter-satellite accelerations gi± z and gi± can be easily related to a set of Stokes coefficients (or to other parameters if the gravity field representation is not the spherical harmonic expanx and sion). In particular, the algorithm to compute gi± z gi± from stockes coefficients can be as follows: (1) Compute the gravitational accelerations at the satellite locations; (2) Compute the inter-satellite differences (“point-wise inter-satellite accelerations”); (3) Apply the averaging filter (Ditmar and Van Eck van der Sluijs, 2004); (4) At each epoch, compute the x and the orthogonal comline-of-sight component gi± z . ponent gi±

3 Implementation Equation (3) can be written as a matrix-to-vector multiplication: Ax = d, where x is the set of gravity field parameters, d is the set of Range-Rate Combinations (RRCs), and A is a design matrix. These equations have to be solved in the least squares sense. The number of unknown parameters (stokes coefficients) in the least-squares adjustment can be up to tens of thousands, and the number of data can reach tens of millions. Therefore, a tailored, numerically efficient adjustment algorithm is advisable. A reasonable choice is an algorithm based on the pre-conditioned conjugate gradient method. Such an algorithm can be split into a number of basic operations, each of which can be efficiently implemented: (1–2) Multiplication of the matrices A and AT to a vector. These steps can be implemented as a fast synthesis and co-synthesis (Ditmar et al., 2003) combined with the application of an averaging filter (Ditmar and Van Eck van der Sluijs, 2004) (3) Multiplication of the inverse data covariance matrix C−1 d to a vector. This step can be implemented as a low-level conjugate-gradient algorithm (Ditmar and Van Eck van der Sluijs, 2004; Klees and Ditmar, 2004). (4) Pre-conditioning, i.e. solving the system of linear equations where the original normal matrix is replaced by its approximation (“a pre-conditioner”). In the proposed functional GRACE B

Z

GRACE B GRACE A GRACE B

Y

ei

e i+1

e i−1

X

GRACE A

GRACE B GRACEA GRACE A

(a)

(b)

Fig. 1. (a): Definition of the working frame; (b): Unit vectors of line-of-sight directions at three successive epochs.

Gravity Field Modeling on the Basis of GRACE Range-Rate Combinations

model, a block-diagonal approximation of the normal matrix can be obtained by making a number of not very unrealistic assumptions (e.g. that the orbit is perfectly circular; the gravity field between the satellites changes linearly; temporal change of the gravity field at a given point in an inertial frame caused by the Earth rotation can be neglected within a sampling interval, etc.)

4 Simulation To test the proposed functional model, a numerical experiment was performed. One-month satellite orbits of two satellites with 5-sec sampling were simulated in compliance with parameters of the GRACE mission. The force model was defined as the gravity field model EIGEN-CG03C truncated at degree 150. The simulated orbits were used to compute both range-rates and the navigation parameters. Next, “observed” RRCs were derived. Furthermore, reference RRCs were computed according to the left-hand side of Equation (3) on the basis of the EGM96 model truncated at degree 150. The residual (“observed” minus “reference”) RRCs were used to compute the corrections to the EGM96 Stokes coefficients by the least-squares adjustment. The obtained gravity field model (reference model + corrections) was compared with the “true” one

19

(see the noise-free case in Figure 2). A perfect agreement between the derived and “true” model can be considered as a proof of validity of the proposed functional model. The remaining small differences between the models presumably stem from a limited accuracy of the orbit integration. The least-squares adjustment took only 31 min (the SGI Altix 3700 super-computing system with eight processing elements was used). The data sets presented above were further used to estimate how noise from different sources propagates into the gravity field model in the proposed approach. Errors of following origins are simulated and added to corresponding quantities: 1. Case I – Noise in range-rates: white noise with a RMS (root-mean-square) of 0.5 ␮m/s. 2. Case II – Noise in satellite orbits: noise with a RMS of 10 mm and an auto-correlation factor of 0.995 (Reubelt et al., 2003) is added to the ‘true’ orbit of the satellite A. Furthermore, noise with a RMS of 10, 1, or 0.1 mm and with the same autocorrelation factor of 0.995 is resepctively added to the baselines of two satellites (3-D differences of the satellite positions). Then, the orbits of satellite B used is computed by adding the noisy baselines to the noisy orbit of the satellite A.

101 10 mm RMS noise in baselines 1.0 mm RMS noise in baselines 0.1 mm RMS noise in baselines 0.5 µm/s RMS noise in range−rates noise free

0

10

−1

10

RMS (m)

−2

10

−3

10

−4

10

−5

10

−6

10

0

10

20

30

40

50

60

70 80 degree

90

100

110

120

130

140

150

Fig. 2. Difference between the obtained models and the “ture” model in terms of geoid height errors per degree (noise-free data, Case I and Case II noisy data are used, respectively).

20

P. Ditmar, X. Liu

Figure 3(a) shows the square root of power spectral density of noise in RRCs for the Case I. The figure clearly displays that computation of RRCs heavily amplifies noise at high-frequencies, therefore, a frequency-dependent data weighting is a must. So that relatively high weights are assigned to low frequencies and low weights to high frequencies. To achieve that, we made use of the noise PSD approximated by an analytic function (Ditmar and Van Eck van der Sluijs, 2004; Ditmar et al., 2007). Figure 3(b) compares the results obtained with and without frequency-dependent data weighting. It can be seen that the frequency-dependent weighting improves the model accuracy both at low and high degrees. Propagation of orbit noise (Case II) into the gravity field model is shown in Figure 2. As seen, the resulting gravity field model is rather sensitive to errors in the baselines of two satellites. This means the orbit noise may become the dominant factor in the error budget. The accuracy of baselines between two satelites is of particular importance. A similar conclusion was aslo made earlier by Jekeli (1999) and Han et al. (2005a) in the context of the energy balance approach.

5 Real Data Processing A GRACE gravity filed model up to degree or order 150 was derived from 101 days of GRACE science data spanning the interval from July 9, 2003 to October 17, 2003. The following main data sets were used: (1) Reduced-dynamic orbit of satellite A (30-sec sampling); (2) Relative baseline vectors between satellite A and B (10-sec sampling); (3) Non-gravitational accelerations (1-sec sampling); (4) Attitude data (5-sec sampling); (5) K-band intersatellite range-rates (5-sec sampling). The items (1) and (2) are kindly provided by Kroes et al. (2005), and the items (3)–(5) are the L1B products which are distributed by JPL PODAAC User Services Office (Case et al., 2004). The principal procedure of real data precessing is similiar to that of simulated data. Additionally, we subtracted temporal variations caused by tides, as well as by atmospheric and ocean mass changes. The data sets used for this purpose, are DE405 numerical ephemerides (Standish, 1998), GOT00.2 ocean tide model (Ray, 1999) and AOD1B atmospheric and ocean de-aliasing product (Case et al., 2004). Daily bias and scale factor in terms of non-gravitational RRCs are estimated altogether with the parameters of gravity field model.

(a)

−6

SQRT−PSD (m/s3/Hz1/2)

10

−7

10

−8

10

−9

10

−10

10

−11

10

−12

10

−4

−3

10

−2

10

−1

10

10

Frequency (Hz)

(b)

1

10

with frequency−dependent data weighting without frequency−dependent data weighting

0

10 RMS (m)

−1

10

−2

10

−3

10

−4

10

−5

10

0

10

20

30

40

50

60

70 80 degree

90

100

110

120

130

140

150

Fig. 3. Power spectral density of noise in RRCs of Case I (a). Difference between the obtained and the “true” model in terms of geiod height errors per degree (b).

Gravity Field Modeling on the Basis of GRACE Range-Rate Combinations

21

100 Deg. dif. between GGM01S and EIGEN_CG03C Cum. dif. between GGM01S and EIGEN_CG03C Formal degree error of GGM01S Formal cumulative error of GGM01S Deg. dif. between our model and EIGEN_CG03C Cum. dif. between our model and EIGEN_CG03C

RMS (m)

10–1

10–2

10–3

10–4 0

10

20

30

40

50

60

70 80 degree

90

100 110 120 130 140 150

Fig. 4. Geoid height difference between our model and EIGEN-CG03C (thin black line), between GGM01S and EIGEN-CG03C (thick grey line): cumulative geiod height difference (dash line) and geiod height difference per degree (solid line). Formal degree error of GGM01S (thick black solid line) and formal cumulative error of GGM01S (thick black dash line). Degree 2 is excluded.

The difference between our model and the stateof-the-art EIGEN-CG03C model (Förste et al., 2005) is shown in terms of geoid heights in Figure 4. For comparison, the difference between GGM01S (Tapley et al., 2004) and EIGEN-CG03C models, as well as the formal error of GGM01S model are also shown. The GGM01S model was produced from 111 days of GRACE data. As can be seen, our model has less than 2-cm geoid height difference up to degree and order 70, and 20-cm difference up to degree and order 120 with respect to the EIGEN-CG03C model. This is comparable with or even better than the GGM01S model. Unfortunately, our model shows a somewhat lower accuracy at low degrees (below 35). There are at least three possible reasons for that. The first reason is temporal gravity filed variations. The data span of the GGM01S model is from April to November, though only 111 days data were selected. Thus, temporal gravity field variations in this model are largely averaged out. The second reason is satellite orbit error. Unlike the developers of the GGM01S model, we have not minimized these errors by adding corresponding nuisance parameters to the list of unknowns at the stage of least-squares adjustement. The third reason to explain low accuracy at low degrees could be the noise in the naviga-

tion parameters. The navigation parameters are computed from the baselines of two satelites, which are determined by GPS data. The navigation parameters certainly contain noise, therefore it could propagate into the RRCs. To what an extent this noise influences of the noise on the final gravity field is not known yet.

6 Conclusion and Future Outlook A new approach has been proposed for gravity field modeling from GRACE KBR data. It is based on usage of so-called range-rate combinations. A numerical study and real data processing prove the validity of the approach. Although the research is by far not complete, it is already clear that the new approach can produce models with an accuracy comparable to that provided by other techniques. In the future, more nuisance parameters have to be estimated in the course of least-squares adjustment. In particular, nuisance parameters related to orbit errors will be incorporated. Furthermore, more modern ocean tides models will be used. It also goes without saying that more GRACE data should be used for computation of a more accurate mean gravity field model.

22

Acknowledgements We thank Ejo Schrama (TU Delft) for his DINGO software, which was used for the orbit simulation. We are extremely grateful to P. Visser and R. Kroes for providing us with GRACE orbits. A part of computations have been done on the SGI Altix 3700 super-computer in the framework of the grant SG-027 provided by Stichting Nationale Computerfaciliteiten (NCF). The support of NCF is acknowledged.

References R. Biancale, G. Balmino, S. Bruinsma, J. M. Lemoine, S. Loyer, and F. Perosanz. GRACE data processing in CNES/GRGS; results and discussion. In J. Ries and S. Bettadpur, editors, Presentation Proceeding of the GRACE Science Team Meeting at Center for Space Research, The University of Texes at Austin, October 13–14, 2005, pages 203– 220. 2005. K. Case, G. Kruizinga, and S. C. Wu. GRACE Level 1B Data Product User Handbook JPL D-22027 GRACE 327–733. 2004. P. Ditmar, R. Klees, and F. Kostenko. Fast and accurate computation of spherical harmonic coefficients from satellite gravity gradiometry data. Journal of Geodesy, 76:690–705, 2003. P. Ditmar, R. Klees, and X. Liu. Frequency-dependent data weighting in global gravity field modeling from satellite data contaminated by non-stationary noise. Journal of Geodesy, 81:81–96, 2007. P. Ditmar and A. A. van Eck van der Sluijs. A technique for Earth’s gravity field modeling on the basis of satellite accelerations. Journal of Geodesy, 78:12–33, 2004. P. Ditmar, and X. Liu. Gravity field modeling on the basis of GRACE range-rate combinations: current results and challenge. The Proceeding of the ’1st International Symposium of the International Gravity Field Service (IGFS): Gravity Field of the Earth’. p 181–186, Aug 28– Sept 1, 2006, Istanbul, Trukey. C. Förste, F. Flechtner, R. Schmidt, U. Meyer, R. Stubenvoll, F. Barthelmes, R. König, K. H. Neumayer, M. Rothacher, C. Reigber, R. Biancale, S. Bruinsma, J.-M. Lemoine, and J. C. Raimondo. 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.

P. Ditmar, X. Liu S.-C. Han, C. K. Shum, and A. Braun. High-resolution continental water storage recovery from low-low satellite-tosatellite tracking. Journal of Geodynamics, 39, 2005a. S.-C. Han, C. K. Shum, C. Jekeli, and D. Alsdorf. Improved estimation of terrestrial water storage changes from GRACE. Geophysical Research Letters, 32, 2005b. L07302, doi: 10.1029/2005GL022382. K. H. Ilk, M. Feuchtinger, and T. Mayer-Gürr. Gravity field recovery and validation by analysis of short arcs of a satellite-to-satellite tracking experiment as CHAMP and GRACE. In Proceedings of the IAG Symposium G02, IUGG General Assembly 2003, Sapporo, Japan. 2003. C. Jekeli. The determination of gravitational potential differences from satellite-to-satellite tracking. Celestial Mechanics and Dynamical Astronomy, 75:85–101, 1999. R. Klees and P. Ditmar. How to handle colored noise in large least-squares problems in the presence of data gaps? In F. Sansò, editor, V Hotine-Marussi Symposium on Mathematical Geodesy. International Association of Geodesy Symposia, volume 127, pages 39–48. Springer, Berlin, Heidelberg, New York, 2004. W. Keller and M. A. Sharifi. Satellite gradiometry using a satellite pair. Journal of Geodesy, 78:544–557, 2005. R. Kroes, O. Montenbruck, W. Bertiger, and P. Visser. Precise GRACE baseline determination using gps. GPS Solutions, 9(1):21–31, 2005. R. D. Ray. A Global Ocean Tide Model From TOPEX/POSEIDON Altimetry: GOT99.2. NASA Technical Memorandum 209478, 1999. C. Reigber, R. Schmidt, F. Flechtner, R. König, U. Meyer, K.-H. Neumayer, P. Schwintzer, and S. Y. Zhu. An Earth gravity field model complete to degree and order 150 from GRACE: EIGEN-GRACE02S. Journal of Geodynamics, 39:1–10, 2005. T. Reubelt, G. Austen, and E. W. Grafarend. Harmonic analysis of the Earth’s gravitational field by means of semicontinuous ephemerides of a low Earth orbiting GPStracked satellite. Case study: CHAMP. Journal of Geodesy, 77:257–278, 2003. R. Rummel. Determination of short-wavelength components of the gravity field from satellite-to-satellite tracking or satellite gradiometry – an attempt to an identification of problem areas. Manuscripta Geodetica, 4(2):107–148, 1979. E. M. Standish. JPL Planetary and Lunar Ephemerides, DE405/LE405. JPL IOM 312.F-98-048, 1998. B. D. Tapley, S. Bettadpur, M. Watkins, and C. Reigber. The gravity recovery and climate experiment: Mission overview and early results. Geophysical Research Letters, 31, 2004. L09607, doi: 10.1029/2004GL019920.

The Torus Approach in Spaceborne Gravimetry C. Xu, M.G. Sideris Department of Geomatics Engineering, University of Calgary, 2500 University Dr., NW, Calgary, Canada T2N 1N4, e-mail: {xuc, sideris}@ucalgary.ca N. Sneeuw Geodätisches Institut, Stuttgart Universität, Stuttgart, Germany, D-70147, e-mail: [email protected] Abstract. Direct global gravity field recovery from the dedicated gravity field satellite missions C HAMP, G RACE and G OCE is a very demanding task. This has led to so-called semi-analytical schemes that are usually classified as either space-wise or timewise under certain approximations and assumptions. Both approaches are efficient in that they provide a block-diagonal structure of the normal matrix for the least squares adjustment. At the same time, both approaches suffer from limitations implied by these approximations and assumptions. In this paper we will focus mainly on the socalled torus approach, that combines the strengths of both time-wise and space-wise approaches, in spaceborne gravimetry. The semi-analytical algorithm will be addressed by comparing the different projection domains: periodic orbit, sphere and torus. Lumped coefficients are obtained from a two-dimensional Fourier analysis of the geopotential functionals on a nominal torus. Subsequently, block-diagonal least squares inversion provides the gravity field spherical harmonic coefficients as output. In addition, important issues, such as interpolation, regularization and optimal weighting will be discussed through the processing of two-year real C HAMP data. Keywords. Torus approach, spaceborne gravimetry, block-diagonality, semi-analytical formulation, spherical harmonic coefficients

1 Introduction Traditional techniques of gravity field determination, e.g. mean gravity anomalies from terrestrial and ship-borne gravimetry, satellite altimetry for ocean areas and satellite orbit analysis, have reached their intrinsic limits. Any advances must rely on space techniques only because they provide global, regular and dense data sets of high and homogeneous quality, cf. (ESA, 1999; Rummel et al., 2002). The dedicated gravity field satellite missions, C HAMP, G RACE and

G OCE represent a new frontier in studies of the Earth and its fluid envelope, such as ocean dynamics and heat flux, ice mass balance and sea level, solid Earth, and geodesy, cf. (NRC, 1997). Spaceborne gravimetry missions provide millions of measurements during their life time. Consequently, direct global gravity field recovery up to a certain maximum spherical harmonic degree from these observations is a very demanding task. For instance, the number of unknowns for G OCE will be close to 100 000 when the maximum degree reaches L = 300. The conventional numerical methods (brute force approach), which are based on the orbit perturbation theory, are unable to solve the huge normal equation system in the least squares inversion because it demands enormous computational time and high memory. This has led to so-called semianalytical schemes that are usually classified as either space-wise or time-wise approach, cf. (Rummel et al., 1993). As a boundary value problem approach to physical geodesy, the former approach treats gravitational observable as a function of spatial coordinates, usually leading to a spherical projection. Rooted in celestial mechanics, the latter treats the measurements as an along-orbit time-series, leading to a one-dimensional Fourier analysis under a repeat orbit assumption, cf. (Sneeuw, 2000a). Both approaches are efficient in that they provide a blockdiagonal structure of the normal matrix for the least squares adjustment under certain approximations and assumptions, cf. Figure 1 and (Koop, 1993). However, both approaches have their intrinsic limitations. The space-wise approach lacks for instance any link to the dynamic orbit configurations, making it very difficult to implement a stochastic model from an error power spectral density ( PSD). The time-wise approach, on the other hand, is very sensitive to data gaps, and the repeat orbit requirement is not always realistic, cf. (Rummel et al., 1993; Sneeuw, 2000a). In this paper we will focus on the so-called torus approach, which combines the strengths of both space-wise and time-wise approaches, 23

24

C. Xu et al. Clm m=0

m=1

Slm m=1

m=2

m=2

e l+1 o 1l+1 1 l 2 2 1l 2

Without any details of the rotation and transformation procedure, the potential V can be rotated and expressed in an orbital system by two orbital variables u (argument of latitude) and (longitude of the ascending node), cf. (Sneeuw, 2000b):

e l

l ∞ l GM R l V (r, I, u, ) = r r

o 1l 2

e

l−1

m=−l k=−l

l=0

o

· K¯ lm F¯lmk (I )e j (ku+m) .

(2)

}1 e o e o

Fig. 1. Order-wise block-diagonal structure of normal matrix, after (Koop, 1993).

cf. (Sneeuw, 2000a,b). Thus, in the following section, the disturbing potential and related geopotential functionals will be represented on a torus. The corresponding gravity field recovery procedure in spaceborne gravimetry using the torus approach will be addressed in section 3. In addition, some critical issues involved in the torus approach, such as interpolation, regularization and optimal weighting will be discussed in this section as well. Section 4 demonstrates that the proposed torus approach is feasible and efficient in spaceborne gravimetry through the processing of two-year real C HAMP data.

2 Gravity Field Representation on a Torus The Earth’s gravitational disturbing potential V is conventionally presented as a spherical harmonic series in a complex-valued expression as follows: L l G M R l ¯ ¯ j mλ K lm Plm e , V (r, φ, λ) = r r m=−l l=0 (1)

in which r, φ, λ, R, G M are the geocentric radius, latitude, longitude, Earth’s equatorial radius and gravitational constant times Earth’s mass, respectively. The normalized spherical harmonic coefficients K¯ lm with degree l and order m up to the maximum degree L. P¯lm is the normalized associated Legendre function.

It introduces the normalized inclination function F¯lmk (I ) with the orbit inclination I , and k is the third index due to the rotation, cf. (Kaula, 1966; Sneeuw, 1992). Note that both orbital coordinates u and ⌳ attain values in the range of [0, 2π) periodically. Topologically, the product of [0, 2π) × [0, 2π) creates a torus, which is the proper domain of a twodimensional discrete Fourier series. Hence, the semianalytical torus approach is sometimes referred to as the two-dimensional Fourier time-wise approach in the frequency domain. It combines the strengths of both space-wise and time-wise approach, because it projects the measurements onto a spatial torus before spectral analysis. Compared to the torus approach, the projection domains for space-wise and time-wise approach are a sphere and a repeat orbit, respectively. Therefore, Figure 2 completes the related classifications and projections of different approaches to recover Earth’s gravity field from spaceborne gravimetry. In order to make the structure of (2) more obvious in terms of Fourier analysis, the “lumped coefficients” Amk and “transfer coefficients” Hlmk are defined, cf. (Sneeuw, 2000a,b):

Spherical Harmonic Analysis

Semi-Analytical

Numerical

Orbit Perturbation

Celestial Mechanics

Physical Geodesy

Brute-Force

Time-Wise

Space-Wise

Freq Domain

Direct

Time Domain

2D Fourier

1D Fourier

Torus

Repeat Orbit

Fig. 2. Relations between different approaches.

Sphere

The Torus Approach in Spaceborne Gravimetry

V Amk (r, I ) =

L l=max(|m|,|k|) l

V (r, I ) = Hlmk

GM r

R r

25

Hlmk K¯ lm ;

(3a)

F¯lmk (I ).

(3b)

Thus the transfer coefficient Hlmk spectrally builds a linear relation between lumped coefficients in spectral domain and spherical harmonic coefficients in spatial domain. Spatially, this is a mapping from sphere to torus. Any realistic orbit is always perturbed by disturbing forces, e.g. air-drag. Consequently, the transfer coefficient is time-dependent with variable heights and inclinations as well, e.g., r (t), and I (t). In order to apply the Fourier analysis, a nominal orbit with constant radius and constant inclination assumptions has to be introduced, then the potential V is only the function of timedependent orbital variables u and ⌳ with constant Amk and Hlmk . It can be expressed concisely into a lumped coefficients formulation: V (u, ⌳) =

L L

(4)

Note that not only the potential, but also its functionals can be represented by a two-dimensional Fourier series. Generally speaking, for any gravitational functional f , the spectral decomposition under a nominal orbit assumption is similar with disturbing potential expression, cf. (Sneeuw, 2000a,b). The only change is that the corresponding transfer coefficients have to be derived for the specific functional f . Sneeuw (2000b) derived several transfer coefficients based on proper differentiations and orbit perturbazz tion theory, for instance, the transfer coefficient Hlmk for gravity gradient tensor in radial direction Vzz is derived as: GM r3

Perturbation Reduction A Taylor expansion series is used to reduce the measurements from the perturbed orbit onto the nominal orbit with respect to the height and inclination variations. Interpolation

V j (ku+m) Amk e .

m=−L k=−L

zz Hlmk =

step, spherical harmonic coefficients K¯ lm for distinct orders m are solved separately by the least squares adjustment based on a linear system in (3a). Again, the big advantage of the torus approach is that the normal matrix shows an m order-wise blockdiagonal structure under the assumption that the Fourier lumped coefficients are independent between different orders. The summarized flow chart of the strategy is shown in Figure 3, and noise information is also included in the presence of PSD. From the torus approach recovery strategy, several important issues, such as perturbation deduction, interpolation, regularization and iteration, have to be considered in order to get a good solution from spaceborne gravimetry.

l R [(l + 1)(l + 2)] F¯lmk (I ). (5) r

The efficiency of the torus approach is given by the block-diagonality (as discussed above) and by applying FFT. Therefore irregularly sampled observations have to be mapped onto a regular (u, ⌳) grid on a nominal torus (constant radius and constant inclination. Interpolation becomes a necessary and powerful tool to accomplish this job. Deterministic approaches, e.g., linear and cubic interpolation are very common and fast. In geodetic applications the statistical approaches least-squares collocation (LSC), cf. (Moritz, 1980) and Kriging method, are very popular since they make use of

Step I

3 Torus Approach Recovery Strategy and Critical Issues Based on the derivation procedure with a nominal orbit assumption, the torus approach basically recovers the Earth’s gravity field by three steps. Firstly, the calibrated observations from spaceborne gravimetry are reduced and interpolated regularly onto the nominal torus. Secondly, the pseudo-observable lumped coefficients are computed by a two-dimensional Fast Fourier Transform (FFT) technique. In the final

Geopotential observations from spaceborne gravimetry

A priori noise model

Reduce height & inclination variations onto nominal torus

Power spectral density (PSD)

Interpolate scattered observations onto grid

Apply discrete 2D FFT to obtain the lumped coefficients

Step II

Iteration

Apply least-squares adjustment order-wisely in a block-diagonal structure

Step III Spherical harmonic coefficients Klm Variance-covariance information

Fig. 3. Flow chart of the torus approach.

Weighting Regularization

26

C. Xu et al.

sill

iterations. Klees et al. (2000) show that an iteration process can reduce the interpolation errors as well. Optimal Weighting

γ(h)

C1

range a nugget

C0

lag (h)

Fig. 4. Empirical fitting of semi-variogram model, after (Burrough and McDonnell, 1998, ch. 6).

the geo-statistical properties in the data and they are able to propagate the error information. The essential parameters, covariance function and semi-variogram have to be fitted empirically among data set, cf. Figure 4. Computational time issues and empirical modeling are of concern for huge data sets. Regularization In the statistical interpolation, the inversion of the covariance function or semi-variogram matrices is typically an ill-posed problem. In addition, if one wants to incorporate a priori information, such as the Kaula’s rule of thumb, or a pre-existing error model in order to stabilize the normal matrix in the spherical harmonic computation, the way of regularization has to be considered as well. The regularization parameter can either be determined by the L-Curve criterion, cf. (Hansen, 1992), or by the Generalized Cross-Validation ( GCV) method, cf. (Golub and von Matt, 1997). Kusche and Klees (2002) conclude that GCV performs slightly better and more robust than the L-Curve criterion when processing satellite gravity gradients. Iteration The beautiful block-diagonal structure, which drastically saves computational time, is only valid for the nominal orbit geometry, i.e. constant inclination and constant radius. This assumption can be overcome by an iteration procedure when orbit eccentricity and variation in inclination are very small, for example, e < 4 · 10−3 and δ I < 0.01◦, cf. (Pail and Plank, 2002), because the torus-based semianalytical approach is a linear system in the leastsquares sense, and a linearization process can always be adjusted more accuracy before convergence due to

C HAMP is sensitive to the low frequencies while G OCE better resolves the high frequencies. Therefore, in order to cover the whole gravity spectrum, a normal matrices from C HAMP and G OCE are necessarily merged in the least squares process using a weighting factor. The optimal weighting factor for a joint solution will be estimated by a parametric covariance method in an iterative way. Optimal weighting is also an important issue for the overall combination from different individual monthly solutions.

4 Gravity Field Recovery from Real C HAMP Data As a demonstration of its feasibility and efficiency, real C HAMP data have been processed using the proposed torus approach. In this test, almost two years (from April 2002 to February 2004) of C HAMP measurements are collected. These measurements are positions, velocities and accelerometer data along the kinematic orbits derived by TU Munich. The in situ gravitational disturbing potentials are calculated and calibrated with the EGM96 model by the energy balance approach, in which calibration parameters (scale and bias) are determined once per day. Tides (astronomic or direct and ocean tides) are taken into account, cf. (Jekeli, 1999; Weigelt, 2006). The calibrated disturbing potentials have been used as input for the test. Least squares collocation method has been employed to create torus grids for the interpolation. The relative interpolation errors are about 1% (4m2/s2 ) compared with the reference values. Monthly solutions of spherical harmonics are calculated up to degree L = 60. Each solution provides its individual normal matrix Ni and observation matrix Ci month by month. Therefore, the overall two-year solution can be determined by a multi-observable model as: K¯ lm =

#months i=1

Ni

−1 #months

Ci

.

(6)

i=1

Choosing the G RACE satellite-only gravity field model GGM 02 S as the reference field, cf. (Tapley et al., 2005), the root-mean-square per degree (RMSl ) of C HAMP two-year overall solution is calculated by (7). RMSl is the average standard deviation to be expected for a specific degree l. Thus the result is

The Torus Approach in Spaceborne Gravimetry

27

Error Degree RMS

10−6

10−7 Kaula’s curve Signal from EIGEN2

10−8

Signal from Torus

10−9 Error from EIGEN2

Error from Torus

10−10

10−11

0

10

20 30 40 spherical harmonic degree

50

60

Fig. 5. Degree RMS of two-year overall solution from C HAMP up to L = 60.

proposed because it links the space-wise and timewise approaches. After generating a regular grid on the torus, Fast Fourier Transform can be used to get the lumped coefficients. The main advantage of this semi-analytical approach is that it provides an orderwise block-diagonal structure in the normal matrix, leading to a very efficient recovery of spherical harmonic coefficients. However, interpolation methods have to be applied to grid the data, and the interpolation errors need to be minimized by iteration. Other aspects, like regularization and optimal weighting are discussed as well. One example from the real two-year C HAMP data is carried out to demonstrate the feasibility, viability and efficiency of the torus approach in spaceborne gravimetry.

Acknowledgements shown in Figure 5. The black solid curves are the signal and error from the torus approach. For comparison, the gray dash-dot curves are the signal and error from E IGEN -2 C HAMP-only models, which is solved based on C HAMP data covering 6 months out of the time periods July–December 2000 and September– December 2001, cf. (Reigber et al., 2003). From the result plot we can see that for the low degrees part the torus approach did not recover the coefficients very well compared to E IGEN -2. The reason might be that the interpolation method filters lower frequencies of the signal. For the degree between 25 ≤ l ≤ 35 and degree above l ≥ 50, the torus approach performs better than the other one, however, more accurate interpolation methods have to be studied to reduce the interpolation errors. Note that no regularization methods and no iteration scheme are involved for the overall solution, furthermore, no optimal weighting factors are determined between different monthly solutions. Therefore, further investigations have to be carried out. l 1

est ref 2 K¯ lm − K¯ lm . (7) σl = 2l + 1 m=−l

5 Conclusions The gravity field recovery from dedicated satellite missions is a computationally demanding task. Several approaches, namely the brute-force, the spacewise and the time-wise approach have been discussed in terms of their individual characteristics. They implicitly or explicitly project the geopotential functional onto different domains: a sphere, a repeat orbit and a torus. The torus approach is

The authors would like to thank TU Munich for providing C HAMP kinematic orbits. Two reviewers are appreciated for their valuable remarks.

References Burrough, P.A. and R.A. McDonnell (1998) ; Princi-ples of Geographical Information Systems - Spa-tial Information Systems and Geostatistics. Ox-ford University Press. ISBN: 0-19-823365-5. ESA, E.S.D. (1999); Gravity Field and Steady-State Ocean Circulation Mission. Tech. Rep. SP-1233, European Space Agency. Golub, G.H. and U.vonMatt (1997); Generalized CrossValidation for Large-Scale Problems. Jour-nal of Computational and Graphical Statistics, vol. 6(1):pp. 1–34. Hansen, P.C. (1992); Analysis of DiscreteIll-Posed Problems by Means of the L-Curve. SIAMReview, vol. 34(4): pp. 561–580. Jekeli, C .(1999); The Determination of Gravitational Potential Differences from Satellite-to-Satellite Tracking. Celestial Mechanics and Dynamical As-tronomy, vol. 75: pp. 85–101. Kluwer Academic Publishers. Kaula, W.M. (1966); Theory of Satellite Geodesy. Blaisdell Publishing Company, Waltham Mas-sachusetts. Klees, R., R.Koop, P.Visser, and J.vanden IJs-sel (2000); Efficient Gravity Field Recovery from GOCE Gravity Gradient Observations. Journal of Geodesy, vol. 74: pp. 561–571. Koop, R. (1993); Global Gravity Field Modelling Using Satellite Gravity Gradiometry. Phddis-sertation, Technische Universiteit Delft, Delft, the Netherlands. New Series 38. Kusche, J. and R. Klees (2002); Regularization of Gravity Field Estimation from Satellite Gravity Gradients. Journal of Geodesy, vol. 76: pp. 359–368. Moritz, H. (1980); Advanced Physical Geodesy. Her-bert Wichmann, Karlsruhe, secondedn. NRC (1997); Satellite Gravity and the Geosphere: Contributions to the Study of the Solid Earth and Its Fluid Envelopes. National Academy Press, United States.

28 Pail, R. and G. Plank (2002); Assessment of Three Numerical Solution Strategies for Gravity Field Recovery from GOCE Satellite Gravity Gradiom-etry Implemented onaParallel Platform. Journal of Geodesy, vol. 76: pp. 462–474. Reigber, C., P. Schwintzer, K. H. Neumayer, F. Barthelmes, R. König, C. Förste, G. Balmino, R. Biancale, J. M . Lemoine, S. Loyer, S. Bru-insma, F . Perosanz, and T. Fayard (2003); The CHAMP-Only Earth Gravity Field Model EIGEN-2.Advances in Space Research, vol. 31 (8): pp. 1883–1888. COSPAR, published by Elsevier Science Ltd. Rummel, R., G. Balmino, J. Johannessen, P. Visser, and P. Wood worth (2002); Dedicated Gravity Field MissionsPrinciples and Aims. Journal of Geodynamics, vol. 33: pp. 3–22. Rummel, R., M. van Gelderen, R. Koop, E. Schrama, F. Sans‘o, M. Brovelli, F. Miggliaccio, andF. Sac-erdote (1993); Spherical Harmonic Analysis of Satellite Gradiometry. New Series 39, Nether-lands Geodetic Commission, Delft, the Nether-lands.

C. Xu et al. Sneeuw, N. (1992); Representation Coefficients and Theiruse in Satellite Geodesy. Manuscripta Geo-daetica, vol. 17: pp. 117–123. Sneeuw, N. (2000a); Dynamic Satellite Geodesy on the Torus: Block-Diagonality from a Semi-Analytical Approach. Gravity, Geoid, and Geo-dynamics 2000, (edited by M. Sideris), vol. 123: pp. 137–142. IAG, Springer-Verlag, Banff, Canada. Sneeuw, N. (2000b); A Semi-Analytical Approach to Gravity Field Analysis from Satellite Obser-vations. Dissertation, der Technischen Univer-sität München, Deutsche Geodätische Kommis-sion. ReiheA, Nr. 527. 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, vol. 79: pp. 467–478. Weigelt, M. (2006); Global and Local Gravity Field Recovery from Satellite-to-Satellite Track-ing. Ph.D. thesis, Department of Geomatics Engineer-ing, University of Calgary.

Gravity Recovery from Formation Flight Missions N. Sneeuw, M.A. Sharifi, W. Keller Institute of Geodesy, Universität Stuttgart, Geschwister-Scholl-Str. 24D, D-70174 Stuttgart, Germany

Abstract. We present a proof-of-concept of gravity field recovery from satellite-to-satellite tracking (SST) in formation flight (FF). Three orbit types will be investigated: GRACE-type SST, co-orbital FF on a 2:1 relative ellipse, and out-of-plane FF on a circular relative orbit. All formations have comparable orbit characteristics: near polar, near eccentric, and short baselines of typically 10 km length. First, we demonstrate that these orbits are sufficiently stable at low altitudes in a realistic gravity field. Next, we perform a closed-loop simulation, in which an input gravity field is used for orbit integration and generation of observations in the forward mode. Subsequently, in the inverse mode, the gravity field is recovered. Comparison between input and output fields demonstrate that gravity recovery based on SST observables from formations containing radial and/or out-of-plane information outperform GRACE-type along-track SST. The gravity fields recovered from the former formation types possess a lower error spectrum and an isotropic error structure. Keywords. Formation flight, gravity field recovery, satellite-to-satellite tracking

1 Introduction The monthly GRACE solutions consistently exhibit typical North–South streaks (Tapley et al., 2004). Although this error behaviour is often associated with the ground-track of the near-polar GRACE satellites, it was argued by Sneeuw and Schaub (2005) that the inherent culprit is the non-isotropic SST observable. G RACE’s K-band range-rate observable is sensitive in the along-track direction and carries no radial or cross-track information in the signal content. It is asserted that any observation geometry that could include radial and/or cross-track gravitational signal would drastically improve the gravity recovery capability.

To support and quantify this assertion a gravity recovery experiment is set up that makes use of the concept of formation flying. Three generic types of Low-Earth Formations (LEF) are simulated: a GRACE-type leader-follower configuration, a Cartwheel formation that performs a 2:1-relative ellipse in the orbital plane, and a LISA-type formation that performs circular relative motion including outof-plane motion. The dynamics of such formations are easily understood in the framework of homogeneous Hill equations, cf. (Sneeuw and Schaub, 2005), although the implementation in a realistic gravity field requires careful attention to many fine details. All three LEF will have a typical baseline length of around 10 km. The names GRACE, Cartwheel and LISA will be used in this paper as placeholders for these generic formations and should not be mistaken for actual missions. The first part of the paper is concerned with demonstrating the feasibility of such FF missions for gravity recovery purposes. In particular the stability of the intersatellite baseline in Low-Earth Formations is monitored over an integration period of one month. In a next step the SST observations are generated. Realistic noise, as derived from the real GRACE mission is then added. Based on these observations the gravity field is recovered for each of the formation scenarios. The results are analysed in the spectral and spatial domains.

2 Formation Stability The gravitational potential for an arbitrary point with spherical coordinates (r, θ, λ) can be represented as: V (r, θ, λ) =

μ + R(r, θ, λ, Clm , Slm ), r

(1)

with μ = G Me , the universal gravitational constant times mass of the Earth. The contribution of the spherical harmonic coefficients Clm and Slm is represented by the disturbing function R. They lead to time variable orbital elements known as osculating 29

30

N. Sneeuw et al.

elements. The equations of motion, describing the time evolution of these elements, are Lagrange’s Planetary Equations (LPE), cf. (Kaula, 1966). The disturbing potential R causes periodic and secular perturbations on the orbital elements. In particular, the degree 2 zonal harmonic J2 , corresponding to the Earth’s flattening, causes strong drift effects, long-term oscillations and short-term oscillations at twice the orbital frequency (2 CPR). For purposes of formation keeping the secular drift—at least the relative drift between the satellites—should be cancelled. For this purpose the short-period terms are averaged out of the J2 disturbing function: (a, e)ημ R¯ 2 (a, e, I ) = 2a

1 1 2 − sin I 3 2

(2)

2 2 4 with (a, e) √= 3 J2 ae /a η and the eccentricity mea2 sure η = 1 − e . Inserting (2) into the LPE gives expressions for the first-order secular perturbation:

a˙ = e˙ = I˙ = 0, ˙ = − (a, e) n cos I, ⍀ 2 (a, e) n(5 cos2 I − 1), ω˙ = 4 (a, e) η(3 cos2 I − 1) . M˙ = n 1 + 4

⎧ ˙ 12 (⌬a, ⌬e, ⌬I ) ⎨ ⌬⍀ ⌬ω˙ 12 (⌬a, ⌬e, ⌬I ) ⎩ ⌬ M˙ 12 (⌬a, ⌬e, ⌬I )

= = =

0 0 0

(5)

Since the differential elements ⌬a, ⌬e and ⌬I are usually small, the nonlinear equations (5) can be replaced by a linearized system of equations ˙

˙

ω, ˙ M J⍀, a,e,I (a1 , e1 , I1 )x = 0.

(6)

T with x = ⌬a, ⌬e, ⌬I . The Jacobian J contains the partial derivatives of the drift toward the metric Kepler elements a, e, I , evaluated in the master satellite. It is a homogeneous system which has at least the trivial solution x = 0. It is a unique solution of the system if rank J = 3. Otherwise the system has many non-trivial solutions. As we will see in the following sections, two different formation types can be realized based on the trivial solution ⌬a = ⌬e = ⌬I = 0. Herein, we will also modify the condition equations to define another type of formation.

(3a)

2.1 GRACE Formation

(3b)

The GRACE-type or the leader-follower formation is the simplest cluster configuration which is obtained from the trivial solution of (6). It will be quite stable even for large baselines. For instance, consider two coplanar satellites with intersatellite separation of 10 km, whose mean differential orbital elements are given in Table 1. Differential elements are zero except the mean anomaly. Since the condition equations (6) are fulfilled we expect a stable formation in a J2 field. Based on this simulation scenario, the orbits of two satellites have been integrated in a realistic gravity field (L = 60) for a one-month time span. The intersatellite baseline is computed and analysed stability. The results for the first and the last day of integration are shown in Figure 1. As expected, the satellites’ relative motion is bounded except for a long-term trend in the intersatellite range. A similar behaviour is observable with the real GRACE mission.

(3c) (3d)

with the mean motion n = μ a −3 . First, the secular rates are independent from ⍀, ω and M. Consequently, they are allowed to be selected freely to form a desired formation. Second, the right ascension of the ascending node ⍀, the argument of perigee ω and the mean anomaly M all experience drift. Therefore, a cluster of satellites retains its relative geometry if the drifts of the satellites are identical. For simplicity, consider a cluster with two satellites whose mean Kepler elements are given as (a1 , e1 , I1 , ⍀1 , ω1 , M1 ) and (a2 , e2 , I2 , ⍀2 , ω2 , M2 ). A formation will be stable in a J2 gravity field if the following constraints are fulfilled: ⎧ ˙2−⍀ ˙1 = 0 ˙ 12 = ⍀ ⎨ ⌬⍀ (4) ⌬ω˙ 12 = ω˙ 2 − ω˙ 1 = 0 ⎩ ⌬ M˙ 12 = M˙ 2 − M˙ 1 = 0 The conditions define a nonlinear system of 3 equations in a1 , a2 , e1 , e2 , I1 , I2 . In formation design one usually fixes the orbital parameters of one satellite. The aim is to find the differential elements ⌬a, ⌬e and ⌬I that realize a stable formation. Hence,

Table 1. Differential mean orbital elements

Elements

GRACE

C ARTWHEEL

LISA

⌬a [km] ⌬e ⌬I [deg] ⌬⍀ [deg] ⌬ω [deg] ⌬M [deg]

0 0 0 0 0 0.08

0 0 0 0 180 180

0 0.001 0 0.15 −180 180

Gravity Recovery from Formation Flight Missions

31

9830 9820 9810

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

29.8

29.9

30

last day of the integrated orbit 9080

ρ [m]

9060 9040 9020 9000 29

29.1

29.2

29.3

29.4

29.5 29.6 time [day]

29.7

Fig. 1. Baseline variations in GRACE-type formation.

2.2 Cartwheel Formation Massonnet (2001) proposed the interferometric Cartwheel as a relatively cheap solution for improving the quality of radar images. The satellites in the formation constitute a large aperture antenna and consequently lead to higher geometrical resolution. The Cartwheel formation is another configuration which fulfils the stability conditions simply by employing the trivial solution (⌬a = ⌬e = ⌬I = 0). Consequently, the formation is geometrically stable. Selecting an identical value for the semi-major axes of all orbits in a cluster leads to synchronized orbits. If the satellites are distributed with regular spacing of the arguments of perigee they will trace out, over one orbital period, an ellipse around a reference point. With identical eccentricities, the satellites sweep the ellipse with a constant angular velocity as viewed from its center (Massonnet, 2001). For simplicity, consider a formation with two satellites. Both satellites have the same orbital elements except the arguments of perigee and the mean anomalies. The latter two elements differ by π. The separation of one satellite’s apogee from the perigee of the other one is equal to the maximum separation of the satellites in radial direction. Moreover, the maximum separation in the alongtrack direction is twice as large as that of the radial component (Schaub and Junkins, 2003). Therefore, the semi-major axis of the relative ellipse corresponds to the maximum along-track separation (ρmax ) whereas the minimum intersatellite range (ρmin ) corresponds to that of the radial component. In other words, the following relationship between the eccentricity and intersatellite separations holds: e=

ρmax ρmin = . 2a 4a

(7)

2.3 LISA Formation A cluster of satellites with out-of-plane motion is obtained only by choosing a non-zero value either for ⌬⍀ or ⌬I . Choosing a non-zero differential

15 10

5 0

−5 −20 y [km]

5

z [km]

9840

0 −5

10

−10

0 −10

−15 −20

x [km]

−10

0 x [km]

−5

0 y [km]

10

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5

5

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ρ [m]

9850

Given a certain eccentricity, a Cartwheel formation, consisting of two satellites, is geometrically defined. For instance, consider the differential mean orbital values given in Table 1. The orbital period of the Cartwheel mission is equal to that of the GRACE mission. Therefore, the number of revolutions and consequently the number of observations are identical in two different configurations. It should be noted here that the minimum separation of the satellites in Cartwheel mission is equal to that of the GRACE formation. Using the given initial values, orbits of the Cartwheel satellites are synthesized in the gravity field of the previous example. The orbits are integrated over one month. Figures 2 and 3 show the relative motion and the intersatellite baseline variations within the integration period respectively. The initial relative orbit is always shown as a solid line, while the path of the remaining orbits is shown as a gray line. The intersatellite baseline is quite stable for the first several revolutions. The oscillation between ρmin and ρmax (= 2ρmin ) at a frequency of 2 CPR is evident. From a theoretical aspect, the orbit is J2 invariant even if the nonlinear stability conditions are employed. However, imposing the constraints does not compensate the differential drifts in the real field. Therefore, an additional frequency (1 CPR) has emerged in the intersatellite range after integrating the orbit for a one-month time span.

z [km]

9860

y [km]

first day of the integrated orbit

9870

0 −5

−5

−10 −15 −20

0

−10

0 x [km]

10

−10

5

10

Fig. 2. Relative motion of two satellites in Cartwheel formation.

32

N. Sneeuw et al. first day of the integrated orbit

25

ρ [km]

20 15 10 5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

29.8

29.9

30

last day of the integrated orbit 25

ρ [km]

20 15 10 5 29

29.1

29.2

29.3

29.4 29.5 29.6 time [day]

29.7

Fig. 3. Baseline variation in Cartwheel formation.

right-ascension results in the maximum separation in the equator whereas the latter case experiences its maximum separation at the maximum latitude. Therefore, an out-of-plane formation with relative circular motion, known as LISA-type formation, is achieved by imposing a set of conditions on the differential elements including ⌬⍀ or ⌬I . A formation with non-zero differential inclination will be J2 -invariant if a non-trivial solution of the linear system (6) exists. In general, the existence of such a solution is not guaranteed. For near-circular orbits, the stability constraints on argument of perigee and mean anomaly can be replaced by a single constraint on the argument of latitude u (Schaub, 2004). ⌬u˙ 12 = ⌬ M˙ 12 + ⌬ω˙ 12 = 0

(8)

Consequently, the linear system (6) is recast into ˙

u˙ J⍀, a,e,I (a1 , e1 , I1 )x = 0.

(9)

This underdetermined linear system has infinitely many non-trivial solutions. In general, two variables can be expressed as multiples of the other one which is selected independently. For instance, one can assume ⌬a as the independent variable and derive ⌬e and ⌬I correspondingly ⌬e = −

˙ a − u˙ a ⍀ ˙I u˙ I ⍀ ⌬a =: α ⌬a ˙ e − u˙ e ⍀ ˙I u˙ I ⍀

(10a)

⌬I = −

˙a ˙e ⍀ ⍀ ⌬a − ⌬e =: β ⌬a. ˙I ˙I ⍀ ⍀

(10b)

A mission with near-polar orbit is ideal for the purpose of the Earth’s gravity field recovery. As the inclination approaches 90 ◦ , the corrections required in eccentricity and semi-major axis to compensate for the J2 effect become too large to be of practical value (Schaub and Alfriend, 2001, Sneeuw and Schaub, 2005). On the other hand, cancelling only the mean latitude rate difference is helpful just for a short time span. Therefore, obtaining a J2 -invariant near-polar orbit requires more studies which is beyond the scope of this paper. Herein, we just consider the relative circular motion requirements for the out-of-plane formation in a central gravity field. The requirements are (Schaub, 2004) ⌬a = 0 (11a) ⌬e =

ρ 2a

⌬I + tan(ω + ⌬ω ± π) sin I ⌬⍀ = 0 ⌬ω + ⌬M + cos I ⌬⍀ = 0 √ 3ρ 2 2 ⌬I + sin i ⌬⍀ = 2a

(11b) (11c) (11d) (11e)

Two differential metric Kepler elements are explicitly defined in (11a) and (11b). However, the necessary conditions for the relative circular motion on the angular Kepler elements plus the differential inclination are given by the last three equations. Therefore, one element out of four can be selected independently. For instance, ⌬ω or ⌬M can be selected such that a particular geometry is obtained. Let us assume ⌬ω = π and derive the respective values for the other elements using the condition equations: √ 3ρ sin ω ⌬I = 2a √ 3ρ cos ω ⌬⍀ = 2a sin I ⌬M = −π − cos I ⌬⍀

(12a) (12b) (12c)

Fulfilling the equations yields a circular motion in the central gravity field. In practice, however, the formation will be affected by the Earth’s flattening as well as the higher order harmonics. For example, two satellites with initial differential elements given in Table 1, are synthesized in the gravity field which has been used in the previous examples. The relative motion of the satellites as well as the intersatellite distance for a one-month span of the orbit are given in Figures 4 and 5.

Gravity Recovery from Formation Flight Missions

33

15 5

z [km]

0 −10 20

10

0 −10

−5

10−7

−10

0 −20 x [km]

original signal GRACE error Cartwheel error Lisa error

0

−15 −20

15

−10

0 x [km]

10

20

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z [km]

10

10−8

10

10

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y [km]

10−6

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10−9

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0 x [km]

10

20

−10

0 y [km]

10

10

15

20

25

30 35 degree

40

45

50

55

60

Fig. 6. Dimensionless degree RMS of the recovered solutions.

Fig. 4. Relative motion of two satellites in the out-of-plane formation.

first day of month

30

ρ [km]

25 20 15 10

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0.1

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25 20 15 10 29

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formation type—with the acceleration difference formulation (Keller and Sharifi, 2005). Due to the relatively low maximum degree, a brute-force scheme was used to recover the spherical harmonic coefficients. The results of these gravity recovery simulations are shown in Figures 6–8. The Cartwheel configuration is seen to outperform the GRACE-type mission both in terms of accuracy and isotropic error behaviour. The same and even marginally better results are obtained by the LISA formation. Moreover, the differences between input and output geoids are computed for the individual solutions. As Figure 8 shows, the Cartwheel mission’s solution is more accurate and homogeneous in error behaviour.

Fig. 5. Baseline variation in LISA formation.

The intersatellite baseline is quite stable for the first revolutions. However, the relative motion has been affected by the high-order harmonics. Since no constraint has been applied on the mean latitude, the relative orbit is thus seen to drift in the negative along-track direction (−x).

differences ΔClm, ΔSlm

standard deviations σlm

−50 0 50 Slm Clm

−50 0 50 Slm Clm

0 20 40 60 0 20 40

3 Gravity Recovery Simulation

60 0

degree

Based on these three basic formation scenarios EGM 96 up to degree and order 60 was used to generate SST observations. Orbits were integrated and intersatellite range, range-rate and rangeacceleration vectors computed. The simulated observations were contaminated with a colored noise sequence comparable to the nominal noise pattern of the real GRACE mission (Reigber et al., 2005). The observation equations were set up—independent of

20 40 60

−9.5

−9

−8.5

−8

−7.5

Fig. 7. Differences and standard deviations of the recovered coefficients (from top: GRACE, Cartwheel and LISA missions).

34

N. Sneeuw et al. 7 6 5 4 3 2 1

1.8 1.6 1.4 1.2 1

isotropic, which is most clearly visualized in the spatial domain. The North-South stripes that are visible in Figure 8 (top), and which are so typical of real GRACE solutions, have disappeared in the Cartwheel solution. Incorporating out-of-plane signal into the observable will even improve on this result, though slightly. We suggest that future gravity field satellite missions should be designed to incorporate at least radial gravity field information. Pure leader-follower configurations are sub-optimal. With the same instrument sensitivity a Cartwheel-type formation attains superior results.

0.8 0.6 0.4 0.2

Fig. 8. The geoid height residuals in meter (top: GRACE bottom: Cartwheel).

4 Conclusion Conditions for satellite formation stability were given. They are typically derived in mean-element space. A J2 based transformation between mean and osculating elements was then used to set up orbit simulations that indeed demonstrate the stability of Low-Earth Formations (LEF). The remaining deviation from exact stability can be explained by the limitations of the mean-to-osculating element transformation used. The long-term stability of LISA type formations needs further careful attention. For all three scenarios SST-observations were generated with an input gravity field model up to degree and order 60 and contaminated with realistic noise. From these observations the gravity field was recovered and analyzed in the spectral and spatial domains. The results of these gravity simulations clearly demonstrate the benefits of incorporating radial information into the observable. The recovered field in the Cartwheel scenario has a significantly lower error spectrum. Moreover, this error spectrum is

References Kaula, W.M., (1966). Theory of satellite geodesy. Blaisdell, Waltham, Mass. Keller, W., M.A. Sharifi (2005) Gradiometry using a satellite pair. Journal of Geodesy 78:544–557. Massonnet, D. (2001). The interferometric Cartwheel: a constellation of passive satellite to produce radar images to be coherently combined. International Journal of Remote Sensing 22:2413–2430. Reigber, C., R. Schmidt, F. Flechtner, R. König, U. Meyer, K.H. Neumayer, P. Schwintzer, S.Y. Zhu (2005). An Earth gravity field model complete to degree and order 150 from GRACE: EIGEN-GRACE02S. Journal of Geodynamics 39:1–10. Schaub, H., K.T. Alfriend (2001). J2 invariant relative orbits for spacecraft formations. Celestial Mechanics and Dynamical Astronomy, 79:77–95. Schaub, H., J.L. Junkins (2003). Analytical mechanics of space systems. AIAA education series Reston VA. Schaub, H., (2004). Relative orbit geometry through classical orbit element differences. AIAA Journal of Guidance, Control and Dynamics 27:839-848. Sneeuw, N., H. Schaub (2005). Satellite Cluster for future gravity field missions. In: Proceedings of IAG symposia ‘Gravity, Geoid and space missions’ Jekeli C Bastos L Fernandes J (eds) Vol. 129, pp 12–17, Springer. Tapley B.D., S. Bettadpur, J.C. Ries, P.F. Thompson, M.M. Watkins (2004) GRACE measurements of mass variability in the Earth system. Science 305: 503–505.

GRACE Gravity Model Derived by Energy Integral Method Z.T. Wang The Institute of Geodesy and Geodynamics, Chinese Academy of Surveying and Mapping, 16 Bei Tai Ping Lu, 100039, Beijing, P.R. China J.C. Li, D.B. Chao The Key Laboratory of Geospace Environment and Geodesy, Ministry of Education, School of Geodesy and Geomatics, Wuhan University, Wuhan 129 Luoyu Road, Wuhan 430079, P.R. China W.P. Jiang School of Geodesy and Geomatics, Wuhan University, 129 Luoyu Road, Wuhan 430079, P.R. China

Abstract. Based on the energy integral equation of satellite orbit-motion, some applied computation formulas for Earth gravity field recovery from satellite to satellite tracking data are presented, in which a strict expression of the difference of kinetic energy between two satellites on the same orbit in terms of KBR range-rate observation value is given. Using GRACE data from the both satellite and energy integral method, a gravity model with max. degree 120 is derived, which named WHU-GM05. The tests of WHU-GM-05 series are performed by multi-comparisons, which include the comparisons between the model series and several analogous international geopotential models with respect to the corresponding degree variances, error spectra and geoidal heights, and comparisons of the model geoidal heights with GPS leveling in the area of U.S. and China (some regions). The results show that the total accuracy of WHU-GM-05 series is near to that of the models used in the comparisons. Keywords. GRACE, gravity filed model, energy conservation

1 Introduction Since the late 1960’s, the theory and method of the Earth gravity field recovery based on satellite to satellite tracking (SST) principle have widely been studied for a long time. In 1990’s, with the development of such new satellite gravity techniques, the studies on the corresponding mathematical models and computational methods for practical uses attracted great interest of geodetic researchers, in the meantime, the concepts on space-wise method and time-wise one with their mathematical models were developed for the recovery of the gravity field from the coming new satellite gravity data (Rummel

et al. 1993, Sünkel 2002), but the main attention was paid to time-wise method including traditional dynamic method for precise orbit determination used to solve geopotential coefficients from SST data. The new satellite gravity missions (CHAMP and GRACE) as the first realization of SST principle were carried out in July 2000 and March 2002 respectively. In recent years, more methods and computational models for solving geopotential coefficients under the concept of time-wise method by using the data of CHAMP and GRACE have been developed, apart from so-called dynamic approaches, which include energy conservation, semi-analytical and satellite acceleration approaches (Jekeli 1999, Han 2003, Sneeuw 2000, Ditmar et al. 2004). Energy conservation method originates Jacobi Integral for study on planetary motion presented in 1836. With the advent of artificial satellite in the last century, some authors tried to apply this energy method in geopotential determination from the observation data of satellite state (O’Keefe 1957, Bjerhammar 1967, Hotine et al. 1969), and their results put forward the further application of the method in the Earth gravity field recovery from SST data. From the point of view of practical uses, this method has the advantage of computational simplicity, therefore, it has been used to solve the observation equation with energy integral for CHAMP gravity model and GRACE one by many authors (e.g., Jekeli 1999, Ilk et al. 2003, Sneeuw et al. 2002, Tscherning 2001, Gerlach et al. 2003, Han et al. 2002). Moreover, the great efforts have been made to investigate the strategy and methodology on numerical solutions of normal equation in least-squares adjustment of satellite gravity observables because the equation system has too high dimensions to be solved by a common computer. One pays attention to an iterative solution for linear equation system, that is the method 35

36

Z.T. Wang et al.

of preconditioned conjugate gradient (PCCG), and it turns out that the method is very efficient for solving the normal equation mentioned above (Sünkel 2000, Ditmar et al. 2003, Han 2003). In this paper, a strict expression of the difference of kinetic energy between two satellites on the same orbit in terms of KBR range-rate observation is presented, and a preconditioned matrix is proposed which can be used in PCCG method to result in convergence of the iterative solutions for normal equation with the most high speed. Finally, a GRACE gravity model with max. degree 120 is derived, and the tests of the models are performed by multi-comparisons with several analogous geopotential models and ground truth.

2 Mathematical Models of Energy Integral Method Based on Energy Conservation Principle, the energy integral equation for single satellite orbital motion in inertial coordinate system can be expressed as (Jekeli 1999) V =

1 2 2 (˙r + r˙iy + r˙iz2 ) − ω(r ¯ i x r˙iy − riy r˙i x ) 2 ix − Vt − ⌬C − E 0

1 V AB = V A − V B = r˙B r˙ AB + |˙r AB |2 2 − ω¯ (rix r˙i y − ri y r˙ix ) A − (rix r˙i y − ri y r˙ix ) B − ⌬Vt AB − ⌬C AB − E 0 AB (4) The first two terms on the right side of equation (4) can strictly be expressed by the range rate ρ˙ AB of KBR observation between the double satellites, omitting the detail derivation, it is as follows:

r˙B r˙ AB +

ρ˙ AB 1 1 |˙r AB |2 = |˙r B | cos β + 2 cos γ 2

ρ˙ AB cos γ

2 (5)

(1)

where the geometric meanings of the angles β and γ are indicated in Figure 1

Or T + E0 =

where the definitions of all notations can refer to related textbooks and literatures, and the potential coefficients ⌬C¯ lm and ⌬ S¯lm here are unknown parameters to be estimated. For GRACE double satellite A and B, the potential difference between them can be expressed as (Jekeli 1999)

1 2 2 + r˙iz2 ) − ω(r ¯ i x r˙iy − riy r˙i x ) (˙r + r˙iy 2 ix − U0 − Vt − ⌬C (2)

where T is disturbing potential, E 0 is integral constant; r and r˙ are position and velocity vectors of satellite; ω¯ is the mean angular velocity of the earth’s rotation; Vt is the total corrections for various tide effects; U0 is the normal gravity potential, and ⌬C is energy loss induced by various unconservative forces. On the right side of equation (2), the first term is kinetic energy of unit mass, the second term is so called “potential rotation”, and all terms in this side can be evaluated with high accuracies. On the left side of the equation, T and E 0 are unknown parameters to be solved. Equation (2) is taken as observation equation in which T is expanded as ∞ l μ R T (r, θ, λ) = ⌬C¯ lm cos mλ (3) R r l=2 m=0 +⌬ S¯lm sin mλ P¯lm (cos θ )

(r A B )x (˙r A B )x + (r A B ) y (˙r A B ) y + (r A B )z (˙r A B )z |r A B | · |˙r A B | (6) (˙r B )x (˙r A B )x + (˙r B ) y (˙r A B ) y + (˙r B )z (˙r A B )z cos β = (7) |˙r B | · |˙r A B |

cos γ =

.

rB β ρ

GRACE

.γ r

rA − rB

AB

rB

rA θ Earth

l+1

Fig. 1. Geometry of GRACE SST.

GRACE

.

rA

GRACE Gravity Model Derived by Energy Integral Method

The corresponding observation equation to equation (4) can be written as

ρ˙ A B 1 ρ˙ A B 2 cos β + T A B + E 0 AB = |˙r B | cos γ 2 cos γ − ω¯ (ri x r˙i y − ri y r˙i x ) A − (ri x r˙i y − ri y r˙i x ) B − ⌬Vt AB − ⌬C A B − U0 AB (8)

where the subscript AB denotes the difference between the corresponding quantities of the two satellites, and T AB can be expressed as

TAB

⎧⎡ l+1 ⎪ R ⎨ P¯lm (cos θ A )cos mλ A ⎢ rA l+1 ⎣ ⎪ R ⎩ − P¯lm (cos θ B )cos mλ B rB lmax l ⎡ l+1 μ R = ¯ Plm (cos θ A )sin mλ A R ⎢ rA l=2 m=0 ⎢ +⎢ ⎣ l+1 − rRB P¯lm (cosθ B )sin mλ B =

⎤ ⎥ ¯ ⎦ ⌬Clm ⎫ ⎪ ⎪ ⎪ ⎬ ⎥ ⎥ ¯ ⎥ ⌬ Slm ⎪ ⎦ ⎪ ⎪ ⎭ ⎤

37

i p , N p j = 0, if i = j . If x i+1 − x i is less than a given limit ε, the iterative process will be ended, otherwise, it should return to step (2). In this paper, we adopt the rule of the arranging in order for potential coefficients as unknown parameters in the observation equation, which was proposed by (Colombo 1981). The arrangement rule will result in a normal matrix in which there is a dominant diagonal blocks (see Figure 2). Figure 3 shows the diagonal block matrix. Obviously, it is approximate to the matrix N, therefore, it is taken as an optimal preconditioned matrix H in our performing of the solution for normal equation with PCCG method. It turns out that the use of this special preconditioned matrix can speed up the convergent rate of iterative solutions very much.

lmax l μ X CA − X CB ⌬C¯ lm + X SA − X BS ⌬ S¯lm R l=2 m=0

(9)

where X c and X s can be calculated with the spherical coordinates of satellite positions at corresponding observing epoch.

3 PCCG Method Used in Solution of Normal Equation The common form of normal equation is N xˆ = b

(10)

Choosing a preconditioned matrix H , it is symmetric and positive definite one being approximate to N, so that the equation H −1 N xˆ = H −1b will be equivalent to equation (10), and the condition number of the matrix M = H −1 N could be lower than that of matrix N itself. The computation scheme of PCCG method is as follows

Fig. 2. The constructional picture of the normal matrix (lmax = 60) produced from real GRACE orbital data.

(1) x 0 = 0,r 0 = −b, p0 = H −1r 0 (2) α i = − (3)

pi

T

H −1 r i

( pi ) N pi T

x i+1 = x i + α i · p i (4) r i+1 = r i − α i N pi (5) pi+1 = −H −1r i+1 +

i+1 T −1 i+1 H r r

(r i )T H −1 (r i )

(11)

pi

where r i = N x i−1 − b is residual vector, α i is factor of iterative step, and pi is direction of correction which satisfies N−conjugate relation, i.e.

Fig. 3. The construction of the diagonal block matrix taken from Fig. 2.

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4 Data Used and Pre-processing 4.1 Data Used in Development of GRACE Gravity Model WHU-GM-05 The released data products of GRACE mission to users comprise about 23 classes. The data used in development of GRACE gravity model WHU-GM05 include the data of four classes: precise orbits, accelerometer, star camera and KBR. The precise orbit data consist of TUM reduced dynamic orbits and kinematic ones (from June first 2003 to October first 2003); the others are GRACE data products of Level-1B released by GFZ and JPL. 4.2 Data Pre-processing For precise orbital data, firstly, the quality of all data sets is identified in order to delete the abnormal data records according to the corresponding qualitative standards. With JPL dynamic orbital data, the qualitative flag (QF) contained in the last byte of each record is used for identification of the data, while with the data of TUM reduced dynamic and kinematic orbits, the a posteriori RMS given in leastsquares adjustment for orbit determination is used to detect the abnormal records to be deleted. Moreover, the comparison between JPL orbits and TUM ones is also used to detect those large differences beyond an acceptable statistical limit, and the corresponding unreliable orbit records will be deleted. Secondly, according to the accuracy information given in the released data files, the accuracies of the orbital data are estimated for setting up the weight matrix of observables in adjustment system. For accelerometer data (ACC1B) contained in GRACE Level-1B, the data provide 3 linear acceleration components and 3 angular ones of test mass in the direction of 3 coordinate axes with 1 Hz sampling rate. Firstly, the quality control of ACC1B data is carried out with QF given in each observation record to delete the unqualified records. Secondly the reliability of ACC1B data, i.e. a capability of anti-disturbance against a strong instant disturbing impulse, is analyzed, in which the data are transformed into inertial coordinate system with the transform matrix provided by star camera data. It turns out that ACC1B data retain their reliability very well. This analysis also shows that GRACE accelerometer has high sensitivity. The final step is to calibrate the systematic errors of ACC1B data. The following calibrating model for one direction of coordinate axis is adopted (Bettadpur 2003): anew = bias + scale · aACC1B

(12)

where a is acceleration, bias and scale are calibrating parameters. The preliminary calibration is firstly performed using the initial estimates of the two parameters released by GRACE data center. The further calibrating is based on a given geopotential model and equation (2). In equation (2), disturbing potential T is calculated with geopotential model, and energy loss term ⌬C is expressed as a linear expression including unknown parameters of bias and scale. Taking the differentiation between two observation equations of epoch ti and epoch ti−1 , the differential observation equation in which the unknown E 0 is excluded is formed for estimating the parameters of bias and scale by least-squares adjustment. The geopotential models including EGM96, OSU91A1F, TEG4 and EIGEN2 are used for the calibration and comparison. The results show that this method for the calibration of GRACE ACC1B data is efficient, and it is also showed that the results are usually not sensitive to priori geopotential models with consistent and higher accuracy level; for example, the difference in parameter bias between EGM96 and other model is very small, but the difference in parameter scale between EGM96 and OSU91A1F appears as the maximum (0.13) because of OSU91A1F model with lower accuracy level than others. For KBR data, the data file is KBR1B included in GRACE Level-1B with 10 Hz data sampling rate. KBR1B contains three classes of the data: biased range (BR), range rate (RR) and range acceleration (RA). Three terms of corrections for ionosphere delay, light time and geometry (the casual bias of phase center of antenna) should be added to KBR1B data. The correction for ionosphere delay has been made in released KBR1B data, and the correction for light time and geometry, i.e. LTC and GC, will be made by users with the correction parameters provided by GRACE data center. After making these corrections mentioned above, the quality control of KBR1B data is then performed to detect those abnormal data records beyond specified indexes of quality control according to QF in the last byte of KBR1B data record, and the abnormal records should be deleted in editing of data file. It should be noted that the abnormal values of signal-noise rate (SNR) about 340 which is less than its admissible value (NSNR ≥ 450) were detected in KBR1B data records of GRACE A (before May 8, 2003) and GRACE B (before February 3, 2003), but it had turned out that the values were false, and they should be taken as normal ones (Frank 2003). Moreover, the compatibility between the pre-processed KBR data and the adopted precise orbit data of GRACE satellites is


39

evaluated in terms of the comparison of the RR data in KBR1B with the computed range rate from the precise orbital data corresponding to KBR sampling epoch, and the results of the comparisons show that the two classes of range rate is consistent very well at corresponding accuracy level.

5 GRACE Gravity Model WHU-GM-05 and Tests Based on energy conservation principle and energy integral method, a GRACE gravity model with max. degree 120 is developed. The involved main computational work include: pre-processing of GRACE precise orbital data to obtain the updated ones in a unified time system consisting of JPL dynamic orbits with 60’s sampling rate and TUM reduced dynamic orbits with 30’s sampling rate; calibrating of accelerometer data (ACC1B) with the geopotential model EGM96 and coordinate transforming of the data to obtain the updated observations in both scientific reference frame and inertial reference system; pre-processing of KBR data (KBR1B) to obtain the corrected observations; computing the time series of the disturbing potentials along single GRACE satellite orbit and the time series of the disturbing potential differences between double GRACE satellites based on energy integral method; computing of design matrices of energy observation equations with respect to the disturbing potential of single satellite and their differences of double satellites as observables based on energy conservation principle as well as forming and solving of the corresponding normal equations for disturbing potential coefficients based on PCCG method. The geopotential model WHU-GM-05 has been obtained, which is the gravity model of double satellites derived from the disturbing potential difference observations of two months (August–September, 2003) with max. degree 120 (see Figures 4 and 5). The tests of WHU-GM-05 series are performed through multi-comparisons, which include the com-

Fig. 5. A anomaly field of WHU-GM-05 (unit: 10–5 m/s2 ).

parisons between the model series and several analogous international geopotential models with respect to the corresponding degree variances, error spectra and geoid heights, and the comparisons of the model geoid heights with GPS leveling. The models used for the comparisons include EGM96 (360/120), EIGEN-GRACE02S (150), EIGEN-CHAMP03S (140), GGM02S (160) and WDM94 (360) (developed by former Wuhan Technical University of Surveying and Mapping (WTUSM), 1994), and the GPS leveling geoid heights used for comparisons come from three local

70°E

90°E

100°E

110°E

120°E

130°E

140°E 70°N

60°N

60°N

50°N

50°N

40°N

40°N

30°N

30°N

20°N

20°N

10°N

10°N

0° 70°E

130°E 60°N

80°E

90°E

120°W

110°W

100°E

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110°E

90°W

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80°W

130°E

70°W

0° 140°E

60°W 60°N

50°N

50°N

40°N

40°N

30°N

30°N

20°N

20°N

130°E

Fig. 4. Geoid of WHU-GM-05 (unit: m).

80°E

70°N

120°W

110°W

100°W

90°W

80°W

70°W

60°W

Fig. 6. Locations of GPS leveling networks used for comparisons.

40

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GPS leveling networks (307 points) in China and GPS leveling (A order and B order) networks (2723 points) in U.S. (see Figure 6). The following Figures 7–13 and Tables 1–3 will show the results of the tests.

6 Conclusions The results of the tests show that the characteristics of GRACE gravity model WHU-GM-05 are near to that of the models used in the comparisons.

Fig. 10. Differences in geoid heights between WHU-GM-05 and GGM02S (unit: m).

Fig. 11. Differences in geoid heights between WHU-GM-05 and EIGEN-GRACE02S (unit: m).

Fig. 7. Comparison of degree variance of WHU-GM-05 to Kaula rule and the degree variance difference between the models.

Fig. 12. Differences in geoid heights between EIGENGRACE02S and GGM02S (unit: m). Fig. 8. Differences in geoid heights between WHU-GM-05 and EGM96 (unit: m).

Fig. 13. Differences in geoid heights between WHU-GM-05 and EIGEN-CHAMP03S (unit: m). Fig. 9. Differences in gravity anomaly between WHU-GM05 and EGM96 (unit: 10–5 m/s2 ).


41

Table 1. Tests by GPS leveling networks in China (unit: m)

Model

Degree

Sampling duration (day)

EGM96 EGM96 WDM94 GGM02S EIGEN-GRACE02S EIGEN-CHAMP03S WHU-GM-05

360 120 360 160 150 140 120

– – – 363 110 975 60

Mean −0.043 −0.058 −0.533 −0.617 −0.260 −0.838 0.567

Max.

Min.

2.929 −3.163 3.086 −4.887 2.809 −5.277 4.439 −5.732 3.157 −6.367 4.657 −11.147 3.339 −4.417

RMS 0.660 0.887 0.639 1.943 0.827 2.085 1.243

Table 2. Tests by GPS leveling networks in U.S. (unit: m)

Model

Degree

EGM96 EGM96 WDM94 GGM02S EIGEN-GRACE02S EIGEN-CHAMP03S WHU-GM-05

360 120 360 160 150 140 120

Sampling duration (day) – – – 363 110 975 60

Mean

Max.

Min.

−1.062 −1.074 −0.775 −1.077 −1.077 −1.064 −1.092

1.668 2.668 2.068 2.358 2.648 3.087 2.588

−4.879 −5.739 −4.639 −4.294 −5.169 −6.049 −6.169

RMS 0.525 0.798 0.581 1.045 0.760 0.896 1.042

Table 3. Difference in geoid heights between various models and EGM96 (unit: m)

Model

Degree

Sampling duration (day)

WDM94 GGM02S EIGEN-GRACE02S EIGEN-CHAMP03S WHU-GM-05

360 160 150 140 120

– 363 110 975 60

With respect to the accuracy of the model, from Table 2, we can see that EGM96 has the highest accuracy with 0.52 m RMS. for satellite models, EIGENGRACE02S has the highest accuracy with 0.76 m, EIGEN-CHAMP03S with 0.90 m, and GGM02S and WHU-GM-05 have the same accuracy with 1.04 m. Therefore, we come to the preliminary conclusion that the real accuracy of the model WHU-GM-05 is about one metre order, and near to the accuracies of other analogous international geopotential models mentioned above at 1 ∼ 2 decimetres level. Moreover, the tests also show that the energy integral method based on energy conservation principle is efficient for solving GRACE gravity model.

Acknowledgements The research was supported by Open Research Fund Program of the Key Laboratory of Geospace

Mean −0.027 −0.005 0.009 −0.011 0.047

Max.

Min.

12.678 7.308 7.801 12.598 6.683

−16.63 −11.872 −11.373 −17.626 −11.399

RMS 1.434 1.144 0.793 1.127 0.887

Environment and Geodesy, Ministry of Education, China (Project No.: 905152533-05-08) and Chinese National Natural Science Foundation Council under grant 40637034, 40704004. Many thanks go to Dr. Drazen Svehla in Technical University of Munich for his GRACE precise kinematic orbit data. We are very grateful to GFZ and JPL for GRACE data.

References Bettadpur, S.:2003, GRACE Product Specification Document, GRACE 327–720. Bjerhammar, A.: 1967, A New Approach to Satellite Geodesy. Research Institute for Geodetic Sciences, 701 Prince Street, Alexandria, Virginia, 22314, USA. Colombo, O. L.: 1981, Numerical Methods for Harmonic Analysis on the Sphere. Department of Geodetic Science Report No. 310, Ohio State University, Columbus.

42 Ditmar, P., and van Eck van der Sluijs, A. A.: 2004, A technique for earth’s gravity field modeling on the basis of satellite accelerations. Journal of Geodesy, 78: 12–33. Ditmar, P., Klees, R., and Kostenko, F.: 2003, Fast and accurate computation of spherical harmonic coefficients from satellite gravity gradiometry data. Journal of Geodesy, 76, 690–705. Frank F.: 2003 GRACE AOD1B product description document. Geoforschungszentrum Potsdam. ˇ 2003, Gerlach, Ch., Sneeuw, N., Visser, P., and vehla, D. S.: CHAMP gravity field recovery using the energy balance approach. Advances in Geosciences, 1, 73–80. Han, S.-C.: 2003, Efficient Determination of Global Gravity Field from Satellite-to-satellite Tracking (SST). Ph. D. Dissertation, The Ohio State Univ, Columbus. Han, S.-C., Jekeli, Ch., and Shum, C. K.: 2002, Efficient gravity field recovery using in site disturbing potential observable from CHAMP. Geophysical Research Letters, 29(16). Hotine, M., and Morrison, F.: 1969, First Integrals of the Equations of Satellite Motion. 41–45. Ilk, K.H. and Löcher, A.: 2003, The use of energy balance relations for validation of gravity field models and orbit determination results. A Window in the Future of Geodesy, pp. 494–499, Proceedings of the International Association of Geodesy IAG General Assembly Sapporo. Japan June 30– July 11.

Z.T. Wang et al. Jekeli, Ch.: 1999, The determination of gravitational potential differences from satellite-to-satellite tracking. Celestial Mechanics and Dynamical Astronomy, 75, 85–101. O’Keefe, J.: 1957, An application of Jacobi’s integral to the motion of an earth satellite. The Astronomical Journal, 62 (1252), 265–266. Rummel, R. et al.: 1993, Spherical harmonic analysis of satellite gradiometry. Netherlands publications on Geodesy. Sneeuw, N.: 2000, A Semi-Analytical Approach to Gravity Field Analysis from Satellite Observations: Ph. D. Dissertation, der Technischen Universität München. ˇ Sneeuw, N., Gerlach, Ch., Svehla, D., and Gruber, Ch.: 2002, A first attempt at time-variable gravity recovery from CHAMP using the energy balance approach. in: IN Tziavos (ed.) Gravity and Geoid 2000, pp 237–242, Proc. 3rd Meeting of the International Gravity and Geoid Commission, Thessaloniki, 26–30. Sünkel, H. (ed.): 2000, From Eötvös to mGal, ESA/ESTEC contract No, 13392/98/NL/GD. Sünkel, H.: 2002, From Eotvos to Milligal. Final Report. ESA. Tscherning, C. C.: 2001, Computation of spherical harmonic coefficients and their error estimates using least-squares collocation. Journal of Geodesy, 75: 12–18.

Robust Estimation and Robust Re-Weighting in Satellite Gravity Modelling J.P. van Loon Delft Institute of Earth Observation and Space System (DEOS), TU Delft, Kluyverweg 1, P.O. Box 5058, 2600 GB Delft, The Netherlands

Abstract. In this paper, we will discuss robust estimation and robust re-weighting techniques for the analysis of data from the space-gravimetric missions (CHAMP, GRACE, and GOCE in the near future). The stochastic and the functional models of these data are not perfectly known beforehand, and the data quality may vary considerably in time. We have used variance component estimation to re-weight the data sets, using a fast Monte-Carlo-type algorithm. In general, the data sets contain a small amount of outliers, which, if not properly treated, will contaminate the spherical harmonic solution. Moreover will they affect an automatic re-weighting scheme, which in turn may wrongly down-weight good data. This study aims at the estimation, validation and improvement of the stochastic model and the spherical harmonic solution, using the VCE technique in combination with a robust treatment of the outliers in the data. Unlike diagnostic methods like the three-sigma rule or data snooping, robust estimation aims not at removing outliers but seeks to minimize their impact. We will assume different probability functions of the data, including Huber’s distribution, to account for the presence of outliers. Special attention will be given to the robustification of the estimation procedure in conjunction with the computation of the variance components, where little theoretical results are known. The method has been tested with real CHAMP satellite gravity data, derived from the energy balance approach, and the results were slightly better than those from conventional diagnostic outlier detection. Finally, the satellite-only solution was combined with a prior model using VCE.

Keywords. Robust estimation, M-estimation, Huber’s distribution, variance component estimation, CHAMP, gravity models

1 Introduction With the launch of the satellite gravity missions CHAMP and GRACE and the expected launch of the GOCE mission in 2007, a new era in the modelling of the Earth’s gravity field has started. Millions of highquality gravity-related observations will be released to the geodetic community. As the measurements are corrupted by many small error contributions, the summation of these errors will, in general, produce normally distributed data (central limit theorem, see e.g. (Cramér, 1946). Under the assumption of normality and with true functional and stochastic models, the Least-Squares (LS) method will produce a Maximum Likelihood (ML) estimate of the vector of unknowns, which is also a Best Linear Unbiased Estimator (BLUE), with best referring to a minimum average variance. However, a set of satellite gravity data is, in general, contaminated by a (small) amount of outliers. As the LS method tries to minimize the square of the residuals, such outliers can have a great effect on the estimation of the vector of unknowns. Moreover, they will affect a possible re-weighting of the data sets. The stochastic properties of the observations change in time, due to e.g. the constellation of the GPS satellites or a malfunctioning of the instruments on board. Several papers, e.g. (Van Loon and Kusche, 2005), have shown the benefits of Variance Component Estimation (VCE) to estimate these stochastic properties and consequently downweight some spurious data sets. The detection of the outliers can either be done in the time domain or in the space domain. The detection in the time domain is performed by fitting the observations to some local model, e.g. polynomials, splines or wavelets. A disadvantage of such a method is that it does not take into account the position of the observation. An anomaly in the field can be detected as an outlier in the time domain. One therefore has to take into account the estimated gravity

43

44

field and search for any large residuals with respect to this field. An overview of several outlier detection techniques which can be applied on the GOCE observations can be found in Kern et al. (2005). Two alternative approaches exist for the treatment of the outliers in the space domain. In the first approach, one tries to detect and subsequently remove outliers using e.g. hypothesis testing, data snooping or the three-sigma rule assuming normality (see e.g. Baarda, 1968), and then, after their elimination, apply the LS method. In Huber (1964), an alternative approach (M-estimation) was proposed for cases, in which the observations have a slightly different distribution from the normal distribution, with a higher probability of large residuals. Several methods are proposed to estimate the vector of unknowns using such an alternative distribution. We will use the Iteratively Reweighted Least Squares method (Beaton and Tukey, 1974), which will down-weight possible outliers and then use the LS methodology. A wide variety of possible distributions exists in literature. We will focus on the most commonly used distribution, i.e. Huber’s distribution (Huber, 1964, 1981), and some modifications of this distribution. In Constable (1988) a method is introduced, which fits a distribution to the standardized residuals, but this method will not be considered in this contribution. Recently, M-estimation techniques have successfully been used e.g. with SLR data (Yang et al., 1999), magnetic Ørsted data (Olsen, 2002) and in relative GPS positioning (Chang and Guo, 2005). However, they assume the stochastic model to be known and compute the robust weights using these prior values. We will use VCE in an iterative way to estimate the stochastic properties of the data, which not only re-weights groups of data, but will also improve the estimation of the robust weights. Attempts to robustify the VCE have been proposed by Fellner (1986) and Dueck and Lohr (2005). The proposed methods have been tested with real CHAMP data, derived from the Energy Balance approach. Comparisons with the diagnostic threesigma method are made and among different distributions. The satellite-only CHAMP-solution is then combined with the prior model EGM96.

J.P. van Loon

y = Xβ + e, and the stochastic model is defined by E(e) = 0

Assuming we have n satellite gravity observations to estimate u unknown parameters (e.g. spherical harmonic coefficients), the functional model reads

;

D(e) = ⌺,

(2)

with n × 1 vector of observations y, n × u design matrix X, u × 1 vector of unknown parameters β, n × 1 vector of stochastic observation errors e. As the observations will be considered uncorrelated throughout this paper, the matrix ⌺ will be a diagonal matrix (⌺ = diag(σ12 . . . σn2 )). All distributions, considered in the M-estimation, can be written, for a suitable function ρ(.), as p(y|β) ∝ exp[−

n

ρ(μ j )],

(3)

j =1

with μ j the standardized observation error, defined by μ j = e j /σ j . Maximization of the probability function p(y|β) results in the Maximum Likelihood ˆ Such a maximization is equal to estimate of β, i.e. β. the minimization of the cost function ρ(μ j ), which can be written as n ⭸ρ(μ j ) j =1

⭸βi

|β=βˆ = 0

for all i = 1, . . . , u.

(4)

Defining the influence function ⌿(μ j ) as ⭸ρ(μ j ) , ⭸μ j

⌿(μ j ) :=

(5)

equation (4) can rewritten as n

⭸μ j X ji = ⌿(μˆ j ) =0 ⭸βi σj n

⌿(μˆ j )

j =1

(6)

j =1

for all i = 1, . . . , u. We will use the Iteratively Reweighted Least Squares method (Beaton and Tukey, 1974) to calculate the Mestimator. Faster methods exist, but we want to prove the validity of the concept here for satellite gravity data. Defining the robust weights as wˆ j := w(μˆ j ) :=

2 M-estimation 2.1 Derivation

(1)

⌿(μˆ j ) , μˆ j

(7)

equation (6) becomes n wˆ j j =1

σ j2

eˆ j X j i = 0

for all i = 1, . . . , u.

(8)

Robust Estimation and Robust Re-Weighting in Satellite Gravity Modelling

which leads to XT Wˆ Xβˆ = XT Wˆ y

(9)

with Wˆ the diagonal weight matrix with the diagonal elements wˆ j /σ j2 . This is an iterative method, as the new estimate βˆ will change the robust weights wˆ j : XT Wˆ (i) Xβˆ

(i+1)

= XT Wˆ (i) y

(10)

2.2 Overview of M-estimators We will now give an overview of several distributions, which have been used for the M-estimation. A broad overview of other distributions can be found in Zhang (1997). 2.2.1 Huber’s Distribution The Huber distribution (Huber, 1964, 1981) assumes the observations to be normally distributed within an interval [−k1, k1 ] and following the Laplace distribution outside this interval, i.e. the cost function continues as a linear function, decreasing the influence of those observations. The distribution is defined by equation (3), where ⎧ 2 μ ⎪ ⎪ ⎨ j if μ j ≤ k1 2 ρ H (μ j ) = 2 ⎪ k ⎪ ⎩k1 · μ j − 1 if μ j ≥ k1 . 2 (11) In Koch (1999), it was stated that if the number of outliers is about 4%, one should use a treshold k1 = 1.5 and a treshold of 2.0 for the 1% level of outliers. Hence, in many cases in satellite gravity modelling, one would set this treshold within this interval. The robust weights will now decrease, when the standardized residual μˆ j is outside the interval [−k1 , k1 ]: 1 if wˆ j = k1 /μˆ j if

μˆ j ≤ k1 μˆ j ≥ k1 .

(12)

In Huber’s M-estimation, no outliers are removed. With each iteration, the weights of the outliers will change (mostly decrease) till convergence. 2.2.2 Tri-weighted M-estimation Observations with a norm of the standardized residual μˆ j above a certain second threshold k2 , with k2 ranging between 3.0 and 8.5 (Yang et al., 2005), are likely to be an outlier. Huber’s M-estimation downweights these observations, instead of removing them. The tri-weighted M-estimation gives zero

45

weights to observations with a standardized residual outside the interval [−k2, k2 ] and assumes a Huber distribution within this interval. The robust weights can now be computed by ⎧ 1 ⎪ ⎪ ⎨

k1 wj = μ ˆ ⎪ ⎪ ⎩ j 0

if

μˆ j ≤ k1

if

k1 < μˆ j ≤ k2

if

k2 < μˆ j .

(13)

In Olsen (2002), the tri-weighted M-estimation was used to re-weight magnetic data from the Ørsted satellite (k1 = 1.5 and k2 = 5.0). 2.2.3 IGG-3 Scheme (Yang) A short-coming of the tri-weighted M-estimation is the discontinuity of the weight and influence functions at the threshold k2 . In Yang (1994) a modification of the tri-weighted M-estimation has been proposed, quite similar to the approach of Hampel (1974), in which the weight and influence functions decay towards 0 in k2 , making the influence function continuous: ⎧ ⎪ 1 if μˆ j ≤ k1 ⎪ ⎪ ⎨ k2 − μˆ j 2 k1 wj = if k1 < μˆ j ≤ k2 · ⎪ μˆ j k2 − k1 ⎪ ⎪ ⎩ 0 if k2 < μˆ j . (14) One could modify this weighting scheme by choosing another power than 2 for the decay function.

3 Robust Estimation of Variance Components In previous work (Kusche and Van Loon, 2004; Van Loon and Kusche, 2005), we have shown that a spherical harmonic solution for CHAMP data could improve considerably if we use VCE methods to improve the stochastic model and, consequently, reweight the individual data sets. We will assume the data to be uncorrelated and divide the data into different data sets, in which the stochastic properties can be considered homogeneous, i.e. we will estimate one variance component γk for each of the p data sets. The a-priori standard deviation σi of obser√ vation i is equal to γk , in which observation yi is a part of group k. 3.1 REML of Huber’s Distribution The Restricted Maximum Likelihood Estimator (see e.g. Koch, 1990) of the p × 1 variance component

46

J.P. van Loon

vector γ (γ1 . . . γ p ) seeks to maximize

p(y|γ ) ∝

∞ −∞

...

∞ −∞

p(y|β, γ )dβ,

4 Test Study: CHAMP Pseudo-Observations (15)

which, under the assumption of uncorrelated observations, can be computed by using p(y|β, γ ) =

p(y j |β, γk ).

(16)

j

The observational probability functions, in case of a Huber distribution, read p(y j |β, γk ) ∝ ⎧ ⎪ ⎪ ⎨

exp [−

μ2j

]

2 ⎪ k2 ⎪ ⎩exp [−(k1 · μ j − 1 )] 2

4.1 Satellite-Only Solution

if μ j ≤ k1

if μ j ≥ k1 . (17) As the shape of the observational probability function depends on μ j and consequently on y j , an analytical expression for p(y|γ ) cannot be derived. 3.2 Fellner’s Approach Although it was not possible to derive equations for the REML in case of a Huber distribution, one can bound the influence of outliers on the estimation of the variance components. The Iterative Maximum Likelihood Estimator (IMLE) assuming a normal distribution reads γˆk =

eˆ kT eˆ k rk

(18)

with rk as the redundancy number of the observation group k and ek as the vector of observation residuals of group k. Fellner (1986) suggested to bound the vector eˆ k , which changes the IMLE to γˆk =

ˆ k )T ⌿(μ ˆ k) γ¯k · ⌿(μ rk

(19)

with γ¯k as the prior estimate of γk . This would, however, be a biased estimate, as normally distributed data will be bounded as well. A quasi-unbiased estimator, therefore, reads γˆk =

ˆ k )T ⌿(μ ˆ k) γ¯k · ⌿(μ h · rk

We have estimated a CHAMP-only global gravity field solution up to degree and order 75, using an equivalent of 232 days of CHAMP pseudoobservations. These observations were derived from ˇ kinematic orbit data (Svehla and Rothacher, 2003) and from accelerometer and quaternion data provided by GFZ, using the energy balance approach (see e.g. Jekeli, 1999 and Visser et al., 2003). A detailed description of the CHAMP data used here, can be found in Van Loon and Kusche (2005). The data are divided into 1250 data sets, of which the variance components are estimated using VCE.

At first, we have estimated a satellite-only solution, in which all observations are given equal weight. The difference in geoid heights between this solution and EIGEN-GL04C can be seen in Figure 1. The EIGEN-GL04C model uses both GRACE and LAGEOS data, combined with 0.5 × 0.5 degrees gravimetry and altimetry surface data (GFZ, 2006) and can, therefore, be used as an independent model to the CHAMP-only solutions. A stripe pattern in the geoid difference due to some spurious data sets is clearly visible in Figure 1. We were able to downweight these data sets using Monte- Carlo Variance Component Estimation (MCVCE), see e.g. (Kusche, 2003). This was done in an iterative procedure, simultaneously with the treatment of the outliers. The variance components are used to compute the standardized residuals μˆ i = eî /σˆ i , which is needed in the treatment of the outliers. The M-estimators are compared with the three-sigma rule, in which all observations with a value for μˆ i above the threshold k = 3 are removed

–180 90

–135

–90

–45

0

45

90

135

180 90

45

45

0

0

–45

–45

–90

–90

(20)

with h = D(⌿(z)), where z ∼ N(0, 1) and D is the dispersion operator.

–2.5

–0.3

–0.2

–0.1

0.0

0.1

0.2

0.3

2.5

Fig. 1. Geoid difference [L = 50], in meters, between the satellite-only solution using equal weights and EIGENGL04C.

Robust Estimation and Robust Re-Weighting in Satellite Gravity Modelling Table 1. Statistics of the geoid differences [m.] of four outlier treatment methods with EIGEN-GL04C L = 50

–180 90

L = 75

Method

|max.|

std.

|max.|

std.

Equal weights VCE-only

2.104 0.616

0.420 0.127

7.600 6.508

1.597 0.994

Sigma Huber Tri-weighted IGG-3

0.507 0.482 0.454 0.477

0.116 0.113 0.112 0.114

6.048 6.049 5.905 5.727

0.948 0.943 0.939 0.941

The satellite-only solution (tri-weighted Mestimation) was combined with EGM96 (Lemoine et al., 1998) up to degree and order 75. Tests have shown (Van Loon and Kusche, 2006) that we need to estimate an extra inconsistency vector for the degrees 2 and 4 coefficients of the CHAMP-only solution. The combined CHAMP-EGM96 solution had an rms geoid difference with EIGEN-GL04C of 8.22 cm (L = 50). The geoid differences can be seen in Figure 3. The good performance of EGM96 over the oceans has improved the combined solution considerably in those regions.

–90

–45

0

45

90

135

180 90

45

45

0

0

–45

–45

–90

–90

–2.5

–0.3

–0.2

–0.1

0.0

0.1

0.2

0.3

–90

–45

0

45

90

135

180 90

45

0

0

–45

–45

–90

–90 –0.3

–0.2

–0.1

0.0

0.1

0.2

0.3

2.5

Fig. 3. Geoid differences [L = 50], in meters, between the CHAMP-EGM96 combined solution and EIGEN-GL04C.

4.2 Combined Solution

–135

–135

45

–2.5

from the data set. Based on the experience with the data sets (number of outliers), we have set k1 = 1.7 and k2 = 4.5. The statistics of the different methods can be found in Table 1. Table 1 shows a slightly better performance of the tri-weighted M-estimation compared with the other three methods. The largest improvement in the result is, however, due to the reweighting of the data sets using MCVCE. The geoid differences between the tri-weighted solution (at convergence) and EIGENGL04C can be found in Figure 2.

–180 90

47

2.5

Fig. 2. Geoid differences [L = 50], in meters, between the tri-weighted M-estimation and EIGEN-GL04C.

5 Summary and Outlook Robust M-estimation techniques seek to minimize the impact of spurious observations instead of removing them. To compute these robust weights one needs to know the stochastic model of the observations. This model can be estimated using Variance Component Estimation (VCE). The estimation of the variance components is, however, contaminated by possible outliers. The treatment of the outliers and the estimation of the variance components are, therefore, linked to each other and must be performed in an iterative way. We have compared several Mestimation techniques with each other and with the conventional three-sigma rule. The tri-weighted Mestimation performed slightly better than the other methods if one compares the geoid differences with the independent EIGEN-GL04C model. A further improvement was made by combining the satelliteonly solution with the EGM96 geopotential model. Earlier tests showed that an inconsistency vector for the degrees 2 and 4 coefficients of the CHAM P-only solution is necessary in the combination procedure. The combined model had a rms geoid difference with EIGEN-GL04C of 8.22 cm up to degree and order 50. In the future, we shall test other M-estimation techniques and test the method of Constable (1988), in which splines are used to estimate the observation probability density function.

Acknowledgements We are grateful to the GFZ Potsdam for providing CHAMP ACC data. Thanks go also to the IAPG, TU Munich, for providing CHAMP kinematic orbits. J.v.L. acknowledges financial support by the GO-2 program (SRON EO-03/057).

48

References Baarda W (1968) A testing procedure for use in geodetic networks, Netherlands Geodetic Commission, Publ. on Geodesy, New Series 2(5). Beaton AE, Tukey JW (1974) The fitting of power series, meaning polynomials, illustrated on band-spectroscopic data, Technometrics, vol. 16, pp. 147–185. Chang XW, Guo Y (2005) Huber’s M-estimation in relative GPS positioning: Computational aspects, Journal of Geodesy, vol. 79, pp. 351–362. Constable CG (1988) Parameter estimation in non-Gaussian noise, Geophysical Journal, vol. 94, pp. 131–142. Cramér H (1946) Mathematical Methods of Statistics, Princeton University Press, Princeton. Dueck A, Lohr S (2005) Robust estimation of multivariate covariance components, Biometrics, vol. 61, pp. 162–169. Fellner WH (1986) Robust estimation of variance components, Technometrics, vol. 28, no. 1, pp. 51–60. GFZ (2006) http://www.gfz-potsdam.de/pb1/op/grace/. Hampel FR (1974) The influence curve and its role in robust estimation, Journal of the American Statistical Association, vol. 69, pp. 383–393. Huber PJ (1964) Robust estimation of a location parameter, Annals of Mathematical Statistics, vol. 35, pp. 73–101. Huber PJ (1981) Robust Statistics, Wiley series in probability and mathematical statistics, New York. Jekeli C (1999) The determination of gravitational potential differences from satellite-to-satellite tracking, Celestial Mechanics and Dynamical Astronomy, vol. 75, pp. 85–100. Kern M, Preimesberger T, Allesch M, Pail R, Bouman J, Koop R (2005) Outlier detection algorithms and their performance in GOCE gravity field processing, Journal of Geodesy, vol. 78, pp. 509–519. Koch K-R (1990) Bayesian inference with geodetic applications, Lecture Notes in Earth Sciences, Springer-Verlag, Berlin, Heidelberg. Koch K-R (1999) Parameter Estimation and Hypothesis Testing in Linear Models, second edition, Springer-Verlag, Berlin. Kusche J (2003) A Monte-Carlo technique for weight estimation in satellite geodesy, Journal of Geodesy, vol. 76, pp. 641–652.

J.P. van Loon Kusche J, Van Loon JP (2004) Statistical assessment of CHAMP data and models using the energy balance approach, in Reigber et al. (ed.) Earth Observation with CHAMP: Results from Three Years Orbit, Springer, Berlin. 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-1998206861, Goddard Space Flight Center, Greenbelt, MD. Olsen N (2002) A model of the geomagnetic field and its secular variation for epoch 2000 estimated from Ørsted data, Geophysical Journal International, vol. 149, pp. 454–462. ˇ Svehla D, Rothacher M (2003) Kinematic and reduceddynamic precise orbit determination of low earth orbiters, Advances in Geosciences, vol. 1, pp. 47–56. Van Loon JP, Kusche J (2005) Stochastic model validation of satellite gravity data: A test with CHAMP pseudoobservations, in Gravity, Geoid and Space Missions, ed. Jekeli C, Bastos L, Fernandes J, IAG Symposia, vol. 129, pp. 24–29, Springer, Berlin. Van Loon JP, Kusche J (2006) Towards an optimal combination of satellite data and prior information, accepted for the proceedings of Dynamic Planet 2005, joint IAG/IAPSO/IABO conference, Cairns. Visser P, Sneeuw N, Gerlach C (2003) Energy integral method for gravity field determination from satellite orbit coordinates, Journal of Geodesy, vol. 77, pp. 207–216. Yang Y (1994) Robust estimation for dependent observations, Manuscripta Geodaetica, vol. 19, pp. 10–17. Yang Y, Cheng MK, Shum CK, Tapley, BD (1999) Robust estimation of systematic errors of satellite laser range, Journal of Geodesy, vol. 73, pp. 345–349. Yang Y, Xu T, Song L (2005) Robust estimation of variance components with application in Global Positioning System network adjustment, Journal of Surveying Engineering, Nov. 2005, vol. 131, no. 4, pp. 107–112. Zhang Z (1997) Parameter estimation techniques: A tutorial with application to conic fitting, Image and Vision Journal International, vol. 15, no. 1, pp. 59–76.

Topographic and Isostatic Reductions for Use in Satellite Gravity Gradiometry F. Wild, B. Heck Geodetic Institute, University of Karlsruhe, Englerstr 7, D-76128 Karlsruhe, Germany, Fax: +49-721-608-6808, e-mail: [email protected]

Abstract. Gravity gradiometry is strongly sensitive to the gravity field induced by the topographic and isostatic masses of the Earth. The downward continuation of the gravitational signals from satellite height to sea level is rather difficult because of the high frequency behaviour of the combined topographicisostatic effect. Therefore a topographic-isostatic reduction is proposed in order to smooth the signals. Based on different isostatic models (Airy-Heiskanen, Pratt-Hayford, Airy-Heiskanen/Pratt-Hayford), the generalized Helmert model and the crust density model via CRUST2.0 the topographic-isostatic effects are calculated for a GOCE-like satellite orbit. Using tesseroids modelled by Gauß-Legendre cubature (3D) leads to high numerical efficiency. For the Marussi tensor of the gravitational potential the order of magnitude of both topographic and isostatic components is about ±8 E.U., while the combined topographic-isostatic effect varies from ±0.08 E.U. (Helmert II), ±0.8 E.U. (Airy-Heiskanen, PrattHayford, Airy-Heiskanen/Pratt-Hayford, Helmert I) and ±4 E.U. (crust density model). In this paper, the focus is put on the gravitational effect of massive bodies in regard to the comparison between the classical isostatic models, the condensation models of Helmert and the crust density model. Keywords. Satellite gravity gradiometry, topographic reduction, isostatic reduction (AiryHeiskanen model, Pratt-Hayford model, Helmert’s condensation models, crust density model), gravitational effect of massive bodies

1 Introduction The effects of the topographic and isostatic masses are visible in the gravitational signals, e.g. in the satellite gravity gradiometry (SGG) observations of the GOCE mission. Because of the rough gravity field the downward continuation is rather complex and challenging. The smoothing of the data using

a topographic-isostatic reduction based on different isostatic and condensation models – as presented in this paper in Chap. 2 – is one possibilty to make the downward continuation easier (see Makhloof and Ilk, 2005). The remove-compute-restore (RCR) technique as described in e.g. Forsberg and Tscherning (1997) is another possibility for smoothing the gravity field. The modelling of the masses, the topographic masses on the one hand and the isostatic balance masses on the other hand, is a central issue in physical geodesy. Two methods can be differentiated: the analytical and numerical methods in the space domain and the spherical harmonic expansion in the frequency domain. In this paper the gravitational effects of massive bodies are described (see Chap. 3). In practice, the volume elements are often approximated by rectangular prims as e.g. described in Mader (1951), Nagy (1966), Forsberg (1984), Tsoulis (1999) and Nagy et al. (2000, 2002). It is also possible to use point masses, mass lines and mass layers (see e.g. Grüninger, 1990; Kuhn, 2000) or a radial integration of the volume elements and a numerical solution of the resulting 2D-integral (see e.g. Martinec, 1998; Novák et al., 2001). In Seitz and Heck (2001) an alternative mass element, the tesseroid, is presented. With the aid of a simulation the different mass elements and methods are analysed and a comparison of the topographic, isostatic and condensation models is performed in Chap. 4.

2 Topographic and Isostatic Reduction The effect of the topographic and isostatic masses can be described using the approximation of the geoid as a sphere g of radius R. r is the radius of the computation point Q; the geocentric radius r is the radius of a running point on the surface of the Earth. ψ is the spherical distance between the radius vectors of Q and P (see Figure 1 in Wild and Heck, 2004). The potential of the topographic and isostatic masses can be described by the Newton integral in 49

50

F. Wild, B. Heck

Fig. 1. Minimum and maximum density information per layer of the CRUST2.0 model.

spherical coordinates which is in general V (Q) = G σ

⎡ ⎢ ⎣

ξ=ξ2

ρ

ξ =ξ1

⎤ ξ2

⎥ dξ ⎦dσ,

(1)

= r 2 + ξ 2 − 2r ξ cos ψ, cos ψ = sin ϕ sin ϕ + cos ϕ cos ϕ cos(λ − λ ). (2) (r, φ, λ) and (ξ, φ , λ ) denote the spherical coordinates of the computation point and the variable integration point, respectively, related to a terrestrial reference frame. G is the gravitational constant, ρ the local mass density and d = ξ 2 · dξ dσ the volume element (dσ : surface element of the unit sphere). The parameters of the topographic, isostatic and condensation models used in this paper are described in Heck and Wild (2005) and Wild and Heck (2005) and are shown in Table 1. These models represent the

crust which corresponds to the reality at some places while extreme deviations exist in other regions. The CRUST2.0 model (see Tsoulis, 2004) takes the known density information into consideration which plays an important role in the compensation of global crustal structures. The CRUST2.0 model as an updated version of the CRUST5.1 model (see Mooney et al., 1998) consists of seven layers: (1) ice, (2) water, (3) soft sediments, (4) hard sediments, (5) upper crust, (6) middle crust and (7) lower crust (see Figure 1). For these seven layers and the mantle, the density (ρi , i = 1...8) and the velocity of the compressional and shear waves is provided, in particular the top of the water and the depths of the seven layers are available (ti , i = 0...7). The depth of the seventh layer reflects the Moho discontinuity. The first version of modelling the Moho can be defined using the depth of the seventh layer analogously to the theory of the Airy-Heiskanen model with a density of the crust ρ0 = 2670 kg · m−3 and the mantle ρm = 3270 kg·m−3 . The standard column has a depth T = 25 km. The second possibility uses the variation of the density according to Kuhn and Featherstone (2003) where the density of the crust is ρ0 = 2861 kg · m−3 and the density of the mantle is ρm = 3381 kg · m−3 . The use of the complete density information in terms of a computation of the mean density of each 2◦ × 2◦ column via weighted calculation of the average density and the consideration of the density of the mantle is a third version of the modelling of the Moho. In Tsoulis (2004) a forth version is listed which is a refinement of the third version; the isostatic effect is computed per layer with a depth Ti . The density contrast of each 2◦ ×2◦ element is ⌬ρi = ρi+1 − ρi , for i = 1...7.

Table 1. Parameters of topographic, isostatic, crust density and condensation models

3D-integrals Topographic model Isostatic model of

Airy-Heiskanen Pratt-Hayford Airy-Heiskanen/ Pratt-Hayford (Claessens, 2003)

Crust density model Condensation model of

Helmert I Helmert II

Radial component ξ

Density ρ

ξ1 = R, ξ2 = r ξ1 = R − T − t , ξ2 = R − T ξ1 = R − D, ξ2 = R Land area (Airy-Heiskanen) Ocean area (Pratt-Hayford) ξ1 = R − T − t , ξ2 = R − T

ρ = ρ0 ρ = ⌬ρ = ρm − ρ0 ρ = ⌬ρ = ρ0− ρ

2D-integrals ξ = Rc,I ξ = Rc,II

ρ = kI ρ = kII

Topographic and Isostatic Reductions for Use in Satellite Gravity Gradiometry

3 Gravitational Effect of Massive Bodies on SGG Data To model the topographic and isostatic masses a segmentation into volume elements i is made where the density ρi is assumed to be constant: V (Q) = G

i

ρi ⍀i

d⍀

(3)

The coordinates of the Marussi tensor with respect to the local (North/East/Up) triad at the computation point Q as second derivatives of the potential are given by Tscherning (1976). The triple integral of each volume element is analytically solvable for the prism and its approximation by the point mass, the mass line and the mass layer. In case of the tesseroid – bounded by geographical grid lines and concentric spherical shells of constant height (see Heck and Seitz, 2007) – no analytical solution exists in general; one variant to solve the triple integral is the evaluation by pure numerical methods, e.g. the Gauß-Legendre cubature (3D). A second alternative is provided by a Taylor series expansion of the integrand, where the term of zero order is equivalent to the point mass derived from the prism. Another possibility to compute the elliptic integral is the decomposition into a one-dimensional integral over the radial parameter ξ for which an analytical solution exists and a two-dimensional spherical integral which is solved by quadrature methods, especially the Gauß-Legendre cubature (2D). In case of the second radial derivative, the respective formulae can be found in Heck and Wild (2005). The approximation of the tesseroids by prisms, postulating mass conservation, also provides an option for the solution of the triple integral; this procedure requires a transformation of the coordinate system of the prism into the local coordinate system at the computation point.

4 Results The formulae for the different mass elements and computation methods have been checked by assuming a synthetic topography of constant height over a spherical cap and the position of the computation point situated on the polar axis. For this special situation an exact analytical solution for the tesseroid exists and a comparison between the analytical solution of a spherical cap and the modelling of different mass elements is possible. Especially the second radial derivative is calculated for a synthetic topography of constant height (1000 m), when the

51

Table 2. Maximum error for a concentric ring zone (⌬ψ = 5 ) /total error for a spherical cap (ψc = 10◦ ) concerning different mass elements and tesseroid height 1000 m

Mass element

Mzz max (absolute value) [E.U.]

Mzz [E.U.]

Point mass Mass line Mass layer Prism Tesseroid-Taylor

5.06505e-005 5.09955e-005 4.52783e-005 4.55422e-005 2.28325e-008

−4.56971e-004 −4.56912e-004 −9.22607e-004 −9.22550e-004 1.31502e-007

dimension of the tesseroids is 5 × 5 and the computation point Q is situated on the polar axis. The height of the point Q is 260 km, approximately equivalent to the flight height of the forthcoming GOCE mission. Other parameters are: R = 6378137 m, G = 6.672· 10−11 m3 · kg−1 · s−2 and the constant density ρ0 = 2670 kg·m−3 . In Tables 2–4 the maximum errors for concentric ring zones and the total errors for the sum of the different mass elements and methods in comparison to the analytical solution of a spherical cap are listed. The numerical results demonstrate that the tesseroid modelled by different methods provides the most accurate approach compared to the exact solution of a spherical shell. The modelling by an analytical integration in r + Gauß-Legendre cubature (2D) or a Gauß-Legendre cubature (3D) is practically equivalent for n = m = p = 2. The modelling by a Taylor series expansion is worse by a factor of 2. The computation time for these three cases is nearly equivalent (see Figure 2a–c). The 2D-cubature method is slightly worse, the 3D-cubature method is slightly better than the Taylor series expansion. The mentioned topographic-isostatic models and their influence on the gravity gradients at the height

Table 3. Maximum error for a concentric ring zone (⌬ψ = 5 ) /total error for a spherical cap (ψc = 10◦ ) concerning the radial analytical integration + Gauß-Legendre cubature (2D) for 0 ≤ n = m ≤ 5 and tesseroid height 1000 m

n=m


Mzz [E.U.]

0 1 2 3 4 5

5.10129e-005 1.04617e-008 5.57690e-012 5.57690e-012 5.57680e-012 5.57670e-012

−4.56932e-004 5.59172e-008 −8.47530e-010 −8.46780e-010 −8.46570e-010 −8.46570e-010

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A-H/P-H Helmert I Helmert II Crust

Table 4. Maximum error for a concentric ring zone (⌬ψ = 5 )/total error for a spherical cap (ψc = 10◦ ) concerning the Gauß-Legendre cubature (3D) for 0 ≤ n = m = p ≤ 2 and tesseroid height 1000 m

n=m= p


Mzz [E.U.]

0 1 2

5.06505e-005 1.04628e-008 5.57680e-012

−4.59729e-004 6.00153e-008 5.0600e-0012

T = 25 km/D = 100 km d = 25 km d = 0 km T = 25 km

The density of the crust and the mantle is ρ0 = 2670 kg · m−3 and ρm = 3270 kg · m−3 in the case of the classical isostatic models and the generalized Helmert model. The density parameters in the crust model are listed in Chap. 2. By variation of the depth of the standard column of the crust model, it is possible to demonstrate that there are large differences (about 40 km) in the roots of AiryHeiskanen (T = 25 km) compared to those of the crust density model (see Tsoulis, 2004). T = 25 km as depth of the standard column for the crust density model approximates best the Airy-Heiskanen model (see Table 5). Figure 3 displays the Marussi tensor of the topographic masses with the order of magnitude of about ±8 E.U. The second radial

of the GOCE satellite are compared by means of the magnitude of the topographic-isostatic effect based on the tesseroid modelling using the Gauß-Legendre cubature (3D) for n = m = p = 1 (8 nodal points per element). This method is favoured because of the sufficient precision and the low computation time (see Tables 2–4, Figure 2a–c). In the case of the condensation models of Helmert, the surface integral is evaluated by the Gauß-Legendre cubature (2D) for n = m = 1 (4 nodal points per element). For this simulation, the topographic/isostatic effects are computed over the whole Earth. The parameters R, G and the satellite height have the same value. The digital terrain model JGP95E “rockequivalent” has the resolution 1◦ × 1◦ . The depths of the standard column of each isostatic model are: Airy-Heiskanen T = 25 km Pratt-Hayford D = 100 km

Table 5. Statistics of the differences between the roots of Airy-Heiskanen (T = 25 km) and the CRUST2.0- layer 7

T = 22.97 km T = 25.00 km T = 30.00 km

max(abs) [km]

min(abs) [km]

rms(abs) [km]

44.141 42.111 37.111

0 0 0

8.677 8.469 9.905

a)

b)

c)

Fig. 2. Comparison of different mass elements and methods regarding the computation time.

Topographic and Isostatic Reductions for Use in Satellite Gravity Gradiometry

53

Fig. 3. Marussi tensor related to the topographic masses.

Fig. 4. Airy-Heiskanen isostatic reduction [⭸2 VA−H (Q)/⭸r 2 ], topographic-isostatic reduction [⭸2 Vt (Q)/⭸r 2 − ⭸2 VA−H (Q)/⭸r 2 ].

derivative ⭸2 VA−H (Q)/⭸r 2 of the isostatic potential of Airy-Heiskanen also has the order of magnitude of about ±8 E.U. (see Figure 4). The combined topographic-isostatic effect of the Airy-Heiskanen model is one order of magnitude smaller (see Wild and Heck, 2005). Figure 5 displays the second radial derivative of the crust density model – version 2; the order of magnitude is also about ±8 E.U. The combined topographic-isostatic effect is about ±4 E.U. In Table 6 the statistics of the topographicisostatic effect of Airy-Heiskanen are listed; in Table 7 the rms values of the Marussi elements of the topographic-isostatic effect for different isostatic and condensation models are shown – excluding the fourth version of Tsoulis (2004). Comparing

Table 6. Statistics of the Marussi tensor of the topographic-isostatic effect (Airy-Heiskanen)

Mxx Mxy Mxz Myy Myz Mzz

max [E.U.]

min [E.U.]

rms [E.U.]

0.4623 0.2917 0.7697 0.4131 0.6054 0.9649

–0.6476 –0.4392 –0.7818 –0.7853 –0.7940 –0.5270

0.0678 0.0351 0.0784 0.0704 0.0778 0.1158

the isostatic and condensation models with respect to Airy-Heiskanen, it is only possible to make relative statements because of the lack of GOCE data. The topographic-isostatic effect of the Pratt-Hayford

Fig. 5. Crust model reduction [⭸2 VCrust−v2 (Q)/⭸r 2 ], topographic-isostatic reduction [⭸2 Vt (Q)/⭸r 2 − ⭸2 VCrust−v2 (Q)/⭸r 2 ].

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Table 7. Statistics of the rms [E.U] of the Marussi tensor of the topographic-isostatic effect (different isostatic and condensation models) Pratt-Hayford A-H/P-H Mxx Mxy Mxz Myy Myz Mzz

0.1214 0.0628 0.1407 0.1259 0.1389 0.2075

0.1071 0.0554 0.1241 0.1097 0.1216 0.1819

Helmert I

Helmert II

Crust – v1

Crust – v2

Crust – v3

0.0561 0.0293 0.0650 0.0584 0.0646 0.0959

0.0051 0.0025 0.0058 0.0047 0.0054 0.0081

0.8729 0.4100 0.9823 0.8932 0.9873 1.5407

0.7784 0.3613 0.8636 0.7756 0.8509 1.3592

1.0250 0.4619 1.1647 1.0550 1.1517 1.8143

model is larger by a factor of 2, the combined A-H/PH model deviates by a factor of 1.5 from the AiryHeiskanen model. The first Helmert model nearly equals the Airy-Heiskanen model; the topographicisostatic effect of the second Helmert model is one order of magnitude smaller. The crust density model deviates in all three versions by a factor of ten from the Airy-Heiskanen model. Regarding the first version the factor is 12, with the parameters of Kuhn and Featherstone (2003) the factor is 11; the third version deviates by a factor of 15.

5 Conclusions The numerical results demonstrate that the tesseroid modelled by different methods provides the most accurate approach compared to the exact solution of a spherical shell. The topographic and isostatic effects are significant at satellite height. The order of magnitude is about ±8 E.U. for each constituent. The combined topographic-isostatic effect varies for the different isostatic and condensation models. In case of the classical isostatic models the effect has an order of magnitude of about ±0.8 E.U.; the results for the first Helmert model are practically equivalent to those of the Airy-Heiskanen model. The effect of the Helmert II model is one order of magnitude smaller; the different versions of the modelling of the crust density are larger by a factor of ten in comparison to the Airy-Heiskanen model. At present it is not possible to recommend a model for the reduction of GOCE-data; therefore only relative statements concerning the smoothness can be done.

Acknowledgement The authors thank Dr. D. Tsoulis and an anonymous reviewer for the valuable comments.

References Claessens S (2003) A synthetic earth model. Analysis, implementation, validation and application. Delft university of Technology, Delft University press (Delft, The Netherlands).

Forsberg R (1984) A study of terrain reductions, density anomalies and geophysical inversion methods in gravity field modelling. Report 355, Department of Geodetic Science, The Ohio State University, Columbus, OH. Forsberg R, Tscherning CC (1997) Topographic effects in gravity modelling for BVP. In: Sansò F, Rummel R (eds) Geodetic Boundary Value Problems in View of the One Centimetre Geoid. Lecture Notes in Earth Sciences, vol. 65. Springer, Berlin/Heidelberg/New York, pp 241–272. Grüninger W (1990) Zur topographisch-isostatischen Reduktion der Schwere. PhD thesis, Universität Karlsruhe. Heck B, Seitz K (2007) A comparison of the tesseroid, prism and point-mass approaches for mass reductions in gravity field modelling. J Geod 81: 121–136 DOI: 10.1007/s00190006-0094-0. Heck B, Wild F (2005) Topographic reductions in satellite gravity gradiometry based on a generalized condensation model. In: Sansò F (ed) A Window on the Future of Geodesy. Springer, Berlin/Heidelberg/New York, pp 294–299. Kuhn M (2000) Geoidbestimmung unter Verwendung verschiedener Dichtehypothesen. Reihe C, Heft Nr. 520, Deutsche Geodätische Kommission, München. Kuhn M, Featherstone WE (2003) On a construction of a synthetic earth gravity model (SEGM). In: Tziavos IN (ed): Gravity and Geoid 2002, Proceedings of the 3rd Meeting International Gravity and Geoid Commission, pp 189–194. Mader K (1951) Das Newtonsche Raumpotential prismatischer Körper und seine Ableitungen bis zur dritten Ordnung. ¨ ¨ Osterr Z Vermess Sonderheft 11, Osterreichische Kommission für Int. Erdmessung. Makhloof A, Ilk KH (2005) Far-zone topography effects on gravity and geoid heights according to Helmerts’s methods of condensation and based on Airy-Heiskanen model. Proceedings of the 3rd Minia International Conference for Advanced Trends in Engineering. Martinec Z (1998) Boundary-value problems for gravimetric determination of a precise geoid. Lecture notes in Earth Sciences 73. Springer, Berlin/Heidelberg/New York. Mooney WD, Laske G, Masters TG (1998) CRUST 5.1: A global crustal model at 5◦ × 5◦ . J Geophys Res 103: 727–747. Nagy D (1966) The gravitational attraction of a right rectangular prism. Geophysics 31: 362–371. Nagy D, Papp G, Benedek J (2000) The gravitational potential and its derivatives for the prism. J Geod 74: 552–560 DOI: 10.1007/s001900000116.

Topographic and Isostatic Reductions for Use in Satellite Gravity Gradiometry Nagy D, Papp G, Benedek J (2002) Corrections to “The gravitational potential and its derivatives for the prism”. J Geod 76: 475 DOI: 10.1007/s00190-002-0264-7. Novák P, Vanićek P, Martinec Z, Véronneau M (2001) Effects of the spherical terrain on gravity and the geoid. J Geod 75: 691–706 DOI: 10.1007/s00190-005-0435-4. Seitz K, Heck B (2001) Tesseroids for the calculation of topographic reductions. Abstracts “Vistas for Geodesy in the New Millenium”. IAG 2001 Scientific Assembly 2–7 September 2001, Budapest, Hungary, 106. Tscherning CC (1976) Computation of the Second-Order Derivatives of the Normal Potential Based on the Representation by a Legendre Series. Manuscr Geodaet 1: 71–92. Tsoulis D (1999) Analytical and numerical methods in gravity field modelling of ideal and real masses.

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Reihe C, Heft Nr 510, Deutsche Geodätische Kommission, München. Tsoulis D (2004) Spherical harmonic analysis of the CRUST 2.0 global crustal model. J Geod 78: 7–11 DOI: 10.1007/s00190-003-0360-3. Wild F, Heck B (2004) Effects of topographic and isostatic masses in satellite gravity gradiometry. Proceedings of the Second International GOCE User Workshop GOCE, The Geoid and Oceanography, ESA-ESRIN, (ESA SP – 569, June 2004), CD-ROM. Wild F, Heck B (2005) A comparison of different isostatic models applied to satellite gravity gradiometry. In: Jekeli C, Bastos L, Fernandes J (eds) Gravity, Geoid and Space Missions. Springer, Berlin/Heidelberg/New York, pp 230–235.

Gravity Change After the First Water Impoundment in the Three-Gorges Reservoir, China S. Sun, A. Xiang, C. Shen Institute of Seismology, CEA, Wuhan 430071, P.R. China P. Zhu Royal Observatory of Belgium, Belgium B.F. Chao College of Earth Sciences, National Central University, Taiwan; NASA Goddard Space Flight Center, USA Abstract. This paper presents the high-precision gravity survey for the gravity changes resulting from the water load, crustal deformation and underground water level change, before and after the first (2003) water impoundment of the Three-Gorges Reservoir. We find: (i) Close to the dam, the gravity effect is the most prominent with maximum of about 200 ␮gal. The gravity effect of the crustal deformation does exist, but is localized to within a few km. Gravity change resulting from underground water level change, and the impact of rainfall should not be neglected. (ii) In the area around the reservoir, the maximum gravity change is near XiangXi. Monitoring the variation of gravity and further study of various relevant data should continue in the future. Keywords. The Three-Gorges reservoir, the first water impoundment, gravity change

1 Introduction By 2009, behind one of world’s highest artificial dams standing 180 m tall, China’s Three-Gorges Reservoir will be holding 40 km3 of water, flooding a stretch of middle Yangtze River about 600 km in length at a width of 1–2 km. The impoundment process is implemented in three phases, in 2003, 2006 and 2009. During the early construction of the Three-Gorges Reservoir, annual monitoring of gravity was conducted by the Three-Gorges gravity observation network established around the reservoir (section between the towns of Sandouping and Badong). Seismic activities were also monitored. However, since the 2003 first water impoundment and the subsequent infiltrating of underground water, the annual examination with the existing moderate observation stations would hardly be sufficient to monitor potential 56

seismic hazard zones and the effects of water load in this area. Thus, with governmental supports we have significantly improved the observation network and scheme in recent years, increasing the spacetime repetition of observation. This paper describes the high-precision gravity survey before and after the 2003 water impoundment and its primary results, particularly the gravity changes observed during the process of impoundment.

2 The Improvement and Complementarity of the Monitoring Networks The Bureau of Earthquakes of the Hubei Province established a seismicity/gravity observation network in the Three-Gorges area in 1982, also to ensure the security of the Gezhouba dam (an existing secondary dam some distance downstream) by monitoring the regional gravity changes, especially in some potential seismic hazard zones such as the Fairy Mountain area. In 1998, the network was enlarged in order to detect and monitor earthquakes that would possibly be triggered by the impoundment water load. Three steps were taken: first to expand the original network to encompass the potential seismic hazard zone of Xing Mountain-Badong; second to revitalize some sub-standard observation stations; and third to add two extra gravity stations of high national standard and other gravity stations co-located with some existing GPS stations for vertical ground observation. Figure 1 depicts the regional distribution of the networks in the area; all gravity observation lines were in superposition with vertical deformation network (Canfei et al. 2002), some of them were incorporated directly into the vertical deformation network. The network after reconstruction consists of three rings and four lines.

Gravity Change After the First Water Impoundment in the Three-Gorges Reservoir, China

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Fig. 1. Map of network station distribution of gravity and GPS, around the Three-Gorges Reservoir area. Tectonic structure lines are also indicated.

Simulation calculations (Wang et al. 2002, Du et al. 2004a) have shown that the gravity change due to the water load of the first water impoundment would reach ∼ 2 mgal(1 gal = 10−2 ms−2 ), most evident in the basin in front of the dam but attenuates very rapidly with distance and practically disappears beyond ∼ 10 km. Thus two additional short section observation lines were planned before the water impoundment. One traversed across the basin area with a span of 20 km with 10 stations. Another was alongside the south bank of the basin centered at the dam, about 10 km in length with 5 stations, among which three were shared with the cross-basin line. Stations are selected in accordance with basin width, possible surface subsidence, and in relation to a well network of the Three-Gorges Project (Yongtai et al. 2002).

3 High-Precision Gravity Survey and Data Processing The first water impoundment happened in late May – mid-June of 2003. We carried out three resurveys at all regional stations within the gravity network – in first half of April, first half of July, and second half of October, in order to compare the situation beforeand-after. Note that while the direct gravity change resulting from the water load would happen instantaneously, the indirect gravity effects by underground

water infiltration and crustal deformation presumably lag behind. The results will be presented in Section 4. For the two short-section observations lines near the dam, 7 observation sessions were conducted during a shorter time period. The first observation period was May 25th 2003 when the water level in dam region reached about 80 m. Afterwards, we measured gravity every time the water level increased by 20 m, until the last measurement on July 11th. The results will be presented in Section 5. All the gravity observations were done by LCR G gravimeters with precision of ∼ 10 ␮gal. We have improved our processing method in precision and in more detail over reported before (Sun et al. 2002). We chose the comparatively stable Huangling anticline area as the benchmark for gravity calculation, and adopted the Kriging method, which grids the gravity change of the observation stations that scattered irregularly within the study area. We used 7 × 7 distance power matrix to sieve the waves. We set the weight of the observation station located in the center of the matrix as 1, and the weight decreases by 0.25 with each grid distance. The gravity changes on the borderline of the map were neglected because most of them were probably the result of local extrapolation. After sieving the waves, we could eliminate the disturbance of haphazard errors, superficial surface of the crust and regional sources of anomaly, accentuating the influences of deeper crust and regional sources of anomaly. In addition, we also eliminated

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some signals not continuous and directional with gravity change in space and time.

4 Gravity Change in the Region Figure 2 shows the pattern of relative changes of gravity in the Three-Gorges area before (Figure 2a), during (2b), and after (2c) the 2003 water

impoundment. Figure 2a shows the gravity change in April 2003 compared with half a year earlier. Figure 2b shows the change right before and right after the impoundment – July relative to April, accentuating the gravity effect of the water load. We found that the maximum centers on XiangXi. Figure 2c is for the few months after the impoundment. It is interesting to note the contrast between Figures 2c and 2a, which suggests that the positive gravity change shifted from the middle and south of the network to the north due to the extra water load. But more detailed studies are needed to fully understand this finding, in terms of underground water movement, land deformation, and rainfall (see below).

5 Gravity Change near the Dam

a (2002.10–2003.04)

b (2003.04–07)

c (2003.07–10)

Fig. 2. Observed relative changes of gravity around the Three-Gorges Reservoir before, during, and after the 2003 water impoundment.

We shall now focus on the short section observation lines during the impoundment. Figure 3 shows the local gravity changes during and right after the 2003 water impoundment, benchmarked against the observed value for the first session (see above). We can see that before June 11th 2003, the gravity increase focused near the dam, and not too much to the south and north, where the gravity effect resulting directly from the water load is more obvious with maximum of about 200 ␮gal. Furthermore, the indirect gravity effects resulting from the underground water infiltration and crustal deformation do exist, although not as strong. After June 10th 2003, when the water impoundment is complete, the gravity increase obviously extended to the two sides (by about 5 km off bank during that period), which illustrates the intensifying indirect gravity effects. Yang’s (Guangliang et al. 2005) numerical simulation of gravity effects is basically consistent with the overall pattern of Figure 3, although differing somewhat in amplitude and range. At the same time, it also proves that the short section observation lines we laid out and the observation scheme we chose are effective. This result is also consistent with the vertical displacement change near the dam observed by large area leveling and GPS (Du et al. 2004b). Figure 4a, drawn on the basis of observations done by the Bureau of Synthetic Reconnaissance under the Yangtze River Water Conservancy Commission, presents the vertical deformation of the basin area around the Three-Gorges dam, with the maximum surface subsidence of less than 20 mm focused largely on the center area of the basin and decreasing with distance outwards. The speed of decrease is faster in the north than in the south.

Gravity Change After the First Water Impoundment in the Three-Gorges Reservoir, China

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Fig. 3. Observed variation of local gravity before and after the 2003 water impoundment near the Three-Gorges Dam.

The regional bedding lithology is generally basic to neutral volcanics with low porosity. So. we can neglect the crustal density change due to deformation. The relation between vertical deformation and gravity change can be regarded as free space gradient δ g = −0.3086δH. Figure 4b shows the resultant gravity change caused by the vertical deformation. The land subsidence by the water loading contributes little to the overall gravity change, the maximum of which is about the observation precision of the gravimeter. According to the study of the Three-Gorges well network (Yongtai et al. 2004), the water impoundment would affect the underground water level. The area is characterized by geology such that only the

weathered surface layer is water penetrable while other rock layers are not, except when fractured (Qingyun et al. 2003). Of the four wells near the dam, water level changes in 2 wells on the northern bank were more obvious, and the maximum was more than 3.7 m. The two wells on the southern bank saw little change. To quantitatively explain these observations requires detailed information about the geological conditions of stratum configuration near observation stations and the distribution of rock fractures. Such information is currently lacking. Besides water impoundment, soil humidity change caused by rainfall during the period as part of the regular weather can also change gravity. This natural contribution is not negligible in our high-precision

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Fig. 4. Observed vertical deformation and the resultant gravity change after the 2003 water impoundment near the Three-Gorges Dam, according to the Yangtze River Water Conservancy Commission report.

gravity observation. For example, during the period of intense observation between May 25th and July 11th 2003, the total rainfall was about 20 cm (according to the record of Maoping station). The rainfall mainly happened in late June and early July, after the impoundment. Therefore, the weather background of Figures 3a and 3b are quite different. We can estimate rainfall’s influence using δg = 2πGσ p, where ␴ is water density, p is rainfall. The result is as much as 8 ␮gal, not large but not negligible either.

6 Discussions We report the spatial pattern and temporal evolution of the gravity change observed by regional as well as local station networks around the ThreeGorges Reservoir before, during, and after the first water impoundment in May–June of 2003. We compare the quantitative results and discuss the various contributions to such gravity change, including the direct water mass, possible water infiltration to underground, vertical crustal deformation due to the water load, and the weather-related rainfall. We also report the contrast and relations of the gravity change with the observed vertical deformation of the crust due to the water impoundment. We note that understanding these mechanisms is pre-requisite knowledge in trying to understand possible triggering of

seismic activities as a complex consequence of the water impoundment. Further monitoring and more detailed analysis are needed to advance these studies. An effective means is to introduce further and independent observation in type, for which we should mention the application of the new space-borne time-variable gravity measurements provided by GRACE and future missions (such as GOCE and GRACE follow-on). An example was reported in the literature (Boy and Benjamin 2002), where GRACE observation was simulated to demonstrate its ability to detect the long-wavelength time-variable gravity due to the total water impoundment of the Three-Gorges Reservoir. Further studies (Carabajal et al. 2006) based on actual observations by the space-borne laser altimetry ICESat together with GRACE timevariable gravity data have also yielded definite and interesting evidences potentially useful to monitoring and towards our overall understanding of the geophysical consequences of the Three-Gorges water impoundment.

References Boy, Jean-Paul, and Benjamin F. Chao, Time-variable gravity signal during the water impoundment of China’s Three-Gorges Reservoir, Geophys. Res. Lett, 2002, 29(24): 531–534, doi:10.1029/2002GL016457.

Gravity Change After the First Water Impoundment in the Three-Gorges Reservoir, China Canfei, Xing,, Gong Kaihong and Du Ruilin, Crustal deformation monitoring network for Three-Gorges Project on Yangtze River, Journal of Geodesy and Geodynamics, 2002, 23(1):114–118. Carabajal, C., D. Harding, J-P. Boy, S. Luthcke, D. Rowlands, F. Lemoine and B. Chao, ICESat observations of hydrological parameters along the Yangtze River and Three-Gorges Reservoir paper presented at the Spring Meeting, Amer. Geophys. U., 2006. Guangliang, Yang, Shen Chongyang, Wang Xiaoquan, Sun Shaoan, Liu Dongzhi and Li Hui, Numerical simulation of gravity effect of water-impoundment in Three-Gorges Projection Reservior, Journal of Geodesy and Geodynamics, 2005, 25(1):19–23. Qingyun, Wang, Zhang Qiuwen and Li Feng, Study on risk of induced earthquake in reservoir head region of ThreeGorges Projection on Yangtze River, Journal of Geodesy and Geodynamics, 2003, 23(2):101–106. Du, Ruilin, Qiao xuejun, Xing Canfei, et al. Vertical deformation after first impoundment of Three-Gorges project, Journal of Geodesy and Geodynamics, 2004a, 24(1):41–45.

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Du, Ruilin, Xing Canfei, Wu zhonghua, et al. Crustal deformation of Three-Gorges area, Journal of Geodesy and Geodynamics, 2004b, 24(2):23–29. Sun Shaoan, Xiang Aimin and Li Hui. Changes of gravity field and tectonic activity in front region of the ThreeGorges Project, Journal of Geodesy and Geodynamics, 2002, 22(3):56–59. Wang, H., H.T. Hsu, Y.Z. Zhu. Prediction of surface horizontal displacements, and gravity and tilt changes caused by filling the Three-Gorges Reservoir. Journal of Geodesy, 2002, 76: 105–114. Yongtai, Che, Yu Jinzi, Liu Wuzhou, Yi Lixin, Xu Feng, Li Jiecheng and Sun Tianlin, Arrangement of well network and establishment of observation well at Three-Gorges of the Yangtze River, Journal of Geodesy and Geodynamics, 2002, 24(3):423–431. Yongtai, Che, Liu Wuzhou, Yan Ping, Liu Xilan and Liu Chenglong, Subsurface behaviors and its changes in Three-Gorges Well network before and after reservoir impounding, Journal of Geodesy and Geodynamics, 2004, 24(2):14–22.

Continental Water Storage Changes from GRACE Line-of-Sight Range Acceleration Measurements Y. Chen, B. Schaffrin, C.K. Shum Geodetic Science, School of Earth Sciences, The Ohio State University, 125 S. Oval Mall, 275 Mendenhall Lab., Columbus, Ohio 43210, USA, e-mail: [email protected]

Abstract. Spaceborne gravimetry such as GRACE provides a unique opportunity to observe basin-scale surface and subsurface global water storage changes with unprecedented accuracy and temporal and spatial coverages. The contemporary methodology to process GRACE data for hydrologic studies is in terms of monthly spherical harmonic geopotential solutions with a spatial resolution longer than 600– 800 km (half-wavelength), after proper smoothing. Alternative methods include the direct processing of satellite-to-satellite K-Band Range (KBR) rate data and to estimate the localized gravity field, and the so-called mascon methods. In this study we estimate monthly continental water thickness changes over the Amazon Basin for one year (2003) by calculating insitu Line-Of-Sight (LOS) gravity differences inferred by the GRACE KBR range acceleration. A regional inversion method based on Bayesian statistics is used to estimate water thickness changes from the LOS gravity difference observations. Power Spectral Density (PSD) comparison with the spherical harmonic monthly solutions at both 600 and 800 km resolutions indicates that the LOS gravity solution has more power at degree 13 or higher, implying that it is a viable technique for potentially enhancing spatial resolutions of GRACE solutions at these frequencies. Keywords. Continental water storage, GRACE lineof-sight gravity differences, Amazon hydrology

1 Introduction Accurate measurements of continental water storage changes for both the surface and subsurface water over basin-scales are useful for hydrologic and climate modeling. However, the temporal and spatial variations of continental water storage are imprecisely known due to the difficulty of direct measurements. The difficulty has been alleviated by the data from the dedicated satellite gravity mission, the Gravity Recovery and Climate Experiment (GRACE), which was 62

launched on March 17, 2002. Its objective is to map the global gravity field with a spatial resolution of 400– 40,000 km every thirty days. It consists of two identical satellites in near-circular orbits at ∼500 km altitude and 89.5◦ inclination, separated by approximately 220 km along-track, and linked by a highly accurate inter-satellite, dual-frequency K-Band microwave ranging system, it also carries high precision Global Positioning System (GPS) receivers, attitude sensors and 3-axis accelerometers (Tapley et al., 2004a). It has been shown by simulation that GRACE should be able to recover changes in monthly continental water storage and ocean bottom pressure at scales of 800 km or longer, with accuracies approaching ±2 mm in equivalent water thickness over land, and ±0.1 mbar or better in ocean bottom pressure (Wahr et al., 1998). Rodell and Famiglietti (1999) stated that GRACE will likely detect changes in water storage in most of the basins on monthly or longer temporal sampling. Rodell and Famiglietti (2001) further concluded that the detectability is possible for a 200,000 Km2 or larger area. In 2004, two years after the GRACE satellites were launched and operational, Tapley et al. (2004b) and Wahr et al. (2004) concluded that the GRACE mission can provide a geoid height accuracy of ±2 to ±3 mms at a spatial resolution as fine as 400 km (half-wavelength) or longer, and explained that geoid variations observed over South America can be largely attributed to surface water and groundwater changes. Such observations will help hydrologists to connect processes at traditional resolutions (tens of kms or less) to those at regional and global scales. Han et al. (2005b) adopted an alternative method using GRACE satellite-to-satellite tracking and accelerometer data to obtain the along-track geopotential differences and directly estimate the terrestrial water storage at monthly and sub-monthly resolutions. This method was tested on the estimation of hydrological mass anomaly over the Amazon and Orinoco river basins, and as opposed to the spherical

Continental Water Storage Changes from GRACE Line-of-Sight Range Acceleration Measurements

harmonic methods, the spatial extent of the estimated GRACE water storage anomaly achieved finer resolution and they follow more closely boundaries of the river basins. The global method to process GRACE data is in terms of monthly geopotential (in spherical harmonics, Bettadpur, 2004) estimates with a spatial resolution longer than 600–800 km (the so-called Level 2 or L2 data products), after proper Gaussian smoothing (Wahr et al., 2004). The regional methods include the direct processing of Level-1 B (L1B) satelliteto-satellite K-Band Range (KBR) rate data and the use of the energy conservation principle to estimate the localized gravity field. Han et al. (2005a), studied the estimation of continental water storage regionally, and claimed that higher frequency information of the continental water storage has been acquired by fully exploiting the high accuracy of the KBR range rate (±0.1 ␮m/sec). Other notable regional methods include the so-called mascon methods (Rowlands et al., 2005; Yuan & Watkins, 2006). In this study we developed a regional method to estimate continental water changes by using in-situ Line-Of-Sight (LOS) gravity differences inferred by the KBR range acceleration data and applied it the Amazon Basin by processing one year of GRACE data (January–December 2003). A regional inversion method based on Bayesian statistics is used to estimate water thickness changes from the LOS gravity difference observations. The resulting LOS solution is compared with global solutions at various resolutions to assess the potential usefulness of the developed technique.

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and satellite 2, respectively. a1 and a2 are the nongravitational forces on the satellite 1 and satellite 2, respectively. P r12 is the relative velocity vector between the two satellites. e12 is the Line-Of-Sight (LOS) vector, and e12 = r12 / |r12 |. The quantity (g2 − g1 )e12 is defined as the LOS gravity difference, and will be denoted as gLOS . With a-priori inter-satellite orbits and KBR range acceleration measurements we use (1) as a stochastic constraint to estimate gLOS as well as inter-satellite orbit vectors. Figure 1 is the flowchart of the procedure to calculate LOS gravity differences. Initial orbit vectors including position and velocity vectors are given, and KBR range acceleration measurement is used as stochastic constraint to refine the orbital vectors at each epoch. Specifically, we estimate arc-dependent accelerometer biases, relative orbital vectors, and other empirical parameters. The next step is to calculate the LOS gravity difference. The relatively well-known perturbations on the GRACE satellites, including the N-Body perturbations, solid earth tides, ocean tides, atmospheric and oceanic barotropic dealiasing are forward modeled, based on the current best models. In this study, we hydro log y can thus be calculated ignored pole tides. gLOS from the following equation, N-Body earth ocean gLOS = gmean + gLOS + gtides LOS LOS + gLOS atmosphere

+gLOS

hydro log y

+ gLOS

A-priori intersatellite orbits

+ gothers LOS .

(2)

KBR range acc. measurements

2 Method The low–low Satellite-to-Satellite Tracking (SST) includes the measurement of the differences in satellite orbit perturbations over baselines of a few hundred kilometers. In our investigation we develop a method to use the GRACE KBR acceleration data, precise orbits, and 3D accelerometer measurements in a Bayesian regional inversion for water thickness changes. Let r1 and r2 represent the position vectors of the two GRACE satellites, and |r12 | is the range between the two satellites. We have 2 |˙r12 |2 − ρ˙12 , |r12 | (1) where ρ˙12 and ρ¨12 are inter-satellite range rate and range acceleration measurements, respectively. g1 and g2 are the gravitational forces on satellite 1

ρ¨12 = (g2 − g1 )e12 + (a2 − a1 )e12 +

Gauss-Markov Model with stochastic constraints

r12,i − r12,i−1 < ε ?

Adjusted orbits Acc. & Att measurements

Various models

Estimate KBR empirical parameters and accelerometer biases Calculate in-situ LOS gravity difference

Fig. 1. Procedure of calculating in-situ LOS gravity difference.

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Y. Chen et al. hydro log y

The relationship between gLOS tal water storage is hydro log y

gLOS ×

N×M i=1

and continen-

(r1 , θ1 , λ1 ; r2 , θ2 , λ2 ; t) = Gρw (R⌬θ)(R⌬λ) 1 1 i i Cn,1 sin θi h i · e12 , ∇ i − Cn,2 ∇ i l1 l2 (3a)

l1i = l2i =

R 2 + r12 − 2Rr1 cos ψ1i ,

(3b)

R 2 + r22 − 2Rr2 cos ψ2i ,

(3c)

cos ψ1i = cos θi cos θ1 + sin θi sin θ1 cos(λi − λ1 ), (3d) cos ψ2i = cos θi cos θ2 + sin θi sin θ2 cos(λi − λ2 ), (3e) where r1 , θ1 and λ1 and r2 , θ2 and λ2 are radius, co-latitude and longitude for satellite 1 and for satellite 2, respectively, G is the gravitational constant, ρw is the density of fresh water (1000 kg/m3 ). R is the mean Earth radius, and (R⌬θ )(R⌬λ)sin θi represents the horizontal area of a rectangular prism at the location (θi , λi ). h i is the mean water thickness i per unit area at the location (θi , λi ) and time t. Cn,1 i and Cn,2 are the transformation matrices from navigation frame (North–East-Down) to inertial frame. N and M are the numbers of the grid intervals, ⌬θ , ⌬λ, along latitude and longitude, respectively. ∇ is the gradient operator. Using (3a) through (3e) we infer the water storage hydro log y , change from the in-situ gravity difference, gLOS using a regional inversion technique based on Bayesian statistics by introducing a stochastic model for the unknown quantity, it’s a-priori information in the form of a covariance matrix (Han et al., 2005a). The final step is to correct the estimated water storage h i considering the loading effect:

p/L θ , and E–W (longitude) frequency, q/L λ . Furthermore, k f is the load Love number at the mean (isotropic) frequency f = ( p/L θ )2 + (q/L λ )2 . We used the relationship of n = 2π R f for the conversion between the spherical harmonic degree, n, and the planar frequency, f (Jekeli, 1981). Note that h(θk , λl ) in (4a) and (4b) corresponds to h i in (3a) for a particular combination of k and l.

3 Results We have chosen the South America region as our test area including the Amazon and the Orinoco basins and applied our method using one year (January– December 2003) of GRACE data. Figure 2 shows the 2◦ × 2◦ solution grids in the region and an example month (July 2003) of GRACE ground track coverage. The boundaries of the Amazon basin and Orinoco basin are also shown. The upper region is the Orinoco basin and the lower region is the Amazon basin. The ground tracks cover the study area fairly homogeneously. It is evident that the N–S direction sampling is better than the W–E direction sampling, as the orbit is non-repeating. The orbital data are given at the accuracy of ±2 cm, the accuracy of the accelerometer data is ±10−10m/sec2 and the accuracy of ρ˙12 and ρ¨12 are ±0.1 ␮m/sec and ±10−10 m/sec2 , respectively. We hydro log y estimate that the corresponding error for gLOS −8 2 is ∼ 10 m/sec . From sensitivity analyses we can conclude that the accuracies of accelerometer measurements, KBR range rate and range acceleration

N−1 M−1 1 h(θk , λl ) Hpq = Lθ Lλ k=0 l=0 √ 2π p 2πq θk + λl (4a) ×exp − −1 Lθ Lλ N−1 M−1 1 1 h(θk , λl ) = Hpq Lθ Lλ (1 + k f ) p=0 q=0 √ 2π p 2πq × exp −1 θk + λl (4b) Lθ Lλ

where, L θ = N⌬θ and L λ = M⌬λ. Hpq is a 2-D Fourier coefficient at N–S (latitude) frequency,

Fig. 2. GRACE ground tracks in July 2003, the 2◦ × 2◦ grid points, and the boundaries of the Amazon and the Orinoco basins.

Continental Water Storage Changes from GRACE Line-of-Sight Range Acceleration Measurements

65

Fig. 4b. Continental water storage changes (in terms of water thickness change in cm) from the GRACE LOS range acceleration measurements. hydro log y

Fig. 3. The differences of estimated gLOS using two different initial orbits, one is from CSR and the other one is from GFZ. The abscissa is the day number of July 2003. The ordihydro log y , nate is the daily mean difference of the estimated gLOS and its unit is 10−8 m/sec2 .

measurements can satisfy the required accuracy of hydro log y ±10−9 m/sec2 for gLOS , while the a priori relative orbital vectors, r12 and P r12 , could not satisfy the accuracy requirement. The one way to solve this problem is to estimate them simultaneously with hydro log y by exploiting the high precision of KBR gLOS measurements. Figure 3 shows the differences of hydro log y estimated gLOS when using two different precise orbits, one from CSR and the other one from hydro log y are small and ∼ GFZ. The differences in gLOS 10−10 m/sec2 , indicating that the use of either orbit is sufficiently accurate for the computation. The results of the LOS solutions are compared to those from GRACE mean monthly gravity field estimates (L2 products). Figure 4a shows the March and April 2003 monthly water storage changes with respect to the GGM01C field using the Gaussian

Fig. 4a. Continental water storage changes (in terms of water thickness change in cm) from the GRACE monthly mean gravity field estimates (using Gaussian smoothing with radius of 600 km).

smoothing technique with radius 600 km. Figure 4b shows the continental water storage changes with respect to the GGM01C field using the range acceleration method over the same time periods. Both figures show a negative anomaly in the northern area and a positive anomaly in the southern area. However, the LOS solutions (Figure 4b) show more spatial resolutions than the global solution (Figure 4a). In addition, the positive anomaly in Figure 4b clearly lies inside the boundary of the Amazon basin, while the anomaly from the global solution is more dispersive. We further compared the global solutions using the 800 and 600 km radii Gaussian smoothing cases, and the LOS gravity solution over the entire year (January–December 2003) in the spectral domain. We compute the averaged monthly Power Spectral Density (PSD) for the three cases over the study region using one year of GRACE data, and the results are shown in Figure 5. For the regional LOS solution, we used n = 2π R f to convert the planar frequency

Fig. 5. Square root of the PSD of estimated water thickness changes (average of monthly PSD’s over one year).

66

f to the spherical harmonic degree n (Jekeli, 1981). Figure 5 shows that the LOS gravity solution (dashed line with circle) has less power than the 600 km Gaussian smoothing case (solid line with crosses) and the 800 km Gaussian smoothing case (solid line with diamonds) for low degrees (< 13), but with more power for degree 13 or higher.

4 Conclusions A new regional method has been developed to estimate the continental water storage changes by calculating the in-situ GRACE LOS gravity difference using KBR range acceleration, and applied to the Amazon Basin to study continental water storage, January–December 2003. The results show that the LOS method achieved higher spatial resolution than the global solution, over the Amazon study region. This study indicates that the use of LOS observations from GRACE is a viable regional solution technique.

Acknowledgements This research is supported by NASA (NNG04GF01G & NNG04GN19G), and NSF’s CMG Program (EAR0327633). Additional computing resources are provided by the Ohio Supercomputer Center. The NASA/GFZ GRACE data products are provided by JPL PODAAC. We thank S. Bettadpur for providing higher sampled precise GRACE orbits. We thank Shin-chan Han for useful discussions.

References Bettadpur, S. (2004), Level-2 gravity field product user handbook, GRACE 327–734, Center for Space Research, University of Texas at Austin. Han, S., C. Shum, and A. Braun (2005a), High-resolution continental water storage recovery from low-low satellite-tosatellite tracking. J. Geodyn., 39(1), 11–28.

Y. Chen et al. Han, S., C. Shum, C. Jekeli, and D. Alsdorf (2005b), Improved estimation of terrestrial water storage changes from GRACE. Geophys. Res. Lett., 32, L07302, doi:10.1029/2005GL022382. Jekeli, C. (1981), Alternative methods to smooth the earth’s gravity field. Technical report #327, Geodetic Science, Ohio State University, Department of Geodetic Science and Surveying, 1958 Neil Avenue, Columbus, Ohio 43210, USA. Rodell, M., J. Famiglietti (1999), Detectibility of variations in continental water storage from satellite observations of the time variable gravity field. Water Resour. Res., 35(9), 2705–2723. Rodell, M., J. Famiglietti (2001), An analysis of terrestrial water storage variations in Illinois with implications for the Gravity Recovery and Climate Experiment (GRACE). Water Resour. Res., 37(5), 1327–1339. Rowlands, D., S. Luthcke, S. Klosko, F. Lemoine, D. Chinn, J. McCarthy, C. Cox, and O. Anderson (2005), Resolving mass flux at high spatial and temporal resolution using GRACE intersatellite measurements. Geophys. Res. Lett., 32, L04310, doi:10.1029/2004GL021908. Tapley, B., S. Bettadpur, M. Watkins, and C. Reigber (2004a), The gravity recovery and climate experiment: Mission overview and early results. Geophys. Res. Lett., 31, L09607, doi:10.1029/2004GL019920. Tapley, B., S. Bettadpur, J. Ries, P. Thompson, and M. Watkins (2004b), GRACE measurements of mass variability in the earth system. Science, 305, 503–505. Wahr, J., F. 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(B12), 30205–30229. Wahr, J., S. Swenson, V. Zlotnicki, and I. Velicogna (2004), Time variable gravity from GRACE: First results. Geophys. Res. Lett., 31, L11501, doi:10.1029/2004GL019779. Yuan D., and M. Watkins (2006), Recent Mascon solutions from GRACE, Proceedings of Hotine-Marussi Symposium, Wuhan, China.

Atmospheric De-Aliasing Revisited T. Peters Institute of Astronomical and Physical Geodesy, Technische Universität München, Arcisstr. 21, D-80290 München, Germany, Tel.: +49-89-28923193, Fax: +49-89-28923178, e-mail: [email protected]

Abstract. In temporal gravity variations, the variations in the atmospheric mass distribution are one of the most prominent signals next to tides. Since they superimpose the desired signals in the case of the current satellite gravity missions GRACE and GOCE, they are considered noise and removed using models. Therefore, errors in these models directly propagate to the results of the missions and may lead to misinterpretations. Considering this background we revisit the forward modelling of atmospheric mass variations in order to derive an optimal strategy for the de-aliasing for the GOCE mission. Starting from basic principles, the parametrization and especially the radial discretization are investigated using operational data from ECMWF. The impact of model updates is discussed in a case study. Finally, a comparison with data from NCEP is used to assess the uncertainty of the mass variations derived from atmospheric models. Keywords. Atmospheric gravity, forward modelling, GOCE de-aliasing

1 Introduction Modelling the mass of the atmosphere has quite a long tradition in geodesy (see e.g. Ecker and Mittermayer (1969); Moritz (1980)). Since the accuracy of the geodetic measurement techniques was increased over the years these models needed to be improved to meet the demands. This process led from the use of standard static atmospheric models to sophisticated descriptions of the state of the atmosphere computed from the output of global meteorological circulation models like those of NCEP or ECMWF with a high spatial and temporal resolution. This is currently used in the GRACE processing (Flechtner, 2005). The satellite mission GRACE aims at the detection of temporal gravity variations from which one wants to learn about mass transports in the Earth system. A

number of studies (e.g. Tapley et al., 2004; Velicogna and Wahr, 2006) showed results for the hydrology and cryosphere based on GRACE gravity models with a monthly resolution. Most of the atmospherical variations take place on daily to seasonal timescales and are fairly well-known due to the large amount of observed meteorological parameters and sophisticated atmospherical models. Since these variations superimpose the desired signals and are not correctly sampled by the satellites because the high-frequency variations alias into the monthly solutions, it is necessary to correct the measurements for the atmospheric signal. Errors in this so-called “de-aliasing” procedure directly propagate to the resulting gravity models and may lead to geophysical misinterpretation. Similar problems arise in the case of the GOCE mission, which will be not capable of measuring temporal gravity variations. Again, corrections computed from atmospheric models are required. Therefore, a review of the forward modelling of atmospheric gravity variations is carried out in Section 2. Case studies with ECMWF data give an impression of the effect of some modelling versions in Section 3. A special focus is set to operational aspects and the influence on the GOCE mission aiming at the establishment of a suitable de-aliasing procedure. Finally, a comparison between NCEP and ECMWF is used for a rough assessment of the errors of the gravity variations derived from these two meteorological models.

2 Forward Modelling The gravity variations are described in the conventional way in spherical harmonics with timevariable coefficients Cnm , Snm depending on the three-dimensional density distribution ρ over the Earth. As usual, the vertical integral In is written separately and loading effects are included via the load Love numbers kn (e.g. Swenson and Wahr, 2002): 67

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T. Peters

Cnm (t) Snm (t)

=

(1 + kn ) 3 · (2n + 1) 4π Rρ cos mλ In (t)Pnm dσ (1) sin mλ σ

with

+∞ n+2 r ρ(t)dr . In (t) = R

(2)

0

Pnm are the fully normalized Legendre functions, ␴ is usually a sphere with radius R and ρ the mean density of the Earth. Time dependent quantities are generally indicated by t in brackets. There are different methods for the computation of equation (2), cf. Swenson and Wahr (2002); Flechtner (2005) or Boy and Chao (2005). We concentrate only on two of them, namely the simple spherical approximation and the most realistic threedimensional case. Both of them assume hydrostatic equilibrium, which gives the link between pressure p and density: d p(t) = −gρ(t)dr.

(3)

2.1 Thin Layer on a Spherical Earth In a very rough approximation, the atmosphere is considered a thin layer on a spherical Earth. Together with equation (3) the vertical integral in (2) reduces to ps (t) (4) In (t) = g0 with the surface pressure ps and some mean surface gravity g0 for a spherical Earth. 2.2 Vertical Integration on a Realistic Earth Air density and air pressure are connected with each other by the gas equation of state (cf. Gill, 1982) ρ(t) =

p(t) , Rd Tv (t)

(5)

where Rd is the gas constant for dry air and Tv is the virtual temperature, which includes the effects of water vapour, see below equation (8). Inserting this together with hydrostatic equilibrium, equation (2) becomes 0 n+2 r Rd Tv (t) d p(t), In (t) = − R p(t)g

(6)

p= ps

with the integration running now over the pressure.

The meteorological models provide certain discrete data, which can be used in the following way (after ECMWF, 2003). The vertical discretization is realised by the model levels, which follow the topography without intersections. They are defined by coefficients Ak+1/2 , Bk+1/2 at the interfaces between the levels. While temperature T and the specific humidity q are given for each level, pressure and virtual temperature follow from pk+1/2 (t) = Ak+1/2 + Bk+1/2 · ps (t) Tv,k (t) = Tk (t)(1 + 0.6078 · qk (t))

(7) (8)

with k indicating the number of the level. The discrete hydrostatic equation delivers the geopotential ⌽ for each level: max(k)

p j +1/2(t) . p j −1/2(t) j =k+1 (9) The geopotential as the potential energy due to gravity of a mass over the geoid can be converted into the geopotential height by division with g0 and subsequently into a geometric height h, which then also refers to the geoid. Consequently, the radial coordinate r consists of ⌽k+1/2 (t) = ⌽s +

Rd Tv, j (t) ln

r = R + N + h

(10)

with an ellipsoidal geocentric radius R and a geoid height N. In the literature, some differences in the details of this modelling approach can be found. They can mainly be reduced to the following three questions: (1) what is the relationship between the geopotential and the geometric height; (2) which values should be used for the constants g0 , R and is the approximation R = R sufficient; and (3) how should the gravity g in equation (6) be approximated. One could also think of using different topography data sets for the geopotential, but this was not tested for reasons of consistency with the topography inherent in the atmospheric models.

3 Case studies with Operational ECMWF data The numerical tests were carried out on the basis of operational ECMWF data from January to April 2006 on a 1◦ × 1◦ grid, 60 and 91 vertical layers respectively and a temporal resolution of 6 hours. Since we are interested in gravity variations, we have to subtract the mean of every time series. In general,

Atmospheric De-Aliasing Revisited

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this reduces the effects we are interested in. Therefore, the total signal is considered in the case of the modelling aspects, which acts like an upper bound, since it will be reduced when subtracting the corresponding mean. 3.1 Modelling Aspects For the three modelling aspects mentioned in Section 2.2, we will focus on the effect in In , which typically varies in the range of ±30 hPa, with increasing spherical harmonic degree n. Following Gill (1982), there are mainly two different g0 values used for the conversion of the geopotential into geopotential height: 9.80 and 9.80665 m/s2 . This acts as a very small scaling in In in the range of 1–2 Pa. Also the composition of the radial coordinate with an ellipsoidal radius or a spherical one has only a small influence on In of 2–20 Pa. Nevertheless, the geoid height is normally defined above an ellipsoid and not a sphere and the effect is systematic with the latitude. Larger effects can be seen when the geopotential height is converted to a geometric height and a latitude dependent (normal) gravity is introduced for g instead of only a height dependent one. Simple relationships are used by Swenson and Wahr (2002) and Flechtner (2005): R·h ⌽ = g0 R+h

(11)

g = g0

2 R r

(12)

More detailed and more accurate expressions are given in Boy and Chao (2005) and result in differences up to a few hPa for In . Again, the deviations are systematic with latitude and increasing with n. In agreement with Boy and Chao (2005), this difference should not be neglected. 3.2 Practical Issues Concerning GOCE De-Aliasing As discussed by Swenson and Wahr (2002) and Boy and Chao (2005), the difference between vertical integration and simple spherical approximation partly exceeds the accuracy of the GRACE mission. This holds also for the GOCE mission, shown in Figure 1. In terms of mean RMS geoid amplitudes taken from 2 months of data, this difference crosses the expected GOCE sensitivity (Rummel, personal communication, 2006) around degree 6–8. The largest differences arise in areas of high latitude up to 6 mm geoid height (not shown). The near real-time processing of the GOCE data and consequently the use of operational ECMWF data raises the question of the influence of model updates at ECMWF on the atmospheric gravity corrections. As an example, the change-over from 60 to 91 vertical layers on February 1, 2006 is investigated. The difference in the corresponding RMS

Fig. 1. RMS geoid amplitudes in [m] for expected GOCE sensitivity (thick solid line), mean difference between spherical and realistic modelling (line with dots) and mean effect of model update (dotted line).

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T. Peters

Fig. 2. Variation of C00 (solid line) and C10 (dashed line) in terms of geoid height [m]; left axis valid for C00 , right axis for C10 .

geoid amplitudes for January computed with both model versions proves to be not relevant for the GOCE mission (Figure 1). Also the spatial distribution of the geoid differences shows no critical areas. But if one considers for instance a 10 year time series of monthly mean fields and concentrates on the low degree coefficients, there are sometimes steps and jumps, which can be related to model updates of the meteorological services (cf. Figure 2). One has to pay attention to this, at least when the mean fields for the static atmosphere are computed.

4 Comparison Between ECMWF and NCEP The accuracy of the meteorological data can be another limiting factor in the de-aliasing procedure. For a rough estimation of this precision, a comparison between the output of the NCEP and the ECMWF model is carried out. Here, we only use the thin layer approximation on a sphere and a reduced horizontal resolution of 2.5◦ × 2.5◦ grid due to the computational cost. Six hourly data from January to March 2006 are included in this comparison. Figure 3 gives the RMS geoid amplitudes of the difference in the mean fields and a mean of the remaining variations after removal of the mean difference. The mean atmospheric gravity fields of the two models show in the spatial domain large discrepancies. The maximum values of 15 and 0.8 mm respectively are restricted in both cases to regions with poor observations like Antarctica and

the Himalayas (Figure 4). Obviously, there is also an offset in the geoid between the models. The picture for the remaining gravity variations looks similar but with values of around 20 times smaller (not shown). This comparison should only give an idea of the precision of the models. A more detailed study comparing also with ground stations was carried out by Trenberth and Olson (1988) or Velicogna et al. (2001). A real error assessment must include the fact that both models are correlated due to the use of mostly the same observation stations. For a detailed comparison with the satellite missions, the sampling characteristics should also be taken into account.

5 Conclusions and Discussion Modelling the atmospheric gravity variations is one of the critical points in the processing of the new gravity satellite missions. It should be as realistic as possible and avoid larger approximations. The threedimensional approach including a latitude and altitude dependent gravity and a correct conversion from geopotential to geometric height is recommended. It will be used with operational ECMWF data for the de-aliasing of the GOCE mission. Attention must be paid to model updates, especially for the computation of the mean atmospheric gravity. The effect of the atmosphere is partly compensated by the oceanic response due to loading, known as the inverted barometer (IB) behaviour. This reduces the atmospheric signal discussed above to a certain

Atmospheric De-Aliasing Revisited

71

Fig. 3. RMS geoid amplitudes for the typical monthly differences between ECMWF and NCEP (solid line with dots) and the mean variations after removal of the different mean fields (dashed line) and GOCE sensitivity (thick solid line).

Fig. 4. Typical monthly mean difference between ECMWF and NCEP expressed in geoid heights [mm].

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extent. But it is known that there are deviations from IB and the signal over the continents is not affected at all. Since this study concentrates only on the atmosphere, there is no ocean model included in the discussion. Assessing the accuracy of the gravity fields derived from meteorological data is a difficult task. It is not meteorological services’ duty to provide such information. Comparisons with ground stations can only be done locally and are influenced by other phenomena, while comparisons between models from different institutes suffer from correlations, since they use similar observation networks. Ideas for a validation with other sensors are given in Gruber and Peters (2003). Further research in this field seems necessary; the differences between models are large especially in areas with little observations. If the precision of the meteorological data is known, a next step could be the implementation of an error propagation in the gravity field estimation of the satellite missions. This includes the effects of the sampling characteristics and leads to more realistic error estimates of the gravity models derived from the satellite missions.

References Boy, J.-P., and B.F. Chao (2005), Precise evaluation of atmospheric loading effects on Earth’s timevariable gravity field, J. Geophys. Res., 110, B08412, doi:10.1029/2002JB002333. Ecker E., and E. Mittermayer (1969), Gravity corrections for the influence of the atmosphere, Boll di Geofisica teor ed appl, XI(41–42), pp. 70–80.

T. Peters ECMWF Research Department (2003), IFS Documentation – Part III: Dynamics and Numerical Procedures (cy28r1), www.ecmwf.int. Flechtner, F. (2005), AOD1B product description document,in: GRACE Project Documentation, JPL 327–750, rev 2.1, JPL, Pasadena, CA. Gill, A.E. (1982), Atmosphere – Ocean Dynamics, Academic Press, London. Gruber, T., and T. Peters (2003), Time variable gravity field: Using future Earth observation missions for high frequency de-aliasing, in: Proc IERS Workshop on Combination Research and Global Geophysical Fluids. Richter B, Schwegmann W, Dick WR (eds.) IERS Technical Note No. 30, BKG, Frankfurt/Main, pp. 157–160. Moritz, H. (1980), Advanced Physical Geodesy, Wichmann, Karlsruhe, pp. 422–425. Swenson, S., and J. Wahr (2002), Estimated effects of the vertical structure of atmospheric mass on the time-variable geoid, J. Geophys. Res., 107(B9), 2194, doi:10.1029/2000JB000024. Tapley, B.D., S. Bettadpur, J.C. Ries, P.F. Thompson, and M.M. Watkins (2004), GRACE measurements of mass variability in the Earth system, Science, 305, pp. 503–505. Trenberth, K.E., and J.G. Olson (1988), An evaluation and intercomparison of global analyses from the National Meteorological Center and the European Centre for MediumRange Weather Forecasts, Bull. Am. Meteor. Soc., 69, pp. 1047–1057. Velicogna, I., J. Wahr, and H. van den Dool (2001), Can surface pressure be used to remove atmospheric contributions from GRACE data with sufficient accuracy to recover hydrological signals? J. Geophys. Res., 106, pp. 16,415–16,434. Velicogna, I., and J. Wahr (2006), Measurements of timevariable gravity show mass loss in Antarctica, Science, 311(5768), pp. 1754–1756, doi: 10.1126/science. 1123785.

First Results of the 2005 Seismology – Geodesy Monitoring Campaign for Volcanic Crustal Deformation in the Reykjanes Peninsula, Iceland J. Nicolas, S. Durand Laboratoire de Géodésie et Géomatique (L2G), Ecole Supérieure des Géomètres et Topographes (ESGT/CNAM), 1 Boulevard Pythagore, F-72000 Le Mans, France S. Cravoisier, L. Geoffroy Laboratoire de Géodynamique des Rifts et des Marges Passives (LGRMP), Université du Maine, UFR Sciences et Techniques, Bât. Géologie, Avenue O. Messiaen, F-72085 Le Mans Cedex 09, France C. Dorbath Institut Physique du Globe de Strasbourg (IPGS), 5 Rue René Descartes, F-67084 Strasbourg Cedex, France Abstract. The Reykjanes Peninsula, Iceland, presents en-echelon volcano-tectonic systems trending oblique to the normal to the NAM-EUR separation. To better understand the tectonic of the Reykjanes Peninsula’s oblique spreading, a combined geodetic/seismologic campaign was performed from April to August 2005. For the seismologic part, 18 3-component seismometers of short period (1, 2 Hz) were deployed in an area of 30 km E–Wx20 km N–S. To detect small tectonic displacements less than 1 mm across postulated active faults, we used classical topometric measurements instead of GPS measurements, with high precision instrumentation: tacheometer, corner cube retro-reflectors, and centering system with hydraulic locking. The geodetic measurements were performed on 4 local networks in the studied area, on both sides of the fractures with points of about 10–100 m apart. In this contribution, we focus on the geodetic part of the campaign. We first present the geodetic networks and the reference frame setting used. Then, we describe the processing strategy, and show first results of the observed crustal deformations and time series analysis. Finally, a first comparison with the seismologic results is given. Keywords. Crustal deformation, geodesy, topometric measurements, seismological measurements

1 Introduction The objectives of this study are to characterize and to model at different time scales the dynamics of an array of oblique spreading rift segments in the Reykjanes Peninsula, Iceland. We realized a combined geodetic and seismological campaign from April to

August 2005. The aim is to localize the active faults that accommodate the oblique spreading in order to establish the relation between volcanic activity at depth, micro-seismicity, and the crustal deformation of the en-echelon rift segments. Since 2005, the L2G (Laboratoire de Géodésie et Géomatique), the LGRMP (Laboratoire de Géodynamique des Rifts et Marges Passives), and the IPGS (Institut de Physique du Globe de Strasbourg) are involved in this investigation with the help of the IMO (Icelandic Meteorological Office, Iceland) and the University of Iceland. The experiment consisted in monitoring the small coseismic or post seismic displacements over the segments on Reykjanes. A laser tacheometer was used for four small geodetic networks across the faults. At the same time, 18 seismic stations were set up along the rift in addition to the Icelandic seismic permanent network SIL in South Iceland Lowland.

2 The Reykjanes Peninsula The Reykjanes Peninsula (Figure 1) in southwest Iceland is the onland structural continuation of the Mid-Atlantic ridge. It connects to the western volcanic zone and the south Icelandic seismic zone in the east (Einarsson 1991). The relative velocity between these two areas is 2.1 ± 0.4 cm/yr in direction N117 ± 11◦ E (Sigmundsson et al., 1995) at a small angle from the calculated NNR-NUVEL 1A velocity vector (∼ N100◦E). Sturkel and Sigmundsson (1994) followed by Arnadottir et al. (2006) suggest, to explain GPS observations, a model consisting of a mostly left-lateral transform plate boundary, locked down at a certain depth (5–11 km). The upper crust would be elastically deforming over this locking depth whereas the lower 73

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Fig. 1. Map showing the Reykjanes Peninsula localization. Black lines indicate surface faults of Holocene age. The indicated areas correspond to individual fissure swarms with associated central volcanoes (from Arnadottir et al. 2004).

crust (beneath the locked area) would be continuously sheared in direction ∼ N76◦ E. Arnadottir et al. (2006) suggest that a component of opening also exists in the Reykjanes Peninsula. The very important seismicity in the Reykjanes Peninsula suggests that the elastic strain in the upper-crust is continuously released through discrete displacement along faults. Most of these faults have been mapped (Clifton and Kattenhorn, 2006). However, the permanent GPS and seismic stations networks do not allow showing precisely which faults are currently active either seismically or aseismically in the upper crust. This motivated our high precision geodetic and seismic survey in the area. We settled the geodetic and seismological networks above the Krisuvik and Grindavik magmatic dormant centers. We studied specifically the displacements across well-located normal faults trending ∼ N040 and displaying significant displacements of the most recent historical lava flows.

3 The Geodetic Campaign As our objective is to detect small crustal displacements (< 1 mm) in a small area, we used classical topometric measurements with high precision instrumentation instead of GPS measurements for this campaign. Indeed, accurate GPS positioning does not allow reaching such a level of displacement measurement. Centering systems with hydraulic locking and attachment bolds were used to ensure a centering precision of about 0.1 mm both for the tacheometer (Leica TCA1800) and for the corner cube centering spheres (retro-reflectors), as shown in Figure 2.

Four local networks were set up in the studied area as indicated in Figure 3 (squares). These networks are located on visible active faults, showing large visible displacements (about 2 m) on recent lava flows less than 1100 years old. The local network sizes are about 70 × 120 m2. Note that networks A and D are on the same fault, which presents a direction change. The geodetic measurements were performed on both sides of the faults with points of 10–100 m apart. These measurements are the slope distance, the horizontal direction (azimut), and the zenithal direction (angle relative to the vertical) between the tacheometer and the target. To optimize the geometrical configuration of each network and the number of stations (maximizing the precision with a minimum number of stations), simulations were performed before the field experiment. These simulations were performed using the Geolab adjustment software (Steeves, http://www.msearchcorp.com) computing the reachable accuracy for various geometry configurations, various distances between the points, and various numbers of stations and targets. The optimum configuration found with a reasonable number of stations and targets compatible with the amplitude of crustal displacement expected and field experiment not too expensive was the one drawn in Figure 4. The results indicated the possible detection of very small displacements (< 1 mm) as required for this study. This configuration was tested with the instrument and the targets before the network installation in Iceland. Each network consists of 6 reference points, located on both sides of the fault (Figure 4): 3 attachment bolds where only a corner cube retro-reflector can be installed in (triangles) and 3 reference cylinders where either the tacheometer or a retro reflector can be installed in (circles). The local tie was performed using differential GPS relatively to 5 reference points of the Icelandic

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Fig. 3. (a) Map showing the study area, the location of the 4 geodetic networks (squares), and the 2 GPS permanent stations REYK and VOGS (stars). (b) Detailed map showing the topography, the Kleifarvatn lake in the middle, the 4 geodetic networks (squares labeled A–D), the permanent seismometers from SIL (circles), and the temporary seismometers (triangles).

network and few permanent GPS stations present not too far from the studied area (REYK and VOGS).

4 The Seismological Campaign For the seismological part of the campaign, in addition to the permanent seismic Icelandic network SIL (circles on Figure 3), 18 short seismic stations (1, 2 Hz) were deployed in a dense network (30 × 50 km2 ) covering the whole studied area (triangles on Figure 3). These stations recorded continuously on the 3 components.

5 Preliminary Results 5.1 Geodesy Each network was observed at least once per week from April to August 2005. The observations (slope distances, horizontal directions, and zenithal angles) were processed using the least squares method (GeoLab software) in order to compute at each observation epoch the local coordinates of each reference point of each network. We only present in this contribution preliminary results for the C network. Figures 5 and 6 show for the East, North, and Up components the times series

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for the displacements of 2 points of the network (C4 and C6) located at the same side of the fault. The peak-to-peak amplitude of the signal is about 2 mm for the horizontal and for the vertical component. The first thing we can notice is the similar behavior of both points. Computing a linear trend for each component for each point for the whole campaign gives the following mean velocities: 0.6 ± 1, 0.2 ± 0.8, and 0.8 ± 1 mm/yr for the North, East, and vertical components, respectively. Computing different values for

different periods as before and after the main seismic events that occurred during the campaign (days of year 2005 numbers 173 and 193, see part 5.2) gives almost the same results for the North component. Concerning the East component, the trend value changes from 1.2 ± 2.6 to −1.4 ± 2.8 mm/yr. The most interesting velocity variation concerns the vertical component with a velocity changing from −4.14±2.8 to +7.1±7.1 mm/yr. These results could indicate mainly a vertical displacement linked to the seismic events. This is not surprising considering that the monitored fault has a dip-slip throw of about 10 m for less than ∼ 1000 yrs. Nevertheless, further investigations are required to confirm these results. For this we will have to analyze carefully longer time series over few years at the same place in order to be able to distinguish the true crustal displacements caused by seismic or a-seismic phenomena from the inherent noise.

5.2 Location of the Seismicity A great number of seismic events occurred in the studied area during the campaign. We picked, by hand, about 900 events with magnitude between 1 and 3.5. After processing, we kept 724 well-located events. The localization of the different events is indicated in Figure 7. Two major swarms occurred in the circled area indicated on Figure 7:

Fig. 5. North, East, and Up time series for point C4 of network C, in millimeters.

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Fig. 6. North, East, and Up time series for point C6 of network C, in millimeters.

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• Up to 130 events on June 22th (day of year 2005 number 173) under the Kleifarvatn lake, • Up to 70 events on July 12th (day of year 2005 number 193) at the southwest of the study area.

is supposed to be larger, just near the Kleifarvatn Lake.

Three additional little swarms occurred in April. The majority of the events are around magnitude Ml = 1 and are located between 3 and 6 km in depth. About 40 events are deeper than 8 km. Although data are currently processed, most of the events seem to be aligned with the direction of the volcano-tectonic segments (ie ∼ N040) and being related to normal and strike-slip faulting mechanisms (Cravoisier et al. 2006). The preliminary inversion of the data suggests that the minimum stress acting across the Reykjanes peninsula is nearly perpendicular to the volcano-tectonic segments as formally proposed in earlier studies (Einarsson 1991) and close to the trend of NOAM and EURA plates’ relative displacement. Such stress system is thought to promote in the uppermost crust horizontal dilatation and vertical displacement across the N040 fault system that we monitored.

The authors wish to acknowledge Samuel Pitiot, Pierre Dejonghe, for their contribution to the geodetic measurements, Michel Frogneux, Henri Haessler, for their help during the installation of the seismometers, the IMO team, and the University of Iceland for the local facilities.

6 Conclusion and Prospect The preliminary results of this combined campaign are promising. A new geodetic campaign has been performed in summer 2006 to revisit the same networks and to confirm the results obtained from the first campaign. The data processing is currently under investigation. Indeed, after 1 year we should be able to distinguish between the crustal deformation signal and noise. A new local network has been installed on the most active fault indicated by the seismological data processing where the local surface displacement

Acknowledgements

References Arnadottir T., Geirsson H., and Einarsson P., Coseismic stress changes and crustal deformation on the Reykjanes Peninsula due to triggered earthquakes on 17 June 2000, J. Geophys. Res., Vol. 109, B09307, doi:10.1029/2004JB003130, 2004. Arnadottir T., Jiang W., Feigl K., Geirsson H., and Sturkell E., Kinematic models of plate boundary deformation in southwest Iceland derived from GPS observations, J. Geophys. Res., Vol. 111, B07402, doi:10.1029/2005JB003907, 2006. Clifton A. and Kattenhorn S., Structural architecture of a highly oblique divergent plate boundary segment, Tectonophysics, Vol. 49, 27–40, 2006. Cravoisier S., Dorbath C., Geoffroy L., Durand S., and Nicolas J., Crustal deformation monitoring of active volcanic rift segments in the Reykjanes Peninsula, Iceland, EGU2006, Vienna, poster EGU06-A-01905. Einarsson P., Earthquakes and present day tectonism in Iceland, Tectonophysics, Vol. 189, 261–279, 1991. Sigmundsson F., Einarsson P., Bilham R., and Sturkell E., Rifttransform kinematics in south Iceland : Deformation from Global Positioning System measurements, 1986 to 1992, J. Geophys. Res., Vol. 100, No. B4, 6235–6248, 1995. Steeves Robin R., The Geolab Adjustment Software, Microsearch Corp, http://www.msearchcorp.com. Sturkell E., and Sigmundsson F., Strain accumulation 1986–1992 across the Reykjanes Peninsula plate boundary, Iceland, determined from GPS measurements, Geophys. Res. Lett., Vol. 21, No. 2, 125–128, 1994.

The Statistical Analysis of the Eigenspace Components of the Strain Rate Tensor Derived from FinnRef GPS Measurements (1997–2004) in Fennoscandia J. Cai, E.W. Grafarend Institute of Geodesy, Universität Stuttgart, Geschwister-Scholl-Str. 24, D-70174 Stuttgart, Germany, e-mail: [email protected] H. Koivula, M. Poutanen Department of Geodesy and Geodynamics, Finnish Geodetic Institute, Geodeetinrinne 2, FI-02430 Masala, Finland Abstract. In the deformation analysis in geosciences (geodesy, geophysics and geology), we are often confronted with the problem of a twodimensional (or planar and horizontal), symmetric rank-two deformation tensor. Its eigenspace components (principal components, principal direction) play an important role in interpreting the phenomena like earthquakes (seismic deformations), plate motions and plate deformations among others. With the new space geodetic techniques, such as VLBI, SLR, DORIS and especially GPS, positions and change rates of network stations can be accurately determined from the regular measurement campaign, which is acknowledged as an accurate and reliable source of information in Earth deformation studies. This fact suggests that the components of deformation measures (such as the stress or strain tensor, etc.) can be estimated from the highly accurate geodetic observations and analyzed by means of the proper statistical testing procedures. We begin with discussion of the geodynamic setting of the selected investigated regions: Fennoscandia. Then the regular GPS observations in Finnish permanent GPS network (FinnRef) and the related data preparation are introduced. Thirdly the methods of derivation the two-dimensional geodetic strain rates tensor from the surface residual velocities and the newly developed estimates BLUUE of the eigenspace elements and BIQUUE of its variancecovariance matrix are reviewed. In the case study both BLUUE and BIQUUE models are applied to the eigenspace components of two-dimensional strain rate tensor observations in Fennoscandia, which are derived from 1997 to 2004 annual station positions and velocities of FinnRef. Further detailed analysis of the results is also performed with respect to geodynamical and statistical aspects.

Keywords. Strain rate tensor, eigenspace components, crustal deformation analysis, postglacial isostatic rebound, FinnRef

1 Introduction With the new space geodetic techniques, such as VLBI (Very Long Baseline Interferometry), SLR (Satellite Laser Ranging), DORIS (Doppler Orbitography and Radiolocation Integrated. by Satellite and especially GPS (Global Positioning System), positions and change rates of network stations can be accurately determined from the regular measurement campaign, which is acknowledged as an accurate and reliable source of information in Earth deformation studies. Therefore the components of deformation measures (such as the stress or strain tensor, etc.) can be estimated from the highly accurate geodetic observations and analyzed by means of the proper statistical testing procedures. The distribution of the eigenspace components of the ranktwo random tensor has been investigated by Xu and Grafarend (1996a,b), which is significantly different from the commonly used Gauss-Laplace normal distribution. Xu (1999) developed also the general distribution of the eigenspace components of the symmetric, rank-two random tensor. In our recent papers about statistical analysis of random deformation tensor (Cai et al. 2005, Cai and Grafarend 2007a) we have achieved the complete solution to the statistical inference of eigenspace components of a 2-D random tensor, which includes the best linear uniformly unbiased estimation (BLUUE) of the eigenspace elements and the best invariant quadratic uniformly unbiased estimate (BIQUUE) of its variance covariance matrix together with the design of a linear hypothesis test. This solution proposes a multivariate model to statistically process random tensors in the eigenspace and, in particular, apply the 79

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technique to analyse the repeated geodetic measurements. Unlike most of geophysical applications of the same kind, this solution focuses on the statistical inference of significance of the computed random tensors to support their interpretations. With these models we have successfully performed the statistical inference of the eigenspace components vector and the variance-covariance matrix of the GaussLaplace normally distributed observations of a random deformation tensor derived from the ITRF data sets in central Mediterranean, Western Europe and Fennoscandia (Cai and Grafarend 2007a,b). With the benefit of the continuous observations of the permanent networks, such as the regional permanent GPS network FinnRef – the permanent GPS network in Finland with 13 permanent GPS stations, we can now derive the strain rate tensor observations and estimate the eigenspace component parameters of these random tensor samples from the more dense time series observations with our newly developed solution. Although there are more and more applications of the strain and strain rate tensor in geodesy and geodynamics, the statistical analyses of them have not yet been paid enough attention, especially in the regional Fennoscandia, for example the newly published paper by Haas et al. (2001) and Scherneck et al. (2002b). Cai and Grafarend (2007b) preformed a statistical analyse of geodetic deformation (strain rate) derived from the space geodetic measurements of BIFROST (Baseline Inferences for Fennoscandian Rebound Observations Sea Level and Tectonics) Project in Fennoscandia only with the sparse ITRF data sets and stations in this region. In this paper we begin with the geodynamic setting of the Earth and especially the selected investigated region – Fennoscandia will be discussed. Then the regular GPS observations in Finnish permanent GPS network (FinnRef) in the frame of the BIFROST Project and data preparation are introduced. Thirdly the main results of statistical inference of eigenspace components of the twodimensional symmetric, rank-two random tensor (“random matrix”), i.e., BLUUE of the eigenspace elements and BIQUUE of its variance–covariance matrix are reviewed with one scheme. Furthermore the methods of derivation for the two-dimensional geodetic stain rates are introduced and applied to derive these strain rates from the residual velocities. In the case study the estimates are applied to the eigenspace components of the two-dimensional strain rate tensor observations in the area of Finland, which are derived from 1997 to 2004 annual

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station positions and velocities of FinnRef. Further detailed analysis of the results is also performed with respect to geodynamical and statistical aspects.

2 Geodynamic Setting and the Finnish Permanent GPS Network (FinnRef) 2.1 Geodynamic Setting As pointed out by Grafarend and Voosoghi (2003), the European and Mediterranean area is known as an extraordinary natural laboratory for the study of seismotectonic processes. This area is geologically, geophysically, and geodetically one of the best-studied regions on the Earth’s surface and can be divided into three main subregions with distinct geodynamic features, namely Western Europe, Northern Europe and the Alpine-Mediterranean sub-regions. The Northern Europe subregion, i.e. Fennoscandia, covers the national territories of Finland, Sweden and Norway, as well as part of the Kola Peninsula and Russian Karelia. In this region the crust is rising continuously since the deloading of the ice sheets at the end of the Ice Age; a small contribution also comes from the deloading of sea water due to the crustal uplift itself. This phenomenon is commonly accepted as an isostatic readjustment of the crust and the upper mantle towards equilibrium, i.e. the socalled glacial isostatic adjustment (GIA). It is mainly an isostatic rebound governed by the viscous properties of the mantle, to some extent modified by the presence of the elastic crust. Only the Fennoscandian uplift is fairly well determined by geodetic observations since 1858. Gregersen (1992) pointed out that in all Fennoscandia the orientations of maximum horizontal stress are internally consistent, and a dominating stress field orientation very close to NWSE compression has been found. These data together with stress measurements made through other methods demonstrate that the whole region is dominated by a uniform compressive stress field controlled by forces related to Mid-Atlantic plate spreading and European-African collision. The geodetic two and three-dimensional crustal motion results are analyzed and compared with the GIA model predictions in Fennoscandia, for more detail see Milne et al. (2001, 2004), Scherneck et al. (2001, 2002a), Johansson et al. (2002), Wahr and Davis (2002) and Marotta et al. (2004). Until now most publications are focused on kinematical analysis with the station velocity results in this area. The dynamical analysis with strain or strain rate results is just studied recently by Scherneck

Statistical Analysis of the Eigenspace Components in Fennoscandia

et al. (2002b), in which the statistical analysis of the strain rate has not been carried out. Cai and Grafarend (2006b) preformed a statistical analyse of geodetic deformation (strain rate) derived from the space geodetic measurements of BIFROST Project in Fennoscandia only with the sparse ITRF data sets and stations in this region. Since there is only one published crustal motions result – BIFROST Solution 2001 (Johansson et al. 2002) available until now in this region, we have to choose the other sourse as our geodetic observations. As a part of the Fennoscandian Regional Permanent GPS Network the Finnish permanent GPS network (FinnRef) was established by the Nordic Geodetic Commission in response to the initiative of the directors of the Nordic Mapping agencies. FinnRef is maintained by the Finnish Geodetic Institute and creates a backbone for the EUREF-FIN coordinate system and provides a connection to international and national coordinate systems, which are fortunately dense and accurate with continuous and stable observations in Fennoscandia. We will review FinnRef and choose the appropriate sites to perform the statistical analysis of the eigenspace components of strain rate tensor observations. 2.2 The Finnish Permanent GPS Network (FinnRef) and Data Preparation FinnRef-the permanent GPS network in Finland consists of 13 permanent GPS stations. Metsähovi is the oldest one in early 1990a. FinnRef data are used both in geodynamical studies and in maintaining reference frames. The stations are equipped with geodetic dual frequency receivers and choke ring antennas that are mounted on concrete pillars or steel grid masts. Most of the installations are made on the solid bedrock, see Figure 1. The network is the backbone of Finnish realization of the EUREF frame, EUREFFIN. Four stations in the FinnRef network are part of the EUREF permanent GPS-network (EPN), and one station belongs to the network of the International GPS Service (IGS). FinnRef creates a seamless connection with the global reference frames through these stations. Since 1997 these 13 GPS stations are fully operated. FinnRef data has been processed in the FGI with Bernese 4.2 Software (Hugentobler et al. 2001) based on data differencing. As input the IGS precise orbit products, FinnRef GPS data in 24-hour sessions and the IRTF coordinates of Metsähovi at the epoch of the observation are used. Processing is done on a daily basis and the normal equations are saved. Later they are combined into the weekly solutions, which

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Fig. 1. Finnish permanent GPS network – FinnRef.

are our primary result. The parameter estimation process by means of least squares adjustment is done in three steps. First the quality of the data is checked with an ionosphere free L3 solution. In this solution, the ambiguities are not solved for and coordinates are not of our major interest. Only the a posteriori rms error of the solution is checked. Typically, this is of the order of 1–2 mm. If the first L3 solution was good the ambiguities are resolved. This is done baselineby-baseline using the QIF algorithm in least squares adjustment. With QIF algorithm the L1 and L2 ambiguities are directly resolved. When the ambiguities of all baselines are resolved, the final least squares adjustment is made. This time all the baselines are processed in the same run and the normal equations of the solution are saved. In this run an ionosphere free linear combination is used and none of the stations are fixed. In spite of fixing the coordinates the Metsähovi coordinates are constrained by giving them sigmas of 0.0001 m for all three components. The previously resolved ambiguities are introduced as known values. Stacking the normal equations then creates weekly solutions. The final product is a set of weekly coordinates. In this study the annual velocities were obtained by solving the velocity trends for weekly GPS time series

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on annual basis. These velocities are with respect to Metsähovi station. In order to get the residual velocities we have calculated them in the following: Vres (J J ) = Vres (J J ) + VMet−absoulte(J J ) − Vplate (1997). 2.3 Derivation of the 2-D Geodetic Strain Rate Tensor The 2-D strain rate tensor can be derived from the horizontal residual velocity data sets with the methods introduced in Cai and Grafarend (2007a, b). Until 2004 there are eight annual velocity data sets, which will be used in deriving the annual strain rate tensor in the region FinnRef. Here we will apply the Finite Element Method to do the linear interpolation within each triangle, which is optimally generated by the Delaunay-triangulation method among the 13

stations. Since the bad geometric condition of these triangles in the north part of FinnRef we choose only the seven stations Oulu, Tuorla, Vaasa, Virolahti, Joensuu, Romuvaara and Kivetty to constructed these Delaunay-triangles, see Figure 4. For every triangle we select the centroid as the reference point, from which it is very easy to derive the velocity gradient tensor. With these velocity gradients the geodetic strain rates of every Delaunaytriangle for eight FinnRef annual realizations can be computed, and their eigenspace components (eigenvalues and eigendirection) together with the maximum shear strain rate and the second strain rate invariant can be successively calculated. 2.4 Representation of 2-D Strain Rate Now we can present the horizontal residual velocities and the principal strain rates of every triangle for


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Fig. 3. Time series of Joensuu station with respect to Metsähovi station. In the right column are the series of baseline length, height difference, East and North components. In the left column are the periodograms of the time series.

eight FinnRef annual realizations derived above. The pattern of the principal strain rates of 6 Delaunaytriangles and the associated residual velocities of the eight annual solutions in the FinnRef are illustrated in Figure 5. Extension is represented by symmetric arrows pointing outwards and contraction is represented by symmetric arrows pointing inwards. The residual velocities are represented by single arrows.

3 Statistical Inference of the Estimates of Eigenspace Component Parameters 3.1 Review of Statistical Inference of the Estimates of Eigenspace Component Parameters With these estimates of the eigenspace components of the random strain rate tensor and their dispersion matrix and under the assumption that the observations of a symmetric rank-two random strain rate

tensor are Gauss-Laplace normally distributed, the following univariate and multivariate hypothesis tests can be performed. We review here the statistical inference and analysis of 2-D, symmetric rank two deformation tensors in the following scheme, more detail can be referred to as developed by Cai (2004) and Cai et al. (2005). 3.2 The Estimates of the Eigenspace Components from the Strain Rates Observations of Six Epochs In this section, as a case study, both model and hypothesis tests will be applied to the observations of random strain rate tensors, derived above for every Delaunay-triangle of the selected FinnRef sites at eight epochs. With the two-dimensional strain rate tensor observations, calculated by the eight epoch FinnRef

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residual velocities, we can now estimate the eigenspace components (eigenvalues and eigendirections) of the two-dimensional strain rate tensors, variance-covariance component matrix of type BIQUUE, and their estimated dispersion matrix, and successively make hypothesis tests. The detailed

results of all 6 Delaunay-triangles in the study region are illustrated in Figure 7 together with their 95% confidence intervals, which provide us with a visual presentation of the possible magnitude and the directions of the extension and contraction of the strain rate. This is important for the prediction of the tectonic activity, including the possible deformation trend and its directions. Furthermore we can benefit from the statistical information from the estimates of the eigenspace component parameters and their related dispersion matrix from the strain rate observations of six epochs reflect the statistical average information of the random strain rate tensor and utilize the advantage of the longer time span. With them we can successively perform the statistical inference, since the limitation of the content, it is omitted here. The above analysis shows that the estimation theory about the two-dimensional, symmetric rank-two random tensor is practical to be applied and produces not only estimates of eigenspace components but also their successive hypothesis tests, which complete the statistical inference of eigenspace component parameters of a two-dimensional, symmetric rank-two strain rate tensor. From Figure 7 we can see that the strain rate field in the central area of Fennoscandia is dominated by the extension in the order of 5 nanostrain/yr, where the general characterstic of the pattern of the principal strain rates is bi-axial extension with a main extensional direction of NNE. In the surounding area,

Residual velocity and Strain rate of FinnRef97

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Fig. 5. Pattern of the principal strain rates of 6 triangles and the associated residual velocities of the selected FinnRef 97–2004 sites in the studying region. Extension is represented by symmetric arrows pointing out and contraction is represented by symmetric arrows pointing in. The residual velocities are represented by single arrows.


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Fig. 5. (continued)

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4 Conclusion and Further Studies

Linearized with Taylor expansion by Jacobi matrix

Nonlinear G-M model

Linearized G-M model

Theorems BLUUE of eigenspace components BIQUUE

Hypothesis tests of the estimates of eigenspace components

Inference and analysis Case study

Fig. 6. The scheme of inference and analysis of the eigenspace components of a two-dimensional random tensor.

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Fig. 7. Eigenspace components (eigenvalues and eigendirections) of the two-dimensional strain rate tensors and their 95% confidence intervals, estimated from the strain rate observations of FinnRef97 to FinnRef2004 series in the seven triangle sites in the study region. Extension is represented by symmetric arrows pointing outward and contraction is represented by symmetric arrows pointing inward.

especially the south-east part in Fennoscandia, the main feature is compression in the order of −6 to −1 nanostrain/yr. These different features of strain rate field reflect the fact of higher uplift rate of postglacial rebound in central Fennoscandia and smaller uplift rate in south and east surrounding area.

With the newly developed models we have successfully performed the statistical inference of the eigenspace components and the variance-covariance matrix of the Gauss-Laplace normally distributed observations of a random deformation tensor (case study: 2-D, symmetric rank two strain rate tensor). In the case study both estimates BLUUE and BIQUUE have been applied to the eigenspace components of 2-D strain rate tensor observations in the area of Fennoscandia, which are derived from FinnRef 97-2004 series station positions and velocities. We could benefit from the statistical information derived from the estimation procedure. For example the 95% confidence intervals for the estimates of eigenvalues and eigendirection, illustrated in Figure 7, provides us with a visual presentation of the possible magnitude and the directions of the extension and contraction of the strain rate, which is important for the prediction of the tectonic activity including the possible deformation trend and directions. They lead to the statement to be cautious with data of type extension and contraction as well as with the orientation of principal stretches. The onging crustal deformation due to postglacial isostatic rebound in Fennoscandia is confirmed and analysed in statistical sense. The general characterstic of the pattern of the principal strain rates is bi-axial extension in the order of 5 nanostrain/yr in central Fennoscandia and compression in the south–east surrounding area. The uplift rate in Fennoscandia is also accompanied by lateral strain in the order of 1 ∼ 2 nanostrain/yr because of the curvature of the Earth’s surface. For further studies we should (1) use more continuous GPS observation stations with better distribution in Fennoscandia to perform strain-rate tensor analysis; (2) compare the geodetic strain-rate field with the geodynamic features in more detail in Fennoscandia;

Acknowledgements This work is the result of the Project Based Personal Exchange Program with Finland DAADD/05/26060 of German Academic Exchange Service.

References Cai J. (2004): Statistical inference of the eigenspace components of a symmetric random deformation tensor, Dissertation, German Geodetic Commission (DGK) Series C, No. 577, 138 pages, München, 2004. Cai J., E. Grafarend and B. Schaffrin (2005): Statistical inference of the eigenspace components of a two-dimensional,

Statistical Analysis of the Eigenspace Components in Fennoscandia symmetric rank-two random tensor, Journal of Geodesy, Vol. 78 (7/8), 425–436. Cai J. and E. Grafarend (2007a): Statistical analysis of the eigenspace components of the two-dimensional, symmetric rank-two strain rate tensor derived from the space geodetic measurements (ITRF92-ITRF2000 data sets) in central Mediterranean and Western Europe, Geophysical. Journal International, Vol. 168, 449–472. Cai J. and E. Grafarend (2007b): Statistical analysis of geodetic deformation (strain rate) derived from the space geodetic measurements of BIFROST Project in Fennoscandia, Journal of Geodynamics, Vol. 43(2), 214–238. Grafarend E. and B. Voosoghi (2003): Intrinsic deformation analysis of the Earth’s surface based on displacement fields derived from space geodetic measurements, case studies: Present-day deformation patterns of Europe and of the Meditterranean area (ITRF data sets), Journal of Geodesy, Vol. 77, 303–326. Gregersen, S. (1992): Crustal stress regime in Fennoscandia from focal mechanisms, Journal of Geophysical Research, Vol. 97, 11,821–11,827. Haas, R., H.-G. Scherneck, J.M. Johansson, S. Bergstrand, G.A. Milne, J.X. Mitrovica and R. Arvidsson (2001): BIFROST Project: The Strain-Rate Field In Fennoscandia, abstract in EGS XXVI General Assembly, Nice, France. Hugentobler U., S. Schaer and P. Fridez (eds.) (2001): Bernese GPS Software Version 4.2. Astronomical Institute, University of Berne. 515 pages. Johansson, J.M., J.L. Davis, H.-G. Scherneck, G.A. Milne, M. Vermeer, J.X. Mitrovica, R.A. Bennett, G. Elgered, P. Elósegui, H. Koivula, M. Poutanen, B.O. Rönnäng, and I.I. Shapiro (2002): Continuous GPS measurements of postglacial adjustment in Fennoscandia, 1. Geodetic results, Journal of Geophysical Research, Vol. 107 (B8), 2002, DOI 10.1029/2001JB000400. Marotta, A.M, J.X. Mitrovica, R. Sabadini and G.A. Milne (2004): Combined effects of tectonics and glacial isotatic adjustment on intraplate deformation in central and northern Europe: Applications to Geodetic Baseline Analyses. Journal of Geophysical Research, Vol. 109, B01413, 22 pages: doi:10.1029/2002JB002337. Milne, G.M., J. L. Davis, Jerry X. Mitrovica, H.-G. Scherneck, J. M. Johansson, M. Vermeer and H. Koivula (2001):

87 Space-geodetic constraints on glacial isostatic adjustment in Fennoscandia, Science, Vol. 291, 2381–2384. Milne, G.M., Jerry X. Mitrovica, H.-G. Scherneck, J. L. Davis, J. M. Johansson, H. Koivula and M. Vermeer (2004): Continuous GPS measurements of postglacial adjustment in Fennoscandia, 2. Modeling results, Journal of Geophysical Research, Vol. 109, B02412, 18 pages, DOI 10.1029/2003JB002619. Scherneck, H.-G., J.M. Johansson, M. Vermeer, J.L. Davis, G.A. Milne, J.X. Mitrovica (2001): BIFROST Project: 3-D crustal deformation rates derived from GPS confirm postglacial rebound in Fennoscandia. Earth Planets Space, Vol. 53, 703–708. Scherneck, 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 (2002a): BIFROST: Observing the three-dimensional deformation of Fennoscandia, in: Ice sheets, sea level and the dynamic earth, Mitrovica, J.X. and B.L.A. Vermeersen (eds), pp. 69–93 AGU Geodynamics Series, Vol. 29, American Geophysical Union, Washington, D.C. Scherneck, H.-G., J.M. Johansson, R. Haas, S. Bergstrand, M. Lidberg and H. Koivula (2002b): BIFROST: Project: From geodetic positions to strain rates, in: Proceedings of the 14th Genreal Meeting of the Nordic Geodetic Commission, M. Poutanen and H. Suurmäki (eds), pp. 62–65, Nordic Geodetic Commission, Kirkkonummi. Wahr, J. M. and J. L. Davis (2002): Geodetic constraints on glocial isostatic adjustment, in: Ice Sheets, Sea Level and the Dynamic Earth, Mitrovica, J.X. and B.L.A. Vermeersen (eds), pp. 3–32, AGU Geodynamics Series, Vol. 29, American Geophysical Union, Washington, D.C. Xu P. (1999): Spectral theory of constrained second-rank symmetric random tensors, Geophysical. Journal International, 138: 1–24. Xu P. and Grafarend E. (1996a): Probability distribution of eigenspectra and eigendirections of a two-dimensional, symmetric rank-two random tensor, Journal of Geodesy 70: 419–430. Xu P. and Grafarend E. (1996b): Statistics and geometry of the eigenspectra of three-Dimensional second-rank symmetric random tensors, Geophysical. Journal International. 127: 744–756.

GPS Research for Earthquake Studies in India M. N. Kulkarni Department of Civil Engineering, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India, e-mail: [email protected]

Abstract. With the recent major earthquakes in the South East Asian region, and the devastation caused by the tsunami effect, the importance of earthquake prediction has been re-emphasized. Monitoring the Geodynamics, including the crustal deformations, using advanced space geodetic techniques like the Global Positioning System (GPS) with the ultimate aim of predicting the natural disasters such as earthquakes, volcanic eruptions, landslides, avalanches, is one of the greatest challenges faced by the scientists in this century. In India, an extensive high precision Geodetic and Geophysical control network has been established by Survey of India (SOI), the national mapping agency of Govt. of India, for the primary purpose of national mapping, through dedicated efforts of over two centuries. More recently, various national organizations and institutions have taken up geodetic, geophysical and geological surveys for a variety of applications. The extensive horizontal and vertical geodetic and geophysical control network established through these collaborative efforts, and the huge amount of valuable data thus generated, has significant contribution towards monitoring the crustal dynamics of the Indian sub-continent. These efforts have now been augmented through an extensive National Programme on GPS and Geodetic Studies for Earthquake Hazard Estimation launched by the Department of Science & Technology (DST), Government of India, since 1998. The programme, GPS network consisting of about 50 permanent, 700 semi-permanent and several hundreds of campaign mode field GPS stations, being implemented in a phased-manner by DST has now reached a significant stage. Several GPS research groups are engaged in the study of specific regions under this national programme. An overview of the recent developments in this programme, along with the present status and future plans have been presented here. Two case studies, being carried out by the IIT Bombay GPS research group, under this National Programme: one in the Koyna region of

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central India, and the other in the Bhuj region of Gujarat, are also briefly described. Keywords. GPS, national programme, earthquakes, research permanent stations

1 Introduction In India, an extensive GPS-based national programme for earthquake studies has been launched recently. The Himalayan region is known to be seismically active, and needs to be monitored for earthquake hazard. The Peninsular Shield of India can also no longer be considered to be seismically inactive, as shown by the Koyna, Latur and Jabalpur earthquakes, hence it is necessary to monitor the crustal movements in the entire country on a regular basis. This was also emphasized by the UN Experts Committee after the 1993 Latur earthquake. This committee recommended establishment of a dense GPS and precise vertical control for this region monitoring the seismotectonic activities. Based on this original intension and the proposal to World Bank by the Department of Science & Technology (DST), a GPS programme was formulated for the peninsular shield which is being implemented through many participating institutions, and major augmentation of resources, including instrumentation, manpower, training, budgetary allocation, etc. is being done. GPS surveys for Latur area and Koyna area have been taken up since 1995. However, in order to evolve a ‘National Programme on GPS for Geodynamic Studies in India’, by integrating the GPS control network for Peninsular Shield and other existing GPS stations to cover the entire country, DST set up a GPS Expert Group, comprising 14 scientists/ experts from 10 Institutions, in 1997. The Expert Group has evolved an extensive ‘National GPS for Geodynamics Plan’, as described in the Report of the Group, compiled by the author, and submitted to

GPS Research for Earthquake Studies in India

DST in February 1998 (DST, 1998). This National GPS Network for Geodynamics, recommended by the Expert Group, is now being implemented in a phased manner under a National Project by DST. The entire network consists of about 50 permanent GPS stations established for this purpose, a total of about 600 semi-permanent (campaign-mode) GPS stations in the Himalayan regions and the peninsular shield of India, and many more field GPS stations, being established and monitored for local campaigns. A National GPS Data Centre has also been established at Dehra Dun, under this project. The extensive geodetic data being generated through these networks will contribute significantly in understanding the complex earthquake phenomenon, and for monitoring the earthquake hazard in the country.

2 Achievements As highlighted above, since 1998, DST has taken up the implementation of the National GPS Programme for Earthquake Hazard Assessment recommended by the GPS Expert Group. An extensive network of permanent, semi-permanent, and field GPS stations is being established by the participating organizations, and a National GPS Data Centre has been established. Specifications of the GPS instrumentation for

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this purpose have also been formulated. A Committee has been constituted for overseeing the programme. A brief report on these activities with some of the significant results, present status, and future plans, is given in a brochure published in February, 2005 (DST, 2005). 2.1 Establishment of Permanent GPS Stations Permanent GPS Stations at about 50 locations all over the country has been set up under the programme (Figure 1). The detail list of the permanent stations and the Institutions responsible for maintaining these stations is given in DST (2005). 2.2 Campaign-Mode Studies In order to study the local crustal deformation process in seismically active areas, several campaign mode studies have been taken up by various research groups under this programme. A list of different GPS Campaigns and the responsible institutions is given in DST (2005). 2.2.1 GPS Studies by Indian Institute of Technology Bombay Team The GPS team at Department of Civil Engineering, IIT Bombay, with the author as its Principal Investigator, has taken up crustal deformation studies in

Fig. 1. Permanent GPS stations network set up under the programme.

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Fig. 2. GPS network in Koyna Dam area.

the Koyna region of Western Maharashtra and Bhuj region of Gujarat, under this programme. i. GPS Data Collection And Processing Under the DST-funded projects, two GPS networks have been set up since 2000–2001: a local

Fig. 3. GPS network in Bhuj area.

network in the Koyna region (Figure 2) and a regional network in the Bhuj region (Figure 3). The objective of the local GPS network at Koyna is to understand the deformations induced in the area, in the vicinity of the Koyna dam and in the dam structure due to various factors such as


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Fig. 4. Regional crustal strains in W. India, estimated using GPS.

crustal movements. The network consists of 34 GPS stations located in the vicinity of the dam, including 8 stations established for monitoring a fault close to the dam connected to a GT station for reference. Till today, ten sets of observations over this local network have been completed. A regional network (Figure 4) in the Western Maharashtra and coastal areas of Konkan has also been established in collaboration with IIG Mumbai consisting of 22 GPS stations, well distributed over the region. GPS Data has been collected by these 22 stations during two field campaigns carried out in April–May 2004 and September– October 2004. The regional GPS network established in Bhuj area, after the 2001 Bhuj earthquake, comprises 14 stations, covering the earthquake-affected

region. This has been observed over three epochs, during 2001, 02 and 03. The data obtained from the entire field campaigns have been processed and analyzed using Bernese 4.2 and GAMIT softwares, and some significant results have been obtained. A permanent GPS station has also been established in 2002 in the campus of IITB, which is being continuously operated with data being processing to estimate the crustal deformations and strain rates. ii. Data Analysis & Preliminary Results (A) Western India Region: From the detailed analysis of the two regional campaigns in 2004, the average strain rate for the

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Western Maharashtra and Konkan region in W. India is estimated to be −1.146 × 10−8 over the year (Figure 4), thus indicating a compression of about 0.1 ppm. An average regional crustal movement of about 45 mm in NNE direction is also estimated. Extended observations over next few years are required to confirm these rates. From the analysis of the local network data, we also sought to correlate the local deformations of the base station with the water level, in order to understand the pattern of deformation of the dam with respect to change in water level. For this purpose, the standardized water level (z) is computed. (Kulkarni et al., 2003). At Koyna, earlier studies done by experts reported that the deflection of the dam increases gradually for water levels up to 625 meters and increases rapidly for water levels beyond that. Keeping this factor in mind, a polynomial in the standardized water level (⌬z) is fitted to the GPS-estimated local deformations. A polynomial in ⌬z of third degree fits the data well. It is estimated that this relationship has a correlation of 0.99, which clearly shows that the major part of the deformation of the dam is caused by the varying water table. From the detailed

Fig. 5. Monumentation of IITB permanent GPS station.

analysis, it is also postulated that the high rate of loading and longer duration of retention of high water levels may cause seepage of water into the fault line, thereby weakening it and progressively lowering the stress needed to trigger an earthquake. These findings require more repeated observations. (B) Bhuj Region, Gujarat: A network was set up in the Bhuj region of Gujarat after January 2001 Bhuj earthquake with the objective of estimating the post-earthquake deformations. A total of 14 GPS stations have been established, out of which five are old Great Trigonometrical (GT) Triangulation stations. Three sets of observations in February 2001, 2002, and 2003 have been completed. Examining the deformation pattern estimated from the detailed analysis of the GPS data, it can be concluded that significant amount of extension is taking place across the fault line. It is possible that there is a definite geological structure (oblique slip fault) in Bhuj area which needs extensively further studies. iii. Permanent GPS Station A permanent reference GPS station (Figure 5) has also been set up at Civil Engineering


Department of IIT Bombay, as a part of the National; Network. It is running continuously since January 2002 and the data is being analyzed periodically. The general horizontal deformation of the station, reflected in the N–E direction, is due to the Indian plate motion. The horizontal deformation observed is about 2.4 cm in North and about 4.8 cm in East direction over a year, which is in close agreement with the estimated plate motion of about 5.5 cm per year in NNE direction (Kulkarni et al., 2004a,b). iv. GPS Training Training programmes in GPS-related areas are also being conducted periodically. Three short training programmes were conducted during 2000–2003, with support from DST, and over 200 students were also trained in GPS during vacations. A GPS Laboratory has been established and GPS has been introduced in the B. Tech. and M. Tech. level courses at the Institute, and several student research projects in GPS applications have been completed.

3 Thrust Areas for National GPS Programme The following thrust areas have been identified for this programme: – Determination of strain fields around different tectonic blocks:- i. e. Bhuj, Son-Narmada Lineament, North Western and Eastern part of Himalaya, Khandwa region of Madhya Pradesh, Ongole area of Andhra Pradesh, Great Boundary fault etc. – Stability of South Indian Peninsula. – Crustal deformation studies along major shear zones. – Motion and active deformation of India. – Geodynamic behavior of Himalayas. – Crustal deformation studies along Eastern and Western Ghat regions.

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– Neotectonic movements, study of active faults, landslides etc. – Quantitative geomorphology. – Ionospheric modeling. – Manpower development in GPS technology, SAR Interferometry and ALTM

4 Summary The National GPS Programme for Earthquake Hazard Assessment, launched by DST, Govt. of India and implemented in a phased manner with participation of various GPS organizations and Institutions since 1998, is in an advanced stage of implementation presently. The programme will continue to contribute significantly to our understanding of the plate motions and crustal deformations in the region and the development of models for earthquake hazard assessment.

References DST (1998) Report on the National Programme for GPS and Geodetic Studies, Expert Group on GPS, Department of Science & Technology, Govt. of India. DST (2005) Brochure (Revised) on the National Programme on GPS for Earthquake Hazard Assessment, Edited M.N. Kulkarni, Department of Science & Technology, Govt. of India, January, 2005, also available at: http://www.civil.iitb.ac.in/∼kulkarni/DSTBroFinal.pdf. Kulkarni, M. N., S. Likhar, V.S. Tomar, P. Pillai (2004a). Estimating The Post-Earthquake Crustal Deformations in Gujarat Region of India Using The Global Positioning System, Survey Review, International Journal of Surveying, Vol. 37, No. 292, pp 490–497, January. Kulkarni, M. N., Rai, D., Pillai, P., Tomar V. S. (2004b), Establishment of a GPS permanent ref. station at Dept. of Civil ENgg., IIT Bombay, India, Acta Geodetica Geophysica, Journal of Hungarian Academy of Sciences, Vol. 39(1), pp 55–59. Kulkarni, M.N., Nisha R., Tomar, V.S., Pillai, P. (2003), Monitoring Deformations of Koyna Dam using GPS, World Conference on Disaster management,- DMIC-2003, Hyderabad, 10–12 November, 2003, pp GRD7: 1–2.

Preliminary Results of Subsidence Measurements in Xi’an by Differential SAR Interferometry C. Zhao, Q. Zhang College of Geological Engineering and Geomatics, Chang’an University, No. 126 Yanta Road, Xi’an, Shaanxi, P.R. China X. Ding, Z. Li Department of Land Surveying and Geo-Informatics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, P.R. China Abstract. Differential Interferometric Synthetic Aperture Radar (D-InSAR) is used to monitor the land subsidence of Xi’an City, China over the period of 1992–1993. Four different interferometric pairs are used to calculate land subsidence rates over the monitored area. A high resolution external DEM and precise satellite orbit data are adopted to remove the topographic effects on interferograms in the D-InSAR processing. Historical leveling measurement results are used to assess the D-InSAR results by overlaying the 2-D subsidence patterns from D-InSAR and subsidence profiles from leveling measurements.

In the past two decades or so, D-InSAR has been widely used in monitoring ground deformations associated with earthquakes (e.g. Massonnet et al., 1993), volcanoes (e.g. Lu et al., 2003) and land subsidence (e.g. Li et al., 2004). In this paper several ERS synthetic aperture radar (SAR) data covering Xi’an and the surrounding areas are selected to calculate the vertical deformation in areas where there are evidences of some subsidence funnels. Section 2 provides information on the study area and the data sets used. Section 3 covers details of D-InSAR data processing, calibration and comparison with leveling results.

Keywords. Subsidence, differential interferometric SAR (D-InSAR), monitoring, leveling

2 Test Area and Data Sets

1 Introduction Xi’an City, the capital city of Shaanxi Province, lies in northwestern China and occupies the main part of Fenwei basin. The City is famous for its historic sites, such as Dayan pagoda, terracotta warriors and especially the city wall that can be clearly detected in LANDSAT-TM image (Figure 1) and SAR image (Figure 2). Xi’an has experienced serious land subsidence and land fissures since 1960s mainly due to the uncontrolled groundwater withdrawal. The leveling monitoring results show that up to 1992, the areas with cumulative subsidence of more than 100 mm were over 105 km2 , and the maximum subsidence was 1940 mm. The average ground subsidence rate was 80–126 mm/year, and the maximum rate was up to 300 mm/year. Eight subsidence funnels can be identified now in the southern, eastern and southwestern suburbs of Xi’an (Yan 1998). Point-based measurement methods such as leveling and GPS have been used to gather land subsidence information. However, these methods are limited caused by their high labor cost, low efficiency and low point density. 94

The study will focus on an area of 18 × 22 km2 (nearly 400 km2 ) that has eight identified funnels. The study area is highlighted by the black rectangle boxes in the LANDSAT-TM image (Figure 1) and the SAR image (Figure 2). Table 1 shows the ERS SAR data acquired during 1992 and 1993. Figure 3 shows the high resolution external DEM with 25 m resolution, which is used in the two-orbit D-InSAR method. Figure 4 gives the annual vertical deformation rate contour map from leveling measurements (Zhu et al., 2005), which is the best data so far collected from the previous works although the accuracy and consistency of the data sets are debatable.

3 Differential InSAR Processing and Results Analysis 3.1 Procedure and Error Sources of D-InSAR Processing Two-orbit D-InSAR method (Massonnet and Feigl, 1998) is applied to calculate the differential interferograms. A number of error sources in the differential InSAR processing may exist including baseline decorrelation, temporal decorrelation, satellite orbit errors, external DEM errors, atmosphere effects and phase unwrapping errors

Preliminary Results of Subsidence Measurements in Xi’an by Differential SAR Interferometry

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from coregistration and flat-earth phase removal (Scharroo et al., 1998). Careful baseline refinement and linear phase ramp removal procedures are also included in the interferometric processing (Atlantis Scientific Inc., 2003). Baseline decorrelation and temporal decorrelation effects are considered by careful selection of interferometric pairs described in the next part of this section. Again, some post-processing are made to filter the interferograms and eliminate gross errors that maybe introduced in phase unwrapping. Finally, the D-InSAR change map is overlaid onto the leveling contour map to show the discrepancy between the results. 3.2 Data Selection and Results Analysis Fig. 1. LANDSAT-TM image showing the study area.

(Zebker et al., 1992). A high resolution DEM is used in this investigation to reduce the errors associated with DEM errors. Precise ERS-1 orbit data from the Delft University of Technology are used to reduce errors

Fig. 2. ERS amplitude SAR image of the study area.

Fig. 3. High resolution (25 m) DEM in Xi’an area.

By considering the interferometric baseline lengths, four interferometric pairs of SAR images of Xi’an City acquired in 1992 and 1993 are selected for the study. Information on the spatial and temporal baselines of the various possible combinations of the images is given in Table 1. Pair 1 with 35-day interval has high coherence as excepted and the derived ground change map (Figure 5) from this pair does not show significant ground subsidence. It is interesting and useful to compare the subsidence maps derived from pairs 2, 3 and 4 that all have the same time intervals of 210 days. Results from pair 4 that has only a perpendicular baseline of 19 m are compared with results from leveling measurements. Figure 6 shows (a) the coherence, (b) interferogram, (c) change map and (d) change map overlaid with LANDSAT TM image. It is obviously to see from Figure 6a that the coherence is high. As a result the interferogram (Figure 6b) and the change map (Figure 6c) are both of good quality. In

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Fig. 4. Vertical deformation rate contour map from leveling (1988–1991). Table 1. Baseline information of possible SAR pairs formed by ERS SAR data of track 390 and frame 2925 over Xi’an Area (Unit: m for spatial baseline and days for temporal baselines) Slave

19920807

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(a) Coherence map

(b) Interferogram

(c) Chang map

(d) Overlay map

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Figure 6c, profiles in X and Y directions are selected for detail comparison. From Figure. 6c–d, an annual subsidence rate of 5–10 cm is detected for most of the area, while the maximum annual subsidence rate is up to 20 cm. Three funnels at Hujiamiao, Fangyuan and Xiaozhai with subsidence rate of over 13 cm/y can be clearly seen. To assess the accuracy of the D-InSAR results, profiles in the X and Y directions of the change map Figure 6d are displayed in Figure. 7a and 7b respectively. About 80 percent of the points in the profiles have residual within ±2.5 cm per year.

nal DEM and the precise satellite orbit data are necessary for improving the accuracy and reliability of D-InSAR results. Most of the areas had an average subsidence rate of 5–10 cm/y while the rate was over 13 cm/y near the 3 funnels. The results are consistent with the in-situ leveling measurement results. However, the contribution of atmospheric effects needs estimated quantitatively. The deformation evolution process should be further researched in the future.

Acknowledgement 4 Conclusions The study has shown that D-InSAR is an effective method for studying ground subsidence in Xi’an. The auxiliary data such as the high resolution exter-

The research is funded by a key project of the Natural Science Fundation of China (NSFC) (project No: 40534021), a key project of the Ministry of Land & Resources, China (project No: 1212010440410) and

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a general project of the Natural Science Fundation of China (NSFC) (project No: 40672173). SRTM DEM, DELFT precise orbit data and SNAPHU phase unwrapping software were used in data processing. Finally we would like to thank Dr. Shusun Li for his constructive comments on the manuscript.

References Atlantis Scientific Inc. (2003). Ev-InSAR Version 3.0 User’s Guide: Atlantis Scientific Inc., Ontario, Canada, 257p. Li, T., Liu, J.N., Liao, M.S., Kuang, S.J. Lu, X. (2004). Monitoring City Subsidence by D-InSAR in Tianjin Area. IEEE IGARSS2004 Proceeding, Anchorage Alaska USA 20. Sep, 3333–3336. Lu, Z., Patrick, M., Fielding, E.J., Trautwein, C (2003). Lava volume from the 1997 eruption of Okmok volcano, Alaska, estimated from spaceborne and airborne interferometric synthetic aperture radar. IEEE Trans. Geosci. Remote Sens, 41 (6), 1428–1436.

C. Zhao et al. Massonnet, D., Feigl, K.L. (1998). Radar interferometry and its application to changes in the Earth’s surface. Rev. Geophys., 36 (4), 441–500. Massonnet, D., Rossi, M., Carmona, C., Adragna, F., Peltzer, G., Feigl, K., Rabaute, T. (1993). The displacement field of the Landers earthquake mapped by radar interferometry. Nature, 364, 138–142. Scharroo, R., Visser, P.N. A.M., Mets, G.J. (1998). Precise orbit determination and gravity field improvement for ERS satellites, J. Geophys. Res., 103 (C4), 8113–8127. Yan W.Z. (1998). Analysis on the origin of land subsidence and its countermeasures of control in Xi’an. Chin. J. Geol. Hazard and Control, 119 (12), 27–32. Zebker, H.A., Villasenor, J. (1992). Decorrelation in interferometric radar echoes, IEEE Trans. Geosci. Remote Sens, 30, 950–959. Zhu, Y.Q., Wang, Q.L., Cao, Q.M., Xu, Y.M. (2005). A study of the space-time change characteristics of ground subsidence in Xi’an and their mechanism. Acta Geoscientica Sinica, 26 (1), 67–70.


Part II

Reference Frame, GPS Theory, Algorithms and Applications

Accuracy Assessment of the ITRF Datum Definition Z. Altamimi, X. Collilieux Institut Géographique National, LAREG, 6–8 Avenue Blaise Pascal, 77455 Marne-la-Vallée, France C. Boucher Conseil Général des Ponts et Chaussées, tour Pascal B, 92055 La Défense, France Abstract. One of the main objectives of the International Terrestrial Reference Frame (ITRF) is to provide a standard global reference frame having the most attainable accuracy of its datum definition in terms of its origin, scale and the time evolution of its orientation. This latter should satisfy, by convention, the no net rotation condition. The accuracy of the ITRF datum specifications are obviously dependent on the quality and the internal consistency of the solutions contributing to its elaboration and definition. In this paper, we examine and review the quality of the current ITRF datum definition with an accuracy assessment based on the ITRF2005 results and by consistency evaluation with respect to ITRF2000. The availability of time series of station positions and Earth Orientation Parameters, used now as input for the current ITRF construction, will facilitate the accuracy assessment. When rigorously stacking the time series of a given technique to estimate a longterm frame solution, the 7 transformation parameters of each individual temporal set of station positions are also estimated. By applying dynamically internal constraints (equivalent to minimum constraints approach) over the time series of the 7 parameters, we then preserve some physical “natural” parameters as for instance the scale and the origin from VLBI and SLR, respectively. Our conservative evaluation of the estimated accuracy of the ITRF datum definition is that the origin and its rate are accurate at the level of 5 mm and 2 mm/yr, the scale and its rate are at the level of 1 part per billion (ppb) and 0.1 ppb/yr and the No-Net-Rotation condition implementation is at the level of 2 mm/yr. Keywords. Reference systems, reference frames, minimum constraints, ITRF

1 Introduction The basic idea behind combining Terrestrial Reference Frame (TRF) solutions estimated by several space geodesy techniques is to gather the strength of all these techniques in one global frame. The International Terrestrial Reference Frame (ITRF) as a

multi-technique combined frame is more and more utilized in diverse Earth science applications and as a consequence, the users are more and more demanding of quality and accuracy. For instance, the current scientific requirement in terms of accuracy and stability over time of the origin and scale of the ITRF is believed to be at the level of 0.1 mm/yr. This is far from being achieved (or even achievable) nowadays as it will be demonstrated in this paper, due mainly to the degradation of the network of space geodesy techniques. Given the fact that the ITRF origin is defined uniquely by SLR, the assessment of its absolute accuracy is hard to evaluate. However we try to address in this paper the accuracy issue by evaluating the origin and scale coherence between ITRF2000 and ITRF2005, both using the same techniques (i.e. SLR and VLBI) to define these parameters. Moreover, the use of time-series of station positions to construct the ITRF (starting with the ITRF2005) allows to evaluate the internal consistency of each analyzed solution. Unlike the previous versions of the ITRF, the ITRF2005 is constructed with input data under the form of time series of station positions and Earth Orientation Parameters (EOPs). The ITRF2005 input time-series solutions are provided in a weekly sampling by the IAG International Services of satellite techniques (the International GNSS ServiceIGS, the International Laser Ranging Service-ILRS and the International DORIS Service-IDS) and in a daily (VLBI session-wise) basis by the International VLBI Service (IVS). Each per-technique timeseries is already a combination, at a weekly basis, of the individual Analysis Center (AC) solutions of that technique, except for DORIS where two solutions are submitted by two ACs, namely the Institut Gographique National (IGN) in cooperation with Jet Propulsion Laboratory (JPL) and the Laboratoire d’Etudes en Geophysique et Oceanographie Spatiale (LEGOS) in cooperation with Collecte Localisation par Satellite (CLS), designated hereafter by (LCA). In section 2 of this paper we recall the ITRF datum definition as conventionally described in the International Earth Rotation and Reference Systems Services (IERS) Conventions 2003 (McCarthy and 101

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Petit, 2003) as well as its practical implementation. In section 3 we describe the combination model used in the analysis of time series, while section 4 elaborates the TRF implementation for time series stacking. Two methods are reviewed here: the minimum constraints approach and a newly implemented method based on dynamically internal conditions over the stacked time series.

2 ITRF Datum Definition Following the IERS Conventions, (McCarthy and Petit, 2003), the ITRF as a realization of the International Terrestrial Reference System (ITRS) should tend to satisfy the following requirements: • The origin should be the Center of Mass of the Earth system. The ITRF presently uses SLR results to achieve this condition. • The scale should be consistent with the Geocentric Time Coordinate (TCG). Up to the ITRF2000, the scale is obtained through a weighted mean of SLR and VLBI data. Note however that the scale of the ITRF2000 and most likely that of the ITRF2005 are expressed in the Terrestrial Time (TT)-frame. This decision was adopted in order to satisfy space geodetic analysis centers using the ITRF results in their routine products and they use a time scale consistent with TT-frame. Nevertheless users can obtain TCG consistent values from ITRF product by applying simple formula (McCarthy and Petit, 2003). The two timescales differ in rate by (TCG – TT)/TCG ≈ 0.7 × 10−9 , (Petit, 2000). • The orientation is consistent with that of Bureau International de l’Heure (BIH) at 1984.0 and its orientation time evolution should satisfy the No-Net-Rotation Condition. By continuity, the ITRF orientation is maintained by alignement of its successive versions. The No-NetRotation Condition is ensured implicitly by aligning the ITRF rotation rate to that of NNR-NUVEL-1A model (DeMets et al., 1990, Argus and Gordon, 1991, DeMets et al., 1994) using the following equation of minimum constraints: (A T A)−1 A T ( X˙ ITRF − X˙ NNR−NA1) = 0

(1)

where A is the design matrix of partial derivatives of the three rotation rates and X˙ denotes the velocity vector. Note that equation (1) involves only the three rotation parameters and is applied over a set

of core stations located at rigid parts of tectonic plates (Altamimi et al., 2002, 2003).

3 CATREF Combination Model The equations implemented in CATREF software related to station positions and velocities were detailed in (Altamimi et al. 2002). For completeness, we summarize below the main equations. We assume in general that for each individual solution s, and each point i , we have position X si at epoch tsi and velocity X˙ si , expressed in a given TRF k. The combination consists in estimating: • Positions X ci at a given epoch t0 and velocities X˙ ci expressed in the combined frame c; • Transformation parameters Pk at an epoch tk and their rates P˙k , from the combined TRF c to each individual frame k. The general combination model used is given by the following equation (2): ⎧ i Xs ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ ˙i Xs

=

X ci + (tsi − t0 ) X˙ ci i i +Tk + Dk X c + Rk X c i i +(ts − tk ) T˙k + D˙ k X c + R˙ k X ci

=

X˙ ci + T˙k + D˙ k X ci + R˙ k X ci

(2) where for each individual frame k, Dk is the scale factor, Tk the translation vector and Rk the rotation matrix. The dotted parameters designate their derivatives with respect to time. The translation vector Tk is composed by three origin components, namely T 1, T 2, T 3, and the rotation matrix by three small rotation parameters: R1, R2, R3, according to the three axes, respectively X, Y , Z . tk is the conventional reference epoch of the 7 transformation parameters. For more details regarding the derivation of the above combination model (2), the reader may refer to (Altamimi et al., 2002). It should also be noted that some unit conversion factors are omitted in equations (2) and (3), e.g., the rotation parameters are in radian unit whereas the pole coordinates are usually expressed in milli-arc-seconds and UT1 in milliseconds. In addition to equation (2) involving station positions (and velocities), the EOPs are added by the folp lowing equations, making use of pole coordinates x s , p ys and universal time U Ts as well as their daily time p p derivatives x˙s , y˙s and L O Ds :

Accuracy Assessment of the ITRF Datum Definition

⎧ p xs ⎪ ⎪ ⎪ p ⎪ ys ⎪ ⎪ ⎪ ⎨ U Ts p x˙s ⎪ ⎪ ⎪ p ⎪ y˙s ⎪ ⎪ ⎪ ⎩ LOD s

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= x c + R2k p = yc + R1k = U Tc − 1f R3k p ˙ k = x˙c + R2 p ˙ k = y˙c + R1 ˙ k = L O Dc + ⌳f0 R3

(3)

where f = 1.002737909350795 is the conversion factor from U T into sidereal time. The last line of equation (3) is derived from the relation between t +⌳ L O D and U T , that is L O D = t 0 dU T . Given dU T the assumption that dt is constant in the interval [t, t + ⌳0 ], then L O D = −⌳0 dUdtT . ⌳0 is homogenous to time difference, so that ⌳0 = 1 day in time unit. The derivation of these equations is based on the transformation formula between celestial and terrestrial reference systems, see for instance (Zhu and Mueller, 1983). The above combination model is well adapted for two types of input data, namely: • data under the form of time series of station positions, e.g. weekly from satellite techniques and daily from VLBI and daily EOPs. In this case it is essential to consider one epoch (ts ) per station, since not all the stations within the week refer to the same epoch, say the middle of the week. ts should in fact rigourously refer to the central epoch of the observations used for station s. Using this type of input data (restricted to station positions with no velocities), the second (velocity) part of equation (2) is not used. The combination here is in fact a stacking of the time series where the outputs are as specified in the next section; • data under the form of global long-term solutions (comprising station positions and velocities) as well as EOPs.

4 Time Series Stacking and the Frame Datum Definition We consider here as input to the analysis of time series, solutions under the form of weekly or daily station positions as well as daily EOPs. The latter includes, Polar Motion components, their rates, Universal Time and Length of Day (LOD). The outputs of the time series analysis are: • A long-term solution of station positions at a chosen reference epoch t0 and velocities; • Daily EOPs, fully consistent with the underlying TRF;

• Time series of the 7 transformation parameters between each weekly (daily) solution and the long-term solution; • Time series of post-fit station position residuals. In the following we will specify the main approaches allowing to define the combined frame in case of the time series stacking. The currently adopted principles for the datum definition are: • Defining the underlying combined TRF at a chosen reference epoch t0 which is usually selected to be the central epoch of the time-span of the analyzed time-series; • Defining or adopting linear (secular) law of time evolution. The above two principles involves 14 degrees of freedom to be fixed/defined, corresponding to the similarity of 14 Helmert transformation parameters. In this approach, we assume linear station motion, but break-wise modelling is added to account for discontinuities in the time series. Moreover, non-linear station motions are then investigated and assessed using the time series of the post-fit residuals of station positions. A more refined combination model allowing to take into account the non-linear part of station motion and frame parameters needs certainly more investigation and is still in the research area. An interesting mathematical discussion and proposal regarding the modelling of the non-linear ITRF is published by Dermanis (2006). There are several ways which could be used to implement the above two principles. We select here two main approaches: the classical method of minimum constraints (Sillard and Boucher 2001) and a method imposing internal conditions over the time series of the transformation parameters. 4.1 TRF Implementation Using Minimum Constraints Using an external reference frame (e.g. a given ITRF solution) with position and velocity vector X R , one efficient way to defined the combined frame (as results from the time series stacking) is the use of minimum constraints with respect to X R over a set of reference stations. The concept of minimum constraints approach is based on the minimization of the transformation parameters between X R and our combined frame X C . We start by writing the 14parameter relatioship between the two, that is under the form

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X R − X C = Aθ where A is the design matrix of partial derivatives of the 14 parameters, constructed by the positions of X R and θ is the vector of the 14 transformation parameters. For the compete structure of A, the reader may refer to (Altamimi et al., 2004). Nullifying θ between the two frames amounts to writing the following minimum constraints equation given by (A T A)−1 A T (X R − X C )

= 0

(4)

Equation (4) allows to express the long-term cumulative solution in the same frame as X R , over the 14 datum parameters (via the selected set of stations). The resulting estimated time series of the transformation parameters are then expressed with respect to that external frame and consequently any drift or shift in the transformation parameters will also be expressed with respect to the external frame. In addition, the possible drift and shift are heavily dependent on the network geometry of the used reference set of stations and might also reflect biases of either the reference or/and the time series of the cumulative solution. 4.2 TRF Implementation Using Internal Constraints

(5)

where t0 is the selected conventional reference epoch of the combination. Note that equation (5) derives from the application of our parameter evolution model Pk (t) = Pk + (t − tk ). P˙k to the epoch t = t0 . Given the fact that in our combination model we solve for transformation parameters between each individual weekly (daily) frame k and the long-term cumulative solution, it is then possible to impose some conditions on these transformation parameters allowing to define the combined frame of the longterm solution at epoch t0 and its time evolution by using the minimal/intrinsic constraints ⎧ ⎨ Pk (t0 )

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(tk − t0 )

k∈K

(7) where the unknowns are Pk (t0 ) and P˙k . Imposing the two conditions implied in equation (6) and taking into account the normal equation system given by equation (7), the implementation of the above conditions in our combination model is dynamically achieved through the following two summations: ⎧ Pk ⎪ ⎪ ⎪ ⎪ ⎨ k∈K ⎪ Pk ⎪ ⎪ ⎪ ⎩ (tk − t0 )−1

=

0 (8)

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Given our combination/stacknig model implied in equation (2), assuming linear time evolution both for station positions and transformation parameters, we can write for any one of the 7 transformation parameter Pk at any epoch tk : Pk = Pk (t0 ) + (tk − t0 ). P˙k

The time series of each transformation parameter as described by equation (5) is in fact linked to a regression line that can easily be solved by least squares adjustment yielding the following normal equation system

(6)

The two parts of equation (8) complete in fact the rank deficiency of our normal equation system constructed over the combination model given by equation (2) and in the mean time define the frame at epoch t0 and its time evolution. These internal constraints should be regarded as equivalent to any type of inner constraints (Blaha, 1971) because of their independency of any external information, at least for the frame physical parameters as the scale and the origin. Because of the nature of these internal constraints, the two part of equation (6) are, by analogy, similar to those of Meissl “inner constraints”, as quoted by Dermanis (2000). This approach is not only completely independent from any external frame, it has also the property to preserves the weekto-week variations of the transformation parameters as does the approach of minimum constraints described in the previous sub-section. In addition, the time series of the post-fit residuals, being independent from the reference frame definition, are still the same using either one of the two methods. The only difference between these two approaches would be a shift at the reference epoch t0 and a drift between the two corresponding long-term solutions which


are fully estimable via the classical 14-parameter formula. This approach could be regarded as defining a physically natural reference frame free from any external information. For instance, the preserved origin and scale of satellite techniques as well as the scale of VLBI determined using this approach would be considered as the physical parameters inherent and proper to those techniques. Therefore any bias in these parameters which might be due to some systematic errors of a particular technique or solution will be transferred to the cumulative long-term solution.

5 Case of ITRF2005 In this section we discuss the accuracy of the ITRF datum definition by using the results of ITRF2005 analysis as well as by comparison to ITRF2000 results. The submitted time series to the ITRF2005 are summarized in Table 1, indicating for each one: the data-span, the type of solution (normal equation or estimated solution with variance-covariance information), the applied constraints and the estimated EOPs. 5.1 ITRF2005 Analysis Strategy The strategy adopted for the ITRF2005 generation consists in the following steps: • Introduce minimum constraints equally to all loosely constrained solutions: this is the case of SLR and DORIS solutions. We applied an equation of type (4), restricted to the orientation datum parameters, thus preserving the origin and the scale of each weekly solution; • Apply No-Net-Translation (NNT) and No-NetRotation (NNR) conditions to IVS solutions provided under the form of Normal Equation. The NNT/NNR conditions are achieved by applying equation (4) where the design matrix A is reduced to the translation and rotation partial derivatives, thus preserving the intrinsic VLBI scale.

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• Use as they are minimally constrained solutions. This is the case of IGS weekly solutions, where the used constraints are also of type (4), applied over the 7 parameters and where the external solution is the ITRF2000; • Form per-technique combinations (TRF + EOP), by rigorously staking the time series, solving for station positions, velocities, EOPs and 7 transformation parameters for each weekly (daily in case of VLBI) solution w.r.t the per-technique cumulative solution. In order to preserve the intrinsic SLR origin and scale and the VLBI intrinsic scale, equation (8) was used in the stacking of the ILRS SLR and IVS VLBI time series and applied over all the 7 parameters. Of interest to the final ITRF2005 combination are of course the translation and scale parameters, the rotation parameters are being conventionally defined using this approach; • Identify and reject/down-weight outliers and properly handle discontinuities using piece-wise approach; • Combine if necessary cumulative solutions of a given technique into a unique solution: this is the case of the two DORIS solutions; • Combine the per-technique combinations adding local ties in co-location sites. In this step, the specifications of the ITRF2005 origin, scale, orientation are imposed as described in the following section. This final step yields the final ITRF2005 solution comprising station positions, velocities and EOPs. Note that EOPs start in the early eighties with VLBI; SLR and DORIS contributions start on 1993 and GPS on 1999.5. The quality of the early VLBI EOPs is not as good as the combined EOPs starting on 1993. 5.2 ITRF2005 Datum Specifications The ITRF2005 origin, scale, orientation and their time evolutions are specified as follows:

Table 1. Submitted solutions to ITRF2005 TC

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Constraints

EOPs

IVS ILRS IGS IDS-IGN IDS-LCA

1980.0 – 2006.0 1992.9 – 2005.9 1996.0 – 2006.0 1993.0 – 2006.0 1993.0 – 2005.8

Normal Equation Variance-Covariance Variance-Covariance Variance-Covariance Variance-Covariance

None Loose Minimum Loose Loose

Polar Motion, rate, LOD, UT1 Polar Motion, LOD Polar Motion, rate, LOD (from 1999.5) Polar Motion, rate, LOD Polar Motion

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• Origin: The ITRF2005 origin is defined in such a way that there are null translation parameters at epoch 2000.0 and null translation rates between the ITRF2005 and the ILRS SLR time series. • Scale: The ITRF2005 scale is defined in such a way that there are identity scale factor (D = 0) at epoch 2000.0 and null scale rate ( D˙ = 0) between the ITRF2005 and the IVS VLBI time series. As it will be seen in this paper, the ITRF2005 combination revealed a scale bias of about 1 ppb at epoch 2000.0 and a scale rate slightly less than 0.1 ppb/yr between VLBI and SLR time series. VLBI scale selection to define that of ITRF2005 is justified by the availability of the full VLBI history of observations (26 years versus 13 for SLR) embedded in the submitted time series. • Orientation: The ITRF2005 orientation is defined in such a way that there are null rotation parameters at epoch 2000.0 and null rotation rates between the ITRF2005 and ITRF2000. These two conditions are applied over a core set of stations, using equation (4). 5.3 Accuracy Assessment As the word accuracy is generally considered as synonym of the “truth” which of course does not exist in terms of a “true” reference system, it becomes then necessary to have redundancies in our observations and estimation of the considered geodetic parameters. We therefore refer to “accuracy” the level of consistency between different estimates or solutions.

Accuracy of the ITRF Origin Although it is hard to assess the origin accuracy of the single ILRS solution that is submitted to ITRF2005, we attempt however to evaluate its consistency with respect to ITRF2000. Figure 1 displays the weekly translation parameters of the ILRS SLR solution with respect to ITRF2000 as results of the applications of the minimum constraints equation (4), whereas Figure 2 is obtained by making use of the internal constraints equation (6). Figure 1 results are based on the usage of a reference set of 12 stations. The criteria selection of these 12 stations is based on their observation history and good performance. They are the only stations that are usable to link the combined SLR TRF resulting from the stacking of the time series to the ITRF2000 frame. Because the estimated transformation parameters are heavily sensitive to the network geometry, the distribution of the reference set of 12 stations is far from optimal, only two of them are in the southern hemisphere (Yaragadee, Australia, and Arequipa, 50

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Accuracy of the ITRF Scale The accuracy of the ITRF scale could be evaluated by (1) assessing the consistency with the ITRF2000 of the “natural” scales of VLBI and SLR as obtained from the stacking of their respective time series and by (2) the level of agreement between the two technique long-term solutions via ITRF2005 combination. We recall here that the ITRF2000 scale was defined in such a way that there is zero scale and scale rate between ITRF2000 and the average of 3 VLBI and 5 SLR solutions (Altamimi et al., 2002). We note however that the results of ITRF2000

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ILRS Weekly # of points

25 # of points

Peru). Apart from the seasonal variations that could be estimated over the translation parameters, the linear trends are of great importance to the ITRF origin stability over time. As it could be seen from Figure 1, there are clear drifts of the translation parameters and more significantly of the Tz component of about 1.8 mm/yr. This drift together with a shift of about 5 mm at epoch 2000.0 will therefore exist between ITRF2000 and ITRF2005. They could be regarded as the current level of the origin accuracy as achieved by SLR, taking into account the poor geometry of the SLR network. From that figure we can also distinguish a “piece-wise” behavior of the Z-translation: between respectively 1993–1996; 1996–2000 and 2000–2006. In our opinion, this is completely related to and correlated with the change of the ILRS network geometry over time. In particular, the drift over the Tz-translation is mostly (if not totally) induced by the poorly distributed SLR network. In order to illustrate that effect, we plotted on Figure 3 the number of SLR stations available in each weekly solution. From this plot, one can easily see the decreasing tendency of the number of stations starting around 2000, which should be correlated with the Tz component that starts to significantly drifting at this same epoch (see Figure 1). In addition, among the approximately 90 SLR stations available in the ITRF2005, approximately 20 of them have sufficient time-span of observations to be considered as core stations for useful and comprehensive analysis.

20 15 10

ILRS Week 5 1994 1996 1998 2000 2002 2004 2006

Fig. 3. Number of stations available in the ILRS SLR Weekly time series.

108

Z. Altamimi et al. 50 40

VLBI

SLR

30 20 10 0 –10 –20 –30 –40

IVS VLBI vs ILRS SLR Scale (mm) wrt ITRF2005 –50 1980 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006

Fig. 4. IVS VLBI (gray triangle) versus ILRS SLR (black circle) scales with respect to ITRF2005 Unit in millimeter (by conversion 1 ppb ≈ 6 mm).

combination showed closer and better agreement between the 3 VLBI solutions and the ITRF2000 than the agreement with and between the 5 SLR solutions (Altamimi et al. 2002, Figure 4, page 2–10). Regarding the SLR scale consistency with the ITRF2000, the right bottom plot of Figure 1 illustrates the weekly SLR scale behavior with respect to ITRF2000, based on the usage of the minimum constraints equation (4) over the selected poorly distributed 12 reference stations. From that figure we can easily see that the scale plot is not clearly linear over time, having “piece-wise” behavior as mentioned above for the TZ and in particular the drifting tendency after epoch 2000.0. Comparing the longterm SLR solution obtained by the usage of internal constraint equation (6) to ITRF2000 over the 12 selected stations yields a scale factor of about 0.5 ppb at epoch 2000.0 and a scale drift of about 0.07 ppb/yr. In our opinion, the poor distribution of 12 reference stations as well as the poor configuration of the SLR network over time (Figure 3) are probably a major limitation of the assessment of the scale between SLR and ITRF2000. Similarly, the direct comparison of the VLBI long-term solution to ITRF2000 using 20 well-observed stations yields a scale factor of about 0.3 ppb at epoch 2000.0 and insignificant scale rate. The ITRF2005 combination (making use of local ties in co-location sites) revealed a scale bias of about 1 ppb between VLBI and SLR solutions at epoch

2000.0 and a scale drift of 0.08 ppb/yr. Given (1) the availability of VLBI time series covering its full history of observations (26 years of observations for VLBI, versus 13 years for SLR) and its scale stability over time compared to the non-linear behaviour of SLR scale, it was then decide that the final and official ITRF2005 scale to be entirely defined by VLBI. Figure 4 displays ILRS SLR and IVS VLBI scale variation with respect to ITRF2005. The accuracy assessment of the ITRF scale is not easy to evaluate, being dependent on several factors, as for instance, the quality and distribution of the local ties, the SLR range bias effect and the tropospheric modelling in case of VLBI. However, given the level of consistency mentioned above between VLBI and SLR scales, and to be more conservative, we can postulate that the current accuracy level of the ITRF scale is around 1 ppb and 0.1 ppb/yr. The No-Net-Rotation Condition The No-Net-Rotation condition is implicitly ensured for ITRF2000 by minimizing the three rotation rate parameters between ITRF2000 and NNR-NUVEL1A using a core set of stations (Altamimi et al. 2002). The ITRF2005 is aligned to ITRF2000 over the same three rotation rate parameters using a reference set of about 70 stations. In past analysis (Altamimi et al. 2003), we demonstrated that the level of accuracy of the NNR realization is around 2 mm/yr. Using recent available NNR models: APKIM2005P

Accuracy Assessment of the ITRF Datum Definition Table 2. Non-zero transformation ITRF2005 to ITRF2000

± ±

109

parameters

from

Tx mm T˙ x mm/y

Ty mm T˙ y mm/y

Tz mm T˙ z mm/y

D ppb D˙ ppb/y

0.1 0.3 −0.2 0.3

−0.8 0.3 0.1 0.3

−5.8 0.3 −1.8 0.3

0.40 0.05 0.08 0.05

(Drewes, personal communication) and GSRMNNR-2 of Kreemer et al. (2006), we estimated the three rotation rates between these two recent models and the ITRF2000 (aligned to NNR-NUVEl-1A) over 70 selected reference stations. The resulting rotation rates we found are of the same magnitude of those published in (Altamimi et al. 2003), suggesting that the accuracy level of the NNR condition implementation is still about 2 mm/yr. Transformation Parameters Between ITRF2000 and ITRF2005 Table 2 lists the non-zero transformation parameters between the ITRF2000 and the ITRF2005, estimated using 70 core stations available at the ITRF2005 web site (http://itrf.ensg.ign.fr/ITRF solutions/2005/). Using different set of stations may yield different values of the transformation parameters depending on the network geometry.

6 Conclusion Despite the optimistic accuracy estimate of the ITRF2000 datum definition as stated in (Altamimi et al., 2002), the refined analysis of the new data submitted to ITRF2005, showed a sort of degradation of the accuracy of the ITRF origin and scale. The ITRF2005 origin and its rate are evaluated to be at the level of 5 mm (at epoch 2000.0) and 1.8 mm/yr, by comparison to ITRF2000, due mainly to the weakness of the translation along the Z -axis. The ITRF2005 scale and its rate are probably determined at the level of 1 ppb (at epoch 2000.0) and 0.1 ppb/yr, based on the level of agreement between VLBI and SLR solutions submitted to ITRF2005 by their respective services. The No-Net-Rotation condition is still at the level of 2 mm/yr. We attribute a significant part of the degradation of the accuracy in origin and scale to the network degradation, especially that of SLR as well as the current poor VLBI and SLR co-location sites. We intend in the near

future to investigate and monitor the ITRF2005 origin and scale stability over time, by integrating new data foreseen to be available at the weekly (daily for VLBI) basis. In addition, more investigation is certainly needed on the seasonal (annual and or semiannual) variations both over station components as well as the frame parameters. In particular, we will explore theoretical and practical ways to address and evaluate the correlation between the station seasonal variations and those of the frame parameters and to attempt to quantify the impact of these seasonal variations on the ITRF long-term stability of its time evolution.

Acknowledgement Athanasios Dermanis reviewed the first draft of the paper with useful and helpful comments that improved the content of the manuscript.

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:10.1029/2001JB000561. Altamimi, Z., P. Sillard and C. Boucher (2003). The impact of a no-net-rotation condition on ITRF2000, Geophys. Res. Lett., 30(2), 1064, doi:10.1029/2002GL016279. Altamimi Z., P. Sillard and C. Boucher (2004). ITRF2000: From Theory to Implementation. In: Sansò F (ed.), V Hotine-Marussi Symposium on Mathematical Geodesy, International Association of Geodesy Volume 127. Springer, Berlin Heidelberg New York, pp 157–163. Argus, D.F. and R.G. Gordon (1991). No-net rotation model of current plate velocities incorporating plate motion model NUVEL-1, Geophys. Res. Lett., 18, 2038–2042. Blaha G. (1971), Inner adjustment constraints with emphasis on range observations, Department of Geodetic Science, Report 148, The Ohio State University, Columbus. Dermanis, A. (2000), Establishing Global Reference Frames. Nonlinear, Temporal, Geophysical and Stochastic Aspects, In: Sideris, (ed.), Gravity, Geoid and Geodynamics 2000, IAG Symposium Vol. 123, pp. 35–42. Dermanis, A. (2008) The ITRF Beyond the Linear Model Choices and Challenges, this issue. DeMets, C., R.G. Gordon, D.F. Argus and S.Stein (1990). Current plate motions. Geophys. J. Int., 101, 425–478. DeMets, C., R. G. Gordon, D. F. Argus, and S. Stein (1994). Effect of recent revisions of the geomagnetic reversal timescale on estimates of current plate motions, Geophys. Res. Lett., 21(20), 2191–2194. Kreemer, C., D. A. Lavallee, G. Blewitt and W. E. Holt (2006), On the stability of a geodetic no-net-rotation frame and its implication for the international terrestrial reference frame, Geophys. Res. Lett., 33, L17306, doi:10.1029/2006GL027058. McCarthy D. and G. Petit, 2003. IERS Conventions 2000, http://maia.usno.navy.mil/conv2000.html.

110 Petit, G., Report of the BIPM/IAU Joint Committee on relativity for space-time reference systems and metrology, in Proc. of IAU Colloquium 180, In: Johnston K. J., D.D. McCarthy, B.J. Luzum and G.H.Kaplan, (eds.), U.S. Naval Observatory, 275–282, 2000.

Z. Altamimi et al. Sillard, P. and C. Boucher, (2001) Review of algebraic constraints in terrestrial reference frame datum definition, J. Geodesy , 75, 63–73. Zhu and Mueller, (1983) Effects of adopting new precession, nutation and equinox corrections on the terrestrial reference frame, Bulletin Godsique Vol 57 n 1. pp 29–41.

The ITRF Beyond the “Linear” Model. Choices and Challenges A. Dermanis Department of Geodesy and Surveying, Aristotle University of Thessaloniki, University Box 503, 54124 Thessaloniki, Greece

Abstract. The current solution to the choice of a reference system for the coordinates of a global geodetic network is based on a linear model for the time evolution of station coordinates. The advantages and disadvantages between a mathematical approach and a physical approach to the optimal definition of a reference system for the International Terrestrial Reference Frame (ITRF) are examined. The optimality conditions are derived for a general class of models, consisting of linear combinations of a system of base functions which is closed under differentiation and multiplication. The general results are then applied and elaborated for polynomial and Fourier Series models. Finally the problem of how these conditions should be implemented in practice is investigated. Keywords. Reference systems, reference frames, International Terrestrial Reference Frame, ITRF

1 Introduction The current solution to the choice of a reference system for the coordinates of a global geodetic network (ITRF) is based on a linear model for the time evolution of station coordinates (Altamimi, 2005, Drewes, 2006b). All deviations from such a model are attributed to “noise”, of either observational or of geophysical origins. The improvement in data quality and quantity, as well as the larger time span covered by these data, will soon or later necessitate more elaborate models. Therefore the problem of the optimal definition of the reference system for non-linear models must be addressed. When dealing with a deformable body or a deformable point network, a reference system consists of a choice for every epoch t of (a) a point O(t) as the origin and (b) three directed lines and a unit of length which constitute the vectorial basis e(t) = [e 1 (t)e 2 (t)e 3 (t)]. This choice must be smooth in time and optimal in the sense that the motion of the body masses or the network points with respect to

the reference system is minimized in some sense that remains to be specified. Thus the reference system separates the total motion with respect to the inertial background into two parts: (a) the translational motion and the rotation of the body/network, as represented by the reference system origin and axes, respectively and (b) the remaining “apparent deformation” represented by relative motion with respect to the reference system. An optimality criterion for the choice of the reference system is associated with a particular choice of a measure of the “apparent deformation”, which should be minimized. The optimality criterion should in fact be applied to the International Terrestrial Reference Frame (ITRF), which is an operational realization of the terrestrial reference system, involving the coordinates of a selected global terrestrial network, expressed as functions of time xi = [x i1 x i2 x i3 ]T = xi (t, ai ),

i = 1, 2, . . . , N, (1) defined through a set of 3N model parameters ai , 3 for each network point Pi with position vector x i = → O Pi = ei xi . Within the current linear model xi = x0i + t vi , the model parameters are the initial coordinates x0i and the velocities vi . Therefore the optimality criterion must be realized by means of a set of mathematical conditions Fk (a1 , a2 , . . . , a N ) = 0,

k = 1, 2, . . . , L < 3N, (2)

Such conditions define only the dynamical behavior of the reference system and must be augmented with conditions which choose a particular reference system out of all optimal dynamically equivalent solutions, e.g. by arbitrarily selecting the reference system at a reference epoch t0 . Two reference systems with coordinates xi and xi , respectively, are dynamically equivalent when 111

112

A. Dermanis

xi (t) = Rxi (t) + d,

(3)

for some constant orthogonal rotation matrix R and a constant translation vector d.

2 The Mathematical and Physical Approaches to the Reference System Choice for the ITRF

h R (t) =

Earth

while the “Tisserand” axes are defined as the ones for which the relative angular momentum vanishes [x(t)×]

dx (t)dm x(t ) = 0, dt

∀t, (5)

Earth

or equivalently the ones for which the relative kinetic energy is minimized TR (t) ≡

1 2

dx (t) dt

T

dx (t)dm x(t ) = min, dt

∀t.

Earth

(6) In the mathematical approach we may imitate (and not approximate) the above definitions, by considering the ITRF network stations as mass points, with unity mass for the shake of simplicity. Thus the origin is defined in relation to the network barycenter by N 1 xi (t) = m = constant, N i=1

∀t,

N i=1

There are two different ways to approach the problem of reference system definition, for the ITRF. The “mathematical approach” aims at a consistent definition taking into account our knowledge of the variation of the point network shape with time, as obtained by geodetic positioning techniques. On the other hand the “physical approach” aims at a (geo)physically meaningful definition for the whole earth and it is based mainly on geophysical knowledge and/or hypotheses, on which geodetic positioning and gravity field information serve merely as external constraints. The definition of the reference system for the whole earth has already been considered in geophysics. In the most convenient choice, proposed by Tisserand (Munk & MacDonald, 1960) and used exclusively in theories of rotation for a nonrigid earth, the origin is defined as the geocenter, i.e. such that x(t)dm x(t ) = 0, ∀t. (4)

h R (t) ≡

where the non-zero constant m serves for sifting the origin to a desired position, e.g. the estimated mean position of the geocenter. The axes are analogously defined by

(7)

[xi (t)×]

dx (t) = 0, dt

∀t.

(8)

We shall call the conditions (7) and (8) the “discrete Tisserand” conditions, instead of the widely used inappropriate terms “no net translation” for (7) and “no net rotation” for (8). The last term derives from the absurd idea that every point has its own rotation vector! For a rotating rigid body or network each point has a (true) velocity due only to rotation with components with respect to a reference system ␻×]xi , where attached to the body given by vi = [␻ ␻ are the components of the rotation vector. Here the concept of the rotation vector follows the basic principle of mathematical physics, to explain a lot by means of little. This principle is severely violated when, given the coordinates xi and the velocity components vi of any point, one fills the mathematical slot by inserting the unique vector ␻ i sat␻i ×] xi . These “rotation” vectors are isfying vi = [␻ deprived of any physical meaning. Points may move but not rotate! In fact there exists no such thing as the rotation of a deformable body or the deformable earth. When we refer to the rotation of the earth we merely mean the rotation of a reference system which is attached to the earth in an optimal way. The physical approach has the great advantage that it provides a physically meaningful reference system depending on the constitution and internal motions of the earth, which is furthermore compatible with the reference system appearing in theories of earth rotation and earth orbital motion. The disadvantage is that it requires additional geophysical information and hypotheses beyond the observational information of geodetic techniques. Any errors in the geophysical data directly affect the definition of the reference system. The advantages and disadvantages of the mathematical approach are exactly the inverse of those in the physical one. It suffers from lack of physical meaning and is not compatible with earth rotation and orbital motion theories. On the other hand it provides ITRF coordinates, which suffer only from errors in estimating the time variation of the network shape, without additional errors arising from uncertainties in the position of the reference system origin and axes with respect to the network. It is

The ITRF Beyond the “Linear” Model

113

a pure geodetic-positional approach free from any geophysical hypotheses and free from errors in the geodetic estimation of the geocenter position, as the mean position of foci of oscillating ellipses of analyzed satellite orbits. We propose that both approaches must be used. The mathematical one will provide an operational ITRF appropriate for positioning purposes. The physical one will give a reference system that is an approximation to the Tisserand system of earth rotation theories. This approximation will become better in the future as geophysical information about earth density, internal mass motions and plate motion and deformation improves, also with the help of geodetic observations, which are more accurate and more densely distributed over the earth surface. The conversion of a mathematically defined ITRF into a (geo)physically meaningful one has been already treated in Dermanis, 2001, 2006. For an alternative approach see Engels & Grafarend, 1999, Drewes, 2006a. We shall therefore concentrate here on the mathematical approach.

3 The Linear Model for the ITRF Currently the ITRF implements a linear model xi (t) = x0i + (t − t0 )vi

(9)

involving reference epoch coordinates x0i = xi (t0 ) and constant velocities vi . It is wrong to assume that there exists in this case an underlying “linear deformation” model, independent from the definition of the reference system. Indeed a time dependent rotation R(t) and translation d(t) leads to a model with respect to a new equally legitimate reference system x (t) = R(t)x(t) + d(t) which is not anymore linear in time x i (t) = R(t)x0i + d(t) + (t − t0 )R(t)vi .

(10)

The same holds true for a spectral analysis (Fourier series) model xi (t) = a0 + [ak cos(ωkt) + bk sin(ωkt)] (11) k

which does not preserve the Fourier series form in a new reference system, where it becomes x i (t) = R(t)a0i +

M k=1

+R(t)bki

x i (t) = [Rx0i +d]+(t −t0 )[Rvi ] = x 0i +(t −t0 )v i , where the linear form is preserved. In the same case equation (11) becomes x i = [Ra0i + d] +

M

[Raki ] cos(kωt) + [Rbki ] sin(kωt)

k=1

= a i + 0

M

k k a i cos(kωt) + b i sin(kωt)

k=1

where the Fourier-series form is preserved. This allows us to provide an ad-hoc definition of a linear deformation model: We say that a point network or body deform in a linear way if the coordinates of any point with respect to any Tisserand reference system are linear functions of time. In a similar way we may speak about the spectral analysis of the deformation of a network or body, meaning the spectral analysis of the coordinate functions of any point with respect to any Tisserand reference system.

4 Beyond the Linear Model: Conditions for the Optimal Choice of the Reference System, Using a General Class of Base Functions The extended time period covered by any ITRF and the particular behavior of some network stations necessitate the use of richer time-evolution models, the first choices being polynomials, Fourier series and splines. We shall first address the problem in a more general way by considering models expressed as linear combinations of base functions xi (t) =

M

cik φk (t)

(13)

k=1

R(t)aki cos(kωt) +

sin(kωt) + d(t)

Therefore there exists no Fourier analysis independent from the choice of the reference system. Frequency components appearing in one coordinate system are different from those in another one, where also the contributions of frequencies in R(t) and d(t) are present. However, if x(t) and x (t) are both referring to systems satisfying the Tisserand conditions, then x i (t) = Rxi (t) + d with constant R and d, in which case equation (10) becomes

(12)

with the sole restriction that the system of base functions {φk (t)} is closed under differentiation and multiplication

114

A. Dermanis

dφm = Anm φn dt M

(14)

n=1

φk φn =

L

j

Bk,n φ j

(15)

j =1 j

Setting Ck,m =

M

5 Optimality Conditions for Polynomial Models

j

Anm Bk,n it follows that

n=1

In the case of polynomial models

j dφm = φk Ck,m φ j . dt L

(16)

j =1

With xi = k cik φk , dtd xi = k cik dtd φk and (16) the discrete relative angular momentum of equation (8) becomes h R (t) =

N M M L j =1

j Ck,m [cki ×]cm i

φ j (t) = 0.

= 0,

j = 1, 2, . . . , L,

q

cki φk (t) = c0i + c1i t + · · · + ci t q , (22)

k=0

the base functions are φk (t) = t k , k = 0, . . . , q, with q−1

dφ0 dφm dxi = 0, = mφ , = (m + 1)cm+1 φm m−1 i dt dt dt and the axes condition becomes hR =

q q−1 N

(m + 1)[cki ×]cm+1 t k+m = 0 i (23)

Using the summation lemma ak,m t k+m =

q r

N

j

j = 1, 2, . . . , L. (19) The origin is defined from the condition (7) for the preservation of the network barycenter, which is satisfied when 1 dm = dt N

1 dxi = dt N

i=1

i=1 m=1

cm i

M

j

Am φ j = 0.

j =1

(20) The corresponding “dynamic constraints” for the origin definition are N M

j

A m cm i = 0,

ar− j, j t r

N

[cki ×]cm+1 , we obtain the i

r− j

( j + 1)[ci

(24)

j +1 r

×]ci

t

r=0 j =0 i=1

i=1 k<m

N M

condition

q−1

r=q+1 j =r−q

i=1

q r N

j

(Ck,m − Cm,k )[cki ×]cm i = 0,

N

with ak,m = (m + 1)

2q−1

ar− j, j t r +

r=0 j =0

k=0 m=0

i=1 k=1 m=1

(18) which can also be written in a form avoiding repetim k tions due to [cki ×]cm i = −[ci ×]ci , i.e.

m=0

i=1 k=0 m=0

q−1 q

j Ck,m [cki ×]cm i

q

xi (t) =

i=1 k=1 m=1

(17) The above equation holds for any epoch t only if all the coefficients of the base functions φ j (t) vanish, thus leading to the “dynamic constraints” for the axes definition M M N

constraints” or “discrete Tisserand constraints” (and not no-net-translation and no-net-rotation constraints!) are in fact a generalization (Dermanis, 1995, 2000, 2003) of the familiar Meissl’s inner constraints (Meissl 1965, 1969) used for defining the reference system for a rigid network.

j = 1, 2, . . . , M.

(21)

i=1 m=1

The constraints (18) and (21) defining the optimal reference system, which we may call “dynamic

+

2q−1

q−1 N

r− j

( j + 1)[ci

j +1 r

×]ci

t = 0.

r=q+1 j =r−q i=1

(25) Equating all coefficients of t r to zero we obtain the axes dynamic constraints r =0, . . . , q : r N

r− j

( j + 1)[ci

j +1

×]ci

= 0,

(26a)

j =0 i=1

r =q + 1, . . . , 2q − 1 : 2q−1

q−1

N

r− j

( j + 1)[ci

j +1

×]ci

= 0.

r=q+1 j =r−q i=1

(26b)


115

Since [a×]b = −[b×]a and [a×]a = 0, the above constraints contain duplicate and vanishing terms. A modified computer-friendly form is N

r = 0, 1 :

(r + 1)[c0i ×]cr+1 = 0, i

M

C 0 m S −m[a0i ×]am i φm + m[ai ×]bi φm

m=1 M M C C k m C S −m[aki ×]am + i φk φm + m[ai ×]bi φk φm k=1 m=1 M M S C k m S S + −m[bki ×]am i φk φm + m[bi ×]bi φk φm .

i=1

r = 2, 3, . . . , q − 1 : N

=

k=1 m=1

(31)

(r + 1)[c0i ×]cr+1 i

i=1

+

N [r/2]

With the use of the trigonometric relations j

r+1− j

(r + 1 − 2 j )[ci ×]ci

= 0,

C C C 2φkC φm = φ|k−m| + φk+m ,

i=1 j =1

r = q, q + 1, . . . , 2q − 2 : N

[r/2]

C C 2φkS φmS = φ|k−m| − φk+m j

r+1− j

(r + 1 − 2 j )[ci ×]ci

= 0, (27)

i=1 j =r−q+1

where [r/2] denotes the largest integer ≤ r/2. The dynamic constraints for the origin are N 1 0 ci = m N

cri = 0,

r = 1, . . . , q.

k−m S |k−m| φ|k−m|

C S 2φkS φm = φk+m +

k−m S |k−m| φ|k−m|

(32)

we arrive at the form

(28a)

M M 2 C S hi = 2m[a0i ×]bm φ − 2m[a0i ×]am i m i φm + ω m=1

i=1

N

S 2φkC φmS = φk+m −

(28b)

+

i=1

M M

m=1

m k C m [aki ×]bm i + [ai ×]bi φ|k−m| +

k=1 m=1

6 Optimality Conditions for Fourier Series Models

+

M

[aki φkC (t) + bki φkS (t)]

+ (29)

k=1

with basis functions φkC (t) = cos(kωt) and φkS (t) = sin(kωt), where ω = 2π/T , with T being the time period of analyzed data, preferably an integer number of years to adjust to the natural frequencies of geophysical processes. Since dφkC /dt = −kωφkS and dφkS /dt = kωφkC , it follows that M

(30)

k=1

The axes condition h R = ⌺i [xi ×]dt xi = ⌺i hi = 0, can also be written in the form ω−1 h R = ⌺i ω−1 hi = 0, where ω−1 hi = ω−1 [xi ×]

M M

dxi = dt

k k m S m [am + φk+m i ×]ai + [bi ×]bi

k=1 m=1

+

dxi =ω k[−aki φkC (t) + bki φkS (t)] dt

m k C m [aki ×]bm − [a ×]b i i i φk+m +

k=1 m=1

The Fourier series model has the form xi (t) = a0i +

M M

M M

k−m |k−m| m

S k m k [bi ×]bm i − [ai ×]ai φ|k−m|

k=1 m=1

(33) Collecting the contributions to the base functions φ0C = 1, φrC , φrS , summing over all stations and setting equal to zero we arrive at the dynamic constraints for the axes. After extensive calculations involving the combination of terms having common cross products and the removal of vanishing terms, we arrive at the following computer-friendly equations akm ≡ [aki ×]am i ,

φ0C

:

bkm ≡ [aki ×]am i , M N i=1

k=1

ckm ≡ [aki ×]bm i , (34)

2kckk

= 0,

(35)

116

A. Dermanis

φrC , r = 1, 2 : N M−r (2k + r )ckk+r 2r c0r + i=1

k=1 M

+

+

i=1

(36)

k=1

+

⎝

i=1 [(r−1)/2]

i=1

⎞

(r − 2k)ckr−k ⎠ = 0

(37)

k=[r/2+1]

φC M : ⎛ ⎞ [(M−1)/2] N ⎝2Mc0M + (M − 2k)ckM−k ⎠ i=1

+

N

⎛ ⎝

i=1

k=1 M−1

i=1

i=1

⎞

k=[M/2+1]

k=r−M

M

⎞ (r

⎠ − 2k)cr−k k

= 0,

k=[r/2+1]

(39)

φrS , r = N i=1

M−r

(2k + r )(akk+r + bkk+r ) = 0,

k=1

(40)

φrS , r = 3, . . . , M − 1 : N M−r k+r k+r r (2k + r )(ak + bk ) −2r a0 − i=1

aki

= 0,

N

bki = 0,

k = 1, . . . , M

(51)

i=1

7 How to Implement the Optimality Conditions There are two different approaches for implementing the dynamical constraints (discrete Tisserand constraints) in the ITRF, each having its own pros and cons: a. A-priori at the level of data analysis for the ITRF construction b. A-posteriori by transformation from an arbitrary reference system to the optimal one.

1, 2 :

−2r a0r −

where [x] denotes the largest integer not exceeding the real number x. The above constraints for the axes need further elaboration in order to remove duplicate and vanishing terms. The origin condition (7) upon differentiation becomes ⌺i (dxi /dt) = 0, which in view of (30) yields the dynamical constraints for the origin N

(M − 2k)ckM−k ⎠ = 0, (38)

φrC , r = M + 1, . . . , 2M : ⎛ [(r−1)/2] N ⎝ (r − 2k)cr−k + k

k=r−M

(43) (r − 2k)ckr−k

k=1

r−1

+

k=1

φrS , r = M + 1, . . . , 2M : ⎞ ⎛ [(r−1)/2] N ⎝ (r − 2k)(−akr−k + bkr−k )⎠ = 0,

⎞

(2k − r )ckk−r ⎠

⎛

S : φM ⎞ ⎛ [(M−1)/2] N

M−k M−k ⎠ ⎝−2Ma M + (M − 2k) −ak + bk = 0, 0

(42)

k=r+1 N

(r − 2k)(−akr−k + bkr−k )⎠ = 0,

k=1

i=1

φrC , r = 3, . . . , M − 1 : N M−r (2k + r )ckk+r 2r c0r + M

⎞

(41)

k=r+1

+

⎝

[(r−1)/2]

⎞

(2k − r )ckk−r ⎠ = 0

i=1

N

⎛

k=1

The a-priori approach requires single epoch data xi (tk ) (daily or weekly solutions). The data enter free of reference system as shapes of sub-networks. They are transformed to the optimal reference system by introducing rotation and translation nuisance parameters and imposing the optimality constraints. While the implementation of the constraints is straightforward, the disadvantage is that it is not possible to introduce time-evolution models at the preprocessing stage (at the data analysis centers


dealing with regional sub-networks). Furthermore, significant sub-network overlap is required to secure successful “patchwork” into a single global shape, before introducing the optimal reference system by imposing the relevant constraints. In the a-posteriori approach the data analysis is first performed in any convenient reference system. The coordinates are then converted to the optimal ones by solving for appropriate rotation R(t) and translation d(t), such that the optimality conditions are satisfied. An advantage is that time-evolution models can be introduced during the preprocessing level at the analysis centers of regional sub-networks. On the other hand optimality can be only approximately achieved, because preservation of the time evolution model requires restriction of the unknown rotation R(t) and translation d(t) to the same model class (linear combinations of the same base functions).

8 Conclusions and Open Problems The derived dynamical constraints provide the means for establishing an optimal reference system, in the sense that the ITRF network barycenter remains constant, while its axes satisfy the discrete Tisserand principle of maintaining minimal kinetic energy when the stations are treated as unit mass points. They are quadratic constraints and they need to be linearized for use in the familiar adjustment methods with linear models and constraints. This necessitates iterations to secure that optimality has been achieved to a high degree of approximation. In addition the usual Meissl inner constraints, with respect to a set of reference (approximate) station coordinates must be applied, in order to select a particular one out of all dynamically equivalent optimal reference systems. It is also possible to allow for change of scale in the transformation to the optimal system, in which case additional dynamical constraints for scale are imposed. We have not discussed this option here due to space limitations. The purpose of any model fitting to the temporal variation of station coordinates is to retain the essential information present in the data and remove the model residuals as noise or of no relevance. Of particular interest is the spectral analysis performed by using Fourier series models. Spectral analysis is typically used for isolating estimation errors (noise) from signal: High frequencies are attributed to noise and are removed, while low frequencies are retained as signal. The difficult point is the treatment of intermediate frequencies. The question whether intermediate frequencies in network deformation are real is very difficult to answer. Systematic errors in

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space-techniques have atmospheric origins with spectral characteristics related to the annual cycle. Real intermediate frequency deformations should have origins with the similar spectral characteristics, thus making it hard, or even impossible, to distinguish between the two. The only hope lies in the future better monitoring of the atmosphere and the effective removal of systematic errors. Once an ITRF has been derived using a particular time evolution model, the ultimate question is whether to retain all the present deformation components, or to use some smoothed version instead. Even if intermediate frequencies are believed to be real it is not clear whether they should be retained or removed leading to a “smoothed” ITRF. Note that already the effects of earth tides, though real, are already removed at the data preprocessing level. The relevant decision depends on the particular intended use of the ITRF. An ITRF with all periodic terms removed, is more appropriate for the study of secular deformation associated with plate tectonics. A similar choice has been made when UT2 has been introduced as a smoothed version of UT1. On the other hand an ITRF with all deformation components, except those attributed to noise, is more appropriate when confronted with the analysis of raw data in order to test some particular geophysical hypothesis. A particular problem is the treatment of discontinuous episodic deformations which must be modeled by step functions and removed from the final solution.

References Altamimi, Z., (2005). Status of the ITRF 2004. Journées 2005 Systèmes de Référence Spatio-Temporels “Earth Dynamics and Reference Systems: Five Years After the Adoption of the IAU 2000 Resolutions”, Warsaw 19–21 September 2005. Proceedings in print. Dermanis, A., (1995). The Non-Linear and the SpaceTime Datum problem. Paper presented at the Meeting “Mathematische Methoden der Geodaesie”, Mathematisches Forschungsinstitut Oberwolfach, October 1–7, 1995. Dermanis, A., (2000). Establishing Global Reference Frames. Nonlinear, Temporal, Geophysical and Stochastic Aspects. IAG Inter. Symp. Banff, Alberta, Canada, July 31–Aug. 4, 2000. In: M.G. Sideris, ed. “Gravity, Geoid and Geodynamics 2000”, IAG Symposia vol. 123, pp. 35–42, Springer, Berlin 2002. Dermanis, A., (2001). Global Reference Frames: Connecting Observation to Theory and Geodesy to Geophysics. IAG 2001 Scientific Assembly “Vistas for Geodesy in the New Millennium” 2–8 September 2001, Budapest, Hungary. Dermanis, A., (2003). On the maintenance of a proper reference frame for VLBI and GPS global networks. In: E. Grafarend, F.W. Krumm, V.S. Volker, eds. Geodesy – the Challenge of the 3rd Millennium, pp. 61–68, Springer Verlag, Heidelberg, 2003.

118 Dermanis, A., (2006). The Definition of a Geophysically Meaningful International Terrestrial Reference System. Problems and Prospects. Presented at the EGU General Assembly, Vienna, April 3–7, 2006. Drewes, H., (2006a). Implementation of the Kinematical NNR Condition for the Terrestrial Reference Frame. Presented at the EGU General Assembly, Vienna April 3–7, 2006. Drewes, H., (2006b). Objectives and perspectives for the realization of geodetic reference frames. In this volume. Engels J., and E. Grafarend (1999). Zwei polare geodätische Bezugsysteme: Der Referenzrahmen der mittleren

A. Dermanis Oberflächenvortizität und der Tisserand-Referenzrahmen. Mitteilungen der Bundesamtes für Kartographie und Geodäsie, Band 5, 100–109, Frankfurt am Main. ¨ Meissl, P., (1965). Uber die innere Genauigkeit dreidimensionaler Punkthaufen. Zeitschrift für Vemessungswesen, 90, 4, 109–118. Meissl, P., (1969). Zusammenfassung und Ausbau der inneren Fehlertheorie eines Punkthaufens. Deutsche Geodätische Kommission, Reihe A, Nr 61, 8–21. Munk, W.H., and G.J.F. MacDonald (1960). The Rotation of the Earth. A Geophysical Discussion. Cambridge University Press, Cambridge.

Approach for the Establishment of a Global Vertical Reference Level L. Sánchez Deutsches Geodätisches Forschungsinstitut, DGFI, Alfons-Goppel-Str. 11, D-80539, Munich, Germany

Abstract. This paper concentrates on the empirical evaluation of the so-called Neumann geodetic boundary-value problem, also known as the fixed gravimetric boundary-value problem, to determine a reference geopotential value W0 as a basis for the global vertical datum unification. The boundary surface is assumed as the mean sea surface derived from satellite altimetry, and the boundary condition is given by the gravity disturbances obtained from a global gravity model in combination with a reference ellipsoid (here GRS80). In order to establish the reliability of the numerical computation, several global gravity models and mean sea surface models are applied. All of them lead to very similar W0 values, the discrepancies of which are less than ∼ ±0.05 m2 s−2 . The largest variation of W0 (∼ 1.2 m2s−2 ) is caused by the reduction of the latitudinal coverage from the range of altimetry (∼ ±80◦ ) to the ice-free ocean limit (∼ ±60◦ ). The W0 value obtained from this approach differs considerably from the theoretical potential U0 of the GRS80 ellipsoid and from the numerical standard included in the IERS conventions. Keywords. W0 determination, global vertical reference level, world height datum, geodetic boundaryvalue problem

1 Introduction One of the current challenges of Geodesy is to establish a gravity field related vertical reference system, the precision of which reaches the same order (10−9) as the geometrical reference system (i.e. ITRS/ITRF) world-wide. Since the existing physical heights are associated to local or regional vertical datums, their integration (or transformation) into a unified global vertical frame is a main component of this challenge. The basic approach is to refer all physical heights (or geopotential numbers) to one and the same equipotential surface. This so-called vertical

datum problem corresponds then to the determination of the geopotential differences between the selected reference surface and the reference points (normally, tide gauges) of the individual height systems; e.g. Rapp (1994). The usual reference surface for heights is the geoid and its potential value W0 establishes the corresponding reference level. Since it is not possible to define an equipotential surface by providing simultaneously both, its potential value and points belonging to it, we can interpret the geoid as the equipotential surface passing through a given point P j , which is the origin of the height system j . So, each high resolution geoid (derived from satellite data in combination with terrestrial gravity data referred to a specific height system) realizes a different equipotential surface, i.e. a local geoid, which is lying very close to sea surface ( 1 are neglected. Once the anomalous potential T P is estimated, the corresponding gravity potential values W P can be obtained through W P = U0 − γ P h P + T P .

(9)

P stands for each point describing the sea surface, which, because of the SSTop, does not coincide with the geoid and, hence, W0 remains unknown. One alternative to estimate this value, is to assume that SSTop ≈

W0 − W P , γP

(10)

and to define W0 through the minimization of the SSTop over all ocean areas (Sacerdorte and Sanso, 2001): ⭸ ⭸W0

SSTop2 dσ =

⭸ ⭸W0 WP

W0 − W P γP

2 dσ = 0,

dσ

γ P2 1 dσ γ P2

.

(11)

The geometrical representation of the level surface W0 (i.e. the geoid) with respect to the reference ellipsoid is then given by introducing equation (6) into the equation (2). In this case, δW = 0.

4 Practical Determination of W0

1 . sin(ψ/2)

(8)

r=1

i.e. W0 =

∞ R G n S (ψ)dσ + 4π n=1

121

(7)

ψ denotes the spherical distance related to the computation point. The first addend in equation (6) takes into account the difference between the geocentric

The gravity disturbances required in equation (6) are obtained by subtracting the normal gravity value generated by the GRS80 ellipsoid from the real gravity provided by a GGM at the same computation point. The anomalous potential (equation (6)) is calculated by the Least Square Collocation approach

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contained in the GRAVSOFT software (Tscherning et al., 1992), including the corresponding modification of the kernel function to take into account equation (7). The ellipsoidal coordinates of the evaluation points are taken from a MSS model with a spatial resolution of 1◦ × 1◦ . The ellipsoidal heights (i.e. sea surface) shall not show any significant temporal variations detectable in the observational satellite altimetry data in order to be, as far as possible, ‘quasi-stationary’. All those effects (tides, sea state bias, etc.) are reduced. Equations (6) and (9) are then evaluated by combining different GGMs with several MSS models: Global gravity models: EGM96 (Lemoine, 1998), TEG4 (Tapley, 2001), GGM02 (Tapley, 2005), EIGEN-CG03C (Förste, 2005), and EIGEN-GL04S (GRGS/GFZ, 2006). Mean sea surface models: GSFC00.1 (Koblinsky et al., 1999), CLS01 (Hernandez and Schaeffer, 2001b), KMS04 (Knudsen, 2004), and yearly DGFI mean sea surface models between 1993 and 2001 derived from the satellite altimetry data available at the MGDR’s Version C AVISO altimetry project (AVISO, 1996). Since both, the mean sea surface and the Earth’s gravity field vary with time, the year 2000.0 is appointed as a reference epoch, and GGM’s coefficients and MSS-heights are appropriately reduced to that epoch. Additionally, the GGM and the gravity reference field are transformed into the same tide system to which the MSS models refer, i.e. the zero tide system. Finally, equation (6) is evaluated with the satellite-only component of the GGMs (n < 150) in order to avoid the discrepancies between the individual vertical datums included in the terrestrial gravity anomalies. Since the precision of the satellite altimetry data decreases in coastal areas, the determination of the anomalous potential in these areas should be considered as a part of the regional solution of the GBVP (local geoid) (see Lehmann, 2000), and the computation points located there are not included in the evaluation of equation (11). In the same way, the seasonal variations of the sea surface should be avoided, principally those generated by the sea ice cycle and glaciations and melting effects in the Polar Regions. The uncertainties generated by these phenomena on the sea surface heights are clearly identifiable through the comparison of the available MSS models, the largest discrepancies of which (> ±2 cm) take place at high latitudes (ϕ > ±60◦ ), e.g. (Andersen, 2004, Hernandez and Schaeffer, 2001a).

L. Sánchez Table 1. W0 values derived from combining the mean sea surface model CLS01 with the global gravity models EGM96 and EIGEN-GL04S

MSS CLS01 Latitude range

W0 [m2 s−2 ] (EGM96, n = 150)

W0 [m2 s−2 ] (EIGEN-GL04S, n = 150)

82◦ N. . .78◦ S

62 636 854.39 ±0.03 σ( P=1) = ±6.41

62 636 854.38 ±0.03 σ( P=1) = ±6.49

67◦ N. . .67◦ S

62 636 853.85 ±0.03 σ( P=1) = ±6.25

62 636 853.83 ±0.03 σ( P=1) = ±6.38

60◦ N. . .60◦ S

62 636 853.16 ±0.03 σ( P=1) = ±5.52

62 636 853.11 ±0.03 σ( P=1) = ±5.69

The W0 values obtained from combining the GGMs with all mentioned MSS models under the same characteristics (identical latitudinal extension, GGM harmonic degree n, MSS model spatial resolution, etc.) are very similar; they disagree with each other by less than 0.05 m2s−2 . Therefore, we present here a few numerical examples only (Table 1). The largest differences between the W0 values are caused by the variation of the latitudinal extension of the computation area for evaluating equation (11) from ϕ = ∼ ±80◦ until ϕ = ±60◦. The standard deviation for the unit weight (σ( P=1) in Table 1), which indicates the variation of (W0 − W P ) in a quadrangle of (1◦ × 1◦ ) at the equatorial region (i.e. ϕ = 0◦ ), shows that the lower the latitudinal extension the more reliable the realization of the reference level W0 . Figure 1 presents the SSTop after evaluating equation (10). The largest values (less than −2 m and more than +2 m) are located in those regions where the satellite altimetry has still problems with the measurement accuracy (island and coastal areas, and northern and southern of ϕ = ±60◦), and therefore, where the MSS models strongly disagree, e.g. (Andersen, 2004, Hernandez and Schaeffer, 2001a). In order to avoid these uncertainties, it would be advisable to limit the latitudinal coverage of the computation area by a middle latitude; for instance by ϕ = ±60◦ .

5 Other W0 Computations Table 2 summarizes some previous computations of W0 in a global context. The first two values follow the traditional approach of determining a best fitting ellipsoid and its potential U0 as a function of the geocentric gravitational constant (GM), the Earth’s

Approach for the Establishment of a Global Vertical Reference Level –180 90

–150

–120

–90

–60

–30

0

123 30

60

90

120

150

180 90

60

60

30

30

0

0

–30

-30

–60

-60

–90 –180

-90 –150

–120

–90

–60 < –2

–30 –2 to –1

0 –1 to 0

30 0 to 1

60 1 to 2

90

120

150

180

> 2 [m]

Fig. 1. Geographical distribution of the residuals [W0 − W ] on marine areas. The geopotential differences are divided by the normal gravity to be represented in metrical units.

Table 2. Previous computations of global W0 values

Id

W0 Value [m2 s−2 ]

1 2 3 4 5

U0 = 62 636 860.850 (GRS80) Moritz (1980) W0 = 62 636 856.88 Rapp (1995) W0 = 62 636 857.5 Nesvorny and Sima (1994) W0 = 62 636 856.0 e.g. Bursa (2002) W0 = 62 636 853.4 Sánchez (2007)

angular velocity (ω), the semi-major axis (a), and the dynamical factor ( J2 ). Then, W0 is defined identical to U0 . Values 3 and 4 are derived by applying the condition (13) (W − W0 )2 dSO = min SO

on marine areas. The geopotential values W are calculated through W =

∞ n GM a n 1+ (Cnm cos mλ r r n=1 m=0 (14) +Snm sin mλ) Pnm (cos θ ) 1 + ω2r 2 cos2 (90◦ − θ ) 2

by introducing a GGM and the sea surface heights obtained from averaging the satellite altimetry observations at the crossover points between ascending and descending passes of some missions (e.g. T/P, ERS1, etc.). Since these intersection points have a

non-regular distribution, different weighting functions are introduced to compute the corresponding mean W0 value, e.g. Bursa (1999). Value 5 is also determined by evaluating equations (13) and (14), with the difference that proper mean sea surface models (homogeneously distributed data) are applied and weighting functions are not required. The discrepancy of about 3 m2 s−2 between values 4 and 5 can be due to the different analysis of the satellite altimetry data. Value 4 is presently included in the numerical standards of the International Earth Rotation and Reference Systems Service (IERS) (McCarthy and Petit, 2004). The differences between the values presented in Tables 1 and 2 indicate that the uniqueness, reliability, and repeatability of a global reference level defined by a fixed W0 value can only be guaranteed by specific conventions, such as a mean sea surface model and its spatial resolution, a global gravity model and its spectral resolution, a reference epoch, a tide system, etc. These conventions shall be defined and officially recommended by the International Association of Geodesy (IAG) in order to be adopted by all ‘W0 users’. Otherwise, it will continue existing as many vertical reference systems as W0 values.

Closing Remarks The unification of classical vertical datums within a world height system is feasible through the estimation of the potential differences between their local reference levels and the global reference surface. In the last three decades, many proposals about the reliable determination of these differences has been

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presented, but in general, all of them point out that the global (unified) reference level can arbitrarily be selected, and as a consequence, it becomes a secondary role in the establishment of a global vertical reference system. Keeping in mind that the definition of a global height system in terms of potential quantities (W0 as a reference level and geopotential numbers as coordinates) is more precise (indeed, univocal) than in terms of heights (geoid or quasigeoid and the corresponding physical heights), it is convenient that the selected W0 value agrees with the numerical results obtained from analysing the modern geodetic observations. Moreover, the determination of this W0 value must appropriately be documented and completely reproducible. Today it is still very inaccurate to derive geopotential numbers from satellite positioning in combination with global gravity models; however, the new satellite geodetic techniques and the permanent improvement of their accuracies and analysis open this possibility in the near future. Therefore, if a new global reference level should be introduced, it has to be able to support expeditiously and precisely the determination of physical heights, not only from spirit levelling, but also from satellite positioning and precise global gravity models.

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). AVISO user handbook. Merged Topex/Poseidon products (GDR-MS). CLS/CNES, AVINT-02-101-CN. 3rd Ed., July. Bursa, M., J. Kouba, M. Kumar, A. Müller, 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. Förste, C., F. Flechtner, R. Schmidt, U. Meyer, R. Stubenvoll, F. Barthelmes, R. König, K.H. Neumayer, M. Rothacher, Ch. Reigber, R. Biancale, S. Bruinsma, 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. GRGS/GFZ (2006). Global gravity model EIGEN-GL04S. Available at http://bgi.cnes.fr:8110/geoid-variations/ SH models /EIGEN GL04S.

L. Sánchez Heck, B., R. Rummel (1990). Strategies for solving the vertical datum problem using terrestrial and satellite geodetic data. IAG Symposia 104: 116–128, Springer. Hernandez, F., Ph. Schaeffer (2001a). The CLS01 mean sea surface: a validation with the GFSC00.1 surface. www.cls.fr/html/oceano/projects/mss/cls 01 en.html. Hernandez, F., Ph. Schaeffer (2001b). MSS CLS01 http:// www.cls.fr/html/oceano/projects/mss/cls 01 en.html. Hofmann-Wellenhof, B., H. Moritz (2005). Physical Geodesy. Springer Verlag, Wien. Koblinsky et al. (1999). NASA ocean altimeter pathfinder project, report 1: data processing handbook, NASA/TM 1998-208605, April. Koch, K.R. (1971). Die geodätische Randwertaufgabe bei bekannter Erdoberfläche. Zeitschrift für Vermessungswesen. 96: 218–224. Knudsen, P., A.L. Vest, O. Anderssen (2004). Evaluating mean dynamic topography models within the GOCINA project. In: Proceedings of the 2004 Envisat & ERS Symposium, 6–10 September, 2004. Salzburg, Austria. (ESA SP-572, April 2005). Published and distributed by: ESA Publications Division. Lehmann, R. (2000). Altimetry-gravimetry problems with free vertical datum. Journal of Geodesy 74: 327–334. Springer. 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. Mather, R.S. (1978). The role of the geoid in four-dimensional geodesy. Marine Geodesy, 1: 217–252. McCarthy, D.D., G. Petit (Eds.) (2004). IERS technical note No. 32: IERS Conventions (2003). Verlag des Bundesamts für Kartographie und Geodäsie. Frankfurt am Main. Moritz, H. (1980). Geodetic Reference System 1980. Bulletin Geodesique. 54(3): 295–405. Nesvorny, D., Z. Sima (1994). Refinement of the geopotential scale factor R0 on the satellite altimetry basis. Earth, Moon and Planets, 65: 79–88. Rapp, R. (1994). Separation between reference surfaces of selected vertical datums. Bull. Géod. 69: 23–31. Rapp. R. (1995). Equatorial radius estimates from Topex altimeter data. Publication dedicated to Erwin Groten on the ocassion of his 60th anniversary. Publication of the Institite of Geodesy and Navigation (IfEN), University FAF Munich. pp. 90–97. Rummel, R., P. Teunissen. (1988). Height datum definiton, height datum connection and the role of the geodetic boundary value problem. Bull. Géod. 62: 477–498. Sacerdorte, F., F. Sanso (2001). W0: A story of the height datum problem. In: Wissenschaftliche Arbeiten der Fachrichtung Vermessungswesen der Universität Hannover. Nr. 241: 49–56. Sacerdorte, F., F. Sanso (2004). Geodetic boundary-value problems and the height datum problem. IAG Symposia 127: 174–178. Springer. Sánchez, L (2007). Definition and realization of the SIRGAS vertical reference system within a globally unified height

Approach for the Establishment of a Global Vertical Reference Level system. IAG Symposia, 130: 638–645, Springer Verlag, Berlin, Heidelberg. Sanso, F. S. Usai (1995). Height datum and local geodetic datums in the theory of geodetic boundary problem. Allgemeine Vermessungsnachrichten (AVN). 102 Jg. Heft 8–9: 343–355. Sanso, F., G. Venuti (2002). The height/geodetic datum problem. Geophys. J. Int. 149: 768–775. Tapley, B., M. Kim, S. Poole, M. Cheng, D. Chambers, J. Ries (2001). The TEG-4 Gravity field model. AGU Fall 2001. Abstract G51A-0236.

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Tapley, B., 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. Journal of Geodesy, 79: 467–478. Tscherning C., R. Forsberg, P. Knudsen (1992). Description of the GRAVSOFT package for geoid determination. In: Holota, P., M. Vermeer (Eds.) Proceedings of the 1st continental workshop on the geoid in Europe. Research Institute of Geodesy, Topography and Cartography. Prague: 327–334.

The Research Challenges of IAG Commission 4 “Positioning & Applications” C. Rizos School of Surveying and Spatial Information Systems, University of New South Wales, Sydney, NSW 2052, Australia

Abstract. The terms of reference of IAG Commission 4 “Positioning & Applications” give some hint as to the range of technologies and applications with which its entities are concerned. The TOR are conveniently summarised in the statement: To promote research into the development of a number of geodetic tools that have practical applications to engineering and mapping. Recognising the central role that Global Navigation Satellite Systems (GNSS) plays in many of these applications, the Commission’s work focuses on several Global Positioning System (GPS)-based techniques. These include precise positioning, but extending beyond the applications of reference frame densification and geodynamics, to address the demands of precise, real-time positioning of moving platforms. Several sub-commissions deal with precise kinematic GPS positioning technology itself (alone or in combination with other positioning technologies such as inertial navigation sensors), as well as its applications in surveying and engineering. For example, there are sub-commissions that interest themselves in the R&D of the technology to address ground or structural deformation, mobile mapping or imaging (from land or airborne platforms). Recognising the role of networks of permanently operating GPS reference stations, research into non-positioning applications of such geodetic infrastructure is also being pursued, such as atmospheric (ionosphere and troposphere) sounding. Finally, geodetic applications of interferometric synthetic aperture radar (InSAR) are also a topic of study for one sub-commission. The research challenges for the sub-commissions, working groups and study groups of Commission 4 will be described.

1 Introduction IAG Commission 4 “Positioning & Applications” is organised around five sub-commissions: • SC4.1 Multi-sensor D. Brzezinska) 126

Systems

(chair:

• SC4.2 Applications of Geodesy in Engineering (chair: H. Kahmen) • SC4.3 GNSS Measurement of the Atmosphere (co-chairs: S. Skone, H. van der Marel) • SC4.4 Applications of Satellite & Airborne Imaging Systems (chair: X. Ding) • SC4.5 Next Generation RTK (chair: Y. Gao) Each sub-commission has one or more working group. There are also four study groups: • SG4.1 Pseudolite Applications in Positioning & Navigation (chair: J. Wang) • SG4.2 Statistics & Geometry in Mixed Integer Linear Models, with Applications to GPS & InSAR (joint with ICCT, chair: A. Dermanis) • SG1.1 Ionospheric Modelling and Analysis (joint with Commission 1 & COSPAR, chair: M. Schmidt) • SG1.2 Use of GNSS for Reference Frames (joint with Commission 1 & IGS, chair: R. Weber) The Steering Committee comprises: President (C. Rizos), Vice-President (P. Willis), the Chairs of the five sub-commissions, and two Membersat-Large (R. Neilan, M. Santos). The commission web site URL is: http://www.gmat.unsw.edu.au/iag/ iag comm4.htm. What’s so special about Commission 4? • Very “applied” – not one of the “Three Pillars of Geodesy”. • Close relations with IAG sister organisations – e.g. FIG, ISPRS, etc. • Many activities concerned with high precision GNSS “navigation”. • Close links to GNSS communities and national/international Institute of Navigation type organisations. Each of the sub-entities of Commission 4 (subcommissions, working groups and study groups) were contacted and asked to provide information

The Research Challenges of IAG Commission 4 “Positioning & Applications”

on “research challenges”. The following sections present this information.

2 Sub-Commission 4.1 “Multi-Sensor Systems” SC4.1 consists of three working groups: • WG4.1.1 Advances in Inertial Navigation & Error Modelling Algorithms • WG4.1.2 Indoor & Pedestrian Navigation • WG4.1.3 Advances in MEMS Technology & Applications The web site is at http://www.ceegs.ohiostate.edu/IAG-SC41/. SC4.1 recognises that there is a paradigm shift from static positioning and orientation applications to kinematic ones, and from point coordination to image capture. The theoretical and technical challenges therefore are: • Automated data fusion: ◦ sensor-level fusion (rather settled) ◦ feature-level fusion ◦ decision-making-level fusion • Real-time operations while maintaining the positioning/navigation accuracy • Exploration of new/available “signals of opportunity”, and their integration with GNSS/Inertial Measurement Unit (IMU) technology • Seamless navigation algorithms for indoor/ outdoor navigation • Improvement of the MEMS (Micro-Electro Mechanical Systems) gyro technology – facilitating MEMS use in land-based or airborne mapping systems • Autonomous navigation • New algorithmic approaches: ◦ Improved algorithmic approaches to integration (alternatives to Extended Kalman Filter) ◦ Learning systems supporting traditional Kalman filtering ◦ Signal processing and advanced error modelling In the case of WG4.1.1 the challenges are: • Better stochastic models for inertial sensor errors. • Assessment of alternative filters to the Extended Kalman Filter (KF), such as Unscented KF and Particle filter. • Other approaches for INS/GPS integration and error compensation, such as Neural Networks and Fuzzy Logic.

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• Efficient & continuous evaluation of new inertial sensor technologies coming to the market (especially MEMS). • Real-time navigation with high accuracy. • Advanced signal processing techniques for dealing with large sensor noise, such as wavelets. In the case of WG4.1.2 the challenges are: • R&D in multi-sensor integrated systems for indoor and pedestrian navigation. • Selection of appropriate sensor data, & integrate data when and where needed. • Integration of medium to low-accuracy MEMS, and other positioning sensors. • Investigation & testing of emerging indoor location techniques, such as WiFi, UWB and RFID. • Seamless positioning between indoor and outdoor areas. The research “drivers” for SC4.1 and its subentities can be summarised as: • Data fusion methodologies & algorithms. • Real-time operations – reliability/QC, automation, sensor calibration. • New/improved navigation sensor technologies – MEMS IMUs, terrestrial ranging systems. • Improved: ◦ ◦ ◦ ◦

observation modelling sensor error characterisation signal processing parameter estimation

• New applications – pedestrian/indoor navigation/positioning. • Interface to the Int. Society for Photogrammetry & Remote Sensing (ISPRS) for photogrammetric/image processing.

3 Sub-Commission 4.2 “Applications of Geodesy in Engineering” SC4.2 consists of four working groups: • WG4.2.1 Measurement Systems for the Navigation of Construction Processes • WG4.2.2 Dynamic Monitoring of Buildings • WG4.2.3 Application of Knowledge-based Systems in Engineering Geodesy • WG4.2.4 Monitoring of Landslides & System Analysis The web site is at http://info.tuwien.ac.at/ingeo/ sc4/sc42.html. SC4.2 focuses on two main topics:

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Optical 3D Measurement Techniques, and Geodesy for Geotechnical and Structural Engineering. Conferences with these two themes are run in conjunction with ISPRS and FIG respectively. The research areas of interest are: • New measurement techniques and calibration devices • Laserscanning and handling of point clouds • Theodolite-based measurement techniques • GPS and pseudolite applications in engineering • Close range imaging and metrology • Target recognition and tracking • Image sequence analysis • Knowledge-based systems for sensor fusion & data analysis • Machine guidance and navigation of construction processes • Geodesy on large construction sites, such as monitoring and setting out of dams, tunnels, bridges • Geodesy for open pit, gas and oil industries • Monitoring of local dynamic processes, such as landslides, mudflows, etc. • Mobile mapping systems (overlap with SC4.1) • Visualization, animation and VR techniques In the case of WG4.2.1 the goals are: • To promote research, and stimulate new ideas and innovation for integrating geodetic measuring systems and concepts, into the navigation and control of construction processes, including: ◦ Better understanding of geometrical requirements of construction processes. ◦ Development of adequate sensor systems. ◦ Development of algorithms for real-time applications. ◦ Interaction between geometrical information & the navigation/steering process. ◦ Definition of interfaces. In the case of WG4.2.2 the challenges are: • Moving from “Initial Performance Assessment” to true “Structural Health Monitoring”. • In-situ extensometers fail after 10 years, mainly due to epoxy bond failure, alternatives? • Emerging MEMS-based laser scanners and interferometers may aid in long-term monitoring. • Issues are crack propagation and deflection in multiple scenarios, such as earthquake, inundation of foundations, high winds, snow/ice loading, etc.

C. Rizos

In the case of WG4.2.4 the challenges are: • Development of multi-sensor systems (geodetic, geophysical, hydrological, etc.). • Development of systems for monitoring processes leading to landslides. • Investigation and interpretation of landslide monitoring data. • Development mathematical models for the investigation of the interaction between the different processes. • Development of early alert systems. The research “drivers” for SC4.2 and its subentities can be summarised as: • Data fusion methodologies & algorithms for increasingly integrated systems. • Real-time operations – reliability/QC, automation, sensor calibration, alert/alarm. • New/improved measurement technologies – laser scanners, optical systems/Robotic Total Stations, terrestrial ranging systems, etc. • Improved: ◦ ◦ ◦ ◦

observation modelling sensor calibration signal processing parameter estimation/knowledge-systems

• New applications in “engineering geodesy”. • Interface to FIG & ISPRS for “applications” research.

4 Sub-Commission 4.3 “GNSS Measurement of the Atmosphere” SC4.3 consists of three working groups: • WG4.3.1 Ionospheric Scintillation • WG4.3.2 Performance Evaluation of Ionosphere Tomography Model • WG4.3.3 Numerical Weather Predictions for Positioning The web site is at http://www.gmat.unsw.edu.au /iag/iag sc43.htm. The Terms of Reference of SC4.3 define the range of interests of its sub-entities. Over the past decade, significant advances in GPS technology have enabled the use of GPS as an atmospheric remote sensing tool. With the growing global infrastructure of GPS reference stations, the capability exists to derive high-resolution estimates of total electron content and precipitable water


vapour in near real-time. Recent advances in tomographic modelling and the availability of spaceborne GPS observations has also allowed 3-D profiling of electron density and atmospheric refractivity. Future plans for the Galileo system will allow further opportunities for exploiting GNSS as an atmospheric remote sensing tool. The focus of this SubCommission is to facilitate collaboration and communication, and support joint research efforts, for GNSS measurement of the atmosphere. In the case of WG4.3.1 the challenges are:

• Improved:

• Assessing Phase Lock Loop tracking errors associated with scintillation, and their impact on precise positioning. • Impact of scintillation on Galileo and new GPS signals. • Mitigating impact of scintillation through development of robust receiver tracking technologies. • Developing new software receiver capabilities for such research (simulations and real data analysis).

5 Sub-Commission 4.4 “Applications of Satellite & Airborne Imaging Systems”

In the case of WG4.3.2 the challenges are: • Development of a global assimilative ionosphere model (GAIM) approach, in which GPS ground and spaceborne observations, ionosonde measurements, and others are included in 3-D electron density model. • Resolving computational and real-time processing limitations associated with the GAIM approach. • Defining and meeting reliability and accuracy requirements for precise positioning applications. In the case of WG4.3.3 the challenges are: • Define positioning/navigation user needs for NWP products, and assess current capabilities to meet accuracy requirements. • Link meteorological and geodetic communities with standardised terminology and procedures. • Develop criteria for assessing quality of troposphere products derived from NWP for positioning applications. • Testing and validating ray-tracing methods applied to NWP data. The research “drivers” for SC4.3 and its subentities can be summarised as: • Understanding the troposphere and ionosphere. • Real-time operations – reliability/QC, automation, sensor calibration, etc. • Assimilation of GPS-derived results into NWP & other meteorological applications.

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◦ observation modelling ◦ receiver-satellite geometries – ground & satellite ◦ calibration/comparison with traditional met sensors ◦ parameter estimation efficiencies • Applications in “improved” GNSS positioning. • Interface to atmospheric science communities & meteorological offices.

SC4.4 consists of four working groups: • WG4.4.1 Permanent Scatterer / Corner Reflector / Transponder InSAR • WG4.4.2 Atmospheric Effects in InSAR / InSAR Meteorology • WG4.4.3 InSAR in Polar Regions • WG4.4.4 Imaging Systems for Ground Subsidence Monitoring The web site is at http://www.gmat.unsw.edu.au /iag/iag sc44.htm. The focus has been on satellite Interferometric Synthetic Aperture Radar (InSAR), and theoretical and technical challenges therefore are: • Future sensor systems that overcome problems/difficulties of existing systems • New and more effective methods for geodetic observations based on satellite & airborne imaging systems • Better understanding and modelling of atmospheric effects on satellite & airborne imaging systems • Theories that better bridge the gap between geodetic observations & geophysical phenomena of the Earth In the case of WG4.4.1 the challenges take Advantage of what the technology now has to offer, but address the Limitations: • Advantages: ◦ Suitability for wide area monitoring ◦ Capability of providing deformation data relative to single targets ◦ Millimetric precision ◦ Possibility to investigate past phenomena (e.g. exploiting the ERS, JERS archives)

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C. Rizos

• Limitations: ◦ Need for Permanent Scatterer targets ◦ Phase ambiguity (slow deformation only) ◦ One-dimensional displacement only In the case of WG4.4.2 the following drives the research agenda: • Atmospheric effects can be misleading in InSAR results. • Atmospheric effects can be corrected using: ◦ CORS data ◦ MODIS ◦ etc • InSAR-derived tropospheric delay can be used for weather forecasting. As far as WG4.3.3 is concerned the focus will be on: • Collaboration with other researchers during International Polar Year 2007-2008. • Development applications for the Japan satellite ALOS’s PALSAR data. For WG4.3.4 the research challenges relate to the Large Displacement case and the Small Displacement case: • Large displacements within a small area >> very dense fringes (e.g. mine subsidence). • Very small displacements over a large area >> need large temporal separation >> temporal decorrelation (e.g. urban subsidence due to underground water extraction). The research “drivers” for SC4.4 and its subentities can be summarised as: • New SAR missions. . .new frequencies, signal polarisations. • New “remote sensing” geodetic tool. . .integrate with other technologies. • Improved: ◦ ◦ ◦ ◦

observation modelling/phase unwrapping tropospheric bias mitigation automation algorithm efficiencies

• Many geodetic (& other) applications. • Interface to ISPRS for “applications” research, IEEE for basic RF research.

6 Sub-Commission 4.5 “Next Generation RTK” SC4.5 consists of four working groups: • WG4.5.1 Network RTK • WG4.5.2 Carrier Phase Based Precise Point Positioning • WG4.5.3 High Precision Positioning of Buoys & Moving Platforms • WG4.5.4 Multiple Carrier Phase Ambiguity Methods & Applications The web site is at http://www.ucalgary.ca /∼ygao/iag.htm. The theoretical and technical challenges are: • Integration of Precise Point Positioning (PPP) and network RTK • Fast ambiguity convergence methods for PPP: parameterisation, decorrelation, bias modelling, processing algorithm • New methods for ambiguity resolution with undifferenced carrier phase observations: initial phase fractional offset problem, ambiguity search method, ambiguity validation method • New methods for carrier phase ambiguity resolution with multiple GNSS signals: error mitigation/modelling, observation combination, processing algorithm • Impact of modernized GPS, Glonass and Galileo on RTK • Improved quality control methods in RTK In the case of WG4.5.2 the challenges are: • Reduce initialisation periods from tens of minutes to minutes, by means of improved processing algorithms and / or incorporating external data. • Develop a mechanism (procedures and protocol) to provide real-time precise GPS satellite ephemerides and clock offset estimates to users via the Internet. In the case of WG4.5.3 the challenges are: • Precise positioning algorithms on the moving platforms. • Multipath effects on water surfaces. • Data fusion of GNSS and other ocean environment sensors. In the case of WG4.5.4 the challenges are: • Standardised geometry-free and geometry dependent approaches to Multiple Carrier Ambiguity Resolutions.


• Developing Local, Regional and Global RTK services based on three or more carrier GNSS signals. • Improved undifferenced carrier phase ambiguity estimation from multiple carriers by means of multiple reference stations. • Precise ionospheric modelling using ambiguityfixed phase measurements from multiple reference stations. • Exploring performance potentials of three carrier signals for scientific applications, including GNSS orbit determination and atmospheric sounding (e.g. slant path delay estimation), etc. The research “drivers” for SC4.5 and its subentities can be summarised as: • Precise Point Positioning. • Real-time operations – reliability/QC, minimise constraints, etc. • New GNSS signals/frequencies. • Improved: ◦ observation/stochastic modelling ◦ measurement quality metrics/indicators ◦ parameter estimation/AR • New applications in “precise navigation” • Interface to FIG & ISPRS for “applications” research.

7 Commission 4 Study Groups 7.1 Study Group 1.2 “Use of GNSS for Reference Frames” This study group is jointly convenved with the IAG’s Commission 1 “Reference Frames”, and has the The work program of the study group is: • Investigate the ties and their time evolution between GNSS Broadcast Frames like WGS84, PZ-90 and the upcoming Galileo Reference Frame, and the ITRF. • Examine deficiencies in the stability of the global GNSS station network, especially focussing on stations contributing to the upcoming ITRF2005 catalogue. • Study the ties of regional and local frames realised by an increasing number of active real-time GNSS networks. • Explore the challenges of processing hybrid GNSS data during the Galileo – IOV Phase. Website: http://mars.hg.tuwien.ac.at/Research/Satellite Techniques/GNSS WG IGS/gnss wg igs.html.

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7.2 Study Group 4.1 “Pseudolite Applications in Positioning & Navigation” The research challenges of SG4.1 are: • Developing tropospheric error models for pseudolite (PL) measurements. • Optimal design of PL locations: strengthening the positioning geometry (especially for pseudoliteonly positioning scenarios) & minimising impact of residual errors on positioning results. • Modelling PL signal penetrating delays in indoor positioning scenarios. • Improve understanding of PL signal dynamics under different positioning environments. • Integrating PLs with INS and/or imaging/navigation sensors. • Novel uses of PLs (e.g. navigating spacecraft, on Mars/Moon). Website: http://www.gmat.unsw.edu.au/pseudolite/. 7.3 Study Group 4.2 “Statistics & Geometry in Mixed Linear Models, with Applications to GPS & InSAR” The research challenges of SG4.2 are: • Quality assessment problem: ◦ New formulations and results departing from simple unrealistic models, considering observational errors with non-zero mean and correlations. ◦ Study of the effect of model biases on integer ambiguity resolution and ultimately on coordinate accuracy. • Computational efficiency: ◦ More efficient algorithms for real-time applications, possibly exploiting new techniques beyond the conditional search for the optimal integer ambiguity vector. • Observational design problem: ◦ Consideration of optimal observational configurations taking into account not only DOP measures reflecting formal non-realistic models, but also robustness with respect to model biases and consequent unsuccessful integer ambiguity resolution. Website: http://der.topo.auth.gr/icct/WGS/7-Dermanis.htm.

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8 Summary Remarks Espite the great variety of technologies and research agends of the Commission 4 sub-entities, there is remarkable commonality of the “research drivers” for the sub-commissions, working groups and study groups. However, one can summarise the research challenges and opportunities in the following list: • Expanded GNSS: GPS, Glonass & Galileo. . .

C. Rizos

• • • • •

New navigation sensor technologies. . . New (deformation) measurement technologies. . . Increasing number of SAR missions. . . Drive to real-time processing/operations. . . Improvements in observation modelling & parameter estimation. . . • New applications in “precise navigation”, “geodetic remote sensing”, “engineering geodesy”. . . • Linkages with FIG, ION, IEEE & ISPRS. . .

Integrated Adjustment of LEO and GPS in Precision Orbit Determination J.H. Geng, C. Shi, Q.L. Zhao, M.R. Ge, J.N. Liu GNSS Research Center, Wuhan University, Wuhan, Hubei, P.R. China

Abstract. At present, usually only the ground stations are used for the precise orbit determination (POD) of GPS satellites. The precise orbits of Low Earth Orbiters (LEO) are determined with GPS orbits and clocks fixed. The integrated adjustment of LEO and GPS for POD that is first reported in (Zhu et al., 2004), which means the orbits of GPS satellites and LEOs are determined simultaneously, is introduced here. At first, the contribution of LEOs to GPS POD is illuminated. On the one hand, the data from 43, 32 and 21 ground stations besides three LEOs are used. In the case of 21 ground stations plus three LEOs, about 5.0 cm accuracy from IGS final orbit can be achieved with only one day’s data. On the other hand, the data from one, two and three LEOs besides 43 stations are employed. It is shown that the improvement can reach 31% when only one LEO is added to 43 ground stations. These achievements are mainly attributed to the strengthened geometry after LEOs are included. Then, the improvement of LEO orbits in the integrated adjustment is presented. When 43 ground stations are used, the improvement can get to 17% and 32% without accelerometer data for CHAMP and GRACE, respectively. So it is demonstrated that the LEO orbit accuracy is improved simultaneously with the GPS orbits in the integrated adjustment. Keywords. Integrated adjustment, precision orbit determination, GPS, LEO

1 Introduction Over the past decade, Low Earth Orbiter (LEO) missions, such as CHAllenging Minisatellite Payload (CHAMP) and GRAvity Climate Experiment (GRACE), aimed at various Earth science experiments like static gravity recovery and temporal variations of gravity field. For the goals, precise LEO orbit is necessary. Among all the space-borne data, the most important one for precise orbit determination (POD) is the measurement from the space-borne GPS receiver.

The approaches for POD include dynamic, kinematic and reduced-dynamic methods. Dynamic method is the only choice for GPS satellites, while kinematic and reduced-dynamic methods are only applied to LEOs (Byun 2003; Rim et al., 1996; Yunck et al., 1990). In this paper, dynamic method is employed. At present, the so-called ‘two-step’ approach is widely adopted for the restitution of LEO orbits from GPS data. In the first step, the precise orbits and clock corrections of GPS satellites are adjusted with the GPS data of globally distributed fiducial ground stations. Then in the second step, LEOs’ orbits are estimated using space-borne GPS data with the GPS orbits and clocks fixed (Moore et al., 2003; Kang et al., 2003; Lee et al., 2004). An alternative POD approach is called the ‘onestep’ method, where the orbits of LEOs and GPS satellites are estimated in one simultaneous least squares adjustment by processing all ground and space-borne data together (Zhu et al., 2004; König et al., 2005). It has been demonstrated that the orbit accuracies of LEOs and GPS satellites are improved at the same time. In this paper, the authors investigate the impact of the integrated geometry of GPS stations and LEOs on GPS and LEO orbits. After a short description of the mathematical formulae of one-step method, the software and data processing strategy are introduced briefly. Then the dependency of the orbit improvement of GPS satellites and LEOs on the network geometry is presented and discussed in details before the conclusions are drawn.

2 One-Step Precision Orbit Determination Given the motion equations of the GPS satellites and the LEOs, we can get the reference orbits by numerical integration. Meanwhile, we can also obtain the transition matrices ⌿G (ti , t0 ), ⌿ E (ti , t0 ) and the sensitivity matrices SG (ti , t0 ), SE (ti , t0 ) after integrating 133

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the variation equations (Montenbruck and Gill, 2000). They satisfy the state equation of the form x Gi = ⌿G (ti , t0 ) x G0 + SG (ti , t0 ) f G (1) x Ei = ⌿ E (ti , t0 ) x E0 + S E (ti , t0 ) f E where x Gi and x Ei are the state vectors of GPS satellites and LEOs, while x G0 and x E0 are the initial epoch states, respectively. f G and f E are the force model parameters for them. Now assume the observation equations of ground stations and LEOs in the two-step method z Gi = G(x Gi , pGi , si , yG , ti ) + vGi

(2)

z Ei = E(x Ei , p Ei , y E , ti ) + v Ei

(3)

where vGi , v Ei ∼ N(0, σ 2 I ). pGi and p Ei denote the time-dependent parameters related to ground stations or LEOs such as receiver clock error while si are time-dependent ones related to GPS satellites such as GPS satellite clock error. yG and y E represent the time-independent parameters like carrier phase ambiguities. In the two-step method, first x Gi is obtained from (2) and then x Ei is computed through (3) with x Gi fixed. However, in the one-step method, the GPS orbit is not fixed, but estimated together with the LEO orbit. Then (3) is modified and takes the form z Ei = E(x Gi , x Ei , p Ei , si , y E , ti ) + v Ei

(4)

Linearize both (2) and (4) at the reference orbits and apply (1), then yield ⎧ ⎪ δz = ⭸x⭸GGi ⌿G (ti , t0 )δx G0 + ⭸x⭸GGi SG (ti , t0 )δ f G ⎪ ⎪ Gi ⎪ ⎪ ⎪ ⭸G ⭸G ⎪ + ⭸G ⎪ ⭸si δsi + ⭸ pGi δpGi + ⭸yG δyG + vGi ⎪ ⎨ δz Ei = ⭸x⭸EGi ⌿G (ti , t0 )δx G0 + ⭸x⭸EGi SG (ti , t0 )δ f G ⎪ ⎪ ⎪ ⎪ + ⭸x⭸EEi ⌿ E (ti , t0 )δx E0 + ⭸x⭸EEi S E (ti , t0 )δ f E ⎪ ⎪ ⎪ ⎪ ⎪ ⭸E ⭸E ⎩ + ⭸E ⭸si δsi + ⭸ p Ei δp Ei + ⭸y E δy E + v Ei (5) where sign δ means the parameter is the correction of its approximate value. δz Gi and δz Ei are observation minus computation (OMC). So the one-step method is actually a least squares estimation for the initial epoch states and the force model parameters of the GPS satellites and LEOs with (5).

3 Data Processing 3.1 Software All the results are computed with PANDA (Positioning And Navigation Data Analyst) that is developed

at the GNSS Research Center of Wuhan University. It is a wide-oriented tool for analyzing data from satellite positioning and navigation systems and is now able to serve as a powerful platform for the scientific study in the related areas and as well to fulfill the urged requirement of a number of ongoing and planned satellites applications projects in China (Liu and Ge, 2003 and Liu et al., 2004). 3.2 Data The GPS data at 43 globally distributed ground stations and three LEOs (CHAMP, GRACE A and GRACE B) from day 154 to 159 in 2002 are chosen for the investment. The star camera data of GRACE satellites are used for attitude computation. It should be noted that the result of GRACE B on day 157 is not computed correctly since there is a possible orbit maneuver on that day which causes orbit abnormity (GFZ, CSR, DLR, NASA, 2002). So it is not used in the following experiments. 3.3 Strategy In order to study the impact of the geometry of ground stations and LEOs on GPS and LEO orbits, integrated estimation of the following six networks are analyzed, 43S, 43S+3L, 43S+3L, 21S+3L, 43S+1L and 43S+2L, where S means ground stations and L means LEO. LEO orbits are also estimated with the two-step approach for comparison with the result of the one-step approach. The arc length used for orbit recovery is 24 hours. No ambiguity fixing is applied. The estimated parameters are listed in Table 1 and are kept the same for all the network configurations.

Table 1. Parameter estimation strategy in orbit recovery. ZTD means zenith tropospheric delay

Parameter

A Priori σ

GPS satellite initial position GPS satellite initial velocity LEO initial position LEO initial velocity Fiducial ground station location Satellite clock: white noise Receiver clock: white noise ZTD: random walk Polar motion: x, y UT1 rate Solar radiation pressure for GPS Atmospheric drag for LEO Empirical parameter for LEO Ambiguity

10 m 0.1 m/s 1000 m 2 m/s IGS weekly estimation 30000 m 9000 m 0.2 + 0.02 m/sqrt(hour) 3 arcsec 0.02 sec/day 10%, ROCK4+BERN every 6 h, DTM94 every 1.5 h, 1 cpr 10000 cycle

4.1 GPS POD Table 2 presents the total RMS with respect to the IGS final products that are combined from multiday orbits to improve the orbit quality to a great extent. Thus we choose them as an external reference. In Case 1, only 43 IGS ground stations are utilized for GPS POD. The RMS are in the range of 7–9 cm that is much larger than 5 cm accuracy of the IGS final orbit (IGS homepage, 2005). The averaged RMS over the six days listed in the last column in Table 1, shows clearly the contribution of LEOs to the POD of GPS satellites. 4.1.1 Impact of Ground Stations and LEOs In Case 2, three LEOs are added to the ground network of Case 1. From Table 2, the RMS of the GPS orbits decrease by 20%–52% and the improvement is 40% on average. Moreover, in Case 3 and 4, where the number of fiducial stations is reduced from 43 in Case 1 to 32 and 21 respectively, the orbits are slightly worse than that of Case 4, but still 39% and 34% improvement compared with that of Case 1, respectively. All of these are clearly illustrated with Figure 1. From the orbit comparison of the above four cases, GPS orbits are improved considerably by utilizing LEOs, especially. With the augmentation of LEOs, similar orbit accuracy can be obtained from a dense or sparse ground network. The improvement of GPS orbits that is gained from the space-borne observations of three LEOs is mainly because of that LEOs observe the GPS satellites in a different geometry layer from the ground stations, which makes the geometry of tracking network for GPS satellites strengthened to a great extent. Furthermore, the force model for the LEOs constrains the orbits of GPS satellites indirectly. Moreover, the space-borne GPS data of the LEOs are not affected by the troposphere, so the zenith tropospheric delays that exhibit high correlations with the

135 8 6 4 2 0

43S

43S + 3L

32S + 3L

21S + 3L

Fig. 1. Improvement of GPS orbit quality by including the observations from three LEOs (L) in addition to observations from different numbers of ground stations (S).

radial component of GPS orbit does not need to be estimated (Zhu et al., 2004). Finally, the multi-path effect in the observations at LEOs is rather weak, so the low-elevation observations of the LEOs have bigger weight than the counterpart of the ground stations, which strengthens the geometry further. Figure 2 shows the improvement of GPS orbit quality by including different numbers of LEOs. In Case 5 where only GRACE A is added, the improvement is about 31% compared with Case 1. In Case 6, where both CHAMP and GRACE A are used, the mean RMS is decreased further and only 1 mm larger than that of Case 2 where three LEOs are included. That means the three LEOs contribute differently to GPS POD. CHAMP is in a different orbit and has no relation with GRACE. But the twin GRACE satellites are in the same orbit and depart not far away from each other (220 km), resulting in the strong correlation between their observations. So including twin GRACE satellites or one of them contribute nearly the same to the POD of GPS satellites.

Total RMS (cm)

4 Results and Discussion

Total RMS (cm)

Integrated Adjustment of LEO and GPS in Precision Orbit Determination

8 6 4 2 0

43S

43S + 1L

43S + 2L

43S + 3L

Fig. 2. Improvement of GPS orbit quality by including the observations from different numbers of LEOs (L) to the observations from 43 ground stations (S).

Table 2. Total RMS of GPS orbits from different network configurations compared with the IGS final orbits, in cm

DOY

154

155

156

157

158

159

Mean

1. 43S 2. 43S+3L 3. 32S+3L 4. 21S+3L 5. 43S+1L 6. 43S+2L

7.7 4.9 5.1 4.9 5.7 5.1

7.9 4.0 4.5 5.1 4.6 4.1

7.0 5.7 5.9 6.3 6.0 5.7

7.8 5.4 5.3 5.8 5.9 5.4

9.1 4.4 4.2 4.6 5.4 4.6

8.7 4.3 4.3 4.9 5.2 4.4

8.0 4.8 4.9 5.3 5.5 4.9

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4.1.2 Improvement in the along, Cross and Radial Components

Difference of RMS (cm)

16

Along

Case 1

15 Along (cm)

Figure 3 describes the improvement of Case 2 with respect to Case 1 in the along, cross and radial directions for each GPS satellite on day 158, 2002. The improvement is the difference of the RMS of Case 1 and Case 2. From Figure 3, the along direction presents the most improvement, 6.4 cm on average, among the three. And the next is the cross direction, 3.6 cm on average. From the point of view of geometry, the velocity of LEOs in the along direction is much larger than those in the other two directions. Then the geometry changes quickly especially in the along-cross plane of the GPS orbit relative to the LEOs. Obviously, it leads to much more significant orbit improvement in the along and cross components. It should be noted that we get obvious negative improvement in Case 2 with respect to Case 1 only on day 158, for example, for PRN20 in the along direction as shown in Figure 3. The orbit residuals of Case 1 and 2 compared with IGS final orbit are plotted in Figure 4. It is due to the fact that the ground stations, the GPS satellites and the LEOs do not form a stable closed loop in the integrated adjustment. That means there are no observations between the ground stations and the LEOs in these cases. Furthermore, the force model for the LEOs is possible to be inappropriate, resulting in the negative impact on the orbits of GPS satellites indirectly. Consequently, the orbits of GPS satellites are more likely to absorb improperly the error of the LEO orbits in the integrated adjustment, which yields worse results than that when only ground stations are used. For instance, in Figure 4, the result in Case 2 (dashed line) departs from that (solid line) in Case 1 evidently for some systematic error.

20 Case 2

10 5 0 –5

1

11

21

31

–10

41

51

61

71

81

91

Epoch (900s)

Fig. 4. Orbit residuals of PRN20 on day 158, 2002 related to IGS final products for Case 1 (43S) and Case 2 (43S+3L).

4.2 LEO POD The daily total RMS of LEOs’ orbits estimated from the one-step and two-step approaches are listed in Table 3, where the precise orbits provided by GFZ (GeoForschungsZentrum) are used as external reference. Here 43 ground stations are used in the one-step method. In the two-step method, the GPS orbits and clocks are fixed to that from Case 1. In practice, however, more accurate GPS orbits and clocks such as the IGS final products should be

Table 3. Total orbit RMS of LEOs from one-step and twostep approaches compared with the precise orbits at GFZ, in cm

CHAMP

GRACE B

GRACE A

DOY

2-step

1-step

2-step

2-step

154 155 156 157 158 159

12.3 12.3 8.7 12.3 11.6 12.9

11.9 8.7 7.9 11.1 8.7 9.9

Cross

5.8 9.8 6.8 8.0 7.1 8.0

1-step 4.8 4.8 5.9 5.3 3.8 6.6

5.6 9.9 6.6 – 7.1 7.8

1-step 4.6 4.5 5.5 – 3.4 6.2

Radial

14 12 10 8 6 4 2 0 –2

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 PRN of GPS satellite (6/7/2002)

Fig. 3. RMS improvement of Case 2 (43S+3L) relative to Case 1 (43S) on day 158, 2002.

Integrated Adjustment of LEO and GPS in Precision Orbit Determination

employed. So the tests presented here only serve as relative comparisons to illuminate the impact of the GPS orbit accuracy on the LEO orbits. From Table 3, the RMS decreases on average by 17% for CHAMP, and by 32% for GRACE. Actually, the accuracy of the LEO orbit depends on the accuracy of the GPS orbit (Rim et al., 2002), whereas the orbit accuracies of the GPS satellites and the LEOs are improved simultaneously in the one-step method.

5 Conclusion It has been demonstrated that the one-step method can achieve more accurate orbits of GPS satellites and LEOs than the two-step method. On the one hand, the geometry for GPS POD is extraordinarily strengthened when the LEOs are included as moving tracking stations. This means that we can utilize LEOs to improve the accuracy of GPS orbits considerably when there are not enough ground stations. So even if there are only 21 globally distributed stations plus 3 LEOs, the orbits of GPS satellites are still more accurate than when only 43 stations are used. On the other hand, the improvement of GPS orbits is related to the orbit configuration of LEOs. So including twin GRACE satellites or one of them contributes nearly the same to the orbits of GPS satellites. Finally, the authors conclude that integrated processing compensates the weakness of either kind of satellite by the strength of the other. So the orbit accuracy of each kind of satellite is improved considerably. However, systematic errors may still remain in the results because of the improperly absorption of each other’s error. Despite this, it is still significant that the integrated adjustment is able to yield more precise orbits for GPS satellites and LEOs, which are crucial in Earth science research. Furthermore, it is also beneficial to the geocenter estimation and in maintaining the terrestrial reference system (Zhu et al., 2004).

Acknowledgement This work is supported by the 973 Program No. 2006CB701301 and China NSFC No. 40574005.

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References Byun, S.H. (2003). Satellite orbit determination using tripledifferenced GPS carrier phase in pure kinematic mode. Journal of Geodesy. Vol. 76, pp: 569–585. International GPS Service (2005). IGS home page. Available at http://igscb.jpl.nasa.gov/components/prods.html. Accessed in May. GFZ, CSR, DLR, NASA (2002). GRACE Newsletter No. 1: GRACE orbits and maneuvers. http://www,gfzpotsdam.de/grace. August 1, 2002. Kang, Z., Tapley, B., Bettadpur, S., Nagel, P., Pastor, R. (2003). GRACE Precise orbit determination. Advances in the Astronautical Sciences. Vol. 114, pp: 2237–2244. König, R., Reigber, Ch., Zhu, S.Y. (2005). Dynamic model orbits and earth system parameters from combined GPS and LEO data. Advances in Space Research. Vol. 36, No. 3, pp: 431–437. Lee, S., Schutz, B.E., Abusali, A.M. (2004). Hybrid precise orbit determination strategy by global positioning system tracking. Journal of Spacecraft and Rockets. Vol. 41, No. 6, pp: 997–1009. Liu, J.N., Ge, M.R. (2003). PANDA software and its preliminary result for positioning and orbit determination. Wuhan University Journal of Natural Science, Vol. 8, No. 2B, pp: 603–609. Liu, J.N., Zhao, Q.L., Ge, M.R. (2004). Preliminary result of CHAMP orbit determination with PANDA software. December 6–8, 2004, the International Symposium on GPS/GNSS, Sydney, Australia. Montenbruck, O., Gill, E. (2000). Satellite Orbits: Models, Methods, and Applications. Springer-Verlag, Berlin, pp: 233–243. Moore, P., Turner, J.F., Qiang, Z. (2003). CHAMP orbit determination and gravity field recovery. Advances in Space Research. Vol. 31, No. 8, pp: 1897–1903. Rim, H.J., Davis, G.W., Schutz, B.E. (1996). Dynamic orbit determination for the EOS laser altimeter satellite (EOS ALT/GLAS) using GPS measurements. The Journal of Astronautical Sciences. Vol. 44, No. 3, pp: 409–424. Rim, H.J., Yoon, S.P., Schutz, B.E. (2002). Effect of GPS orbit accuracy on CHAMP precision orbit determination. AAS 02-213, AAS/AIAA Space Flight Mechanics Meeting, San Antonio, pp: 1411–1415. Yunck, T.P., Wu, S.C., Wu, J.T., et al. (1990). Precise tracking of remote sensing satellites with the global positioning system. IEEE Transactions on Geoscience and Remote Sensing. Vol. 28, No. 1, pp: 108–116. Zhu, S., Reigber, C., König, R. (2004). Integrated adjustment of CHAMP, GRACE, and GPS data. Journal of Geodesy. Vol. 78, No. 1, pp: 103–108.

Reduced-Dynamic Precise Orbit Determination Based on Helmert Transformation J. Chen Department of Surveying and Geo-informatics, Tongji University, Siping Road 1239, 200092 Shanghai, P.R. China; GeoForschungsZentrum Potsdam, Telegrafenberg A17, 14473 Potsdam, Germany J. Wang Department of Surveying and Geo-infomatics, Tongji University, Siping Road 1239, 200092 Shanghai, P.R. China Abstract. Four fundamental approaches in the Precise Satellite Orbit Determination (POD) are Dynamic POD, Reduced-Dynamic POD, Kinematic POD, and Reduced-Kinematic POD (Svehla & Rothacher, 2003, Liu et al. 2004). Generally, two kinds of information are considered: dynamic parameters and kinematic factors. In the dynamic method, the orbit precision mainly depends on the initial orbits and the dynamic models. Using the dynamic models recommended by IERS, the 12-hour extrapolated orbits reach a precision of several centimeters. Therefore the main task for dynamic POD is to get the precise initial orbits. On the other hand, in the kinematic method, the main problem is how to reduce influence of weak geometry and phase breaks. We suggest a reduced-dynamic POD model based on the Helmert transformation. The model connects the dynamic integrated orbits and the precise kinematic orbits. Using GPS data, we get the results that the orbit-precision of none-corrected orbits is at the level of decimeter, while the corrected orbit-precision using the proposed model is several millimeters. Keywords. Dynamic orbit, kinematic orbit, precise orbit determination, helmert transformation

1 Introduction Satellite Precise Orbit Determination is important because of the advance of satellite technology. In the latest gravity satellite mission of CHAMP(CHAllenging Minisatellite Payload) and GRACE(Gravity Recovery And Climate Experiment), precise satellite orbit is the key for gravity recovery, atmospheric research and magnetic field research, etc. The POD method can be divided into dynamic method and kinematic method according to the theory and observations it used. In the dynamic method, the orbit precision mainly 138

depends on the initial orbits and the dynamic models used in the movement equation. Since the dynamic models improve significantly, the 12-hour extrapolated dynamic orbits reach a precision of several centimeters (Chen, 2006). Therefore the main task for dynamic POD is to get the precise initial orbits. On the other hand, in the kinematic method, the main problem is how to reduce influence of weak geometry and phase breaks. Data screening is one of the most important works. The dynamic integrated orbit is available almost at any time depending on the integration interval. Its precision depends on the initial orbit precision and the dynamic model. In reduced-dynamic method, the kinematic orbits may be used as pseudo-observations. The mathematical model can be treat simply as Gause-Markov procedure. We suggest a model based on Helmert transformation, which connects the dynamic integrated orbits and the kinematic orbits. Using GPS data, we get the results that the orbit-precision of none-corrected orbits is at the level of decimeter, while the corrected orbit-precision using the proposed model is several millimeters.

2 Orbit Integration According to dynamic POD theory, the satellite movement equation and the initial value of satellites’ orbit can be written as, x˙ = F(x, t) (1) x|t 0 = x 0 T where x = r r˙ p is the initial orbits including the position, velocities and dynamic parameters (such as the solar radiation pressure parameters) of the satellites. And F(x, t) are the modeling equations of the complete set of factors acting on an orbiting satellite (Wang, 1997 & IERS 2003). With proper integration methods based on these initial orbits x 0 , such as Adams-Cowell numerical integration, the dynamic integrate orbit x ∗ can be computed.

Reduced-Dynamic Precise Orbit Determination Based on Helmert Transformation

The satellite orbit at any time can be expressed by its initial value through the transition matrix. In equation (1), define δ = x − x ∗, where x ∗ is the orbit from integration. Using the Taylor expansion, then, ⭸F(x, t) ˙δ = δ (2) ⭸x ∗ Assuming that the solution of (2) can be expressed as, δ = ⌿(t, t0 )δ0 (3) where δ0 = x 0 −x ∗ is the corrected value at the initial time. Substitute (3) into (2), we have the following equation,

˙ t0 ) = ⭸F ⌿(t, t0 ) ⌿(t, ⭸x ⌿(t0 , t0 ) = I

(4)

where I is the unit matrix, ⌿(t, t0 ) is the transition matrix. It can be expressed in detail as, ⎞ ⎛ ⭸r ⭸r ⭸r

⌿ t, t0

⎜ ⎜ =⎜ ⎝

⭸r0 ⭸r˙ ⭸r0 ⭸p ⭸r0

⭸r˙0 ⭸r˙ ⭸r˙0 ⭸p ⭸r˙0

⭸p ⭸r˙ ⭸p ⭸p ⭸p

⎟ ⎟ ⎟ ⎠

(5)

During the integration we can get the transition matrix as well as integrated orbits x ∗ . The transition matrix relates integrated orbits x ∗ and the initial orbits value x 0 , and is used to correct the orbits x ∗ by correcting the initial orbits values x 0 in the dynamic POD.

axis of the coordinate frame. The structure of matrix R is according to IERS standard. For simplify, (6) can also be written as, x = F(s, t) = T0 + (1 + K ) · R · x

(7)

in which x is the function of satellites’ initial orbits value x 0 and transition matrix ⌿(t, t0 ). Since most the kinematic orbit is given in the earth-fixed frame, we need to transfer it to the inertial frame. Consequently, the earth orientation parameters can also be added in this model. So, all parameters to be solved in this model are, s = (eop, trans, x 0 )T

(8)

in equation (8), T eop = X p , Y p , U T 1, X˙ p , Y˙ p , U˙ T 1 is the earth orientation parameters, trans = (⌬X, ⌬Y, ⌬Z , ␣, ␤, ␥, K ,)T is the transformation parameters , and x 0 = (r0 , r˙0 , P)T is the correction of orbits initial values. Using the Taylor expansion, equation (7) can be written as, ⭸F(s, t) δs (9) x = x 0 + ⭸s And, ⭸F(s, t) = ⭸s

3 Orbit-Corrected Model Based on Helmert Transformation Normally, Helmert transformation is mostly used in the transformation between different coordinate frames, such as the transformation between ITRF97 and ITRF2000. It considers the movement of origin and rotation. Considering the frames differences between dynamic orbits and kinematic orbits, we build up the transformation between the dynamic integrated orbits and the kinematic orbits. It can be expressed in Helmert transformation as, ⎛ ⎞ ⎞ ⎛ ⎞ ⎛ X ⌬Z X ⎝ Y ⎠ = ⎝ ⌬ Y ⎠ + (1 + K )R (α, β, γ ) ⎝ Y ⎠ Z ⌬Z Z (6) Where (X , Y , Z ) is the precise kinematic orbits x , and (X, Y, Z ) is the dynamic integrated orbits x, α, β, γ is the three rotation angles based on the three

139

⭸x ⭸x ⭸x , , ⭸eop ⭸trans ⭸x 0

⭸x ⭸trans

can be derived easily accord-

ing to equation (7), and

can derived as following,

In which

⭸x ⭸eop

and

⭸x ⭸x 0

⭸x ⭸x ⭸x · = ⭸x 0 ⭸x ⭸x 0

⭸x where ⭸x ⭸x = (1 + K ) · R. And ⭸x 0 is the transition matrix of the orbit integration ⌿(t, t0 ). Then, (9) can be written as

v = Aδs0 − l

(10)

(s,t ) , l = x − x 0 where A = ⭸F⭸s Using the least-squares estimation (Jin et al. 1994), we get the adjusted value ␦s = (␦eop, ␦trans, ␦x 0 ).

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After we get the adjusted value of initial orbits, the dynamic orbit integration runs again using the new initial values.

4 Data Processing In the reduced-dynamic method introduced above, the data processing procedure mainly contains the following two parts: • kinematic orbit determination • Reduced-dynamic orbit determination The procedure can runs in iteration. It starts with kinematic orbit determination without apriori orbits. Then, the reduced-dynamic orbit determination will be performed to provide better orbits. This reduceddynamic orbits can be used as a new apriori orbits in the kinematic orbit determination. The contribution of reduced-dynamic orbits to kinematic orbit determination may exist in the data screening. As an example, we carry out an experiment using GPS data. We assume GPS precise orbits (SP3 from IGS) are the kinematic orbits, and use this SP3 orbits as observations in the reduced-dynamic orbit determination. The other parameters of the experiment are listed below, – – – –

time: GPS DOY (Day Of Year) 2005.062; Interval of integration: 10 s; Interval of printing out: 15 min; Estimated parameters: corrections of initial orbits (coordinates and velocities), 9 parameters of sun radiation pressure model (Springer et al. 1998); – Initial dynamic orbits: initial coordinates taken from SP3 file (at the epoch of 12:00), but

5-centimeter error is added. Initial velocities also computed from SP3 file, with 5 cm/h error added. The orbit integration starts at 12:00 (GPS time), and the integration running backward and forward for 12-hours respectively from the initial epoch. The following figures show the results of satellites PRN02. Where, Figure 1 shows the difference between the dynamic integrated orbits and SP3 orbits, while Figure 2 shows the difference between the corrected orbits (using the model introduced in this paper) and SP3 orbits.

5 Results and Conclusion From the results we see that 12-hour extrapolated dynamic orbits have the difference about 1 meter, the average difference dynamic integrated orbits is about decimeters. This is caused by the errors added in the initial orbits and errors of the dynamic model. In reduced-dynamic orbits, the 5-centimeter error of the initial orbits is removed and the average difference between corrected orbits and IGS orbits is about centimeters. We also carry out statistic analysis of all the satellites in the same day. The following pictures show the results. It covers the average difference between the integrated orbits and IGS orbits (Figure 3) and the standard deviation of the difference (Figure 4), the average difference between the corrected orbits and IGS orbits (Figure 5) and also the standard deviation of the difference (Figure 6). The statistics show that the difference between the integrated orbits and IGS orbits is about decimeters, and the difference between corrected orbits and IGS orbits is about millimeters. The standard deviation of the difference between the corrected orbits and IGS

X-axis

1

Orbit Difference (m)

0 –1 2 Y-axis 0 –2 2 Z-axis 0 –2 0

5

10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 Epochs (Interval is 15min)

Fig. 1. The orbit difference between dynamic integrated orbits and IGS orbits of PRN02.

Orbit Difference (cm)

Reduced-Dynamic Precise Orbit Determination Based on Helmert Transformation

4 2 0 –2 –4

141

X-axis

4 2 0 –2 –4

Y-axis

4 2 0 –2 –4

Z-axis

0

5

10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 Epochs (Interval is 15min)

Fig. 2. The orbit difference between corrected orbits and IGS orbits of PRN02.

Difference (m)

Difference from Precise Orbits 1.6 1.2 0.8 0.4 0 –0.4 –0.8 –1.2

X-Axis

Y-Axis

Z-Axis

Fig. 3. Average of the difference between dynamic integrated orbits and IGS orbits.

Std. Error of the Difference 4.8

X-Axis

Y-Axis

Z-Axis

Std. Error (m)

4 3.2 2.4 1.6 0.8 PR N PR 01 N0 PR 2 N PR 03 N PR 04 N PR 05 N PR 06 N PR 07 N PR 08 N0 PR 9 N PR 10 N PR 11 N PR 13 N1 PR 4 N PR 15 N PR 16 N1 PR 8 N PR 19 N PR 20 N PR 21 N2 PR 2 N PR 23 N2 PR 4 N PR 25 N PR 26 N PR 27 N2 PR 8 N PR 29 N PR 30 N3 1

0

Fig. 4. Standard error of the difference between dynamic integrated orbits and IGS orbits.

Difference (m)

Difference from Precise Orbits 0.4 0.3 0.2 0.1 0 –0.1 –0.2 –0.3 –0.4 –0.5

X-Axis

Y-Axis

Fig. 5. Average of the difference between corrected orbits and IGS orbits.

Z-Axis

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J. Chen, J. Wang Std. Error of the Difference

Std. Error (m)

3.2

X-Axis

Y-Axis

Z-Axis

2.4 1.6 0.8 PR N PR 01 N PR 02 N0 PR 3 N0 PR 4 N PR 05 N PR 06 N PR 07 N0 PR 8 N0 PR 9 N PR 10 N PR 11 N PR 13 N PR 14 N1 PR 5 N PR 16 N PR 18 N PR 19 N PR 20 N2 PR 1 N PR 22 N PR 23 N PR 24 N2 PR 5 N PR 26 N PR 27 N2 PR 8 N PR 29 N PR 30 N3 1

0

Fig. 6. Standard error of the difference between corrected orbits and IGS orbits.

orbits is less than 3 centimeters, while that of integrated orbits reaches decimeters.

Acknowledgement The paper is substantially supported by ChangJiang Scholars Program, Ministry of Education, People’s Republic of China. The author would like to thank ChangJiang Scholars Professor Li RongXing for his help.

References McCarthy, D.D. and Gérard Petit (eds.): IERS conventions (2003), IERS Technical Note, 32, 2004. Drazen Svehla & Markus Rothacher. Kinematic, ReducedKinematic, dynamic and reduced-dynamic precise orbit

determination in the LEO orbit, Second CHAMP Science Meeting, GeoForschungsZentrum Potsdam, September 1–4, 2003. Jiexian WANG. Precise Positioning and Orbiting for GPS. Press of Tongji University. 1997. Junping CHEN. The Models of Solar Radiation Pressure in GPS Precise Orbit Determination, ACTA ASTRONOMICA SINICA, 2006. Guoxiong JIN, Dajie LIU, Yiming SHI. The Application of GPS and GPS Data Processing. Press of Tongji University. 1994. Jinnan LIU, Qile ZHAO, Xiaohong ZHANG. Geometric Orbit Determination of CHAMP Satellite and Dynamic Models’ Compensation During Orbit Smoothing. Journal of Wuhan University, 2004, 29, 1–6. T.A. Springer, G. Beutler, M. Rothacher. A new solar Radiation Pressure Model for the GPS Satellites, CODE papers, GPS Solution, 1998.

GNSS Ambiguity Resolution: When and How to Fix or not to Fix? P.J.G. Teunissen, S. Verhagen (invited paper) Delft Institute of Earth Observation and Space systems (DEOS), Delft University of Technology, Kluyverweg 1, 2629 HS Delft, The Netherlands, e-mail: P.J.G. [email protected]

Abstract. In order to facilitate rapid and precise GNSS positioning, the integer carrier phase ambiguities need to be resolved. Since wrong integer ambiguity estimates may result in fixed position estimates which are worse than their float counterparts, very high success rates (i.e. high probabilities of correct integer estimation) or very low failure rates are required when performing ambiguity resolution. We discuss two different approaches of ambiguity resolution, a model-driven approach and a data-driven approach. The first is linked to the theory of integer estimation and the second is linked to the theory of integer aperture estimation. In the first approach, the user chooses an integer estimator and computes on the basis of his/her model the corresponding failure rate. The decision whether or not to use the integer ambiguity solution is then based on the thus computed value of the failure rate. This approach is termed model-driven, since the decision is solely based on the strength of the underlying model and not dependent on the actual ambiguity float estimate. This approach is simple and provides a priori information on the outcome of the decision process. A disadvantage of the model-driven approach is that it does not provide the user of any control over the failure rate. This disadvantage is absent when one uses the more elaborate data-driven approach of integer aperture estimation. With this approach the user sets his/her own failure rate (irrespective of the strength of the underlying model), thus generating an aperture space which forms the basis of the decision process: the integer solution is chosen as output if the float solution resides inside the aperture space, otherwise the float solution is maintained. Although more elaborate, the data-driven approach is more flexible than the model-driven approach and can provide a guaranteed failure rate as set by the user. In this contribution we compare the model-driven and data-driven approaches, describe the decision making process of when and how to fix (or not to fix) and also give the optimal data-driven approach. We

also show how the so-called ‘discrimination tests’, in particular the popular ‘ratio test’, fit into this framework. We point out that the common rationales for using these ‘tests’ are often incorrectly motivated in the literature and we show how they should be modified in order to reach an overall guaranteed failure rate for ambiguity resolution. Keywords. Integer least-squares, integer aperture estimation

1 GNSS Ambiguity Resolution 1.1 The GNSS Model As our point of departure we will take the following system of linear(ized) observation equations y = Aa + Bb + e

(1)

where y is the given GNSS data vector of order m, a and b are the unknown parameter vectors respectively of order n and p, and where e is the noise vector. In principle all the GNSS models can be cast in this frame of observation equations. The data vector y will usually consist of the ‘observed minus computed’ double-difference (DD) phase and/or pseudorange (code) observations accumulated over all observation epochs. The entries of vector a are then the DD carrier phase ambiguities, expressed in units of cycles rather than range. They are known to be integers, a ∈ Z n . The entries of the vector b will consist of the remaining unknown parameters, such as for instance baseline components (coordinates) and possibly atmospheric delay parameters (troposphere, ionosphere). They are known to be real-valued, b ∈ R p. The procedure which is usually followed for solving the GNSS model (1), can be divided into three steps. In the first step one simply disregards the integer constraints a ∈ Z n on the ambiguities and 143

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P.J.G. Teunissen, S. Verhagen

applies a standard least-squares adjustment, resulting in real-valued estimates of a and b, together with their variance-covariance (vc-) matrix aˆ Q aˆ Q aˆ bˆ , (2) Q bˆ aˆ Q bˆ bˆ This solution is referred to as the ‘float’ solution. In the second step the ‘float’ ambiguity estimate aˆ is used to compute the corresponding integer ambiguity estimate a. ˇ This implies that a mapping S : R n → n Z , from the n-dimensional space of reals to the ndimensional space of integers, is introduced such that aˇ = S(a) ˆ

(3)

Once the integer ambiguities are computed, they are used in the third step to finally correct the ‘float’ estimate of b. As a result one obtains the ‘fixed’ solution (aˆ − a) ˇ bˇ = bˆ − Q bˆ aˆ Q −1 aˆ

(4)

This three-step procedure is still ambiguous in the sense that it depends on which mapping S is chosen. 1.2 The Class of Integer Estimators If one requires the output of the map S to be integer, S : R n → Z n , then S will not be one-to-one due to the discrete nature of Z n . Instead it will be a manyto-one map. This implies that different real-valued vectors will be mapped to one and the same integer vector. One can therefore assign a subset Sz ⊂ R n to each integer vector z ∈ Z n : Sz = {x ∈ R n | z = S(x)},

z ∈ Zn

(5)

The subset Sz contains all real-valued vectors that will be mapped by S to the same integer vector z ∈ Z n . This subset is referred to as the pull-in region of z. It is the region in which all vectors are pulled to the same integer vector z. Since the pull-in regions define the integer estimator completely, one can define classes of integer estimators by imposing various conditions on the pull-in regions. One such class was introduced in (Teunissen, 1999a) as follows. Definition 1 (Integer estimators). The mapping aˇ = S(a), ˆ with S : R n → Z n , is said to be an integer estimator if its pull-in regions satisfy ⎧ ⎨ ∪z∈Z n Sz = R n Int(Su ) ∩ Int(Sz ) = ∅, ∀u, z ∈ Z n , u = z ⎩ Sz = z + S0 , ∀z ∈ Z n

This definition is motivated as follows. The first condition states that the pull-in regions should not leave any gaps and the second that they should not overlap. The third condition of the definition follows from the requirement that S(x + z) = S(x) + z, ∀x ∈ R n , z ∈ Zn. Using the pull-in regions, one can give an explicit expression for the corresponding integer estimator a. ˇ It reads 1 if aˆ ∈ Sz aˇ = zsz (a) ˆ with sz (a) ˆ = 0 if aˆ ∈ / Sz z∈Z n (6) The three best known integer estimators are integer rounding, integer bootstrapping and integer leastsquares. The latter is shown to be optimal, cf. (Teunissen, 1999a), which means that the probability of correct integer estimation is maximized. The integer least-squares (ILS) estimator is defined as aˇ LS = arg minn ||aˆ − z||2Q aˆ z∈Z

(7)

with the squared norm || · ||2Q = (·)T Q −1 (·). In contrast to integer rounding and integer bootstrapping, an integer search is needed to compute aˇ LS . The ILS procedure is mechanized in the LAMBDA method, which is currently one of the most applied methods for GNSS carrier phase ambiguity resolution, see e.g. (Teunissen, 1993; Hofmann-Wellenhoff and Lichtenegger, 2001; Strang and Borre, 1997; Teunissen, 1998; Misra and Enge, 2001). As mentioned above, the ILS estimator maximizes the probability of correct integer estimation, referred to as the success rate, which is given by: f aˆ (x)dx (8) Ps,ILS = P(aˇ = a) = Sa

where P(aˇ = a) is the probability that aˇ = a, and f aˆ (x) is the probability density function of the float ambiguities, for which in practice the normal (Gaussian) distribution is used. Hence, the success rate can be evaluated without the need for actual data, since it can be computed once the vc-matrix Q aˆ is known.

2 Model-Driven Approach The precision of the fixed baseline solution can be shown to be much better than the precision of its float counterpart, provided that the success rate is close to 1 and thus the failure rate, P f = 1− Ps , must be close

GNSS Ambiguity Resolution: When and How to Fix or not to Fix?

to 0. Hence, it can be argued that the fixed solution should only be used if this is known to be the case. The model-driven approach to integer estimation is a three-step procedure: 1. Apply an integer map such that aˇ = S(a) ˆ 2. Evaluate the failure rate P f = P(aˇ = a) = 1 − P(aˇ = a) 3. Decision: Pf ≤ use aˇ If (9) Pf > use aˆ where is a user-defined threshold, e.g. = 10−2 . Obviously, this model-driven approach is simple and valid, since the failure rate can be determined prior to the actual integer estimation step. Hence, also the decision whether or not to fix can be made a priori. The choice on how to fix is also clear: the integer least-squares estimator is optimal since it minimizes the failure rate. An additional advantage is that an overall and rigorous quality description of the ˇ is available, cf. (Teunissen, fixed solution (aˇ and b) 1999b, 2002). Disadvantages of the model-driven approach are that the sample aˆ has no influence on the decision and that the user has no control over the failure rate except for strengthening the model. The first disadvantage implies that there might be one or more other integer candidates which is/are almost as likely to be the correct solution. Therefore in practice often a discrimination test is applied as an (additional) decision tool, see section 3.1. Strengthening the model may only be an option in the design phase of a measurement campaign, but not after the data is collected.

3 Data-Driven Approach 3.1 Discrimination Tests In literature several discrimination tests have been proposed in order to decide whether or not to fix the ambiguities. A review and evaluation of the tests can be found in Verhagen (2004). Well-known examples are the ratio test, distance test and projector test. Among the most popular tests is the ratio test, see e.g. (Euler and Schaffrin, 1991; Wei and Schwarz, 1995; Han and Rizos, 1996; Leick, 2003), where the decision is made as follows: 1. Apply ILS to obtain aˇ and aˇ 2 2. Evaluate ratio

||a− ˆ a|| ˇ 2Q

aˆ

||a− ˆ aˇ 2 ||2Q

aˆ

145

3. Decision: If

||aˆ − a|| ˇ 2Q aˆ ||aˆ −

aˇ 2 ||2Q aˆ

≤δ >δ

use aˇ use aˆ

(10)

where aˇ 2 is the second-best integer solution in the ILS sense. Obviously, the decision is data-driven. Note that in practice the reciprocal of the test statistic is mostly used. It is, however, not clear what role is actually played by this ratio test. The common motivation for using it is that it is a validation test, i.e. that it tells the user whether or not the fixed solution is true or false. This is not correct. Moreover, the current ways of choosing the threshold value δ are ad hoc or based on false theoretical grounds, see (Teunissen and Verhagen, 2004). Often a fixed value of 12 or 13 is used. A problem with the ratio test, or any other discrimination test, is that the model-driven failure rate is not applicable anymore, since the test implicitly introduces a probability of not fixing. Another implication is that the overall quality of the fixed solution cannot be evaluated when the ratio test is included, whereas with the model-driven approach this is possible. 3.2 Integer Aperture Estimation In practice, a user will decide not to use the fixed solution if either the probability of failure is too high, or if the discrimination test is not passed. This gives rise to the thought that it might be interesting to use an ambiguity estimator defined such that three situations are distinguished: success if the ambiguity is fixed correctly, failure if the ambiguity is fixed in correctly, and undecided if the float solution is maintained. This can be accomplished by dropping the condition that there are no gaps between the pullin regions, so that the only conditions on the pull-in regions are that they should be disjunct and translational invariant. Then integer estimators can be determined that somehow regulate the probability of each of the three situations mentioned above. The new class of ambiguity estimators was introduced in Teunissen (2003a, c), and is called the class of Integer Aperture (IA) estimators.

Definition 2. (Integer aperture estimators) The integer aperture estimator, a, ¯ is defined as: a¯ =

z∈Zn

zωz (a) ˆ + a(1 ˆ −

z∈Zn

ωz (a)) ˆ

(11)

146


with the indicator function ωz (x) defined as: ωz (x) =

1 0

if x ∈ ⍀z otherwise

aˆ ∈ ⍀a aˆ ∈ ⍀\⍀a aˆ ∈ /⍀

(12)

The ⍀z are the aperture pull-in regions, which have to fulfill the following conditions (compare with Definition 1): ⎧ ⍀ =⍀ ⎪ ⎪ ⎨ z∈Zn z Int(⍀u ) Int(⍀z ) = ∅, ⎪ ⎪ ⎩ ⍀ z = z + ⍀0 ,

success: correct integer estimation failure: incorrect integer estimation undecided: ambiguity not fixed

The corresponding probabilities of success (s), failure ( f ) and undecidedness (u) are given by: Ps = P(a¯ = a) = Pf =

∀u, z ∈ Zn , u = z ∀z ∈ Zn

z∈Zn \{a}⍀

faˆ (x)dx ⍀a

f aˆ (x)dx

(13)

z

Pu = 1 − Ps − P f = 1 −

f ˇ (x)dx ⍀0

⍀ ⊂ Rn is called the aperture space. From the first condition follows that this space is built up of the ⍀z . The second and third conditions state that these aperture pull-in regions must be disjunct and translational invariant. Figure 1 shows a two-dimensional example of aperture pull-in regions that fulfill the conditions in Definition 2, together with the ILS pull-in regions. So, when aˆ ∈ ⍀ the ambiguity will be fixed using one of the admissible integer estimators, otherwise the float solution is maintained. This means that indeed the following three cases can be distinguished:

The first two probabilities are referred to as success rate and failure rate respectively. Note the difference with the ILS success rate given in equation (8), where the integration is over the ILS pull-in region Sa ⊃ ⍀a . The expression for the failure rate is obtained by using the probability density function of the ambiguity residuals ˇ = aˆ − a: ˇ f ˇ (x) =

f aˆ (x + z)s0 (x)

(14)

z∈Zn

with s0 (x) = 1 if x ∈ S0 and s0 (x) = 0 otherwise, cf. Teunissen (2002); Verhagen and Teunissen (2004). 3.3 Fixed Failure Rate Approach

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−1

−0.5

0

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1

1.5

Fig. 1. Two-dimensional example of aperture pull-in regions (ellipses), together with the ILS pull-in regions (hexagons).

As mentioned in the beginning of this section, for a user it is especially important that the probability of failure, the failure rate, is below a certain limit. The approach of integer aperture estimation allows us now to choose a threshold for the failure rate, and then determine the size of the aperture pull-in regions such that indeed the failure rate will be equal to or below this threshold. So, applying this approach means that implicitly the ambiguity estimate is validated using a sound criterion. However, there are still several options left with respect to the choice of the shape of the aperture pull-in regions. It is very important to note that Integer Aperture estimation with a fixed failure rate is an overall approach of integer estimation and validation, and allows for an exact and overall probabilistic evaluation of the solution. With the traditional approaches, e.g. the Ratio Test applied with a fixed critical value, this is not possible.

GNSS Ambiguity Resolution: When and How to Fix or not to Fix?

147

3.4 ILS + Ratio Test is Integer Aperture Estimator

1

0.8 success rate

Despite the criticism on the Ratio Test given in section 3.1 it is possible to give a firm theoretical basis for this test, since it can be shown that the procedure underlying the Ratio Test is a member from the class of integer aperture estimators. The acceptance region or aperture space is given as:

0.6

0.4

0.2

⍀ = {x ∈ Rn | x − xˇ 2Q aˆ ≤ δ x − xˇ2 2Q aˆ , 0 < δ ≤ 1}

0

(15)

⍀z = ⍀0 + z, ∀z ∈ Zn

⍀z ⍀=

(16)

z∈Zn

The proof was given in Teunissen (2003b). The acceptance region of the Ratio Test consists thus of an infinite number of regions, each one of which is an integer translated copy of ⍀0 ⊂ S0 . The acceptance region plays the role of the aperture space, and δ plays the role of aperture parameter since it controls the size of the aperture pull-in regions. Compared to the approach in section 3.1 an important difference is that δ is now based on the modeldriven failure rate. Hence, before the decision step, an additional step is required to determine δ based on the user-defined choice P f = β. This implies that a probabilistic evaluation of the solution can be made, see section 3.3. As an illustration of the difference between the traditional and the Integer Aperture approach with the Ratio Test, five dual-frequency GPS models are considered. Based on Monte-Carlo simulations the success and failure rates as function of δ are determined for each of the models, see Figure 2. It can be seen that with a fixed value of δ = 0.3 (close to the value of 13 often used in practice) for most of the models considered here a very low failure rate is obtained, but that this is not guaranteed. This seems good, but at the same time also the corresponding success rate is low. If the threshold value would have been based on a fixed failure rate of e.g. 0.005, the corresponding

0.2

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1

0

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0.7

0.8

0.9

1

0.04 failure rate

∀z ∈ Zn \ {0}}

0.1

0.05

with xˇ and xˇ2 the best and second-best ILS estimator of x. Let ⍀z = ⍀ Sz , i.e. ⍀z is the intersection of ⍀ with the ILS pull-in region. Then all conditions of Defintion 2 are fulfilled, since: ⍀0 = {x ∈ Rn | x 2Q aˆ ≤ δ x − z 2Q aˆ ,

0

0.03

0.02

0.01

0

δ

Fig. 2. Success and failure rates as function of the threshold value δ for 5 GPS models.

δ would have been very different for each of the models, and in most cases larger than 0.3, and thus a higher success rate and higher probability of a fix (= Ps + P f ) would be obtained. Hence, the integer aperture approach with fixed failure rate is to be preferred. 3.5 Optimal Integer Aperture Estimation The approach of integer aperture estimation with a fixed failure rate has two important advantages. The first is that IA estimation can always be applied, independent of the precision, since the user does not have to be afraid that the failure rate is too high. The second advantage is that for the first time sound theoretical criteria are available for the validation of the estimates. For that purpose, the Ratio Test can be used. However, it will now be shown that also an optimal integer aperture (OIA) estimator exists. As with integer estimation, the optimality property would be to maximize the success rate, but in this case for a fixed failure rate. So, the optimization problem is to determine the aperture space which fulfills:

148


max Ps

⍀0 ⊂S0

subject to:

Pf = β

(17)

where β is a chosen fixed value for the failure rate. The solution of the optimization problem is given by, cf.Teunissen (2003c, 2004): ⍀0 = {x ∈ S0 | f aˆ (x + z) ≤ μf aˆ (x + a)} (18) z∈Zn

The best choice for S0 is the ILS pull-in region. Using equations (14) and (18) the optimal datadriven approach follows as: 1. Apply ILS to obtain aˇ 2. Evaluate the probability densities f ˇ (aˆ − a) ˇ and ˇ f aˆ (aˆ − a) 3. Determine μ based on the user-defined P f = β 4. Decision: f ˇ (aˆ − a) ˇ ≤μ use aˇ If (19) f aˆ (aˆ − a) ˇ >μ use aˆ Compare this result with the approach using the Ratio Test in section 3.4. In both approaches the test statistics are defined as a ratio. In the case of the Ratio Test, it only depends on ||aˆ − a|| ˇ 2Q aˆ and ||aˆ − 2 aˇ 2 || Q aˆ , whereas from equation (18) it follows that the optimal test statistic depends on all ||aˆ − z||2Q aˆ , z ∈ Zn if it is assumed that the float solution is normally distributed. In Teunissen and Verhagen (2004) the performance of the Ratio Test and Optimal integer aperture estimator were compared. It followed that often the Ratio Test performs close to optimal, provided that the fixed failure rate approach is used. Furthermore, with integer aperture estimation a shorter time to first fix can be obtained as compared to the model-driven approach of section 2 and the traditional Ratio Test with fixed critical value δ, while at the same time it is guaranteed that the failure rate is below a userdefined threshold.

References Euler, H. J. and Schaffrin, B. (1991). On a measure for the discernibility between different ambiguity solutions in the statickinematic GPS-mode. IAG Symposia no. 107,

Kinematic Systems in Geodesy, Surveying, and Remote Sensing. Springer-Verlag, New York, pages 285–295. Han, S. and Rizos, C. (1996). Integrated methods for instantaneous ambiguity resolution using new-generation GPS receivers. Proceedings of IEEE PLANS’ 96, Atlanta GA, pages 254–261. Hofmann-Wellenhoff, B. and Lichtenegger, H. (2001). Global Positioning System: Theory and Practice. Springer-Verlag, Berlin, 5 edition. Leick, A. (2003). GPS Satellite Surveying. John Wiley and Sons, New York, 3rd edition. Misra, P. and Enge, P. (2001). Global Positioning System: Signals, Measurements, and Performance. Ganga-Jamuna Press, Lincoln MA. Strang, G. and Borre, K. (1997). Linear Algebra, Geodesy, and GPS. Wellesley-Cambridge Press, Wellesley MA. Teunissen, P. J. G. (1993). Least squares estimation of the integer GPS ambiguities. Invited lecture, Section IV Theory and Methodology, IAG General Meeting, Beijing. Teunissen, P. J. G. (1998). GPS carrier phase ambiguity fixing concepts. In: PJG Teunissen and Kleusberg A, GPS for Geodesy, Springer-Verlag, Berlin. Teunissen, P. J. G. (1999a). An optimality property of the integer least-squares estimator. Journal of Geodesy, 73(11), 587–593. Teunissen, P. J. G. (1999b). The probability distribution of the GPS baseline for a class of integer ambiguity estimators. Journal of Geodesy, 73, 275–284. Teunissen, P. J. G. (2002). The parameter distributions of the integer GPS model. Journal of Geodesy, 76(1), 41–48. Teunissen, P. J. G. (2003a). Integer aperture GNSS ambiguity resolution. Artificial Satellites, 38(3), 79–88. Teunissen, P. J. G. (2003b). Theory of integer aperture estimation with application to GNSS. MGP report, Delft University of Technology. Teunissen, P. J. G. (2003c). Towards a unified theory of GNSS ambiguity resolution. Journal of Global Positioning Systems, 2(1), 1–12. Teunissen, P. J. G. (2004). Penalized GNSS ambiguity resolution. Journal of Geodesy, 78(4–5), 235–244. Teunissen, P. J. G. and Verhagen, S. (2004). On the foundation of the popular ratio test for GNSS ambiguity resolution. Proceedings of ION GNSS-2004, Long Beach CA, pages 2529–2540. Verhagen, S. (2004). Integer ambiguity validation: an open problem? GPS Solutions, 8(1), 36–43. Verhagen, S. and Teunissen, P. J. G. (2004). PDF evaluation of the ambiguity residuals. In: F Sansò (Ed.), V. Hotine-Marussi Symposium on Mathematical Geodesy, International Association of Geodesy Symposia, Vol. 127, Springer-Verlag. Wei, M. and Schwarz, K. P. (1995). Fast ambiguity resolution using an integer nonlinear programming method. Proceedings of ION GPS-1995, Palm Springs CA, pages 1101–1110.

Probabilistic Evaluation of the Integer Least-Squares and Integer Aperture Estimators S. Verhagen, P.J.G. Teunissen Delft Institute of Earth Observation and Space Systems (DEOS), Delft University of Technology, Kluyverweg 1, 2629 HS Delft, The Netherlands, e-mail: A.A. [email protected]

Abstract. The carrier phase observations start to act as very precise pseudorange observations once the ambiguities are resolved as integers. However, the integer ambiguity estimates should only be used if the reliability of the integer solution is high. The question is then how to assess this reliability. A wellknown a-priori reliability measure is the ambiguity success rate. But even with a high success rate, integer ambiguity validation remains indispensable in order to check whether or not a specific integer solution is sufficiently more likely than any other integer candidate. A solution to the integer validation problem is the use of integer aperture estimation. With this approach an aperture space is defined such that only float samples that fall into this space are fixed to the corresponding integer estimates, otherwise the float solution is maintained. The aperture space is built up of translationally invariant aperture pull-in regions centered at all integers. The size of these pullin regions is determined by the condition of a fixed failure rate. In this contribution, we will present the probabilistic measures that can be used to assess the reliability of the integer least-squares and the integer aperture ambiguity estimators, as well as the reliability of the corresponding baseline estimators. These probabilities will also be evaluated in the presence of a bias in order to study the bias-robustness of the integer ambiguity estimators. A case study is carried out with several GNSS models, which shows that the integer aperture estimator has some favorable probabilistic properties as compared to integer least-squares estimation, both in the unbiased and in the biased case.

difference (DD) carrier phase ambiguities, a. The first group is referred to as the baseline unknowns, b. The ‘float’ estimators of the unknown parameters and their variance-covariance (vc-) matrix are obtained after a standard least-squares adjustment:

Keywords. Integer least-squares, integer aperture estimation, bias-robustness

(aˆ − a) ˇ bˇ = bˆ − Q bˆ aˆ Q −1 aˆ

1 Integer Estimation A GNSS model generally contains real-valued and integer-valued parameters. The latter are the double

aˆ bˆ

;

Q aˆ Q aˆ bˆ

Q aˆ bˆ Q bˆ

(1)

In the next step, the float ambiguities are fixed to integer values, which is referred to as ambiguity resolution: aˇ = S(a) ˆ

(2)

where S : Rn −→ Zn is the mapping from the n-dimensional space of real numbers to the n-dimensional space of integers. The optimal result in the sense of maximizing the probability of correct integer estimation (success rate) is obtained using integer least-squares (ILS), cf. Teunissen (1993, 1999). The ILS ambiguity estimator is given by: aˇ = arg min ||aˆ − a|| ˇ 2Q aˆ

(3)

An efficient implementation of ILS estimation is provided by the LAMBDA method, see e.g. (Teunissen, 1993; Hofmann-Wellenhoff and Lichtenegger, 2001; Strang and Borre, 1997; Teunissen, 1998; Misra and Enge, 2001). Finally, the float baseline estimator is adjusted by virtue of its correlation with the ambiguities, giving the ‘fixed’ baseline estimator: (4)

Ambiguity resolution should only be applied when there is enough confidence in its results. A wellknown reliability measure is the success rate. The success rate equals the probability of correct integer estimation: 149

150

S. Verhagen, P.J.G. Teunissen

PS = P(aˇ = a) = P(aˆ ∈ Sa )

(5)

with a the true, unknown integer vector. Sz is the pull-in region centered at the integer z; it contains all real-valued vectors that will be mapped to the same integer z. The pull-in region of any admissible integer estimator should fulfill the following conditions: ⎧ Sz = Rn ⎪ ⎨ z∈Z n Int(Su ) Int(Sz ) = ∅, ⎪ ⎩ Sz = z + S0 ,

∀u, z ∈ Zn , u = z ∀z ∈ Zn (6)

where ‘Int’ denotes the interior of the subset. It is generally required that the success rate of integer estimation should be very close to one, or equivalently the failure rate, P f = 1− Ps , should be close to 0. However, there is no general rule which states how large the success rate should be in order to guarantee a reliable fixed solution. What a user essentially would like to know is the probability that the fixed ˇ is better than the float solution, baseline solution, b, ˆb, and the effect on the baseline estimate if the ambiguities are fixed incorrectly. Hence, evaluation of the probability that the fixed estimator is closer to the true but unknown b than the float estimator, P(||bˇ − b|| ≤ ||bˆ − b||),

(7)

is not enough, since even if this probability is larger than 0.5 it does not tell how large the position error can become in the cases that bˇ is not better than ˆ Therefore, also the probabilities that the position b. error is larger than a certain value, β, should be considered: P(||bˇ − b|| ≥ β)

and

P(||bˆ − b|| ≥ β) (8)

2 Integer Aperture Estimation In practice, a user may decide not to use the fixed solution if the failure rate is too high. This gives rise to the thought that it might be interesting to use an ambiguity estimator defined such that three situations are distinguished: success if the ambiguity is fixed correctly, failure if the ambiguity is fixed incorrectly, and undecided if the float solution is maintained. This can be accomplished by dropping the condition that there are no gaps between the pull-in regions, so that the only conditions on the pull-in regions are that they should be disjunct and translationally invariant. Then integer estimators can be determined that somehow regulate the probability of each of the three situations mentioned above. The new class of ambiguity estimators was introduced in Teunissen (2003), and is called the class of Integer Aperture (IA) estimators. The IA ambiguity estimator is given by:

a¯ =

aˇ aˆ

if aˆ ∈ ⍀ otherwise

(9)

with ⎧ ⍀ =⍀ ⎪ ⎪ ⎨ z∈Zn z

Int (⍀u ) ∩ Int (⍀z ) = ∅, ⎪ ⎪ ⎩ ⍀ z = z + ⍀0 ,

∀u, z ∈ Zn , u = z ∀z ∈ Zn (10) ⍀ ⊂ Rn is called the aperture space. It follows that this space is built up of the Ωz , which will be referred to as aperture pull-in regions. Besides the success and failure rates, in this case also the undecided rate, Pu , must be considered: Ps = P(aˆ ∈ ⍀a ) P f = P(aˆ ∈ ⍀\ {⍀a })

(11)

Pu = P(aˆ ∈ / ⍀a ) = 1 − Ps − P f Unfortunately, none of the probabilities in equation (7) and (8) can be evaluated exactly. In Verhagen (2005) it was empirically shown that it can be ˇ ˆ expected that P(||b−b|| ≥ β) > P(||b−b|| ≥ β) for small β, but that for large β the opposite is true. This is an indication that the position error due to incorrect ambiguity fixing may be very large. This can be further evaluated by considering the probabilities that the position errors are larger than β if the ambiguities are known to be fixed incorrectly. For instance, if the failure rate P f is considered large and P(||bˇ − b|| ≥ ˆ β | aˆ ∈ / Sa ) > P(||b−b|| ≥ β | aˆ ∈ / Sa ) for all values of β, then the float solution should be preferred.

The approach of integer aperture estimation allows us to choose a threshold for the failure rate, and then determine the size of the aperture pull-in regions such that indeed the failure rate will be equal to or below this threshold. So, applying this approach means that implicitly the ambiguity estimate is validated using a sound criterion. However, there are still several options left with respect to the choice of the shape of the aperture pull-in regions. In the class of integer estimators the ILS estimator is optimal as it maximizes the success rate. Likewise, the Optimal IA estimator can be defined as the one which maximizes the

Probabilistic Evaluation of the Integer Least-Squares and Integer Aperture Estimators

success rate on the condition that the failure rate is equal to a fixed value α, cf. Teunissen (2005). Hence, the aperture pull-in region, ⍀0 is obtained by solving: subject to P f = α

max Ps ⍀0

Ps Ps + P f

(13)

It can be easily shown that with Optimal IA estimation also the conditional success rate is maximized. Obviously, Ps + P f equals the probability of an integer outcome. Hence, the conditional success rate will be close to one when the failure rate is chosen close to zero and Ps >> P f . This implies that one can have a very high confidence indeed in the correctness of the integer outcomes of the OIA estimator, even for modest values of the unconditional success rate. With ILS estimation such a high confidence can only be reached once the ILS success rate is close to one, cf. Section 1. In order to compare the reliability of the integer ¯ with that of the float aperture baseline estimator, b, and fixed estimators, probabilities equivalent to those in equations (7) and (8) can be evaluated: P(||b¯ − b|| ≤ ||bˆ − b||) and P(||b¯ − b|| ≥ β) (14)

3 Effect of a Bias So far, the reliability of integer estimation has only been considered under the assumption that no model error is present. However, incorrect ambiguity fixing can also be due to the presence of a bias. Assume that the observations are biased, and the corresponding biased float solution is given as:

bˆ∇ aˆ ∇

=

bˆ + ∇ bˆ aˆ + ∇ aˆ

The corresponding biased fixed solution will be denoted as aˇ ∇ and bˇ∇ . Note that: ˆ + S(∇ a) ˆ aˇ ∇ = aˇ + ∇ aˇ = S(aˆ ∇ ) = S(a)

(16)

(12)

Note that for α ≥ P(aˆ ∈ / Sa ), OIA estimation is identical to ILS estimation. The unconditional success rate, P(a¯ = a), is not very useful as a reliability measure for integer aperture estimation, since there is always a probability of undecidedness. Hence, the unconditional success rate will always be lower than or, in very favorable conditions, equal to the success rate of ILS estimation, cf. equation (5). But at the same time the failure rate is always smaller or equal. Instead of the unconditional success rate one should therefore consider the conditional success rate, which is the probability that an integer outcome is correct: = P(a¯ = a | aˆ ∈ ⍀) = Ps|a∈⍀ ˆ

151

(15)

Hence, the effect of the bias on the fixed ambiguities, ∇ aˇ does not only depend on the bias itself, but also on the float solution. The bias in the fixed baseline solution follows as: (∇ aˆ − ∇ a) ˇ (17) ∇ bˇ = bˇ∇ − bˇ = ∇ bˆ − Q bˆ aˆ Q −1 aˆ Of course it is not possible to make general statements on how a bias in the float solution will affect the fixed baseline solution, since that depends on the type and the size of the bias as well as on the normal uncertainty of the float solution. The first question is how the float solution itself is affected by the bias: are the ambiguities and baseline both severely affected, or not? The next question is how the bias in the float ambiguities affects the probability of correct fixing. The probability of correct ambiguity fixing in the presence of a bias is called the bias-affected success rate, cf. Teunissen (2001), and can be computed once the bias in the float ambiguities, ∇ a, ˆ is known: Ps∇ = P(aˇ ∇ = a) = P(aˆ + ∇ aˆ ∈ Sa )

(18)

If the bias-affected success rate is close to the success rate, obviously the ambiguity estimator is not sensitive to the bias. However, this does not tell us how the fixed baseline solution is affected. As can be seen in equation (17), the bias in the fixed baseline solution ˆ ∇ aˆ and ∇ a. depends on ∇ b, ˇ It is interesting though, to evaluate the probabilities in equations (7) and (8) for the biased float and fixed estimators, and to compare these probabilities with those of the unbiased equivalents. Finally, it is interesting to study the bias sensitivity of the OIA estimator. The bias-affected success rates, both the conditional and the unconditional, will be lower than in the unbiased situation. This becomes clear from the two-dimensional example in Figure 1, where the contours of the PDF of the float ambiguities are shown for the unbiased situation on the left, and for two biased cases in the center and on the right. The smaller regions inside the ILS pull-in regions (hexagons) depict the aperture pull-in regions corresponding to a failure rate of 0.005 (in the unbiased case). It can be expected that if the precision of the float solution is not so good, and consequently the integer least-squares success rate is low, that relatively

152


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Fig. 1. Probability distributions of the biased float ambiguities. The true integer is a = 0, the bias is depicted with the arrow. The aperture pull-in regions and ILS pull-in regions (hexagons) are shown. Left: no bias; Center: bias of [0.5 0]; Right: bias of [1 0].

small biases in the float ambiguities will be absorbed in the noise. In that case there is namely a high probability of large ambiguity residuals, and thus a large undecided rate with IA estimation. This remains unchanged in the presence of the bias. The opposite may be true for stronger models. In that case it really depends on the size of the bias whether or not the IA estimator is sensitive to it. If the bias is such that almost all probability mass of the float ambiguities is transferred to the wrong pullin region, as in the right panel of Figure 1, this will certainly not lead to more rejected fixed ambiguities. In fact, there is a very high probability then that the ambiguities are fixed incorrectly. If the bias causes a high probability that the float ambiguity will be close to the boundary of a pull-in region, as in the middle panel of Figure 1, there is some sensitivity to the bias, as more samples will be rejected. At the same time, the bias-affected success rate will be lower and the bias-affected failure rate will be higher. If the bias is such that most of the probability mass is still in the correct aperture pull-in region ⍀a there is a low sensitivity to the bias, but that is good, since then the bias-affected success rate will be close to the success rate in the unbiased situation. Apart from the conditional and unconditional success rates, also the probabilities in Eq.(14) should be evaluated for the biased IA estimator.

4 Case Study The probabilities presented in this paper are evaluated for a dual-frequency GPS model, with four satellites visible. The results are representative for practical situations; other models have been considered but the results are not shown here. For the biased situation, an unmodeled ionospheric delay is

considered; the resulting bias in the float baseline solution is ∇ bˆ = 4 cm. The results are shown in Table 1 and Figure 2. Some important conclusions will be discussed here. 1. If Ps >> P f the fixed baseline estimator is better than the float baseline estimator, but with the fixed estimator there is a higher probability of a very large position offset than with the float estimator due to its multi-modal distribution. From the probabilities in Table 1 follows that in most cases bˇ − b ≤ bˆ − b . Also from the graphs in Figure 2 it follows that there is indeed a ˆ but there high probability of bˇ being better than b, ˇ is also a higher probability that b is very far off the true b. It follows that if aˆ ∈ / Sa there is a risk of a very large position offset bˇ − b as compared to

bˆ − b . 2. Ambiguities should be fixed if one can have enough confidence in their correctness. In all cases that the ambiguities are fixed correctly

bˇ − b will be very small, as follows from the Table 1. ILS and OIA probabilities, with Pb = P( b¯ − b ¯ − b ≤ b∇ ˆ −b ≤ bˆ − b | aˆ ∈ ⍀) and Pb∇ = ( b∇ | a∇ ˆ ∈ ⍀). Note that for ILS ⍀ = Rn

ILS

OIA

Pf Ps Ps|a∈Ω ˆ Pb

0.183 0.817 0.817 0.862

0.050 0.655 0.929 0.946

0.005 0.330 0.985 0.989

P∇ f Ps∇ Ps|∇a∈Ω ˆ

0.689 0.311 0.311

0.275 0.169 0.381

0.056 0.046 0.450

0.549

0.604

0.653

Pb∇

Probabilistic Evaluation of the Integer Least-Squares and Integer Aperture Estimators

1

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OIA, Pf = 0.005

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biased float biased fixed biased fixed, amb. wrong biased fixed, amb. correct

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float fixed fixed, amb. wrong fixed, amb. correct

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20

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Fig. 2. Baseline probabilities P( be − b ≥ β), where be is either the float, fixed or OIA baseline estimator. The probabilities conditioned on the ambiguities being fixed correctly and incorrectly are also shown for the float and fixed estimators in the left panels. Top: unbiased situation. Bottom: biased situation.

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graph of the probability P( bˇ − b ≥ β | aˆ ∈ Sa ) in Figure 2 (top left). This conclusion confirms the importance of integer ambiguity validation and the need for appropriate reliability measures for the fixed estimator, since a user does not want to take the risk of using a fixed position which is much further away from the true position than the float estimate. IA estimation provides the means for using the fixed estimates only if there is enough confidence in their correctness. The graphs in Figure 2 show that indeed the risk of a very large position offset due to incorrect fixing is smaller than with the fixed estimator. At the same time there is a lower probability of

b¯ − b being small than of bˇ − b being small. The IA estimator can be considered as a good compromise between the float and fixed estimator. Furthermore it follow that b¯ is generally closer to the true b than bˆ if aˆ ∈ ⍀, and that P(||b¯ − b|| ≤ ||bˆ − b|| |aˆ ∈ ⍀)

(19)

increases with a decreasing failure rate. Hence, the probability in equation (19) will be lower than with ILS estimation if the ILS failure rate is larger than the user-defined value α. Apparently, one can be quite sure that if the fixed solution is accepted it is indeed better than the float solution. Finally note that the conditional success rates are higher than the integer least-squares success rate. 3. If the float solution is biased, the probability of the fixed baseline estimator being better than the float estimator becomes lower than in the unbiased situation. The baseline probabilities are all lower in the biased situation, except for the probability P( bˇ − b ≤ bˆ − b | aˆ ∈ / Sa ), which means that there is a somewhat higher probability that the fixed baseline estimator is better than the float if the ambiguities are fixed incorrectly. 4. Integer Aperture estimation does provide some protection, but is not a safeguard against biases. The bias in the float baseline solution is small (4 cm) and not visible in the corresponding graph in Figure 2 (bottom left). However, there is a much higher probability of a large offset bˇ∇ − b as compared to the unbiased situation. If, on the other hand, the ambiguities are fixed correctly, the fixed baseline estimator still performs much better than its float counterpart.

Hence, also in the biased situation IA estimation may offer a good compromise between the float and fixed estimator, since the undecided rate will be higher. As expected the probability that b¯∇ − b ≤ bˆ∇ − b if a¯ = aˇ is smaller than in the unbiased situation, but still is higher than the probability that

bˇ∇ − b ≤ bˆ ∇ − b . Note that in the biased situation the failure rate is larger than the threshold value due to the incorrect model assumption. But, of course, the bias-affected failure rate is still lower than the corresponding biasaffected ILS failure rate. In Figure 2 it can be seen that also in the biased situation, the probabilities P( b¯∇ − b ≥ β) are in between those of the float and fixed estimators. The probability is almost equal to that of the float estimator when the failure rate is chosen small, since then the bias-affected success rate is small too, and thus the undecided rate is large. Obviously, the biased fixed estimator has a higher probability of a large position offset as compared to the biased IA estimator. The difference is even larger than in the unbiased situation.

References Hofmann-Wellenhoff, B. and Lichtenegger, H. (2001). Global Positioning System: Theory and Practice. Springer-Verlag, Berlin, 5 edition. Misra, P. and Enge, P. (2001). Global Positioning System: Signals, Measurements, and Performance. Ganga-Jamuna Press, Lincoln, MA. Strang, G. and Borre, K. (1997). Linear Algebra, Geodesy, and GPS. Wellesley-Cambridge Press, Wellesley MA. Teunissen, P. J. G. (1993). Least squares estimation of the integer GPS ambiguities. Invited lecture, Section IV Theory and Methodology, IAG General Meeting, Beijing. Teunissen, P. J. G. (1998). GPS carrier phase ambiguity fixing concepts. In: PJG Teunissen and Kleusberg A, GPS for Geodesy, Springer-Verlag, Berlin. Teunissen, P. J. G. (1999). An optimality property of the integer least-squares estimator. Journal of Geodesy, 73(11), 587–593. Teunissen, P. J. G. (2001). Integer estimation in the presence of biases. Journal of Geodesy, 75, 399–407. Teunissen, P. J. G. (2003). Integer aperture GNSS ambiguity resolution. Artificial Satellites, 38(3), 79–88. Teunissen, P. J. G. (2005). GNSS ambiguity resolution with optimally controlled failure-rate. Artificial Satellites, 40(4), 219–227. Verhagen, S. (2005). The GNSS integer ambiguities: estimation and validation. Ph.D. thesis. Publications on Geodesy, 58, Netherlands Geodetic Commission, Delft.

The Evaluation of the Baseline’s Quality Based on the Probabilistic Characteristics of the Integer Ambiguity R. Xu, D. Huang, C. Li, L. Zhou, L. Yuan Center for Geomation Engineering, Southwest Jiaotong University, Chengdu, P.R. China

Abstract. The ‘fixed’ baseline estimator is much more rigorous in studying and evaluating the probabilistic properties of the GPS baseline than the ‘conditional’ used in common practice, while it is very complicated to implement in practice because of its non-normal distribution. In order to simplify its application, the effects of variance and correlation coefficient on the baseline estimator’s PDF (Probability Distribution Function) were studied in this paper. The studied results show that the PDF of the ‘fixed’ baseline estimator and the ‘conditional’ baseline estimator can approach consistent by choosing the two parameters reasonably. Keywords. GPS, probability distribution function, evaluation, baseline’s quality, consistency

1 Introduction As was pointed out in Teunissen (1996, 1999), the estimated ambiguities have their own probability distribution, despite their integerness. For a proper evaluation of the GPS baseline, the random characteristics of the integer ambiguity should be taken into account as well. However, because of the non-normal distribution of the ‘fixed’ baseline estimator’s PDF (probability distribution function), it is not a trivial job to implement in common practice. In order to overcome its complication in application, the effects of variance and correlation coefficient on the PDF were studied in this paper to uniform the ‘fixed’ baseline estimator with the ‘conditional’ baseline estimator, a simple one being widely used. This paper is organized as follows. In Sect. 2 we review the three kinds of the baseline estimators. In Sect. 3 the closed-form expression of the PMF (Probability Mass Function) of the ‘fixed’ baseline and the PDF of the other two baseline estimators are introduced, and these result are then used in Sect. 4 to study how the distributions of the ‘fixed’ baseline and ‘conditional’ baseline can approach consistent

between each other. In Sect. 5 some useful conclusions and remarks are given.

2 Three Baseline Estimators The relationship between the baseline estimator B and the estimated ambiguity A can be cast in the following equation: (aˆ − A) B = bˆ − Q bˆ aˆ Q −1 aˆ

(1)

where aˆ is the float solution of the ambiguity, bˆ is the float solution of the baseline; Q aˆ and Q bˆ aˆ are their corresponding variance-covariance matrix. As was introduced in Teunissen (1998, 1999, 2000, 2001, 2002), three different baseline estimators can be obtained from equation (1), which read that: ⎧ ˆ ⎪ solution), ⎨ b(‘float’ bAmb=z (‘conditional’ solution), B= ⎪ ⎩ b(‘fixed’ solution),

A = aˆ A=z

A=a (2) where z and a are both the integer solutions of the ambiguity, while the integer vector a is random and z is not.

3 The Distributions of the ‘Fixed’ Baseline Let p be the PDF of the ’fixed’ baseline, q be the b (x) dimension of the baseline, then according to Teunissen (1998, 1999, 2000, 2001, 2002), we have: P (x) = b

z∈Z n

p (x|y = z)P(a = z) b|a

(3)

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where the conditional distribution: p = pbˆ aˆ (x, y)/ p (y = z) a b |a 1

=

exp

1q det Q (2π ) 2 b|a

⎧ ⎨ ⎩

− 12 x − b

⎫ ⎬

2

a =z Q ⎭ b|a

P(bˆ b|ˆ aˆ ∈ R)P(aˆ = a) ≤ P(bˆ ∈ R) ≤ P(bˆ b|ˆ a=a ∈ R) ˆ (8) where P(bˆ ∈ R) = P(χ 2 (q, 0) ≤ β 2 ). ˆ a =a b|

Other important remarks and details have been given in Teunissen (1999, 2001, 2002), and we will just use these useful results for our further research.

(4)

with conditional mean b|a=z = b − Q bˆ aˆ Q −1 ˆ aˆ (a − = Q bˆ − z), conditional variance matrix Q b| ˆ aˆ −1 2 T −1 Q bˆ aˆ Q aˆ Q aˆ bˆ and · Q = (·) Q (·). Additionally, the p(a = z) in equation (3) is the success rate (Teunissen, 2000) of the integer ambiguity and it could be computed by simulation if the LAMBDA method was used, Verhagen (2003). The corresponding PMF of the above PDF can be obtained for an admissible confidence region R:

P(b ∈ R) =

p dx b (x) = pb|ˆ aˆ (x|y = z) P(a = z) R

R

z∈Z n

(5) As is proposed in Teunissen (1999, 2001), R can take the form as follows: R = {x ∈ R q |(x − b)T Q −1 (x − b) ≤ β 2 } ˆ b|aˆ

(6)

where the size of the confidence region can be varied by the confidence level β 2 . With this submitted to equation (5), then

P(b ∈ R) =

z∈Z n

=

R

4 Test Results 4.1 The Sketch Maps of the Distributions For convenience, a uniform format was used in the following part of the paper to explain the precision of the float solutions of the ambiguity and the baseline. Example (0.5, 0.5, 0.75) represents that the accuracy of the float solutions is (0.5, 0.5) and their correlation coefficient is 0.75. Additionally, their expectations are assumed to be zero. To obtain the probability distribution of the ‘fixed’ baseline estimator, the joint distribution of the ‘fixed’ solutions f (u, w) should be depicted at first as b ,a shown in Figure 1. Once f (u, w) is derived, the PDF of the ‘fixed’ b ,a baseline f (u) can be obtained by the integral of b f (u, w) along the w-axis. The integral result is b ,a shown in Figure 2. Figure 3 shows the probability distribution of the three estimators. Where, the full curve, the dashed curve and the dash dotted curve represent the distribution of the ‘fixed’, the ‘float’ and the ‘conditional’ baseline estimator respectively. From Figure 3, it can be seen that the peak of the ‘fixed’ baseline’s PDF is greater than that of the ‘float’ one. But it can not be concluded that the precision of the ‘fixed’ baseline

Pb| ˆ aˆ (x|y = z) P(a = z)

χ 2 (q, λz ) ≤ β 2 P(a = z)

z∈Z n

(7) where χ 2 (q, λz ) denote the non-central Chi-square distribution with p degrees of freedom and noncentrality parameter λz , and 2 λz = ∇ b z

Q bˆ aˆ

and ∇ bˆ z = Q bˆ aˆ Q −1 (z − a) aˆ

It is difficult to evaluate this infinite sum exactly, while the bounds interval can be easily obtained as is shown in Teunissen (1999):

Fig. 1. PDF of f (u, w). b,a

The Evaluation of the Baseline’s Quality Based on the Probabilistic Characteristics of the Integer Ambiguity

Fig. 2. PDF of f (u).

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Fig. 3. PDF of the three baseline estimators.

b

is absolutely better than that of the ‘float’ solution because there are fluctuations on its slide-lobes. In addition, the relatively low success rate of the integer ambiguity leads to a significant difference between the ‘conditional’ baseline estimator’s PDF and that of ‘fixed’ estimator, which makes it not suitable for the variance to evaluate the precision of ‘fixed’ baseline as usual. Otherwise, fatal results would be derived. In common practice, it is often expected that the variance components can be used to evaluate the quality of the baseline straightly rather than with the PMF of the ‘fixed’ estimator. After all, evaluating the PMF is not easy. So, in the next section, we will put more efforts into studying the effects of variance and correlation coefficient on these two probability distributions to realize their consistency. 4.2 The Realization of the Consistency of the Two Distributions Two methods can be used to realize the consistency of the two probability distributions as mentioned before.

Method 1: leave the variance and the confidence level β 2 unchanged and decreased the correlation coefficient of the float solutions. The results are shown in Table 1. In addition, the results when both the parameters were improved are also shown in the last line of Table 1. The three sub figures in Figure 4 record the change of the three distributions when the correlation coefficient of float solutions ranges from 0.95 to 0.05. It can be seen from the Table 1 that method 1 can be used to realize the consistency of the two probability distributions. We can also find that the probability distribution of the ‘float’ baseline estimator finally approaches uniform with the aforementioned two distributions. From a theoretical point of view, keeping the variance component unchanged and decreasing the correlation coefficient means to reduce the covariance of the float solutions. And with the decrease of covariance, the precision of the ‘float’ baseline and the ‘conditional’ baseline approach equal and only little influence of the resolved ambiguities can act on the baseline solution. So, when the PDF of these two estimators approaches uniform,

Table 1. The concentration of the baseline solution with different correlation coefficient of the float solutions

Confidence level

Accuracy of the float solution

Correlation coefficient

Success rate

concentration

Upper bound

Lower bound

0.99 0.99 0.99 0.99 0.99 0.99

(1, 1) (1, 1) (1, 1) (1, 1) (1, 1) (0.5, 0.5)

0.95 0.75 0.55 0.35 0.05 0.75

0.383 0.388 0.384 0.381 0.378 0.689

0.533 0.903 0.966 0.987 0.989 0.940

0.379 0.384 0.380 0.377 0.374 0.682

0.99 0.99 0.99 0.99 0.99 0.99

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Fig. 4. Comparisons of the three distributions with the correlation coefficient of float solutions ranging from 0.95 to 0.05.

the ‘fixed’ solution in the ‘middle’ position also approaches consistent with them. However, it should be noted that, with the reduction of the correlation coefficient, the peaks of the two distributions both reduce dramatically until they approach uniform with the ‘float’ baseline estimator. Because the ‘float’ baseline estimator is the most inaccurate one among the three estimators, we can conclude that the PDF of the ‘fixed’ baseline estimator goes towards the ‘worse’ direction to realize the consistency with the PDF of the ‘conditional’ one. In essential, with the decrease of correlation coefficient, the corresponding conditional variance of the ambiguity increases rapidly. On that condition, if the β 2 is kept unchanged, the integral region R will expand rapidly. Therefore, the increase of concentration P(b˘ ∈ R) in Table 1 is resulted from the enlargement of its integral region essentially. So, Method 1 is not suitable. Method 2: leave the correlation coefficient and confidence level unchanged and improve the accuracy of the float solutions. The results are shown in Table 2. The results when both the parameters were improved are also shown in the last line of the table. Figure 5 record the change of the three distributions when the variance of float solutions was decreased from (0.9, 0.9) to (0.1, 0.1). It can be

observed that, when the precision of the float solutions is improved, the peak value of the probability distribution of the ‘fixed’ baseline increase from 2.0 to about 43.0, and it is almost the same to that of the ‘conditional’ baseline. So, in contrast to the method 1, method 2 can make the two distributions go towards the ‘better’ direction to realize their consistency and it should be an ideal method. In essential, this is because that improving the precision of the ambiguity means to improve the success rate of the integer ambiguity estimation, and when the success rate is close to 1, the expressions of the two baseline estimators’ PDF approach equal. In theory, it should also be the essential reason why we often strive to improve the accuracy of the float solutions in common practice.

5 Concluding Remarks Although the PDF of the ‘fixed’ baseline estimator could be a rigorous measure of the GPS baseline’s quality, it is somewhat complicated to implement in practice because of its non-normal distribution. Therefore, it is often expected that the ‘fixed’ baseline estimator could be replaced by the ‘conditional’ one to evaluate the quality of the ambiguity resolved baseline. It is also the main research subject of this paper.

Table 2. The concentration of the baseline solution with different variance of the float solutions

Confidence level

Accuracy of the float solution


Success rate

concentration

Upper bound

Lower bound

0.99 0.99 0.99 0.99 0.99 0.99

(0.9, 0.9) (0.7, 0.7) (0.5, 0.5) (0.3, 0.3) (0.1, 0.1) (0.9, 0.9)

0.95 0.95 0.95 0.95 0.95 0.75

0.423 0.518 0.683 0.900 1.000 0.430

0.548 0.577 0.690 0.892 0.990 0.914

0.419 0.513 0.676 0.891 0.990 0.426

0.99 0.99 0.99 0.99 0.99 0.99

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Fig. 5. Comparisons of the three distributions with the variance of float solutions ranging from (0.9, 0.9) to (0.1, 0.1).

In this paper, two methods were tried to realize the consistency of the two baseline estimators, the ‘conditional’ baseline estimator and the ‘fixed’ baseline estimator. As was shown in the test results, improving the correlation coefficient and the precision of the float estimations both can realize our goals, while only the later one could be an ideal choice.

References Teunissen PJG (1996), The Mean and the Variance Matrix of the ‘Fixed’ GPS Baseline. LGR. Delft University of Technology.

Teunissen PJG (1998), On the integer normal distribution of the GPS ambiguities. Artificial satellites 33(2):1–13. Teunissen PJG (1999), The probability distribution of the GPS baseline for a class of integer ambiguity estimators. J Geod 73:275–284. Teunissen PJG (2000), The success rate and precision of GPS ambiguities. J Geod 74: 321–326. Teunissen PJG (2001), The probability distribution of the ambiguity bootstrapped GNSS baseline. J Geod 75:267–273. Teunissen PJG (2002), The parameter distribution of the integer GPS model. J Geod 76:41–48. Verhagen (2003), On the approximation of the integer least squares success rate: which lower or upper bound to use? J Global Positioning Syst 2:117–124.

Kinematic GPS Batch Processing, a Source for Large Sparse Problems M. Roggero Politecnico di Torino, Department of Land Engineering, Environment and Geotechnologies, Corso Duca degli Abruzzi 24, 10129, Torino, Italy, e-mail: [email protected]

Abstract. In kinematic observation processing the equivalence between the state space approach (Kalman filtering plus smoothing) and the least squares approach including dynamic has been shown (Albertella et al., 2006). We will specialize the proposed batch solution (least squares including dynamic), considering the case of discrete-time linear systems with constant biases, a case of practical interest in geodetic applications. A discrete-time linear system leads often to large sparse matrices, and we need efficient matrix operation routines and efficient data structure to store them. Finally, constant biases are estimated using domain decomposition methods. Simulated and real data examples of the technique will be given for kinematic GPS data processing.

1 Introduction Kinematic surveying and navigation usually maintain different approaches for the optimal parameters estimation: while Kalman filtering is the typical method for real time navigation, Kalman filtering plus Kalman smoothing is the method generally used to post process kinematic data. Has been demonstrated that, alternatively to a Kalman filtering plus Kalman smoothing solution can be used a different approach, called batch solution by the authors (Sansò et al., 2006). This method, with some algebraic expedient, allows obtaining least squares solutions equivalent to Kalman solutions with a comparable computational load. Has been numerically shown by the authors the equivalence between the state space approach and the batch solution. State space approach presents advantages and disadvantages, as already discussed in (Colomina and Blàzquez, 2004). The main advantages are the realtime parameter estimation capability and the need to invert only small matrices, but the connectivity of parameters through static observation or constrain equations is not supported. Then filter divergence 160

is possible, and last the computation of covariance matrices for all the state vectors cannot be avoided. On the other hand, we will take advantage of some characteristics of the batch solution: the support for connectivity of parameters regardless of time and the possibility to compute the covariance of a limited number of selected parameters. We have the disadvantage of very large systems of linear equations, but this will be overcome thanks to domain decomposition and to the sparse structure of matrices. However, real-time parameter estimation is not feasible in general. We move now from the proposed geodetic solution, to study discrete-time linear systems with constant biases, a case of practical interest: integer ambiguities in carrier phase observations, constant offset and drift in accelerometers and gyros, lever arm and bore sight angles in direct photogrammetry are only some examples quite common in geodetic applications. A discrete-time linear system leads often to large sparse matrices, and we need to perform efficiently some matrix operation, especially the inversion. Moreover we need to study efficient data structure to store them. The native data structure for a matrix is a two dimensional array. Each entry in the array represents an element ai, j of the matrix and can be accessed by the two indices i and j . For a n×m matrix we need at least (n · m)·8 bytes to represent the matrix when assuming 8 bytes for each entry. A sparse matrix contains many (often mostly) zero entries. The basic idea when storing sparse matrices is to only store the non-zero entries as opposed to storing all entries. Depending on the number and distribution of the non-zero entries, different data structures can be used and yield huge savings in memory when compared to a native approach (Saad, 2000). Finally, constant biases are estimated using domain decomposition methods, that allows to estimate a subset of the unknown parameters, requiring only the inversion of small or banded matrices.

Kinematic GPS Batch Processing, a Source for Large Sparse Problems

An application of the technique to real data will be given for kinematic GPS data processing, where float ambiguities are estimated via Schur decomposition, and where system dynamic strengthen ambiguities fixing. After fixing the integer carrier phase ambiguities by means of LAMBDA method, we will come back to observation domain, to solve for the other non constant system unknown parameters.

Let us consider a discrete-time linear system described by a finite state vector x and constant biases b evolving with known dynamics trough the epochs t (t = 1, . . ., T ): x t +1 = Tt +1 x t + Bt +1bt + νt +1 (1)

bt +1 = bt where y are the observations and the last equation shows that the bias vector b is a constant. The matrix B and C link the bias vector to the system dynamic and to the observations respectively. Then we have also: T transition matrix (dynamics) H design matrix ν system noise with covariance matrix Rνν ε observation error with covariance matrix Rεε A new state vector z can be defined including the constant bias vector, so the state and observation equation can be expressed as z t +1 = Z t +1 z t + Gνt +1 yt +1 = L t +1 z t +1 + εt +1

(2)

where

zt =

xt bt

p n ↔ ⎤ ⎡↔ Tt Bt Z t = ⎣ ---- ---- ⎦ np 0 I

p n ;

Lt =

p

↔

Ht

n

↔

Ct

These equations yield to the system Dz = ω Mz + ε = y

(3)

⎡

⎡

I .. .

x1 ⎢ .. ⎥ ⎢ ⎥ z = ⎢ . ⎥; ⎣ xT ⎦ b ⎡ ⎢ ⎢ M=⎢ ⎣

I p G= 0 n

s

(4)

(5)

−B1 −B2 .. .

..

. −TT ⎡

⎤

⎢ ⎢ ω=⎢ ⎣

I

⎥ ⎥ ⎥; ⎦

ν1 + T1 X 0 + B1 b0 ν2 .. .

⎤ ⎥ ⎥ ⎥ ⎦

νT

H2

⎡

⎤

−BT

H1

⎤

y1 ⎢ .. ⎥ y = ⎣ . ⎦; yT

..

C1 C2 .. .

.

HT ε1 ⎢ .. ε=⎣ . ⎡

⎤ ⎥ ⎥ ⎥; ⎦

CT ⎤ ⎥ ⎦

εT

The system (6) has the optimal solution −1 zˆ = D T Wω D + M T Wε M M T Wε y

(7)

with covariance matrix of the estimation error e = x − xˆ −1 Ree = D T Wω D + M T Wε M

(8)

It is interesting for us to note that the normal matrix (that is D T Wz D + M T Wε M) has a quasi-triangular Schur form. This reduced form will be very useful to solve for the constant bias vector, ignoring the other non constant unknown parameters. Before computing the normal matrix, we note that also the weight matrix Wz for the time t can be partitioned, because of

(6)

The matrix D and M, with the related vectors z, ω, y and ε, are ordered and partitioned as follows I ⎢ −T2 ⎢ D=⎢ ⎣

2 Discrete-Time Linear System with Constant Biases

yt +1 = Ht +1 x t +1 + Ct +1 bt +1 + εt +1

161

Wz = Rz−1 =

Rω 0 ---------0 Rb

−1

=

Wω 0 ----- ----0 Wb

(9) and that Wω and Wb are diagonal with dimensions p and n. Fortunately, the large coefficient matrix to

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be inverted has a bordered block or band-diagonal structure, so the large matrix can thus be blockwisely inverted by using the following analytic inversion formula:

A C

B D

−1

=

A−1 + A−1 B S −1 C A−1

− A−1 B S −1

−S −1 C A−1

S −1

(10)

where A, B, C and D are generic matrix sub-blocks of arbitrary size. This strategy is particularly advantageous if Ais large and block or band diagonal matrix and S = D − C A−1 B (the Schur complement of A) is a small matrix, since they are the only matrices requiring to be inverted. The structure of the normal matrix is given in Table 1. As the terms in the sum (D T Wz D and M T Wε M), also the normal matrix N = D T Wz D + M T Wε M is in the quasi triangular Schur form, and can be partitioned Wz =

Rz−1

=

Nx Nxb --------T Nxb Nb

−1 (11)

The optimal solution can be written as −1 zˆ = D T Wω D + M T Wε M M T Wε y = N −1 U (12) where U is the known normalized term ⎡ ⎢ ⎢ ⎢ ⎢ U = M T Wε y = ⎢ ⎢ ⎢ ⎢ ⎣

H1T Wε1 y1 H2T Wε2 y2 .. . HTT WεT yT − − − − − − − − − − −− C1T Wε1 y1 + · · · + C TT WεT yT

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ = ux ⎥ ub ⎥ ⎥ ⎦ (13)

p·T

with dimension U1 . To estimate the constant bias vector, we can apply the Schur decomposition bˆ = Sb−1 gb

(14)

where Sb is the Schur complement to Nb T Sb = Nb − Nxb Nx−1 Nxb T gb = u b − Nxb Nx−1 u x

(15)

Note that we need only to invert Nx , that is large because has dimension ( p · T ) × ( p · T ) but is tridiagonal, and Sb that has dimension n × n.

3 Application to GPS Observations The case with B = 0 has practical interest in GPS observations processing. We need to make the hypothesis that the bias vector has no direct connection to the parameters, but affect directly only the observations (B = 0). The bias vector can be constant or constant with steps, that is the case of carrier phase ambiguities affected by cycle slips; steps are taken in account by matrix C. The matrix C, that link the bias vector to the observations, must be known a priori. So we need an algorithm to detect cycle slips before writing this matrix. We expect to smooth the estimated parameters according with a dynamic (stochastic) model, and to fix the integer ambiguities reducing the search area. The system is now x t +1 = Tt +1 x t + νt +1 yt +1 = Ht +1 x t +1 + Ct +1 bt +1 + εt +1 bt +1 = bt

(16)

and the last n row and columns of the matrix D T Wz D became 0, with the consequence to reduce the number of matrix sums and multiplications necessaries to compute N. Anyway, matrix storage and inversion time are not changed. Thanks to domain decomposition is possible to estimate the float ambiguities with their variancecovariance matrix, without solving for the other unknown parameters (i.e. trajectory). Then, this result is used as input to search for the integer ambiguities through LAMBDA method. Finally, the double differenced GPS observations can be corrected for the carrier phase integer ambiguities and the system can be solved for the non costant unknowns (positions, velocities, etc.); this last step does not require to invert the sub matrix Nx that has already been inverted to compute the Schur complement to Nb .

4 Test Description The algorithm has been tested on GPS single frequency kinematic observations, over short baselines (2 – 10 km), so is possible to neglect the residuals of the tropospheric and ionospheric delays in the double differences observations. The GPS observations have been acquired at 1 s data rate, by a moving antenna mounted on a boat, and the results in the height component of the trajectory will be shown. To take account of the dynamic constraint and of the constant biases requires large matrices. As example: a data set of 15m at 1s data rate and double precision (8 byte) GPS single frequency

Geometry

Dynamic

(

T

T

p

p

n

n p

C1T Wε 1 A1

H Wε 1 A1

T 1

B Wω 2T2 − B Wω 1

CTT Wε T AT

H TT Wε T AT

p

n

+ CTT Wε T CT

H TT Wε T CT

H Wε 1C1

T 1

T T −1

−TT WωT

)

B WωT TT − B WωT −1 T T

C1T Wε 1C1 +

−T2Wω 2

p

T 1

Wω 1 + T Wω 2T2

p

T 2

−T Wω 2 Wω 2 + T3T Wω 3T3

T 2

M Wε M =

T

DT Wz D =

p T 2

p

Contribution to the normal matrix N = D Wω D + M Wε M

Table 1. Structure of the normal matrix

− B WωT T T

WωT

−TTT WωT

n

+ BTT WωT BT

− BT WωT B Wω 1 B1 + T 1

TTT WωT BT − BT −1Wω (T −1)

T Wω 2 B2 − B1Wω 1 T 2

Kinematic GPS Batch Processing, a Source for Large Sparse Problems 163

164

M. Roggero

observations, produce a design matrix of ∼ 120 Mbyte. Is necessary to treat the matrices as sparse. Matrices are stored in CSR (Compact Sparse Row) format or equivalent specific compact format. In this way the design matrix reduces to ∼ 1 Mbyte. The dynamic model is described by the transition matrix T and by the stochastic model of the system noise ν. A constant velocity model has been implemented in the test: ⎡ ⎢ ⎢ ⎢ Tt = ⎢ ⎢ ⎢ ⎣

1

1 1

⎤ ⎥ ⎥ 1 ⎥ ⎥; ⎥ ⎥ ⎦ 1 1

⎡

1

1 1

xt

⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎣

Xt Yt Zt X˙ t Y˙t Z˙ t

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

Fig. 2. Autocovariance of the process (height component of the trajectory). The solid line represents the batch solution with the dynamic constraint, while the dotted one represents the batch solution without the dynamic constraint.

(17) The stochastic model of the system is given by ⎡

Rνν

⎢ ⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎣

σ X2

⎤ σY2

σ Z2

σ X2˙

σY2˙

σ Z2˙

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(18)

The batch solution has been implemented in two versions, one including the dynamic model and one without taking account of the dynamic, so the effect of the dynamic constraint can be analyzed in the results. One can note that the estimated trajectory is smoothed in the batch solution including dynamic constraints, according to the dynamic model; however this effect depends on the a priori stochastic model of the process, that has not been calibrated on the observations (see Figure 1).

The decorrelation time of the process (height component of the moving antenna) is 25 s, and can be verified numerically that the decorrelation time is not changed applying the dynamic constraint (see Figure 2). So we can conclude that the a priori stochastic model of the dynamics fits with the process, even if a procedure to calibrate the model on the observations have not been applied. This aspect is important because the batch approach can be applied to a large set of different applications, such as the determination of the trajectory of a moving object, or the slow motion of a deforming structure; in other words, the stochastic model of the dynamics depends on the application, and can be studied with some a priori assumption, or either can be calibrated on the observations. Verifying the autocovariance function, we have only shown that no wrong information have been added to the process constraining the dynamic. The variance of the process (the first value of the autocovariance function) is 0.048 m2 without constraining the dynamic, and decrease to 0.037 m2 if we introduce the dynamic constraint.

5 Conclusions

Fig. 1. Estimated trajectory of a moving antenna: the height component is shown. The least square solution obtained with and without considering the contribution of the dynamic model have been compared.

Constraining dynamic in Least Square estimation of float ambiguities the search volume for LAMBDA method have been reduced, and the robustness of cycle slip fixing have been increased. The effect of noise on the estimated trajectory have been reduced, according to the kinematic model. The computational load is comparable to Kalman filtering plus smoothing, thanks to the structure of the normal matrix and to domain decomposition.

Kinematic GPS Batch Processing, a Source for Large Sparse Problems

References Kailath T., Sayed A. H., Hassibi B. (2000). Linear estimation, Prentice Hall, New Jersey. Saad Y. (2000). Iterative methods for sparse linear systems, Second edition with corrections. Teunissen P. (2001). Dynamic data processing, Delft University Press.

165

Colomina I., Blàzquez M. (2004). A unified approach to static and dynamic modeling in photogrammetry and remote sensing, In: Altan, O. (ed.), Proceedings of the XXth ISPRS Congress, Istanbul, pp. 178–183. Albertella A., Betti B., Sansò F., Tornatore V. (2006). Real time and batch navigation solutions: alternative approaches, In: Bollettino SIFET, n. 2 – 2006.

Optimal Recursive Least-Squares Filtering of GPS Pseudorange Measurements A.Q. Le, P.J.G. Teunissen Delft Institute of Earth Observation and Space Systems (DEOS), Delft University of Technology, The Netherlands

Abstract. Code pseudorange measurement noise is one of the major error sources in Precise Point Positioning. A recursive least-squares solution with proper functional and stochastic modelling would help to exploit in addition the ultra high precision of the carrier phase measurement. Analyses of different methods, including phase smoothed, phase connected and phase adjusted pseudorange algorithm, will show that the phase adjusted pseudorange algorithm is statistically optimal. Static and kinematic experiment results also support the conclusion with more than 30% of improvement by going from the phase smoothed to the phase adjusted algorithm. Keywords. Recursive least-squares filtering, PPP

1 Introduction In standalone positioning such as standard GPS positioning, Wide Area Differential GPS (WADGPS) positioning or Precise Point Positioning (PPP), the prime observation type is the code pseudorange. The main sources of error in this positioning mode include satellite orbits and clocks, the ionosphere, the troposphere, and the pseudorange errors (noise and multipath). While the first three error sources can be mitigated by augmentation corrections or products from a network of reference stations (as done in WADGPS and PPP), the pseudorange errors cannot since they are local effects. In this case, the extremely precise carrier phase measurement can come to rescue. The noise can be reduced with carrier phase by using the popular Hatch smoothing algorithm. However, this is not an optimal solution as it is based on a channel-by-channel basis. Instead, a recursive leastsquares filter which can be proven to be statistically optimal will be deployed, namely the phase-adjusted pseudorange algorithm. The algorithm takes both pseudorange and carrier phase measurements in one integral least-squares solution where carrier phase ambiguities are considered as constant but unknown parameters. The 166

positioning parameters together with carrier phase ambiguities are estimated recursively. This processing scheme minimises computational load and all information is preserved. The algorithm can be applied to any kind of GPS positioning where both pseudorange and carrier phase are involved. In this paper, Precise Point Positioning will be implemented with the phase-adjusted pseudorange algorithm and a comparison to other smoothing approaches will be made. Although the results are post-processed due to the current availability of global data products, the processing engine is purely kinematic (no dynamic assumption needed) and suitable for real-time applications. In fact real-time operation has been emulated in this paper. Results to be presented in the paper are of longterm static data from several stations around the world with different GPS receivers, as well as from kinematic experiments. In general, the obtained accuracy is at sub-metre level worldwide. Under favourable conditions, it can reach 40 centimetres horizontally and 60 centimetres vertically (at the 95% level). The results also show a large improvement going from the classical Hatch smoothing algorithm to the phase-adjusted pseudorange algorithm. It will be shown that the 95% positioning error can be improved by about 30–50%.

2 Filtering Methods Using Carrier Phase Measurements 2.1 Classical Phase Smoothing Algorithm The classical phase smoothing algorithm1 was introduced by Hatch (1982) and is still widely used 1 Strictly, smoothing implies the computation of estimates for unknowns parameters (e.g. position coordinates) pertaining to epoch tk , using observations from the whole data collection period, i.e. [t1 , tl ] with 1 ≤ k ≤ l; the data period extends beyond epoch tk . Filtering refers to estimates for parameters at epoch tk , using solely data up to and including epoch tk , i.e. [t1 , tk ]. Filtering allows real-time operation and smoothing does not. In this paper, we continue to refer to ‘phase smoothing’, as commonly done, but strictly filtering is meant instead.

Optimal Recursive Least-Squares Filtering of GPS Pseudorange Measurements

nowadays due to the simplicity and flexibility of the algorithm. The recursive formula of the algorithm reads: k−1 + δ⌽k,k−1 k = 1 Pk + k − 1 P P k k

If the denote x i and Ai are the vector of unknown parameters and its design matrix at epoch i , and ∇ is the vector of ambiguities (unchanged over time), the observation equations at epoch i can be written: ⎧⎛ ⎞⎫ ⎛ ⎞⎛ ⎞ ⎬ ⎨ P A 0 x i i ⎠ ⎠ ⎝ i ⎠ (3) =⎝ E ⎝ ⎭ ⎩ ⌽ A I ∇

(1)

k the phase-smoothed pseudorange at epoch with P tk ; Pk the pseudorange observation at epoch tk ; k−1 the phase-smoothed pseudorange at epoch tk−1 ; P δ⌽k,k−1 = ⌽k − ⌽k−1 the time-differenced carrier phase observation; ⌽k the carrier phase observation at epoch tk ; ⌽k−1 the carrier phase observation at epoch tk−1 . Note that all the carrier phase observations are in units of range. The same smoothed pseudorange equation can be expressed in a different form as a linear combination of the previous epochs’observations, including both pseudorange and carrier phase: 1 k = 1 Pi − ⌽i + ⌽k P k k k

k

i=1

i=1

167

i

i

From (2) and (3), we have: k } = E{ P

1 1 Ai x i − (Ai x i + ∇) k k k

k

i=1

i=1

+ Ak x k + ∇ = Ak x k

(4)

with E{.} the mathematical expectation operator. As shown in (4), through the linear combination, the design matrix for the smoothed pseudorange is preserved when no cycle slips occur. However, the variance matrix is no longer (block) diagonal and hence, recursive computation is not possible for this model.

(2)

⎛

⎞ P1

⎛

1 P

⎞

⎛

I ⎟ ⎜ ⎜ ⎜ ⎜ ⎟ ⎜ I ⎜ P2 ⎟ ⎜ 2 ⎟ ⎜ .. ⎜ .. ⎟ ⎜ ⎜ . ⎟=⎜ . ⎟ ⎜ ⎜ ⎟ ⎜ ⎜ ⎟ ⎜ I ⎜P ⎝ k−1 ⎠ ⎜ ⎝ k−1 k I P k

⎛ ⎧⎛ ⎞⎫ QP ⎪ ⎜ 1 ⎪ P ⎪ ⎪ ⎪ ⎜ ⎪ ⎜ ⎪ ⎟⎪ ⎜ Q ⎪ ⎜ ⎟⎪ ⎪ ⎪ ⎜ P ⎪ P ⎜ ⎪ ⎪ ⎜ 2 ⎟⎪ 2 ⎪ ⎨⎜ ⎬ ⎜ ⎟⎪ ⎜ .. ⎟ . D ⎜ . ⎟ =⎜ ⎜ .. ⎪ ⎜ ⎪ ⎟ ⎪ ⎪ ⎜ ⎪ ⎜ ⎟⎪ ⎜ ⎪ k−1 ⎟⎪ ⎪ ⎜P ⎪ ⎜ QP ⎪ ⎪ ⎪ ⎪ ⎝ ⎠ ⎜ (k−1) ⎪ ⎪ ⎪ ⎪ ⎩ P k ⎭ ⎝ Q P

k

0

0

0

...

...

0

− 2I

I 2

I 2

0

...

0 .. .

..

.

I − k−1

...

...

I k−1

(k−2)I k−1

0

− kI

...

...

I k

− kI

I k

QP 2

...

QP (k−1)

.. .

QP 2

.. .

.. .

..

QP (k−1)

...

QP (k−1)

QP k

...

QP k

.

⎞

QP k ⎟

⎜ ⎟ ⎜ ⎟ ⎜ ⌽ ⎟ ⎜ 1 ⎟ ⎟ ⎞⎜ ⎜ P ⎟ 0 ⎜ 2 ⎟ ⎟ ⎟⎜ ⎟ ⎟⎜ 0 ⎟ ⎜ ⌽2 ⎟ ⎟ ⎟⎜ ⎜ . ⎟ .. ⎟ ⎜ ⎟ . . ⎟⎜ . ⎟ ⎟ ⎟ ⎟⎜ ⎟ ⎟⎜ 0 ⎟ ⎜ P k−1 ⎟ ⎟ ⎠⎜ ⎜ ⎟ (k−1)I ⎜⌽ ⎟ ⎜ k−1 ⎟ k ⎜ ⎟ ⎜ P ⎟ ⎜ k ⎟ ⎝ ⎠ ⌽k

(5)

⎛

⎞

⎜ ⎜ .. ⎟ ⎟ ⎜ . ⎟ ⎜ ⎟ ⎜ ⎜ .. ⎟ ⎟ ⎜ . ⎟+⎜ ⎟ ⎜ ⎟ ⎜ QP ⎟ ⎜ ⎜ k ⎟ ⎠ ⎝

0

QP k

0

0 Q⌽ 2

..

. (k−2)Q ⌽ (k−1) (k−1)Q ⌽ k

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

(6)

168

A.Q. Le, P.J.G. Teunissen

It can be seen clearly when taking the first k epochs into account. The smoothed pseudoranges on the left form the system in (5). Assuming no (cross and time) correlation between the original code and phase observations and that code and phase variances are Q P and Q ⌽ at every epoch, application of the propagation law gives the variance of the smoothed pseudoranges in (6). Based on the smoothed pseudoranges, subsequent recursive processing is usually carried out to obtain estimates for the receiver position, epochafter-epoch. However, there is clearly (extremely) high time-correlation between the smoothed pseudoranges, which prevents the model to work recursively. The correlation is obviously ignored in the smoothing algorithm.

2.2 Phase-Connected Pseudorange Algorithm Another newly developed algorithm using the carrier phase to smooth the pseudorange was from Bisnath et al (2002), in which differenced carrier phase measurements between epochs are used as additional observation next to the pseudorange. As long as no cycle slips occur, ambiguity parameters are absent.

⎧⎛ ⎞⎫ ⎛ ⎪ ⎪ P A1 ⎪ ⎪ 1 ⎪ ⎜ ⎜ ⎪ ⎟⎪ ⎪ ⎪ ⎪ ⎜ ⎜ ⎪ ⎟ ⎪ ⎪ ⎪⎜ δ⌽21 ⎟⎪ ⎪ ⎜−A1 ⎪ ⎜ ⎪ ⎟⎪ ⎪ ⎜ ⎪ ⎜ ⎪⎜ ⎟⎪ ⎪ ⎪ ⎪ P ⎜ ⎪ ⎟ ⎪ ⎜ 2 ⎪ ⎨⎜ ⎟⎪ ⎬ ⎜ ⎜ ⎜ ⎟ E ⎜ δ⌽32 ⎟ = ⎜ ⎜ ⎜ ⎪ ⎟ ⎪ ⎪ .. ⎪⎜ ⎜ ⎪ ⎟⎪ ⎪ ⎪ ⎜ ⎪ ⎜ ⎪ ⎟ . ⎪ ⎜ ⎪ ⎜ ⎪ ⎟⎪ ⎪ ⎪ ⎪ ⎜ ⎪ ⎜ ⎟ ⎪ ⎪⎜δ⌽k,k−1 ⎟⎪ ⎪ ⎜ ⎪ ⎪ ⎪ ⎪ ⎝ ⎝ ⎠ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ P

For a single epoch, the observation equations are given: ⎧⎛ ⎨ E ⎝ ⎩ ⎧⎛ ⎨ D ⎝ ⎩

Pk δ⌽k,k−1

⎞⎫ ⎛ ⎬ ⎠ =⎝ ⎭

⎞⎛ 0

Ak

−Ak−1

Ak

⎞⎫ ⎛ ⎬ QP ⎠ =⎝ k ⎭ δ⌽k,k−1 0

⎠⎝

⎠

xk ⎞

0

Pk

⎠

(7)

Q ⌽k,k−1

where P k the linearised pseudorange observation; δ⌽i,i−1 = ⌽i − ⌽i−1 the linearised time-differenced carrier phase observation. By linearised we mean ‘observed minus computed’ following from the linearisation of the original non-linear functional relation. Again, taking k epochs together, the full model of observation equations is given (8). Vector x k primarily contains the receiver position coordinates but other parameters could be included as well, as for instance the receiver clock error. With the same assumption of no correlation between (undifferenced) pseudorange and carrier phase measurements as above, the variance matrix can be derived for the observation vector in (9) with Q ⌽i,i−1 = Q ⌽i + Q ⌽i−1 . ⎞

A2 A2 −A2

A3 ..

. −Ak−1

k

⎟ ⎟⎛ ⎞ ⎟ ⎟ x1 ⎟⎜ ⎟ ⎟⎜ ⎟ ⎟ ⎜x 2 ⎟ ⎟⎜ ⎟ ⎟⎜ . ⎟ ⎟ ⎜ .. ⎟ ⎟⎝ ⎠ ⎟ ⎟ x ⎟ k Ak ⎟ ⎠ Ak

(8)

⎛ ⎧⎛ ⎞⎫ ⎜ ⎪ ⎪ P1 ⎪ ⎪ ⎜ ⎪ ⎜ ⎟⎪ ⎪ ⎪ ⎜ ⎪ ⎜ ⎟⎪ ⎪ ⎪ ⎜ ⎪ ⎜ δ⌽21 ⎟⎪ ⎪ ⎪ ⎜ ⎪ ⎜ ⎟⎪ ⎪ ⎪ ⎜ ⎪ ⎜ ⎟⎪ ⎪ ⎪ ⎜ ⎪ ⎪⎜ P 2 ⎟ ⎪ ⎪ ⎜ ⎪ ⎨⎜ ⎬ ⎜ ⎟⎪ ⎜ ⎜ ⎟ D ⎜ δ⌽32 ⎟ = ⎜ ⎜ ⎜ ⎪ ⎟⎪ ⎜ ⎪ .. ⎜ ⎪ ⎪ ⎟⎪ ⎜ ⎪ ⎪ ⎪⎜ ⎟⎪ . ⎜ ⎪ ⎪ ⎪ ⎜ ⎟⎪ ⎜ ⎪ ⎪ ⎪ ⎜ ⎟⎪ ⎜ ⎪ ⎪ ⎪ ⎜δ⌽k,k−1 ⎟⎪ ⎜ ⎪ ⎪ ⎪ ⎝ ⎠⎪ ⎜ ⎪ ⎪ ⎪ ⎪ ⎜ ⎩ ⎭ Pk ⎝

⎞ x k−1

⎞ Q P1

0 Q ⌽21

0

−Q ⌽2

Q P2 −Q ⌽2

0

Q ⌽32

0 ..

−Q ⌽3 . ..

−Q ⌽k−1 0

.

0

Q ⌽k,k−1 Q Pk

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

(9)


Similar to that of the Hatch smoothing algorithm, this matrix is not (block) diagonal though the correlation is not as heavy as that of the first algorithm. Strictly speaking, the system (8) and (9) cannot be recursively solved to obtain estimates for the position coordinates. One assumption has been made that all the resulting time correlation (due to the differenced carrier phase observations) is neglected. In this case, the system can be solved recursively (see Le (2004)).

The update for the ambiguities is also needed to fulfill the recursive solution: ∇ˆ k = ∇ˆ k−1 + Q ∇ˆ

Q xˆk =

+ [Q ⌽k + Q ∇ˆ k−1 ]

⎧⎛ ⎪ ⎪ ⎪ ⎜ ⎪ ⎪ ⎜ ⎪ ⎪ ⎜ ⎪ ⎪ ⎜ ⎪ ⎪ ⎜ ⎪ ⎪ ⎜ ⎪ ⎪ ⎪ ⎨⎜ ⎜ D ⎜ ⎜ ⎪ ⎜ ⎪ ⎪ ⎜ ⎪ ⎪ ⎜ ⎪ ⎪ ⎜ ⎪ ⎪ ⎪ ⎜ ⎪ ⎪ ⎜ ⎪ ⎪ ⎝ ⎪ ⎪ ⎩

P1 ⌽1 P2 ⌽2 .. . ⌽k−1 Pk ⌽k

−1

)Ak ]

⎞⎫ ⎛ ⎪ ⎪ Q ⎟⎪ ⎪ ⎜ P1 ⎜ ⎟⎪ ⎪ ⎪ ⎜ ⎟⎪ ⎪ ⎜ ⎟⎪ ⎪ ⎟⎪ ⎪ ⎜ ⎜ ⎟⎪ ⎪ ⎜ ⎟⎪ ⎟⎬ ⎜ ⎟ =⎜ ⎟⎪ ⎜ ⎟⎪ ⎪ ⎜ ⎜ ⎟⎪ ⎪ ⎟⎪ ⎪ ⎜ ⎜ ⎟⎪ ⎪ ⎟⎪ ⎪ ⎜ ⎜ ⎟⎪ ⎪ ⎝ ⎠⎪ ⎪ ⎪ ⎭ 0

[Q Pk + Q ∇ˆ

k−1

]−1

(11) Q ∇ˆ k = Q ∇ˆ k−1 − Q ∇ˆ k−1 [Q ⌽k + Q ∇ˆ k−1 ]

−1

Q ∇ˆ k−1

+ Q ∇ˆ k−1 [Q ⌽k + Q ∇ˆ k−1 ]−1 × Ak Q xˆk AkT [Q ⌽k + Q ∇ˆ k−1 ]−1 Q ∇ˆ k−1

The optimal solution would be a model where all the observations (including carrier phase measurements) are put into a unique model of observation equations. This is the model where all the information should be preserved and the unknowns at each epoch can be computed by a recursive least-squares solution. Based on this criterion, the phase-adjusted pseudorange algorithm was developed by Teunissen (1991). In this model, all original (undifferenced) pseudorange and carrier phase measurements are the basic observations; the unknowns including ambiguities and positioning parameters are recursively estimated (as seen in (12, 13) with P i the vector of linearised pseudoranges at epoch ti ; ⌽i the vector of linearised carrier phases at epoch ti ; x i the vector of unknown parameters at epoch ti ; Ai the linearised design matrix at epoch ti ; and ∇ the vector of unknown ambiguities (assumed to be timeinvariant for simplicity in this explanation); Q Pi variance matrix of code measurements at epoch ti and Q ⌽i variance matrix of carrier phase measurements). The recursive equations for the position parameters (and possibly others as well) can be given as (10): xˆ k = Q xˆk AkT (Q −1 Pk P k [ AkT (Q −1 Pk

k−1

× [⌽k − ∇ˆ k−1 − Ak xˆ k ]

2.3 Phase-Adjusted Pseudorange Algorithm

+ [Q ⌽k + Q ∇ˆ k−1 ]−1 [⌽k − ∇ˆ k−1 ])

169

The initial epoch’s parameters xˆ 1 and ∇ˆ 1 with Q xˆ1 and Q ∇ˆ 1 follow from a least-squares solution based on P1 and ⌽1 . ⎧⎛ ⎞⎫ ⎛ ⎞ ⎪ ⎪ A P 0 ⎪ ⎪ 1 1 ⎪ ⎜ ⎪⎜ ⎟⎪ ⎪ ⎟ ⎪ ⎜ ⎪ ⎪ ⎟⎪ ⎟⎛ ⎞ ⎪ ⎪⎜ ⎜ ⎜ ⎪ ⎪ ⎟ ⌽1 ⎟⎪ I ⎟ A1 ⎪ ⎜ ⎜ ⎪ ⎪ ⎟ x1 ⎪ ⎜ ⎪⎜ ⎟⎪ ⎪ ⎟⎜ ⎟ ⎪ ⎪ ⎜ ⎪ ⎜ ⎪ ⎜ ⎟ ⎟ ⎪ P 2 ⎟⎪ 0 ⎟ A2 ⎜ ⎪ ⎜ ⎪ ⎟ ⎜x 2 ⎟ ⎪ ⎬ ⎜ ⎨⎜ ⎟⎪ ⎟⎜ ⎟ ⎜ ⎜ ⎟ ⎟⎜ . ⎟ I ⎟ ⎜ .. ⎟ A2 E ⎜⌽2 ⎟ = ⎜ ⎜ ⎜ ⎪ ⎪ ⎟ ⎟⎜ ⎟ ⎪ ⎜ ⎜ . ⎟⎪ ⎪ ⎪ ⎟⎜ ⎟ .. .. ⎪ ⎪ . ⎜ ⎜ ⎪ ⎪ ⎟ ⎟ ⎜x k ⎟ . . ⎪ . ⎟⎪ ⎜ ⎜ ⎪ ⎪ ⎟⎝ ⎠ ⎪ ⎪ ⎜ ⎜ ⎪ ⎪ ⎟ ⎟ ⎪ ⎪ ⎜ ⎜ ⎪ ⎪ ⎟ ⎟ ∇ ⎪ ⎪ A 0 P ⎜ ⎜ ⎪ ⎪ ⎟ ⎟ k k ⎪ ⎪ ⎪ ⎪ ⎝ ⎝ ⎠ ⎠ ⎪ ⎪ ⎪ ⎪ ⎩ ⌽ ⎭ I A k k (12) This algorithm is optimal from a statistical point of view since it properly treats the model as a whole (with all observations of all epochs). No further assumption is made beside the assumption of no time correlation between epochs. The algorithm is purely kinematic as no dynamic model is needed for the receiver. On the other hand, if such information is available, it can be easily incorporated.

(10) −1

⎞ 0 Q ⌽1 Q P2 Q ⌽2

..

. Q ⌽k−1 Q Pk Q ⌽k

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3 Experimental Results with Precise Point Positioning In standalone positioning, the main error sources are satellite orbits and clocks, the ionosphere, the troposphere and the pseudorange noise. The first three sources can be compensated for in Precise Point Positioning (PPP) by using publicly available products, such as precise ephemerides, Global Ionosphere Maps (GIM) and a precise tropospheric model. Not as such, the code noise cannot be eliminated in a similar way and is significantly large in this mode of positioning. Hence, PPP benefits a lot from filtering using carrier phase measurements to mitigate the noise of pseudorange measurements as outlined in the previous section. Various experiments were carried out with the phase-adjusted pseudorange algorithm, both static and kinematic. In all experiments of this paper, the singlefrequency PPP approach using publicly available products was implemented. The corrections/models in use include precise orbits and clocks from International GPS Services (IGS), GIMs from Center for Orbit Determination in Europe (CODE) and the Saastamoinen tropospheric model. An elevation angle dependent weighting scheme is used for both code and phase measurements. Le (2004) describes the approach in more detail. 3.1 With Static Receivers An extensive static test of one week was performed with 4 stations, namely DELF, EIJS, DUBO and HOB2. The first two stations are part of the AGRS.NL network in the Netherlands while the other two belong to the IGS network, one (DUBO) in Canada and one (HOB2) in Australia. Data were collected with 30-second interval. Table 1 shows the results for the phase-adjusted pseudorange algorithm. In general, the accuracy is about half a metre horizontally and one metre vertically. Better

Table 1. Extensive static test results [m]. 95%-value (95th percentile) of position errors in local North, East and Up coordinates (with respect to known coordinates of the markers) with one week of data at 30-second interval for 4 different locations around the world

North East Up

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DUBO

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0.41 0.42 0.82

0.78 0.59 1.01

HOB2 0.53 0.72 1.39

Table 2. Static test results at DELF [m]. 95%-value of position estimates in local North, East and Up coordinates with one day of data at 30-second interval for 4 different approaches (Phase smoothed algorithm with different window lengths - 3, 5, 8 epochs) Phase smoothed No smooth North 0.71 East 0.49 Up 1.10

3

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0.59 0.59 0.67 0.49 0.37 0.36 0.39 0.32 0.95 0.95 1.13 0.76

Phase adjusted 0.43 0.42 0.74

results are obtained in Europe and (possibly) in North America thanks to better quality of the GIMs. A comparison between the three algorithms was made from 24 hours data collected at the DELF station. Again, a 30-second interval was used. Table 2 shows about 30% improvement by going from the Phase smoothing to the Phase adjusted algorithm in the North and vertical direction. The accuracies of all algorithms in the East component are comparable. The horizontal scatters and time-series of the three components, North, East and Up are plotted in Figures 1 and 2. The ‘no smooth’ solution is also included for reference (using solely pseudorange observations). Note that the phase smoothed algorithm was implemented with different window lengths. It can be seen that for large sampling intervals (e.g. 30 s), the phase smoothed algorithm is sensitive to the window length choice. As shown in Table 2, it should be about 5 epochs (at 30-second interval) or equivalent to 2–3 min. 3.2 With Kinematic Receivers A maritime kinematic experiment was carried out with a small boat on Schie river (between Delft and Rotterdam, the Netherlands). Nearly 3 hours (1 Hz) of kinematic data from 2 receivers, namely Ashtech Z-XII3 and Leica SR530 were collected. The cmaccuracy reference trajectories were computed in a (dual-frequency carrier phase) differential GPS solution with a reference station nearby (only few kilometres away). Again, the three algorithms’ results are included in Table 3. The window length used in the phase smoothed algorithm is 100-second. In the kinematic results, the accuracy is improved by more than 50% in the North component. Significant differences also can be seen in other components of about 30–50% (see Figures 3 and 4), especially with the Ashtech receiver.

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4 Conclusions The Phase-adjusted pseudorange algorithm, statistically optimal, is a fully kinematic filter. It has been demonstrated to work robustly in various circumstances, from static to kinematic, over short time spans and long time spans. The accuracy of its application in single-frequency PPP, in general, can be confirmed at half a metre horizontally and one metre vertically (at the 95% level). It proves to have a better accuracy than that of the phase smoothed approach, by about 30% to 50%. In favourable conditions, the

Table 3. Kinematic results [m]. 95%-value of position estimates in local North, East and Up coordinates with 2 receivers for 3 different approaches, 3 hours of data at one-second interval (Phase smoothed algorithm with 100-epoch window length) Receiver

Phase smoothed

connected

Phase adjusted

Ashtech

North East Up

1.12 0.39 1.20

0.54 0.31 0.88

0.45 0.29 0.84

Leica

North East Up

0.79 0.36 0.83

0.48 0.28 0.76

0.39 0.34 0.56

accuracy gets close to 4 decimetres horizontally and 6 decimetres vertically (95%), and does not depend on the receiver’s dynamics. At this level of accuracy, other sources of errors should be accounted for. They are solid earth tides, ocean loading, phase wind-up and others. The full correction of satellite antenna phase centre also should be applied. All those corrections/modelling might bring the accuracy close to sub-decimetre level since the errors are at a few decimetres level in total. Moreover, the GIMs cannot completely eliminate the ionospheric errors. Due to residual ionospheric delays, ionospheric divergence occurs in all the three algorithms. Note that the ionospheric delay was not included in the vector of unknown parameters. In the stochastic model, the code noise is assumed to be white noise, i.e. no time-correlation between epochs. However, in practice, not all receivers provide white noise pseudoranges as discussed in Bona (2000) (the Trimble 4700 used in the static test was found to have white noise pseudoranges [ibid]). For a receiver without white noise characteristic, i.e. there is time correlation between epochs, current modelling is still sub-optimal. Multipath is also a significant error source that needs to be considered since it is not included in the model. In certain aspects and depending on the time

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scale, it could be regarded as both a functional and stochastic error as it contains both a bias and random component.

References Bona, P. (2000). Precision, cross correlation and time correlation of GPS phase and code observations. GPS Solutions, Vol. 4, No. 2, pp. 3–13. Bisnath, S.B., T. Beran and R.B. Langley (2002). Precise platform positioning with a single GPS receiver. GPS World, April 2002, pp. 42–49.

Hatch, R. (1982). The Synergism of GPS code and carrier measurements. Proceedings of the 3rd International Geodetic Symposium on Satellite Doppler Positioning, Vol. 2. Las Cruces - New Mexico, 1982, pp. 1213–1231. Le, A.Q. (2004). Achieving Decimetre Accuracy with Single Frequency Standalone GPS Positioning. Proceedings of the ION GNSS 17th International Technical Meeting of the Satellite Division, 21–24 September 2004, Long Beach, CA, pp. 1881–1892. Teunissen, P.J.G. (1991). The GPS phase-adjusted pseudorange. Proceedings of the 2nd International Workshop on High Precision Navigation. Stuttgart/Freudenstadt, 1991, pp. 115–125.

A Comparison of Particle Filters for Personal Positioning D. Petrovich, R. Piché Institute of Mathematics, Tampere University of Technology, P.O. Box 553, 33101 Tampere, Finland

Abstract. Particle filters, also known as sequential Monte Carlo methods, are a convenient and popular way to numerically approximate optimal Bayesian filters for nonlinear non-Gaussian problems. In the literature, the performance of different filters is often determined empirically by comparing the filter’s conditional mean with the true track in a set of simulations. This is not ideal. Because these filters produce approximations of the optimal Bayesian posterior distribution, the comparison should be based on the quality of this approximation rather than on an estimate formed from the distribution. In this work, we apply a multivariate binning technique to compare the performance of different particle filters. In our simulation, we find that the conclusions of the distribution comparison are similar to the conclusions of a root mean square error analysis of the conditional mean estimate. Keywords. Sequential monte carlo, particle filter, bayesian estimation

1 Introduction Particle filters (PFs) implement the recursive Bayesian filter with Monte Carlo (MC) simulation and approximate the posterior distribution by a set of samples with appropriate weights. This is most attractive in nonlinear and non-Gaussian situations where the integrals of Bayes’ recursions are not tractable. In the literature, many PF simulations focus on the MC variation of the mean estimates, i.e. the randomness introduced by the MC algorithm that can be observed from the empirical mean estimates. Better PFs vary less. In this work, we develop a method to compare PF performance that uses the distribution approximation itself rather than a single estimate formed from it. A distribution analysis could be more useful than an estimate analysis. As examples, two filters could

give similar mean estimates although one of them has a distribution “closer” to the true distribution. Also, in a bimodal case, the mean could be between the two modes in a region of the state-space where there is little probability of the target being located; in a case such as this, the mean is less interesting to analyze and we are more concerned if our filter appropriately characterizes this bimodal characteristic. We will discuss one proposed method for comparing distributions and then apply this method to the comparison of four PFs (SIR1, SIR2, SIR3, SIR4) described in the Appendix.

2 Comparing Distributions with χ 2 -Tests An interesting application of distribution comparisons was given by Roederer et al. (2001) in the field of cytometry. Test samples, i.e. sets of multidimensional data, are to be ranked according to their similarity to a control sample, which is a sample of data chosen to represent some known behavior. A multivariate data-dependent binning technique was proposed that adaptively constructed bins according to the control sample, followed by the use of a test statistic to quantify the difference between the test and control sample. Baggerly (2001) provides a more theoretical discussion of this approach with the recommendation to use the standard two-sample χ 2 -test statistic. The form of the computed test statistic is (Baggerly (2001)) ψ=

2 B (oi j − ei j )2 , ei j i=1 j =1

where B is the number of bins, oi j is the observed count of the j th bin of the i th sample set, and ei j is the expected count of the j th bin of the i th sample set given by 173

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Fig. 1. Probability binning in 2 with B = 32.

ei j = n i

(o1 j + o2 j ) , (n 1 + n 2 )

where n i is the number of samples in set i . The algorithm of Roederer et al. (2001) that is used for constructing the bins is called probability binning and is as follows. The variance of the control sample along each of the d dimensions is computed and the dimension with the largest variance is chosen to be divided. The sample median value of the chosen dimension is then chosen as the point at which to partition the state-space in two. This is then repeated for each partitioned subspace, continuing until the desired number of bins has been reached. The result is a set of d-dimensional hyper-rectangular bins with sides parallel to the coordinate axes and each bin containing roughly the same number of control samples, see Figure 1. Assuming then, that a test sample is from the same distribution as the control sample, each bin will roughly have the same expected frequency. The two-sample test statistic is 2 distributed, i.e. approxithen approximately χ B−1 mately distributed according to a χ 2 distribution with B − 1 degrees of freedom.

3 Application to Linear and Gaussian Filtering Scenarios Assume for a moment, that we are able to generate IID samples from the true marginal posterior. We can then apply the probability binning procedure at each time step so that the state-space n x is partitioned into bins using a sample from the true marginal posterior. It might be reasonable then to assume that the quality of a PF approximation of the marginal posterior can be assessed using the two-sample χ 2 -test, where the hypothesis is that the samples from the true marginal posterior and the samples from the PF approximation are from the same distribution.

Roughly speaking, we might expect better PFs to give better test scores, where “better test scores” refers to realizations of a random variable with distribution closer to the assumed χ 2 distribution. We check this empirically by repeating the simulation 1000 times and comparing the mean of the realized test scores to that from the assumed χ 2 distribution. 2 For a χ B−1 distribution, the mean is B − 1. All of our simulations use B = 64. The binning of the PF samples is done as follows. At each time step, we simulate M = N = 104 samples using multinomial sampling from the importance distribution, evaluate weights, and skip the final resampling step. The binning at each time step uses the weighted samples. We draw 104 samples using systematic resampling from this weighted discrete distribution, bin the equally-weighted samples, compute the test score, and discard these resampled points. If the weights are equal, then the binning can proceed without resampling. To verify that the test score analysis is meaningful, we can compare our conclusions to those made from a root mean square error (RMSE) analysis of the mean estimates. The RMSE is given as RMSEk =

2 E μˆ k − μk ,

(1)

where μˆ k is the random posterior mean estimate and μk is the true posterior mean. The expectation in equation (1) is approximated by repeating the simulation 500 times and using the sample mean. Due to lack of space, we have omitted our linearGaussian simulations and proceed directly to the general filtering scenario. The interested reader is referred to Petrovich (2006).

4 Application to General Filtering Scenarios We would like to apply this distribution analysis to PFs in the general filtering scenario where we do not have an analytic form for the posterior. The difficulty is then to determine a control sample that can be used to partition the state-space. Ideally, we would have some algorithm that was known to produce IID samples from the marginal posterior that we could use to produce the control sample. In the absence of such an ideal algorithm, we have chosen one PF algorithm as a reference. The partitioning of the state-space using a reference PF is done as follows. We have used the SIR3 importance distribution and, at each time step, draw M = 107 samples from the importance distribution,

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evaluate weights, and then resample N = 104 samples from this weighted distribution. The importance sampling uses deterministic sampling (103 samples from each mixture component) and the resampling uses systematic sampling. The binning at each time step uses the M weighted samples. We draw 106 samples from this weighted discrete distribution, perform the probability binning on these equally-weighted samples, and discard these resampled points. It should be mentioned that the simulation scenario of this section is the same simulation scenario as that in Heine (2005), i.e. the same signal and measurement models and the same realized measurement and signal process. We have reproduced the results (i.e. the relative RMSE plots) and review the conclusions of that publication here for convenience. The contribution of this work is the application of the probability binning method to the comparison of PFs. 4.1 Simulation Description For this simulation, the state is in 4 with two position coordinates rk and two velocity coordinates u k , i.e. T x k = rke rkn u ek u nk . The evolution of the state in continuous-time can be described by a white noise acceleration model, see e.g. Bar-Shalom and Li (1998). The discretized state equation for each [rk u k ]T coordinate is written as x k = Fk−1 x k−1 + vk−1 ,

Fk−1 =

1 0

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,

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1 3 3 ⌬tk 1 2 2 ⌬tk

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γ.

We simulate a 200 second trajectory with a constant time step of ⌬tk = 1 second and ␥ = 3 m2 /s3 . The state’s initial distribution Px0 is Gaussian and known. Three base stations are used and, at each time step, one base station produces a range measurement. The range measurement of the form yk = r b − rk + z k , z k ∼ N(0, σ 2 ), where r b is the known position of a base station. In the simulation of the observation process, the probability of a base station producing a measurement is inversely proportional to the squared distance to the target.

For each filter, four separate simulations were run, each using a different variance σ 2 for the noise in the measurement model: 104 , 25, 1, 0.1 m2 . The true posterior mean μk in the RMSE analysis, see equation (1), is approximated using the same reference PF that was used for creating the bins. 4.2 Results We first consider an analysis of the RMSE of the mean estimates. The RMSE values are divided by the SIR1 RMSE values and plotted in Figure 2. These relative RMSE values show how the MC variation of mean estimates for the different filters compare to SIR1: values lower than one indicate improved performance compared to SIR1 (less MC variation) while values above one indicate worse performance (more MC variation). With the largest noise variance (σ 2 = 104 m2 ), all the filters seem to have similar MC variance. With σ 2 = 25 m2, the SIR2 and SIR4 filters clearly have less MC variation, while for cases with smaller measurement noise variance (σ 2 = 1 and 0.1 m2), the SIR3 and SIR4 filters have less MC variance. Also, the SIR2 behaves erratically at the smaller measurement noise variance. Due to weights summing to zero in the PF algorithm on the σ 2 = 0.1 m2 case, SIR1 is averaged over only 998 realizations and SIR2 is averaged over only 365 realizations for that case. In Figure 3, the test score mean is plotted over the whole simulation. It is interesting to note that the test scores rarely resemble the χ 2 theoretical distribution. However, there appears to be different behavior for the different filters. Note that all the results are rather similar with large σ 2 , and the differences become more apparent as σ 2 decreases. Also note that for σ 2 = 104 and 25 m2, there is near identical results for the SIR1 and SIR3 filters and for the SIR2 and SIR4 filter. For σ 2 = 0.1 m2, the SIR2 score means are quite large and are outside of the plotted region. The intention of these plots was not to display the actual score means, but instead to show the relative performance of the different filters. For cases with larger measurement noise variances (σ 2 = 104 and 25 m2), the two filters using alternative weights in the importance distribution (SIR2 and SIR4) have smaller means. For cases with smaller measurement noise variances (σ 2 = 1 and 0.1 m2), the two filters using alternative components in the importance distribution (SIR3 and SIR4) show smaller means, while the SIR2 results behave erratically. It might be reasonable then to conclude that the filters having test scores closer to the theoretical

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5 Conclusions In this work, we applied a multivariate binning technique from Roederer et al. (2001) to the comparison of PFs. This was described for a linear and Gaussian filtering scenario, where we have an analytical form

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of the marginal posterior, and also for a nonlinear and Gaussian filtering scenario, where we estimated the optimal solution with a PF. The conclusions resulting from the proposed test were similar to the conclusions from an RMSE analysis of the mean estimates. We have not offered a detailed discussion of the practical, implementation aspects of such a test. It should be mentioned that our implementation of the test, i.e. the construction of the bins and the actual binning of the samples, used data structures similar to Kd-trees, see e.g. de Berg et al. (2000), and was computationally feasible for the cases that we considered. The literature on the χ 2 -test is vast and admittedly, our treatment of the test has been brief. In this section, we point out some questionable aspects

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of our use of this test for comparing PFs. First, we should question the use of the χ 2 -test itself. We are testing whether the two samples are from the same distribution, although it was already noted in Pitt and Shephard (1999) that the methods will not produce IID samples from the true posterior due to the finite mixture approximation. In spite of this, we have still considered the test scores as a way to empirically quantify the difference between distributions. Second, we should question how the χ 2 -test was actually performed. The number of bins and the construction of the bins in χ 2 -tests is not always straightforward and is therefore debatable. The resampling that is carried out before binning the PF samples is also questionable; due to the “extra” resampling, our test

is then comparing an approximation of the PF approximation, which complicates the analysis. Finally, our use of a large sample PF to approximate the optimal posterior, as well as our choice of importance distribution for this PF, is not properly justified; other methods such as rejection sampling or MCMC might result in a better approximation of the posterior. The intention of the distribution comparison was to devise better ways of comparing PFs and possibly other Bayesian filters. The binning procedure that was described is, of course, limited to sample-based methods. Other possible methods could include integrating the posterior distribution over a finite number of regions and then using some distance, e.g. Kullback Leibler divergence or total variation distance,

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on the resulting finite state-space to quantify the distance between the distributions. However, the choice of distance is then quite arbitrary and there is little reason to prefer one distance over another. This seems to be an interesting area for future work.

References Baggerly, K.A. (2001). Probability Binning and testing agreement between multivariate immunofluorescence histograms: Extending the chi-square test. Cytometry, Vol. 45, pp. 141–150. Bar-Shalom, Y., and X.R. Li, Estimation and Tracking: Principles, Techniques, and Software. Ybs Publishing, 1998. De Berg, M., M. van Kreveld, M. Overmars, and O. Schwartzkopf (2000). Computational Geometry: Algorithms and Applications. Springer, 2nd Ed. Heine K. (2005). Unified framework for sampling/importance resampling algorithms. In Proceedings of Fusion 2005. Pitt, M.K. and N. Shephard (1999). Filtering via simulation: Auxiliary particle filter. In Journal of the American Statistical Association, Vol. 94, No. 446, pp. 590–599. Petrovich D. (2006). Sequential Monte Carlo for Personal Positioning. M.Sc. Thesis, submitted to: Tampere University of Technology. Roederer, M., W. Moore, A. Treister, R. Hardy, and L.A. Herzenberg (2001). Probability binning comparison: a metric for quantitating multivariate differences. Cytometry, Vol. 45, pp. 47–55.

Appendix In this section, we briefly describe the four different importance distributions used for the PFs tested in this work. The system equations are typically given as x k = f k−1 (x k−1 ) + vk−1 yk = h k (x k ) + z k , where x k and yk are the values of the signal and observation stochastic process, respectively, at time k. Furthermore, vk−1 denotes the state of the signal noise process at time k − 1 and z k denotes the state of the observation noise process at time k. We follow the “auxiliary” formulation of Pitt and Shephard (1999). Assuming that we have N particles approximately distributed according to Px0 , the algorithm is as follows: For k = 1, 2, . . . • For i = 1:N, assign 1st-state weights βki and normalize

• For i = 1:M, sample index j i from discrete distribution of 1st-stage weights • For i = 1:M, sample state x ki conditioned on index j i and all received measurements y1:k • For i = 1:M, evaluate 2nd-stage weights and normalize • Resample N particles The general form of the unnormalized secondstage importance weight can be written as

w(x k , j ) ∝

p(x k , j |y1:k ) q(x k , j |y1:k ) j

∝

j

wk−1 p(yk |x k ) p(x k |x k−1 ) j

βk q j (x k )

,

j

where j is the auxiliary variable, βk is the j th firststage weight, and q j is the j th importance density. Note that for notational convenience, we have dropped the conditioning on the measurements for the first-stage weight and importance density. The four different importance distributions result j from different choices of βk and q j . SIR1 results j j j from using βk = wk−1 and q j = p(x k |x k−1 ). SIR2 is an example from Pitt and Shephard (1999) that j j j j uses βk ∝ wk−1 p(yk |ξk ) and q j = p(x k |x k−1 ), j j where ξk is the mean of the distribution of x k |x k−1 . Note that in the literature, this example is often referred to as the auxiliary particle filter. SIR3 again j j uses βk = wk−1 , but now uses an EKF for each j j importance distribution, i.e. q j = ν(x k ; μk , Ck ) where ν is a density of a Gaussian distribution and j j the mean μk and covariance Ck are given by the posterior of the j th EKF. SIR4 uses the same importance distribution as SIR3, but has different firstj j j stage weights given by βk = wk−1 ck , where

j ck = ν yk ; m, ⌶ j

m = h k ( f k−1 (x k−1 )) j j = Hˆ k ⌺vk−1 ( Hˆ k )T + ⌺zk , j where Hˆ k is the Jacobian matrix of h k evaluated at j f k−1 (x k−1 ), ⌺vk−1 is the covariance matrix of the signal noise vk−1 , ⌺zk is the covariance of the observation noise z k .

An Effective Wavelet Method to Detect and Mitigate Low-Frequency Multipath Effects E.M. Souza, J.F.G. Monico, W.G.C. Polezel Department of Cartograph, São Paulo State University – UNESP, Roberto Simonsen, 305, Pres. Prudente, SP, Brazil A. Pagamisse Department of Mathematics, São Paulo State University – UNESP, Roberto Simonsen, 305, Pres. Prudente, SP, Brazil Abstract. To ensure high accuracy results from GPS relative positioning, the multipath effects have to be mitigated. Although the careful selection of antenna site and the use of especial antennas and receivers can minimize multipath, it cannot always be eliminated and frequently the residual multipath disturbance remains as the major error in GPS results. The high-frequency multipath from large delays can be attenuated by double difference (DD) denoising methods. But the low-frequency multipath from short delays is very difficult to be reduced or modeled. In this paper, it is proposed a method based on wavelet regression (WR), which can effectively detect and reduce the low-frequency multipath. The wavelet technique is firstly applied to decompose the DD residuals into the low-frequency bias and high-frequency noise components. The extracted bias components by WR are then directly applied to the DD observations to correct them from the trend. The remaining terms, largely characterized by the high-frequency measurement noise, are expected to give the best linear unbiased solutions from a least-squares (LS) adjustment. An experiment was carried out using objects placed close to the receiver antenna to cause, mainly, lowfrequency multipath. The data were collected for two days to verify the multipath repeatability. The “ground truth” coordinates were computed with data collected in the absence of the reflector objects. The coordinates and ambiguity solution were compared with and without the multipath mitigation using WR. After mitigating the multipath, ambiguity resolution became more reliable and the coordinates were more accurate. Keywords. Low-frequency regression, GPS

multipath,

wavelets

1 Introduction The multipath effect distorts the signal modulation and the carrier phase degrading the accuracy of

absolute and relative GPS positioning. Furthermore, this effect can impede the ambiguities solution, or lead to incorrect one. However, multipath signals are always delayed compared to line-of-sight signal because of the longer travel paths caused by the reflection. If the multipath effect is from long delays, it is characterized by high frequencies; otherwise, it is of low frequencies. The low-frequency multipath from short delays causes the largest errors and it is very difficult to be reduced or modeled. Some works to reduce low-frequency multipath have been proposed in relation to the receiver technology. Sleewaegen and Boon (2001) proposed posprocessing estimation of the tracking errors that seems to be a good performance but a scaling factor, depending on multipath environment, has to have used to link the signal amplitude with the range error. Other technique, a virtual multipath based technique, was proposed by Zhang and Law (2005). This method could provide a 2 m maximum range error absolute for C/A code. There are also several postreception methods to mitigate multipath. Xia (2001) used the wavelet transform to separate multipath from the DD carrier phase observation. Satirapod et al. (2003) have used wavelets to mitigate multipath of permanent GPS stations. Souza (2004) used the wavelets multiresolution analysis to reduce the highfrequency multipath from the DD temporal series. The low-frequency multipath is very difficult to be reduced directly from the DD. In this paper, the WR is used to estimate the low-frequency multipath effect from the pseudorange and carrier phase DD residual temporal series, which is obtained from the LS solution. Then, the low-frequency multipath components are directly applied to the DD observations to correct them from this effect. The idea of this method is that the LS estimation is based on the formulation of a mathematical model consisting of the functional and the stochastic model. If the function model is adequate, the 179

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residuals obtained from the LS solution should be randomly distributed (Satirapod et al., 2003). However, the GPS observations are contaminated by several types of biases. A double-differencing technique is commonly used for constructing the functional model as it can eliminate or reduce many of the GPS biases (atmospheric, orbital, receiver and satellite clock biases) for short baselines. However, the multipath error is not eliminated because it depends of the geometry and environment of each station. Therefore, multipath is a major residual error source in the double-differenced GPS observables. This warrants that the effect estimated using WR is, mainly, due to multipath. Furthermore, the multipath repeatability could be verified by repeating the process during different days. In Section 2 the wavelet transform and the mainly aspects related to its implementation and computation using the pyramidal algorithm are described. In Section 3, the wavelet regression to estimate and correct the low-frequency multipath is outlined. The experiment and results are present in Section 4. Finally, the conclusions are given in Section 5.

The equation (3) creates an orthonormal family in L 2 (), φ j,k (t) = 2− j/2 φ 2− j t − k ,

ψ (t) =

x −b ; a

ψ j,k (t) = 2

− j/2

ψ 2− j t − k ,

√ hk = 2

is an orthonormal basis (Chui, 1992). But a remaining question is how to obtain the wavelet ψ. Daubechies (1988) developed a approach to construct this wavelet using the scaling function that is the solution to the equation: φ (t) =

√ 2 h k φ (2t − k) . k

√

2

(3)

∞

φ (t) φ (2t − k) dt and

(6)

ψ (t) φ (2t − k) dt.

(7)

−∞ ∞

−∞

Thus, considering the orthonormal system {φ j,k (t), ψ j,k (t), j, k ∈ Z}, f (t) ∈ L 2 () can be written by

f (t) =

c J,k

φ J,k (t) +

d j,k

ψ j,k (t),

(8)

j≤J k

k

where c J,k = d j,k =

(2)

(5)

where gk = (−1)k h 1−k is the quadrature filter relation. In fact, h k and gk are coefficients of the lowpass and high-pass filters also called quadrature filters. These filters are used to compute de Discrete Wavelet Transformation (DWT) and are given by

a, b ∈ R, a = 0

(1) where a represents the dilation parameter and b the translation parameter. For some very special choices of ψ, a and b, the ψa,b constitute an orthonormal basis for L 2 (). In particular, with the choice a = 2 j and b = k2 j , with j , k ∈ Z , then there is ψ, such that

√ 2 gk φ (2t − k) , k

2 Wavelet Transform and Pyramidal Algorithm

1 ψa,b (x) = √ ψ |a|

(4)

In these conditions, ψ can be obtained by

gk =

Wavelets are building block functions and localized in time or space, or both. They are obtained from a single function ψ ∈ L 2 (), called the mother wavelet, by translations and dilations (Daubechies, 1992):

j, k ∈ Z .

∞

−∞ ∞ −∞

f (t) φ J,k (t) dt,

(9)

f (t) ψ j,k (t) dt.

(10)

To work with discrete signals, the remaining question is how to compute, in practice, the d j,k e c j,k . coefficients. They are in fact computed by the pyramidal algorithm developed by S. Mallat (1989), what is being described below. Using the scaling φ (equation 3) and the ψ wavelet definition (equation 5), the coefficients 9 and 10 can be written by

An Effective Wavelet Method to Detect and Mitigate Low-Frequency Multipath Effects

d j,k = f, ψ j,k L 2 =

gn f, ϕ j −1,2k+n L 2 , (11)

n∈Z

d j,k =

c j,k = f, ϕ j,k

gn−2k c j −1,n ,

n∈Z

L2

=

(12)

h n f, ϕ j −1,2k+n

L2

, (13)

181

3 Wavelet Regression to Estimate and Mitigate the Low-Frequency Multipath The DD residuals (⌬∇v) can be modeled by: ⌬∇vi = f (ti ) + εi ,

i = 0, . . . , n − 1,

(15)

n∈Z

c j,k =

h n−2k c j −1,n

(14)

n∈Z

with h k e gk given by the equations (6) and (7), respectively. These four equations above are the base of pyramidal algorithm (Mallat, 1998; Gilbert and Keller, 2000). In these equation one can note that the algorithm computes c j,k e d j,k . using the coarse coefficients of the level j − 1. As stated before, the transformation (14) can be interpreted as low-pass filter and the transformation (12) as high-pass filter. This means that the high frequencies components (details or noise) of the signal are separated from the low frequencies ones (lowfrequency multipath) in many resolution levels. The name multiresolution analysis comes from this idea. Furthermore, the equations (14) and (12) can be seem as a convolution process followed by a downsampling by two (↓ 2) that applied to a sequence implies that the even or odd samples are eliminated. Consequently, the level j has half of the coefficients of the level j − 1, indicating the pyramidal name. Figure 1 illustrates this procedure. The Inverse DWT (IDWT) essentially perform the same operations associated with the DWT in opposite direction. Instead of downsampling, the signal is interpoted: zeros are add among the coefficients (upsampling, ↑ 2). More details of pyramidal algorithm can be found in Chui (1992), Strange and Nguyen (1996) and Mallat (1998). The great advantage of pyramidal algorithm is that for the wavelet transformation only O(N) operations are necessary.

H 0 = ∑ hn − 2 k c 0, n

H 1 = ∑ hn − 2 k c1, n

n

n

c0,k

↓2

c1,k

c2,k ...

↓2

G 0 = ∑ g n − 2 k c 0, n

G1 = ∑ g n − 2 k c1, n

n

n

↓2

↓2

d1,k Fig. 1. Decomposition pyramidal algorithm.

d 2,k ...

where ti = i /n, f is the function desirable to be estimated, that is, the low-frequency multipath effect in the DD, εi ∼ N(0, σ 2 ) is a Gaussian white noise and σ is the noise level. A wavelet regression estimator works as follows: 1. Find the DWT of DD residuals ⌬∇v to obtain the wavelet coefficients using the pyramidal algorithm as explained in Section 2; 2. Modify the wavelet coefficients by thresholding; 3. Reconstruct the function f (low-frequency multipath effect) with the IDWT, as described in Section 2. The crucial part of the regression is step 2 (Nason, 1994). Nearly all the relevant works on thresholding rules and optimal thresholds are contained within Donoho and Johnstone (1994). The thresholds presented in this work were analyzed by Souza and Monico (2004). So, in this paper, only the thresholds that presented the best performance for GPS signal will be described. Let di , i = 1, . . ., n the wavelet coefficients. The hard threshold function (TλH ) given by

TλH (di ) =

0, |di | < λ di , |di | ≥ λ

(16)

where λ is the universal threshold parameter λ = σˆ 2 log n

(17)

that provides a fast and automatic thresholding. The σˆ is the observation noise level and should be estimated from each temporal series of DD residuals in the first decomposition level (the finest scale). As the empirical wavelet coefficients at the finest scale are, with a small fraction of exceptions, essentially pure noise, Donoho and Johnstone (1994) proposed the following estimator:

σˆ = median d J −1,k : 0 ≤ k < n/2 /0, 6745, (18) where J − 1 is the finest scale. In relation to the mother wavelet, Symmlets with eight vanishing moments was used. Souza and

182

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Monico (2004) showed that, for GPS signal, this mother wavelet is better than other Symmlets and Daubechies wavelets. Therefore, the undesired parts (noise) are removed by modifying the wavelet coefficients and the wavelet regression can reconstruct the multipath effect from DD residual temporal series. Then, the low-frequency multipath components are directly applied to the DD observations to correct them from this effect.

4 Experiments Description In order to test the proposed method based on WR to estimate the low-frequency multipath an experiment was conducted in the Takigawa Company (TAK) at Presidente Prudente, Brazil on September 16, 17, 20 and 21, 2003. During the two first days, an Astech ZXII receiver was placed on the ground, at a distance of about 6 m from one aluminum cart of 13 × 2.5 m and from 0,5 m of a zinc plate of 0.30 × 1.10 m (Figure 2). From these two days the multipath repeatability can be analyzed because the survey geometry remained unchanged. Due to the short distance of the objects, the main multipath caused is the low-frequency one from short delays. In the last two days (20 and 21), the objects were removed. From these two days the best coordinate could be estimated because of the absence of the multipath effect. The data was collected at a sample rate of 15 seconds with a cut off elevation angle of 5◦ . A permanent GPS station (UEPP) was used as reference station, where there was a Trimble 4000 SSI receiver with a choke ring antenna centered in a concrete pillar of 3 m height. This station can be considered as multipath-free site. Since the baseline length is about 1900 m, errors resulting from ionospheric, tropospheric and orbits are assumed to be insignificants. Therefore, the double-differenced pseudorange and carrier phase estimated residuals may exhibit, mainly, multipath and observation noise. The pseudorange and carrier phase DDs were processed using the GPSeq software (FORTRAN

language), which is under development at UNESP, Presidente Prudente (Machado and Monico, 2002). WR method used to mitigate low-frequency multipath effect was also implemented in this software. It was chosen for processing a data session of 25 minutes (100 epochs) which was sufficient to observe the multipath errors. As base satellite, PRN 23 was chosen (elevation angle 70◦–58◦). The results and analyses are presented in the next section.

5 Experiments Results and Analyses To make sure that the errors in the residuals are due to multipath, the repeatability of this effect was analyzed in the days 16 and 17. An example of a DD residual temporal series obtained from the LS solution can be visualized in Figure 3. Once the multipath repeatability was verified, the wavelet regression was applied to each DD residual temporal series to estimate the low-frequency multipath effect as described in Sections 2 and 3. Then, the low-frequency multipath components were directly applied to the DD observations to correct them from this effect. The LS was performed again with the corrected DD observations. The LS with the wavelet regression is denoted by LSWR. The DD residual temporal series obtained from LS and LSWR were compared. The estimated pseudorange and carrier phase of the cases with less (DD 23–15) and more (DD 23–21) multipath effect in the DD residuals are plotted in Figures 4((a) and (b)) and 5((a) and (b)), respectively. One can see in Figures 4 and 5 that the multipath trend was significantly mitigated for the carrier phase and the pseudorange using the LSWR. This fact is confirmed in Table 1, where one can observe that the improvement reached up 99% and 90% in the pseudorange and carrier phase, respectively. To compare the quality of the pseudorange and carrier phase DD observations model, before (LS) and after (LSWR) the low-frequency multipath

Astech ZXII Fig. 2. Survey environment with objects – Sep 16 and 17.

Fig. 3. Multipath repeatability on DD 23–21 residuals.

An Effective Wavelet Method to Detect and Mitigate Low-Frequency Multipath Effects

183

GOM

15 10

LS LSWR

5 0,107

0,350

0 Day16 (a) Pseudorange

Day17

Fig. 6. GOM test statistic.

(b) Carrier Phase

Fig. 4. DD 23–15 residuals – Day 16.

mitigation, the Global Overall Model (GOM) test statistic (Teunissen, 1998) was used (Figure 6). From Figure 6 it can be verified that GOM statistic improved after using the LSWR to correct the low-frequency multipath, showing that this systematic effect was significantly mitigated.

It is important to analyze the ambiguity solution, because multipath effects can affect the real-valued float ambiguity solution, conducting to an incorrect integer (fixed) ambiguity solution. The quality of the ambiguity solution was also compared. The success (Ps ) and fail (P f ) probabilities of Ratio Test of Integer Aperture (RTIA) proposed by Teunissen and Verhagen (2004) were implemented and computed using 500,000 simulations. With this test it is possible to verify the correctness of the ambiguities with a firm theoretical foot. The fixed solution was accepted if P f was lower than 0.025 (user-defined), otherwise it was rejected and the float solution used. The Ps for LS and LSWR are shown in Figure 7. It can be seen in Figure 7 that the Ps was very low using the LS. This was expected because the multipath influences the ambiguity solution very much. After the low-frequency multipath mitigation (LSWR) the Ps became very high, consequently, the

Sucess Pobability

(a) Pseudorange

(b) Carrier Phase

Fig. 5. DD 23–21 residuals – Day 16.

0,976

0,976

1,0 0,8 0,6

LS LSWR

0,4 0,2 0,0

0,005

0,045

Day16

Day17

Fig. 7. Success probabilities of RTIA.

Table 1. RMS of the DD residuals – Days 16 and 17

Pseudorange DD Residuals DD 23–02 DD 23–03 DD 23–14 DD 23–15 DD 23–16 DD 23–18 DD 23–21 Average

Carrier Phase

LS

LSWR

Improvement

LS

LSWR

Improvement

0.51 0.30 0.93 0.41 0.57 0.42 2.19 –

0.13 0.048 0.011 0.118 0.133 0.079 0.13 –

−75% −84% −99% −71% −77% −81% −94% −83%

0.009 0.01 0.009 0.009 0.008 0.011 0.009 –

0.003 0.001 0.003 0.002 0.003 0.003 0.004 –

−67% −90% −67% −78% −63% −73% −56% −70%

(m)

184

E.M. Souza et al. 0,04 0,03 0,02 0,01 0,00 –0,01 –0,02 –0,03

LS LSWR

E

N

u

E

Day 16

N

u

Day 17

more reliable and the success probabilities increased significantly (Figure 7). The accuracy of the coordinates after applying the LSWR improved up to 16 mm, and 10 mm on average. Further experiments will be carried out in order to confirm the results presented so far.

Acknowledgments

Fig. 8. Coordinate component discrepancies.

The authors thank FAPESP for the financial support (03/12770-3 and 04/02645-0).

0,015

References

(m)

0,01 LS

0,005

LSWR

0 E

N Day 16

h

E

N

u

Day 17

Fig. 9. Coordinate component standard deviations.

P f was smaller than 0.025, indicating the fixed solution should be accepted with reliability. Now, in order to compare the coordinates using the methods LS and LSWR, the “ground truth” coordinates were estimated using data collected in the absence of the reflector objects (Days 20, 21). The coordinate discrepancies between the “ground truth” coordinates and those estimated with the LS and LSWR were computed for the days 16 and 17 (Figure 8). The respective standard deviations are shown in Figure 9. After multipath mitigation using the LSWR, discrepancies between the coordinates were the smallest for the three components: E, N and u (Figure 8). The standard deviations were also significantly improved using the LSWR. Therefore, one can conclude that multipath was the main error affecting the ambiguity resolution and the accuracy of the coordinates and that the LSWR reduced significantly this error.

6 Conclusions From these preliminary results, without doubt, LSWR is a powerful method to mitigate lowfrequency multipath effects in GPS applications. The low-frequency multipath trend in DD residual temporal series was significantly corrected. The RMS of the residuals improved up to 99% using the LSWR method. Furthermore, after low-frequency multipath mitigation using LSWR, the ambiguity solution became

Chui CK (1992) An introduction to wavelets. Boston: Academic Press. Daubechies I (1992) Ten Lectures on Wavelets. Vol 61 Regional Conference, SIAM, Philadelphia, PA. Daubechies I (1988) Orthonormal bases of compact supported wavelets. Communications on Pure and Applied Mathematics, Vol. 41, pp. 909–996. Donoho DL, Johnstone IM (1994) Ideal spatial adaptation by wavelet shrinkage. Biometrika, Vol 81, pp. 425–455. Gilbert A, Keller W (2000) Deconvolution with Wavelets and Vaguelettes. Journal of Geodesy, Vol. 74, pp. 306–320. Machado WC, Monico, JFG (2002) Utilizaça˜ o do software GPSeq na Soluça˜ o rápida das ambigüidades GPS no Posicionamento Relativo Cinemático de Bases Curtas. Pesquisas em Geociências, Vol. 29, No. 2, pp. 89–99. Mallat S (1998) A Wavelet Tour of Signal Processing. United States of America: Academic Press, p. 577. Nason GP (1994) Wavelet Regression by Cross-Validation, TR 447, Department of Statistics, Stanford University. Satirapod C et al. (2003) Multipath mitigation of permanent GPS stations using wavelets. In: Int. Symp. on GPS/GNSS, Tokio, Japan. 15–18 November. Sleewaegen JM, Boon F (2001) Mitigating short-delay multipath: A promising new technique. In: ION GPS, Salt Lake City, UT, pp. 204–213. Souza EM (2004) Multipath reduction from GPS double differences using wavelets: How far can we go? In: ION GNSS, 17, Long Beach, CA, pp. 2563–2571. Souza EM, Monico JFG (2004) Wavelet shrinkage: High frequency multipath reduction from GPS relative positioning. GPS Solutions. Vol 8, No. 3, pp. 152–159. Strang G, Nguyen T (1996) Wavelets and Filter banks, Wellesley-Cabridge Press. Teunissen, PJG (1998) Quality control and GPS. In: Teunissen, PJG.; Kleusberg, A. GPS for Geodesy. 2 ed. Berlin: Springer Verlag, pp. 271–318. Teunissen PJG, Verhagen S (2004) On the foundation of the popular ratio test for GNSS ambiguity resolution. In: ION GNSS, 17. Long Beach, CA, pp. 2529–2540. Xia L (2001) Approach for multipath reduction using wavelet algorithm. In: ION GPS, Salt Lake City, UT, pp. 2134–2143. Zhang Z, Law CL (2005) Short-delay multipath mitigation technique based on virtual multipath. In: IEEE Antennas and Wireless Propagation Letters, Vol. 4, pp. 344–388.

Regional Tropospheric Delay Modeling Based on GPS Reference Station Network H. Yin Earthquake Administration of Shandong Province, Jinan 250014, P.R. China Department of Geomatic Engineering, Southwest Jiaotong University, Chengdu 610031, P.R. China D. Huang, Y. Xiong Department of Geomatic Engineering, Southwest Jiaotong University, Chengdu 610031, P.R. China Abstract. Global Positioning System (GPS) satellite signals suffer ranging errors when traveling through the neutral atmosphere. Tropospheric delay consists of a hydrostatic component depending on air pressure and temperature, and a wet delay depending on water vapor pressure and temperature. Due to the strong spatial inhomogeneity and temporal variability of atmospheric density, especially for water vapor, accurate modeling of path delay in GPS signals is necessary in high-accuracy positioning and meteorological applications (climatology and weather forecasting). A newly developed tropospheric delay model is proposed to simulate regional tropospheric zenith delay using multiple reference stations. The new model is proposed based on the relationship between tropospheric zenith delays and the elevation of GPS stations. To evaluate the performance of the new model data from both GPS reference station network of Sichuan and weather station in Chengdu were used. Experimental results show that the accuracy of GPS tropospheric zenith delay can be improved (less than 1 cm) using the new method both in static and rover stations. It is very useful to enhance the effectiveness and reliability of the tropospheric delay resolution process for regional GPS network users.

redundant observations of several reference stations improve the reliability of the reference station data. On the other hand, the covered areas of the network are much wider than that of standard single reference technique using the same number stations. One way to use the full information of multiple reference stations network is the so-called virtual reference station (VRS) technique. An optimum set of code and carrier phase observations of a virtual reference station is obtained by using data from multiple reference stations (Bevis et al., 1992, Businger et al., 1996, see Figure 1). Since some of the observation errors depend on the horizontal position and height differences between the rover receiver and the reference stations, (Dodson et al., 1996), the rover’s approximate position has to be known. Several processing steps have to be performed in order to transform the carrier phase observations of real reference stations to that of a virtual reference station (Wanninger, 1997, see Figure 2).

Keywords. GPS, tropospheric delay, altitude, real time

1 Introduction With the establishment of active GPS-reference networks with station distances from 30 to 100 km or more, a new era of cm-accurate GPS positioning for long baseline has begun. Centimeter-accurate fast static or real-time kinematic (RTK) positioning will no longer be performed relative to one single reference receiver but relative to multiple references. This enables faster and more reliable ambiguity resolution and more accurate coordinates. Furthermore,

Fig. 1. Virtual GPS reference station in a regional network. 185

H. Yin et al.

Fig. 3. Distribution of reference stations.

OPRD

OAES

LDES

EWPP

DSSC

CTMS

CDMT

BSRY

BILL

0

BBRY

500

Fig. 4. Altitude of reference stations.

South California Integrated GPS Net (SCIGN), which consists of 11 reference stations (see Figure 3). The altitude of the reference stations are shown in Figure 4. GAMIT software (Version 10.2) is used for data processing and analyzing. The cut-off elevation angle was chosen as 15◦ . Testing results of tropospheric delay are shown in Figure 6. From the Figure 4 and Figure 5, we can see that tropospheric zenith delay is strong correlated with the altitude of station, and tropospheric zenith delay is inversely proportion to the altitude of station. By analyzing the test data, the author found that the relationship of tropospheric zenith delay and the altitude of station appeared to be a negative exponent. With the help of above finding, we present two regional tropospheric fitting models, i.e. EH1QX1 and EH1QM3. For GPS reference station network, the fitting model EH1QX1 has only one geographical factor, the height dT = a0 e−a1 h + a2

(1)

where h is the height of station in meter, ai (i = 0, 1, 2) is unknown fitting coefficients, which can be estimated by least-squares criteria. The model is applicable in small and middle areas. With the extending of coverage of the GPS network, the correlation between multiple stations will decrease, and the impact of the horizontal positioning of stations to tropospheric delay will increase. 2.5 2.3

Fig. 5. Tropospheric zenith delay in reference stations.

OPRD

OAES

LDES

EWPP

DSSC

CTMS

1.9

CDMT

2.1

BSRY

Tropospheric Zenith Delays/m

In order to analyze the relationship of tropospheric delay to the position of station, we use data (collected on May 4, 2005) from a subnet of

1000

BILL

2 Modeling Regional Tropospheric Delay Using GPS Data of Multi-Reference Stations

1500

BBRY

The observations of the virtual station are computed from the observations of multiple real reference stations (Brunner and Gu, 1991). In Figure 1, the blue areas are the range of traditional RTK technology service, and the yellow areas are the range of VRS network. If the tropospheric delay can be estimated or predicted in some ways, GPS accuracy performance can be enhanced significantly. Accurate modeling and estimating of tropospheric delays becomes critical to the improvement of the positioning accuracy (Ichikawa, 1995, Guoping, 2005). The redundant information available through the availability of multiple reference stations should intuitively be useful in improving the prediction accuracy of tropospheric delays.

2000

AGMT

Fig. 2. Working flow of VRS.

2500

AGMT

Altitudes of stations/m

186

Regional Tropospheric Delay Modeling Based on GPS Reference Station Network

187

For these reasons, we developed a model called EH1QM3, which can be applied to larger areas (more than 300 × 300 km), that is dT = a0 e−a1 ·h + a2 B + a3 L + a4

(2)

where B and L are latitude and longitude of stations respectively, the other parameters in (2) are the same to (1).

3 Test in GPS Reference Station Network of Chengdu, China In order to test the regional tropospheric fitting models, a experiment is conducted using the data from GPS reference station network of Sichuan on the day of 295 in 2005 which was established by Bureau of Sichuan Province in 2001, 5 stations are consisted of (see Figure 6). The tropospheric zenith delays in reference and rover stations in the network are estimated using GAMIT software (Version 10.2), which are used as true values to test the new tropospheric models. In order to mitigate orbital errors, precise orbits from IGS were used in the data processing. The cut-off angle is 15◦ . Precise coordinates were predetermined for reference stations (see Table 1) using GAMIT software. One tropospheric zenith delay parameter is estimated for per a two-hour period and one azimuthal gradient parameter for per twelvehour period. To evaluate the accuracy performance of the new fitting models, the Saastamoinen tropospheric model

Fig. 6. Location of the network stations. The sign denotes the reference stations, and the sign • denotes temporary station.

is used for comparison, which is expressed by the following equation (Saastamoinon, 1972) dh = 0.2277 × 10−2 × P/ f (B, H ) dw = 0.2277 × 10−2 (1.255 × 103 /T + 0.005) × e/ f (B, H ) (3) −2 f (B, H ) = 1 − 0.266 × 10 × cos(2B) − 0.28 × 10−3 × H where B is latitude of stations, H (kilometer), is the height of station, and the meteorological data (P, e

Table 1. Coordinates of GPS reference station network of Sichuan (∗ denotes temporary stations) Station Name

Latitude (deg min sec)

Longitude (deg min sec)

Height (m)

CHDU PIXI RENS ZHJI QLAI XINJ∗ DMSH∗ JDSI∗ YCHA∗ HSWA∗ CJWG∗ YANE∗ JTAN∗ DANI∗ DSHU∗ FHSH∗ LQUA∗ JTUN∗

30 38 21.84152 30 54 36.54516 30 12 1.375535 31 0 22.62147 30 21 15.74566 30 26 38.28696 30 53 7.083955 30 35 52.12911 31 18 49.03464 30 34 0.29242 30 49 29.13622 30 32 15.52041 30 52 10.90216 30 35 23.82297 30 22 27.54517 30 43 55.05001 30 33 14.69398 30 53 55.99534

104 3 52.16706 103 45 25.1146 104 6 10.33951 104 32 44.82645 103 18 21.4763 103 53 10.37569 103 59 20.89385 103 57 15.94407 103 52 5.93502 103 15 35.29551 103 44 33.38949 104 4 32.43237 104 23 3.01375 104 11 10.67608 103 50 38.32064 104 4 38.66296 104 18 7.57601 104 8 54.15575

491.8164 617.6185 791.06145 897.0281 732.11735 455.32635 503.29455 454.20865 1233.3604 752.5536 565.0644 442.6038 429.8798 496.0949 407.967 490.1799 628.6879 470.5673

H. Yin et al.

Not only the horizontal positioning but also altitudes of reference stations should be considered when establishing the network.

JTUN

JDSI

LQUA

FHSh

DSHU

DAMI

JTAN

YANE

CJWG

HSWA

DMSH

Acknowledgements

XINJ

0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 –0.01

YCHA

Mean square Error/m

188

Fig. 7. Mean square Errors of tropospheric zenith delay for the new fitting model.

and T ) in the station is the practical measurement on the day.

4 Results and Analysis Then we compute the regional tropospheric zenith delay using the new simulated model (1). The tropospheric zenith delay in the rover station can be computed in real-time using the new model. The mean square errors for the new model on a whole day are shown in Figure 7. In Figure 7, we can see that all the mean square errors are less than 1 cm apart from station YCHA and DSHU. Why the accuracy of the station YCHA and DSHU is lower than others? The reason is that the heights of the two stations are beyond the height range of all reference stations in the network. So we concluded that the distributing of the height of reference stations should be considered when establishing the network.

5 Conclusions and Suggestions New regional fitting tropospheric models: EH1QX1 and EH1QM3 are proposed in this paper. The new model was estimated and tested in the Chengdu reference station network, the results showed that the accuracy of all rover stations whose heights were not beyond the reference stations height range were less than 1 cm.

The authors acknowledge with thanks to IGS for the precise orbit products and reference station data and Chengdu weather station for meteorological data. Special thanks are also given to Dr. Guoping Li for his work involved.

References Bevis M., Businger S., et al. (1992). “GPS meteorology: Remote sensing of atmospheric water vapor using the Global Positioning System”, Journal of Geophysical Research, Vol. 97, No. D14, pp. 15787–15801. Businger S., Chiswell S.R., Bevis Michael (1996). “The promise of GPS in the atmospheric monitoring”, Bulletion of American Meteorology Society, Vol. 77, No. 1, pp. 1–18. Dodson A.H., Shardlow P.J., Hubbard L.C.M., Elegered G., and Jarlemark P.O.J. (1996). “Wet tropospheric effects on precise relative GPS height determination”, Journal of Geodesy, Vol. 70, No. 4, pp. 188–202. Brunner F. K. and Gu M. (1991). “An improved model for the dual frequency ionospheric correction of GPS observations”, Manuscripta Geodetica, Vol. 16, pp. 205–214. Saastamonien J. (1972). “Atmospheric correction for the troposphere and stratosphere in radio ranging of satellites”, The Use of Artificial Satellites for Geodesy, Geophysics Monograph Series, Vol. 15, pp. 245–251. Ichikawa R. (1995). “Estimations of atmospheric excess path delay based on three-dimensional, numerical prediction model data”, Journal of Geodesy Society (Japan), Vol. 41, pp. 379–408. Li G., Huang D., Liu B. (2005). Diurnal Variation of GPS Precipitable Water Vapor in Chengdu Plain during the Warm Season , The International Association of Meteorology and Atmospheric Sciences (IAMAS) 2005 Scientific Assembly, Beijing, China, 2–12. Wanninger, L. (1997). “Real-time differential GPS error modeling in regional reference station networks”, In Proceedings of the International Association of Geodesy Symposia, Vol. 118. Advances in Positioning and Reference Frames (pp. 86–92). Riode Janeiro, Brazil.

Research on GPS Receiver Antenna Gain and Signal Carrier-to-Noise Ratio J. Liu, J. Huang, H. Tian, C. Liu School of Geodesy and Geomatics, Wuhan University, Wuhan, China

Abstract. In this paper, the gain and gain pattern of GPS receiver’s antenna and the carrier-to-noise ratio of received signal are described. The concept of C/N0 pattern is introduced. The relationship between the C/N0 pattern and gain pattern is derived. The time variation and direction variation of atmospheric loss were determined initially. The concept of C/N0 anomaly and the C/N0 anomaly pattern were introduced. It would be shown that the orientation of passive interference could be determined with the C/N0 anomaly pattern. Keywords. GPS, antenna, gain pattern, carrier-tonoise ratio, C/N0 anomaly

1 Introduction Gain pattern is an important technical specification of GPS receiver antenna. It is elevation dependent. In generally, the determination of antenna’s gain pattern should be carried out in Anechoic Chamber with specialized equipment. C/N0 (Carrier-to-Noise Ratio, in dB–Hz) is the signal quality indicator recorded by the GPS receiver. It is dominated by the receiver antenna’s gain pattern. In recent years, GPS receiver antenna’s gain pattern is applied in GPS data analysis to improve accuracy and reliability of GPS positioning, navigation or to determine the attitude of vehicle. In this paper, the concept of C/N0 pattern is introduced. Relationship between gain pattern and C/N0 pattern is derived. The time variation and direction variation of atmospheric loss are evaluated. The concept of C/N0 Anomaly is introduced also.

given antenna as compared to an isotropic or dipole antenna. It can only be achieved by making an antenna directional, that is, with better performance in one direction than in others. It’s usually expressed in dB. For GPS receiver antenna gain, it describes its sensitivity over some range of elevation and azimuth angles, its ability to discriminate against low-elevation signals. It dependents on incident signal direction, signal frequency and polarization etc. Gain pattern is a model which describes antenna gain characteristics in different spatial direction. This model can be presented as a figure or a formula. Figure 1 is examples of gain pattern from two different viewing angles: horizontal and vertical:

3 C/N 0 and C/N 0 -Pattern C/N(Carrier-to-Noise Ratio) and S/N(Signal-toNoise Ratio) are two measurements of GPS signal quality. C/N is the ratio between carrier power and noise power at the input of receiver while S/Nis the ratio between signal power and noise power at the output of receiver. The relationship of them is as follows (Roddy, 2002): S/N = C/N + G p

(1)

2 Antenna Gain and Gain Pattern GPS receiver antenna is used to capture the transmitted satellites signal, and then convert it into the electric current for receiver to process. Definition of antenna gain is that: The raitio of energy density, received or transmitted by a

vertical

horizontal

Fig. 1. Examples of gain pattern from two different viewing angles. 189

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In (1), G P is the processing gain of receiver. So it is obviously that S/N is also related to the method of processing signal in GPS receiver. For example, when measuring the L2 carrier, different receiver would use different method to do that such as Cross Correlation Z-tracking and Square method (Hofmann-Wellenhof et al., 2001). Because of related to the bandwidth, C/N should be standardized to C/N0 in per Hz bandwidth, which is represented as dB–Hz. Relationship between C/N0 and C/N is (Roddy, 2002): C/N0 = C/N − BN

(2)

BN is effective noise bandwidth. The un-jammed C/N0 for received GPS satellites signal is as follows (Kaplan et al., 1996): C/N0 = Pr −10 Log10 (Tsys )+L NF +L I −228.6 (3) In (3),Pr : received power at the input of the low noise amplifier, it is related to effective isotropic radiated power (EIRP), atmosphere losses (AL), receiving antenna gain (G r ); Tsys: system noise temperature; L NF : noise figure of receiver; L I : losses in analog to digital (A/D) conversion. EIRP is dependent on GPS satellites transmission power and transmitted antenna gain. GPS transmission gain is used to provide relatively uniform power on the surface of the earth. This gain variation is worked as that: Increasing the L-Band antenna gain from the nadir level to a maximum at an antenna theta angle of 10◦ and decreasing slowly at 16◦. The gain increased from nadir to edge-to-earth (EOE) is approximately 2 dB to compensate the free space losses due to extra path traveled (Morea et al., 2002, and Parkinson et al., 1996). For GPS signal, AL in different season climate condition and direction is usually less than 1 dB. Its maximum is less than 2 dB. This can be proved by our following experiments. G r varies a lot in different elevation. From 0◦ to ◦ 90 , gain roll-in can achieve 15–20 dB. For example, based on our experiment, the gain roll-in of Trimble Choke Ring can be about 20 dB. Values of L NF and L I can maximumly achieve 5–6 dB, however what they affect C/N0 perform stochastic fluctuations. From above analysis, factors which affect C/N0 can be classified into three kinds: (1) Factors which is relatively changeless and minimal such as EIRP and AL; (2) Factors which has stochastic fluctuations

such as L NF and L I ; and (3) Factors which varies a lot in C/N0 and are systematic such as G r and others. As for the GPS receiver on the surface of the earth, generally when there are no interference and no stochastic fluctuations, C/N0 has a trend dominated by the receiver antenna gain pattern as follows: [C/N0 ] ≈ [G r ]

(4)

in (4), [C/N0 ] = [C/N0 ] − [C/N0 ]0 , [G r ] = [G r ] − [G r ]0 , [C/N0 ]0 and [G r ]0 are the values of [C/N0 ] and [G r ] in the direction of antenna boresight vector. So when using C/N0 raw data, we can build a relative gain pattern which can describe the dispersion from bore-sight vector to different directions. If a known antenna gain pattern is used as reference, we can also build a absolute antenna gain pattern. In practical application, the predicted values of signal C/N0 are just what we need, it can be provided by a model which describes characteristics of received signal C/N0 in different direction using specific combination of receiver-antenna. And this model is just what we call C/N0 -Pattern.

4 Antenna Coordinate System Antenna gain pattern and C/N0 -Pattern of received signal should be built in antenna reference coordinate frame. An antenna coordinate system is defined as follows (see Fig. 2): Origin = The antenna’s mean phase center X-Axis = direction of the orientation indicator Z-Axis = Vertical to the antenna’s base board, Y-Axis completes a right-handed orthogonal coordinate system. It’s obviously that when the antenna is horizontal and the orientation indicator is towards the north, the antenna coordinate system is coincided with the topcentric coordinate system of the antenna’s mean phase center. Also, a polar coordinate form antenna coordination system can be defined. When the antenna is skew, we must determine attitude parameter of it. The procedure of coordinate transformation as followings (see Fig. 3): 1. One antenna is horizontal placed with orientation indicator pointed to the north. And the other is skew placed with orientation indicator pointed to the ground. Both of them should be centered to the same point P. So, the attitude of skew antenna

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5 Experiment of Received Signal C/N0 Determination Four periods of experiments (In July 8th–9th 2004, December 9th–20th 2004, July 11th–17th 2005 and April 22th–23th 2006) were carried out to determine the variation of C/N0 according to different conditions, such as different antenna orientations and different observing environments etc. The experimental field was on the top of building of School of Geodesy and Geomatics (see Fig. 4). The equipments and settings were as follow Cartesian Coordinate System

• Receiver: (1) Trimble 4000SSi (Period 1–3) and (2) Trimble 5700 (Period 4) • Antenna: (1) Trimble Compact L1/L2 with Ground Plane, (2) Trimble Choke Ring, and (3) Trimble Zephyr • Settings: (1) Period 1–3 – interval: 15 s, cut-off elevation: 10◦ ; and (2) Period 4 – interval: 15 s, cut-off elevation: 0◦ . Experimental method were (1) horizontal placed antenna; (2) skew placed antenna; and (3) horizontal placed antenna with metal plate nearby to induce Polar Coordinate System Fig. 2. Antenna coordinate system.

can be derived the baseline P O1, which is represented as elevation (e) and azimuth (a) from O1 to P. 2. The transformation from the topcentric coordinate system of O1 (XTPC ) to the antenna coordinate system of that skew antenna (XANT ) can be done with the following equation π R3 (a)X TPC X Ant = R2 e − 2

Fig. 3. Determination of attitude of skew antenna.

(5)

Fig. 4. Experimental field.

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a C/N0 model (C/N0 Pattern) which is only dependent on E Ant can be constructed as follow (Herring, 1999):

C/N0 (E Ant ) =

k

ai sini (E Ant)

(6)

i=0

CN0(EAnt) =

The following figures, equations and tables are the L1 signal C/N0 Patterns of Trimble Compact L1/L2 with Ground Plane antenna (See Figs. 5 and 6). The Fig. 6 is derived from the data collected by a horizontal placed antenna, and the Fig. 5 is derived from a skew placed antenna. It’s obviously that the two C/N0 Patterns are almost of the same.

39.41 + 31.42sin(EAnt)– 44.59sin2(EAnt) + 67.68sin3(EAnt)–38.65sin4(EAnt)

Fig. 5. Skew antenna, July 12th 2005.

7 Evaluation of Atmosphere Loss Variation 7.1 Time Variation

multipath etc. For horizontal antenna, its orientation indicator should be pointed to the north; for skew antenna, its attitude should be determined (This will be described in Sect. 5).

6 Modelling of C/N0 Pattern In normal condition, there is a strong correlation between C/N0 and satellites elevation and weak correlation between C/N0 and satellites azimuth. So,

C/N0(EAnt) = 40.68 + 15.67sin(EAnt)– 10.61sin2(EAnt) + 0.59sin3(EAnt)–12.25sin4(EAnt)

Fig. 6. Horizontal antenna, July 17th 2005.

We can evaluate the time variation of atmosphere loss with the difference between C/N0 contours that derived from the data collected by the same type of antenna in the different periods. Figure 7 is one of this kind result. The data were collected by the Trimble 4000ssi receiver with Compact L1/L2 with Ground Plane antenna. It’s shown that time variation of atmosphere loss is not obviously and the variation is almost less than 1 dB.

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8 C/N0 Anomaly and its Applications

Fig. 7. Difference between C/N0 contours in different periods.

7.2 Directional Variation Also, we can evaluate the direction variation of atmosphere loss with the difference between contours of C/N0 which derived from the data collected by two misaligned antennas, that one is horizontal and the other is skew. Figure 8 is one of this kind result. The data were collected by the Trimble 5700 receiver with Zephyr antenna. In the area where exist observed data, the variation is almost less than 1 dB. It should be noted that all the contours of C/N0 are in the antenna coordinate system.

Fig. 8. Difference between C/N0 contours of two misaligned antennas.

C/N0 Anomaly is the difference between observed value and modeled value of C/N0 . It is a function of inference. So, we can refine the stochastic model with the C/N0 Anomaly in parameter estimation. C/N0 Anomaly Pattern is the model or figure which describes the C/N0 -Anomaly in different spatial direction. C/N0 Anomaly pattern could be plotted by the data of C/N0 Anomaly. The Figs. 9 and 10 are some C/N0 Anomaly patterns in different periods using different antenna type by different experimental conditions. For clarified, only those absolute values greater than 5 dB are marked as black dot.It’s shown from these figures that in specific direction GPS signal is obviously interfered. By site inspection, there are lightning rods and metallic brackets (This is described in Sect. 2). This result indicates that metal objects such as metallic rods affects GPS signal significantly. Meanwhile, there is an area with large C/N0 Anomaly values. Those Anomaly values (in Fig. 10) are in the direction about 50◦ azimuth in C/N0 anomaly pattern of July 17th 2005. It was caused by a high building under constructing. For the passive interference source (such as metal objects, reflection, diffraction, obstruction), C/N0 Anomaly pattern could be used to locate their orientation. For autonomous sources, it is hard to locate them because they don’t have obvious directivity L1.

Fig. 9. C/N0 anomaly pattern of a site with a metal plate nearby the antenna.

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• Normally, for GPS signal, time variation and direction variation of atmosphere loss is not obvious. • C/N0 Anomaly can be used to be a measurement of the interference effect on GPS signal. With C/N0 Anomaly pattern, the orientation of passive interference sources can be determined.

10 Future Works

July 8th 2004

• An In-depth study of the signal transmitting power and transmitting antenna gain pattern for various GPS satellites should be taken. • The accurate direction variation of atmosphere loss should be determined to improve precision of C/N0 Pattern. • An In-depth study of the relationship between signal C/N0 Anomaly and the precision of range observation (such as pseudo-range and carrier phase) should be taken. • A study of possibility of obtaining atmosphere physical characteristics utilizing the signal C/N0 should be taken.

Acknowledgements The project was supported by Fund of LIESMARS (State Key Lab of Information Engineering in Surveying, Mapping and Remote Sensing) No. (03)0201.

References

July 17th 2005 Fig. 10. C/N0 anomaly pattern of the sites in normal condition.

9 Conclusion • For the GPS receiver on the surface of Earth, the variation of C/N0 of received signal is mainly dependent on antenna gain pattern. With observed values of C/N0 , the C/N0 Pattern can be obtained. And the C/N0 Pattern can be transformed to the antenna’s gain pattern.

Herring T., 1999, Source code of program spcsnr, http://www-gpsg.mit.edu/∼tah/snrprog/source/spcsnr.f., 1999, Accessed Feb. 2006. Hofmann-Wellenhof B., Lichtenegger H., and Collins J., Global Positioning System Theory and Practice, Fifth, revised edition, Springer-verlag Wien New York, 2001. Kaplan, E. D. (ed), Understanding GPS: Principles and Applications, Artech House Inc., Boston & London, 1996. Morea, M. C., Davis, E. P., and Carpenter, J. R., Results from the GPS Flight Experiments on the High Earth Orbit AMSAT OSCAR-40 Spacecraft, presented at ION GPS-02, 2002. Parkinson, B. W. and Spilker Jr., J. J. (ed), Global Positioning System: Theory and Applications Volume I., Progress in Astronautics and Aeronautics, Vol. 163, AIAA, 1996. Roddy, D., Satellite Communications, 3rd Edition, McGrawHill, New York, 2002.

Optimal Combination of Galileo Inter-Frequencies B. Li, Y. Shen Department of Surveying and Geo-informatics, Tongji University, 1239 Siping Road, Shanghai 200092, P.R. China

Abstract. There are four different frequencies in Galileo system, which can form more combinations than that of GPS. Therefore, it is worth, not only theoretically but also practically, to find the best or at least good combination for fast positioning. In this paper we first introduce the criteria for optimally combining the four frequencies based on 5 kinds of cost functions or constraints, then solve the correspondent coefficients for combinations. At last, we propose an algorithm for ambiguity resolution using optimally combined observables. Keywords. Galileo, GPS, combination, optimization

1 Introduction The linear combinations of dual-frequency GPS observables have played an important role in computing GPS baseline for a long time, because the combined observables have many advantages such as longer wavelength, reduced ionospheric delay and so on (Melbbourne, 1985; Han, 1995; Teunissen, 1997; Zhang et al., 2003; Schlötzer and Martin, 2005). Galileo system will be established by European Union in recent years. Since 4 frequencies of Galileo observables can form more combinations than that of GPS, the phase ambiguities can be determined much more efficiently, and the systemic delay of some combined observables can also be significantly reduced. A lot of papers have discussed the combinations of Galileo inter-frequencies. For example, Wang and Liu (2003) gave out several combinations with relatively longer wavelength or smaller ionospheric delay, unfortunately most of the combinations possessing longer wavelength will have either bigger ionospheric delay or larger combined noise. Nevertheless, the best combination of Galileo interfrequencies is still not discussed in the literatures until now. In the present contribution, we will investigate the optimal combinations based on different

criteria and subjected to different constraints. These optimally combined observables are used to fix the phase ambiguities of uncombined observables. This paper is organized as follows. Section 2 will present the general form of inter-frequency combination. Section 3 will put forward 5 criteria of optimal combinations and solve combined coefficients. Section 4 will discuss the phase ambiguity resolution with optimal combined observables.

2 General Form of Inter-Frequency Linear Combination 2.1 Definition of Combined Observables In the dual-difference model, the observation equation of the i th frequency can be expressed as, ϕi = ρ λi − Ni + T λi − Ii λi + εi λi

(1)

where, ϕi is differenced phase observable in cycle;ρ is the satellite-to-receiver range; Ii is ionospheric delay; T is tropospheric delay; λi ,Ni and εi denote the wavelength, ambiguity and noise of observable, respectively. The general form of the combined observables can be represented as, ϕC = n 1 ϕ1 + n 2 ϕ2 + n 3 ϕ3 + n 4 ϕ4

(2)

where, ϕC is combined observable; n 1 , n 2 , n 3 and n 4 are the coefficients of combination. The subscript “C ” here and in the following sections denotes combined variables. Substituting ϕi = f i t into (2), we obtain the following equation (Han, 1995; Hofmann et al., 2001), ϕC = n 1 f 1 t + n 2 f 2 t + n 3 f 3 t + n 4 f 4 t = f C t (3) with combined frequency, fC = n 1 f 1 + n 2 f 2 + n 3 f 3 + n 4 f 4 .

(4) 195

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The combined ambiguity is integer, if and only if the combination coefficients are integers, and the combined ambiguity is as follows, NC = n 1 N1 + n 2 N2 + n 3 N3 + n 4 N4

(5)

If the combined frequency is not equal to zero, the wavelength of combined observable can be computed as, λCm =

c c = fC n 1 f 1 + n 2 f 2 + n 3 f3 + n 4 f 4

(6)

where, c denotes the speed of light, the additional subscript “m ” indicates the unit in meter. 2.2 Definition of Basic Combined Variables Constrained to Non-Zero Combined Frequency The ionospheric delay, tropospheric delay and the noise of combined observables can be derived from the original observables constrained to the condition that the combined frequency is not equal to zero. Because the ionospheric delays are inversely proportion to square of frequencies, they can be expressed as, Ii = f 12 I

f i2 (i =1, 2, 3, 4)

(7)

where, I is the ionospheric delay of the observable in first frequency. Therefore, we can derive the ionospheric delay of combined observable as follows, n 1 + n 2 f 1 f 2 + n 3 f1 f 3 + n 4 f1 f 4 I ICm = n 1 + n 2 f 2 f 1 + n 3 f3 f 1 + n 4 f4 f 1 (8) We can similarly derive the tropospheric delay of combined observable as, TCm =

n 1 f 1 + n 2 f 2 + n 3 f 3 + n 4 f4 T =T n 1 f 1 + n 2 f 2 + n 3 f3 + n 4 f

(9)

This indicates that the tropospheric delay is invariable for the non-zero combined frequency. If the observation noise is one percent of wavelength (Hofmann et al., 2001; Wang and Liu, 2003), and σ1 , σ2 , σ3 and σ4 denote the noises offour frequencies in meter, respectively, i.e. σi = λi 100(i = 1, 2, 3, 4). According to the law of error propagation, we obtain the noise of combined observable as follows,

σCm =

c n 21 + n 22 + n 23 + n 24 100 (n 1 f 1 + n 2 f 2 + n 3 f 3 + n 4 f 4 )

(10)

2.3 Ratio Factors for Evaluating Combined Observables Ratios of the combined variables over corresponding original ones are used as the factors to evaluate the quality of combined observables. We define four kinds of ratio factors as follows, βλ =

λCm f4 = λ4 n 1 f 1 + n 2 f 2 + n 3 f3 + n 4 f 4 βT = TCm T ≡ 1

(11)

(12)

n 1 + n 2 f1 f 2 + n 3 f1 f 3 + n 4 f1 f 4 ICm = βI = I n 1 + n 2 f2 f 1 + n 3 f3 f 1 + n 4 f4 f 1 (13) f 4 n 21 + n 22 + n 23 + n 24 σCm (14) = βσ = σ4 n 1 f 1 + n 2 f2 + n 3 f 3 + n 4 f where, βλ , βT , β I and βσ represent the ratio factors of wavelength, troposheric, ionospheric and noise, respectively.

3 Optimal Combinations Based on Different Criteria The optimal criterion is usually to minimize or maximize the cost function subjecti to some constraints. Three types of optimal combination models will be presented in this section in the following three cases: (1) combinations with non-zero combined frequency; (2) combinations with free tropospheric delay; and (3) combinations with much smaller ionosphere delay. Corresponding to each of the three kinds of constraints, not only the optimal combination but also the combinations with relatively better properties are given out in order to provide alternative choice for specific application. 3.1 Optimal Combination Models with Non-Zero Combined Frequency In the first optimal combination model, the cost function is to maximize the wavelength of combined observable subjecting to the following three constraints. The first one is the wavelength of combined observable is greater than any of original wavelengths (See equations (6) and (10), and the second one is that the ratio of wavelength over noise is greater than 3. The criterion of the first optimal problem is represented as follows,

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Table 1. Three combinations with the longest wavelength and relatively smaller noises n1 n2 n3 n4 βI σCm λCm σCm 0 0 0

–2 1 4

7 –3 –13

–5 2 9

–3.72 –0.77 2.17

2.59 1.10 4.78

11.3 26.7 6.13

max : λCm = c (n 1 f 1 + n 2 f 2 + n 3 f 3 + n 4 f4 ) (15a) ⎧ ⎨ λCm > λ4 constr ai nts : λCm σCm > 3 (15b) ⎩ n1, n2, n3, n4 ∈ Z There are enormous combinations fitting to the constraints (15b), and some of which combinations can get the same maximal wavelength 29.305 m. In other words, the solution of the optimal problem (15) is not unique. Fortunately, there is only one combination that has the smallest ionospheric delay and the smallest noise. Therefore, if the smallest ionospheric delay and the smallest noise are introduced as the additional constraint, it exists unique solution as shown in the second row of Table 1. In order to provide an alternative selection, we list 3 combinations with the longest wavelength and relatively smaller noises in Table 1. In the second optimal combination model, the cost function is to maximize the ratio of combined wavelength over the noise subject to the same constraints as that in the first one. The cost function is as follows, n 21 + n 22 + n 23 + n 24 max : λCm σCm = 100 (16) The solution of this optimal problem is also not unique, and there are total 3 solutions with the maximal ratio λCm σCm = 70.71 as listed in Table 2. Introducing anadditional condition of maximizing the ratio λCm β I , we get the unique solution as shown in the first row of Table 2. In the third optimal combination model, the cost function is to minimize the ratio factor of ionospheric

Table 2. Three combinations with the maximal ratio factor of wavelength n1 n2 n3 n4 λCm βI λCm β I σCm 0 0 0

0 1 1

1 –1 0

–1 0 –1

9.768 4.187 2.931

–1.75 –1.61 –1.65

5.59 2.60 1.78

0.138 0.059 0.041

Table 3. Three combinations with smaller ionospheric delay n1

n2

n3

n4

λCm

βI

σCm

λCm σCm

3 5 6

–6 –19 –19

–11 10 0

14 4 13

1.221 0.837 0.666

–0.00068 –0.00278 0.00244

0.232 0.188 0.159

5.26 4.46 4.20

delay subject to the same constraints as the first and second problem. The cost function reads, n1 + n2 f1 f2 + n3 f1 f3 + n4 f1 f4 min : β I = n1 + n2 f2 f1 + n3 f3 f1 + n4 f4 f1 (17) There is unique solution with the coefficients of combination (3, −6, −11, 14) and the ratio factor is β I = −0.00068. Three kinds of combinations with relatively smaller ionospheric delay are enumerated in Table 3. The optimal combination in the first row of Table 3 significantly reduces the ionospheric delay. 3.2 Optimal Combination Models with Free Tropospheric Delay If the frequency of combined observable equals to zero, the wavelength will be infinite and the tropospheric delay will be completely eliminated. In fourth optimal combination model, the first constraint is that the frequency of combined observable is equal to zero. In this case, the noise of the combined observable can only be expressed in cycle since the wavelength is infinite. The second constraint is that the noise of combined observable is smaller than 0.3 cycles. Since the noise can be expressed as, σC =

n 21

+ n 22

+ n 23

+ n 24

100

(18)

The criterion of the fourth optimal problem is as follows,

fc = 0 n 21 + n 22 + n 23 + n 24 < 100 × 0.3

(19)

There are also a lot of combinations solved by the optimal problem (19). Among these combinations, we need to select some better combinations with comparatively smaller ratio factor of ionospheric delay. In order to compare the ratio factors among different combinations, an arbitrary wavelength, which is normally selected as λ1 , is used to substitute the infinite wavelength. Then, the ratio factor of

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Table 4. Three better troposphere-free combinations N1

n2

n3

n4

λC

βI

σC

–4 1 1

5 –6 –3

–3 7 –3

3 –2 5

∞ ∞ ∞

2.26 0.07 0.08

0.077 0.095 0.066

ionospheric delay can be derived and expressed as follows, β I = n 1 + n 2 f1 f2 + n 3 f 1 f3 + n 4 f 1 f4 (20) We enumerate 3 combinations with comparatively smaller noise and ionosphere delay in Table 4. The combination in the last row of Table 4 has bigger ratio factor of ionospheric delay, but much smaller noise, than that in the second row. Therefore, the combination in the last row of Table 4 is taken as the optimal combination. 3.3 Optimal Combination Models with Smaller Ionospheric Delay In the fifth optimal combination model, the cost function is to maximize the wavelength of combined observable subject to two constraints. The first one is that the ratio factor of ionospheric delay is less than 0.001, which indicates that the ionospheric delay of combined observable is so small that it can be neglected. The second constraint is that the ratio factor of wavelength must be greater than 3. Therefore, the criterion of the fifth optimal combination model can be summarized as, max : λCm = c n 1 f 1 + n 2 f 2 + n 3 f 3 + n 4 f 4 (21a) ⎧ < 0.001 < β ⎨ −0.001 I λCm σCm > 3 (21b) constraints : ⎩ n1, n2 , n3, n4 ∈ Z There is the unique solution for the optimal problem (21) with the wavelength as long as 1.221 meters shown in the first row of Table 5. Although there are many combinations fitting the constraints (21b), all of the wavelengths are less than 15 centimeters except for the optimal one. The other two comparatively better combinations with relatively smaller ionospheric delay and relatively longer wavelength are also listed in the table.

Table 5. The combinations with relatively greater ratio factor of wavelength n1 n2 n3 n4 λCm ICm λCm σCm 3 2 3

–6 3 3

–11 10 0

14 –14 –5

1.22 0.116 0.112

–0.068 –0.060 0.020

5.26 5.68 15.33

4 Ambiguity Resolution with the Combined Observables In this section, we will discuss the application of the above optimal combinations in fixing the phase ambiguities, and these combinations are listed in the Table 6. The tropospheric delay can be efficiently corrected via models, and the ionospheric delay could be significantly reduced via proper combination of observables. The ionospheric delay of the first three combinations in Table 6 have been greatly reduced so that their influences can be ignored, and the ratios of λCm σCm are all larger than 5. Although the fourth combination in Table 6 will enlarge the ionospheric delay, it amplifies the wavelength much more significantly, its ratio of phase wavelength over β I is equal to 29.33, which is much larger than 3.53 for the wide-lane combination in GPS. Therefore, the ambiguities of the combined observables can be determined much easier than that of the wide-lane combination in GPS. The algorithm for fixing phase ambiguities can be summarized as follows, 1. Computing four combined phase observables in Table 6, e.g. l1,−3,−3,5, l3,3,0,−5 , l0,1,−1,0 and l0,0,1,−1, according to li, j,k,m = i ϕ1 + j ϕ2 + kϕ3 + mϕ4 . 2. The dual-difference observation equation for the i th satellite at the j th epoch is given as, j

li =

j

Ai

I

b ai

(22)

Table 6. The combinations used to fix phase ambiguity n1 n2 n3 n4 λCm βI λCm σCm 1 3 0 0

–3 3 1 0

–3 0 –1 1

5 –5 0 –1

∞ 0.112 4.187 9.768

0.0800 0.002 –1.61 –1.7500

15.08 15.33 70.71 70.71

Optimal Combination of Galileo Inter-Frequencies

T is the baseline

x y z i i i = N1,−3,−3,5 N3,3,0,−5 N0,1,−1,0

where, b =

vector; ai T i N0,0,1,−1 is the vector of 4 dual-difference j

combined ambiguities, Ai is the 4 × 3 design matrix for the baseline coordinates and coefficient matrix I, respecting to combined ambij guities, is a 4 × 4 identity matrix. li = T l1,−3,−3,5 l3,3,0,−5 l0,1,−1,0 l0,0,1,−1 is the vector of combined observables. If there are n+1 satellites simultaneously tracked by two stations for m epochs, we get the observation equation as,

b L= A B (23) a

T T T T , where, A = A12 · · · Am A11 n B = em ⊗ I4n , em is m column unit vector and I4n is 4n × 4n T iden, and tity matrix, ; a = a1T · · · anT 1 1 T m L = l1 l2 · · · ln . 3. Solving (23) with the least squares adjustment. The estimates and their covariance matrix are,

Qbˆ Qbˆ aˆ bˆ (24) Qaˆ bˆ Qaˆ aˆ where, Qbˆ and Qaˆ are respectively the covariance matrix of bˆ and aˆ , and Qbâˆ is the covariance between bˆ and aˆ . The integer ambiguities of combined observables are searched based on the following optimal problem, T a − aˆ with a ∈ Z n (25) min : a − aˆ Q−1 aˆ Since the optimally combined observables have better properties than the original observables, the covariance matrix Qaˆ of combined ambiguities is less correlated and therefore the combined ambiguities can be solved much more efficiently than original ambiguities. Once four combined ambiguities for each satellite are obtained, the original ambiguities can be easily calculated by inversely multiplying the 4 × 4 squared matrix of combination coefficients. This procedure is just like the decorrelation algorithm (Teunissen, 1995, 1997; de Jonge and Tiberius, 1996; Xu, 2001, 2006). In fact, the decorrelation is to find better combination for all ambiguities to be searched, while the combinations discussed in this paper are among the different frequency observables from the same station to the same satellite.

199

5 Concluding Remarks The 4 optimal combinations are derived for Galileo 4 frequency observables based on the 5 optimal criteria. An algorithm is proposed for ambiguity resolution with the optimally combined observables. We will carry out simulation experiments to numerically study the efficiency of ambiguity resolution with the combined observables and compare this algorithm with LAMBDA or decorrelation algorithm in our next contribution.

Acknowledgement The authors thank to the National Natural Science Funds of China for the substantially support of this paper (Project number: 40674003, 40474006).

References de Jonge P., Tiberius C., 1996, The LAMBDA method for integer ambiguity estimation: implementation aspects, LGRSeries, Publications of the Delft Geodetic Computing Centre, No.12. Han S.W., 1995, Theory and application of the combinations of GPS dual frequency carrier phase observations. Acta Geodaetica et Cartographica Sinica. vol 24(2):8–13 (in Chinese with English abstract). Hofmann-Wellenhof B., Lichtenegger H., Collins J., 2001, Global positioning system: theory and practice 5th, revised edition, Springer Wien New York. 92–94. Melbbourne W. G., 1985, The case for ranging in GPS- based geodetic systems. Proceeding 1st international symposium on precise positioning with Global Positioning System, 15–19 April, Rockville, pp 373– 386. Schlötzer S., Martin S., 2005, Performance study of multicarrier ambiguity resolution techniques for galileo and modernized GPS, ION GPS/GNSS 2003, Portland, USA, pp 142–151. Teunissen P.J.G., 1995, The least-squares ambiguity decorrelation adjustment: a method for fast GPS integer ambiguity estimation, JOG, 70:65–82. Teunissen P.J.G., 1997, On the GPS widelane and its decorrelating property, JOG, 71:577–587. Wang Z.M., Liu J.B., 2003, Model of inter-frequency combination of galileo GNSS, Wuhan University Journal (Natural Science), (6):723–727.(in Chinese with English abstract). Xu P.L., 2001, Random simulation and GPS decorrelation, JOG, 75: 408–423. Xu P.L., 2006, Voronoi cells, probabilistic bounds, and hypothesis testing in mixed integer linear models, IEEE Transactions on information theory, vol.52. No.7, July 2006:3122–3138. Zhang W., Cannon M. E., Julien O., Alves P, 2003, Investigation of combined GPS/GALILEO cascading ambiguity resolution schemes. Proceedings of ION GPS/GNSS 2003, Portland, USA, pp 2599–2610.

Closed-Form ADOP Expressions for Single-Frequency GNSS-Based Attitude Determination D. Odijk, P.J.G. Teunissen, A.R. Amiri-Simkooei Delft Institute of Earth Observation and Space Systems (DEOS), Delft University of Technology, Kluyverweg 1, 2629 HS Delft, The Netherlands, e-mail: [email protected]

Abstract. Integer ambiguity resolution is a prerequisite to high-precision real-time GNSS-based attitude determination. The ADOP is a well-known scalar measure to infer whether ambiguity resolution can be expected successful or not. To compute ADOP it is sufficient to have knowledge about the measurement setup and the measurements noise characteristics; hence it can be used as a planning tool. In this contribution we present closed-form expressions for the ADOP in case of attitude determination. Using these expressions one may infer the impact of GNSS design aspects such as number of satellites, choice of frequency and the precision of the phase and code observables. In addition, they are useful to quantify the influence of the number of antennas in the configuration and the use of geometric constraints, such as the lengths of the baselines and/or the angles between the baselines in the configuration. In this article the behavior of the ADOPs as function of these design aspects will be evaluated for several GPS attitude determination scenarios. Keywords. GNSS, attitude determination, baseline constraints, LAMBDA method, ADOP

1 Introduction Crucial to GNSS-based attitude determination of vehicles or platforms is the resolution of the integer ambiguities of the relative carrier-phase observations. A widely used approach for this is the integer least-squares search as implemented in the LAMBDA method, see e.g. Han et al. (1997), Park and Teunissen (2003), Dai et al. (2004) and Li et al. (2005). For successful ambiguity resolution it is required that the probability that the estimated integer solution coincides with the correct integers (the ambiguity success rate) is sufficiently high. One way to get insight into this is to evaluate the ADOP (Ambiguity Dilution Of Precision). This ADOP measure is derived from the variance matrix of the float ambiguity solution and is thus purely based on the assumptions in the mathematical model underlying. 200

In Teunissen and Odijk (1997) closed-form expressions were derived for the ADOP in a range of GPS positioning scenarios. These ADOP expressions are then a function of the model assumptions, such as number of satellites and receivers, number of frequencies and epochs, and the assumptions concerning the stochastic properties of the observations. In this article we will present closed-form expressions for the ADOP in case of attitude determination. Hereby we take the following restrictions into account. First, we only discuss instantaneous attitude determination, based on a single epoch of singlefrequency phase and code (pseudo-range) data. We only present expressions for two- or three-antenna GNSS attitude determination systems. Although these antennas are usually connected to a common oscillator, implying that there are no receiver clock errors, in this article however receiver clock errors are taking into account because of the presence of unknown line biases, i.e. differential errors caused by differences in cable lengths between the antennas and oscillator. These line biases show up as receiver clock errors and as a consequence the ambiguities to be estimated are double-differenced (DD). It is finally assumed that atmospheric errors do not play a role since the distances between the antennas are very short, and that errors due to multipath are absent. Using the expressions presented in this article one will also be able to analyze the influence of geometric constraints on ADOP in case of attitude determination. These constraints on the baseline lengths and/or angle between baselines will be treated as stochastic constraints, to serve two goals: (i) in case of a rigid antenna platform these constraints can be applied ‘hard’ by setting the standard deviations of the constraints to zero, (ii) in case of more flexible platforms (e.g. in airplanes or ships, when the baselines can be longer, e.g. a few m) the constraints can be applied more loosely by setting their standard deviations to certain appropriate values. The paper is set up as follows. In Sect. 2 the ADOP concept is reviewed. Sect. 3 reviews the ADOP expressions in case of positioning, while Sect. 4 presents the expressions for attitude determination.

Closed-Form ADOP Expressions for Single-Frequency GNSS-Based Attitude Determination

Examples are given in Sect. 5 and finally in Sect. 6 the conclusion follows. We remark that the results in this paper are given without proof. For proofs we refer to Odijk and Teunissen (2008).

GNSS models for fast and precise relative applications can all be cast into the following model of linear(ized) observation equations: D{y} = Q y

(1)

where E{·} denotes the expectation operator and D{·} the dispersion operator. The vector y denotes the normally distributed GNSS data vector (‘observed-minus-computed’ in case of a linearized model), whereas vectors a (of order n) and b (of order o) denote the unknown parameter vectors, for which A and B are the corresponding design matrices. Note that a contains the integer carrierphase ambiguities, a ∈ Zn , and b the remaining (real-valued) parameters, b ∈ Ro . The stochastic properties of the observations are included in Q y , the variance matrix. The procedure to solve the model in equation (1) is usually divided into three steps. In the first step we disregard the integer constraints on the ambiguities and perform a standard least-squares adjustment. As a result, we obtain the (real-valued) estimates of a and b, together with their variance-covariance matrix: aˆ Q aˆ Q aˆ bˆ , . (2) Q bˆ aˆ Q bˆ bˆ This solution is referred to as the ‘float’ solution. In the second step, the float ambiguity estimate aˆ is used to compute the corresponding integer ambiguity estimate, denoted as a: ˇ aˇ = F(a) ˆ

(3)

with F : Rn → Zn , a mapping from the real to the integer space. Once the integer ambiguities are computed, they are used in a third step to correct the float estimate of the real-valued parameters b. As a result we obtain the ‘fixed’ solution: ˆ aˇ = bˆ − Q ˆ Q −1 (aˆ − a) ˇ bˇ = b| baˆ aˆ

the fixed solution can be described by the following variance matrix (in which the integer ambiguities are assumed non-stochastic): = Q bˆ − Q bˆ aˆ Q −1 Q aˆ bˆ Q bˇ Q b|a ˆ aˆ

2 Ambiguity Dilution of Precision

E{y} = Aa + Bb,

201

(4)

If the ambiguity success rate, i.e. the probability that the estimated integers coincide with the true ambiguities, is sufficiently close to one, the precision of

(5)

The success of ambiguity resolution depends on the quality of the float ambiguity estimates: the more precise the float ambiguities, the higher the probability of estimating the correct integer ambiguities. A simple measure to infer the float ambiguity precision is the Ambiguity Dilution of Precision (ADOP) defined as (Teunissen, 1997): 1

ADOP = |Q aˆ | 2n

[cyc]

(6)

Advantage of this scalar measure is that by taking the determinant we capture information not only on the variances but also on the covariances between the ambiguities. By raising the determinant to the power 1/(2n), the scalar is, like the ambiguities themselves, expressed in cycles. The ADOP is linked to the ambiguity success rate as follows (Teunissen, 1998): PADOP =

2

1 2ADOP

−1

n (7)

x with Φ(x) = −∞ √1 exp − 12 v 2 dv. Although 2π this ADOP-based probability PAD O P is an approximation of the true success rate of integer leastsquares, in Verhagen (2005) it was by means of simulations demonstrated that they agree reasonably well. Figure 1 shows PADOP as function of ADOP for varying levels of n (n = 1, . . . , 20). It can be seen that the ADOP-based success rate decreases for increasing ADOP and this decrease is steeper the more ambiguities are involved. In general, Figure 1 shows that if ADOP is larger than about 0.12 cyc PADOP becomes significantly smaller than 1. As mentioned in the introduction, in case of attitude determination it is common to include constraints on (some of) the parameters. The model incorporating stochastic constraints on the realvalued parameters, denoted as c, reads: E{

y c

}=

A 0

B C

a b

(8)

202

D. Odijk et al. 1 1 0.998

n=1 0.6

PADOP

0.8

0.996 0.994

PADOP

0.992

n=2

0.99 0.1

0.12

0.4

0.14 0.16 0.18 ADOP [cyc]

0.2

n=3 0.2 n = 20 0

0

0.2

0.4 0.6 ADOP [cyc]

0.8

1

Fig. 1. PADOP versus ADOP for varying n.

The stochastic model, extended for the variance matrix of the constraints (denoted as Q c ), reads: D{

y c

Qy }= 0

0 Qc

(9)

The ADOP of the model in presence of constraints then follows as, see Odijk and Teunissen (2008):

ADOP = ADOP(∞)

|Q c +Q cˇ (∞)| |Q c +Q cˆ (∞)|

1

2n

3 Positioning ADOP In this section the closed-form ADOP expressions for GNSS-based positioning are reviewed. Let us first consider a single baseline, i.e. two GNSS antennas tracking m satellites. In that case, there are n = m − 1 DD ambiguities. The closedform expression for the single-frequency singleepoch ADOP was derived in Teunissen (1997) as 1

pos

(10)

with ADOP(∞) the ADOP of the model without constraints. Moreover, the float and fixed variance matrices of the constrained parameters are computed as Q cˆ (∞) = C Q bˆ (∞)C T and Q cˇ (∞) = C Q bˇ (∞)C T , respectively, where Q bˆ (∞) and Q bˇ (∞) are the float and fixed variance matrices of the real-valued parameters in absence of constraints. It is thus shown that in presence of constraints the ADOP can be directly computed from the ADOP in absence of constraints. It can be proved that the ratio in equation (10) as raised to the power 1/(2n) is always smaller than or equal to one. Consequently, ADOP ≤ ADOP(∞). This is understandable, since addition of constraints makes the model stronger. In the limiting case, if Q c = 0, the constraints maximally contribute to the ambiguity precision; they have become hard constraints. On the other hand, if Q c = ∞, the constraints do not have any weight and do not contribute at all to the ambiguity precision; i.e. ADOP = ADOP(∞).

ADOPr=2 = m 2(m−1)

σφ λ

1+

σ p2

3 2(m−1)

(11)

σφ2

with m the number of satellites tracked, λ the wavelength, σφ the standard deviation of the singledifferenced phase observables and σ p the standard deviation of the single-differenced code observables. Usually, in case of GPS, σ p2 /σφ2 ≈ 104, which implies that the term between the brackets in equation (11) is relatively large. In case we have more than two – say r – antennas simultaneously tracking the same m satellites, then the ADOP for the (r − 1)(m − 1) DD ambiguities (r ≥ 2) in the network can be related to the singlebaseline ADOP as (Teunissen and Odijk, 1997) pos

ADOPr

=

1 2

√

1

pos

2 r 2(r−1) ADOPr=2

(12)

It can be seen that for the purpose of ambiguity resolution the contribution of an additional receiver/antenna is low: when going from two to three antennas the ADOP of the network is only 0.93 times its single-baseline counterpart.


4 Attitude Determination ADOP The ADOP expressions for positioning turn out to play a role in the expressions for attitude determination. This relation will be revealed in the current section for attitude determination based on two and three antennas, respectively. 4.1 Two Antennas In two-antenna attitude determination one tries to solve for pitch/elevation α and heading/azimuth/yaw γ of the baseline, plus its length l, which are related to the local East-North-Up coordinates as, see Figure 2: ⎡ ⎤ ⎡ ⎤ E 12 l cos α sin γ ⎣ N12 ⎦ = ⎣ l cos α cos γ ⎦ (13) l sin α U12 In absence of any constraint, it can be shown that the ADOP in case of two-antenna attitude determination equals its single-baseline position counterpart (i.e. a reparametrization of the position vector does not affect ADOP). In presence of a constraint on the length of the baseline, having standard deviation σl , the ADOP can be easily obtained using equation (10), as att ADOPr=2

=

pos ADOPr=2

σl2 +σ ˇ2 (∞)

1 2(m−1)

(14)

l σl2 +σ ˆ2 (∞) l

pos ADOPr=2

with as in equation (11) and σlˆ(∞) and σlˇ(∞) the standard deviation of the float and fixed baseline lengths in absence of the constraint, respectively. To obtain σ ˆ2 (∞) and σ ˇ2 (∞) we need to linl l earize 2 + N2 + U2 (15) l(∞) = E 12 12 12

coordinates. Application of the the variance propagation law results in: σ ˆ2 (∞) = μT Q gˆ μ, σ ˇ2 (∞) = μT Q gˇ μ l

l

4.2 Three Antennas If a third antenna is added to the system, such that the baseline between antennas 1 and 3 is non-collinear with the baseline between 1 and 2, in addition to the pitch and heading also the roll angle β can be determined (see Figure 3). With three antennas it becomes possible to determine the attitude by introducing a body frame (denoted using u, v, w) fixed to the vehicle, with the first antenna chosen as the origin of the body frame and the plane through the three antenna positions defining the (u, v)-plane. The w-axis is perpendicular to this plane. Both frames are connected as (for i = 1, 2, 3): ⎡ ⎤ ⎤ ui E 1i ⎣ N1i ⎦ = Rw (γ )Ru (−α)Rv (β) ⎣ vi ⎦ (17) U1i wi ⎡

with Ru , Rv and Rw rotation matrices. The body coordinates of the antennas are (u 1 , v1 , w1 ) = (0, 0, 0), (u 2 , v2 , w2 ) = (0, l1 , 0) and (u 3 , v3 , w3 ) = (l2 sin ϕ, l2 cos ϕ, 0), with l1 and l2 the baseline lengths and ϕ the angle between both baselines.

U

γ

v 2

β N

U

l1 l2cosφ

2

1

l

α

N12 γ

E

3

l2sinφ

1 E12

φ

l2

N

U12

E

Fig. 2. Attitude determination based on a single baseline.

(16)

0 0 0 T with μ = l10 (E 12 , N12 , U12 ) a unit vector based on the (approximated) position coordinates and g = (E 12 , N12 , U12 )T the position vector itself, with Q gˆ and Q gˇ its float and fixed variance matrices, respectively.

w

i.e. the baseline length in absence of the constraint, completely determined by the East-North-Up

203

α u

Fig. 3. Attitude determination based on three antennas.

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In absence of constraints on the baseline lengths and angle, there are now 6 unknowns: the three Euler angles (α, β and γ ), plus the two baseline lengths and baseline angle. Thus the 6 position coordinates are reparametrized into 6 other parameters. As in the dual-antenna case, it can be proved that the three-antenna attitude ADOP equals the threeantenna positioning ADOP, since in the latter case there are also 6 unknowns (coordinates). In presence of constraints on both baseline lengths, the ADOP becomes, using equation (10):

pos

att (σ =∞)=ADOP ADOPr=3 ϕ r=3

|Ql +Q ˇ(∞)| l |Ql +Q ˆ(∞)| l

1 4(m−1)

(18)

with Q l the variance matrix of the two length constraints and Q lˆ(∞) extracted from: ⎡ ⎣

Q lˆ(∞) Q ϕˆ lˆ(∞)

Q lˆϕˆ (∞) σϕ2ˆ (∞)

⎤ ⎦=

μT Q gˆ μ η T Q gˆ μ

μT Q gˆ η η T Q gˆ η

(Note: Q lˇ(∞) is computed analogously based on Q gˇ (∞)). Here μ = blkdiag(μ1 , μ2 ) where μ1 and μ2 are unit vectors based on the (approximated) coordinates of antennas 2 and 3, g = (E 12 , N12 , U12 , E 13 , N13 , U13 )T and η a 6×1-vector obtained when the following expression is linearized: ϕ(∞) = arccos

E 12 E 13 +N12 N13 +U12 U13 l1 l2

(19)

where use is made of the inner product between the two baseline vectors. Now assume – in addition to the baseline length constraints – also a constraint on the baseline angle, having a standard deviation σϕ . Then the ADOP in presence of both types of constraints can be computed from the ADOP in presence of baseline length constraints only, see equation (18): att =AD O P att (σ =∞) ADOPr=3 r=3 ϕ

σϕ2 +σ 2 (σϕ =∞) ϕˇ σϕ2 +σ 2 (σϕ =∞) ϕˆ

1 4(m−1)

(20)

with σϕ2ˆ (σϕ = ∞) computed as σϕ2ˆ (σϕ = ∞) = σϕ2ˆ (∞)− −1 Q ϕˆ lˆ(∞) Q l + Q lˆ(∞) Q lˆϕˆ (∞)

5 Examples As an illustration we computed instantaneous ADOPs for GPS-based attitude determination, using the presented closed-form expressions. In these computations we used the receiver-satellite geometry of permanent GPS station Delft (52.0◦N, 4.4◦E), the Netherlands, for 1 January 2003 (00–24 h UTC; 30s sampling interval; cut-off elevation: 15deg). Singlefrequency (L1) phase and code data were assumed having√(single-differenced)√standard deviations of σφ = 2· 3 mm and σ p = 2· 30 cm. In the dual-antenna case the computations have been conducted using α 0 = γ 0 = 0, and with this 0 0 0 choice the a priori coordinates are (E 12 , N12 , U12 )= 0 (0, l , 0). Note that this choice only affects ADOP in presence of a (soft) baseline length constraint, through vector μ in equation (14). It was numerically verified that with other choices for α 0 and γ 0 ADOP is hardly changed. Concerning the a priori baseline length l 0 , this is set to l 0 = 1 m in the computations, although this choice does not affect ADOP at all since it gets eliminated in vector μ (which equals (0, 1, 0)T in this case). In the triple-antenna case the a priori Euler angles were set to α 0 = β 0 = γ 0 = 0, which means that the body frame coincides with the local EastNorth-Up frame. It was numerically verified that this a priori attitude hardly affects ADOP. The two baseline lengths were chosen as l10 = l20 = 1 m and this choice only affects ADOP in case of a constraint on the baseline angle. It should be realized that since both σϕˆ (σϕ = ∞) and its fixed counterpart are inversely proportional to the baseline length (see equation (21)). This implies that the standard deviation of the baseline angle constraint is allowed to be larger in case the baseline lengths are shorter (for example if σϕ is set to 5 deg in case the baseline lengths are 1 m, then the same ADOP level is reached in case the baseline lengths are 0.1 m, but with σϕ taken equal to 50 deg!). The a priori baseline angle was in all computations set to ϕ 0 = 90 deg, since it turned out that ADOP is hardly changed by other choices of this angle. 5.1 Two Antennas

(21)

The ambiguity-fixed σϕ2ˇ (σϕ = ∞) is computed analogously by considering the fixed variance matrices.

In the dual-antenna case, we first computed the ADOPs and ADOP-based success rates in absence of a baseline length constraint (σl = ∞). These ADOPs, corresponding to their counterparts in case of single-baseline positioning, are plotted for the day in Figure 4. It can be seen that the ADOP can be large; almost 3 cycle in case of 4 satellites. The

# satellites

1000

PADOP

1500

2000

1500

2000

1000

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5 500

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1500 epoch [30s]

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Dual−antenna: σl = ∞

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Dual−antenna: σl = 10 cm

0.5

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PADOP

500 3 Dual−antenna: σl = 0 2 1 0 500 10

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# satellites

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1500

PADOP

3 2 Dual−antenna: σl = ∞ 1 0 500 1000 3 Dual−antenna: σl = 10 cm 2 1 0 500 1000 3 Dual−antenna: σl = 1 cm 2

PADOP

ADOP [cyc] ADOP [cyc] ADOP [cyc] ADOP [cyc]


10

500

1000

Dual−antenna: σl = 1 cm

500 Dual−antenna: σl = 0

5 1500 epoch [30s]

Fig. 4. Dual-antenna case. Shown are the ADOPs (left) and ADOP-based success rate PADOP (right). The first row shows the results for σl = ∞ (no constraint); the second row the results for σl = 10 cm; the third row those for σl = 1 cm, while the fourth row shows the results for σl = 0 (hard constraint). The last row gives the number of satellites during the day.

# satellites

ADOP

P P P 2000

2500

1000

1500 epoch [30s]

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2500

0.5

1

Triple−antenna: σl = σϕ = ∞

500

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Triple−antenna: σl = 1 cm σϕ = ∞

500

1000

Triple−antenna: σl = 1 cm σϕ = 5 deg

0.5 0

2500 ADOP

2000

1

0

2500 ADOP

2000

1 0.5 0

2500 ADOP

2000

5 500

In the triple-antenna case, the variance matrix of the two baseline lengths is assumed as a scaled identity matrix: Q l = σl2 I2 . We first computed ADOPs in absence of any constraints (σl = σϕ = ∞), see Figure 5. As a result we obtain ADOPs that correspond to those of network-based positioning. Compared to the dual-antenna case, the ADOPs are decreased only marginally. Addition of two baseline length constraints with σl = 1 cm lower the ADOPs and hence increase the ADOP-based success rates significantly during the day. If we add a constraint on the angle between both baselines of σϕ = 5 deg as well, the success rates are close to 1, except during

P

3 Triple−antenna: σl = σϕ = ∞ 2 1 0 500 1000 1500 3 Triple−antenna: σl = 1 cm σϕ = ∞ 2 1 0 500 1000 1500 3 Triple−antenna: σl = 1 cm σϕ = 5 deg 2 1 0 500 1000 1500 3 Triple−antenna: σl = σϕ = 0 2 1 0 500 1000 1500 10

5.2 Three Antennas

1

500

1000

Triple−antenna: σl = σϕ = 0

0.5 0

# satellites

ADOP [cyc] ADOP [cyc] ADOP [cyc] ADOP [cyc]

corresponding success rate is close to zero. Only with a large number of satellites, at least 9, ADOP is sufficiently small such that the success rate approaches to 1. Better results are obtained in case the baseline length is constrained. Using a standard deviation of σl = 10 cm, though the ADOPs are smaller than without constraint, for many parts of the day the ADOP-based success rate is still insufficient. With a standard deviation of the constraint of 1 cm however, for many times (with at least 6–7 satellites) the ADOP is sufficiently small such that the success rate approaches to 1. Even better results are obtained when the baseline length is incorporated as hard constraint (σl = 0).

500

1000

500

1000

10 5 1500 epoch [30s]

Fig. 5. Triple-antenna case. Shown are the ADOPs (left) and ADOP-based success rates PADOP (right). The first row shows the results for σl = σϕ = ∞ (no constraints); the second row those for σl = 1 cm and σϕ = ∞, while the third row shows the results for σl = 1 cm and σϕ = 5 deg. Finally, the fourth row shows the results using hard constraints: σl = σϕ = 0 and the last row gives the number of satellites during the day.

206

periods with 5 satellites or less. The best results are of course obtained if the baseline lengths and angle may be hard constrained (σl = σϕ = 0). In that case only during the (short) periods of having 4 satellites the ADOP-based success rate is not close to 1. Note especially in this scenario the benefits of tripleantenna attitude determination: in the dual-antenna case still for some considerable time of the day the success rate is not sufficiently close to 1, despite the hard constrained baseline length.

6 Conclusion In this article closed-form expressions have been presented for GNSS-based attitude determination. It was shown that the ADOP of attitude determination corresponds to the ADOP of positioning in absence of constraints on baseline lengths and/or baseline angle. It was also demonstrated that the contribution of constraints to ADOP becomes immediately clear since they appear as additional (scaling) factors in the closed-form ADOP expressions.

References Dai, L., K.V. Ling, and N. Nagarajan (2004). Real-time attitude determination for microsatellite by Lambda method

D. Odijk et al. combined with Kalman filtering. Proceedings of the 22nd AIAA ICSSC and Exhibit 2004, Monterey, CA, USA, May 9–12. Han, S., K. Wong, and C. Rizos (1997). Instantaneous ambiguity resolution for real-time GPS attitude determination. Proceedings of the International Symposium on Kinematic Systems in Geodesy, Geomatics and Navigation, KIS97, Banff, Canada, June 3–6, 409–416. Li, Y., K. Zhang, and R. Grenfell (2005). Improved Knight method based on narrowed search space for instantaneous attitude determination. Navigation, 52(2): 111–119. Odijk, D., and P.J.G. Teunissen (2008). Ambiguity Dilution Of Precision: Diagnostic for Global Navigation Satellite System attitude determination. Submitted to Journal of Guidance, Control, and Dynamics. Park, C., and P.J.G. Teunissen (2003). A new carrier phase ambiguity estimation for GNSS attitude determination systems. Proceedings of the International Symposium on GPS/GNSS, Tokyo, Japan, November 15–18. Teunissen, P.J.G. (1997). A canonical theory for short GPS baselines. Part IV: Precision versus reliability. Journal of Geodesy, 71: 513–525. Teunissen, P.J.G., and D. Odijk (1997). Ambiguity Dilution Of Precision: Definition, properties and application. Proceedings of ION GPS-1997, Kansas City, MO, USA, September 16–19, 891–899. Teunissen, P.J.G. (1998). Success probability of integer GPS ambiguity rounding and bootstrapping. Journal of Geodesy, 72: 606–612. Verhagen, S. (2005). On the reliability of integer ambiguity resolution. Navigation, 52(2): 99–110.

Safety Monitoring for Dam Construction Crane System with Single Frequency GPS Receiver W. Wang, J. Guo, B. Chao, N. Luo The School of Geodesy and Geomatics, Wuhan University, Wuhan, P.R. China, e-mail: wangwei [email protected]

Abstract. The construction site for large dam is very complex. Many huge construction cranes forms the key part for the construction. These huge cranes may move in some common working areas. The 3-dimensional movements of these machines are very complicated and the speed is relatively high. It’s very dangerous and disastrous if these machines collide each other with huge inertia. Therefore, dynamic and real time monitoring the movements of the machines and ensuring their safety running are very important. This paper’s background is based on LongTan hydropower station project in GuangXi Province. With an effective combination of GPS raw data collection, wireless network communication, data computing and analyzing, and alarming sets, a dynamic monitoring system was established. It’s very effective to prevent collisions among the working cranes with relative low cost. It has been proved that this dynamic monitoring system is functional and feasible for the construction of the dam. Keywords. Safety monitoring, GPS, wireless network communication

1 Introduction The dynamic monitoring system has being in operation for safety protection for the dam construction since July, 2005. It composed of five primary units: GPS raw data collection unit, wireless network communication unit, data computing and analyzing unit, alarming unit and interface unit. All these five units combined together and made a whole. There are seven huge machines equipped with GPS raw data collection units called roving points, adding the base point which is used to generate differential correction to the roving points. Totally, eight GPS raw data collection units are in this dynamic monitoring system. The seven roving points collect the raw binary GPS data every 0.1 s round-theclock, and send the data back to the computing and

analyzing unit via wireless network communication. The GPS receiver located at the base station in linked directly to the computer in the analysis center. All the raw data will be processed at the analysis center with a processing software developed by us. If any two cranes are too close, an alarming signal will be sent to the corresponding machine control rooms by wireless network communication, the operators will act according to the instructing signals. Red signal is for immediate braking, yellow signal is for slowing down and mentioning others, green signal is normal. Therefore, machine collision can be avoided with the help of the monitoring system based on single frequency GPS receivers. The system working structure is showed in Figure 1. For this project, the top speed of the moving machine can reach 7 m/s. In order to have a high distant recognition resolution and send out a precise foretell to the crane controller, tracking the cranes’ movement every 0.1 ms is indispensable. Under this consideration, eight single frequency Ashtech DG14TM OEM boards, with 10 Hz output rate of raw data ability, are integrated in the system. All the processes are done within 100 ms by two Pentium IV 3.4 GHz PCs. One is for receiving and computing, and the other is used to analyze and send alarm signal back to controller.

2 GPS Raw Data Collection Unit In this unit, the GPS OEM board is linked to the wireless communication network through a RS-232 serial port. This unit includes GPS antenna, GPS Ashtech DG-14TM OEM board, Serial Port Server, wireless network bridge, omni-directional microwave antenna and built-in battery. The structure of this unit is showed in Figure 2.

3 Wireless Network Communication Unit Totally 21 wireless bridges were used to form the wireless network. Nine of them were for alarming 207

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W. Wang et al. GPS Raw Data Collecting Unit ROVER

GPS Raw Data Collecting Unit BASE

Wireless Network Communication Unit

No: 1

C T R L

No:2

Alarming And Interface Unit No: 1

R O O M

No: 3

Wireless Network

No: 7

Computing And Analyzing Unit

No: 7

Fig. 1. The monitoring system structure.

signals transmission, and the rest were for data transmission. Two kinds of different carrier frequency bridges (900 and 2.4 GHz) were used because of the complex construction environment. 2.4 GHz wireless bridge was for eyesight communication and 900 MHz was for non-eyesight communication. The GPS raw data transmitted at the band rate of 115200 bps.

√

4 Methodology and Result In the data computing unit located in the analysis center, single difference of code smoothed by carrier phase is formed as the major observation. It is expressed by equation (1). 1 i k−1 i i ρ (tk )+ [ρ¯ (tk−1 )+δρphase (tk , tk−1 )] k k (1) Where ρ¯ i (tk ), ρ¯ i (tk−1 ) are the carrier smoothed pseudorange at epoch tk and tk−1 ; ρ i (tk ) is the original pseudorange observation at epoch tk ; i δρphase (tk , tk−1 ) is the carrier phase difference between epoch tk and epoch tk−1 ;k represents the ρ¯ i (tk ) =

GPS Antenna

GPS OEM BOARD

number of epoch smoothed; superscript i is the satellite number. The computation of the rover positions is based on the Least Square method but with different weight contribution on each corrected pseudorange. Equation (2) gives out the weighting scheme for pseudorange, which related to the satellites’ Signal Noise Ratio, the satellite elevation angle, and the smooth number.

Microwave Antenna

Serial Port Server

POWER

Fig. 2. The GPS raw data collection unit.

Wireless Bridge

P(i ) = e(c0 ×sin E+c1 ×

n/100+SNR(i)×c2 )

+ c3

(2)

Where c0 , c1 , c2 , c3 are positive constants, P(i ) is the weight for the pseudorange of satellite i , E is the satellite elevation angle, n is the smooth number for pseudorange, SNR(i ) is the Signal Noise Ratio, and i is the satellite number. In order to get a better rover position, the Kalman filter can be utilized (C. Hu, et al., 2003). Assuming that the noise is Gaussian white noise, considering the general linear system: x(t + 1) = φt +1,t x(t) + ω(t)

(3)

z(t + 1) = Ht +1 x(t + 1) + v(t + 1)

(4)

Where x(t) = [E, v E , a E , ξ E , N, v N , a N , ξ N , U, vU , aU , ξU ]T is the state vector of epoch t; E, v E , a E , N, v N , a N , U, vU , aU are the vector of position, velocity, acceleration in EastNorth-Up Cartesian coordinates respectively; ξ E , ξ N , ξU are the position error in East-NorthUp respectively; this error generally considers as one-order Markov process; ω(t) = [0, 0, ξae , ξe , 0, 0, ξan , ξn , 0, 0, ξau , ξu ]T is the vector of dynamic model noise; ξe , ξn , ξu are the equaled errors in the position components caused

Safety Monitoring for Dam Construction Crane System with Single Frequency GPS Receiver

by all kinds of reasons and ξae , ξan , ξau are Gaussian white noises; φt +1,t is the state transmit matrix; z(t + 1) is the observation vector; v(t) = [ε Le , εve , ε Ln , εvn , ε Lu , εvu , ]T is the observation noise; ε Le , ε Ln , ε Lu , εve , εvn , εvu are the position and velocity noise in East-North-Up respectively; Ht +1 is the design matrix for observation. ⎡ ⎤ 2 ⎤ 1, T, T2 , 0 φe , 0, 0 ⎢ 0, 1, T, 0 ⎥ ⎢ ⎥ Set φt+1,t = ⎣ 0, φn , 0 ⎦ , andφe = ⎢ ⎥, 0, 0, 1, 0 ⎣ ⎦ 0, 0, φu T 0, 0, 0, e− τe ⎡ ⎡ ⎤ ⎤ 2 2 1, T, T2 , 0 1, T, T2 , 0 ⎢ 0, 1, T, 0 ⎢ 0, 1, T, 0 ⎥ ⎥ ⎢ ⎢ ⎥ ⎥ φn = ⎢ ⎥ , φu = ⎢ ⎥, ⎣ 0, 0, 1, 0 ⎣ 0, 0, 1, 0 ⎦ ⎦ T T 0, 0, 0, e− τn 0, 0, 0, e− τu ⎤ ⎡ 110100000000 ⎢ 000100000000 ⎥ ⎥ ⎢ ⎢ 000011010000 ⎥ ⎥ andHk+1 = ⎢ ⎢ 0 0 0 0 0 0 0 1 0 0 0 0 ⎥. ⎥ ⎢ ⎣ 000000001101 ⎦ 000000000001 ⎡

Where T is the sampling interval; and τe , τn , τu are the time constants of one-order Malkov process. Adaptive Kalman filter estimation can be expressed as follows (Leuven K.U.):

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In the dynamic monitoring system, setting the sampling interval as 100 ms, setting ⎡

⎡ ⎤ ⎤ 0.003, 0, 0 0.3, 0, 0 Q 1 = ⎣ 0, 0.005, 0 ⎦ , R1 = ⎣ 0, 0.4, 0 ⎦ , r = 1.03 0, 0, 0.010 0, 0, 1.5

The output roving positions will be used to analyze the possible collision of the cranes. We tested the above method with a static data set and the errors for the epoch by epoch position solutions are showed in Figure 3. From Figure 3, we can see that the fluctuation of the filtered North and East component are within 0.5 m and the Up component is within 0.8 m; this accuracy can satisfy the determination of avoiding the cranes’ collisions. All the filtered positions of the seven roving units are sent to the second computer which specially used to analysis the collision possibilities. In this unit, all kinds of collision possibilities are considered, and crane system collision thresholds are set according to the cranes’ moving velocities and its real time positions. If the practical relative positions of the cranes exceed the threshold, an alarming signal will

⎧ x k+1/k = φk+1,k x k ⎪ ⎪ ⎪ T ⎪ ⎨ pk+1/k = Sφk+1,k pk φk+1,k + Q k T T +R −1 K k+1 = pk+1/k Hk+1 (Hk+1 pk+1/k Hk+1 k+1 ) ⎪ ⎪ ⎪ x = x + K (Z − H x ) k+1,k k+1 k+1 k+1 k+1/k ⎪ ⎩ k+1 pk+1 = (I − K k+1 Hk+1 ) pk+1/k (5)

Where pk is the variance matrix of x k ; pk+1/ k is the prediction variance matrix of x k ; x k+1/ k is onestep prediction of x k ; x k+1 is updating;K k+1is the gain of epoch k+1; Q k is the variance matrix of system noise; Rk+1 is the observation noise matrix; S is the adaptive factor which can be calculated by the following equations: Set the one-step prediction residual is Vk+1 , Vk+1 = Z k+1 − Hk+1 X k+1/ k ; and set: M(k + T T 1) = Hk+1 φk+1/ k pk+1 φk+1/ N(k + 1) = k Hk+1 ; T C0 (k + 1) − Hk+1 Q k+1 Hk+1 − Rk+1 ; And the adaptive factor S can be expressed as (J. Fang, et al., 1997/1998): S(k + 1) = max(1, r × tr [N(k + 1)]/tr [M(k + 1)]) where r is an adjustable positive rational number, and ⎧ T ⎨ S(k+1)vk+1 vk+1 k≥1 1+S(k+1) C0 (k + 1) = 1 T v v k=0 ⎩2 1 1

Fig. 3. The error for simulated roving position (X:Northing, Y:Easting, Z:Up, Pink is for unfiltered result and blue is for the filtered result).

210

be issued and sent to the machine controller via wireless communication.

5 Conclusions The operation safety for the huge construction cranes is improved with the help of this GPS based monitoring system. It is appreciated by the crane operators working at LongTan hydropower station. The veracity and trust probability is above 99 percent according to the statistic during its practice testrun period. It ensures the dam safety construction and avoids huge cranes’ collisions by an effective way, but relative low costs.

W. Wang et al.

References Leuven K.U., Adaptive Kalman Filter For Noise Identification, Pma, Celestijnenlaan 300b, 3001 Heverlee, Belgium. Congwei Hu, Wu Chen, Yongqi Chen, Dajie Liu, Adaptive Kalman Filter For Vehicle Navigation, Journal of Global Positioning System Vol. 2, No. 1, 2003, 42–47. Jiancheng Fang, Gongxun Shen, Dejun Wan, Bailing Zhou, Adaptive Kalman Filter Research In GPS dynamic Position, Signal Procssing, Vol. 14. No. 2, June 1998, 97–103. Jiancheng Fang, Dejun Wan, Qiuping Wu, Gongxun Shen, A New Kinematic Filtering Method of GPS Positioning for Moving Vehicles, China Inertial Technology Learned Journal, Vol. 5 No. 2, 1997, 4–10.

PPP for Long-Range Airborne GPS Kinematic Positioning X. Zhang, J. Liu School of Geodesy and Geomatics, Wuhan University, 129, Luoyu Road, Wuhan 430079, P.R. China R. Forsberg Geodynamic Department, Danish National Space Center, Juliane Maries Vej 30, 2100 Copenhagen, Denmark Abstract. Kinematic GPS positioning has been widely used, but the available commercial software systems most are only suitable for processing the short or medium range kinematic baseline. However, airborne surveying typical has few hundreds kilometer, even more than one thousand kilometers kinematic baseline as logistic limitation in some extreme case, e.g. airborne surveying in polar region. It is a real challenge to the traditional kinematic GPS software based on double differenced model. Since Zumbergre et al. demonstrated the perfect performance of Precise Point Positioning (PPP) in both static and kinematic applications, the PPP attracted a lot attentions and opened a new alternative to precise kinematic positioning. In this paper software, named TriP, based on PPP technology, developed by the first author will be introduced briefly. And then in-situ GPS data from airborne survey, in the Arctic region, was analyzed. Both static simulation test and flight kinematic test demonstrate that cm to dm RMS could be achieved with TriP in kinematic case. The accuracy could satisfy kinematic positioning in airborne survey applications. Keywords. GPS, long-rang kinematic positioning, precise point positioning, comparison

1 Introduction GPS kinematic positioning in the post-processed or in the real-time mode is now increasingly used for many surveying and navigation applications on land, at sea and in the air. Techniques range from the robust pseudo-range-based differential GPS (DGPS) techniques capable of delivering accuracies at the meter level, to sophisticated carrier phase-based centimetre accuracy techniques, such as RTK and VRS. The distance from the mobile receiver to the nearest reference receiver may range from a few kilometers to hundreds of kilometers. A vast literature exists on the topic of airborne kinematic positioning (e.g., Cannon et al. 1992; Colombo and Evans, 1998; Han, 1997;

Han and Rizos, 1999). Recently there are two representatives of them that fully review and summarize airborne kinematic positioning progress (Castleden et al. 2004; Mostafa, 2005). However, as the distance from rover to base increases, the problems of accounting for distance-dependent biases grow in airborne kinematic positioning. For carrier phase-based techniques reliable ambiguity resolution becomes an even greater challenge. Table 1 shows the typical airborne DGPS positioning errors. In order to derive reliable and accurate estimates of the trajectory of a survey aircraft without the establishment of dense GPS base stations, four approaches have been used as Castleden et al. (2004) and Mostafa (2005) summarized: The first approach is to make use of data available from existing Continuously Operating Reference Stations (CORS) networks to estimate the position of the aircraft. While such stations are often at a considerable distance from the survey area (e.g. 50–500 km), they are often large in number and their data is usually freely available. More experience and results are required to make a definitive statement about using CORS data certainly the potential is there (Mostafa, 2005). The second approach is using virtual reference station (VRS) concept. The VRS approach can deliver single coordinate accuracies of a few centimetres for a network of reference stations separated by only 50–70 km presently. The third approach is using the satellite-based differential corrections available in real-time, but only sub-meter positioning over most land areas worldwide can be achievable, an example of these systems is the NavCom. The fourth approach is to use the IGS products, where the precise orbits and the satellite clock corrections are obtained after the fact and used in a single point positioning mode, i.e. PPP. Since Zumbergre et al. (1997) demonstrated subdecimetre level accuracy could be achieved using PPP technology in kinematic case irrespective of baseline length. PPP is now attracting much attention internationally (Kouba, 2001; Gao and Shen, 2001, Bisnarth et al., 2002). PPP requires only one dualfrequency carrier-phase GPS receiver and thus avoids 211

212 Table 1. Typical (Mostafa, 2005)

X. Zhang et al. airborne

DGPS

positioning

errors

GPS error source

Typical relative positioning error (PPM)

Positioning error for a 50 km baseline (m)

Orbital (SA is on) Ionospheric Tropospheric Multipath Receiver Noise Total Error

1 1∼10 2 0.01 0.001 2.5∼10.25

0.05 0.05∼0.50 0.10 0.05 < 0.025 0.1∼0.5

the expense and logistics of deploying a network of GPS receivers surrounding the survey area, as is needed for the MRS and VRS techniques. Geodynamic Department of Danish National Space Center (DNSC) has undertaken airborne surveys for many years in the Arctic region, especially in Greenland. This kind of airborne survey has two important objectives: measuring the gravity acceleration of the Earth and determining topography of sea ice or land ice, determining the geoid in the Arctic area as well. Because the aerogravimeter (or accelerometer) and the altimeter are firmly attached to the aircraft, kinematic positioning using GPS plays a key role for determining the flight acceleration, velocity and flight trajectory, as well as orientation of the aircraft. The high sensitivity of the sensors requires high quality aircraft positioning and attitude. Even above referred literatures demonstrated subdecimeter level kinematic positioning accuracy could be achieved in their experiments and test with typically tens kilometres kinematic baselines length. In practice, the above accuracy is not always ensured, certainly the potential is there. However, one of the logistic limitations commonly faced when using GPS for airborne surveying is the requirement for continuous data collected by a GPS receiver at one or more base stations in the survey area (e.g. having a station within 30–50 km of the aircraft at all times). While the use of such dedicated base station is a means of meeting the accuracy requirements of survey applications, establishing a base station is often difficult in the Arctic where airborne surveying projects are usually undertaken in remote areas where geodetic control is sparse and ground access is problematic, and the airborne survey of one flight possible cover 1000 km distance away from the reference station. Furthermore, even careful mission planning is mandatory for high accuracy airborne surveying applications in order to keep good PDOP in GPS data, but it could be unrealizable in the hostile Arc-

tic region, that means the data quality is not always guarantee. Such extreme long-range kinematic data processing as mentioned above is a great challenge to the available GPS software systems around us. In most case, GPSurvey was used to process long-range kinematic GPS data routinely in DNSC, and now the GrafNav software is available too. But it was never recommended to process such long-range kinematic baseline in their manuals. Therefore, in this paper, PPP method to meet such case will be investigated in detail, static simulation and a real flight were conducted to assess the ability of kinematic GPS positioning with PPP for long-range, carrier phase-based positioning in support of DNSC projects in Arctic.

2 PPP Concept and Software By using the IGS precise orbit products and precise satellite clock information as the known information, and combining the GPS carrier phase and pseudorange data, geodetic users achieve precise positioning with single dual-frequency receiver on the earth anywhere. Such method is called precise point positioning, abbreviated as PPP. In PPP, the following ionosphere-free combined observations are generally used to form the observation equations. l P = ρ + c(dt − dT ) + M · zpd + ε P

(1)

lΦ = ρ + c(dt − dT ) + amb + M · zpd + εΦ (2) where: l P is code ionosphere-free combination of P1 and P2;lΦ is phase ionosphere-free combination of L1 and L2; dt is the receiver clock offset; dT is the satellite clock offset; c is the speed of light; amb is ambiguity of the phase ionosphere-free combination (non-integer); M is the mapping function; zpd is the zenith tropospheric delay correction; ε P and εΦ are noise of combined observations; ρ is geometric range between receiver (X r , Yr , Z r ) and satellite (X S , Y S , Z S ): ρ = (X S − X r )2 + (Y S − Yr )2 + (Z S − Z r )2 Linearization of observation equations (1) and (2) around the a-priori parameters (X 0 ) and observations equation becomes, in matrix form: V = Aδ X + W Where A is the design matrix; δ X is the vector of corrections to the unknown parameters (X) included receiver position, receiver clock offset, ambiguity of the phase ionosphere-free combination and zenith

PPP for Long-Range Airborne GPS Kinematic Positioning

tropospheric delay correction; W is the misclosure vector and V is the vector of residuals. GPS satellite clock offsets and orbits come from IGS products. The detail model about PPP can be found in the following references such as Zumbergre et al. (1997), Kouba and Heroux (2001), etc. We will not focus on the details of PPP as many literatures can be referred. TriP, developed by the first author, is a software package for PPP, which overcomes the need for a dedicated base station in GPS positioning service. TriP is based on the processing of un-differenced code and carrier phase data from a single GPS receiver, integrated with widely-available precise GPS orbit and clock products from IGS. It could provides global consistency and globally-attainable positioning accuracy at the centimeter to decimeter level for kinematic positioning and millimeter to centimeter level for static positioning case.

3 Tests and Results 3.1 Static Simulation Test In kinematic position, the true trajectory is not always well known, which make the evaluation to the solution is argued. However, the static solution with few centimeters accuracy on a fix point is always believable to user. If using such static GPS data does the force kinematic solution with TriP, and then compare the difference between coordinates from TriP and the known coordinates epoch by epoch. In this way, we can investigate how accurate the TriP kinematic solution could achieve. Example of such comparison has been made as following. The example data was taken from reference station of SYY1 on July 5, 2003 in the Arctic supported by Geoid and Ocean Circulation in the North Atlantic (GOCINA) project. Its known coordinates came from Auto GIPSY solution. Figure 1 shows the difference between the TriP solution and known coordinates in NEU epoch by epoch. The statistic comparison of simulation (cf. Table 2) indicates impressive accuracy of the TriP kinematic solution could achieve, 3D RMS is better than 5 cm.

213

Fig. 1. Comparison between TriP kinematic solution and known coordinates epoch by epoch (NEU) by using static GPS data of site SYY1 on July 5, 2003.

time is there. The flight started from Stornoway, Hebrides and ended at Hornafjordur, Iceland (cf. Figure 2) The discrepancies of the flight trajectories from PPP and double-difference solution were made. The results were given in Figure 3. For doubledifferenced processing, we used the GPSurvey software as far as possible in the mode recommended by its author or commonly applied respectively. PPP kinematic solution of the aircraft is computed by using only the airborne GPS observations, integrated with widely-available precise GPS orbit and clock products from IGS. This analysis does not include a comparison of the physical models involved or their realizations in the software. The differences of trajectory in NEU were made between TriP’s solution and GPSurvey’s solution Figure 3 shows the flight trajectory differences between each other. Figure 3 demonstrates that the coincidence between PPP solution and doubledifferenced solution is better than decimetre in such case even discrepancies are there. The discrepancies mainly come from positioning errors of both PPP and double-differenced solution.

3.2 Kinematic Test In order to investigate the long-range airborne kinematic positioning ability, a long-range flight airborne data has been used. The flight of airborne survey was conducted on July 5, 2003 in the Arctic supported by Geoid and Ocean Circulation in the North Atlantic (GOCINA) project. The flight was flown with Air Greenland’s Twin-Otter OY-POF. Flight took off at 12:46 (UTC) and landed at 16:29. 3 h 42 min flight

Table 2. Statistic comparison of simulation test/m Vector

N

E

U

Bias Standard deviation RMS Max. Min.

–0.022 0.014 0.026 0.064 –0.042

–0.004 0.010 0.011 0.042 –0.033

–0.013 0.029 0.032 0.081 –0.109

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X. Zhang et al.

Fig. 2. Flight trajectory.

Fig. 3. Differences between TriP’s solution and GPSurvey’s solution epoch by epoch on July 5, 2003.

Furthermore, another test was done as following to investigate the accuracy again. On the aircraft, two GPS receivers (AIR1 and AIR2, AIR1 is Trimble, AIR2 is Javad) were set up and shared the same GPS antenna. Ideally, if there is no observation error and model errors, using the GPS observations from AIR1 or AIR2 should produce the same trajectory as they shared the same antenna. That means the differences between trajectories from AIR1 and AIR2 reflect the kinematic positioning accuracy and ability to some extent. Figure 4 shows such differences between each other. The differences mainly came from the observation noise of different GPS receiver. Different receivers have different data quality, with different cycle slip happening and multipath. The above kinematic tests demonstrate that decimeter accuracy could be achieved. Generally, static GPS data quality is better than airborne GPS kinematic data, so static data simulating kinematic solution makes the accuracy looks better than real kinematic positioning.

PPP for Long-Range Airborne GPS Kinematic Positioning

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fact, even using double differenced mode the integer ambiguities also can not be fixed in such cases.

Acknowledgements This study was supported by National 973 Program of China (No: 2006CB701301) and National High Technology Research and Development Program of China (No: 2006AA12Z325). The authors are grateful to the many individuals and organizations worldwide who contribute to the international GPS service.

References Fig. 4. Differences between AIR1 and AIR2 from the same antenna with different GPS receivers.

4 Conclusions and Remarks We have processed the real flight of airborne GPS data using GPSurvery, TriP softwares respectively. The obtained results show that PPP solution could be used for long-range kinematic position. It indicates that decimeter accuracy can be achieved for long rang kinematic positioning, certainly more accurate is potential, but not always optimistic in practice. More tests and validations should be done with real ground truth data. Although the IGS products are available, the products accuracy and quality is however to be investigated. This topic has not been studied although it is crucial in the airborne surveying field since the entire surveying mission depends on the quality of the GPS data. Therefore, it is strongly recommended to study the quality control aspects when implementing the IGS products for aiborne surveying applications if PPP is used. More experiences and results are required to make a definitive statement about using PPP without dedicated base stations. PPP is an alternative way to solve the long range kinematic positioning when no reference station is available, and almost the same accuracy could be achieved in such case with any dedicated base station. Quality control should be concentrated when PPP was used in very long-range kinematic solutions, because no integer ambiguities can be fixed in PPP mode presently, in

Bisnath S.N., Beran T., Langley R.B. (2002) Precise platform positioning with a single GPS receiver, GPS World, 13(4): 42–49. Cannon M.E., Schwarz K.P., Wei M. (1992) A consistency test of airborne GPS using multiple monitor stations, Bulletin Géodésique, 66(1): 2–11. Colombo O.L., Evans A.G. (1998) Testing decimeter-level, kinematic, differential GPS over great distances at sea and on land, Proceedings of ION GPS 1998, 15–18 Sept, Nashville, Tennessee, USA, pp 1257–1264. Castleden N., Hu G.R., Abbey, D.A., Weihing D., Ovstedal O., Earls C.J., Featherstone W.E. Recent results of long-range airborne kinematic GPS positioning research at the Western Australian Center for Geodesy, The 2004 International Symposium on GNSS/GPS, Sydney, Australia, 6–8 December 2004. Gao Y., Shen X. (2001) Improving ambiguity convergence in carrier phase-based precise point positioning. Proceedings of ION GPS 2001, 11–14 September, Salt Lake City, Utah, USA, pp 1532–1539. Han S. (1997) Carrier phase-based long-range GPS kinematic positioning. PhD Dissertation, School of Geomatic Engineering, The University of New South Wales, Sydney, Australia. Han S., Rizos C. (1999). Sea surface determination using longrange kinematic GPS positioning and Laser Airborne Depth Sounder techniques. Marine Geodesy, 22, 195–203. Kouba J., Héroux P. (2001) PPPusing IGS orbit and clock products. GPS Solutions 5(2): 12–28. Mostafa M.R. (2005): Precise airborne GPS Positioning alternatives for the aerial mapping practice. In: Proceedings of FIG working week 2005 and GSDI-8, Cairo, 16–21 April 2005. Zumbergre J., Heflin M., Jefferson D., Watkins M., Webb F. (1997) PPPfor the efficient and robust analysis of GPS data from large networks. Journal of Geophysical Research 102, 5005–5017.

The Uniform Tykhonov-Phillips Regularization α -weighted S-homBLE) and its Application (α in GPS Rapid Static Positioning J. Cai, E.W. Grafarend Department of Geodesy and GeoInformatics, University of Stuttgart, Geschwister-Scholl-Str. 24, D-70174 Stuttgart, Germany, e-mail: [email protected] C. Hu, J. Wang Department of Surveying and Geo-informatics, Tongji University, Siping Road 1239, 200092 Shanghai, P.R. China

Abstract. In high accuracy GPS positioning the conventional least-squares method is widely applied in processing of carrier phase observation. But it will not be always succeed in estimating of unknown parameters, in particular when the problem is illposed, for example, there is the weak multicollinear problem in the normal matrix with shorter period GPS phase observation. Here the newly developed method of determining the optimal regularization parameter α in uniform Tykhonov-Phillips regularization (α-weighted S-homBLE) by A-optimal design (minimizing the trace of the Mean Square Error matrix MSE) is reviewed. This new algorithm with A-optimal Regularization can be applied to overcome this kind problem in both GPS rapid static and real time kinematic positioning with single or dual frequency measurements, especially for the shorter period observation. In the case study, both the estimate methods are applied to process the twoepoch L1 data in single frequency GPS rapid static positioning. A detailed discuss about effects of the initial coordinate accuracy will also be presented. The results show that newly algorithm with optimal regularization can significantly improve the reliability the GPS ambiguity resolution in shorter observation period. Keywords. Integer least-squares, ambiguity resolution, regularization, GPS rapid static positioning, (α-weighted S-homBLE, A-optimal design

1 Introduction High precision relative global positioning system (GPS) positioning is based on the very precise carrier phase measurements, where the resolution of the phase ambiguity is a crucial problem. When 216

the integer ambiguity is rightly determined the positioning results with millimeter accuracy can be achieved. Until now there are many algorithms on the GPS ambiguity determination, which can be classified as three categories (Hatch and Euler 1994), the first is ambiguity resolution in measurement domain which uses the dual frequency C/A code or P code and carrier phase to calculate the ambiguity directly (Hatch and Euler 1994), the second one is the search technique within coordinate domain which uses the fractional part of carrier phase measurement and search the maximum point with coordinate values, namely the Ambiguity Function Method (AFM) (Remondi 1984, Han and Rizos 1996), because of the heavy computational burden, this kind of search method is not widely adopted now. The third one is the ambiguity search technique in ambiguity domain. There are so many search technique developed in ambiguity domain, such as FARA (Frei and Beulter 1990), LSAST (Hatch 1990), the modified Cholesky decomposition method (Euler and Landau 1992), LAMBDA (Teunissen 1993), FASF (Chen and Lachapelle 1995) and modified LLL Algorithm (Grafarend 2000, Lou and Grafarend 2003); Xu (2001) studied the GPS decorrelation techniques with random simulation and proposed an inverse integer Cholesky decorrelation method. The first and second kinds can be used for single epoch positioning and usually the dual frequency data are preferred. The third kind is mostly based on multi-epoch float least squares estimates and an integer least-squares problem due to the integer-constraint for the ambiguity parameters (Kim and Langley 2000). Moreover the efficiency, validation and quality issues are also important aspects in GPS ambiguity resolution. The more data sets and dual frequency data are used, the more efficiency and reliable ambiguity solutions can be obtained. In some applications, only single frequency GPS

Uniform Tykhonov-Phillips Regularization and Application in GPS Rapid Static Positioning

receivers are used and a real time or near real time observations are required as for the cost effectiveness concerns, such as those monitoring system in dam, mountain sliding and volcano. Therefore some of the existing ambiguity search techniques can not be applied and some others have to be improved. When short period single frequency data are used, the least squares float solution may cause large biases due to the weak multicollinearity of the normal equation and thus make the ambiguity search techniques fail to work. Until now Ou and Wang (2004), Shen and Li (2005) and Wang et al. (2006) have discussed this kind problem. But ever since Tykhonov (1963) and Phillips (1962) introduced the hybrid minimum norm approximation solution (HAPS) of a linear improperly posed problem there has been left the open problem to evaluate the weighting factor α between the least-squares norm and the minimum length norm of the unknown parameters. In some applications of Tykhonov-Phillips type of regularization the parameter (weighting factor) is determined by heuristic methods, such as by means of or the Cp -Plot (Mallows 1973) or L-Curve (Hansen 1992) which produces over-smoothed results as pointed out by Xu (1998). A comprehensive commented references on Tykhonov-Phillips regularization with emphasis on the determination of the weighting factor can be found in the joint paper of first and third authors with B. Schaffrin in 2004. Here the newly developed method (Cai et al. 2004) of determining the optimal regularization parameter α in uniform TykhonovPhillips regularization (α-weighted S-homBLE) by A-optimal design (minimizing the trace of the Mean Square Error matrix MSE) is reviewed. Then the reasons of the weak multicollinear problem by processing of shorter period GPS phase observation are discussed. And the solution about the weak multicollinear problem in both GPS rapid static and realtime kinematic positioning with single or dual frequency GPS measurements, especially for the shorter period GPS observation by the new algorithm with A-optimal Regularization are proposed. In the case study, both the conventional and the new algorithm are applied to process the two-epoch L1 data in single frequency GPS rapid static positioning, which was collected by 15 minutes with 5 seconds interval, totally about 180 epochs and the baseline is about 9.2 km. A detailed discussion about effects of the initial coordinate accuracy to the determination of the A-optimal regularization parameter is also performed. At end some conclusions and outlooks are presented.

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2 Review of the α -weighted S-homBLE and A-Optimal Design of the Regularization Parameter The biased estimation is a special inverse problem, also related to Tykhonov-Phillips regulator or ridge estimator. Ever since Tykhonov (1963) and Phillips (1962) introduced the hybrid minimum norm approximation solution (HAPS) of a linear improperly posed problem there has been left the open problem to evaluate the weighting factor α between the least-squares -norm and the minimum length norm of the unknown parameters. Cai et al. (2004) and Cai (2004) and Grafarend (2006) have developed the solution of determining the regularization parameter in uniform Tykhonov-Phillips regularization (αweighted BLE) by minimizing the trace of the Mean ˆ ( A-optimal design) in Square Error matrix MSE{␰} the general case for the Gauss–Markov model. Here we presented a review of the main results of Aoptimal regularization with α-weighted S-homBLE.

Box 2.1 Special Gauss–Markov model y = A␰ + e 1st moments A␰ = E{y}, A ∈ Rn×m , E{y} ∈ R(A), rkA =m

(1)

2nd moments Σy = D{y} ∈ Rn×n , Σy positive definite, rkΣy = n␰, y − E{y} = e unknown, Σy unknown, but str uctur ed. (2)

Theorem 1. (␰ˆ BLUUE of ␰): Let ␰ˆ = Ly be Σy - BLUUE ␰ˆ of ␰ in the special linear Gauss–Markov model (1), (2). Then ␰ˆ = Ly = (A y−1 A)−1 A y−1 y

(3)

␰ˆ = Σ␰ˆ A Σy−1 y

(4)

subject to the related dispersion matrix −1

ˆ = Σ ˆ = (A Σ A)−1 . D{␰} y ␰

(5)

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In the case of linear improperly posed problem or ill-conditional problem we always meet the multicollinearity of normal matrix A y A. Exact or strict multicollinearity means |A Σ y A| = 0, For weak multicollinearity in the senses of |A Σ y A| ≈ 0, the smallest eigenvalue or the so-called condition number k = (

λmax 1 )2 λmin

(6)

is used for diagnostics (Weisberg 1985). Definition 2. ( ␰ˆ homogeneously linear α-weighted hybrid min var-min bias solution, or α- homBLE) A m×1 vector ␰ˆ is called homogeneously linear α-weighted hybrid minimum variance-minimum bias estimate (α-homBLE) of ␰ in the special linear Gauss–Markov model with fixed effects of Box 2.1, if and only if (1st) ␰ˆ is a homogeneously linear form ␰ˆ = Ly,

−1

−1

−1

ˆ = (A Σ y A + αS−1 )−1 A Σ y A(A Σ y A + αS−1 )−1 D{␰} (10)

by the bias vector ˆ −␰= ␤ := E{␰} −1 = −[Im − (A Σ y A + αS−1 )−1 A Σ −1 y A] ␰ −1 −1 −1 −1 = −α(A Σ y A + αS ) S ␰ (11) ˆ and by the Mean Square Error matrix MSE{␰} ˆ = E{(␰ˆ − ξ )(␰ˆ − ␰) } = D{␰} ˆ + ␤␤ MSE{␰} −1

= (A Σ y A + αS−1 )−1 −1

−1

× A Σ y A(A y A + αS−1 )−1 −1

+ [(A Σ y A + αS−1 )−1 αS−1 ] ξ ξ −1

× [αS−1 (A Σ y A + αS−1 )−1 ] −1

= (A Σ y A

(12)

−1 + αS−1 )−1 [A Σ y A + (αS−1 ) ξ ξ (αS−1 )] −1 × (A Σ y A + αS−1 )−1 .

(7)

(2nd) in comparison to all other homogeneously linear estimates ξˆ has the minimum variance-minimum bias property in the sense of the α-weighted hybrid norm ˆ 2= ||MSEα, S {␰}|| 1 + α tr (Im − LA) S (Im − LA) = 2 2 = L + α1 (Im − LA) S = min tr LD{y}L y

complemented by the dispersion matrix

L

(8) in particular with respect to the special assumption α ∈ R+ , dim R(SA ) = rkSA = rkA = m ⇒ ˆ 2 estabS−1 exists. The hybrid norm ||MSE␣,S {␰}|| lishes the Lagrangean L3 (L) = tr LΣy L + α1 tr (Im −LA)S(Im −LA) = min

ˆ = min, or • tr D{␰} α • tr ␤␤ = min, or α ˆ = min . • tr MSE{␰}

L

for ␰ˆ as α-homBLE of ␰. ˆ Theorem 3. (␰α-homBLE, also known as: ridge estimator or Tykhonov-Phillips regulator) Let ␰ˆ = Ly be α-homBLE of ␰ in the special linear Gauss–Markov model with fixed effects of Box 2.1. Then equivalent representations of the solutions of the normal equations are −1 −1 ␰ˆ = (A Σ y A + αS−1 )−1 A Σ y y

The interpretation of the very important estimator α-homBLE ξˆ of ␰ is as follows: ␰ˆ of type (9), also called ridge estimator or Tykhonov-Phillips regulator, contains the Cayley inverse of the normal equation matrix which is additively composed of A Σ −1 y A and α S−1 . The weight factor α balances the first observational weight and the second bias weight within the inverse. While the experiment informs us y , the of the variance-covariance matrix y , say Σ weight of the bias weight matrix and the weight factor α are at the disposal of the analyst. For instance, by the choice S= Di ag(s1 , ..., sm ) we may emphasize an increase or decrease of certain bias matrix elements. The choice of an equally weighted bias matrix is S = Im . In contrast, the weight factor α can be alternatively determined by the A-optimal design of type

(9)

α

In the first case we optimize the trace of the varianceˆ type (10) . Alternatively by covariance matrix D{␰}of means of tr ␤␤ = min we optimize the quadratic α bias where the bias vector β of type (11) is chosen, regardless of the dependence on ξ . Finally for the third case – the most meaningful one – we minimize ˆ the trace of the Mean Square Error matrix MSE{␰} of type (12), despite of the dependence on ξ ξ . Here

Uniform Tykhonov-Phillips Regularization and Application in GPS Rapid Static Positioning

we concentrate on the third case and the main result is summarized in theorem 4 below. Theorem 4. (A-optimal design of α) ˆ of αLet the Mean Square Error matrix MSE{␰} ˆ homBLE ξ with respect to the linear Gauss–Markov model be given by ˆ trMSE{␰}

−1 = tr (A Σ y A + αS−1 )−1 −1 −1 × A Σ y A(A Σ y A + αS−1 )−1 −1 + tr [(A Σ y A + αS−1 )−1 αS−1 ] ξ ξ −1 × [αS−1 (A Σ y A + αS−1 )−1 ],

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ˆ Theorem 6. (␰α-homBLE, also known as: ridge estimator or Tykhonov-Phillips regulator) Let ␰ˆ = Ly be α- homBLE of ␰ in the special linear Gauss–Markov model with fixed effects of Box 2.1. Then equivalent representations of the solutions of the normal equations are −1 −1 ␰ˆ = (A Σ y A + αS−1 )−1 A Σ y y

−1 −1

= (A PA + λS

)

(17)

A Py

complemented by the dispersion matrix −1

−1

−1

ˆ = (A Σ A + αS−1 )−1 A Σ A(A Σ A + αS−1 )−1 D{␰} y y y

(18) = σ 2 (A PA + λS−1 )−1 A PA(A PA + λS−1 )−1

then αˆ follows by A-optimal design in the sense of ˆ = min trMSE{␰} if and only if αˆ =

trA Σy−1 A(A Σy−1 A + αS−1 )−2 S−1 (A Σy−1 A + αS−1 )−1 ξ S−1 (A Σy−1 A + αS−1 )−2 A Σy−1 A(A Σy−1 A + αS−1 )−1 S−1 ξ (13)

For the independent, identically distributed (i.i.d) observations Theorem 4 will be simplified as: Corollary 5. (A-optimal design of α for the special Gauss–Markov model with i.i.d. observations) For the special Gauss–Markov model Aξ = E{y}, P−1 σ 2 = y = D{y}, Im σ 2 = S (14) of independent, identically distributed (i.i.d.) observations with variance σ 2 and an analogous substitute weight matrix S scaled by the variance σ 2 , an A-optimal choice of the weighting factor α is ˆ 2 Im )−3 tr A PA(A PA + ασ αˆ = 2 −2 ξ (A PA + ασ ˆ Im ) A PA(A PA + ασ ˆ 2 Im )−1 ␰ (15)

With introduction of the Tykhonov-Phillips regularization factor λ λ = σy2 /α −1 = ασy2 , we get an A-optimal choice of the parameter λ (λ-Aoptimal): λˆ =

ˆ m )−3 σ 2 tr A PA(A PA + λI ˆ m )−2 A PA(A PA + λI ˆ m )−1 ␰ ξ (A PA + λI (16)

by the bias vector ˆ −␰ ␤ = E{␰} −1 −1 −1 = −[Im − (A Σ −1 y A + αS ) A Σ y A] ␰ −1 −1 −1 = −α(A Σ −1 y A + αS ) S ␰ −1 −1 = λ(A PA + λS ) S−1 ␰ (19) ˆ and by the Mean Square Error matrix MSE{␰} ˆ = E{(␰ˆ − ξ )(␰ˆ − ␰) } = D{␰} ˆ + ␤␤ MSE{␰} −1 −1 −1 = (A Σ y A + αS ) A Σ −1 y −1 −1 × A(A −1 y A + αS )

−1 −1 −1 + [(A Σ −1 y A + αS ) αS ] ξ ξ

−1 −1 × [αS−1 (A Σ −1 y A + αS ) ] 2 −1 −1 = σ (A PA + λS ) A PA(A PA + λS−1 )−1 + [(A PA + λS−1 )−1 λS−1 ] ξ ξ × [λS−1 (A PA + λS−1 )−1 ] = (A PA + λS−1 )−1 [σ 2 A PA+(λS−1 )ξ ξ (λS−1 )] × (A PA + λS−1 )−1

(20)

3 Methodology The high accuracy GPS relative positioning is usually based on the double-differenced (DD) carrier phase observables. When considering short baseline (less than 20 km), the linear model for DD phase may be simplified to ␰ ␩ y = Aξ + B␩ + e = [A B] +e η n m m 1 2 (21) y∈R , ␰∈R , η∈Z , A ∈ Rn×m 1 , B ∈ Rn×m 2 ␩, D{y} = Σy . E{y} = Aξ + B␩ Given the column vector y ∈ Rn of DD phase observations we fit to it the linear model Aξ + Bη where

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A ∈ Rn×m 1 , B ∈ Rn×m 2 are n × m1 , n × m2 first order design matrices. The unknown parameters ␰ ∈ Rm 1 , η ∈ Zm 2 , are the real-valued coordinate and the integer-valued ambiguity, respectively. The parameters estimation in models (21) has been usually processed as following steps:

factor λ with (16) of Corollary 4: λˆ = f (σy2 , [ξ η ] , [A, B]), wi th y = σy2 P−1 (24) The new float solution with A-optimal regularization is performed instead of step 1

1. The parameters η were treated as not integer but real. When proceeding (21) by least-squares method, the ambiguity parameters are estimated as real or floating values, the so-called the float ambiguity solution;

␰ˆ 0 = {[A B] y−1 [A B]}−1 [A B] Σy−1 y ηˆ 0 ␰ˆ 0 ∈ Rm 1 , ηˆ 0 ∈ Rm 2 , (22) together with the variance-covariance matrix D

␰ˆ 0 ηˆ 0

Σ ␰ˆ 0 Σ ηˆ ␰ˆ

␰ˆ ηˆ 0 0 Σ ηˆ 0

= Σ⎡ ˆ ⎤ = ⎣ ␰0 ⎦ 0 0 ηˆ 0 −1 −1 A Σ y A A Σ −1 y B = B Σ −1 B Σ −1 y A y B (23)

2. Using different search techniques (e.g. LAMBDA, FARA, LLL) to determinate the ambiguity resolution, where the search criterion is usually based on the variance-covariance matrix (23) of the float estimated parameters. The search procedure is also completed with ambiguity resolution validation based on certain statistical information. 3. The integer ambiguities are fixed as known quantities and a related least-squares adjustment to determine the final coordinate parameter is performed. As we have introduced in Section 1 in rapid static or kinematic positioning, especially with single frequency and shorter period observations, the normal matrix may be weak multicollinear since the GPS satellite’s positions and geometries are changed slightly. Therefore the least squares float solution may cause large biases due to the weak multicollinearity and thus make the ambiguity search techniques fail to work. In this case the A-optimal Tykhonov-Phillips regularization described above can be applied. We estimate firstly the regularization

␰ˆ = {[A B] P [A B] + λˆ Im }−1 [A B] P y ␩ˆ ␰ˆ ∈ Rm 1 , ηˆ ∈ Rm 2 , (25) complemented by the variance-covariance matrix

␰ˆ ␩ˆ

Σ⎡

⎤=

Σ ␰ˆ Σ ηˆ ␰ˆ

␰ˆ ηˆ Σ ␩ˆ

= ␰ˆ ⎦ ␩ˆ σ 2 {[A B] P [A B]+ λˆ Im }−1 [A B] P [A B]{[A B] ˆ m }−1 . P [A B] + λI (26) Then we can continue the step 2 and3 to complete the estimation of the parameters in model (21). D

=

⎣

4 Case Study and Analysis A GPS test data was used to evaluate the new algorithm with A-optimal regularization described above. The data used was collected by 15 minutes with 5 seconds interval (totally about 180 epochs) at a baseline of 9.2 km. We process the L1 phase observables firstly with conventional least squares method and then with the A-optimal uniform TykhonovPhillips regularization. In order to compare the performance of the new algorithm, we just make use of two-epochs’ data each time as one observation set, so that there are 90 (totally 180 epochs data) estimates can be obtained for each process. In addition the initial coordinate accuracy is assigned to different bias as 0.1, 0.3, 0.5 and 1.0 m, respectively. In Table 1 the average condition numbers from the normal matrix of the 90 float solutions are listed. Since only two epochs’ data are used, they are all at the level about 3.2×1010 when proceeding with least-squares method for different initial coordinate biases. After the A-optimal regularization method is applied the correspondent condition numbers decreased to the level of 104 ∼105 respectively. Figure 1 shows the float ambiguity solution bias with conventional least squares method for all satellite pairs at different two-epochs with respect to the true value calculated the accurate coordinate when initial position error is set to 0.3m. Since the weak

Uniform Tykhonov-Phillips Regularization and Application in GPS Rapid Static Positioning Table 1. Average condition number of the normal matrix Initial position error 0.1 m Conventional method A-optimal regularization

0.3 m

0.5 m

1.0 m

3.2×1010 3.2×1010 3.2×1010 3.2×1010 3.7×104

9.6×104

3.4×105

2.5×105

multicollinearity the float ambiguity bias for different satellite pairs at different epoch varies while the maximum bias may reach more than 100 cycles. Figure 2 illustrates the float ambiguity bias estimated with A-optimal regularization. The results are improved greatly and fall into a few cycles and the variation of the ambiguity bias for each satellite at different epoch is within 1 cycle. The rooted mean squares (RMS) of the float ambiguity solution bias are also calculated and presented in Table 2. The RMS statistics for float ambiguity bias estimated by conventional least squares method for different initial position errors are almost in the same order of 31.45 cycles, while the RMS with A-optimal regularization reduces to a few cycles and increases with initial position errors increases. In the case study for the each observation data sets we apply the A-optimal Tykhonov-Phillips regularization described above, where the regularization factor λ is estimated with (16) and new float solution is processed, then the step 2 and 3 is followed to

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complete the estimation of the parameters in model (21). Figure 3 illustrates the trace of the MSE functions for the λ-weighted S-homBLE estimates as functions of λ by one of the observation sets, where the minimum trace of MSE is arrived at 0.00037 ˆ (right dot line). The regularization factor λestimated by the A-optimal design (Corollary 4) (left dot line) is about 0.00030. There is no significant difference between the minimum point of trace of MSE and regularization factor λˆ , which demonstrates that the estimate of Tykhonov-Phillips regularization factor λˆ with A-optimal design is proper and practical in application. Finally we compare the overall performance for the ambiguity resolution with, for example, LAMBDA method by making use of the float solution information from least squares method and Aoptimal regularization. Table 3 summaries the successful rate from the total 90 data set processes. The ambiguity resolution fails for all the data sets processed by least squares method, while the success rates with A-optimal regularization are 100%, 94%, 57% and 0%, with respect to the initial positional errors are 0.1, 0.3, 0.5 and 1.0 m. Consequently the ambiguity resolution success rate improves greatly when the a priori position error is within a certain value. And there is no improvement when the initial position error is larger than 1.0 m, which shows that even thought the regularization method can greatly improve the GPS float estimation the integer ambiguity resolution can only be improved when the initial position error is also within a limited value.

150

Ambiguity bias (cycle)

100 Sat 1 Sat 2 Sat 3 Sat 4 Sat 5 Sat 6 Sat 7 Sat 8 sat 9

50 0 –50 –100 –150

1

11

21

31

41

51

61

71

Epoch No.

Fig. 1. Ambiguity bias estimated with conventional LS method(initial position error: 0.3 m).

81

91

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Ambiguity bias (cycle)

2 Sat 1 Sat 2 Sat 3 Sat 4 Sat 5 Sat 6 Sat 7 Sat 8 Sat 9

1 0 –1 –2 –3

1

10

19

28

37

46 55 Epoch No.

64

73

82

91

Fig. 2. Ambiguity bias estimated with A-optimal regularization (initial position error: 0.3 m).

5 Conclusions and Outlook

Table 2. Float ambiguity solution bias (RMS) in cycle

Based on the review of the newly developed method of determining the regularization parameter in uniform Tykhonov-Phillips regularization (αweighted S-homBLE) by A-optimal design (minimizing the trace of the Mean Square Error matrix MSE) we have applied this A-optimal regularization in dealing with the weak multicollinear problem in GPS rapid static and real-time kinematic positioning with single or dual frequency phase measurements, especially for the shorter observation span. The case

Initial position error

0.1 m

0.3 m

0.5 m

1.0 m

Conventional method A-optimal regularization

31.45 0.78

31.45 1.71

31.45 2.67

31.45 5.35

study shows us that the A-optimal regularization improves the RMS of the GPS float ambiguity solution from about 31 cycles to a few cycles with the initial position errors less than 1.0 m. The success

4

3 tr MSE (in cycles2) 2

1

0

1

2

3

4

5

λ

Fig. 3. The trace of the MSE functions for the .λ-weighted homBLE estimates of the unknown as functions of .λ,where the minimum trace of MSE is arrived 0.00037 (right dot line). The regularization factor λˆ estimated by the A-optimal design (Corollary 5) (left dot line) is about 0.00030.

Uniform Tykhonov-Phillips Regularization and Application in GPS Rapid Static Positioning Table 3. Ambiguity resolution successful rate Initial Position error

0.1 m

0.3 m

0.5 m

1.0 m

Conventional method A-optimal regularization

0% 100%

0% 94%

0% 57%

0% 0%

rate of ambiguity resolution improves from 0% with conventional LS method to 100%, 94%, 57% with regularization when initial position errors are 0.1, 0.3 and 0.5 m, respectively. Through A-optimal regularization method can improve the GPS float estimation greatly the integer ambiguity resolution can only be improved when the position error is within a limited value (1 m). There is a good agreement between the A-optimal regularization factor estimate and the factor with the minimum trace of MSE, which proves that the estimate of Tykhonov-Phillips regularization factor λˆ with A-optimal design is proper and practical in the application of GPS rapid static and real-time kinematic positioning. For the further researches we should study the reliability and the relation between the initial error and final success results for A-optimal regularization and develop the determination method about multi-regularization parameters with A-optimal design:

␰ˆ ␩ˆ

= +

A PA B PA

A PB B PB

λˆ 1 Im 1 0

0 λˆ 2 Im 2

−1

A Py B Py

.

Acknowledgements This work presents part of the research results of the DFG project GR 323/44-2 “Statistical estimation and hypothesis tests for the mixed integer linear model with GPS phase measurements”.

References Cai J. (2004): Statistical inference of the eigenspace components of a symmetric random deformation tensor, Dissertation, Deutsche Geodätische Kommission (DGK) Reihe C, Heft Nr. 577, München, 2004. Cai J., E. Grafarend and B. Schaffrin (2004): The A-optimal regularization parameter in uniform Tykhonov-Phillips regularization – α-weighted BLE, IAG Symposia 127. In F. Sanso (ed.) V Hotine-Marussi Symposium on Mathematical Geodesy, pp. 309–324, Springer, Berlin, Heidelberg, New York.

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Chen D. and G. Lachapelle (1995): A comparison of the FASF and least-squares search algorithms for on-the-fly ambiguity resolution, Navigation: Journal of the Institute of Navigation, Vol. 42, No. 2, pp. 371–390. Euler H.-J. and H. Landau (1992): Fast GPS ambiguity resolution on-the-fly for real-time application. In Proceedings of Sixth International Geodetic Symposium on Satellite Positioning, Columbus, OH, pp. 650–659. Frei E., and G. Beulter (1990): Rapid Static Positioning Based on the Fast Ambiguity Resolution Approach ‘FARA’: Theory and First Results, Manuscripts Geodaetica, Vol.15, pp. 325–356. Grafarend E. (2000): Mixed integer-real valued adjustment (IRA) problems, GPS Solutions, Vol. 4, pp. 31–45. Grafarend E. (2006): Linear and Nonlinear Models – Fixed Effects, Random Effects and Mixed Models, Walter de Gruyter, Berlin, New York. Han S. and C. Rizos (1996): Improving the computational efficiency of the ambiguity function algorithm, Journal of Geodesy, Vol. 70, No. 6, pp. 330–341. Hansen P. (1992): Analysis of discrete ill-posed problems by means of the L-curve, SIAM Review, Vol. 34, pp. 561–580. Hatch R. (1990): Instantaneous ambiguity resolution. In Proceedings of KIS’90, Banff, Canada, September 10–13, pp. 299–308. Hatch R. and H.-J. Euler (1994): Comparison of several AROF kinematic techniques. In Proceedings of ION GPS-94, Salt Lake City, Utah, September 20–23, pp. 363–370. Kim D and R.B. Langley (2000): GPS Ambiguity Resolution and Validation: Methodologies, Trends and Issues, the 7th GNSS Workshop, International Symposium on GPS/GNSS, Seoul, Korea. Lou L. and E. Grafarend (2003): GPS integer ambiguity resolution by various decorrelation methods, Zeitschrift für Vermessungswesen, Vol. 128, No. 3, pp. 203–210. Mallows C.L. (1973): Some comments on Cp, Technometrics, Vol. 15, pp. 661–675. Ou J. and Z. Wang (2004): An improved regularization method to resolve integer ambiguity in rapid positioning using single frequency GPS receivers, Chinese Science Bulletin, Vol. 49, pp. 196–200. Phillips D.L. (1962): A technique for the numerical solution of certain integral equations of the first kind, Journal of the Association for Computational Machinery, Vol. 9, pp. 84–96. Remondi B.W. (1984). Using the Global Positioning System (GPS) phase observable for relative geodesy: Modeling, processing and results, Ph.D. Dissertation, Center for Space Research, University of Texas at Austin. Shen Y. and B. Li (2005): A new approach of regularization based GPS fast ambiguity resolution in rapid GPS positioning, GNSS Hong Kong 2005 Conference, Hong Kong, PR China; 08.12.2005 – 10.12.2005; in: “GNSS Hong Kong 2005”. Teunissen P.J.G. (1993): Least-squares estimation of the Integer GPS ambiguities, Invited lecture, Section IV: Theory and Methodology, IAG General Meeting, Beijing, China. Tykhonov A.N. (1963): The regularization of incorrectly posed problem, Soviet Mathametical Doklady 4, 1624–1627.

224 Wang Z., C. Rizos and S. Lim (2006): Single epoch algorithm based on TIKHONOV regularization for dam monitoring using single frequency receivers, Survey Review, Vol. 38, pp. 682–688. Weisberg S. (1985): Applied Linear Regression, 2nd ed, Wiley, New York.

J. Cai et al. Xu P. (1998): Truncated SVD methods for discrete linear illposed problems, Geophysical Journal International, Vol. 135, pp. 505–514. Xu P. (2001): Random simulation and GPS decorrelation, Journal of Geodesy, Vol. 75, pp. 408–423.


Part III

Statistical Estimation: Methods and Applications

Collocation with Integer Trend P.J.G. Teunissen Delft Institute of Earth Observation and Space systems (DEOS), Delft University of Technology, Kluyverweg 1, 2629 HS Delft, The Netherlands, e-mail: [email protected]

Abstract. Collocation is a popular method in geodesy for combining heterogeneous data of different kind. It comprises adjustment, interpolation and extrapolation as special cases. Current methods of collocation apply however only if the trend parameters are real valued. In the present contribution we will generalize the theory of collocation by permitting the trend parameters to be integer valued. It will be shown how the collocation formulae change when the integerness of the trend parameters is taken into account. We will also address the problem of evaluating the quality of the collocation results. The quality of the collocation results is usually described by the so-called error covariances. We will show how the error covariances change due to the integerness of the trend. But we also show that the approach based on error covariances does not give an adequate quality description of the collocation results in case of an integer trend. How this approach needs to be generalized is also presented. Keywords. Collocation, trend-signal-noise model, integer least-squares

1 Introduction Least-squares collocation is a popular method in geodesy for combining heterogeneous data of different kind (Krarup, 1969, Moritz, 1973, Dermanis, 1980, Sanso, 1986). It comprises adjustment, interpolation and extrapolation as special cases. The underlying model of least-squares collocation consists in its general form of three terms: a trend, a signal and a noise term. The trend is then often further parametrized in a set of unknown parameters. This so-called trend-signal-noise model is quite general and it encompasses many of the conceivable geodetic measurements (Moritz, 1980). The current collocation methods are however only applicable if the trend parameters are real valued. In the present contribution we will generalize the theory of collocation by

permitting some or all of the trend parameters to be integer valued. We first give a brief review of the method of collocation when the trend parameters are real-valued. This includes the part where the observable vector is separated in the trend, the signal and the noise, as well as the part in which an unobservable vector, such as the signal for instance, is predicted. We then show how the collocation formulae change when the integerness of the trend parameters is taken into account. It is shown that the general structure of collocation remains unaffected, but that an additional computational step based on the principle of integer leastsquares needs to be inserted. We also address the problem of evaluating the quality of the collocation results. In the classical case the quality of the collocation results is described by the so-called error covariances. We show how the error covariances change due to the integerness of the trend. But we also show that the approach based on error covariances does not give an adequate quality description of the collocation results in case of an integer trend. Instead one will have to make use of the joint probability density function of the collocation error. The error distribution will not be normal even if the input data are normally distributed. The distribution of the collocation error will also be given.

2 Collocation 2.1 Trend, Signal and Noise Model Separation of Trend, Signal and Noise In the trend-signal-noise model the observable vector y is written as a sum of three terms, y = t + s + n, with t a deterministic, but unknown trend, s a zero-mean random signal vector, and n a zeromean random noise vector. The trend is usually further parametrized in terms of an unknown p × 1 parameter vector x as t = Ax. The signal and noise vector are assumed to be uncorrelated and their variance matrices are given as Q ss and Q nn , respectively. 227

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P.J.G. Teunissen

Thus we have

2.2 Error Variance Matrices y = Ax + s + n

(1)

with Q yy = Q ss + Q nn . We assume the variance matrices to be positive definite and matrix A to be of full column rank. Application of the least-squares collocation minimization principle (Moritz, 1973), gives xˆ = (A T (Q ss +Q nn )−1 A)−1 . A T (Q ss + Q nn )−1 y

(2)

−1

sˆ = Q ss (Q ss +Q nn ) (y − A x) ˆ ˆ nˆ = Q nn (Q ss +Q nn )−1 (y − A x) Note that the separation of trend, signal and noise is reflected in the identity y = A xˆ + sˆ + n. ˆ Also note that eˆ = sˆ + nˆ = y − A xˆ is the least-squares residual vector. Predicting an Unobservable Vector Often one can extend the trend-signal-noise model so as to hold true for an unobservable vector y0 as well. This gives y0 = A0 x + s0 + n 0

(3)

in which A0 is a given m 0 × p matrix and s0 and n 0 are uncorrelated zero-mean random vectors, with variance matrices Q s0 s0 and Q n0 n0 , respectively. The two signal vectors, s0 and s, are assumed correlated (Q s0 s = 0), whereas the two noise vectors, n 0 and n, are (usually) assumed to be uncorrelated (Q n0 n = 0). Application of the least-squares collocation principle gives ˆ yˆ0 = A0 xˆ + Q s0 s (Q ss + Q nn )−1 (y − A x) ˆ sˆ0 = Q s0 s (Q ss + Q nn )−1 (y − A x)

(4)

nˆ 0 = 0 Note that A0 xˆ is the least-squares estimator of the mean of y0 . Thus if s0 and s are uncorrelated, then the predictor of y0 coincides with the estimator of its mean. Also note that the predictor of the trend plus signal, A0 x + s0 , is identical to the predictor of y0 . Both are given as A0 xˆ + sˆ0 . In general this is not the case. In the present situation, the two predictors coincide since the noise vector n 0 was assumed to be uncorrelated with s and n. For the same reason the predictor of n 0 is identically zero.

In order to judge the prediction quality of collocation, we need to consider the prediction errors. The prediction error of yˆ0 is defined as ˆ0 = y0 − yˆ0 . It is a zeromean random vector, E(ˆ0 ) = 0. Thus the predictor yˆ0 is unbiased. The variance matrix of ˆ0 is called the error variance matrix of yˆ0 . It will be denoted as Pyˆ0 yˆ0 and it should not be confused with the variance matrix of yˆ0 . To determine Pyˆ0 yˆ0 , we first write ˆ0 = y0 − yˆ0 as ˆ0 = (y0 − Q y0 y Q −1 yy y) − (A 0 − A) x. ˆ Note that the first bracketed term is Q y0 y Q −1 yy uncorrelated with y. Since xˆ is a linear function of y, it follows that the first bracketed term is also uncorrelated with x. ˆ Application of the error propagation law gives therefore Pyˆ0 yˆ0

= Q y0 y0 − Q y0 y Q −1 yy Q yy0 +

−1 T (A0 − Q y0 y Q −1 yy A)Q xˆ xˆ (A 0 − Q y0 y Q yy A) (5)

ˆ The three in which Q xˆ xˆ is the variance matrix of x. terms on the right hand side of this expression can be understood as follows. Would x be known and y be absent, the error variance matrix would be given as Pyˆ0 yˆ0 = Q y0 y0 . In this case the uncertainty is completely due to the uncertainty of y0 . But with the observable vector y present and x still known, the error variance matrix gets reduced to Pyˆ0 yˆ0 = Q y0 y0 − Q y0 y Q −1 yy Q yy0 . The uncertainty reduces due to the contribution of y. But since x is unknown, and has to be estimated, the error variance matrix gets enlarged by the third term. With Q y0 y0 |y = Q y0 y0 − Q y0 y Q −1 yy Q yy0 and −1 A0|y = A0 − Q y0 y Q yy A, we can write the error variance matrix in compact form as T Pyˆ0 yˆ0 = Q y0 y0 |y + A0|y Q xˆ xˆ A0|y

(6)

When A0 = 0, we can obtain an alternative expression for the error variance matrix and one which is expressed in the variance matrix of the least-squares residual vector. It is given as −1 Pyˆ0 yˆ0 = Q y0 y0 − Q y0 y Q −1 yy Q eˆ eˆ Q yy Q yy0

(7)

where Q eêˆ = Q yy − AQ xˆ xˆ A T . This result applies, for instance, when s0 plays the role of y0 . A complete probabilistic description of the prediction error can be given once its probability distribution is known. If we assume that y and y0 are normally distributed, then – since all relations are linear - also the prediction error is normally distributed. Its distribution is then given as ˆ0 ∼ N(0, Pyˆ0 yˆ0 ).

Collocation with Integer Trend

229

3 Collocation with Integer Trend 3.1 Trend, Signal and Noise Model We now extend the model of the previous section so as to include the option that the trend parameter vector x is integer valued, x ∈ Z p . We will assume that all trend parameters are integer valued. The results of this and the following sections can be generalized however to the case that some but not all of the trend parameters are integer valued. For the separation of trend, signal and noise, application of the least-squares collocation principle, but now with the integer constraints included, gives xˇ

= arg minz∈Z n xˆ − z 2Q xˆ xˆ

sˇ = Q ss (Q ss + Q nn )−1 (y − A x) ˇ ˇ nˇ = Q nn (Q ss + Q nn )−1 (y − A x)

(8)

with · 2M = (·)T M −1 (·). Compare with (2) and note that now y = A xˇ + sˇ + n. ˇ For the prediction of y0 , s0 and n 0 , we get ˇ yˇ0 = A0 xˇ + Q s0 s (Q ss + Q nn )−1 (y − A x)

ˇ sˇ0 = Q s0 s (Q ss + Q nn )−1 (y − A x) nˇ 0 = 0

(9)

Compare with (4). Note that the structure of the collocation results remains the same. That is, the above results can be obtained form those of (2) and (4) by replacing xˆ by x. ˇ For information on how the integer least-squares solution xˇ can be computed, we refer to (Teunissen, 1993, 1995, de Jonge and Tiberius, 1996). To see the above equations at work, consider the following two examples. Example 1. Consider the single equation y = ax + s + n, with scalar a given, x an unknown integer, and s and n the zero mean signal and noise, respectively. The integer least-squares estimator of x is then given as xˇ = [y/a], in which ‘[·]’ denotes rounding to the nearest integer. For the signal and the noise, we σs2 σn2 ˇ and nˇ = σ 2 +σ ˇ obtain sˇ = σ 2 +σ 2 (y − a x) 2 (y − a x), s n s n respectively. Thus fractions of the residual y −a xˇ are assigned to sˇ and n. ˇ They get an equal share of the residual vector if the two variances are equal. Example 2. As a trend-signal-noise model, we consider the single-frequency, single epoch, geometryfree GPS equations, based on double-differenced (DD) carrier phase and pseudorange. The carrier

phase and pseudorange equations are given as y1 = λx + ρ + s + n 1 and y2 = ρ − s + n 2 , with x the unknown integer DD carrier phase ambiguity, λ the known wavelength of the carrier phase, ρ the unknown DD range, s the residual ionospheric signal, and n 1 and n 2 the noise of the carrier phase and the pseudorange, respectively. If we eliminate the range ρ by taking the difference of the two equations, we obtain after dividing by two, the single trendsignal-noise equation y = (λ/2)x + s + n with y = (y1 − y2 )/2 and n = (n 1 − n 2 )/2. The integer least-squares estimator of x is then given as xˇ = [(y1 − y2 )/λ]. Now let s0 be the ionospheric signal at another time instant. Then its predictor sˇ0 = Q s0 s (Q ss + Q nn )−1 (y − A x) ˇ works out as sˇ0 =

2σs0 s 4σs2

+ σ12 + σ22

(y1 − y2 − λx) ˇ

(10)

where σs2 denotes the variance of the ionospheric signal, σs0 s is the covariance between s0 and s, and σ12 and σ22 are the variances of the DD carrier phase and pseudorange, respectively.

3.2 Error Variance Matrices The prediction error of yˇ0 is defined as ˇ0 = y0 − yˇ0 . Note that it is a zero-mean random vector, provided that E(x) ˇ = x holds true. Thus if the integer leastsquares estimator xˇ is an unbiased estimator of x, then ˇ = x holds E(ˇ0 ) = 0. It can be shown that E(x) true when y is normally distributed (Teunissen, 1999). The variance matrix of ˇ0 is the error variance matrix of yˇ0 . It will be denoted as Pyˇ0 yˇ0 . To determine the error variance matrix, we write the prediction error similarly as before as ˇ0 = (y0 − −1 Q y0 y Q −1 ˇ But now it is yy y) − (A 0 − Q y0 y Q yy A) x. not generally true anymore that the two terms on the right hand side of this expression are uncorrelated. This is due to the fact that xˇ is a nonlinear function of y. Hence, we need to make some additional assumptions on the distributional properties of y and y0 . In order to obtain a result which in structure is comparable to our previous result (6), we will assume that y and y0 are normally distributed. Then the first bracketed term is independent of y. And since xˇ is a function of y, it follows that the first bracketed term is also independent of x. ˇ Application of the error

230

P.J.G. Teunissen

propagation law gives therefore

The prediction error can be expressed as

T Pyˇ0 yˇ0 = Q y0 y0 |y + A0|y Q xˇ xˇ A0|y

(11)

with Q xˇ xˇ the variance matrix of x. ˇ Compare with (6) and note that Q xˆ xˆ has been replaced by Q xˇ xˇ . When A0 = 0, we can obtain an expression which in structure is similar to (7). It is given as T −1 Pyˇ0 yˇ0 = Q y0 y0 −Q y0 y Q −1 yy (Q yy −AQ xˇ xˇ A )Q yy Q yy0 (12) But note that the term within brackets is now not the variance matrix of the integer least-squares residual vector, Q eˇeˇ = Q yy − AQ xˇ xˇ A T .

Example 3. We can use (12) to obtain the error variance of the ionospheric predictor of Example 2. It is given as Psˇ0 sˇ0 = σs20 −

4σs2

2σs0 s + σ12 + σ22

2

× 4σs2 + σ12 + σ22 − λ2 σxˇ2 in which σxˇ2 is the variance of the integer estimator. Although it is nice to know the first two moments of the prediction error ˇ0 , this information is not sufficient to capture all the probabilistic properties of the prediction error. This is due to the fact that ˇ is not normally distributed, even if y and y0 are. Hence, with only the error variance matrix available one can only make approximate statements. One may use the multivariate version of the Chebyshev inequality, for instance, to obtain an upper bound on the tail probability of the distribution of ˇ0 . Then for every t > 0, the following bound holds, P ||ˇε0 ||2 ≥ t 2 ≤ trace Pyˇ0 yˇ0 /t 2 where || · || denotes the standard norm. 3.3 Distributional Results A complete probabilistic description of the prediction error can be given once its probability distribution is known. In this section we will present the distribution of ˇ0 . Although a rigorous proof is outside the scope of the present contribution, we will make the result plausible. As before, we will assume y and y0 to be normally distributed.

ˇ0 = y0 − Q y0 y˙ Q −1 yy y − A 0|y xˇ

(13)

It depends on the three random vectors y0 , y and x. ˇ Each one of them contributes to the random behaviour of ˇ0 . Would xˇ be nonrandom and equal to, say, z, then ˇ0 would be normally distributed with mean A0|y (x − z) and variance matrix Q y0 y0 |y . Hence, its probability density function (PDF) would then be given as f ˇ0 |x=z ˇ (v) =

(2π)m 0 /2

1 × detQ y0 y0 |y

1 exp{− ||v − A0|y (x − z)||2Q y y |y } 0 0 2 (14) However, since xˇ is not deterministic, but a random vector, one needs to take its distributional properties into account as well. And since xˇ is a nonlinear function of y, one can not expect the prediction error to be normally distributed. Since xˇ has integer outcomes only, its distribution will be a probability mass function (PMF). Let f xˆ (u) be the normal PDF of x. ˆ The PMF of xˇ can then be shown to be given as f xˆ (u)du , ∀z ∈ Z p (15) P[xˇ = z] = Sz

where Sz is the pull-in region of the integer leastsquares estimator, see (Teunissen, 1999). With these probability masses and the PDF of (14), the PDF of the prediction error follows as the infinite sum

f ˇ0 (υ) = ˇ (υ)P[ xˇ = z]. We therez∈Z p f ˇ0 | x=z fore have f ˇ0 (v) =

z∈Z p

P[xˇ = z] × (2π)m 0 /2 detQ y0 y0 |y

1 exp{− ||v − A0|y (x − z)||2Q y y |y } (16) 0 0 2 This result shows that the PDF of the prediction error is a multimodal distribution. It is an infinite sum of weighted and shifted versions of the PDF of (14). The weights are given by the probability masses of the PMF of x. ˇ Note that the PDF of the prediction error is symmetric with respect to the origin. This confirms that E(ˇ0 ) = 0. Also note, if the probability of correct integer estimation P[xˇ = x] approaches one, that the PDF of ˇ0 approaches the normal distribution with zero mean and variance matrix Q y0 y0 |y .

Collocation with Integer Trend

231

With the PDF of ˇ0 one can now describe the predictive quality of integer trend collocation. For instance, if one wants to obtain confidence regions for the prediction of the signal s0 , one uses (16) with A0 = 0 and Q y0 y0 |y = Q s0 s0 − Q s0 s (Q ss + Q nn )−1 Q ss0 . The confidence region follows then as the set {υ ∈ R m 0 | f ˇ0 (υ) ≥ c}, in which the constant c is taken in accordance with the chosen coverage probability. Example 4. We determine the PDF of the collocation error in the ionospheric prediction of Example 2, cf. (10). To apply (16), we need the PMF P[xˇ = z], the mean A0|y (x − z) and the variance Q y0 y0 |y . The PMF P[xˇ = z] of integer rounding by thefunc is given (1+2(x−z)) (1−2(x−z)) + + 1, tion F(z) = 2σxˆ 2σxˆ x 1 2 1 √ with (x) = −∞ exp{− 2 ω }dω. The function 2π F(z) is symmetric with respect to x and its shape is governed by σxˆ . The smaller this standard deviation is, the more peaked the PMF is. For σxˆ < 0.10, one will have P[xˇ = x] ≈ 1. If we denote the mean A0|y (x −z) as m(z) and the variance Q y0 y0 |y = Q s0 s0 − Q s0 s (Q ss + Q nn )−1 Q ss0 as σ 2 , the error PDF follows as

F(Z ) 1 v − m(z) 2 √ exp{− } f eˇ0 (v) = 2 σ σ 2π z∈Z

with m(z) = − 4σs2 s 0 2 4σs +σ12 +σ22

2λσs0 s (x 4σs2 +σ12 +σ22

− z) and σ 2 = σs20 −

. Note, since the unknown mean x is an

integer, that the PDF of the prediction error is independent of x.

References Dermanis, A. (1980): Adjustment of geodetic observations in the presence of signals. In: Proceedings of the International School of Advanced Geodesy. Bollettino di Geodesia e Scienze Affini. Vol. 38, pp. 419–445. de Jonge P.J., C.C.J.M. Tiberius (1996): The LAMBDA method for integer ambiguity estimation: implementation aspects. Publications of the Delft Computing Centre, LGRSeries No. 12. Krarup, T. (1969): A contribution to the mathematical foundation of physical geodesy. Publ. Danish Geod. Inst. 44, Copenhagen. Moritz, H. (1973): Least-squares collocation. DGK, A 59, Muenchen. Moritz, H. (1980): Advanced Physical Geodesy. Herbert Wichmann Verlag Karlsruhe. Sanso, F. (1986): Statistical methods in physical geodesy. In: Mathematical and numerical techniques in physical geodesy, H. Suenkel Ed., Lecture Notes in Earth Sciences, Springer-Verlag, Vol. 7, 49–156. Teunissen, P.J.G. (1993): Least-squares estimation of the integer GPS ambiguities. Invited Lecture, Section IV Theory and Methodology, IAG General Meeting, Beijing, China, August 1993. Also in: LGR Series, No. 6, Delft Geodetic Computing Centre. Teunissen, P.J.G. (1995): The least-squares ambiguity decorrelation adjustment: a method for fast GPS integer ambiguity estimation. Journal of Geodesy, 70: 65–82. Teunissen, P.J.G. (1999): An optimality property of the integer least-squares estimator. Journal of Geodesy, 73: 587–593.

Multidimensional Statistical Tests for Imprecise Data H. Kutterer, I. Neumann Geodetic Institute, Leibniz University of Hannover, Nienburger Straße 1, D-30167 Hannover, Germany

Abstract. The total uncertainty budget of geodetic data usually comprises two main types of uncertainty: random variability which reflects uncontrollable effects during observation and data processing, and imprecision which is due to remaining systematic errors between data and model. Whereas random variability can be treated by means of stochastics, it is more adequate to model imprecision using Fuzzy-theory. Hence, it is necessary to extend the classical techniques of geodetic data analysis such as parameter estimation and statistical hypothesis testing in a suitable way in order to take imprecision into account. The study focuses on imprecise vector data and on the consistent extension of a multidimensional hypothesis test which is based on a quadratic form. Within the considered approach it is also possible to introduce fuzzy regions of acceptance and rejection in order to model linguistic uncertainties. For the final decision the crisp degree of rejectability for the null hypothesis is computed. Whereas in the onedimensional case this is straightforward, in the multidimensional case the so-called ␣-cut optimization technique has to be applied. The global test in outlier detection and the congruence test of static deformation analysis are considered as application examples. Keywords. Imprecise data, fuzzy data analysis, ␣-cut optimization, multidimensional hypothesis test, outlier detection, global test, congruence test

1 Motivation In geodetic data analysis random variability and imprecision are the two main types of data uncertainty. They are exclusively considered in the following. Note that the complete error budget may contain additional uncertainties such as the imprecision of the geodetic model or deficiencies due to fragmentary or contradictory data (Kutterer, 2001). 232

Random variability describes the deviations of the results of repeated observations due to the laws of probability. It corresponds with the random errors which are assumed to be Normal with zero expectation. Imprecision is considered throughout the paper as a non-stochastic type of uncertainty. It is due to remaining systematic deviations in the data which could not be eliminated. It is possible to assess and to quantify data imprecision by means of a sensitivity analysis of the mostly sophisticated observation models; see Schön (2003) for all relevant terrestrial geodetic observations and, e.g., Schön and Kutterer (2006) for GPS observations. Here, random variability and imprecision of the data are jointly treated by means of Fuzzy-theory which has proven to be adequate for this task (Bandemer and Näther, 1992; Viertl, 1996). The respective concepts which are applied to statistical hypothesis tests for imprecise vector data have been outlined by Kutterer (2004). This earlier work is now extended regarding multidimensional hypotheses and the explicit use of the so-called card-criterion for test decision. Due to the limited space fuzzy-theoretical basics are not given in this paper. The reader is referred to, e.g., Kutterer (2004) or the above-mentioned monographs for further reading. In the following, the formulation and the use of multidimensional tests in Geodesy are shortly reviewed. Afterwards, extended hypothesis tests for imprecise data are presented and discussed. Finally, the global test of geodetic adjustment and the congruence test of static deformation analysis are given as application examples to show the main characteristics.

2 Multidimensional Statistical Tests in Geodesy The basic setting for the following discussion is shortly described as follows. A p-dimensional continuous random vector y is assumed to be Normal with expectation vector

Multidimensional Statistical Tests for Imprecise Data

E (y) = ␮ y ,

233

(1)

and (positive definite or positive semi-definite) variance-covariance matrix (vcm) D (y) = yy = ␴20 Q yy ,

(2)

where ␴20 denotes the variance of the unit weight and Q yy the associated cofactor matrix. In Geodesy, such a random vector is either an observable quantity or a derivable quantity such as the parameters estimated by means of a least-squares (LS) adjustment. A typical test scenario is concerned with the statistical comparison of a given (or model) value with an estimate of the expectation vector. It is described by the null hypothesis H0 : ␮ y = ␮ 0 ,

(3)

and the corresponding alternative hypothesis Ha : ␮ y = ␮ 0

(4)

If ␴20 is known the quadratic form T = yT −1 yy y =

1 T −1 y Q yy y ␴20

(5)

can be used as test statistic. Note that the case of using the estimated variance of the unit weight ␴ˆ 20 instead of ␴20 is not considered in this paper as the numerical solution of the extension to imprecise data has not been solved yet. In case of H0 , T is chi-square distributed as T ∼ χ 2f |H0 ,

yy ) ≤ p f = rank(

(6)

In case of Ha , T follows a noncentral chi-square distribution as T ∼ χ 2f,λ |Ha ,

yy ) ≤ p, f = rank(

(7)

with the non-centrality parameter λ. This kind of test is used in many geodetic applications such as in outlier testing (e.g., global test) or the comparison of two random vectors (describing, e.g., the 2D or 3D positions of a number of points) with respect to identity, congruence, similarity or affinity. For the test decision it is necessary to define the level of significance γ . In case of a two-sided test the regions of acceptance A and rejection R are derived consistently using

1 − γ = P (T ∈ A |H0 ) ,

(8)

γ = P (T ∈ R |H0 ) ,

(9)

with P denoting the probability. Since the regions A and R are defined complementarily, the region R is divided into two subsets which are symmetric by probability. Hence, the borders of A are given by the associated quantile values k1 = χ 2f,␣/2 ,

k2 = χ 2f,1−␣/2 .

(10)

In case of T ∈ R the null hypothesis H0 is rejected, else it is not.

3 Modeling of Imprecision Throughout this paper, imprecision is modeled using L R-fuzzy intervals according to Dubois and Prade (1980). A L R-fuzzy interval is a special case of a one-dimensional fuzzy set x, m A˜ (x) |x ∈ X , m A˜ : X → [0, 1] , (11) with m A˜ (x) denoting the membership function and X a classical set like, e.g., the set of real numbers R. The height of a fuzzy set is defined as the maximum membership degree value, the core of a fuzzy set as the classical set of elements of A˜ with membership degree equal to 1, and the α-cut of a fuzzy set as the classical set of elements of A˜ with membership degree greater or equal α with α ∈ [0, 1]. A L R-fuzzy interval is then defined as a fuzzy set over R with a non-empty core and compact ␣-cuts. Its membership functions are given by monotonously decreasing (left and right) reference functions L and R, respectively. For L and R the range of values is [0,1]. For a graphical sketch see Figure 1. LR-fuzzy intervals can be represented by X˜ = (x m , r x , xl , xr ) L R . The mean point is denoted by x m . The radius of the interval representing the core is r x . Together with the (left and right) spreads xl and xr it serves as a A˜ :=

Fig. 1. LR-fuzzy interval with different reference functions.

234

measure of imprecision. A single element core yields L R-fuzzy numbers X˜ = (x m , xl , xr ) L R . Totally symmetric L R-fuzzy numbers (identical reference functions and spreads) are called L-fuzzy numbers. They are represented by X˜ = (x m , x s ) L with x s denoting the common spread. Based on the membership functions set-theoretical operations can be consistently extended to fuzzy sets: the intersection can be defined as m A∩ ˜ B˜ = min m A˜ , m B˜ and the complement as m A˜ c = 1 − m A˜ . Arithmetic operations for fuzzy quantities can be defined based on the extension principle which allows the generalization of functions with real arguments to functions with fuzzy arguments: B˜ = g˜ A˜ 1 , . . . , A˜ n :⇔ m B˜ (y) = sup ... (x 1 , . . . , x n ) ∈ X 1 × . . . × X n g (x 1 , . . . , x n ) = y . . . min m A˜ 1 (x 1 ) , . . . , m A˜ n (x n ) ∀y ∈ Y (12) For a more detailed introduction see standard references like, e.g., Dubois and Prade (1980) or Bandemer and Näther (1992). Studies of fuzzy data analysis in the geodetic context are presented by, e.g., Kutterer (2004).

4 Extended Tests for Imprecise Data It is obvious that in the scenario described in Section 2 there is both a clear separation between A and R and a strict (and unique) test decision (precise case). Nevertheless, in practical test situations imprecision often superposes (and hence mitigates) this simple procedure (imprecise case). Note that “precise” is used here as the opposite of “imprecise”. Two cases have to be considered: (a) due to its imprecision the test statistic may be element of both A and R, (b) due to the linguistic imprecision or fuzziness of the formulated hypotheses such as “observation i is an outlier” the regions A and R are additionally imprecise. Case (b) is strongly connected with the definition of regions of transition between strict acceptance and rejection of a given hypothesis. Here, a setting is developed which extends the one introduced in Section 2 to multidimensional imprecise data. The test statistic according to equation (5)

H. Kutterer, I. Neumann

serves as the basic quantity which is now superposed by imprecision. Therefore, the extension principle described in equation (12) is applied; the respective optimization problem is considered in Section 5. ˜ In general, this yields a LR-fuzzy interval T = T m , r T , Tl , Tr L R as imprecise test statistic with the stochastic mean point T m ; the underline indicates a random variable. Actually, T˜ is a special case of the so-called fuzzy random variables (Bandemer and Näther, 1992). Without imprecision the precise case is obtained which was described in Section 2. Also the region of acceptance is defined as L R-fuzzy interval A˜ = (Am , r A , Al , Ar ) L R and the region of rejection as its complement R˜ = A˜ c . This situation is illustrated in Figure 2. The considered approach follows the test strategy introduced by Kutterer (2004). It is based on the quantitative evaluation of a precise criterion for the rejectability of H0 . For this purpose the degree of agreement of T˜ and R˜ is defined as card T˜ ∩ R˜ , γ R˜ T˜ := (13) card T˜ with card denoting the cardinality of a fuzzy set m T˜ (x) dx. (14) card T˜ := R

Note that in Kutterer (2004) both the card-criterion and the height criterion were introduced. The cardcriterion is the more natural one as it comprises the complete set and not only its maximum value. Nevertheless, the height-criterion is much easier evaluated numerically. Consistently, the degree of disagreement of T˜ and A˜ is defined as card T˜ ∩ A˜ . (15) ␦ A˜ T˜ := 1 − card T˜

Fig. 2. Test scenario in case of imprecision: the imprecise test statistics is in general element of both the region of acceptance and the region of rejection.


235

The degree of rejectability of T˜ with respect to R˜ (precisely: the rejectability of the null hypothesis H0 under the condition of T˜ ) is finally defined as ␳ R˜ T˜ := min γ R˜ T˜ , ␦ A˜ T˜ .

(16)

Now, the precise and unique decision criterion ≤

Do not reject H0 ␳ R˜ T˜ ␳crit ∈ [0, 1] ⇒ Reject H0 > (17)

can be evaluated. In addition it is possible to calculate the probability of a type I error as ␥ = P ␳ R˜ T˜ > ␳crit |H0 ,

(18)

Fig. 3. Quantitative evaluation of the degree of rejectability based on ␣-cut optimization and the card-criterion.

According to Dubois and Prade (1980, p. 37) the extension principle can equivalently be solved by computing the range of values ␣-cut by ␣-cut as

B˜ ␣ = B˜ ␣,min, B˜ ␣,max

and the probability of a type II error as 1 − β = P ␳ R˜ T˜ ≤ ␳crit |Ha .

(21)

when (19)

The selection of ␳crit for a particular application depends on the purpose of the test. In order to keep as many data as possible for further processing a conservative strategy is typically preferred in outlier testing; then ␳crit = 1 is a proper choice. If, however, the result of the test is safety relevant, ␳crit has to be as small as possible; best it is equal to 0.

5 Numerical Solution Based on ␣-cut Optimization In general there is no analytical solution to the test strategy presented in Section 4. In order to quantitatively evaluate equations (13)–(19) based on the extension principle formulated in equation (12) numerical procedures have to be applied. A straightforward method is the so-called ␣-level or ␣-cut optimization. This presentation follows the outline by Möller and Beer (2004). The main idea is based on the ␣-cut representation of L R-fuzzy intervals which is an equivalent alternative to the representation using membership functions which was introduced in Section 3. Note that the ␣-cuts A˜ ␣ = x ∈ R m A˜ (x) ≥ ␣ with ␣ ∈ [0, 1] (20) of L R-fuzzy intervals A˜ are real intervals which are uniquely described by their upper and lower bounds A˜ ␣,min and A˜ ␣,max , respectively. So A˜ is completely given by the bounds of all ␣-cuts.

B˜ ␣,min = min (y = g (x 1 , x 2 , . . . , x n )) ,

(22)

B˜ ␣,max = max (y = g (x 1 , x 2 , . . . , x n )) ,

(23)

x 1 ∈ A˜ 1,␣ , x 2 ∈ A˜ 2,␣ , . . . , x n ∈ A˜ n,␣ .

(24)

and

This is obviously a constrained optimization problem for the function g which can be solved by standard optimization tools if g is sufficiently differentiable. Because of the monotonicity of the reference functions L and R and their smoothness in practical applications only a limited number of ␣-cuts is considered. For each fuzzy interval the same number of ␣-cuts referring to the same ␣-values has to be provided. and Beer (2004) call this procedure ␣cut discretization. So the numerical evaluation of the relevant measures is reduced to the solution of a sequence of max-min optimization problems. See Figure 3 for a graphical illustration of the procedure of ␣-cut optimization applied to the imprecise test statistics.

6 Application Examples In practice, there are various possibilities for the application of the presented strategy. Here, a monitoring network of a road tunnel beneath the river Weser in Northern Germany is discussed in order to illustrate the main outcomes. The length of the tunnel is 2000 m. The network has been observed in three campaigns. Outside the tunnel differential GPS and

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H. Kutterer, I. Neumann

Fig. 4. Monitoring network for a road tunnel beneath the river Weser in Northern Germany.

precise digital levelling was used. Inside the tunnel terrestrial observations were performed using automatic total stations and precise digital levels as well. See Figure 4 for a network sketch. Because of the strong tidal influence the complete network had to be observed within less than 2 hours. In such a setting the presence of imprecision is expected. Due to the constructional conditions of the tunnel the network is mainly extended in the longitudinal direction of the tunnel. For this reason the error propagation is far from optimum. This holds for the random and (even more because of the immanent linear propagation) for the systematic components. Hence a strong imprecision of the test statistic has also to be expected. Two applications of the presented test strategy are discussed in the following: the global test of the network adjustment of a single epoch (see Figure 5) and the congruence test when comparing the network shapes observed at low tide in two epochs (see

Figure 6). In both cases quadratic forms according to equation (5) are applied as test statistics. The imprecision of the test statistic is derived from the imprecision of the original observations by applying ␣-cut optimization as described in Section 5. Therefore the interval [0, 1] was equidistantly divided into 11 ␣-cut levels. For the regions of acceptance and rejection a transition region is defined between full acceptance of the null hypothesis for an error probability above 5% and full rejection for an error probability below 1%. Hence they are imprecise as well. Note that both error probabilities refer to the precise case of hypothesis testing. For the global test presented in Figure 5 the imprecise test statistic is completely element of the imprecise region of rejection. Although the test statistic seems to be symmetric both reference functions are curved. The left reference function is concave, the right one is convex. The mean point of this fuzzy number equals the value of the test statistic in the precise case; obviously the null hypothesis was clearly rejected in this case.

Fig. 5. Exemplary imprecise test statistic and imprecise regions of acceptance and rejection for the global test of an epoch network adjustment (transition zone between error probabilities 1% and 5%).

Fig. 6. Exemplary imprecise test statistic and imprecise regions of acceptance and rejection for the congruence test between two low tide states (transition zone between error probabilities 1% and 5%).


237

The degree of rejectability according to equation (16) based on the card-criterion equals 0.86. If ␳crit is less than this value, the null hypothesis is rejected according to equation (17). Otherwise, if, e.g., ␳crit = 1 it can not be rejected as due to the present imprecision it is also element of A˜ to some amount. Figure 6 shows the results of a congruence test between two low tide epochs. Here, the asymmetry of the imprecise test statistic is clearly visible. Again the mean point refers to the test statistic of the precise case. The impact of imprecision is now mainly on the higher values. The degree of rejectability of the null hypothesis is 0.08. Hence H0 is rejected for all ␳crit < 0.08.

our opinion it is a well-suited alternative to the other approaches which treat all types of uncertainty in a pure stochastic framework. There is some more work needed concerning the adaptation of the theoretical methods to practical applications. It is certainly worthwhile to discuss the impact of the strategy on the meaning of the type I and type II error. The computational performance with respect to runtime has to be improved as the applied algorithms are not optimized. Besides some other tasks it is necessary in future work to generalize the method in order to use F-distributed test statistics where both enumerator and denominator are influenced by imprecise data.

7 Conclusions

References

Data analysis in Geodesy is often concerned with data which are both randomly varying and imprecise. The classical way is to treat these two individual types of uncertainty as one in terms of stochastics. In contrast, the approach discussed in this paper makes a distinction between these two types to overcome this strong simplification and to be more adequate. It is focussed on accordingly extended statistical hypothesis tests. Therefore the common test strategy for both randomly varying and imprecise data is used which was presented by Kutterer (2004). The new results shown in this paper are as follows. First, the card-criterion is thoroughly used as it is more comprehensive from the theoretical point of view than the height-criterion. Certainly it is also more expensive from the computational point of view. Second, test statistics for imprecise vectors are developed which are evaluated numerically. The possible use of the presented work is manifold. Besides the more classical applications in engineering geodesy and geodetic networks it can be transferred to many other testing and decision problems such as, e.g., in geoinformatics; see, e.g., Kutterer (2006). The presented approach is deeply concerned with the quality of geodetic decisions and products. In

Bandemer, H., Näther, W. (1992): Fuzzy Data Analysis. Kluwer Academic Publishers, Dordrecht. Dubois, D., Prade, H. (1980): Fuzzy Sets and Systems. Academic Press, New York. Kutterer, H. (2001): Uncertainty assessment in geodetic data analysis. In: Carosio, A. and H. Kutterer (Eds.): Proceedings of the First International Symposium on Robust Statistics and Fuzzy Techniques in Geodesy and GIS. Swiss Federal Institute of Technology Zurich, Institute of Geodesy and Photogrammetry – Report No. 295, pp. 7–12. Kutterer, H. (2004): Statistical hypothesis tests in case of imprecise data. In: Sansò, F. (Ed.): V Hotine-Marussi Symposium on Mathematical Geodesy. IAG Symposia 127, Springer, Berlin Heidelberg, pp. 49–56. Kutterer, H. (2006): A more comprehensive modeling and assessment of geo-data uncertainty. In: Kremers, H. (Ed.): ISGI 2005 International CODATA Symposium on Generalization of Information, CODATA Germany, Lecture Notes in Information Sciences, pp. 43–56. Möller, B., Beer, M. (2004): Fuzzy Randomness. Springer, Berlin Heidelberg. Schön, S. (2003): Analyse und Optimierung geodätischer Messanordnungen unter besonderer Berücksichtigung des Intervallansatzes. Deutsche Geodätische Kommission, Reihe C, Nr. 567, München. Schön, S., Kutterer, H. (2006): Uncertainty in GPS networks due to remaining systematic errors – the interval approach. Journal of Geodesy 80 (2006): 150–162. Viertl, R. (1996): Statistical Methods for Non-Precise Data. CRC Press, Boca Raton New York London Tokyo.

Multivariate Total Least – Squares Adjustment for Empirical Affine Transformations B. Schaffrin Geodetic Science Program, The Ohio State University, Columbus, Ohio, USA Y.A. Felus Surveying Engineering Dept., Ferris State University, Big Rapids, Michigan, USA Abstract. In Geodetic Science it occurs frequently that, for a given set of points, their coordinates have been measured in two (or more) different systems, and empirical transformation parameters need to be determined by some sort of adjustment for a defined class of transformations. In the linear case, these parameters appear in a matrix that relates one set of coordinates with the other, after correcting them for random errors and centering them around their mid-points (to avoid shift parameters). In the standard approach, a structured version of the Errorsin-Variables (EIV) model would be obtained which would require elaborate modifications of the regular Total Least-Squares Solution (TLSS). In this contribution, a multivariate (but unstructured) EIV model is proposed for which an algorithm has been developed using the nonlinear EulerLagrange conditions. The new algorithm is used to estimate the TLSS of the affine transformation parameters. Other types of linear transformations (such as the similarity transformation, e.g.) may require additional constraints.

nevertheless, it is erroneously applied occasionally as, e.g., by Kuo (2006), Akyilmaz (2007), or Shum and Kuo (2008). Here, we can avoid this pitfall by rephrasing the underlying relationships as Multivariate EIV Model for which a multivariate TLSS algorithm will be developed in this contribution. For the sake of simplicity, we center both groups of coordinate estimates beforehand so that the determination of shift parameters is no longer necessary. A simple example will show the differences in performance and will allow us to draw some preliminary conclusions.

2 Modeling an Empirical 2-D Affine Coordinate Transformation For an arbitrary point Pi (1 ≤ i ≤ n) let us denote its (estimated) source coordinates by (x i1 , x i2 ) and its (estimated) transformed coordinates by (yi1 , yi2 ), thereby restricting ourselves to the 2-D (twodimensional) case. An affine transformation can now be introduced via

Keywords. Multivariate Total Least-Squares Solution (MTLSS), empirical coordinate transformation, Errors-in-Variables modeling.

1 Introduction In the case that, for a certain set of points, their coordinates are available as estimates with respect to two different reference frames, an empirical transformation formula can be set up and the respective transformation parameters may be estimated by some adjustment approach. In fact, in the affine/linear case, when treating both groups of coordinate estimates as random, the resulting model will be of type “Errors-in-Variables” (EIV) with a structured coefficient matrix which contains all the coordinate estimates twice as well as plenty of zero entries. In this case, the standard Total Least-Squares Solution (TLSS) will not provide correct results; 238

E{yi1 } E{yi2 }

s2 sin(β + ε) s1 cos β = −s1 sin β s2 cos(β + ε) E{x i1 } t1 · E{x i2 } + t2

(1a) ξ E{x i1 } =: + 31 · E{x i2 } ξ32 (1b) where E denotes the “expectation” operator, s1 and s2 represent the two scale factors on the respective axes, β and (β + ε) the two rotation angles for the two axes, and t1 and t2 the shifts of the origin. Here, we shall assume that both groups of estimated coordinates are already centered so that the shift parameters ξ31 = t1 and ξ32 = t2 need not be estimated at this stage. Once estimates for the other four parameters in the vector ξ = [ξ11 , ξ21 , ξ12 , ξ22 ]T are found, their geometric equivalents are readily derived from

E{yi1 } E{yi2 }

ξ11 ξ12

ξ21 ξ22

Multivariate Total Least – Squares Adjustment for Empirical Affine Transformations

s1 =

β = arctan(−ξ12 /ξ11 )

(2a)

β + ε = arctan(ξ21 /ξ22 )

(2b)

In general, 0 would be a symmetric 2 × 2 matrix of (unknown) variance and covariance components, which will be defined here simply by:

(2c)

0 := σ02 I2

2 + ξ2 , ξ11 12

s2 =

2 + ξ2 ξ21 22

Since these are nonlinear relationships, we cannot immediately conclude that any optimality properties of the estimated ξ j k translate into optimal estimates of β, ε, s1 or s2 . The consequences of this fact will be investigated separately. Note that the parameters of a similarity transformation may be obtained by simply introducing the two linear constraints ξ11 − ξ22 = 0,

ξ21 + ξ12 = 0

(3)

which can be done in accordance with Felus and Schaffrin (2005) and Schaffrin (2006). We refer to Felus and Schaffrin (2005) for an alternative approach, based on an idea by Cadzow (1988). By introducing the 2 × 2 parameter matrix ξ11 ξ12 (4) := ξ21 ξ22

⎢ ⎢ X := ⎢ ⎣

x 11 x 21 .. .

x 12 x 22 .. .

x n1

x n2

⎤

⎡

⎥ ⎥ ⎥ ⎦

⎢ ⎢ Y := ⎢ ⎣

y11 y21 .. .

y12 y22 .. .

so that σ02 remains as the only unknown variance component. Obviously, (7a–7b) constitutes a special Multivariate EIV-Model which is further specialized through (9), thereby also classifying it as nonlinear multivariate Gauss-Helmert model. We refer to Pope (1972) for some common pitfalls when treating such nonlinear models by iterative linearization.

3 Total Least-Squares Solution for a Multivariate EIV – Model (with only one variance component) The Total Least-Squares Solution (TLSS) for model (7a–7b) in conjunction with (9) is generated by the objective function (vec EX )T · vec EX + (vec EY )T · vec EY = eX T eX + eY T eY = min(eX , eY , )

(10a)

yn1

yn2

⎤

Y − EY = (X − EX ) · .

⎥ ⎥ ⎥, ⎦

The corresponding Lagrange target function is now readily obtained as

(5) the centered version of model (1a–1b) now reads E{Y} = E{X} ·

(6)

or, after defining random error matrices EX and EY, respectively, Y − EY = (X − EX ) ·

(7a)

The random characteristics may be defined for the vectorized forms of EX and EY as vec EX 0 eX := ∼ , 0 ⊗ I2n eY vec EY 0 (7b) where the “vec” operator stacks one column of a matrix underneath the previous one, proceeding from left to right, and ⊗ denotes the “Kronecker – Zehfuss product” of matrices defined by G ⊗ H := [gi j · H]

(9)

subject to (s.t.)

and the data matrices of size n × 2, namely ⎡

239

if G = [gi j ].

(8)

(10b)

(eX , eY , ξ = vec , λ ) : = := eX T eX +eY T eY + +2λ T vec Y − eY − (I2 ⊗ (X − EX ))␰␰ = stationar y, (11)

resulting in the (nonlinear) Euler-Lagrange necessary conditions 1 ⭸⌽ ⊗ In ) · ␭ˆ =0 · = e˜ X + ( ˙ 2 ⭸eX 1 ⭸⌽ · = e˜ Y − ␭ˆ =0 ˙ 2 ⭸eY 1 ⭸⌽ ˜ X )T ] · ␭ˆ =0 · = −[I2 ⊗ (X − E ˙ 2 ⭸ξξ

(12a)

(12b)

(12c)

1 ⭸⌽ ˆ T ⊗ In ) · (vec X − e˜ X )=0 · = vec Y − e˜ Y − ( ˙ 2 ⭸λ (12d) These can be partly solved as follows; from (12b): ˆ e˜ Y = ␭,

(13a)

240

B. Schaffrin, Y.A. Felus

and from (12a):

1st step :

−1 ˆ ⊗ In ␭ˆ = − · e˜ X = −1 ˆ =− ⊗ In · vecE˜ X

−1 T ˆ = −vec E˜ X · .

(13b)

Then from (12d) in conjunction with (13a–13b): ˆ = vec(Y − X · ) ˆ T ⊗ In ) · e˜ X = = e˜ Y − ( ˆ = = vec(E˜ Y − E˜ X · ) T ˆ ⊗ In · ␭ˆ = ˆ ⊗ In · = ␭ˆ + T ˆ ⊗ ␭ˆ n · ␭ˆ = ˆ = I2n + =

ˆ T ˆ ⊗ In · ␭ˆ I2 +

(14a)

ˆ (0) := 0, N

(16a)

ˆ (1) := (XT X)−1 XT Y,

(16b)

where (16b) obviously represents the standard LEast–Squares Solution (LESS), followed by: 2nd step: ˆ (1) := [I2 + ( ˆ (1)]−1 · ˆ (1) )T N (1) T ˆ ) (Y − X · ˆ (1)), ·(Y − X ·

(17a)

ˆ (2) := (XT X)−1 (XT Y + ˆ (1)N ˆ (1) ),

(17b)

which is iterated until convergence is monotonically achieved by: 3rd step: (k+1) ˆ ˆ (k) − (18) < δ0

from which we further obtain: ˆ T )−1 ⊗ In ] · vec(Y − X ˆ = ) ␭ˆ = [(I2 + ˆ · (I2 + ˆ T ˆ −1 = = vec (Y − X ) ) = vec E˜ Y

(14b)

for some matrix norm and a given threshold δ0 . At the convergence point, we should obviously have the relationship

ˆ T ⊗ I2 ) · vec ˆ = (I2 ⊗ XT X) − (N = vec(XT Y) = (I2 ⊗ XT ) · vec Y

and ˆ · (I2 + ˆ T ˆ T. ˆ −1 · E˜ X = −(Y − X ) )

(14c)

fulfilled which, in first order approximation when ˆ yields: neglecting the randomness of X and N N, ˆ ≈ D[vec ]

Finally, from (12c) we get: ˆ = ˆ T 0 = (X − E˜ X )T · E˜ Y (I2 + ) T ˆ −1 · ˆ + ˆ · (I2 + T = X (Y − X ) ) ˆ T · (Y − X ˆ · (Y − X ) ) (14d) or, alternatively, ˆ ˆ − Y) = ˆ ·N N, XT (X ·

(19)

(15a)

ˆ := (I2 + ˆ T ˆ T (Y − X ˆ ˆ −1 (Y − X N ) ), (15b) ) which obviously represents a nonlinear equation system that may be solved iteratively.

4 An Algorithm for the Multivariate TLSS After having established the nonlinear normal equation system (15a–15b) we may start with the following initial values:

ˆ T ⊗ I2 )]−1 · ≈ ␴20 [(I2 ⊗ XT X) − (N T X][(I ⊗ XT X)−(N ˆ ⊗ I2 )]−1 ˆ ˆ T )⊗X ·[(I2 + 2 (20) For its proper application, we need an empirical value of σ02 which may be taken proportional to the sum of squared residuals which is best computed via: ˜ T ˜ (vec E˜ X)T vec E˜X + (vec EY ) vec EY = T T ˜ ˜ ˜ ˜ = tr E EX + tr E EY = X

Y

ˆ 2 + ˆ −1 (Y − X ˆ T ˆ T· = tr [ (I ) ) T ˆ −1 ˆ T ]+ ˆ 2 + ˆ ) ·(Y − X )(I ˆ T ˆ T· ˆ −1 (Y − X +tr (I2 + ) ) ˆ T ˆ 2 + ˆ −1 = · (Y − X )(I ) ˆ T (Y − X ˆ 2 + ˆ T ˆ −1 ] = tr [(Y − X ) )(I ) ˆ = tr N (21)

Multivariate Total Least – Squares Adjustment for Empirical Affine Transformations

Taking the redundancy (2n–4) as proportionality factor since n points yield 2n observations in 2D, an estimate for the variance component may be given by ˆ N/(2n − 4) ␴ˆ 20 = tr N

(22)

which may or may not turn out to be (at least, weakly) unbiased. The answer to this non-trivial question has to be presented elsewhere.

Let us consider the following situation where a platform, defined by its four corner points in 2-D, is given by their estimated coordinates in two different systems. After centering, the list of coordinates is seen in Table 1. The iterative solution for TLSS according to (15a) is compared with the standard LESS from (16b), assuming homoscedasticity, and the estimated transformation parameters are summarized in Table 2. In addition, the TLSS yields an estimated matrix: ˆ = N

Table 2. Comparison of the Total Least-Squares Solution with the Standard Least-Squares Solution of the four affine transformation parameters for centered coordinate estimates

Est. of

ξ11

ξ21

ξ12

ξ22

TLSS LESS

2.4213 2.4000

1.6418 1.6375

–1.590 –1.583

1.8111 1.8125

Table 3. Comparison of the estimated variance component for the Total Least-Squares and the standard LEast–Squares Solutions, respectively

5 A Simple Example

241

26.1909 −8.1846 −8.4597 2.6436

(23)

from (15b) whose trace represents the sum of all squared residuals following (21), namely: ˆ = tr N ˜ Y + (vec E˜ X )T vec E˜ X = (vec E˜ Y )T vec E = 28.834; (24a) in contrast, the standard LESS results in a sum of squared residuals for the coordinates in Y only, namely: T (24b) e˜ Y L S e˜ Y L S = 281.0 which turns out to be more than nine times as large. Using the redundancy of 2n–4 = 4 as divisor, the respective variance component estimates are compared in Table 3.

Table 1. The two sets of centered 2-D coordinate estimates for the four corner points of a platform

␴ˆ 20

TLSS

LESS

7.20

70.25

Because of space restriction we do not present the full results for E˜ Y , E˜ X , and (˜eY ) L E S S which, however, deserve further analysis indeed.

6 Conclusions and Outlook In this contribution on the empirical affine transformation parameters, we have presented a multivariate version of a Total Least-Squares Solution (TLSS) based on Lagrange theory, thereby avoiding the troublesome problem of a structured EIV-model. The developed algorithm usually needs relatively few iterations until convergence (about seven iterations in the example), but accelerated versions will certainly be investigated in the future. A simple example has been presented which clearly shows the superiority of the TLSS over the standard LEast-Squares Solution (LESS). If this turns out to be also true in the more realistic case of a country-wide network will soon be investigated with a dataset from the Rep. of Korea (cf. Kwon et al., 2008).

Acknowledgements This contribution was finalized while the first author was visiting the Institute of Navigation and Satellite Geodesy, Graz University of Technology/Austria, with Prof. Bernhard Hofmann-Wellenhof as his host. This is gratefully acknowledged. The authors would like to thank Sibel Uzun for identifying a calculation error.

Point No.

xi1 [m]

xi2 [m]

yi1 [m]

yi2 [m]

References

1 2 3 4

–30.00 +30.00 +30.00 –30.00

–40.00 –40.00 +40.00 +40.00

–129.50 –1.50 +145.50 –14.50

–27.50 –117.50 +22.50 +122.50

Akyilmaz O (2007) Total Least Squares Solution of Coordinate Transformation. Survey Review (July issue). Cadzow JA (1988) Signal enhancement – A composite property mapping algorithm. IEEE Trans on Acoustics Speech and Signal Processing, 36(1):49–62.

242 Felus Y, Schaffrin B (2005) Performing Similarity Transformations Using the Error-In-Variables Model. ASPRS Annual Meeting, Baltimore, Maryland, on CD. Kuo C-Y (2006) Determination and Characterization of 20th Century Global Sea Level Rise. OSU-Report No 478, Geodetic Science Program, The Ohio State University, Columbus, OH, April 2006. Kwon J, Lee JK, Schaffrin B, Choi YS, Lee IP (2008) New affine transformation parameters for the horizontal Korean network by multivariate TLS-adjustment. Survey Review accepted for publication. Pope JA (1972) Some pitfalls to be avoided in the iterative adjustment of nonlinear problems. Proceedings

B. Schaffrin, Y.A. Felus of 38th Annual Meeting. ASPRS, Washington, DC, pp. 449–477. Schaffrin B (2006) A note on constrained total leastsquares estimation. Linear Algebra and its Applications, 417(1):245–258. Schaffrin B, Felus Y (2005) On total least-squares adjustment with constraints. In: F. Sanso (Ed.), A Window on the Future of Geodesy, IAG Symposia, Vol. 128. Springer– Verlag, Berlin, pp. 417–421. Shum CK, Kuo C-Y (2008) 20th Century sea level rise: Its determination and cause, paper presented at the 6th HotineMarussi Symp. on Theoret. and Computat. Geodesy, Wulian/PR China, May/June 2006.

Robust Double-kk -Type Ridge Estimation and Its Applications in GPS Rapid Positioning S. Han, Q. Gui, C. Ma Institute of Science, Information Engineering University, No. 62, Kexue Road, Zhengzhou 450001, Henan, P.R. China

Abstract. It is well-known that there are both multicollinearities and outliers in DD (Double Difference) model which is commonly employed in GPS rapid positioning. For the complicated complexion, we present the robust double-k-type ridge estimator (RDKRE) by combining the double-k-type ridge estimator (DKRE) and correlative equivalent weight, and we improve the LAMBDA method through replacing the cofactor matrix computed by the LSE with the cofactor matrix computed by the RDKRE. A new algorithm of GPS rapid positioning based on the RDKRE is proposed. It is concluded after investigations that the RDKRE is highly efficient and reliable to GPS rapid positioning even when both ill-conditioning and outliers exist. Keywords. GPS rapid positioning, ill-conditioning, outliers, robust double k type ridge estimator

1 Introduction Nowadays the Global Positioning System (GPS) has an extensive application outlook in geodetic surveying, engineering surveying, terrain prospecting crustal movement and so on (Hofmann-Wellenhof et al, 1997; Teunissen and Kleusberg, 1998; Horemuz and Sjoberg, 2002; Wang, 2003). For the very high-accuracy applications, one needs carrier phase measurements which are much more precise than the pseudo-range measurements (Teunissen and Kleusberg, 1998; Wang, 2003). For the purpose of ambiguity resolution, GPS data processing is usually carried out in three steps. Firstly, the integer ambiguities are treated as the real-valued (continuous) parameters, which together with other unknown parameters such as the baseline coordinate corrections, can be estimated using different algorithm. The result so obtained is often referred to as the “float solution”. Then, the float solution of the ambiguities is used to compute the corresponding integer

ambiguity values with various search methods such as FARA (Frei and Beutler, 1990), LAMBDA (Teunissen, 1993, 1995a,b, 1996, 1999) and OMEGA (Kim and Langley, 1999) and so on. Finally, the computed integer ambiguities are used to improve the first-step solution for the remaining parameters, like baseline coordinates corrections and so on. These parameters are recomputed again. This final result is referred to as the “fixed solution” and it generally inherits a much higher precision than the previously obtained “float solution”. It is to be noted that when only using severalepoch single frequency carrier-phase data for rapid positioning of a short baseline based on the common DD (Double Difference) model, there must be severe multicollinearities among the columns of the design matrix in DD model, which not only reduce the precision of the float solution, but also have a bad influence to the search efficiency of the integer ambiguities and the precision of the fixed solution. Moreover, there are cycle slips because of operation mistake, instrument trouble and anomalistic drift of survey conditions. This means that there are outliers in carrier phase observation (Wang, 2003). When multicollinearity and outliers exist simultaneously, the existing estimators such as the LS estimator, ordinary ridge estimator (ORE), partial ordinary ridge estimator (PORE), TIKHONOV regularization estimator (TRE), robust estimator (RE) and robust ordinary ridge estimator (RORE) (Hoerl and Kennard, 1970; Gui and Zhang, 1998; Ou and Wang; 2003; Gui and Guo, 2004) are not suitable to compute the float solution. For this combined problem of multicollinearities and outliers, we present the robust double-k-type ridge estimator (RDKRE) and improve the LAMBDA method through replacing the cofactor matrix computed by the LSE with the cofactor matrix computed by the RDKRE. In order to illustrate the good performance of the RDKRE, we also give an example of GPS rapid positioning at last. 243

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2 GPS DD Model In GPS short-baseline positioning, the carrier phase DD observations are commonly employed. If m + 1 GPS satellites are simultaneously tracked by two different GPS single frequency receivers T1 and T2 , and the two receivers continuously observe the same m + 1 satellites for n(≥ 2) epochs, then the DD observation equations can be written as L = AX + B N + Δ = A0

X N

+Δ

(1)

where L is an mn × 1 vector of single frequency carrier-phase DD observations, X is a 3 × 1 vector of unknown coordinate corrections of T2 ,N is an m × 1 vector of unknown integer ambiguities, A and B are the mn × 3 and mn × m design matrixes corresponding to X and N respectively, with A0 = [ A B]andΔ is an m × 1 vector of observation noises. Assume that E(Δ) = 0 and Cov(Δ) = σ02 P −1 , where the weight matrix P is symmetrical and positive-definite, σ02 is an unknown variance component of unit weight (Teunissen and Kleusberg, 1998; Ou and Wang, 2003; Wang, 2003). The LSE, ORE, PORE, TRE, and RORE of the RE X unknown parameters vector N in model (1) can be expressed as

Xˆ L S Nˆ L S

=

AT P A BT P A

AT P B BT P B

−1

A0T P L (2)

(3)

(4)

Xˆ T R (R) Nˆ T R (R) T A PA = BT P A

AT P B BT P B

+R

−1

A0T P L

=

A T P¯ A B T P¯ A

A T P¯ B B T P¯ B

−1

A0T P¯ L

(6)

X¯ R O R (k ) N¯ R O R (k ) T −1 A P¯ A + k I3 A T P¯ B A0T P¯ L (7) = B T P¯ A B T P¯ B + k Im

respectively, where k, k and k are the ridge parameters of the ORE, PORE and RORE respectively, R is a regularizer of the TRE and P¯ is an equivalent weight matrix (Hoerl and Kennard, 1970; Yang, 1993; Zhow et al, 1997; Gui and Zhang, 1998; Ou and Wang; 2003; Gui and Guo, 2004).

3 Double-kk -Type Ridge Estimation Carrying on spectral decomposition to the normal matrix A0T PA0 , we obtain Q T A0T P A0 Q = = diag(␭1 , · · · , ␭m+3 ) where λ1 ≥ · · · ≥ λm+3 is the ordered eigenvalues of the normal matrix A0T P A0 . It can be demonstrated that there always exist three very small eigenvalues in the normal matrix, whereas the others are much bigger than the three (Ou and Wang, 2003). Noticing the characteristics of the multicollinearities of normal matrix, the double-k-type ridge estimator (DKRE) of XN is defined by Xˆ D K R (k1 , k2 ) Nˆ D K R (k1 , k2 ) T −1 A P A + k 1 I3 A T P B = A0T P L B T P B + k 2 Im BT P A (8)

Xˆ P O R (k ) Nˆ P O R (k ) T −1 A P A + k I3 A T P B = A0T P L BT P B BT P A

Xˆ O R (k) Nˆ O R (k) T −1 A P A + k I3 A T P B = A0T P L B T P A B T P B + k Im

X¯ R N¯ R

(5)

where k1 > 0 and k2 > 0 are the two ridge parameters of the DKRE. It is obvious that the LSE and ORE are the special ones of the DKRE, and we can prove that the DKRE is not worse than the LSE in the sense of the MSEM (Gui and Zhang, 1998). It is highly important to suitably choose the two ridge parameters k1 and k2 in the DKRE for application. According to the thought of Hoerl and Kennard (1970) and considering the characteristics of the multicollinearities in the normal matrix, the two ridge parameters k1 and k2 are respectively given as follows

Robust Double-k-Type Ridge Estimation and Its Applications in GPS Rapid Positioning

σˆ 02

k1 =

min

1≤i≤m+3

(9)

αˆ i2

where

σˆ 02

(10)

max αˆ i2

(t ) (t −1) γ11 p¯ 11 ··· =⎝ (t ) (t −1) γ mn1 p¯ mn1

γi(tj ) =

1≤i≤m+3

where VT

PV mn − m − 3

σˆ 02 =

V = A0

Xˆ L S Nˆ L S

γii(t )

(t )

v˜i

4 Robust Double-kk -Type Ridge Estimation

=

X¯ RDKR (k1 , k2 ) N¯ RDKR (k1 , k2 )

A T P¯ A + k1 I3 A T P¯ B B T P¯ B + k2 Im B T P¯ A

−1

⎧ ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

= Q ¯ X RDKR (k1 , k2 ) N¯ RDKR (k1 , k2 )

AT P¯ A B T P¯ A

AT P¯ B B T P¯ B

A0T P¯ L (11)

AT P¯ A + k1 I3 AT P¯ B B T P¯ B + k2 Im B T P¯ A

AT P¯ A + k1 I3 AT P¯ B B T P¯ B + k2 Im B T P¯ A

−1

k0 −k0

v¯i

σ¯ 0(t )

(t )

, σ¯ 0 =

(t ) med v¯i 0.6745

k1(h)

(h)2

σ¯ 0

=

min

−1 (12)

(t +1)

=

A T P¯ (t ) A B T P¯ (t ) A

A T P¯ (t ) B B T P¯ (t ) B

−1

A 0T P¯ (t ) L

(13)

(h)2

α¯ i

(h)2

σ¯ 0

(h)

k2 =

max

1≤ i≤m+3

X¯ R N¯ R

, k0 ∈ (2.0, 3.0)

Assume the iteration times is h, then we get the (h) X¯ R iteration result and P¯ (h) . N¯ R Step 2. Get the two ridge parameters of the RDKRE. We can use the spectral decomposition method ¯ (h) = to obtain Q¯ (h)T A0T P¯ (h) A0 Q¯ (h) = (h) (h) (h) (h) diag(λ¯ 1 , . . . , λ¯ m+3 ) where λ¯ 1 ≥ · · · ≥ λ¯ m+3 are the eigenvalues of A0T P¯ (h) A0 . The two ridge param(h) (h) eters k1 and k2 are

1≤ i≤m+3

A new algorithm of GPS rapid positioning based on the RDKRE is established as follows. Step 1. Compute the RE and take it as the starting value. The calculation of (6) can be made by iterations. (t ) X¯ R Suppose we have obtained the tth RE and N¯ R residual vector V (t ), then from (6) we can get the (t + 1)th RE

(t ) v˜i < k0 (t ) k0 ≤ v˜i ≤ k0 (t ) v˜i > k0

2 (t) k0 −v˜ i

The cofactor matrix of RDKRE is

γii(t ) γ j(tj )

0

(t )

=

k0 (t) v˜ i

and k0 ∈ (4.5, 8.5)

Combining the DKRE and correlative equivalent weight, we get the robustdouble-k-type ridge estiX mator (RDKRE) of as follows N

=

−L

αˆ L S = (αˆ 1 , . . . , αˆ m+3 )T = −1 Q T A0T P L

(t ) (t −1) ⎞ ··· γ1mn p¯ 1mn ⎠ ··· ··· (t ) (t −1) · · · γmn mn p¯ mn mn

⎛

P¯ (t ) k2 =

245

(14)

(h)2

α¯ i

respectively, where T (h) (h) α¯ (h) = (α¯ 1 , . . . , α¯ m+3 )T = Q¯ (h)

(h)

σ¯ 0

=

X¯ R N¯ R

(h)

(h) med v¯i 0.6745

(h) T ) = A0 V¯ (h) = (v¯1(h) , · · · , v¯mn

X¯ R N¯ R

(h)

−L

246

S. Han et al.

Step 3. Compute the float solution of

X N

by

using the RDKRE. Step 4. Search the integer ambiguities by the following improved LAMBDA method. The original LAMBDA method was systematically developed by Teunissen (1993, 1995a, b, 1996, 1999). The core of LAMBDA is decorrelation, which was first demonstrated by Melbourne (1985). A new idea for GPS decorrelation was proposed by Hassibi and Boyd (1998) and Grafarend (2000), based on the integer orthogonal process of Lenstra et al. (1982). Xu (2001) proposed an inverse integer Cholesky decomposition for GPS decorrelation and showed that no decorrelation can work well in the case of high dimension. It is noted that the multicollinearities of the normal matrix in DD model cause the cofactor matrix of LSE become ill-conditioned, and badly affects the accuracy of the integer ambiguity search. Therefore, we improve the LAMBDA method using the cofactor matrix of the RDKRE instead of the cofactor matrix of the LSE. Firstly, transform the float integer ambiguity vector N¯ RDKR (k1 , k2 ) to a new vector zˆ RDKR (k1 , k2 ) by integer Gaussian Transformation (Z-transformation), that is zˆ RDKR (k1 , k2 ) = Z T N¯ RDKR (k1 , k2 )

(15)

Then, search z RDKR (k1 , k2 ) in the transformed search space based on the L DL T transformation method that satisfies

z RDKR (k1 , k2 ) = arg min(ˆz RDKR (k1 , k2 ) − z )T

·Q −1 zˆ

z

RDKR (k1 ,k2 )

(ˆz RDKR (k1 , k2 ) − z )

(16) = Z T Q N¯ RDKR (k1 ,k2 ) Z , where Q zˆ RDKR (k1 ,k2 ) Q N¯ RDKR (k1 ,k2 ) is computed by (12). At last, we use the inverse integer transformation on the integer vector z RDKR (k1 , k2 ) and get the integer vector N RDKR (k1 , k2 ). Step 5. Compute the fix solution by the LSE as

X LS = (A T P A)−1 A T P(L − B N RDKR (k1 , k2 ))

5 Experiments and Analysis The experiment data set was collected at 4 : 14 ∼ 4 : 36 p.m. on June 22, 2000, which was received by two JAVAD LEGACY type single frequency receivers with a sampling interval of 2 s and cut-off angle of 15◦ , and a total observation time was 22 min.

Table 1. The result of coordinate corrections Schemes

Precision (m)

1 2 3 4 5 6 7 8

332400 332400 0.3045 1915.6 0.1036 3803.4 3119400 0.0648

The length of baseline was 1904.194(m). The corresponding five satellite pairs that form the DD model were PRN10-18, PRN18-13, PRN13-24, PRN24-7 and PRN7-4. The total number of epoch is 154. By using the Bernese software, we get the correction of the baseline vector and the fixed integer ambiguities as follows X = (–0.2268, 0.0401, –0.0455) and N = (1, 0, 0, 1, –1) respectively and we take this result as the true values for comparison. Four continuous single frequency epoch observations are chosen at random and the data is computed by using the eight schemes, which are the LSE, ORE, PORE, TRE, DKRE, RE, RORE and RDKRE with the improved LAMBDA method by using the cofactor matrix of them respectively, where the three outliers of 0.7 (m), 0.6 (m) and 0.4 (m) are added to L 2 , L 13 and L 20 respectively. The comparisons of the results among the eight schemes are illustrated in Table 1 where the square of the Euclidean distance between the estimates and the true values is chosen as the criterion of precision. From the comparison of the eight schemes, we can see that Scheme 8 has the high accuracy for the estimation of the coordinate corrections. It indicates that the RDKRE can not only overcome the multicollinearities but also have the ability to resist outliers. So the new solution is more suitable for GPS rapid positioning.

6 Acknowledgements This work has been supported jointly by National Science Foundation of China (No.40474007), National Splendid Youth Science Foundation of China (No. 40125013, No. 49825107), “Basic Geographic Information and Digital Technique” Task of Important Laboratory of Shandong Province (No.SD040202) and Henan Province Nature Science Foundation of China (No. 0511010100).

Robust Double-k-Type Ridge Estimation and Its Applications in GPS Rapid Positioning

References Frei, E. and Beutler, G. (1990). Rapid static positioning based on the fast integer ambiguity resolution approach “FARA”: theory and first results. Manuscr. Geod., 15(3),172. Grafarend, E.W. (2000). Mixed integer-real valued adjustment (IRA) problems: GPS initial cycle ambiguity resolution by means of the LLL algorithm, GPS Solutions, 4: 31–44. Gui, Q.M. and Guo, J.F. (2004). A new ambiguity resolution based on partial ridge estimator. Chinese Journal of Information Engineering University. 5:137–139. Gui,Q.M. and Zhang, J.J. (1998). Robust biased estimation and its applications in geodetic adjustments. Journal of Geodesy. 72: 430–435. Hassibi, A. and Boyd, S. (1998). Integer parameter estimation in linear models with applications to GPS, IEEE Trans.Signal Proc., 46: 2938–2952. Hoerl, A.E. and Kennard, R.W. (1970). Ridge regression: biased estimation for non-orthogonal problems. Technometrics. 12: 55–88. Hofmann-Wellenhof, B., Lichtenegger, H. and Collins, J. (1997). GPS, Theory and Practice, 4th ed. Springer, Berlin Heidelberg New York. Horemuz, M. and Sjoberg, L. E. (2002). Rapid GPS ambiguity resolution for short and long baselines. Journal of Geodesy, 2002, 76, 381–391. Kim, D. and Langley, R. B. (1999). An optimized least squares technique for improving ambiguity resolution and computational efficiency. ION GPS-1999, 1579–1588. Lenstra, A.K., Lenstra, H.W. and Lov’asz, L. (1982). Factoring polynomials with rational coefficients, Math. Ann., 261: 515–534.

247

Melbourne, W. (1985). The case for ranging in GPS-based geodetic systems, Proc. 1st Int. Symp. on Precise Positioning with GPS, Rockville, Maryland, April 15-19, 373–386. Ou, J. K. and Wang, Z. J. (2003). An improved regularization method to resolve integer integer ambiguity in rapid positioning using single frequency GPS receivers. Chinese Science Bulletin. 49(2): 196–200. Teunissen, P.J.G. (1993). Least-squares estimation of the integer GPS ambiguities. Invited Lecture, Section IV, Theory and Methodology, IAG General Meeting, Beijing, China, August, 16. Teunissen, P.J.G. (1995a). The invertible GPS ambiguity transformation, Manuscr. Geod., 20: 489–497. Teunissen, P.J.G. (1995b). The Least-square integer ambiguity decorrelation adjustment: a method for fast GPS integer ambiguity estimation. Journal of Geodesy. 70:65–82. Teunissen, P.J.G. (1996). The LAMBDA method for integer ambiguity estimation: implantation aspects. Delft Geodetic Computing Centre LGR series, No.12, 50. Teunissen, P.J.G. (1999). An optimality property of the integer least-squares estimator. Journal of Geodesy. 73:587–593. Teunissen, P.J.G. and Kleusberg, A. (1998). GPS for Geodesy, 2nd ed. Springer, Berlin Heidelberg New York. Wang, H. N. (2003). Principles and Applications of GPS Navigation (in Chinese). Beijing: Scientific Publishing House. Xu, P.L. (2001). Random simulation and GPS decorrelation, Journal of Geodesy, 75: 408–423. Yang, Y.X., (1993). Robust Estimation and Its Applications (in Chinese). Beijing: Bayi Publishing House. Zhow, J.W., Huang, Y.C., Yang, Y.X. and Ou, J.K. (1997). Robust Least Squares Method (in Chinese). Wunan: Huazhong University of Science and Technology Publishing House.

Adaptive Robust Sequential Adjustment L. Sui, Y.Y. Liu, W. Wang, P. Fan Zhengzhou Institute of Surveying and Mapping, 66th Longhai Middle Road, Zhengzhou 450052, P.R. China, e-mail: [email protected]

Abstract. An adaptive sequential adjustment and an adaptive robust sequential adjustment are presented on basis of the adaptive filtering and the robust estimation principles. The corresponding estimation formulae of the parameters in both the adaptive sequential adjustment and the adaptive robust sequential adjustment are also derived. The calculation results based on a numerical example show that the adaptive sequential adjustment can efficiently resist the abnormal influence from the prior model information and well balance the prior model information and the posterior observation information as well, and the adaptive robust sequential adjustment is not only simple in calculation but also can efficiently resist the bad influences from the abnormal disturbance in the prior information model parameters and the measurement outliers in the estimated parameters. Keywords. Sequential adjustment, adaptive estimation, robust estimation

1 Introduction Sequential adjustment is a recursive data processing method which is usually applied in the adjustments of repeated and dynamic geodetic networks. What we need in the sequential adjustment is to deal with the measurements at the present epoch and the prior parameter estimates with their covariance matrix. Thus the historical measurements and their variances are not needed to be stored. It does not only save the data processing time, but also memory space. Different data processing principles correspond with different sequential adjustment formulae and different data processing quality. If the least squares (LS) principle is applied, the LS sequential adjustment formula is obtained (Jazwinski 1970; Mikhail 1976); if a robust estimation principle is employed, then the robust sequential adjustment 248

estimators can be derived (He and Yang 1998), which can resist the measurement outlier influence. However, the sequential adjustment methods based on the LS principle or robust estimation principle do not consider the outlier influence of prior parameters. The robust M–M sequential adjustment applies the maximum likelihood principle to dealing with both of the measurement errors and the prior parameter errors (Yang 1991; Masreliez and Martin 1977). But if both of the measurements and the prior parameter estimates have contaminated by outliers, it will result in confused estimates, even lead to estimation process divergence. An adaptive sequential adjustment algorithm, thus, can be applied, which is very similar to Yang’s adaptive Kalman filter (Yang et al. 2001). It uses an adaptive factor to balance the contribution of the prior parameters and the measurements to the least parameter estimates. For example, if the sequential adjustment method is applied in the earth crust deformation monitoring, even if the crust has actually changed and moved, the traditional sequential adjustment can not reflect the real crust deformation adequately, for the precise posterior observation is enforced to fit with the deformed prior model parameter and its influence is weakened far below the prior observation data. Inspired by the robust adaptive filtering algorithm (Yang et al. 2001; Ou et al. 2004), the adaptive sequential adjustment and the adaptive robust sequential adjustment are proposed in the article. It can well balance the weights between the prior information and the posterior information by its adaptive factor to get a credible result.

2 Adaptive Sequential Adjustments The observation equation can be written as: Vk = Ak Xˆ k − L k

(1)

where Lk is an n k ×1 observation vector, Ak is an n k × m design matrix. V k is an n k ×1 residual vector. The observation noise variance-covariance matrix is

Adaptive Robust Sequential Adjustment

249

Σ k , and its weight matrix is Pk , namely Pk = Σk−1 ␴20 , ␴20 is variance of unit weight. Xˆ k is the estimated state vector of Xk . After the (k–1) period iterations, the estimated vector Xˆ k−1 and its covariance matrix ΣXˆ k−1 are known. Using the LS estimation, we get: Xˆ k = (AkT Pk Ak + PXˆ k−1 )−1 (AkT Pk L k + PXˆ k−1 Xˆ k−1 ) (2) (3) Σ Xˆ k = (AkT Pk Ak + PXˆ k−1 )−1 ␴20 ␴20 =

VkT Pk Vk + ( Xˆ k − Xˆ k−1 )T PXˆ k−1 ( Xˆ k − Xˆ k−1 ) ␶

(4) where τ is the number of extra observations. In equation (2), the sequential estimated parameter vector Xˆ k is really the weighted mean value of the prior value Xˆ k−1 and the observation vector Lk . It can be seen from equation (3), Xˆ k will be more and more precise with the increase of the observation period. That is, the prior weight PXˆ k−1 will be larger and larger. No matter how precise Lk is, its influence on the sequential adjustment result will be weakened. We use adaptive sequential adjustment to describe the actual state. The score function is: VkT Pk Vk +α( Xˆ k − Xˆ k−1 )T PXˆ k−1 ( Xˆ k − Xˆ k−1 ) = min (5) We will have the result as follow: Xˆ ad = (AkT Pk Ak +␣PXˆ k−1 )−1 (AkT Pk L k +␣PXˆ k−1 Xˆ k−1 ) (6) Σ Xˆ ad = (AkT Pk Ak + ␣PXˆ k−1 )−1 (AkT Pk Ak + ␣2 PXˆ k−1 ) (AkT Pk Ak + ␣PXˆ k−1 )−1 ␴2ad

␴2ad =

(7)

T P V + (X ˆ ad − Xˆ k−1 )T P¯ ˆ ( Xˆ ad − Xˆ k−1 ) Vad k ad X k−1

␶

(8)

where τ is the number of extra observations, V ad is the residual vectors correspond to the observation vector. P¯ Xˆ k−1 = ␣P ,ˆ ␣ is an adaptive factor, 0 ≤ X k−1 α ≤ 1, which can be chosen as (Koch and Yang 1998; Yang et al. 2002): ␣=

1 c |ΔX k |

|ΔX k | ≤ c (9) |ΔX k | > c

where c is a constant, which is chosen as 1.0∼2.5. |ΔX k | = X˜ k − Xˆ k−1 tr(Σ Xˆ k−1 ) (10)

In equation (10), X˜ k = (AkT Pk Ak )−1 AkT Pk L k

(11)

X˜ k is the LS estimated vector from the actual observations.

3 Adaptive Robust Sequential Adjustments The adaptive factor ␣ is used for decreasing the prior weight in the adaptive sequential adjustment. It can reduce the abnormal influence of prior parameter. But ␣ is chosen upon the current observation, thus on the basis of equation (11). Obviously, it cannot resist the outliers, which will lead to ill result. So we add the robust estimation into adaptive sequential adjustment. Based on the error equation (1) and M-Estimation model (Huber 1964, 1981; Andrews 1974; Hampel et al. 1986; Yang 1993), we can derive the following equation: Xˆ R = (AkT P¯k Ak + PXˆ k−1 )−1 (AkT P¯k L k + PXˆ k−1 Xˆ k−1 ) (12) where P¯k is the equivalent weight matrix of Lk , and represents the adaptive estimation of the observation vector. P¯k can be calculated by IGGIII (Institute of Geodesy and Geophysics) scheme (Yang 1993), or bifactor (two shrinking factors acting on the weight element) equivalent weights (Yang et al., 2002). In equation (12), the robust sequential adjustment can resist the outliers by choosing the equivalent weights. But there are still some problems in it. Since the computation of the equivalent weight matrix P¯k is based upon the credible prior data and the computation is a recursive process, it is very difficult to get the creditable P¯k , if there exists great difference between the prior value and the actual estimation of parameter or abnormal disturbances. Therefore, it is necessary to restrict the prior weight on the basis of the robust sequential adjustment, and form the adaptive robust sequential adjustment. If the prior estimation Xˆ k−1 far deviates from the robust sequential estimation Xˆ R , we can replace the PXˆ k−1 by the equivalent weight P¯Xˆ k−1 , P¯Xˆ k−1 = ␣PXˆ k−1

(13)

where ␣ is the same as in equation (6), representing the adaptive factor, which can be chosen as: Δ X¯ k ≤ c 1 (14) ␣= Δ X¯ k > c c Δ X¯ k

L. Sui et al.

where, Δ X¯ k

=

¯ ˆ tr(Σ Xˆ k−1 ), X k − X k−1

X¯ k = (AkT P¯k Ak )−1 AkT P¯k L k , are the robust estimation results. One has the adaptive robust sequential result as: Xˆ Rad = (AkT

P¯k Ak + P¯Xˆ k−1 )

−1

(AkT

P¯k L k + P¯Xˆ k−1 Xˆ k−1 ) (15)

/ Xe–X/ (mm)

250 3 2.5 2 1.5 1 0.5 0

sequential adjustment adaptive sequential adjustment P1

P2

P3

P4

P5

P6

P7

P8

Fig. 2. The differences between the estimated parameters and the true values without additional outliers (for the period 2).

Or Xˆ Rad = Xˆ k−1 + Σ Xˆ k−1 AkT (␣Σk + Ak Σ Xˆ k−1 AkT )−1 · (L k − Ak Xˆ k−1 )

(16)

In equation (15), if the equivalent weight matrix P¯Xˆ k−1 is chosen as IGGIII scheme (Yang 1993), or bifactor equivalent weight function (Yang et al. 2002), the estimation can be regarded as the robust M–M estimation (both observations and the prior parameter values follow contaminated normal distributions) of sequential adjustment (Yang, 1991).

4 Test Computation and Analysis A simulated example is shown in Figure 1, the height of the point A is H A = 31.100 m. Suppose that there are 3 period observations of h (see Table 1). The estimated parameters based on the LS from the first period of measurements can be seen as the prior values, the rests are the periods which need to be studied. We add different outliers to several observation

P7

P8

11

12

10 P2

4 P6

P4

2

values in the second and third periods, +1 cm to h3 , –2 cm to h7 , +1 cm to h4 and –3 cm to h12. Then we study the influence caused by these outliers. The following schemes have been adopted. Scheme 1: Compare the sequential adjustment with the adaptive sequential adjustment without any outliers. The differences between the estimated parameter vector Xe and the true value X are shown in Figure 2 (for the period 2) and Figure 3 (for the period 3). Scheme 2: Compare the sequential adjustment, the robust sequential adjustment (He and Yang 1998), the adaptive robust sequential adjustment and the robust M–M estimation when there are outliers. The differences between the estimated parameters and the true values are shown in Figure 4 (for the period 2) and Figure 5 (for the period 3). The adaptive factor is chosen according to equation (9), where c = 1.5. The equivalent weight functions are chosen as IGGIII scheme (Yang 1993), where k0 = 1.5 k1 = 3.0. The true values of the unknown points are: X = [30.160, 40.173, 36.356, 42.840, 43.316, 44.547, 169.084, 208.926]T [m] According to the test results, the following conclusions can be drawn.

8 A 1

P1

6

9

P3

5

7

/ Xe–X/ (mm)

3

P5

2.5 2 1.5 1 0.5 0

sequential adjustment adaptive sequential adjustment P1

P2

P3

P4

P5

P6

P7

P8

Fig. 3. The differences between the estimated parameters and the true values without additional outliers (for the period 3).

Fig. 1. A Simulated Leveling Network.

Table 1. Observations h (m), S (km) Number

1

2

3

4

5

6

7

8

9

10

11

12

No.1 h No.2 h No.3 h Length S

0.9430 0.9430 0.9400 15

9.0730 9.0740 9.0735 20

10.0140 10.0135 10.0119 10

2.6668 2.6665 2.6668 30

6.1939 6.1948 6.1965 25

6.4843 6.4847 6.4854 20

6.9630 6.9604 6.9618 20

1.7036 1.7064 1.7063 15

1.2275 1.2310 1.2293 5

126.2400 126.2445 126.2434 30

39.8430 39.8434 39.8408 10

164.3800 164.3782 164.3804 25

Adaptive Robust Sequential Adjustment

/ Xe–X/ (mm)

6

sequential adjustment Robust sequential adjustment adaptive robust adjustment M-M estimates

4 2 0

251

P1

P2

P3

P4

P5

P6

P7

P8

/ Xe–X/ (mm)

Fig. 4. The differences between the estimated parameters and the true values with additional outliers (for the period 2).

8

sequential adjustment

6

Robust sequential adjustment adaptive robust adjustment M-M estimates

4 2 0

P1

P2

P3

P4

P5

P6

P7

P8

Fig. 5. The differences between the estimated parameters and the true values with additional outliers (for the period 3).

1. It can be seen from Figures 2 and 3, that the results calculated by the adaptive sequential adjustment are much better than those of the sequential adjustment, because of the adaptive factor. 2. Figures 4 and 5 show that, the traditional sequential adjustment can not resist the outliers in the observations. However, the robust sequential adjustment, the adaptive robust sequential adjustment and robust M–M estimation can all effectively resist the influence of the measurement outliers. 3. Both the adaptive robust sequential adjustment and the robust M–M estimation can resist the outliers of the observations and abnormal perturbing of the prior parameters. The difference lies in the choice of the equivalent weight of the prior parameter value (see Figures 4 and 5). 4. The adaptive factor is used to optimally balance the relative contribution between the posterior observations and the prior parameter values of the model.

5 Concluding Remarks The adaptive sequential adjustment can best utilize the prior parameters and the actual observations, and optimally balance the prior information and the observations through adjusting the effects of

the adaptive factors. Thus the higher precision can be acquired. When the observations are normal, the adaptive sequential adjustment is significantly superior to the sequential adjustment. The adaptive robust sequential adjustment proposed in this paper can not only resist the influence of the measurement outliers and the abnormal prior information, but also can balance the contribution between the updated parameters and the measurements in accordance with their discrepancies magnitude. But when there are outliers in the observations and the prior information, the robust sequential adjustment can also get a credible result of the parameters.

References Andrews, D.F. (1974) A Robust Method for multiple Linear Regression, Technometrics, 16: 523–531. Hampel, F.R., Ronchetti, E.M., Rousseeuw, P.J., Stahel, W.A. (1986) Robust Statistics: the Approach Based on Influence Functions, Wiley, New York. He H, Yang Y (1998) Robust Estimation for Sequential Adjustment (in Chinese), Engineering of Surveying and Mapping, Vol.7, No.1, 36–40. Huber, P.J. (1964) Robust Estimation of a Location Parameter, Annals of Mathematical Statistics, 35: 73–101. Huber, P.J. (1981), Robust Statistics, Wiley, New York. Jazwinski, A.H. (1970) Stochastic Processes and Filtering Theory. New York: Academic press. Koch, K.R., Yang Y (1998) Robust Kalman Filter for Rank Deficient Observation Model. Journal of Geodesy, 72(8): 436–441. Masreliez, C.J., Martin, R.D. (1977) Robust Bayesian Estimation for the Linear Model and Robustifying the Kalman Filter. IEEE Trans, Automat, Contr, Vol. AC-22. Mikhail, E.M. (1976), Observations and Least Squares, New York. Ou J, Chai Y, Yuan Y (2004) Adaptive Filtering for Kinematic Positioning by Selection of the Parameter Weights (in Chinese). Progress in Geodesy and Geodynamics[C]: 816–823. Yang Y (1991) Robust Bayesian Estimation, Bulletin Geodesique, 65(3): 145–150. Yang Y (1993) The Theory and Application of Robust Estimation (in Chinese). Beijing: Bayi Publication House. Yang Y, He H, Xu G (2001) Adaptively Robust Filtering for Kinematic Geodetic Positioning. Journal of Geodesy, 75(2): 109–116. Yang Y, Song L, Xu T (2002) Robust estimator for correlated observations based on bifactor equivalent weights, Journal of Geodesy, 76(6–7): 353–358.

Application of Unscented Kalman Filter in Nonlinear Geodetic Problems D. Zhao School of Geodesy and Geomatics, Wuhan University, Hubei Province, P.R. China Department of Geodesy and Navigation Engineering, Zhengzhou Institute of Surveying and Mapping, Longhai Middle Road, No. 66#, Zhengzhou, Henan Province, P.R. China Z. Cai Global Information Application and Development Center of Beijing, Beijing 100094, P.R. China C. Zhang Department of Geodesy and Navigation Engineering, Zhengzhou Institute of Surveying and Mapping, Longhai Middle Road, No. 66#, Zhengzhou, Henan Province, P.R. China

Abstract. The Extended Kalman Filter (EKF) has been one of the most widely used methods for nonlinear estimation. In recent several decades people have realized that there are a lot of constraints in applications of the EKF for its hard implementation and intractability. In this paper an alternative estimation method is proposed, which takes advantage of the Unscented Transform thus approximating the true mean and variance more accurately. The method can be applied to non-linear systems without the linearization process necessary for the EKF, and it does not demand a Gaussian distribution of noise and its ease of implementation and more accurate estimation features enable it to demonstrate its good performance. Numerical experiments on satellite orbit determination and deformation data analysis show that the method is more effective than EKF in nonlinear problems. Keywords. EKF, unscented transform, satellite orbit simulation

1 Introduction In practical engineering problems, whenever the state of a system must be estimated from noisy sensor information, some kind of state estimator is employed to fuse data from different sensors together to produce an accurate estimate of the true system state. For this purpose, statistical inference method based on Bayes’ Theorem, or Bayesian (re-)sampling method has always played an important role in filtering for many years. In (Koch, 1990) the KalmanBucy filter was derived by recursively applying the Bayes’ Theorem. In (Steven, 1993) the Wiener Filter 252

as well as its relation with the Kalman Filter were discussed. In many nonlinear engineering problems, the EKF is the most widely used method. The EKF applies the Kalman filter to nonlinear systems by simply linearizing all the nonlinear models so that the traditional Kalman filter can be used. However, as people have found, the use of the EKF in practice has two well-known drawbacks. One is that linearization can produce highly unstable filters if the assumptions of local linearity is violated. The other is that the derivation of the Jacobian matrices is nontrivial in most applications and often leads to significant implementation difficulties. For the study of nonlinear filtering, Xu (1999, 2003) presented a completely statistical analysis of bias and higher-order accuracy of nonlinear filters, including the EKF. In addition, (Simon, 1996; Montenbruck and Gill 2000; Shi, 2001) made a full study on numerically computing nonlinear filters. In this paper an alternative estimator is introduced which yields performance equivalent to the Kalman filter for linear systems, yet generalizes elegantly to nonlinear systems without the linearization steps required by the EKF. The method uses the Unscented Transform (UT) proposed by Julier et al. (1995) and Julier and Uhlmann (1997) to produce a set of points, which are called Sigma Points, and are then propagated to accurately approximate the true statistical properties of random variables. It must be pointed out that the method used here does not try to overcome all the problems related with nonlinear filtering, for example, the bias issue, but simply aims to improve the approximation accuracy of random variables after nonlinear transformation, and applications of the method in solving nonlinear geodetic problems

Application of Unscented Kalman Filter in Nonlinear Geodetic Problems

are presented. Here ‘Sigma Point’ is used instead of the word ‘unscented’ for a better understanding.

2 Unscented Kalman Filter 2.1 Sigma Point Transform Sigma Point Transform is actually the Unscented Transform proposed by Julier and Uhlmann (1996) to calculate statistical properties of random variables after nonlinear propagation. Consider problem: propagate a random variable x, whose dimension is dx , through a nonlinear function y = f (x). Assume x has mean x¯ and covariance Px . To compute the statistical properties of y, the following formulas are used χ0 = x¯ √ χi = x¯ + √(dx + λ) Px i i = 1, . . . dx χi = x¯ − (dx + λ) Px i−d i = dx + 1, . . . 2dx x (1) in which χ is a matrix consisting of 2dx + 1 vectors χi (called Sigma Point), with λ = α 2 (dx + κ) − d x being a scaling parameter, and constant α determines the extension of these vectors around x¯ and usually set 1e − 2 ≤ α ≤ 1 (Julier and Uhlmann, 1997). κ is another scaling factor, and √ often set as zero for state estimation problems. (dx + λ) Px i is the i-th column of the square root of the matrix. The acquired Sigma Points are transformed or propagated through the nonlinear function f (·) Yi = f (χi ),

i = 0, 1, . . . 2dx

(2)

to obtain the transformed vectors Yi . And then the mean and covariance of y are approximated using the weighted mean and covariance of the transformed vectors 2dx (m) y¯ ≈ ωi Yi (3) i=0

Py ≈

2dx

(c)

ωi

(Yi − y¯ )(Yi − y¯ )T

(4)

i=0

in which weights ωi are given as ω0(m) = λ (d x + λ) (c) ω0 = λ (d x + λ) + 1 − α 2 + β (5) ωi(m) = ωi(c) = 1 {2 (dx + λ)} i = 1, 2, . . . 2d x And the parameter β contains the a priori information of x(for Gaussian Distribution, β = 2).

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Further analysis (Wan and Nelson, 2001) shows that the statistical properties of random variables after nonlinear function propagation can be approximated with higher accuracy using the Sigma Point Transform than that using the linearization of the EKF. 2.2 Unscented Kalman Filtering For the basic framework of the EKF which involves the estimation of the state of a discrete-time nonlinear dynamic system x k+1 = F (x k , u k , vk )

(6)

yk = H (x k , n k )

(7)

where x k represents the unobserved state of the system and yk is the only observed signal of the system. The outer input u k is known and not a random variable. The process noise vk drives the dynamic system, and the observation noise is given by n k . The system dynamic model F and H are assumed known. In state estimation, the EKF is the standard method of choice to achieve a recursive maximum likelihood estimation of the state x k

xˆ k = (predicted x k )+ K k · yk − (predicted yk ) (8) The Unscented Kalman Filter (UKF) is a straightforward extension of the Sigma Point Transform to the recursive estimation (8), where the random variable has been redefined as the concatenation of the

T original state and noise variables x ka = x kT vkT n kT . The sigma point selection scheme (1) is applied to this new augmented state vector to calculate the corresponding sigma matrix, χka . And the UKF equations are given in Table 1. It must be pointed out that the implementation of the algorithm needs no explicit calculation of Jacobians or Hessians, which are always non-trivial burden of computation. And furthermore, the overall workload of computations is the same order as that of the EKF. As there often appears the special case where the process and measurement noise are additive, the computational complexity of the UKF can be reduced. In such a case, the system state vector need not be augmented with the noise vector, which reduces the dimension of the sigma pints as well as the total number of sigma point used. The covariance matrices of the noise sources are then incorporated into the state covariance using a simple additive procedure.

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Table 1. Unscented kalman filter (zero mean noise case) Initialization xˆ0 = E[x

0] P0 = E (x0 − xˆ0 )(x0 − xˆ0 )T for k ∈ {1, . . . , ∞} Compute Sigma Points:

χk−1 = xˆk−1 xˆk−1 + (dx + λ)Pk−1 xˆk−1

− (dx + λ)Pk−1

Prediction:

∗ = F χk−1 , u k−1 χk|k−1 xˆk− = Pk− =

2d x i=0

∗ Wi(m) χi,k|k−1

2d x i=0

T ∗ ∗ Wi(c) χi,k|k−1 − xˆk− χi,k|k−1 − xˆk− + R v

χk|k−1 = xˆk− xˆk− + (dx + λ) Pk− xˆk− − (dx + λ) Pk−

Yk|k−1 = H χk|k−1 yˆk− =

2d x i=0

(m)

Wi

Yi,k|k−1

Correction: Py˜k y˜k = Px k yk =

2d x i=0 2d x i=0

T Wi(c) Yi,k|k−1 − yˆ k− Yi,k|k−1 − yˆk− + R n

T Wi(c) χi,k|k−1 − xˆk− Yi,k|k−1 − yˆk−

K k = Px k yk Py˜−1 k y˜k xˆk = xˆk− + K k yk − yˆk− Pk = Pk− − K k Py˜k y˜k K kT where λ is the composite scaling factor; dx is the dimension of the state vector; R v is the process noise covariance matrix; R n is the measurement noise covariance matrix; Wi is weight as calculated in equation (5).

system and TT is the time reference datum, the statespace model of satellite dynamics is as follows state vector r (t) y (t) = v(t) dynamic model v(t) + w (t) y˙ (t) = a0 (t) + a J2 (t) + a D RG (t) state prediction − yk+1 = yk+ +

tk

v(t) a0 (t) + a J2 (t) + a D RG (t)

dt + wk

where r (t) is position vector; v(t) is velocity vector; a0 (t) is the Earth’s central body gravitational acceleration; a J2 (t) is the non-spherical gravitational perturbation acceleration of J2 ; a D RG (t) is atmospheric resistance; w (t) is mechanical noise; wk is state noise; As observation is made under the topocentric coordinate system, the satellite coordinates of J2000 should be converted into those under horizontal coordinate system. Disregarding minute corrections such as aberration, atmospheric refraction, etc, the observation equations are set up, including azimuth A, elevation angle E and ρ with the horizontal ⎤ ⎡ distance sE coordinates s = ⎣ s N ⎦ sZ

sE A = arctan + vA sN ⎛ ⎞ s Z ⎠ + vE E = arctan ⎝ 2 s E + s N2

3 Two Numerical Experiments Example 1. Considering a common mode of orbit determination of a LEO (Low Earth Orbit) satellite, observations are horizontal angle, elevation angle and distance. To be simple, only the J2 perturbation of the Earth central body and atmospheric resistance perturbation are taken into account. The true ephemeris is acquired through the DE (Montenbruck and Gill, 2000) algorithm, and the observations are acquired through simulation based on the true ephemeris and the known coordinates of observation station. Provided that J2000 is the spatial reference

tk+1

ρ=

s E2 + s N2 + s 2Z + vρ

in which s E −coordinate component along eastern direction; s N −coordinate component along northern direction;


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s Z −coordinate component along zenith direction; v A −noise of the observed azimuth; v E −noise of the observed elevation angle; vρ −noise of the observed distance; The accurate estimation that is assumed as the true value is ⎤ ⎡ −6345.000e3 ⎢ −3723.000e3 ⎥ ⎥ ⎢ ⎢ −580.000e3 ⎥ ⎥ and ⎢ y0 = ⎢ ⎥ ⎢ +2.169000e3 ⎥ ⎣ −9.266000e3 ⎦ −1.079000e3 ⎡ ⎢ ⎢ ⎢ P0 = ⎢ ⎢ ⎢ ⎣

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 1e − 6 0 0

0 0 0 0 1e − 6 0

The initial value of filtering is ⎡ −6345.000e3 + 100 ⎢ −3723.000e3 − 100 ⎢ ⎢ −580.000e3 + 100 + y0 = ⎢ ⎢ +2.169000e3 − 0.1 ⎢ ⎣ −9.266000e3 + 0.1 −1.079000e3 − 0.1

0 0 0 0 0 1e − 6

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

and ⎡ P0+

⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎣

100 0 0 0 0 0 0 100 0 0 0 0 0 0 100 0 0 0 0 0 0 0.01 0 0 0 0 0 0 0.01 0 0 0 0 0 0 0.01

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

Fig. 1. Position deviations of UKF and EKF.

period T, which is 38048 seconds. And the filtering results with UKF and EKF for 10 periods are given in Figures 3 and 4. Figures 3 and 4 show that EKF demonstrates obvious deviation and divergence, whereas UKF can lead to satisfactory results while maintaining certain accuracy. Example 2. For the prediction and filtering problem in deformation monitoring data processing, we made numerical experiments based on descriptions in (Tor, 2002, 2003). According to (Tor, 2003), Kalman filter is successful in weeding out the sudden surge in the readings even without incorporating a smoothing function and has robustness in avoiding the use of spurious observations. However, in our data processing, our main aim is to test what a performance improvement the UKF method could gain compared to the EKF method when either system dynamic equation or observation equation is or both are nonlinear, so we just use simulated data of height

The initial epoch of observation is 00h:00m:00s, Mar.1, UTC1999. The frequency is 1Hz, and the observation errors are σ A = 0.01◦ × cos(E),

σ E = 0.01◦,

σρ = 1m.

For 100 sets of observations, the filtering results of UKF and EKF are shown in Figure 1. As is shown by Figures 1 and 2, the accuracy of UKF is basically equivalent to that of EKF, which is due to the weak nonlinearity for short interval between observation epochs. To test the performance under strong nonlinearity situation, the observation interval is set to be one

Fig. 2. Velocity deviations of UKF and EKF.

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methods. Figures 5 and 6 shows a sub-segment of the estimates generated by the EKF and the UKF respectively, and Figure 7 shows the difference of estimates between the EKF and the UKF. The superior performance of the UKF is clearly visible.

Height Displacement (mm)

4

displacements for the experiment, and we simplify the state vector as much as possible. The simulated data is a non-periodic and non-convergent time series acquired using a nonlinear auto regression model. By adding Gaussian white noise to this time series a noised observation series can be obtained yk = x k + n k

(9)

The state-space representation is the following state transition equation combined with (9) x k = f (x k−1 , vk−1 )

(10)

where vk−1 is the model parameter. Note that in equation (10) there is no outer input. In the estimation problem, the noisy time series yk is the only observed input to the EKF and UKF

0

–2

–4 clean noisy ekf estimate

–6

–8

0

100 200 300 400 500 600 700 800 900 1000

time

Fig. 5. ekf : Mean square error (MSE) of estimate : 0.3667. 4

Height Displacement (mm)

Fig. 3. Position deviations of UKF and EKF.

2

clean noisy ukf estimate

2 0 –2 –4 –6 –8

0

100 200 300 400 500 600 700 800 900 1000

time

Fig. 6. ukf: Mean square error (MSE) of estimate : 0.0947.

Estimation Difference (mm)

7 ekf ukf

6 5 4 3 2 1 0 0

100 200 300 400 500 600 700 800 900 1000

time

Fig. 4. Velocity deviations of UKF and EKF.

Fig. 7. Estimation difference between ekf and spkf.


4 Conclusion This paper shows that better performance can be acquired using UKF than EKF for nonlinear applications. And the Unscented Kalman Filter has two prominent advantages. One is that it can predict the state of a nonlinear dynamic system with accuracy up to the third order in approximating the true statistical properties of random variables. The other advantage is that it is easy to perform as no analytical calculation of the Jacobian or Hessian matrices of nonlinear system is needed. Comparisons of UKF and EKF in simulation tests demonstrate the superior performance of UKF.

Acknowledgements The work of this paper is funded by the following grants: Special Foundation for the Author of Chinese Excellent Doctoral Degree Dissertation, No. 200344; China National Science Foundation, No. 40774031; Open Foundation of the Dynamic Geodesy Laboratory of the Institute of Geodesy and Geophysics (DGLIGG) of China Science Academy, No. L06-06

References Feng, K. (1978). Methods of Numerical Computation. Publishing House of National Defense Industry, Beijing. Julier, S. and Uhlmann J.K. (1996). A General Method for Approximating Nonlinear Transformations of Probability

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Distributions. Technical report, RRG, Dept. of Engineering Science, University of Oxford. Julier, S., Uhlmann J.K. (1997). A New Extension of the Kalman Filter to Nonlinear Systems. In Proc of AeroSense: The 11th International Symposium on Aerospace/Defence Sensing, Simulation and Controls. Julier, S., Uhlmann, J.K., Durrant-Whyte H. (1995). A new approach for filtering nonlinear systems. In Proceedings of the American Control Conference, pp. 1,628–1,632. Koch, K.P. (1990). Bayesian Inference with Geodetic Applications. Berlin Heidelberg: Springer-Verlag, pp. 92–98. Montenbruck, O., and Gill, E. (2000). Satellite orbits: Models, Methods and Applications. Springer Verlag New York Inc. Shi, Z.K. (2001). Computation Methods of Optimal Estimation. Beijing: Science Publishing House. Simon, H. (1996). Adaptive Filter Theory. Verlag: Prentice Hall, 3rd edition. Steven, M.K. (1993). Fundamentals of Statistical Signal Processing: Estimation Theory (in Chinese). Publishing House of Electronics Industry: Beijing, pp. 301–382. Tor, Y.K. (2002). L1, L2, Kalman Filter and Time Series Analysis in deformation Analysis, FIG XXII International Congress, Washington, D.C. USA. Tor, Y.K. (2003). Application of kalman filter in real-time deformation monitoring using surveying robot, Surveying Magazine: civil engineering research, January, pp. 92–95. Wan E.A. and Nelson A.T. (2001). Kalman Filtering and Neural Networks, chap. Dual EKF Methods, Wiley Publishing, Eds. Simon Haykin. Xu, P.L. (1999). Biases and accuracy of, and an alternative to, discrete nonlinear filters. Journal of Geodesy, Vol. 73, pp. 35–46. Xu, P.L. (2003). Nonlinear filtering of continuous systems: foundational problems and new results Journal of Geodesy, Vol. 77 pp. 247–256.

Order Statistics Filtering for Detecting Outliers in Depth Data along a Sounding Line M. Li, Y.C. Liu Department of Hydrography and Cartography, Dalian Naval Academy, Dalian 116018, P.R. China; Geomatics and Applications Laboratory, Liaoning Technical University, Fuxin 123000, P.R. China; Institute of Surveying and Mapping, Information Engineering University, Zhenzhou 450052, P.R. China Z. Lv Institute of Surveying and Mapping, Information Engineering University, Zhenzhou 450052, P.R. China J. Bao Department of Hydrography and Cartography, Dalian Naval Academy, Dalian 116018, P.R. China Abstract. According to the special requirements in filtering depth data, the models based on order statistics filter are presented to detect outliers in depths along a sounding line, in which, the key problem is to distinguish the outliers from good depths by the value of the differences between a would-be-filtered point and its neighboring points. At the end, the method has been tested using observed data. The results show that the method could remove gross errors as well as zero, and negative sounding values, and preserve consecutive (more than 3) false echoes. And it could help the surveyors find and remove outliers easily and effectively. Keywords. Order statistics filter, sounding outliers, the principle of preferring shallower to deeper (PPSD)

1 Introduction In marine sounding, the sounding data is often affected by random errors, systematic errors and outliers. The outliers may be caused by the unsteady signal of transducer, the reverberation of shallow water, or floats (such as fishes and seaweed) in sound-ray direction. Generally, the outliers can be classified into two types; one type includes isolated points, such as zero or negative data caused by unsteady sonar signal, easy to be recognized, usually regarded as blunders. The other includes consecutive outliers, such as false echoes from floats (fishes or seaweeds) or shallower data from protrudents of sea bottom. In a general way, the consecutive outliers need manual recognition based on the echogram of sonar to judge further if the outliers are due to false echoes or shallower data from the protrudents of sea bottom. 258

Generally, after preprocessed by depth-threshold filtering built in the echo sounder (MacDonald (1984)) and affected by the sounder’s beamwidth (Chen and Liu (1997)), the sounding data is the shallowest in the ensonified zone, on the principle of preferring shallower to deeper (PPSD). The PPSD is mainly taken to meet the requirements for the safety of navigation at sea. In fact, by the PPSD, the deeper soundings are more reliable than the shallower. This will cause the outliers incline to be shallower data. So we will pay main attention to the shallower outliers in this paper. In practice, in order to ‘clean’ the sounding data by the PPSD, there are some concrete requirements to meet: (1) to preserve good sounding data, and to remove zero, negative and abruptly-changed sounding data; (2) to preserve consecutive outliers as doubtful data to verify further while the number of consecutive outlier is more than three; (3) to preserve the shallowest and deepest sounding data, of which the shallowest points is important for the safety of vessel at sea. Because the outliers in soundings will deform or even distort the results of sea-bottom from soundings, they must be removed from the soundings in order to get the ‘clean’ data. Nowadays, the common methods used to filter the soundings could be classified into two types: one is to fit a trend surface to help judge if a big isolated sounding value is an outlier or a good one, the other is to edit soundings by a surveyor with his/her hands in a visual way, see for example Calder and Mayer (2001), Lenk (2003) and Zoraster and Bayer (1992). However, the trend surface-fitted methods need complicated algorithms to construct a trend surface. The manual way to edit soundings is a reliable but time-consuming operation in ‘cleaning’ sounding data. In recent years, the filtering methods based on the linear theory often

Order Statistics Filtering for Detecting Outliers in Depth Data along a Sounding Line

are used in analyzing time-serial data, for example, smoothing data series or detecting outliers from data series, in which a Kalman filter is often used as a basic tool. However, the filtering methods based on the linear theory will come into troubles with the special requirements above. The requirements lie in two following aspects: on one hand, the filter is expected to remove pikes, zeros, and negative sounding values. On the other hand, the filter is expected to preserve shallowest soundings while considering the safety of vessels at sea and to preserve consecutive outliers (maybe protuberance of sea bottom) to confirm further. The two aspects requirements are self-contradictory and make it difficult or even impossible to find a suitable filter in the linear theory. Many method based on order statistics can be found in Gather et al. (1998). Those methods can meet some special requirements for filtering in a combined way. So, we will use order statistics filtering to process the outliers in soundings in this paper.

2 Models of Order Statistics Filtering for Sounding According to the requirements mentioned before for filtering sounding data, the median filter and minimum filter are chosen as basic tools to detect outliers in soundings. 2.1 Median Filtering Model Here, the median filtering model is given as below ˆ d(k) = med{d(k − N), · · · , d(k), · · · , d(k + N)} (1) where d(k) is a sounding data at time k on a survey line, med( r) stands for getting the median of series r. The filtering window W N = {–N, . . ., 1, 0, 1, . . ., N} and its width L = 2N + 1, when ˆ L = 1, we have d(k) = d(k). The median filtering can resist blunders very well. The accuracy of sounding data depends on some factors from the surveying environment, such as depth range, the motion of vessel, which may be different along a sounding line. However, in a short segment of the sounding line, i.e. within the width of the filtering window, the surveying environment could be regarded as unchanged. So it is supposed that the soundings in the filtering window have almost the same variance σ 2 [d(k)] and Gaussian probability density function f . And then the filtered value ˆ ˆ d(k) has its variance ␴2 [d(k)] = 1/[4L · f 2 (0)]] ˆ = (L > 1) (Gather et al. 1998). Especially, σ 2 [d(k)] 2 πσ [d(k)]/10 when L = 5.

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2.2 Minimum Filtering Model Here, the minimum filtering model is given as below ˆ d(k) = min{d(k − N), · · · , d(k − 1), d(k + 1), · · · , d(k + N)} (2) where min( r) stands for getting the minimum of series r. The min filtering could be a good tool in removing zero or false echoes, obtaining shallowest soundings, but the job to obtain the accuracy of the filtered value may be difficult and complicated. Here, ˆ = we give the accuracy of filtering value σ 2 [d(k)] 2 (1−1/π)σ [d(k)] when L = 2, see Wolfram (2005).

3 Determination of Outliers Usually, compared with their neighboring good soundings the outliers changed abruptly and become shallower. According to this, it is easy to recognize outliers from normal data. For the sake of convenience of the discussion later, the three concepts of backward difference, forward difference and for-andaft difference of sounding data should be introduced here. The backward difference of sounding d(k) is defined as Δd L = d(k) − d(k − 1)

(3)

In equation (3), Δd L < 0 when seabottom inclined upwards; if, Δd L > 0 when seabottom downwards; Δd L will be closed to zero while seabottom flat. The forward difference of sounding d(k) is defined as Δd R = d(k) − d(k + 1)

(4)

In equation (4), Δd R > 0 while seabottom inclined upwards; Δd R < 0 while seabottom downwards; Δd R will be closed to zero while seabottom flat. The for-and-aft difference of sounding d(k) is defined as Δd = Δd L + Δd R

(5)

According to equations (3) and (4), if data changes steady, Δd will be closed to zero while seabottom upwards, downwards, or flat. If d(k) changes abruptly Δd may lager or less than zero. So, the sign and value of Δd can be used to judge whether a sounding data is an outlier or not.

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The variance of a filtering window is defined as follows σˆ 2 =

1 2 ¯ [d(k − j ) − d(k)] 2N

(6)

ˆ ˆ − 2), d(k ˆ − 1), d(k), d(k) =med d(k

j ∈W N

¯ where d(k) =

1 2N+1

j ∈W N

d(k − j )

d(k + 1), d(k + 2)}

If Δd > 4σˆ , d(k) could be regarded as a shallow outlier; if –Δd > 4σˆ , d(k) could be regarded as a deep outlier; because a deep outlier do not threaten the safety of vessel at sea, we discuss how to recognize a shallow outlier later.

4 Scheme for Filtering In fact, the sounding data may include random errors, blunders of zero, negative and abruptly-changed data, consecutive outliers, or small gross errors. It is very difficult to meet the requirements above by one kind of filter such as equations (1) or (2). It is needed to make a reasonable filtering scheme for sounding data. If a sounding is regarded as an zero or negative data, its filtered value should be the shallowest depth in its two neighboring data, i.e. if the afterward sounding is still zero or negative, its filtered value should be the faltered value of the forward point; if the afterward sounding is normal, the filter value should be the minimum of the two neighboring sounding in order to preserve shallowest data and remove zero or negative sounding data. Meanwhile, because the filtered value before d(k) is reliable, it can be used to optimize filtering d(k). So, equation (2) could be changed as follows While d(k + 1) > 0, d(k) ≤ 0 ˆ ˆ − 1), d(k + 1)} d(k) = min{d(k

(7)

While d(k + 1) ≤ 0, d(k) ≤ 0 ˆ ˆ − 1) d(k) = d(k

Hence, the width of the median filtering window is chosen as five, i.e. L = 5. If the filtered values before d(k) are used to filter its following soundings, equation (1) can be expressed as

(8)

According to equation (1), its filtered value should be the median of filtering window if a sounding is regarded as an outlier. The median is outlier if there are equal to or more than three outliers in the filtering window when L = 5; meanwhile, the median is shallower sounding while there are less than three outliers in the window when L = 5. So a filter like this can preserve the consecutive outliers when the number of consecutive outliers is more than three and guarantee that the filtered values are the shallowest soundings if there is an outlier at the shallowest point.

(9)

Accordingly, equation (3) could be expressed as ˆ − 1) Δd L = d(k) − d(k

(10)

According to the analysis above, the following filtering scheme could be formed as follows First step: to judge d(k) is zero or negative, if ˆ ˆ − 1); if d(k) ≤ 0 and d(k + 1) ≤ 0, d(k) = d(k ˆ ˆ − 1), d(k) ≤ 0 and d(k + 1) > 0, d(k) = min{d(k d(k + 1)}; if d(k) > 0, then do the second step; Second step: to calculate Δd and σˆ through equations (5) and (6), if –Δd ≥ 4σˆ , d(k) could be ˆ ˆ − regarded as an outlier, and d(k) = med{d(k ˆ 2),d(k − 1),d(k), d(k + 1),d(k + 2)}; if –Δd < 4σˆ , ˆ d(k) is a good sounding and then d(k) = d(k). Third step: to filter the sounding at the start and end of the series. BecauseL = 5, there are two soundings at the start and end of the series respectively to process in a special way. While k = 1, if d(1) ≤ 0, do the first step,; if d(1) > 0, the filtering window can be {d(1), d(2), d(3)} and its standard deviation is σˆ . if –Δd R ≥ 4σˆ , d(1) is an outlier and its filtered value ˆ should be d(1) = med{d(1), d(2), d(3)}; if Δd R < ˆ 4σˆ , d(1) = d(1). While k = 2, if d(2) ≤ 0, do the first step; if d(2) > 0, the filtering window can be ˆ {d(1), d(2), d(3)}, and its standard deviation is σˆ , ˆ if –Δd ≥ 4σˆ , d(2) is an outlier and d(2) = ˆ ˆ med{d(1), d(2), d(3)}, if –Δd < 4σˆ , d(2) = d(2). While k = M–1, and k = M, do as k = 2 and k = 1, where M is the number of the series. After the three steps above, the filtering scheme can detect and remove big blunders in sounding data. It is needed to choose a smaller threshold, i.e. –Δd < 3σˆ , –Δd < 2.5σˆ , or even-Δd < 1.5σˆ , when necessary to remove small gross errors, and redo the three steps above.

5 Examples The study data is from the sounding data along a sounding line in East China Sea, and collected by single-beam echo sounder typed SDH-13D, March 2, 2000. There are 619 soundings alone the line, and

Order Statistics Filtering for Detecting Outliers in Depth Data along a Sounding Line

Fig. 1. Echogram flow (scanned and zoomed out).

the sample interval of time is 1 s. Due to an unsteady sonar signal, there are many zeros in sounding data, and there are false echoes at the middle of the line, proved to be caused by seaweeds later. The sonar echogram flow is shown as Figure 1 (scanned and zoomed out), the digital profile of soundings shown as Figure 2. In order to verify the efficiency of the order statistic filtering scheme, we compare it with manual editing method. Manual editing method is an operation for surveyors to edit and correct the apparent big gross errors by hand according to the echogram flow in actual practice. It is reliable but time-consuming. Scheme 1: manual editing method According to echogram Figure 1, a skilled surveyor spent 10 min in ‘cleaning’ the sounding data, and got the result shown as Figure 3. Scheme 2: with the order statistic filtering presented in this paper, using Matlab, the result was obtained and shown in Figure 4.

Fig. 2. Digital profile of soundings.

Fig. 3. The filtered data by manual editing method.

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The Figure 4 preserves more than three consecutive false echoes for surveyors to recognize further. The differences between the sounding data and their filtered values are shown in Figure 5. From Figures 3–5, the order statistic filtering can detect and remove zero, negative or some isolated blunders effectively and preserve consecutive false echoes, which need to be judge with echogram like Figure 1. Based on Figure 4, the surveyor can edit and correct the soundings and get the filtered result as Figure 3 within 1 min according to Figure 1. It is shown that the order statistic filtering, with a little job of manual editing, can detect and remove outliers fast and effectively.

6 Conclusions The order statistic filtering method presented in this paper can meet the special requirements of filtering sounding data very well. The conclusions can be summarized as follows (1) The method can detect and remove zero, negative and isolated blunders in sounding data fast and effectively. (2) The method can preserve the consecutive outliers and help a surveyor with cleaning the sounding data by comparing to the echogram flow while considering the quality of sounding data. It should be pointed out that the length of the consecutive outlier to preserve depends on the width of the filtering window in practice. So we can change the width of the filtering window to preserve the different length of the consecutive outlier according to the circumstance of sea bottom.

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Fig. 4. The filtered data by order statistic filtering.

Fig. 5. The differences between sounding data and their filtered values.

Acknowledgement Supported by the National Natural Science Foundation of China (No.40071070, 40671161) and by Open Research Fund Program of the Geomatics & Applications Laboratory, Liaoning Technical University (No.200502) E-mail: [email protected]

References Calder, B. R., Mayer L. A. (2001). Robust Automatic Multibeam Bathymetric Processing. U. S. Hydrographic Conference 2001, see also http://www.thsoa.org/hy01/ Chen, Y. C., Liu, Y. C. (1997). Corrections for the Seabed Distortion Caused by the Angular Beam-width of Echo Sounders. The Hydrographic Journal, No.84: 15–19.

Gather, U., Fried, R., Lanius,V. (1998). Robust Detail-preserving Signal Extraction. see also http://www.sfb475.uni-dortmund.de/berichte/tr54.pdf. Lenk, U. (2003). A Detailed Algorithm to Compute (Adaptive) Triangulations and Potential Indications of Data Quality. The Hydrographical Journal, No.107: 3–10. MacDonald, G. (1984). Computer-Assisted Sounding Selection Techniques. The International Hydrographic Review, LXI(1): 93–109. Wolfram (2005). Technical Support on Probability and Statistics. see also http://mathworld.wolfram.com/ extremevaluedistrubtion.html. Zoraster S., Bayer, S. (1992). Automatic Cartographic Soundings Selections. The International Hydrographic Review, LXIX(1).

Stepwise Solutions to Random Field Prediction Problems M. Reguzzoni Italian National Institute of Oceanography and Applied Geophysics (OGS), c/o Politecnico di Milano, Polo Regionale di Como, Via Valleggio, 11, 22100 Como, Italy N. Tselfes, G. Venuti DIIAR, Politecnico di Milano, Polo Regionale di Como, Via Valleggio, 11, 22100 Como, Italy Abstract. The approximation of functionals of second order random fields from observations is a method widely used in gravity field modelling. This procedure is known as collocation or Wiener – Kolmogorov technique. A drawback of this theory is the need to invert matrices (or solve systems) as large as the number of observations. In order to overcome this difficulty, it is common practice to decimate the data or to average them or (as in the case of this study) to produce more manageable gridded values. Gridding the data has sometimes the great advantage of stabilizing the solution. Once such a procedure is envisaged, several questions arise: how much information is lost in this operation? Is rigorous covariance propagation necessary in order to obtain consistent estimates? How do different approximation methods compare? Such questions find a clear formulation in the paper: some of them (the simplest ones) are answered from the theoretical point of view, while others are investigated numerically. From this study it results that no information is lost if the intermediate grid has the same dimension of the original data, or if the functionals to be predicted can be expressed as linear combinations of the gridded data. In the case of a band-limited signal, these linear relations can be exploited to obtain the final estimates from the gridded values without a second step of collocation. A similar result can be obtained even in the case of non-band limited signal, due to the low pass filtering of the signal, along with the noise, performed by the intermediate gridding. Keywords. Wiener-Kolmogorov principle, local gridding, collocation

1 Introduction The immediate motivation behind this study is the data analysis of the GOCE mission (ESA, 1999) and in particular the space-wise approach (Migliaccio et al., 2004). According to this approach the

geo-potential model is estimated by exploiting the gravitational potential, obtained via the energy conservation method (Jekeli, 1999; Visser et al., 2003), as well as its second derivatives measured by the onboard gradiometer. The numerous data are first filtered in time, then they are projected onto a smaller regular intermediate set (grid) and finally spherical harmonic coefficients are estimated by applying a numerically efficient algorithm. This step-wise approach has been successfully tested in realistic simulated scenarios (Migliaccio et al., 2006). The problem under study is now formulated in general terms: consider the vector y 0 of irregularly distributed data: y 0 = y + v,

(1)

which is equal to the signal y plus uncorrelated noise v. The signal vector is given by: y = G (u) ,

(2)

where the functional vector G (·) acts on a random field u. The value z to be predicted is represented by the functional F: z = F (u) .

(3)

The problem of estimating z will be treated in the context of the Wiener-Kolmogorov (WK) principle (Moritz, 1980). The direct application of this principle faces numerical drawbacks, such as the solution of very large systems and possible singularity problems due to irregular data distribution. A solution to these problems is to perform an intermediate step: from the data y 0 the gridded values wˆ are estimated and then the functional zˆ is derived. However some questions arise: does the hypothesis of almost sufficiency of the step-wise procedure hold, or is there information lost with respect to the direct solution? If yes, how 263

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much information is lost and why? Is rigorous covariance propagation necessary to obtain consistent estimates? Which intermediate functionals w and how large dim w to choose? In the following sections, for some of these questions an analytical answer is given, while for others simple numerical examples are provided.

2 Sufficiency of the Stepwise Estimator The direct estimation of the unknown is given by: y . zˆ = C zy0 C y−1 0 y0 0

(4)

The two-step estimation is given by: wˆ = Cwy0 C y−1 y , 0 y0 0

(5)

zˆˆ = C z wˆ Cw−1 w. ˆ ˆ wˆ

(6)

Lemma 1. If w ↔ y, implying dim w = dim y , and if the covariance is correctly propagated, then no information is lost. This means that the estimates zˆ and zˆˆ coincide. Corollary 1. The optimal estimate zˆ can be obtained without computing the covariance of wˆ : −1 z˜ = C zw Cww wˆ = C z wˆ Cw−1 wˆ = zˆ . ˆ wˆ

(7)

According to lemma 1, if there is a bi-univocal correspondence (1:1) between the observables and the intermediate functionals and if the covariance matrices of equation (6) are derived by rigorous propagation, then the two-step estimate is equivalent to the direct WK estimation and no information is lost. This is the case when first the data is filtered and then the final estimate is computed from these filtered values. Note that, under the above hypothesis, the final optimal (linear) estimate can also be obtained without computing the covariance of w. ˆ The already known covariance of w can be used, thus reducing the computational burden. See the appendix for the proof. Lemma 2. If dim w < dim y , then the estimation error of the step-wise procedure eˆˆ = z − zˆˆ increases with respect to that of the direct solution eˆ = z − zˆ : ␴2 eˆˆ ≥ ␴2 eˆ .

(8)

Corollary 2. If dim w < dim y and if z ∈ span w , i.e. ∃λ : z = λT w, then the projection to a smaller spacedoes any loss of not imply information: zˆˆ = zˆ , σ 2 eˆˆ = σ 2 eˆ . According to lemma 2, if the original data is projected on a smaller space, then the estimation error of the step-wise procedure is generally larger than that of the direct solution. However, it can be easily shown that if the functional z to be predicted is a linear combination of the intermediate values w, then the two-step estimator comes out to be completely sufficient with respect to z. See the appendix for the proof. 2.1 Numerical Example 1 Consider a stochastic process y (t), stationary in −τ t ∈ R 1 , with covariance function C y (τ ) = e , τ = t − t . The observations are defined as the sum of signal plus (white) noise with covariance function Cv (τ ) = σv2 δτ 0 . Signal and noise are assumed uncorrelated. Two observations are taken at t = ±1. Two intermediate estimates at t = ±δ with are made δ ≤ 1 (note that dim w = dim y = 2). The value to be predicted is the process itself (evaluation functional) at t = 0. For any δ, either using the covariance of wˆ or that of w, the two-step estimate has the same error variance as the one-step estimate, as expected from lemma 1. If however the covariance function at the second step is approximated (using Cee , where e = w − w, ˆ as an independent error covariance), the error is higher and depends on δ (see Figure 1). 2.2 Numerical Example 2 The previous example is repeated, this time with only one intermediate estimate at t = δ ≤ 1 (note that now dim w < dim y ). The two-step procedure has an error larger than the direct solution, according

σ 2( z − zˆ) Cww ˆ ˆ ≅ Cww + Cee Cww ˆ ˆ ≅ Cww + diag (Cee) zˆ , zˆˆ 0

1

δ

Fig. 1. 1-step estimate zˆ and 2-step estimate zˆˆ error variance (dotted line). 2-step estimate error variance by approximating the intermediate step covariance (solid lines).

Stepwise Solutions to Random Field Prediction Problems

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where σ02 = 10 and a = 0.1. The maximum degree N is 10. The process is stationary in time, with a periodic covariance function:

σ 2( z − zˆ ) zˆˆ zˆ

0

E {y (t) y (t + ␶)} = C y (␶) =

δ

1

to lemma 2. For δ = 0, z = w then zˆ = zˆˆ ; this confirms corollary 2, being z a trivial case of linear combination of w (see Figure 2).

3 Model Coefficients Estimation Lemmas 1 and 2 are valid for infinite dimensional spaces. Consider a finite dimensional model, where the signal y is a linear combination of basis functions (matrix A) with stochastic coefficients x: (9)

Assume to project the observations, by some operator R, on a smaller intermediate vector (dim w < dim y ) that is also expressed by the coefficients x using the matrix B: w = Ry = R Ax = Bx.

(10)

If B is of full rank, with some algebra, it can be easily proved that the step-wise solution coincides with the direct solution: ˆˆ y = C x wˆ Cw−1 wˆ = x. xˆ = C x x A T C y−1 ˆ wˆ 0 y0 0

(11)

−1 T B : xˆ = H w, ˆ In other words, ∃H = B T B so that by simply applying this linear operator to the intermediate estimate, the optimal (linear) solution is obtained without undertaking the computational burden of a second collocation step. 3.1 Numerical Example in 1D Consider a process y (t) in t ∈ R 1 , periodic in [0,1): y0 (t) =

N

␴2n cos (2␲n␶) .

n=0

Fig. 2. 1-step estimate zˆ (dotted line) and 2-step estimate zˆˆ (solid line) error variance as a function of δ.

y 0 = y + v = Ax + v.

N

x nc cos (2␲nt) + x ns sin (2␲nt) + v (t) .

(14)

The noise v is white (but it could be also coloured) and uncorrelated with the signal, with variance ␴2v = 0.25␴20 and covariance function: E {v (t) v (t + ␶)} = ␴2v ␦␶0.

(15)

Gridded values wˆ are computed by collocation, using all the data and with dim w > 2N + 1. The coefficients are then estimated both via collocation and Discrete Fourier Transform (DFT). The two solutions are found completely equivalent. In fact, since the grid sampling is above the Nyquist limit, the matrix H (mentioned before) is just the DFT operator. This solution gives rise to the optimal result without covariance propagation, thus reducing the numerical complexity. 3.2 Local Gridding The gridding itself does not imply loss of information. The problem is that, in some cases, not all the data can be jointly used for the gridding (for GOCE this would be numerically impossible), but each gridded value is estimated from a local patch of data (local gridding). This causes loss of information, depending on the size of the local patches used. This size is typically calibrated on the covariance function of the process, even though, to be more precise, it should be calibrated on the basis of the weighting kernel of the corresponding Wiener filter. Local gridding is applied to the case of the example 3.1. It is clear (see Figure 3) that increasing the dimension of the window around each gridded point (here up to the second zero of the Wiener filter kernel) the solution with local gridding tends to the optimal one, i.e. the one with global gridding. So a proper patch size for the local gridding and then DFT give rise to an almost sufficient estimator.

n=0

(12) The coefficients x nc , x ns of degree n are random variables, independent of one another, with zero mean and known variance: ␴2n = ␴20 e−an , 2

(13)

3.3 Numerical Example in 1D with Aliasing Example 3.1 is examined again: high degree coefficients (n > N) are added (equation 12), but the grid resolution is not increased (dim w = 2N + 1):

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0.3

0.2

0.1

0 0

1

2

3

4

5 6 degree

7

8

9

10

Fig. 3. Estimated coefficients error degree standard deviation from gridding and DFT: the bottom line is the optimal solution using a global gridding, the middle and top lines are solutions with local gridding and window dimension Δ = 0.2 and Δ = 0.1, respectively.

y0a (t) =y0 (t) +

L

x nc cos (2␲nt)

n=N+1

+x ns

sin (2␲nt).

(16)

In this example there is aliasing (L = 20). However, at high frequencies the signal power is smaller than that of the noise. This is typical in a real case. The corresponding Wiener filter is practically zero at those high frequencies and the signal is almost filtered out. In other words, the first step of collocation gridding acts as a low-pass filter, controlling also the aliasing effect and allowing for the subsequent application of the DFT method. 3.4 Analysis on the Sphere The numerical example in 1D is now generalized on the sphere. An isotropic harmonic and homogenous random field is considered: y=

n N

c cos (m␭) P¯nm (cos ␪) x nm

n=0 m=0 s +x nm sin (m␭) ,

(17)

where P¯nm are the normalized Legendre functions, θ and λ are the spherical coordinates. The maximum c , x s of degree degree N is 5. The coefficients x nm nm n and order m are considered independent random variables with zero mean and variance ␴2n given by equation (13) with ␴20 = 3 and a = 0.05. The covariance function of the signal, depending on the spherical distance ψ, is: C y (␺) =

N n=0

(2n + 1) ␴2n Pn (cos ␺),

(18)

where Pn are the Legendre polynomials of degree n. Noise, uncorrelated with the signal, is added, with σv2 = 0.5σ02. This is a situation similar to the GOCE data analysis. After the gridding via collocation, in the space-wise approach, three different harmonic analysis methods are considered: (1) Fast Spherical Collocation (FSC) (Sansò and Tscherning, 2003), which is a numerically efficient collocation. It requires that the noise covariance depends only on the longitude difference. This approximation is unrealistic for the case of GOCE. (2) Numerical integration, which is an approximation of the quadrature formula, based on the orthogonality of the spherical harmonic functions (Colombo, 1981; Migliaccio and Sansò, 1989). Its main disadvantage is the introduction of discretization error. (3) Two dimensional spherical Fourier analysis (Driscoll and Healy, 1994). This is an exact relation between the unknown parameters and the spherical grid. If local gridding is applied, the spherical Fourier solution tends to the optimal one by increasing the radius of the spherical cap around each gridded point (see Figure 4). This result is consistent with the one obtained in the previous 1D example. Here the cap radius of 50 degrees corresponds more or less to the first zero of the covariance function. Note that in a realistic GOCE scenario the correlation length of the second radial derivatives is of the order of some degrees. Therefore the application of the local gridding is feasible. Note also that, in order to reach the optimal solution, the integration analysis requires increasing the grid resolution to reduce the discretization error.

error degree standard deviation

error degree standard deviation

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1.35

1.25

1.15

1.05

0

1

2

3

4

5

degree

Fig. 4. Estimated coefficients error degree standard deviation: the bottom line is the optimal solution; the middle down, middle up and top lines are solutions with local gridding with cap radius Δψ = 20◦ , 40◦ and 50◦ , respectively.

Stepwise Solutions to Random Field Prediction Problems

As for the choice of the intermediate functional, in the GOCE data analysis, if a global gridding were performed any functional could be predicted bearing the same information. However, since a local gridding has to be performed with a consequent loss of information, different functionals may lead to coefficients estimates of different accuracy. Further study should be conducted.

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Finally equation (5) is substituted into (6) and using the five previous equations it is found: −1 C C −1 y zˆˆ = C z wˆ Cw ˆ wˆ wy y0 y0 0 −1 −1 = C zy C −1 Cwy C −1 y0 y0 C yw C wy C y0 y0 C yw y0 y0 y 0 −1 T N C C −1 C N T = C zy C −1 N C yy C −1 yy y0 y0 yy y0 y0 C yy N y0 y0 y 0 T T = C zy C −1 y0 y0 C yy N N

= C zy C −1 y0 y0 y 0 = zˆ .

4 Conclusions A step-wise procedure, passing through an estimate of regularly distributed data, is commonly used to overcome some numerical problems of the direct WK functional prediction. The sufficiency of the step-wise procedure has been analytically derived when some conditions are satisfied. In these cases, full covariance propagation is not required to get the optimal solution. The loss of information in the step-wise procedure is mainly related to the local gridding. This has been studied numerically (in 1D and 2D on the sphere), showing that the dimension of the local patch of data used around the prediction point is critical for obtaining an almost optimal estimate. An exact DFT algorithm has been used for the second step, instead of collocation, reducing the computational burden.

5 Appendix Proof of Lemma 1 A vector of irregularly distributed observations is considered (equation 1). If w ↔ y, then ∃N: w = N y, (19)

−1

−1 −1 N C C −1 y C −1 yy y0 y0 0 yy C y0 y0 C yy N

The last equality is derived from equation (4). Thus lemma 1 is proved. Proof of Corollary 1 Starting from equation (6) and substituting equations (23) and (24), it turns out that: wˆ zˆˆ = C z wˆ Cw−1 ˆ wˆ

−1 −1 C = C zy C y−1 C C C wˆ yw wy yw y0 y0 0 y0 = C zy C y−1 C NT NT 0 y0 yy

−1

−1 −1 −1 C yy C y0 y0 C yy N wˆ −1 −1 −1 = C zy C yy N wˆ = C zy NC yy wˆ −1 −1 = C zy N T NC yy N T wˆ = C zw Cww wˆ = z˜ .

Since from lemma 1 it holds that zˆ = zˆˆ , the corollary (equation 7) is proved. Proof of Lemma 2 The two-step estimate from equations (5), (6) and (24) is: zˆˆ = C zy C y−1 C C −1 C C −1 y . ˆ wˆ wy y0 y0 0 0 y0 yw w

(25)

where N is invertible. This also means that: Therefore the estimation error variance is: C yw = C yy N T .

(20)

The signal and noise are considered uncorrelated so: Cwy0 = Cwy ,

(21)

C zy0 = C zy .

(22)

The covariance matrix of the intermediate estimate wˆ is derived via propagation from equation (5): C T = Cwy C y−1 C . Cwˆ wˆ = Cwy C y−1 0 y0 wy 0 y0 yw

(23)

Taking the mathematical expectation of the product of z and wˆ T and using equation (5), it is found: C z wˆ = C zy C y−1 C . 0 y0 yw

(24)

2 E eˆˆ2 = E z − zˆˆ = σ 2 (z) − 2C zy C y−1 C C −1 C C −1 C ˆ wˆ wy y0 y0 yz 0 y0 yw w

+C zy C y−1 C C −1 C C −1 C C −1 C C −1 ˆ wˆ wy y0 y0 y0 y0 y0 y0 yw wˆ wˆ 0 y0 yw w ×Cwy C y−1 C . 0 y0 yz

Then it holds with equation (23) that: Cwy C y−1 C C −1 C C −1 = I. ˆ wˆ 0 y0 y0 y0 y0 y0 yw w

(26)

Therefore: E eˆˆ2 = ␴2 (z) − C zy C y−1 C C −1 C C −1 C . ˆ wˆ wy y0 y0 yz 0 y0 yw w (27)

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Similarly, the one-step error from equation (4) is: E eˆ2 = ␴2 (z) − C zy C y−1 C . (28) 0 y0 yz

Therefore equation (32) becomes:

The difference between the two errors is: E eˆˆ 2 − E eˆ2 = −1 = C zy C y−1 − C C C C C C y−1 y y yw wy y 0 0 w ˆ w ˆ 0 0 0 y0 yz −1 −1 C = C zy C y−1 − C C C C C y0 y0 yw wy y0 y0 yw wy 0 y0

and this proves the corollary.

×C y−1 C . 0 y0 yz The form in parentheses is known to be positive definite (it has the same structure of the covariance matrix of residuals of a least squares adjustment). Therefore the difference of the errors is larger than or equal to zero, so equation (8) is verified. Proof of Corollary 2 Since z = λT w, then by covariance propagation it holds that: C zy = ␭T Cwy ,

(29)

C z wˆ = ␭T Cwwˆ .

(30)

So now the one-step estimate (equation 4) is: y = ␭T w. ˆ zˆ = ␭T Cwy C y−1 0 y0 0

(31)

Then the two-step estimate (equation 6) is: w. ˆ zˆˆ = ␭T Cwwˆ Cw−1 ˆ wˆ

(32)

Applying covariance propagation through equation (5) and comparing to equation (23), it is found that: (33) Cwwˆ = Cwˆ wˆ .

zˆˆ = λT wˆ = zˆ ,

(34)

References Colombo, O. L. (1981). Numerical Methods for Harmonic Analysis on the Sphere. Report No. 310, Department of Geodetic Science and Surveying, Ohio State University, Columbus, Ohio. Driscoll, J. R., and D. M. Healy (1994). Computing Fourier transforms and convolutions on the 2-sphere. Advances in applied mathematics, 15, pp. 202–250. ESA (1999). Gravity Field and Steady-State Ocean Circulation Mission. ESA SP-1233 (1). ESA Publication Division, c/o ESTEC, Noordwijk, The Netherlands. Jekeli, C. (1999). The determination of gravitational potential differences from satellite-to-satellite tracking. Celestial Mechanics and Dynamical Astronomy, 75, pp. 85–101. Migliaccio, F., and F. Sansò (1989). Data processing for the Aristoteles mission. In: Proc. of the Italian Workshop on the European Solid-Earth Mission Aristoteles. Trevi, Italy, May 30–31 1989, pp. 91–123. Migliaccio, F., M. Reguzzoni and F. Sansò (2004). Spacewise approach to satellite gravity field determination in the presence of coloured noise. Journal of Geodesy, 78, pp. 304–313. Migliaccio, F., M. Reguzzoni and N. Tselfes (2006). GOCE: a full-gradient solution in the space-wise approach. In: International Association of Geodesy Symposia, “Dynamic Planet”, Proc. of the IAG Scientific Assembly, 22–26 August 2005, Cairns, Australia, P. Tregoning and C. Rizos (eds), Vol. 130, Springer-Verlag, Berlin, pp. 383–390. Moritz, H. (1980). Advanced Physical Geodesy, Wichmann Verlag, Karlsruhe. Sansò, F., and C. C. Tscherning (2003). Fast Spherical Collocation: theory and examples. Journal of Geodesy, 77, pp. 101–112. Visser, P. N. A. M., N. Sneeuw and C. Gerlach (2003). Energy integral method for gravity field determination from satellite orbit coordinates. Journal of Geodesy, 77, pp. 207–216.

Maximum Possibility Estimation Method with Application in GPS Ambiguity Resolution X. Wang, C. Xu School of Geodesy & Geomatics, Wuhan University, 129 Luoyu Road, Wuhan 430079, P.R. China; Key Laboratory of Geomatics and Digital Technology of Shandong Province, Shandong University of Science and Technology, 579 Qianwangang, Qingdao 266510, P.R. China; Research Center for Hazard Monitoring and Prevention, Wuhan University, 129 Luoyu Road, Wuhan 430079, P.R. China Abstract. Based on the fuzzy set and the possibility theory, this paper presents the Maximum Possibility Estimation method. The new estimation method is different from the other ones. The principle of the Maximum Possibility Estimation method, using triangular fuzzy number as a membership function, is given in this paper. Then, the hybrid algorithm is discussed for the resolution of the nonlinear programming model. Finally, an example of GPS ambiguity resolution using Maximum Possibility Estimation is presented. The results indicate that this new estimation method is feasible and useful. Keywords. Possibility theory, fuzzy number, ambiguity resolution

1 Introduction The GPS observations are influenced by various effects such as ionospheric effects, tropospheric effects, relativistic effects, earth tide and ocean loading tide effects, clock errors, antenna mass centre and phase center corrections, multipath effects, etc. Some of the effects are difficult to eliminate, such as multipath, antenna mass centre and instrumental biases, etc. So the residual errors of GPS observations may not obey normal distribution. Lan and Tang (2005) analyzed some GPS observations from continuous GPS site of Wuhan and drew the conclusion that the errors of GPS observations obviously do not obey normal distribution. And some other researchers also used the fuzzy set theory or fuzzy logical technique for GPS data processing (Lin et al. 1996). So the authors try to avoid the assumption that the observation errors are all random and use the Possibility Theory to process the surveying data in this paper.

domain is a specified set. As the observations normally surround its true value, we can treat the observations as symmetrical fuzzy number with triangular membership function. Figure. 1 shows the symmetrical fuzzy number and equation (1) is its membership function. In Figure 1 the horizontal axis denotes the observations and the vertical axis denotes the possibility of the observations. The possibility belongs to [0, 1], and δ denotes the uncertainty of the observation. The uncertainty indicates the quality of the observation, a smaller uncertainty means a higher quality. L(x) =

1−

|x−α| c

0,

, −c ≤ x − α ≤ c else

(1)

Possibility Theory has been coined by Zadeh (1978), which is devoted to the handling of incomplete information. It has been applied in a number of areas such as data analysis, scheduling, database querying, diagnosis, belief revision etc (Dubois and Prade 2003). So the authors try to use the Possibility Theory as the estimation method to process the uncertain observations. Maximum Possibility Estimation Theory (MPE) comes from Possibility Theory and treats the surveying data as a symmetrical fuzzy number. Its estimation criterion is “the possibility of the observations to be their true values must be maximum”.

2 Maximum Possibility Estimation Theory A fuzzy number is a quantity whose value is imprecise, and it can be thought of as a function whose

Fig. 1. Symmetrical fuzzy number with triangular membership function. 269

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X. Wang, C. Xu

The equations of observations can be written as: y = f (x).

(2)

where y denotes the observations and x denotes the unknown parameters. To let the possibilities of the observations as their true values be maximum, the principle must read: n

poss(yi ) =

i=1

n 1− i=1

=n−

n i=1

|y i − f (x i )| δi

|y i − f (x i )| δi

(3)

= max

equation (3) is equivalent to: n |y i − f (x i )| = min δi

(4)

i=1

And we need to let the uncertainties of the observations be minimum, that is: n

δi = min

(5)

i=1

Furthermore, the uncertainties of the observations are defined as positive numbers: δi > 0. And a inequalityconstraint can be expressed as: |vi | = |yi − f (x i )| ≤ δi So the general estimation model is: ⎧ n |vi | ⎪ ⎪ ( + δi ) = min ⎪ ⎪ ⎨ i=1 δi subject to ⎪ ⎪ δi > 0 ⎪ ⎪ ⎩ |vi | ≤ δi

(6)

(7)

The model (7) is a nonlinear programming model with inequality-constraints. The resolution of these models usually uses intelligent optimization algorithms, such as simulated annealing, genetic algorithm, artificial immune algorithm and their hybrid algorithms and so on (Xu et al. 2002). The presented estimation model seems to be quite similar to weighted L1-norm minimization method because their forms are similar, and they both get the results by iteration. But they are based on different theories and their estimation models are different too. The L1-norm method sets the weight as 1 when |v| ≤ c, and sets the weight as other small values when |v| > c while the MPE sets the uncertainties of

the observations as unknown values, and subject to |vi | ≤ δi . Moreover, every observation has an uncertainty. So the MPE has more unknown parameters than L1-norm, and the estimation model is a nonlinear programming.

3 The Hybrid Algorithm for Maximum Possibility Estimation Genetic algorithms (GA) were formally introduced in the United States in the 1970s by John Holland at University of Michigan. The continuing performance improvement of computational systems has made them attractive for some types of optimization. Furthermore, the simplex method is a local search technique that uses the evaluation of the current data set to determine the promising search direction (Yen et al. 1998; Xu et al. 2002). It searches for an optimal solution by evaluating a set of points and continually forms new simplex by replacing the worst point. While GA has been shown to be effective for solving a wide range of optimization problems, its convergence speed is typically much slower than simplex method. Many researchers have combined GA with other optimization techniques to develop hybrid genetic algorithms (Yen et al. 1998; Xu et al. 2002). To speed up the rate of convergence and retain the ability of global optimum, the authors integrate GA and simplex method for the resolution of Maximum Possibility Estimation method. The detailed steps for the hybrid algorithm are given as follows: Step 1. Start. Define the parameters of population size iPopSize, unknown parameter numbers N, crossover probability Pc , mutation probabilityPm , simplex step length, and maximum iterative number. Then generate random population of chromosomes (suitable solutions for the problem). Step 2. Fitness. Evaluate the fitness of each chromosome in the population, and according to the fitness value of each chromosome, chromosomes are sorted in a descending order. Step 3. Simplex Method. Add the best chromosome to a new simplex point and then random select N chromosomes from the full population to the simplex point. A new point is generated by using the simplex method and replaced the best chromosome. Step 4. Selection. Select Ns parents from the full population. The parents are selected according to their fitness. (The better fitness the bigger chance to be selected).

Maximum Possibility Estimation Method with Application in GPS Ambiguity Resolution

Step 5. Crossover. For each pair of parents identified in Step 1, perform crossover on the parents at a randomly chosen probability. Step 6. Mutation. With the mutation probability, mutate new offspring at each locus (position in chromosome). Step 7. Fitness and End Test. Compute the fitness values for the new population of N chromosomes. Terminate the algorithm if the stopping criterion is met; else return to Step 2.

The GPS double-differenced carrier phase observation can be expressed as: Vϕi = (∇Δφi − ∇ΔNi )λ − ∇ΔRi

(8)

where ∇ΔN denotes the double-differenced ambiguity, which should be an integer. ∇Δφ is the doubledifferenced carrier phase observation, ∇ΔR is the double-differenced true range from satellite to station. Vϕ is the correction of the double-differenced carrier phase. The coordinates, the double-differenced ambiguity and the uncertainty of observations are treated as unknown parameters. From equation (7), the evaluation function with dual frequency carrier phase can be expressed as: φ=

⎛

i j

vϕ1 ij ij ⎝ + δϕ1 + Ωϕ1 + ij δϕ1 j =1

i=1

ij

vϕ2 ij

δϕ2

parameters of the genetic algorithm and generate the random population of chromosomes from the initial space. (2)Search the true value of the unknown parameters using the genetic algorithm until the stopping criterion is met. (3)Set the integer ambiguity of frequency 1 as known value, and use the hybrid algorithm to search the coordinates and get the result.

5 Experiment

4 The Application in Ambiguity Resolution

E N

271

⎞ + δϕ2 + Ωϕ2 ⎠ ij

ij

(9)

If |vi | > δi , Ωϕ is a large number taken to penalize the evaluation function; otherwise Ωϕ = 0. To assure the double-differenced ambiguity as integer, a rounding function is taken in the crossover operation, mutation operation, and simplex method. Moreover it rejects the chromosome which can not fit the constraint δi > 0. So while the algorithm is iterating, the unknown parameters of ambiguity always are integers and the constraints of |vi | > δi are true all the time. We can get the integer ambiguities and their uncertainties when iteration ends. When the ambiguity uncertainty is smaller, the possibility that it is the true value is higher. The ambiguity resolution is performed by the following steps: (1)Initiate the coordinate and double difference ambiguity space using the positioning result of the C/A code pseudorange observations. Initiate the

To test the proposed method of parameter estimation, the authors did some experiments of fast static data processing, using software package of our own, which is totally created by us. That is, the algorithm is new, not from any of the existing software. One of the experiments use two Leica 1230 GPS receivers, observing synchronously and statically for 10 min. The sample rate is 10 s, and the length of baseline is 1.3 km. Only frequency L1 was used in this experiment and the evaluation function can be expressed as: ⎛

i j

⎞ E N

vϕ1 ij ij ⎝ + δ ϕ1 + Ωϕ1 ⎠ φ= (10) ij δ i=1 j =1 ϕ1 We compared the result with Trimble Geomatics Office (TGO). Satellite 10 was selected as the reference satellite. The different result between Maximum Possibility Estimation and TGO which uses the LAMBDA method (Teunissen et al. 1995) to resolve the ambiguities are shown in the following tables. And the run-time is less than 1 min. It shows that the new method get the correct result. For we have used different estimation methods, the final results are a litte different, too. The parameters of the new algorithm is listed in Table 1. From Table 2, Table 3, we can see that, the

Table 1. The parameters of hybrid algorithm Parameter

Value

Parameter

Value

Ambiguity searching area Coordinate searching area Crossover rate

±5 Cycle

Mutation rate

0.2

±1.5 m

Population size

80

0.3

Evaluation number

2000

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Table 2. The double-differenced ambiguities PRN

MPE (Cycle)

TGO (Cycle)

6 14 16 20 25 30

18 –12 63 –6 1 –11

18 –12 63 –6 1 –11

Table 3. The difference of the coordinates Coordinate

MPE (m)

TGO (m)

Difference (cm)

X Y Z

1131.875 689.394 –271.631

1131.878 689.398 –271.628

–0.3 –0.4 0.3

MPE method can get the correct integer ambiguity, but the coordinates are of little difference. This is caused by the stopping criterion of the hybrid algorithm. If higher position precision is needed, we can set a new stopping criterion. However, it will take a longer time of data processing.

6 Conclusions The Maximum Possibility Estimation method is a new parameter estimation method which is different from the traditional ones. It based on Possibility Theory and treats the observations as fuzzy numbers. In contrast, the traditional parameter estimation methods are based on probability theory, and

treat the observations as random values. The experiment demonstrated that it can be used to process the GPS observations. As a new method for parameter estimation, it still has some problems and needs further study.

Acknowledgements The research project is funded by the National Natural Science Foundation of China (40474003) and Special Project Fund of Taishan Scholars of Shandong Province (TSXZ0502).

References Dubois, D., Prade, H. (2003). Possibility Theory and its Applications: A Retrospective and Prospective View, The IEEE international Conference on Fuzzy Systems, pp. 3–11. Lan, Y., Tang, Y. (2005). The Distribution Test of Observation Error for GPS, International Symposium on GPS/GNSS 2005, Hong Kong, China. Lin, C. J., Chen, Y. Y., Chang, F. R. (1996). Fuzzy processing on GPS data to improve the position accuracy. Fuzzy Systems Symposium, Proceedings of the 1996 Asian 11–14 Dec., pp. 557–562. Teunissen, P. J. G. (1995). The least-square ambiguity decorrelation adjustment: A method for fast GPS integer ambiguity estimation. Journal of Geodesy, pp. 65–82. Xu, J., Arslan T., Wang Q., Wan, D. (2002). GPS attitude determination using a genetic algorithm, IEEE, Volume 1, 12–17 May, pp. 998–1002. Yen, J., Liao, J. C., Randolph, D., Lee B. (1998). A hybrid approach to modeling metabolic systems using genetic algorithm and simplex method, IEEE Trans SMC, 28(2):173–191. Zadeh L. A. (1978). Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems, 1, pp. 3–28.

Variance Component Estimation by the Method of Least-Squares P.J.G. Teunissen, A.R. Amiri-Simkooei Delft institute of Earth Observation and Space systems (DEOS), Delft University of Technology, Kluyverweg 1, 2629 HS Delft, The Netherlands, e-mail: [email protected]

Abstract. Motivated by the fact that the method of least-squares is one of the leading principles in parameter estimation, we introduce and develop the method of least-squares variance component estimation (LS-VCE). The results are presented both for the model of observation equations and for the model of condition equations. LS-VCE has many attractive features. It provides a unified least-squares framework for estimating the unknown parameters of both the functional and stochastic model. Also, our existing body of knowledge of least-squares theory is directly applicable to LS-VCE. LS-VCE has a similar insightful geometric interpretation as standard least-squares. Properties of the normal equations, estimability, orthogonal projectors, precision of estimators, nonlinearity, and prior information on VCE can be easily established. Also measures of inconsistency, such as the quadratic form of residuals and the w-test statistic can directly be given. This will lead us to apply hypotheses testing to the stochastic model. Keywords. Least-squares variance component estimation, BIQUE, MINQUE, REML

1 Introduction Estimation and validation with heterogeneous data requires insight into the random characteristics of the observables. Proper knowledge of the stochastic model of the observables is therefore a prerequisite for parameter estimation and hypothesis testing. In many cases, however, the stochastic model may still contain unknown components. They need to be determined to be able to properly weigh the contribution of the heterogeneous data to the final result. Different methods exist in the geodetic and statistical literature for estimating such unknown (co)variance components. However, the principles on which these methods are based are often unlinked with the principles on which the estimation of the parameters of the functional model is based.

This paper formulates a unified framework for both the estimation and validation problem of the stochastic model. We concentrate on the problem of estimating parts of the stochastic model. The method is based on the least-squares principle which was originally proposed by Teunissen (1988). We will therefore have the possibility of applying one estimation principle, namely our well-known and well understood method of least-squares, to both the problem of estimating the functional model and the stochastic model. We give the results without proof. For proofs we can closely follow Teunissen and Amiri-Simkooei (2007). We present the weighted least-squares (co)variance component estimation (LS-VCE) formula for which an arbitrary symmetric and positive-definite weight matrix can be used. Weighted LS-VCE gives unbiased estimators. Based on the normal distribution of original observations, we present the covariance matrix of the observables in the stochastic model. We can obtain the minimum variance estimators by taking the weight matrix as the inverse of the covariance matrix. This corresponds to the best linear unbiased estimator (BLUE) of unknown parameters x in the functional model. These estimators are therefore unbiased and of minimum variance. In this paper the property of minimum variance is restricted to normally distributed data. Teunissen and Amiri-Simkooei (2007) derived such estimators for a larger class of elliptical distributions. We will make use of the vector (vec) and vectorhalf (vh) operators, the Kronecker product (⊗), and the commutation (K ) and duplication (D) matrices. For a complete reference on the properties and the theorems among these operators and matrices we refer to Magnus (1988).

2 Least-Squares Estimators Consider the linear model of observation equations E{y} = Ax ; D{y} = Q y = Q 0 +

p

σk Q k , (1)

k=1

273

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P.J.G. Teunissen, A.R. Amiri-Simkooei

with y the m × 1 vector of observables (the underline indicates randomness), x the n×1 vector of unknown parameters, A the m ×n design matrix, Q y the m ×m covariance matrix of the observables (Q 0 its known part; the m × m cofactor matrices Q k are also known but their contributions through σk are unknown). The unknowns σk are for instance variance or covariance components. The matrices Q k , k = 1, . . . , p should be linearly independent. The second part of (1) can be written as D{y} = E{(y − Ax)(y − Ax)T }. To get rid of the unknown parameters x in E{(y − Ax)(y − Ax)T }, one can rewrite (1) in terms of the model of condition equations. One can therefore show that (1) can equivalently be reformulated as E{t} = 0; E{t t T } − B T Q 0 B =

p

σk B T Q kB , (2)

k=1

with the b × 1 vector of misclosures t = B T y, the m × b matrix B satisfying B T A = 0. b = m − n is the redundancy of the functional model. The matrices B T Q 1 B, ..., B T Q p B should be linearly independent, which is a necessary and sufficient condition in order for the VCE model to have a unique solution. The first part of (2), i.e., the functional part, consists of all redundant observations as there exists no unknown in this model. The adjustment of this part is trivial because tˆ = 0. We may therefore concentrate on the second part, i.e., the stochastic model. Note also that the condition E{t} = 0, which implies that there is no misspecification in the functional model, has been used in the second part by default because Q t = E{t t T } − E{t}E{t}T .

This results in the following linear model of observation equations (note that both the vh and the E operators are linear): E{y vh } = Avhσ, Wvh or Q vh ,

(3)

with y vh = vh(t t T − B T Q 0 B) the observables in the stochastic model, and Avh a b(b+1) × p (design) 2 matrix of the form Avh = vh(B T Q 1 B) · · · vh(B T Q p B) , (4) T and σ is a p-vector as σ = σ1 σ2 · · · σ p . The b(b+1) × b(b+1) matrix Q vh is the covariance matrix 2 2 of the observables vh(t t T ) and the b(b+1) × b(b+1) 2 2 matrix Wvh is accordingly the weight matrix. This is therefore a standard form of the linear model of observation equations with a b(b+1) 2 -vector of observ× p design matrix and a p-vector of ables, a b(b+1) 2 unknown (co)variance components. Weighted LS Estimators Having established these results, we can now apply the method of least-squares to estimate σ . In other words, if the weight matrix Wvh is known, we can obtain the weighted least-squares estimators of the (co)variance components. The weighted leastsquares estimators of the (co)variance components then read T T σˆ = (Avh Wvh Avh )−1 Avh Wvh y vh = N −1 l,

(5)

T W A , the p × p normal matrix, where N = Avh vh vh T and l = Avh Wvh y vh , a p-vector, are of the forms

Stochastic Model The matrix equation in the second part of (2) can now be recast into a set of b2-number of observation equations by stacking the b-number of b × 1 column vectors of E{t t T } into a b2 × 1 observation vector. Therefore, just like we interpret the functional model E{y} = Ax as a set of m-number of observation equations with the observation vector y, we are going to interpret the stochastic model E{t t T − p T B Q 0 B} = k=1 σk B T Q k B as a set of b2 -number of observation equations with the observation matrix t t T −B T Q 0 B. Since the matrix of observables t t T is symmetric, its upper triangular elements do not provide new information. There are only b(b+1) distinct 2 (functionally independent) elements. We can therefore apply the vh-operator to the second part of (2).

n kl = vh(B T Q k B)T Wvh vh(B T Q l B) ,

(6)

l k = vh(B T Q k B)T Wvh y vh ,

(7)

and

respectively, with k, l = 1, · · · , p. Any symmetric and positive-definite matrix Wvh can play the role of the weight matrix. Weight Matrix From a numerical point of view, an arbitrary weight matrix Wvh in (6) and (7) may not be advisable as it is of size b(b+1) × b(b+1) 2 2 . For this reason, we now restrict ourselves to those weight matrices which computationally are more efficient. One admissible

Variance Component Estimation by the Method of Least-Squares

and, in fact, simple weight matrix Wvh has the following form Wvh = D T (Wt ⊗ Wt )D ,

(8)

where Wt is an arbitrary positive-definite symmetric duplication matrix of size b and D is the b2 × b(b+1) 2 matrix. Using the properties of the Kronecker product one can show that Wvh is in fact positive-definite and therefore can play the role of the weight matrix. Substituting (8) into (6) and (7) gives n kl = tr(B T Q k BWt B T Q l BWt ),

(9)

and lk = t T Wt B T Q kBWt t −tr(B T Q kBWtB T Q 0BWt ).

(10)

respectively. The weighted least-squares (co)variance component estimation was formulated by rewriting the (co)variance component model into a linear model of observation equations. The above formulation of VCE is based on the weighted least-squares method for which an arbitrary weight matrix Wvh (e.g. in form of (8)) can be used. An important feature of the weighted least-squares estimators is the unbiasedness property.

Covariance Matrix of vh(t tT ) In order to evaluate the covariance matrix of (co)variance components, i.e. Q σˆ , we need to know × b(b+1) covariance matrix of vh(t t T ), the b(b+1) 2 2 namely Q vh . In addition, one can in particular choose the weight matrix Wvh as the inverse of Q vh to obtain the minimum variance estimators. Let us first present the covariance matrix of vec(t t T ) which is based on the following theorem: Theorem 1. Let the stochastic vector t be normally distributed with mean zero and covariance matrix Q t , i.e. t ∼ N(0, Q t ), then the covariance matrix of the observables vh(t t T ) is given as Q vh = 2 D + (Q t ⊗ Q t )D +T ,

(11)

where D is the duplication matrix and D + is its pseudo-inverse as D + = (D T D)−1 D T . Proof. Closely follow Teunissen and Amiri-Simkooei (2007). Using the properties of the duplication matrix and the Kronecker product, the inverse of Q vh is obtained as Q −1 vh =

1 T −1 D (Q t ⊗ Q −1 t )D . 2

(12)

275

For normally distributed data, Q −1 vh is thus an element of the class of admissible weight matrices defined in . This is in fact an interesting (8) with Wt = √1 Q −1 2 t result because we can now choose the weight matrix Wvh = Q −1 vh to obtain the minimum variance estimators of the (co)variance components.

Minimum Variance Estimators As with the best linear unbiased estimator (BLUE) in the functional model, the (co)variance components can be estimated according to BLUE with the observables vh(t t T ). One can obtain such estimators by taking the weight matrix Wvh as the inverse of the covariance matrix of the observables, Q −1 vh . Then this linear form of the observables vh(t t T ) can be rewritten as the best (minimum variance) quadratic unbiased estimator of the misclosures t. To obtain the minimum variance in (9) estimators, one needs to substitute Wt = √1 Q −1 2 t and (10). Such estimators are therefore given as σˆ = N −1 l with n kl =

1 T −1 tr(B T Q k B Q −1 t B Q l B Q t ), 2

and lk =

1 T −1 T t Q t B Q k B Q −1 t t, 2

(13)

(14)

in which we assumed Q 0 = 0. Since the covariance matrix Q vh in (11) is derived for normally distributed data, the ‘best’ (minimum variance) property is restricted to the normal distribution.

3 Formulation in Terms of A-Model Weighted LS Estimators The least-squares method to (co)variance component estimation can directly be used, if the matrix B is available (model of condition equations). In practice, however, one will usually have the design matrix A available (model of observation equations) instead of B. We now extend the least-squares method for estimation of (co)variance components to the model of observation equations. We consider again the case that the covariance matrix can be split into a known part (co)variance component model, Q 0 and an unknown p namely Q y = Q 0 + k=1 σk Q k . To apply the weighted least-squares variance component estimation to the model of observation equations we shall therefore have to rewrite (9) and (10) in terms of the design matrix A. Using the relation between elements of the B and A models and also taking into account the trace properties, the matrix N in (9) and the vector l in (10) can be reformulated as n kl = tr(Q k W PA⊥ Q l W PA⊥ ) ,

(15)

276


and

Implementation of LS-VCE (A-model) T

l k = eˆ W Q k W eˆ

− tr(Q k W PA⊥ Q 0 W PA⊥ ) ,

(16)

respectively, where W is an arbitrary m × m positivedefinite matrix, eˆ is the least-squares residuals given as eˆ = PA⊥ y, with the orthogonal projector PA⊥ = I − A(A T W A)−1 A T W . The weighted least-squares estimator is therefore given as σˆ = N −1 l with N and l given by (15) and (16), respectively.

Minimum Variance Estimators To obtain the minimum variance estimators, we should choose the weight matrix as the inverse of the covariance matrix. In an analogous way to Wt = √1 Q −1 t , 2

. If we now subone can use the matrix W = √1 Q −1 2 y stitute W into (15) and (16), we will then obtain n kl =

1 ⊥ −1 ⊥ tr(Q k Q −1 y PA Q l Q y PA ), 2

(17)

Input: 1. design matrix A of observation equations; 2. observation vector y; 3. cofactor matrices Q k , k = 0, ..., p; 4. initial (co)variances σ = σ 0 = [σ10 , ..., σ p0 ]T ; 5. small value for ; begin check for presence of gross errors in observations; set iteration counter i = 0; begin p evaluate matrix Q y = Q 0 + k=1 σk Q k ; calculate N and l from (17) and (18); solve for a new σˆ from equations N σˆ = l; i ← i + 1; update vector σ i ← σˆ ; while σ i − σ i−1 Q −1 > repeat; σˆ end obtain σˆ and its covariance matrix Q σˆ = N −1 . end

and 1 1 −1 −1 ⊥ −1 ⊥ l k = eˆ T Q −1 y Q kQ y eˆ − tr(Q k Q y PA Q 0 Q y PA ), (18) 2 2

Fig. 1. Symbolic algorithm for implementation of leastsquares variance component estimation in terms of linear model of observation equations (A model); σ i is the vector of (co)variance components estimated in iteration i.

−1 AT Q −1 . where PA⊥ = I − A(AT Q −1 y A) y

Implementation Equations (17) and (18)with σˆ = N −1 l show that p we need Q y = Q 0 + k=1 σk Q k in order to compute the estimates σˆ k . But the (co)variance components σk are unknown apriori. The final solution should be sought through an iterative procedure. For this purpose we start with an initial guess for the σk . Based on these values, we compute with σˆ = N −1 l estimates for the σk , which in a next iteration are considered the improved values for σk . The procedure is repeated until the estimated components do not change by further iteration. Figure 1 gives a straightforward iterative algorithm for implementing LS-VCE in terms of the model of observation equations. There are two ways of estimating (co)variance components. The first way is to consider the cofactor matrices as a whole and try to estimate unknown unit factors (scale factors). That is, in each iteration we modify the cofactor matrices by multiplying them with the estimated factors. After a few iterations we expect the factors to converge to ones. In the second way, we consider the cofactor matrices to be fixed. In each iteration, the (co)variance components rather than the cofactor matrices are modified. After some iterations, the modified (co)variance components converge so that their values do not change by further iterations. For example, consider the covariance matrix as Q y = σ1 Q 1 + σ2 Q 2 . At the point of convergence, the above

strategies look as follows: In the first way, we obtain the factors f 1 and f 2 , therefore Q y = fˆ1×σ1 Q 1 + fˆ2× σ2 Q 2 where fˆ1 = fˆ2 = 1 and in the second way we estimate the components σ1 and σ2 , therefore Q y = σˆ 1 × Q 1 + σˆ 2 × Q 2 .

4 Properties of Proposed Method Since we have obtained the least-squares (co)variance estimators based on a model of observation equations, see (3), the following features can easily be established:

4.1 Unification of Methods To obtain the weighted least-squares solutions, no assumption on the distribution of vh(t t T ) is required. Also, we know without any additional derivation that the estimators are unbiased. This property is independent of the distribution of the observable vector vh(t t T ). This makes the LS-VCE method more flexible as we can now use a class of weight matrices as Wvh = D T (Wt ⊗ Wt )D where Wt is an arbitrary positive definite matrix and D the duplication matrix. In a special case where one takes the weight matrix as the inverse of the covariance matrix, i.e. Wvh = Q −1 vh , one can simply obtain the minimum variance estimators. Therefore, LS-VCE is capable of unifying many of the existing VCE methods such as minimum norm quadratic unbiased estimator (MINQUE) (see

Variance Component Estimation by the Method of Least-Squares

Rao, 1971, Rao and Kleffe, 1988, Sjöberg, 1983), best invariant quadratic unbiased estimator (BIQUE) (see Caspary, 1987, Koch, 1978, 1999, Schaffrin, 1983), and restricted maximum likelihood (REML) estimator (see Koch, 1986).

277

of the stochastic model. Further discussion on this topic is beyond the scope of the present contribution. For more information we refer to Amiri-Simkooei (2007).

4.5 Nonlinear Stochastic Model 4.2 Similarity with Standard LS LS-VCE has a similar insightful geometric interpretation as the standard least-squares. Properties of the normal matrix, estimability of (co)variance components, and the orthogonal projectors can easily be established. Also, in an analogous way to the functional model in which one deals with redundancy b = m − n, one can define the redundancy (or here the degrees of freedom d f ) in the stochastic model. From (4) it follows − p, when the design matrix Avh that d f = b(b+1) 2 of the stochastic model is assumed to be of full rank, and with p, as before, being the number of unknown (co)variances components. This implies that the maximum number of estimable (co)variance components is p = b(b+1) 2 , which leads to d f = 0 (see also Xu et al., 2007).

4.3 Covariance Matrix of Estimators Since the weighted least-squares estimators are in a linear form of the observables y vh , applying the T W y error propagation law to σˆ = N −1 Avh vh vh automatically gives us the covariance matrix of the estimated (co)variance components, namely Q σˆ = N −1 M N −1 where the p × p matrix M is given as m kl = 2tr(B T Q k BWt Q t Wt B T Q l BWt Q t Wt ) = 2tr(Q k W PA⊥ Q y W PA⊥ Q l W PA⊥ Q y W PA⊥ ). This equation can therefore provide us with the precision of the estimators. This is in fact an important feature of the least-squares variance component estimation. In case of ), one can minimum variance estimators (W = √1 Q −1 2 y simply show that M = N , and therefore Q σˆ = N −1 .

4.4 Measures of Inconsistency Since the approach is based on the least-squares principle, parts of the standard quality-control theory can be applied to the model in (3). One can in particular apply the idea of hypotheses testing to the stochastic model. For example, one can deal with the w-test statistic and the quadratic form of the residuals in the stochastic model. As an important measure of any least-squares solution, one can compute the quadratic form of the residuals. This also holds true for the LS-VCE. The quadratic form of the residuals is then given as T Q −1 eˆvh vh eˆ vh =

1 T −1 2 (ê Q y eˆ ) − l T N −1 l , 2

LS-VCE has the capability of applying to a nonlinear (co)variance component model, namely Q y = Q(σ ). To overcome the nonlinearity, one can expand the stochastic model into a Taylor series, for which one needs the initial values of the unknown vector σ , namely σ 0 . When expanded into Taylor series, the covariance matrix can be written as Q y = Q(σ ) ≈ Q 0 + p k=1 σk Q k . We can now apply the LS-VCE to estimate σ . The estimated σˆ can then be considered as a new update for σ 0 and the same procedure can be repeated. We can iterate until the estimated (co)variance components do not change by further iterations. The applied iteration is the Gauss-Newton iteration which has a linear rate of convergence (see Teunissen, 1990).

4.6 Prior Information In some cases, we may have prior information about the (co)variance components. Such information can be provided by equipment manufacturers or from a previous process. Let us assume that this information can be expressed as E{σ 0 } = σ ; D{σ 0 } = Q σ0 , which means that the (co)variance components σ 0 are earlier estimators available with the covariance matrix Q σ0 . One important feature of the LS-VCE is the possibility of incorporating such prior information with the observables vh(t t T ). Without additional derivations, one can obtain the least-squares (co)variance estima−1 −1 tors as σˆ = (N + Q −1 σ0 ) (l + Q σ0 σ 0 ). Note that the covariance matrix of these estimators is simply given −1 as Q σˆ = (N + Q −1 σ0 ) .

4.7 Robust Estimation Since we estimated the (co)variance components on the basis of a linear model of observation equations, we can think of robust estimation methods rather than the least-squares. One can in particular think of an L 1 -norm minimization problem. The usual method for implementation of the L 1 -norm adjustment leads to solving a linear programming problem (see e.g. Amiri-Simkooei, 2003). This may be an important alternative if one wants to be guarded against misspecifications in the functional part of the model.

5 Simple Examples (19)

in which we assumed Q 0 = 0. One can also obtain the w-test statistic to identify the proper noise components

Example 1 (Minimum variance estimator). As a simple application of LS-VCE, assume that there is only one variance component in the stochastic model, namely Q y = σ 2 Q. If our original observables y are

278


normally distributed, it follows with (17) and (18) from σˆ = N −1 l that l σˆ = = n 2

1 T ˆ 2e

−1 ˆ Q −1 y QQy e

−1 ⊥ −1 ⊥ 1 2 tr(Q Q y PA Q Q y PA )

,

(20)

Using Q y = σ 2 Q, PA⊥ PA⊥ = PA⊥ , and tr(PA⊥ ) = rank(PA⊥ ) = m − n = b, the preceding equation, its mean, and its variance simplify to eˆ T Q −1 eˆ 2σ 4 ; E{σˆ 2 } = σ 2 ; D{σˆ 2 } = , m−n m −n (21) respectively. These are the well-known results for the estimator of the variance of unit weight. This estimator can thus be obtained from the least-squares residuals without iteration. This estimator is unbiased and of minimum variance. The variance of the estimator was 2σ 4 . simply obtained by D{σˆ 2 } = N −1 = m−n σˆ 2 =

Example 2 (Weighted LS estimator). To see an important application of the weighted LS-VCE, we derive the empirical autocovariance function in a time series (e.g. to estimate the time correlation of a time series). For simplicity we assume that (1) we measure a functionally known quantity (e.g. a zero baseline measured by GPS receivers), and (2) the cofactor matrices are side-diagonal with equal values which implies that the covariance between observations i and j is only a function of time-lag τ = | j − i |, i.e. σi j = στ . The covariance matrix can thus be written as a linear combination of m cofactor matrices as Qy = σ2 I +

m−1

στ Q τ ,

(22)

τ =1

T T where Q τ = m−τ i=1 ci ci+τ + ci+τ ci , τ = 1, ..., m − 1, with ci the canonical unit vector, are some cofactor matrices and σ 2 is the unknown variance of the noise process. We can now apply the weighted least-squares approach to estimate the (co)variance components. One particular choice of the weight matrix W is the unit matrix, W = I . Since the design matrix A is empty, it follows that PA⊥ = I . To estimate the (co)variance components σˆ one needs to obtain N and l from (15) and (16), respectively. One can show that the (co)variance components στ are estimated as m−τ

i=1 eˆ i eˆ i+τ , τ = 0, 1, ..., m − 1, m−τ (23) where eî is the i th least-squares residual, σˆ 0 = σˆ 2 is the variance, and σˆ τ , τ = 1, ..., m − 1 are the covari-

l σˆ τ = τ = n τ,τ

ances. One can also derive the covariance matrix of these estimators using Q σˆ = N −1 M N −1 .

6 Concluding Remarks There are various VCE formulas based on optimality properties as unbiasedness, minimum variance, minimum norm, and maximum likelihood. In this paper we introduced the method of least-squares for estimating the stochastic model for which any symmetric and positive-definite weight matrix can be used. The method is easily understood and very flexible. It can be used for estimation of both variance and covariance components in the A-model and the B-model, both for linear and nonlinear stochastic models. Since the method is based on the least-squares principle, we know without any additional derivation that the estimators are unbiased. One advantage of this technique over other methods of VCE is that the weighted leastsquares solution can be obtained without any supposition regarding the distribution of the data. This holds true also for the property of unbiasedness of the estimators. We then simply presented the minimum variance estimators by taking the weight matrix as the inverse of the covariance matrix of observables. Since we formulated the LS-VCE based on a linear model of observation equations, the proposed method has special and unique features. LS-VCE allows one to apply the existing body of knowledge of least-squares theory to the problem of (co)variance component estimation. With this method, one can (1) obtain measures of discrepancies in the stochastic model, (2) determine the covariance matrix of the (co)variance components, (3) obtain the minimum variance estimator of (co)variance components by choosing the weight matrix as the inverse of the covariance matrix, (4) take the a-priori information on the (co)variance component into account, (5) solve for a nonlinear (co)variance component model, (6) apply the idea of robust estimation to (co)variance components, (7) evaluate the estimability of the (co)variance components, and (8) avoid the problem of obtaining negative variance components.

Acknowledgments The second author would like to kindly acknowledge Christian Tiberius for his valued contribution and comments on earlier versions of this work. We also thank Frank Kleijer for reading a draft version of this paper.

References Amiri-Simkooei, A. R. (2003). Formulation of L 1 norm minimization in Gauss-Markov models. Journal of Surveying Engineering, 129(1), 37–43. Amiri-Simkooei, A. R. (2007). Least-squares variance component estimation: theory and GPS applications. Ph.D.

Variance Component Estimation by the Method of Least-Squares thesis, Delft University of Technology, Publication on Geodesy, 64, Netherlands Geodetic Commission, Delft. Caspary, W. F. (1987). Concepts of network and deformation analysis. Technical report, School of Surveying, The University of New South Wales, Kensington. Koch, K. R. (1978). Schätzung von varianzkomponenten. Allgemeine Vermessungs Nachrichten, 85, 264–269. Koch, K. R. (1986). Maximum likelihood estimate of variance components. Bulletin Geodesique, 60, 329–338. Ideas by A.J. Pope. Koch, K. R. (1999). Parameter estimation and hypothesis testing in linear models. Springer Verlag, Berlin. Magnus, J. R. (1988). Linear Structures. Oxford University Press, London School of Economics and Political Science, Charles Griffin & Company LTD, London. Rao, C. R. (1971). Estimation of variance and covariance components - MINQUE theory. Journal of multivariate analysis, 1, 257–275. Rao, C. R. and Kleffe, J. (1988). Estimation of variance components and applications, volume 3. North-Holland. Series in Statistics and Probability.

279 Schaffrin, B. (1983). Varianz-kovarianz-komponentenschätzung bei der ausgleichung heterogener wiederholungsmessungen. C282, Deutsche Geodätische Kommission, München. Sjöberg, L. E. (1983). Unbiased estimation of variancecovariance components in condition adjustment with unknowns – a MINQUE approach. Zeitschrift für Vermessungswesen, 108(9), 382–387. Teunissen, P. J. G. (1988). Towards a least-squares framework for adjusting and testing of both functional and stochastic model. Internal research memo, Geodetic Computing Centre, Delft. A reprint of original 1988 report is also available in 2004, No. 26, http://www.lr.tudelft.nl/mgp. Teunissen, P. J. G. (1990). Nonlinear least-squares. Manuscripta Geodetica, 15(3), 137–150. Teunissen, P. J. G. and Amiri-Simkooei, A. R. (2008). Least-squares variance component estimation. Journal of Geodesy (in press), doi 10.1007/s00190-007-0157-x. Xu, P. L., Liu, Y. M., and Shen, Y. Z. (2007). Estimability analysis of variance and covariance components. Journal of Geodesy, 81(9), 593–602, doi 10.1007/s00190-006-0122-0.

Noise Characteristics in High Precision GPS Positioning A.R. Amiri-Simkooei, C.C.J.M. Tiberius, P.J.G. Teunissen, Delft Institute of Earth Observation and Space systems (DEOS), Delft University of Technology, Kluyverweg 1, 2629 HS Delft, The Netherlands, e-mail: [email protected]

Abstract. In this contribution we present the results of three different studies in which the method of least-squares variance component estimation (LS-VCE) was used to infer the stochastic properties of GPS data. The three studies cover the GPS geometry-free model, the GPS coordinate time series model, and the GPS zero-baseline model. In the GPS geometry-free model, LS-VCE is applied to assess the precision of different observation types, correlation between observation types on L1 and L2, and satellite elevation dependence of the GPS observables precision. We show, for example, that the precision of code observations (for zero baseline) ranges from 10 to 15 cm depending on the satellite elevation and the type of the receiver used. The LSVCE time series analysis pertains to data of various permanent GPS tracking stations. It reveals that the noise can be divided into two components, namely white noise and flicker noise. We show that both noise components are spatially correlated (e.g. a correlation coefficient of about 0.8 over short distances between permanent stations). Finally, in the (classical) zero-baseline model, nonlinear LS-VCE is applied to assess the noise characteristics of GPS receivers based on covariance functions. Keywords. Variance component estimation, noise characteristics, GPS time series, covariance function

A systematic study of the GPS stochastic model is of course far from trivial. In this contribution we demonstrate this using three GPS applications. Least-squares variance component estimation (LS-VCE) is employed to assess the noise characteristics of GPS data. Consider the following linear model of observation equations with a p-number of unknown (co)variance components E{y} = Ax, D{y} = Q y =

A proper choice of the data weight matrix is of importance for many parameter estimation problems. This also holds true for the many applications of GPS. A realistic description of the GPS data noise characteristics is required to obtain minimum variance estimators through the functional model. In the case of GPS, the functional model is welldeveloped and well documented. The same can not yet be said of the covariance matrix of the GPS data. 280

σk Q k ,

(1)

k=1

where y is the m-vector of observables, x is the n-vector of parameters of interest, and A is the m × n design matrix. The covariance matrix Q y is expressed as an unknown linear combination of the known m×m cofactor matrices Q k ’s. The LS estimator for the p-vector of unknown (co)variance components σ = [σ1 σ2 ... σ p ]T can then be obtained as follows (Teunissen, 1988, Teunissen and AmiriSimkooei, 2007): σˆ = N −1 l with the p × p normal matrix N and the p-vector l as n kl =

1 ⊥ −1 ⊥ tr(Q −1 y PA Q k Q y PA Q l ), 2

(2)

and 1 T −1 ⊥ ⊥ y Q y PA Q k Q −1 y PA y; k, l = 1, ..., p, 2 (3) where the orthogonal projector is given as PA⊥ = −1 T −1 I − A(A T Q −1 y A) A Q y . The estimators obtained by this method are unbiased and of minimum variance. To apply the method, one starts with an initial guess for the (co)variance components and performs iterations. The iterative procedure is repeated until the estimated (co)variance components do not change with further iterations. Since the method is based on the least-squares principle, the inverse of the normal matrix N automatically gives the covariance matrix of the estimated (co)variance lk =

1 Introduction

p

Noise Characteristics in High Precision GPS Positioning

2 GPS Geometry-Free Model The GPS geometry-free observation model (GFOM) is one of the simplest approaches to processing and analysing data from a baseline and to integer GPS double differenced (DD) ambiguity estimation in particular. One advantage of using this model is its ease which stems from the linearity of the observation model and its independence of satellite orbit information. This model will be used in this section as a favorable model for estimation of (co)variance components via LS-VCE. The GFOM consists of two parts: the functional model and the stochastic model. The functional model relates the observations to the parameters of interest whereas the stochastic model describes the precision of and the mutual correlation between the observations. The functional model is based on the non-linearized DD dual frequency pseudo range and carrier phase observation equations. We use the LAMBDA method (see e.g. Teunissen (1993)) to fix the DD ambiguities and then introduce them into the model. For more information we refer to Jonkman (1998). We need to come up with a realistic and adequate covariance matrix of the GPS observables in the case of the linear and simple GFOM. To this end we apply the LS-VCE. The construction of the covariance matrix (for undifferenced observables) starts from a scaled unit matrix per observation type and takes place in different steps. In this application, the following three characteristics of GPS observables will be evaluated: (1) the precision of the phase and code observations, (2) correlation between the observables on L1 and L2 frequencies, (3) satellite elevation dependence of the observables precision. Noise characteristics of the GPS observables have recently been assessed by different authors. We refer to e.g. Bischoff et al. (2005, 2006). A data set was obtained from the Delfland 99 campaign in the Netherlands; one hour of Trimble 4000SSI zero baseline data, 8 satellites with four observation types, namely C1-P1-L1-L2, and a 1 sec interval. In the sequel, the estimated (co)variance components for this receiver, over 3600 epochs divided into 360 10-epoch groups, will be presented. The goal now is to estimate one variance component for each observation type when considering 10 epochs of all observations. We neglect here the satellite elevation dependence of the observables precision, the time correlation, the covariance

between channels, and the covariance between different observation types. The final (co)variances can be obtained by multiplying the estimated (co)variance components with their a-priori values 302 cm2 and 32 mm2 for undifferenced code and phase observables, respectively. Note that the estimation of two individual variance components for L1 and L2 carrier phase data would cause the VCE problem to be ill-posed. The ill-posedness will be removed if we estimate one single variance component instead. Figure 1 shows the groupwise estimates of variance components in the last iteration, using the full hour of the data for the L1/L2 phase and C1 and P2 code observations of Trimble receiver. The estimated factors, if multiplied with their initial values given in the cofactor matrices, give the final estimates. In Figure 1 the estimated variance components grow towards the end of the graph. It is likely because of one satellite, which is setting and has the lowest elevation angle (nearly 10◦). Table 1 gives the standard deviation estimates (square-root of variance components) as well as their precision (in terms of undifferenced observables). The results indicate that the noise of GPS observations is about 0.3 mm, 10 cm, and 16 cm for phase, C1, and P2, respectively. The precision of the estimates are at micrometre and millimetre level for phase and code observations, respectively. In addition to the variances, one can for instance estimate the covariance between C1 and P2. The satellite elevation dependence of the observables precision, the time correlation and the covariance between channels are disregarded. Figure 2 shows the groupwise estimates of the correlation

C1 L1/L2 P2

1 Variance component of observations

components, namely Q σˆ = N −1 which offers us measures of precision for the estimators.

281

0.8 0.6 Mean: 0.2673

0.4

0.2 Mean: 0.0987

0

Mean: 0.0095

0

50

100

150

200

250

300

350

10−Epoch Group

Fig. 1. Variance components groupwise estimated for L1 and L2 phase and C1 and P2 code observables using LS-VCE. Factors are to be multiplied by 302 cm2 and 32 mm2 for code and phase observables, respectively.

282

A.R. Amiri-Simkooei et al.

Table 1. Standard deviation estimates of phase and code observables as well as their precision obtained using LS-VCE

σˆ (mm)

σσˆ (mm)

L1/L2 C1 P2

0.29 94.25 155.09

0.001 0.410 0.510

coefficient. As can be seen, the mean is around ρˆ = 0.44 and the estimates do not average out. In order to test the significance of the correlation coefficient, one needs to know the distribution of ρ. ˆ For some special cases it is possible to determine the distribution. But in general one will have to be satisfied with an approximation using a normal distribution, which is not unrealistic when the redundancy of the model is large. The correlation between C1 and P2 code observations for the 4000SSI turns out to be significant. This is verified when we compare the mean correlation coefficient with its precision, namely σρˆ = 0.007. This can also be simply resulted from the Chebyschev inequality even when one does not specify a distribution for ρ. ˆ To evaluate the satellite elevation dependence of the GPS observables, 3 satellites have been employed. Figure 3 shows the groupwise estimates of variance components using the full hour of the data (C1 code) of the satellites PRN 05, 29 and 09. The variance components computed for satellite PRN 09, with the lowest elevation angle, are larger than those estimated for satellites PRN 05 and 29. Also, as the elevation angle decreases, a positive trend is observed (for the last groups, on average, the estimated variance components are larger than those

Correlation between P2 and C1

1

0.5

0 Mean: 0.44

−0.5

−1 0

50

100

150

200

250

300

350

10−Epoch Group

Fig. 2. Groupwise estimated correlation coefficient between C1 and P2 codes obtained from LS (co)variance estimates.

Variance component of C1

Observation type

PRN 05 mean: 0.0263 PRN 29 mean: 0.0687 PRN 09 mean: 0.1756

1.2 1 0.8 0.6 0.4 0.2 0 0

50

100

150 200 250 10−Epoch Group

300

350

Fig. 3. Groupwise estimated variance components of C1 code estimated for different satellites. Factors are to be multiplied by 302 cm2 for code observables.

of the first groups). Another point is that the variance components estimated for satellites PRN 05 and 29 are negatively correlated. This can also be obtained from the covariance matrix of the estimates Q σˆ = N −1 . The correlation coefficients between satellites PRN 05 and 29 are ρˆφ = −0.70, ρˆc1 = −0.63 and ρˆ p2 = −0.67, for phase, C1, and P2, respectively. This implies that the precision of the data of the satellites PRNs 29 and 05 is not much different. This makes sense since they both have high elevation angles. The numerical results indicate that the noise of satellites PRN 05-29-09 observations is about 0.20.2-0.8 mm, 5-8-13 cm and 4-12-33 cm for phase, C1, and P2, respectively. The precision of these estimates is at a few micrometre level, one millimetre level, and a few millimetre level, respectively.

3 GPS Coordinate Time Series In this section we assess the noise characteristics in time series of daily position estimates for permanent GPS stations using LS-VCE. The precision of these estimates is often assessed by their repeatability defined by the mean squared error (MSE) of individual coordinate components (i.e. north, east, and up) about a linear trend. Except for a significant episodic deformation, such as large earthquakes, a linear trend can be a good representation of the (long term) deformation behavior. Therefore, the site velocities are usually determined by linear regression of individual coordinate components. Previous work reveals the presence of white noise, flicker noise, and random walk noise in the GPS time series (see e.g. Langbein and Johnson, 1997, Zhang et al., 1997). If we now assume that the time series of GPS coordinates are composed of white


noise with variance σw2 , flicker noise with variance 2 , the σ 2f , and random walk noise with variance σrw covariance matrix of the time series can be written as 2 Q Q y = σw2 I + σ 2f Q f + σrw rw where I is the m × m identity matrix, and Q f and Q rw are the cofactor matrices relating to flicker noise and random walk noise, respectively. The structure of the matrix Q y is known (through I , Q f , and Q rw ), but the contributions through σw , σ f , and σrw are unknown. The 2 can now be variance components σw2 , σ 2f and σrw estimated using the method of LS-VCE. We analyze global time series of site positions which are supposed to have more noise than those from a regional solution. The daily GPS solutions of 5 stations, namely KOSG, WSRT, ONSA, GRAZ and ALGO, processed by the GPS analysis center at the Jet Propulsion Laboratory (JPL), are adopted. In geophysical studies, for most available time series, 2 is estimated along with σ 2 only one of σ 2f and σrw w (Mao et al., 1999). One advantage of LS-VCE is the possibility of hypothesis testing with the stochastic model. Using the w-test statistic, we can in fact decide which noise component, in addition to white noise, is likely to be present in the time series; either flicker noise or random walk noise. Based on the results (not included here), a combination of white noise plus flicker noise turns out in general to best characterize the noise in all three position components (see Amiri-Simkooei et al., 2007). Table 2 gives the white and flicker noise amplitudes for two stochastic models. We find in general that the horizontal components are less noisy than the

283

vertical components by a factor of 2–4. Compared to the white noise model only, the amplitude of white noise for the white noise plus flicker noise model is about 30% smaller. There exists significant flicker noise in the data (compare flicker noise amplitudes with their precision). We have obtained that the (formal) standard deviations of the site velocity for white noise plus flicker noise model compared to those for the pure white noise model are larger by factors of 10–15. Therefore, the simple pure white noise model gives too optimistic results for the site velocity uncertainty. A significant and comparable amount of flicker noise (between sites) may reflect a common physical basis, such as seasonal atmospheric mass distribution, atmospheric noise, or second order ionospheric effects. Reduction in time-correlated noise from global solutions to regional solutions suggests that some of the noise is spatially correlated (Williams et al., 2004). An issue related to the noise in global time series is the impact of spatial correlation on the rate uncertainties. Using the LS-VCE, we have estimated the spatial correlation, each time between two stations whereby we obtain one correlation coefficient for each noise component. Table 3 gives the numerical results. The table includes the spatial correlation coefficients of noise components (white and flicker noise). Both noise components seem to be spatially correlated to some extent. The spatial correlation of white noise (absolute values) is less than that of flicker noise, on average, by factors of 0.90, 0.70, and 0.65 for

Table 2. White noise and flicker noise amplitude estimates obtained by LS-VCE for north, east, and up components of site time series for two stochastic models (white noise only: WN, combination of white noise and flicker noise: WN+FN); σ is standard deviation of estimator; N: north, E: east, and U: up component

WN (mm)

WN (mm)

+

FN (mm)

Site Code

N

E

U

N

E

U

N

E

U

KOSG σ

3.34 0.04

3.44 0.04

7.45 0.09

2.45 0.05

2.54 0.05

5.21 0.12

3.41 0.18

3.67 0.18

8.82 0.40

WSRT σ

2.76 0.04

2.82 0.04

7.12 0.11

2.12 0.05

2.35 0.05

5.08 0.14

2.58 0.18

2.37 0.19

8.24 0.47

ONSA σ

3.35 0.04

3.65 0.04

7.85 0.10

2.54 0.05

2.70 0.05

5.28 0.13

3.39 0.18

3.45 0.19

9.62 0.42

GRAZ σ

3.74 0.05

4.75 0.06

9.12 0.11

2.81 0.06

3.62 0.07

6.53 0.15

3.62 0.20

4.16 0.25

9.90 0.49

ALGO σ

3.62 0.04

3.60 0.04

8.22 0.10

2.32 0.05

2.77 0.06

5.20 0.13

3.87 0.17

3.67 0.19

9.71 0.40

284


Table 3. Estimated spatial correlation coefficients of white noise (top) and flicker noise (bottom) components obtained from LS (co)variance estimates (using LS-VCE). Functional model consists of two time series, i.e. between corresponding N: north, E: east, or U: up components, and stochastic model includes four white and flicker noise variance components as well as two covariances of noise components between the two series. This table presents only correlation coefficients. Precision of estimators ranges from 0.01 to 0.10


WN

FN

Between Sites

N

E

U

KOSG KOSG KOSG KOSG WSRT WSRT WSRT ONSA ONSA GRAZ

WSRT ONSA GRAZ ALGO ONSA GRAZ ALGO GRAZ ALGO ALGO

0.85 0.69 0.64 0.16 0.79 0.76 0.30 0.62 0.11 0.14

0.61 0.46 0.38 −0.17 0.51 0.48 −0.13 0.46 −0.18 −0.12

0.70 0.52 0.30 −0.20 0.55 0.49 −0.09 0.39 −0.13 −0.09

KOSG KOSG KOSG KOSG WSRT WSRT WSRT ONSA ONSA GRAZ

WSRT ONSA GRAZ ALGO ONSA GRAZ ALGO GRAZ ALGO ALGO

0.92 0.80 0.63 0.27 0.81 0.75 0.20 0.62 0.27 0.32

0.89 0.69 0.51 −0.45 0.86 0.50 −0.07 0.49 −0.26 −0.35

0.88 0.72 0.69 −0.15 0.79 0.83 −0.28 0.57 −0.22 −0.29

north, east, and up components, respectively. The maximum correlations for both noise components have been obtained between the nearest sites, i.e. between KOSG and WSRT (they are only 100 km apart). This confirms that the noise has a common physical basis. Over the largest station separation (between ALGO and other sites), the spatial correlation is the lowest for the north component. It becomes negative for the east and up components. These all together confirm that the site velocity between stations will be correlated as well. If one treats the time series individually, the correlation between time series should be added after into the covariance matrix of the site velocities.

4 GPS Receiver Noise Characteristics We now consider a nonlinear variance component estimation problem. For this purpose we apply the LS-VCE to covariance functions. Using the Taylor series expansion, it is in principle possible

to linearize the nonlinear stochastic model. When expanded into the Taylor series, the covariance matrix can be written as Q y = Q(σ ) ≈ Q 0 + p k=1 σk Q k . We can therefore apply the LS-VCE to estimate σ (see Teunissen and Amiri-Simkooei, 2007). Since we linearize a nonlinear function, the solution should be sought through an iterative procedure. We can iterate until the estimated (co)variance components do not change by further iterations. The goal, as an example, is to assess the noise characteristics of a GPS Trimble 4000SSI receiver (again data from the Delfland 99 campaign). Our point of departure here is the zero baseline time series. To obtain such baseline components, the single difference phase observation equation is employed. Thedata collected by the 4000SSI were static, but they were processed in kinematic mode (new unknown coordinates for every epoch). Baseline components and differential receiver clock biases along with double difference ambiguities were estimated by least-squares. We used the LAMBDA method to fix the ambiguities (see Teunissen, 1993). We therefore introduced the fixed ambiguities into the model (see Amiri-Simkooei and Tiberius, 2007, Tiberius and Kenselaar, 2000). We have now time series of zero baseline components with one second interval. This is considered as input for further assessment by LS-VCE. We will focus on time correlation. In practice, covariance functions are formed by combining a small number of simple mathematically acceptable expressions or models. We will apply the method to the autoregressive noise (model I) and Gaussian noise (model II) models to the real data of the zero baseline test. The covariance functions related to these models are Q 1 (τ ) = σc2 exp (−ατ ) and Q 2 (τ ) = σc2 exp (−ατ 2 ), respectively. The parameters α and σc2 are the time-scale (time-constant) and variance of the noise process, respectively. Both α and σc2 are assumed to be unknown and should be estimated using the nonlinear LS-VCE. In each noise model, we will also include a white noise variance component, namely σw2 . Therefore, for each model, three unknown parameters are to be estimated; the variances σw2 and σc2 and the time-scale α. The numerical results are given in Table 4. The results show that the 4000SSI is not free from time correlation. The variance components σw2 and σc2 of both stochastic models, when compared to their precision, are significant. This confirms that the time series contain white and colored noise components. The time-scale parameter α is on average 0.16 sec−1 and 0.04 sec−2 for the autoregressive and

North

(a)

285

Empirical and theoretical autocorrelation functions −− Autoregressive noise (b) 1

1

North


0.5

0.5

0

0 0

5

10

15

20

0

25

East

East

0.5

5

10

15

20

15

20

25

5

10

15

20

25

5

10

15

20

25

0.5

0

25

1

1

0.5

0.5

Up

Up

10

0

0

0

0 0

5

1

1

0

Empirical and theoretical autocorrelation functions −− Gaussian noise

5

10

15

20

25

0

Lags (sec)

Lags (sec)

Fig. 4. Empirical (light line) and theoretical (black line) autocorrelation functions. Dashed lines show 95% confidence interval of theoretical autocorrelation function; white plus autoregressive noise model (a); white plus Gaussian noise model (b).

Gaussian noise models, respectively. This implies that there exist time correlation over about 10–20 seconds. One can also verify this by the empirical autocorrelation function which can simply be obtained from the least-squares residuals eî as σˆ τ = ( m−τ e ˆ eî+τ )/(m − τ ), τ = 0, ..., m − 1 (see i i=1 Teunissen and Amiri-Simkooei, 2007). On the other hand, we can define the theoretical autocorrelation function which is obtained from the LS estimates of ˆ Figure 4 shows the empirical and theσˆ w , σˆ c , and α. oretical autocorrelation functions which match each other well. Note that the empirical autocorrelation function resulted from the weighted LS-VCE when the weight matrix is taken as an identity matrix. The presence of a sharp bend in both the empirical and theoretical autocorrelation functions (at time lag of τ = 1 sec) confirms that the noise of the time series Table 4. Estimated variances σw2 and σc2 and time-scale α as well as their precision by LS-VCE; white plus autoregressive noise model (left); white plus Gaussian noise model (right)

model I Component σw2 N α σc2

model II

σˆ 0.039 0.149 0.195

σσˆ 0.003 0.013 0.013

σˆ 0.065 0.042 0.156

σσˆ 0.002 0.003 0.010

E

σw2 α σc2

0.017 0.146 0.058

0.001 0.014 0.004

0.025 0.036 0.047

0.001 0.002 0.003

U

σw2 α σc2

0.098 0.180 0.436

0.007 0.015 0.026

0.164 0.048 0.346

0.005 0.003 0.021

is not all colored but partly white. This is also verified by the numerical results (see Table 4).

5 Conclusions In this study we demonstrated that the general LSVCE can easily handle different models and serve different applications. We presented the results of three different GPS application examples of the method for which linear forms of the functional models were used. The goal was to assess the stochastic properties of GPS data. On the basis of the numerical results obtained, the following conclusions and remarks can be given:

r

r

r

The LS-VCE model is a powerful method for estimation of the stochastic model parameters. It also provides the precision of the estimators. The method has several other attractive features (see Teunissen and Amiri-Simkooei, 2007). This method can therefore be introduced as a standard method for estimation (and also testing) of the (co)variance components in the stochastic model. In the GPS geometry-free model, LS-VCE was used to assess the noise characteristics of GPS observables. As expected, the variance of a GPS observable generally depends on the elevation of the satellite. Also, significant correlation can occur between different observation types, e.g. between the C1 and P2 codes. This is a good motivation to study the GPS stochastic model in more detail. The LS-VCE was applied to data of various permanent GPS tracking stations. It revealed that both white and flicker noise components are significant in GPS coordinate time series. In fact,

286

r


ignoring the colored (flicker) noise, gives too optimistic results for the site velocity uncertainty. We also showed that both noise components are spatially correlated. The largest correlation coefficients were obtained between the nearest stations. This confirms that the noise has a common physical basis on the global time series. Finally, nonlinear LS-VCE was applied to assess the temporal correlation of GPS receivers based on covariance functions. The results showed that the 4000SSI GPS receiver is not free from time correlation. This was verified based on the empirical and theoretical autocorrelation functions.

References Amiri-Simkooei, A. R. and Tiberius, C. C. J. M. (2007). Assessing receiver noise using GPS short baseline time series. GPS Solutions, 11(1), 21–35. Amiri-Simkooei, A. R., Tiberius, C. C. J. M., and Teunissen, P. J. G. (2007). Assessment of noise in GPS coordinate time series: methodology and results, Journal of Geophysical Research, 112, B07413, doi:10.1029/2006JB004913. Bischoff, W., Heck, B., Howind, J., and Teusch, A. (2005). A procedure for testing the assumption of homoscedasticity in least squares residuals: a case study of GPS carrier-phase observations. Journal of Geodesy, 78, 397–404. Bischoff, W., Heck, B., Howind, J., and Teusch, A. (2006). A procedure for estimating the variance function of linear models and for checking the appropriateness of estimated variances: a case study of GPS carrier-phase observations. Journal of Geodesy, 79, 694–704.

Jonkman, N. F. (1998). Integer GPS ambiguity estimation without the receiver-satellite geometry. LGR-Series 18, Geodetic Computing Centre, Delft. Langbein, J. and Johnson, H. (1997). Correlated errors in geodetic time series: Implications for time-dependent deformation. Journal of Geophysical Research, 102(B1), 591–603. Mao, A., Harrison, C. G. A., and Dixon, T. H. (1999). Noise in GPS coordinate time series. Journal of Geophysical Research, 104(B2), 2797–2816. Teunissen, P. J. G. (1988). Towards a least-squares framework for adjusting and testing of both functional and stochastic model. Internal research memo, Geodetic Computing Centre, Delft. A reprint of original 1988 report is also available in 2004, No. 26, http://www.lr.tudelft.nl/mgp. Teunissen, P. J. G. (1993). Least squares estimation of the integer GPS ambiguities. In Invited Lecture, Section IV, Theory and Methodology, IAG General Meeting, Beijing, China. Teunissen, P. J. G. and Amiri-Simkooei, A. R. (2008). Least-squares variance component estimation. Journal of Geodesy (in press), doi 10.1007/s00190-007-0157-x. Tiberius, C. C. J. M. and Kenselaar, F. (2000). Estimation of the stochastic model for GPS code and phase observables. Survey Review, 35(277), 441–454. Williams, S. D. P., Bock, Y., Fang, P., Jamason, P., Nikolaidis, R. M., Prawirodirdjo, L., Miller, M., and Johnson, D. J. (2004). Error analysis of continuous GPS position time series. Journal of Geophysical Research, 109. Zhang, J., Bock, Y., Johnson, H., Fang, P., Williams, S., Genrich, J., Wdowinski, S., and Behr, J. (1997). Southern California Permanent GPS Geodetic Array: Error analysis of daily position estimates and site velocities. Journal of Geophysical Research, 102, 18035–18055.

Helmert Variance Component Estimation-based Vondrak Filter and its Application in GPS Multipath Error Mitigation X.W. Zhou, W.J. Dai, J.J. Zhu, Z.W. Li, Z.R. Zou Department of Survey Engineering and Geomatics, Central South University, Changsha, Hunan Province 410083, P.R. China, e-mail: [email protected]

Abstract. Vondrak filter is a unique smoothing method that aims to find a balance between the smoothness of the filtered data series and the closeness of the filtered series to the original one. The key element of the Vondrak filter is the determination of the smoothing factor, which controls the degree of smoothing. We propose in this paper a new smoothing factor determination method that is based on the Helmert variance component estimation for the Vondrak filter. Experiments with simulated and real datasets indicate that the proposed method can select the optimal smoothing factor, and separate the signals and random noise at different noise levels as long as the noise level is lower than the magnitude of the signals successfully. By exploiting the day-to-day repeating property of GPS multipath errors, we can use the proposed method to correct GPS measurements for the multipath errors. We first use the proposed method to separate the multipath signals from noise, and then use the separated multipath signals to correct the subsequent sidereal day’s multipath errors of GPS survey. The results show that the accuracy of the GPS positioning is improved significantly after applying the proposed methods. Comparisons with some well-known filters are also made. Keywords. Vondrak filter; helmert variance component estimation; GPS; multipath effects

1 Introduction Global Positioning System (GPS) has been used in a wide variety of high precise applications such as real-time positioning and deformation monitoring, due to a number of significant advantages that GPS has over other technologies. However, limited measurement accuracy has refrained from the technology being more widely adopted in such high precise applications. GPS observables are contaminated by various errors, but many of them (such as satellite and

receiver clock biases) can be eliminated by the double differencing approach used in relative GPS positioning. In addition, the distance-dependent errors (ionospheric and tropospheric refraction and delays, orbital errors) can also be nearly canceled by using the double-differencing approach when the baseline is short (< 3 km). Carrier phase multipath however cannot be canceled or mitigated using such a double-differencing approach. So multipath is the most limiting errors in short baseline applications. Significant progress on multipath mitigation has been achieved in recent years. Generally, the methods can be classified into two types: hardware and software methods. Special antenna design is the most common way to prevent multipath signals entering a receiver among the hardware method. Mitigation by software techniques mainly uses digital filtering that deals with the observables in the data processing stage, by exploiting the repeating property of the multipath. The commonly used digital filters include the Band-pass Finite Impulse Response Filter (FIR) (Han and Rizos 1997), the Wavelet Filter (Xia and Liu 2001, Xiong et al. 2004), the Adaptive Filter (Ge et al. 2000), and the Cross-validation Vondrak Filter (CVVF) (Zheng et al. 2005). Vondrak Filter (Vondrak 1967) is a method that aims to find a balance between the smoothness of the filtered data series and the closeness of the filtered series to the original one. Selecting an optimal smoothing factor is the key of the filter. Generally, the methods of selecting an optimal smoothing factor include frequency response method, observation-error method, general crossvalidation (GCV) method (Ding 1998), etc. Cross-Validation Vondrak Filter was developed to mitigate GPS multipath effect by Zheng et al. (2005). Reliable GPS multipath models for positioning series were acquired using this method and then applied to calibrate subsequent days’ multipath errors. In this method, Cross-Validation was used to select the smoothing factor. Its basic concept is to crossvalidate the filtered results derived from different 287

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smoothing factors with that of data samples. The smoothing factor that yields the smallest mean variance is selected as the optimal one. From the viewpoint of computation, CVVF is quite inefficient as it chooses the smoothing factor by the tedious try. Frequency response method needs to know the frequency of the data series and observation error method needs to know the stand variance of the data series (Ding 1998). In order to select the smoothing factor quickly and automatically, we propose a new smoothing factor determination method based on virtual observable and Helmert variance component estimation. The basic idea of this method is to regard the smoothing factor as the weight ratio of two types of observables, and to use the Helmert variance component estimation which is an unbiased estimation (Cui et al. 2001) to derive the optimal weight ratio (i.e., the optimal smoothing factor). For simplicity, we will call the proposed Vondrak filter, which uses virtual observable and Helmert variance component estimation to select smoothing factor as Helmert Vondrak Filter (HVF).

2 Vondrak Filter Method Mathematically the objective function of Vondrak filter is Vondrak (1967): Q = F + λ2 S = min

(1)

with F= S=

n i=1 n−3

absolute smoothing. Let the equation (1) multiplied by ε, where ε = 1/λ2 , we get Q = ε · F + S = min

(4)

In matrix form, equation (4) can be written as: Q = VT PV = min

(5)

with V= P=

V1 V2

=

ε · P1 0

0 P2

B1 B2

X−

f1 f2

(6) (7)

where ⎡ ⎢ ⎢ B1 = ⎢ ⎢ ⎣

⎤

1

⎥ ⎥ ⎥ ⎥ ⎦

1 ... 1

1 n×n √ ⎧ 6 (x i+2 −x i+1 )(n−3)/(x n−1 −x 2 ) ⎪ ⎪ B2 (i, i ) = (xi −x ⎪ i+1 )(x i −x i+2 )(x i −x i+3 ) ⎪ ⎪ √ ⎪ ⎪ 6 (x i+2 −x i+1 )(n−3)/(x n−1−x 2 ) ⎪ ⎪ ⎨ B2 (i, i + 1) = (xi+1 −xi )(xi+1 −xi+2 )(xi+1 −xi+3 ) √ 6 (x i+2 −x i+1 )(n−3)/(x n−1−x 2 ) B2 = ⎪ ⎪ ⎪ ⎪ B2 (i, i + 2) = (xi+2 −xi )(xi+2 −xi+1 )(xi+2 −xi+3 ) ⎪ ⎪ √ ⎪ ⎪ 6 (x i+2 −x i+1 )(n−3)/(x n−1−x 2 ) ⎪ ⎩ B2 (i, i + 3) = (x −x i )(x i+3 −x i+1 )(x i+3 −x i+2 ) i+3 (i = 1, . . . , n − 3)

pi ( y˜i − yi )

2

(2) f1 = [y1 , y2 , . . . , yn ]T ,

3

( y˜i )

2

(3)

f2 = 0,

P1 = P0 = diag [ p1 , p2 , . . . , pn ] ,

P2 = I,

i=1

Where y˜i is the smoothed value corresponding to observation yi ; pi is the weight of yi ; 3 y˜i is the third-order differentiation of smoothed values; and λ2 is a unit less positive coefficient that controls the degree of the smoothness of the filtered series.

By using the least-squares method, equation (5) can be resolved:

X = N−1 fe

n,1

(8)

n,n n,1

where

3 HVF Method

N = N1 + N2 = B1T P1 B1 + B2T P2 B2

From equation (1), it can be seen that when the coefficients λ2 → 0 and F → 0, the filtered values approach the measured ones, namely, ( y˜i − yi ) → 0, a rough curve will result and the operation is called absolute fitting. When λ2 → +∞ and S → 0 then F → min, namely, 3 y˜i → 0, a smooth parabola will be derived, and the operation is called

fe = fe1 + fe2 =

B1T P1 f1

+

B2T P2 f2

(9) (10)

It can be seen from above derivations that Vondrak filter is actually an adjustment problem, with two types of (pseudo-) observables involved. From the theory of adjustment, we know that when there are more than two types of observables in the

Vondrak Filter and its Application in GPS Multipath Error Mitigation

adjustment, their weight ratio must be determined appropriately to get accurate parameter estimation. Examining equation (7), we can see that the problem of determining the optimal smoothing factor ε for Vondrak filter has translated to the problem of determining the optimal weight ratio of different types of observables in the adjustment domain. In the theory of adjustment, the posterior variance component estimation methods including the Helmert variance component estimation are always used to determine the optimal weight ratio. In Helmert variance component estimation, the error vector V is used to estimate the variance, and then to modify the weights of different types of observables. With the new weights, the unknown parameters are re-estimated using least-squares method. These processes are repeated until the correct weight ratio is achieved and the unknowns are estimated optimally. The Helmert variance component estimation can be expressed as follows (Cui et al. 2001): 2

␴0i =

ViT Pi Vi n i − tr (N−1 Ni )

(i = 1, 2)

(11) 2

2

with equation (11), the posterior variances ␴01 , ␴02 of the two types of observables can be estimated. The estimated posterior variances are then used to modify the weights:

2

2

P1 = C/␴01 × P1 P2 = C/␴02 × P2

(12)

where P1 and P2 are posterior weights, P1 and P2 are a priori weights. Cis a custom constant. The new weights are then used to re-adjust the observation equations, and again the posterior variances and new weights are calculated. These processes are repeated until ␴201 = ␴202 . For the case of Vondrak Filter, we can set C = ␴202 , and only modify P1 in the iterative computations. Finally, the smoothing factor can be derived as:

ε = 1/λ2 = P1 (i, i )/P0 (i, i )

(i = 1, . . . , N) (13)

4 Simulation Studies The simulated test data are generated with the following model: (14) u t = yt + ε t Where εt is Gaussian white noise series with normal distribution, and yt is the signal component in the

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‘observable’ sequence u t . The simulated signals consist of three sinusoidal waves, with periods of 250 s, 100 s and 50 s, and a modulation signal with a period of 1000 s added to the sinusoidal wave of 250 s periods. The model for simulating the signals is then yt = 2.0 sin(2πt/1000) × sin(2πt/250) +(2πt/100) + 0.5(2πt/50) The data sampling interval is 1 s and the sample size is 2000. We simulate some signals at different noise levels (e.g. N(0, 0.2), it means the mean value of εt is 0 and the stand variance of εt is 0.2) and select three methods to determine the smoothing factor and filter the data serials respectively (cross-validation method, observation-error method and HVF method). In order to explain the problem more clearly, we give two formulas which are the RMS (root mean square) values of noise component εt and the correlate coefficients between the filtered signals u t and the true signals yt . See equations (15) and (16):

NRMS

N 1 = (u¯ t (i ) − u t (i ))2 N

(15)

i=1

Cov(u¯ t , yt ) R= ␴u¯ · ␴ y

(16)

Where u¯ t is the filtered value of u t . N is the length of the signals. Cov(u¯ t , yt ) is the co-variance between u¯ t and yt , ␴u¯ and ␴ y are the standard variances of u¯ t and yt . And R is the correlation coefficient between u¯ t and yt . It can be seen from equations (15) and (16) that, the better the filter effect the closer the value of NRMS is to the variance of εt and the closer the value of R is to 1. So we can compare these three methods. The performance of the three methods at different noise levels is as follows; see Tables 1–3. It can be seen from the Table 1, Table 2 and Table 3 that, the values of NRMS are close to stand variance of εt and R are close to 1 at different noise levels using these three methods. It means that these three methods can get the better results and that in the same noise level, the result of HVF is a little better than the other two methods. In addition, using the observation-error method need to know the variance of data series which couldn’t be obtained in most case and cross-validation method need to time after time to try. But HVF method for using the ratio of the variances of the data serials which can automatically adjust to determine the smoothing factors at different

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Table 1. The results and smoothing factors determined with cross-validation method Noise level

N(0,0.2)

N(0,0.5)

N(0,1.0)

N(0,1.5)

N(0,2.0)

N(0,3.0)

Smoothing factor NRMS R

1E-4 0.201 0.999

1E-4 0.502 0.997

1E-6 1.052 0.970

1E-6 1.551 0.963

1E-7 2.083 0.926

1E-8 3.136 0.825

Table 2. The results and smoothing factors determined with observation-error method Noise level

N(0,0.2)

N(0,0.5)

N(0,1.0)

N(0,1.5)

N(0,2.0)

N(0,3.0)


1E-4 0.201 0.999

1E-5 0.516 0.995

1E-6 1.052 0.970

1E-7 1.583 0.934

1E-7 2.083 0.926

1E-8 3.136 0.825

Table 3. The results and smoothing factors determined with HVF method Noise level

N(0,0.2)

N(0,0.5)

N(0,1.0)

N(0,1.5)

N(0,2.0)

N(0,3.0)


1.6E-4 0.200 0.999

2.8E-5 0.507 0.997

2.2E-6 1.039 0.977

4.2E-7 1.559 0.956

1.7E-7 2.071 0.938

6.4E-9 3.144 0.833

noise levels without knowing the stand variances of the data series.

5 Real Data Experiments and Analysis In order to test the HVF method, a real data experiment was conducted. The tested datasets are GPSderived height series H of a fixed station collected at 3 consecutive days. Since the baseline is very short

(about 100 m), and the reference station influenced with few multipath errors, the errors of receiver clock and satellite clock and other errors can be ignored in case of double difference and the multipath disturbance is becoming the most important error sources for the tested station (Zheng et al. 2005). Figure 1 shows the original coordinates series of H for the three consecutive days. It is clear that the coordinate series have a significant repeatability. According to this repeating property of multipath, we first filter the first day’s data series to derive the model of multipath, and then use the derived model of multipath to correct the data series of the other two days, using the HVF, db8, haar wavelet respectively (see Figure 2–4). To quantitatively evaluate the three filters, we calculate: (1) the correlation coefficients of the derived multipath model between day 1 and 2, and day 2 and 3, and (2) the RMS errors of the coordiate series before and after correction. The results are listed in Tables 4 and 5, respectively. From Table 4, we can see that Table 4. Correlation coefficients of multipath series After filtering

Fig. 1. Raw coordinates of H for the three consecutive days.

Day

Before filtering

HVF

db8

Haar

1–2 1–3

0.834 0.722

0.970 0.953

0.968 0.945

0.967 0.947

Vondrak Filter and its Application in GPS Multipath Error Mitigation

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Table 5. RMS error of coordinate series (mm) After correction Day

Before Correction

HVF

db8

Haar

2 3

5.99 6.11

3.26 3.50

3.44 3.74

3.47 3.76

Fig. 4. Result of de-noising using haar.

Fig. 2. Result after de-noising using HVF.

the correlation coefficients of the multipath models between different days are increased significantly after filtering with the three filters, though comparatively those by HVF are slightly higher. Table 5 however indicates that after the multipath corrections the RMS errors decrease by 50–60%, and comparatively those by HVF have the lowest RMS errors. Thus, the HVF method outperforms both the db8 and the Haar wavelet filters in the GPS multipath modeling and correcting. Another unfavorable properties for the wavelet filters we must mention is that it is easily influenced by the mother function (in other words, different mother functions can lead to different results), while there is no such constrain for the HVF method. However, HVF can’t care for different scales and easily affected by outliers which are the limits of it.

6 Conclusions

Fig. 3. Result of de-noising using db8.

In this paper, we proposed a new smoothing factor determination method, based on Helmert variance components estimation for the Vondrak filter. The main advantage of the Vondrak filter is that no predefined fitting function is required and it is applicable to data of equal and unequal intervals. Helmert variance components estimation is an unbiased estimation. HVF method just unites the advantages of these two methods. Experiments with simulated data and real GPS data indicate that the method can separate the noise from the signals and better model and thus correct the multipath errors. Future work will

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concentrate on testing the filter with other geodetic data and under different noise models.

Acknowledgments The work presented in this paper is supported by National Science Foundation of China (project No. 40404001). The author also would like to thank Hansjörg Kutterer for kind recommendations and heartily help.

References Cui X.Z. et al. (2001) Generalized Surveying Adjustment (in Chinese). Wuhan Technical University of Surveying and Mapping Press, Wuhan, pp.78–92. Ding Y.R. (1998) Processing methods of astronomical data, 1st edn (in Chinese). Nanjing University Press, Nanjing.

X.W. Zhou et al. Ge L., Han S., Rizos C. (2000) Multipath mitigation of continuous GPS measurements using an adaptive filter. GPS Solutions, Vol. 4, No. 2. pp. 19–30. Han S, Rizos C (1997) Multipath effects on GPS in mine environments. In:10th Int. Congress of the Int. Society for Mine Surveying, Fremantle, Australia, 2–6 November, pp. 447–457. Vondrak J. (1967) A Contribution to the problem of Smoothing Observational Data. Bull Astron Inst Czech, Vol. 20, No. 6., pp. 349–355. Xia L., Liu J. (2001) Approach for Multipath Reduction Using Wavelet Algorith. In: Proc. of ION 2001. pp. 2134–2143. Xiong Y.L., Ding X.L., Dai W.J., Chen W., Huang D.F. (2004) Mitigation of Multipath Effects Based on GPS Phase Frequency Feature Analysis for Deformation Monitoring Applications. In: Proc. of ION 2004. Kazimierz Dolny, Poland, June 14–17. Zheng D.W., Zhong P., Ding X.L., Chen W. (2005) Filtering GPS time-series using a Vondrak filter and cross-validation. Journal of Geodesy, Vol. 79, pp. 363–369.

Statistical Analysis of Negative Variance Components in the Estimation of Variance Components B. Gao, S. Li, W. Li Civil Engineering Department, Shijiazhuang Railway Institute, Shijiazhuang 050043, P.R. China, e-mail: [email protected] S. Li, X. Wang College of Geodesy and Geomatics, Wuhan University, Wuhan 430079, P.R. China, e-mail: [email protected] Abstract. This paper analyzes the problem of negative variance components in the estimation of variance components from the statistical point of view, based on two kinds of estimators of variance components. Our analysis shows that too low relative accuracy and too large errors of the estimated variance components in comparison with the variance component itself could be partly responsible for negative variance components, if the quadratic form of the residuals is indefinite. If initial values are supposed to be measurement- dependent, the estimate of variance components can be biased. The points are demonstrated by using simulated examples. In addition, a method for improving the convergence of the estimation of variance components has been presented and shown to improve the convergence performance of the estimation of variance components. Keywords. Variance component estimation, problem of negative variance components, statistic analysis, relative accuracy

1 Introduction Negative variances may occur in the estimation of variance components. Most of methods would produce negative variances (see, e.g., Searle (1971), unless the condition of positive variances is directly implemented (see, e.g., Schaffrin 1981; Xu et al. 2007). If no constraints are imposed, the estimation of variance components will result in a linear system of equations with variance components as unknowns, which provides no guarantee to obtain a non-negative solution (Wang 1987). In other words, negative variances are unavoidable in this case, although variances should be positive by definition. In order to get non-negative values, one may set ␴ˆ 2i to zero, if. ␴ˆ 2i < 0. Negative variance components were attributed to inadequate data, i.e. the number of data is not sufficient or the quality of

data is not good enough. (see, e.g., Sjöberg 1995). Thus acquisition of more data is necessary. Still other people maintain that the reason that causes negative variance components lies in the used estimation method itself, and once negative variance occurs, some other methods such as MINQUE method, maximum likelihood method should be used in stead of the unsuccessful one (Wu 1992). This practice is likely not applicable. On the other hand, Zhang et al. (2000) concluded that negative variances result mainly from rank deficit coefficient matrix of observation equations that causes the normal matrix to be ill-conditioned or simply rank defect, which likely is not correct, as may be seen in Xu et al. (2006a). This paper will investigate the cause of negative variance components from the statistical point of view. We find that negative variance components are closely related to the relative accuracy and biases of estimated variance components, if the estimators are not positive-definite. We will also study the bias issue of variance component estimation.

2 Canonical and Non-Canonical Estimators of Variance Components All estimators for variance components may be classified into two kinds: canonical and noncanonical. The former always makes positive variance components. For example, the estimated variance of unit weight is always positive in least squares adjustment, while the latter may produce negative values, if they are obtained by solving the linear system of equations. The probability distribution of the first class of estimators will be asymmetric with bound (0, +∞), as shown in Figure 1a. For the second class of estimators, the marginal probability density function f (␴ˆ 2i ) of the estimate ␴ˆ 2i may have a distribution defined over (–∞, +∞), if ␴ˆ 2i cannot be proved to be a 293

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components may be unbiased, which might be seen from the Helmert estimator of variance components. Consider the following Gauss-Markov model:

f ( σˆ i2 )

2

σˆ i

σ

0

ˆ E(L) = B X,

(a) f ( σˆ 2 )

f ( σˆ i2 )

2

2

σˆ i

σ

(b) f ( σˆ i2 )

f ( σˆ 22 ) f ( σˆ 12 )

with

2

σˆ i

σ

0

L=

(c)

Fig. 1. Probability distribution of ␴ˆ 2i .

0 −∞

f (␴ˆ 2i )d ␴ˆ 2i

L1 L2

P1 ,P = 0

0 P2

V =

V1 V2

B=

B1 B2

,

D(L 1 ) = ␴20 P1−1 ,

positive definite quadratic form which can only be defined in (0,+∞). In this case, the probability that ␴ˆ 2i falls into the interval (–∞, 0) is pi =

(2)

where all the measurements L are classified into two independent groups, L 1 and L 2 , with weight matrices P1 and P2 , respectively. Thus (2) can be rewritten as follows: V1 = B1 Xˆ − L 1 (3) V2 = B2 Xˆ − L 2

f ( σˆ 12 )

0

D(L) = ␴20 P −1

D(L 2 ) = ␴20 P2−1 , 1 2 T T N = B P B = B1 P1 B1 + B2T P2 B2 = N1 + N2 , W = B T P L = B1T P1 L 1 + B2T P2 L 2 = W1 + W2

let T =

(1)

which will always be larger than zero, and may depend on the accuracy of ␴ˆ 2i . ␴ˆ 21 and ␴ˆ 22 can be negative, as shown in Figure 1b and c. If ␴ˆ 21 is less accurate than ␴ˆ 22 , and if ␴ˆ 21 and ␴ˆ 22 are both unbiased, as shown in Figure 1b, the probability of ␴ˆ 21 falling into the interval (–∞, 0) is higher than that of ␴ˆ 22 . If both ␴ˆ 21 and ␴ˆ 22 are of the same accuracy, and if ␴ˆ 22 is biased negatively, as shown in Figure 1c, the corresponding probability of ␴ˆ 22 falling into the interval (–∞, 0) is higher than that of ␴ˆ 21 . In summary, from the statistical point of view, negative values of variance components may result from low precision and biases of the estimation, if they are not positive definite.

3 Bias Aspect in Connection of Estimation of Variance Components Methods for estimating variance components, such as the least squares method, MINQUE and maximum likelihood estimation, are equivalent or approximately equivalent (see, e.g., Yu et al. 1993; Grodecki 2001). In order to start the estimation of variance components, initial values have to be given. The estimated values are then used to form the new equations for estimation. The procedure is repeated until convergence is reached. The final estimate of variance

T1

,

T2

θ=

where T1 = E(V1T P1 V1 ), let S=

n 1 − 2tr(N −1 N1 ) + tr(N −1 N1 )2 tr(N −1 N1 N −1 N2

␴201

␴202

T2 = E(V2T P2 V2 ) tr(N −1 N1 N −1 N2 n 2 − tr(N −1 N2 ) + tr(N −1 N2 )2

then we have the expectation relation for the estimate of variance components:

or

Sθ = T

(4)

θ = S −1 T

(5)

substituting θˆ and Tˆ for θ and T becomes θˆ = S −1 Tˆ

(6)

with Tˆ =

Tˆ1 Tˆ2

,

Tˆ1 = V1T P1 V1 ,

θˆ =

␴ˆ 201 ␴ˆ 202

,

Tˆ2 = V2T P2 V2

The solution (6) should be solved iteratively. If the intermediate values are treated as independent

Statistical Analysis of Negative Variance Components in the Estimation of Variance Components

of measurements, the estimate is unbiased; otherwise, due to the dependence of the estimator of variance components on the measurements, the estimate should be biased. Since the weights are iteratively solved, we have a nonlinear system of equations involving all the measurements. By using Taylor expansion, we may derive the biases, which will not be discussed here.

4 Improving the Convergence Rate and Behavior of Estimating Variance Components When estimating variance components, we found that in some cases, convergence highly depends on given initial values of weight of the measurements. In this section, we will try to investigate this dependence and will propose an algorithm to improve the convergence speed and behavior of estimating variance components. We observed from a large number of numerical examples of variance component estimation that the intermediate estimates will be oscillating with iterations. When the estimates are approaching to convergence, the oscillating amplitudes decrease gradually. Otherwise, if the estimates are diverging, the oscillating amplitudes will not decrease with the iteration procedure. Keeping this phenomenon in mind, we will propose a new algorithm by damping the amplitudes of oscillation of the estimated variance components. The algorithm will be developed and described in the following. First of all, suppose that by using a conventional technique of variance component estimation, we have the two estimates at the j th step as follows: j

␴ˆ 201 ␴ˆ 202

j j

j = ␴ˆ 201 =

5 Examples In order to demonstrate the improvement of convergence, we will use an example in Cui et al. (1992) and use the Helmert method to estimate the two variance components, with and without using our new algorithm. Although the two variance components from the two algorithms converge correctly to ␴ˆ 201 = 3.372,

j

j ␴ˆ 20 + ␴ˆ 20 1

2

2

⎫ ⎪ ⎬ ⎪ ⎭

␴ˆ 202 = 3.372

the numbers of iterations are equal to 25 and 21, respectively. Clearly, our algorithm has slightly improved the convergence. Now we simulate a leveling network, which consists of 10 benchmarks and 39 measurements, as shown in Table 1. All the lines ar e supposed to be of the same length. We then divide all these observables into two groups, each with 19 and 20 measurements, respectively. To confirm our convergence and/or divergence analysis, we assume that the standard deviations for the first and second groups of measurements are equal to 3 and 5, respectively. The results with different initial values are shown in Table2. Table 3 is made on the basis of the following three schemes: Scheme I: The standard deviations are assumed to be equal to 3 for the first group and 6 for the second group, respectively. Scheme II: The standard deviations are assumed to be equal to 3 for both groups. Scheme III: The standard deviations are assumed to be equal to 6 for the first group and 3 for the second group, respectively.

␴ˆ 02

To continue the next iteration, we compute the new set of two variance components as follows:

Although (6) and (7) are demonstrated with two variance components only, they can be readily extended to the case with many more variance components, which will not be discussed here.

j

␴ˆ 01 ,

295

The results in Tables 2 and 3 have clearly demonstrated the significant improvement of our method

(7)

from which, we can further compute the weights for the next iteration: ⎫ j +1 ⎪ P1 = 1 ⎪ ⎪ j ⎬ 2 ␴ ˆ 01 j +1 (8) P2 = j ⎪ ⎪ ⎪ ␴ˆ 202 j ⎭ P2

Table 1. Connections of the leveling benchmarks 1 1 2 3 4 5 6

2

3

4

5

6

7

8

9

10

1

2 10

3 11 18

4 12 19 25

5 13 20 26 31

6 14 21 27 32 36

7 15 22 28 33 37

8 16 23 29 34 38

9 17 24 30 35 39

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problem with Helmert method for variance component estimation.

Table 2. Convergence comparison Initial values 0.2

0.3 0.4 0.5 0.6 0.7

Convergence Helmert method D Our method C

D C

D C

C C

D C

D C

Note: C- converged; D-diverged.

Table 3. Convergence dependence on standard deviations Schemes Initial values Convergence

Helmert method Our method

I

II

III

0.2 D C

1.0 C C

4.0 C C

Note: C- converged; D-diverged.

over the conventional Helmert method. In particular, they have shown that our modified method has almost produced correct, positive and convergent variance components.

6 Conclusions To summarize, if an estimated variance component cannot be represented as a positive definite or positive semi-definite quadratic form, such a variance component would highly likely take negative values. In this case, we have analyzed the effect of statistical aspects of variance components estimations on negative values of variance components. We have also proposed an algorithm to improve the convergence

References Cui X., Yu Z., Tao B., Liu D., Yu Z. (1992). General Adjustment of Observation (2nd Ed). Publishing House of Surveying and Mapping, Beijing. Grodecki J. (2001). Generalized Maximum-likelihood Estimation of Variance-Covariance Components with Noninformative Prior. J. Geod., 75, pp. 157–163. Schaffrin B. (1981). Auslgeichung mit Bedingungs-Ungleichungen, AVN, 88, pp. 227–238. Searle S.R. (1971). Linear models. Wiley, New York. Sjöberg L. (1995). The Best Quadratic Minimum Bias Nonnegative Definite Estimator for an Additive Two Variance Component Model. Manuscr. Geodaet., 20, pp. 139–144. Wang G. (1987). Theory and its Applications of Linear Mode. Publishing House of Anhui Education, Hefei. Wu X. (1992). Helmert-WF (weight factor) Variance Component Estimation. J. Wuhan Tech. Univ. Surv. Map., 17(4), pp. 1–10. Xu P.L., Shen Y.Z., Fukuda Y., Liu Y.M. (2006a). Variance Component Estimation in Linear Inverse ill-posed Models, J. Geod. 80, pp. 69–81. Xu P.L., Liu Y.M., Shen Y.Z., Fukuda Y. (2007). Estimability Analysis of Variance and Covariance Components, J. Geod., 81, pp. 593–602. Yu Z., Tao B., Liu D., Zhang F. (1993). The Paper for Theory & Application of Survey Adjustment Models & Model Errors. Publishing House of Surveying and Mapping, Beijing. Zhang F., Feng C., Huang C., Li Y.-Q. (2000). Application of the Improved Helmert Method for Variance Component Estimation in Precise Orbit Determination. Acta Geod. Cartogr. Sinica, 29(3), pp. 217–223.

A Method to Adjust the Systematic Error along a Sounding Line in an Irregular Net M. Li, Y.C. Liu Department of Hydrography and Cartography, Dalian Naval Academy, Dalian 116018, P.R. China; Geomatics and Applications Laboratory, Liaoning Technical University, Fuxin 123000, P.R. China; Institute of Surveying and Mapping, Information Engineering University, Zhenzhou 450052, P.R. China Z. Lv Institute of Surveying and Mapping, Information Engineering University, Zhenzhou 450052, P.R. China J. Bao Department of Hydrography and Cartography, Dalian Naval Academy, Dalian 116018, P.R. China Abstract. It had been discussed how to detect and adjust the systematic error in the data from a regular net before, in which each reference line crosses all main scheme sounding lines. However, in most practical cases, a reference line may not cross all main scheme sounding lines for the reason of irregular survey area, and they may form an irregular net. It is discussed how to determine and adjust the systematic error along a sounding line in an irregular net further in this paper. Firstly, a lot of crossing points and segments of sounding lines should be chosen out from an irregular net. Secondly, a method has been developed, in which the formulae of adjustment are given for different patterns of segments of sounding lines. Lastly, the method has been practiced by using observed data. The results show the method works well to improve the accuracy and efficiency of soundings in traditional single-beam hydrographic surveys. The principle and method in this paper may be widely used in the data processing for other marine survey, such as marine gravitimetry and magnetic survey. Keywords. Systematic error along a sounding line, irregular net, adjustment

1 Introduction Generally, it is required that every reference line should cross all of main scheme lines, which form a regular net in hydrographic surveying, in order to get discrepancies at their crossing points, see for example S44 (1998) and M13 (2005). However, limited with the surveying environment at sea, it is difficult, sometimes impossible to fulfill the requirement that each reference line cross all of main lines, and all of lines become an irregular net. How can systematic

errors in the data be detected and adjusted? How can the accuracy of data be estimated in a hydrographic net? An adjustment method has been established to detect and correct the systematic error along a sounding line in a regular net, see Liu et al. (2002, 2004) and Liu (2003), which can improve the accuracy of sounding data and assess the data quality in a rigorous way of statistic theory. Here, the adjustment method is developed and the models are established to detect systematic error in an irregular net further, in order to assess and improve the accuracy of data from an ordinary surveying area.

2 Irregular Sounding Net 2.1 Lines and Their Crossing Points in a Net With limits of shape of surveying area, a sounding line may be composed of straight and curvilinear segments and lays out disorderly, which may cause the following results: (1) a reference line may not cross all of scheme main lines, or (2) may cross a scheme main line twice or more, and (3) a main line may cross itself or another main line. Under the above circumstance, the net composed of all lines become irregular. The crossing points in an irregular net will be poorly proportioned in the whole surveying area. The sounding lines and their crossing points in an irregular net are shown as Figure 1. From Figure 1, there are straight and curvilinear segments on some sounding lines, and there are no straight or linear segments on other lines. In fact, the plane layout of sounding line may show the effects of errors on the sounding data are changed or not, for the change of the errors often depends on the change of the velocity, motion and heading of survey boat, i.e. the change of surveying environment. In a general way, it is a matter of fact that the surveying environment on a straight segment may be 297

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Fig. 2. Three types of crossing points ( stands for good crossing points, ◦ stands for ordinary crossing points, and • stands for bad crossing points).

crossing points are the points between a straight and curved segments, and the bad crossing points is the crossing points between two curved segments. The three types of segments crossing points are shown as Figure 2. Fig. 1. Regular and irregular segments in an irregular net.

3 Models of Irregular Net steady or invariant, but that on a curvilinear segment is changed. Hence, the systematic error on a straight segment may be treated as a relatively stable constant, and will be embodied in the differences of sounding data at crossing points between main scheme lines and reference lines; and the systematic error on a curved segment may change and could not be regarded as a stable constant to detect. Therefore, the differences of sounding values at crossing points on a straight segment can be used to detect and find the systematic errors in order to improve the sounding data. 2.2 Segments of Line and Classification of Crossing Points Generally, a sounding line can be classified into three types of segments: good, ordinary and bad. The good segment is usually a straight one on which the systematic error is steady. The ordinary segment is linear or near linear on which the systematic error may be invariant. The bad segment is curved disorderly on which the surveying environment is variant and the systematic error is not a constant. For the good and ordinary segments, the systematic error may be steady and embodied in the difference at their crossing points, and can be detected and adjusted by the adjustment theory, see Liu et al. (2002, 2004) and Liu (2003). Consequently, the crossing points on the sounding line are classified into three types: good, ordinary and bad, too. The good crossing points is the points between two straight segments; the ordinary

An irregular net consists of good segments, but the length of good segments may be not the same, which forms an irregular net with four patterns as Figure 3. In Figure 3, each good segment of reference line in the pattern (a) cross all of the good segments of main lines, which is similar to the grid pattern, see Liu et al. (2002, 2004). In patterns (b) and (c), each linear segment of reference line can not cross all of the linear segments of main lines; there are nonexistent crossing points in the net. In pattern (d), there is only one reference segment, and it crosses all main lines. In fact, a layout of main lines and reference lines in a net may include patterns (a), (b), (c) and (d). So, the adjustment models for patterns (a), (b), (c) and (d) are established respectively at first, and then combined into a general model for the irregular net in a surveying area.

(a)

(b)

(c)

(d)

Fig. 3. The patterns of an irregular net.

A Method to Adjust the Systematic Error along a Sounding Line in an Irregular Net

3.1 Model of Pattern (a) In pattern (a), there are m linear segments of main lines, n linear segments of reference lines, and consequently there are (M =)m × n crossing points between them. Let Di, j be the sounding data on i th main segments at the crossing point between i th main segments and j th reference segments, and D˜ i, j the sounding data on the j th reference segments at the same crossing point, the difference in depth L i, j , i.e. L i, j = Di, j − D˜ i, j , where i = 1, . . ., m and j = 1, . . ., n. Hence, all soundings and depth differences at the crossing points in pattern (a) could be shown in matrices as follows ⎡ ⎢ ⎢ L =⎢ ⎣

L 1,1 L 2,1 .. .

L 1,2 L 2,2 .. .

L m,1

L m,2

··· ··· .. . ···

L 1,n L 2,n .. .

⎥ ⎥ ⎥ ⎦

L m,n

(1) m×n

Di, j = di, j + ai + δi, j D˜ i, j = di, j + b j + δ˜i, j

L i, j = Di, j − D˜ i, j = ai − b j − vi, j

(3)

And then the adjustment model for pattern (a) could be obtained as follows, see Liu et al. (2004) ⎧ C M×(m+n) X (m+n)×1 − L M×1 , ⎪ ⎨ VM×1 = −1 2 L = PL σ0 ⎪ ⎩ GT P X =0

(4)

(m+n)×1

V M×1 =(v1,1 , v2,1 , . . ., vm,1 , v1,2 , . . ., vm,2 , . . ., v1,n , . . ., vm,n )

Em

−em η1T −em η2T .. .

⎥ ⎥ ⎥ ⎦

−em ηnT

M×(m+n)

T em = (1, · · · , 1)1×m

j

η Tj = [0, · · · , 0, 1, 0, · · · 0]1×n Em an m × m unit matrix

L M×1 is a column vector obtained by arranging matrix L in equation 1 in columns L is the variances and covariances of L M×1 P L is a weight matrix of L M×1 , usually PL is an unit matrix σ02 is an arbitrary positive value, standing for standard deviation here G T = (1, · · · , 1)1×(m+n) P X is a datum matrix. 3.2 Model of Patterns (b) and (c) In nature, pattern (b) is the same as pattern (c). Suppose that there are m linear segments of main lines, n linear segments of reference lines, and N crossing points between them. Let m k be the number of crossing points on kth reference segment, then N =

n k = 1 m k , where m k ≤ m. Similar to pattern (a), the difference in depth at crossing points could be shown as follows ⎡ ⎢ ⎢ L=⎢ ⎣

L 1,1 L 2,1 .. . ×

where

T

⎢ ⎢ C=⎢ ⎣

(2)

where di, j is the true depth at the crossing points, ai the systematic error on i th main segment, b j the systematic error on j th reference segment; δi, j the random error of i th main segment at crossing point, δ˜i j the random error of j th reference segment at crossing point. Here, suppose δi, j , δ˜i, j independent and submitted to normal distribution N(0, σ 2 ), and let vi, j = δ˜i, j −δi, j , thus vi, j submitted to N(0, 2σ 2 ). And then L i, j in equation 1 can be expressed as below

1×(m+n) X (m+n)×(m+n)

Em Em .. .

⎤

X T = (a1 , a2 , · · · am , b1 , b2 , · · · bn )1×(m+n)

⎤

Usually, L = 0 for there is errors in soundings. So the error models are given as below

299

⎡

L 1,2 × .. . L m,2

··· ··· .. . ···

L 1,n L 2,n .. . L m,n

⎤ ⎥ ⎥ ⎥ ⎦

(5) m×n

where × stands for nonexistent crossing points. Further, the adjustment model for patterns (b) and (c) can be shown as follows ⎧ C N×(m+n) X (m+n)×1 − L N×1 , ⎨ VN×1 = −1 2 L = PL σ0 ⎩ T G 1×(m+n) PX (m+n)×(m+n) X (m+n)×1 = 0

(6)

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where ⎡ ⎢ ⎢ C=⎢ ⎢ ⎣

ωm 1 ,m ωm 2 ,m .. .

ωm n ,m

−em 1 η1T −em 2 η2T .. .

σˆ =

⎤ ⎥ ⎥ ⎥ ⎥ ⎦

T −em n ηm

, M×(m+n)

and ωm k ,m is an m k × m matrix which obtained by removing m −m k rows of nonexistent crossing points from Em . 3.3 Model of Pattern (d) In pattern (d), there are one reference segment and m main segments, and m crossing points. Obviously pattern (d) is a special case of pattern (a) when n = 1. For there is only one crossing point on each main segment, it is impossible to detect systematic error on main segment with one crossing point. Hence, the error model is shown as follows Di,1 = di,1 + δi,1 (7) D˜ i,1 = di,1 + b + δ˜i,1

L 2i, j /S

where σˆ is the mean square root of the differences at crossing points, S the number of crossing points. Usually, an overall net in an area may consist of patterns (a), (b), (c), (d) and bad segments. So the models above should be used reasonably according to the overall shape of the sounding data.

4 Examples There are 144 sounding lines surveyed in South China Sea, on July 1, 2002. The plane layout of sounding lines is not regular, shown as Figure 4. According their layout, the sounding lines can be divided in two parts, regular and irregular. The regular part is shown as Figure 5 and the irregular part shown as Figure 6.

where Di,1 is the sounding data on i th main segment at the crossing points, and D˜ i,1 is the sounding data on reference segment at the same crossing points, b is the systematic error on the reference segment, di,1 is the true depth at the crossing point, δi,1 is the random error on i th main segment at the crossing points, δ˜i,1 is the random error on the reference segment at the crossing points. And the adjustment model can be obtained as Vm×1 = −em b − L m×1 ,

L

= PL−1 σ02

(8)

Fig. 4. The plane layout.

Equation (8) shows, in fact, that m main segments are used to check one reference segment while one reference segment is used to check m main segments in practice of hydrographic survey. Fig. 5. The regular part.

3.4 Model of Irregular Segment For the surveying environment of bad segments is unsteady, it is very difficult to detect systematic errors from their crossing points. Hence, a rough error model is given as follows

Di, j = di, j + δi, j D˜ i, j = di, j + δ˜i, j

(9)

Thus L i, j = vi, j from equation 1, and the following model can be used to assess the quality of bad segments, see Wells and Krakiwsky (1973).

(10)

Fig. 6. The irregular part of survey lines.

A Method to Adjust the Systematic Error along a Sounding Line in an Irregular Net

301

Table 1. Estimates of parameters in the first part No.

1

2

3

4

5

6

7

8

9

Main Ref.

0.03 0.00

0.02 –0.04

0.13 –0.05

0.11 –0.16

–0.01 0.07

–0.03 –

–0.09 –

–0.05 –

–0.18 –

No. Main

10 –0.25

11 –0.20

12 –0.08

13 0.08

14 0.37

15 0.18

16 0.16

17 –0.01

In the regular part there are 22 regular segments of which 17 main segments and 5 reference segments, and 85 crossing points between them. According to the model of the pattern (a), while PX is a unit matrix, the parameter value can be figured out as Table 1. σ1 is the standard deviation of the regular part and σ1 = 0.26 m. It is found that the estimates on the 14th main segment and 4th reference are significant by t-test method, and the sounding data on the two lines should be corrected (Liu et al. 2002, 2004, Liu, 2003). The irregular part of the net contains 122 sounding lines and 845 crossing points, and their layout is irregular and disorderly. However, most of the 122 sounding contains straight or near straight segments, and they could be used to detect systematic errors. So the irregular part can be divided into two sub areas, one of which contains good segments as Figure 7, the other contains many bad segments at and near the edge of the area shown as Figure 8. The first sub area in Figure 8 consists of 44 main segments and 5 reference segments, with the model of the patterns (b) and (c), the estimates of their parameter are obtained as Table 2. σ2 is the standard deviation of the first sub area in Figure 7 and σ2 = 0.24 m, near σ1 . The estimates on main segments 11, 23, 26, 27, 29, 30, 35, 39, 43

Fig. 7. The first sub area.

and 1,7 reference segments are significant by t-test method and the sounding data should be corrected, see Liu et al. (2002, 2004) and Liu (2003).

Fig. 8. The second sub area of the irregular part of the irregular part.

Table 2. Estimates of systematic parameters in the first sub area of the irregular part No.

1

2

3

4

5

6

7

8

9

Main Ref.

–0.01 0.21

–0.06 –0.07

–0.14 0.03

–0.04 –0.06

–0.04 –0.02

–0.08 –0.07

0.01 –0.29

–0.01 –

–0.08 –

No. Main

10 –0.11

11 –0.28

12 –0.24

13 –0.25

14 –0.15

15 –0.15

16 –0.03

17 0.08

18 –0.02

No. Main

19 –0.19

20 –0.08

21 0.02

22 –0.10

23 0.29

24 0.15

24 –0.13

26 0.37

27 0.51

No. Main

28 0.15

29 0.44

30 0.52

31 0.31

32 –0.12

33 0.01

34 0.14

35 0.25

36 –0.04

No. Main

37 0.11

38 –0.11

39 –0.28

40 –0.23

41 –0.17

42 –0.11

43 0.26

44 –0.13

302

In the second sub area in Figure 8, there are 649 crossing points, most of which are at the turn of ship track, and most of their valued is larger than 1 m and much more than the differences in the area of Figure 4. Most of them may be bad crossing points. It is shown that there are some unknown elements from the environments to affect the sounding data at the turning of ship track, or near the shore, and the difference at the bad crossing points is not suitable for assessing the quality of observed data.

5 Conclusions The net in a surveying area may include good, ordinary and bad segments of survey lines and become irregular, which cause troubles in detecting and adjusting systematic error in observed data. By classifying the soundings into three types: good, ordinary and bad according to the layout of the segments of survey lines, the adjustment models for good and ordinary data are given to detect and adjust systematic errors and assess the quality of sounding data. The bad data is not suitable to estimate systematic error in fact.

M. Li et al.

Acknowledgement Supported by the National Natural Science Foundation of China (No.40071070, No.40671161) and by Open Research Fund Program of the Geomatics and Applications Laboratory, Liaoning Technical University (No.200502) E-mail: [email protected]

References IHO (1998). IHO Standards for Hydrographic Surveys (4th Edition, Special Publication No.44, abbr. S44). IHO (2005). Manual on Hydrography (1st Edition, Miscellaneous Publication No.13, abbr. M13). Liu, Y. C. (2003). Space Structure and Data Processing for Marine Sounding. Publishing House of Surveying and Mapping. (in Chinese). Liu, Y. C., Li, M. C., and Huang, M. T. (2002) Rank-defect Adjustment Model for Survey-line Systematic Error in Marine Survey Net. Geo-Spatial Information, No.4: 14–20. Liu, Y. C., Li, M. C. and Xiao, F. M., et al. (2004). A Method for Detecting and Adjusting Systematic Errors of Singlebeam Sounding Data Acquired in a Grid Pattern. International Hydrographic Review, 5 No.1: 34–53. Wells, D. E., and E. J. Krakiwsky (1973): The Method of Least Squares. University of New Brunswick, Canada.

Research on Precise Monitoring Method of Riverbed Evolution J.H. Zhao School of Geodesy and Geomatics, Wuhan University, Wuhan, Hubei 430079, P.R. China H.M. Zhang School of Power and Mechanical Engineering, Wuhan University, Wuhan, Hubei 430072, P.R. China Abstract. Multibeam bathymetric system (MBS) is widely used for the monitoring of riverbed evolution now. However, the sound velocity in surveying water area is a key in MBS survey and sound ray tracing, and heavily influences on sounding accuracy of MBS and the accuracy of the monitoring riverbed evolution. In traditional MBS data processing, sound velocity profile (SVP) near surveying location is often used directly for sound ray tracing, which easily leads to obvious representation errors in depth and therefore influences the accuracy of monitoring riverbed evolution. In order to overcome the drawback of the traditional method, the paper presents a new method in acquiring the actual sound velocity variations by constructing a local sound velocity field (LSVF) in survey area. Before constructing LSVF, all SVPs observed in surveying water area are classified by Self-organizing Feature Maps (SOFM) neural network at first. Then, some representative SVPs are drawn from these classified SVPs for building LSVF. During building LSVF, a polynomial function is constructed with sound velocities and locations of these representative SVPs at a defined depth layer. According to depths of all SVPs, we can define many depth layers. Correspondingly, a cluster of functions in different depth layers can be produced. These functions are also namely LSVF. After constructing LSVF, the sound velocity at any location and depth can be calculated for sound ray tracing in MBS data processing and work for precise monitoring of riverbed evolution. An experiment was implemented for checking the accuracy of constructing LSVF. The traditional method and the new method in this paper are used for sound ray tracing in MBS data processing. The traditional method achieved the accuracy of 13–15 cm in monitoring riverbed evolution, while the new method reached 6 cm. The statistic result shows that LSVF can reflect accurately the actual velocity variation of surveying water area and increases the accuracy of monitoring riverbed evolution.

Keywords. multibeam bathymetric system (MBS), riverbed evolution, sound velocity profile (SVP), local sound velocity field (LSVF)

1 Introduction Riverbed evolution is generally monitored by crosssection method or bathymetric survey method. The former reflects the riverbed evolution by two adjacent crossing-section variations, thus can only reflect roughly the evolution by the underwater topography changes of river segments. The latter reflects riverbed evolution by various heights of underwater points in different periods, thus only monitors the evolution by the underwater topography changes of many squares. But when the scale of bathymetric survey is smaller, the monitoring by the latter also can not reflect the evolution accurately. Now, riverbed evolution can be monitored with multibeam bathymetric system (MBS) which can acquire dense sounding points by swath survey. However, the accuracy of sound velocity in surveying water area often restricts the accuracy of sound ray tracing in MBS data processing (see also Clarke 1996; Dunham et al. 2004; Zhou et al. 2001). In traditional MBS data processing, SVP near survey location is often adopted in sound ray tracing, which easily brings obvious representation errors in depth and therefore decreases the accuracy of monitoring riverbed evolution (see also Jensen et al. 1994). In order to weaken this effect, this paper presents a new method for reflecting actual sound velocity changes in surveying water area and fulfils the precise monitoring of riverbed evolution. This new method is namely the construction of local sound velocity field (LSVF). In the following, the paper mainly introduces relative theories and methods on constructing LSVF and its applications in sound ray tracing of MBS data processing as well as the precise monitoring of riverbed evolution. 303

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2 Classifications of Sound Velocity Profiles Before constructing LSVF, we need to choose characteristic or representative SVPs from all SVPs observed in surveying water area. Thus the first problem that we face is how to classify these SVPs and draw out representative SVPs from classified SVPs. Generally, SVPs are classified by artificial judgment, which easily causes artificial errors in the classifications and finally influences the sounding accuracy. Self-organizing Feature Maps (SOFM) neural network can analyze the features of input vectors, and organize automatically input vectors with similar feature by learning and training, and classify these input vectors finally (see also Demartines and Herault 1997; Kohonen 1997; Zhang 1993; Nasrabadi and Feng 1988). Thus SOFM is introduced to the classification of SVPs. In order to use SOFM in the work, we need to define an appropriate vector structure to depict SVP firstly. SVP can be illustrated with sound velocity at corresponding depth or the gradient of sound velocity. The latter not only includes all information of SVP, but also meets with the normalization of input vector in SOFM network, and therefore is used as the format of SVP input vector in the construction of SOFM neural network during classifying SVPs. The classification based on SOFM needs to resort to a well-organized SOFM neural network. An ideal SOFM network can be built by continuous training and learning with TFCN, DFCN, OLR, OSTEPS and TLR. These parameters are depicted in Table 1. Generally, a SOFM network needs a higher learning rate so as to adjust the network structure quickly during network ordering, while a lower learning rate in the period of network convergence. By many experiments, we think it is appropriate to set TFCN, DFCN, OLR, OSTEPS and TLR as grid topography function, Link distance function, 0.9, 1000 and 0.1 respectively in training the SOFM network used for classifying SVPs. Table 1. Parameters of SOFM Name

Explanation

TFCN

Topology function, decides the distributions of neurons and network structure. Distance function, decides the distributions of neurons and network structure. Ordering phase learning rate, influence the training times and the network performance Ordering phase steps, influence the training times and the network performance Tuning phase learning rate, influence the training times and the network performance

DFCN OLR OSTEPS TLR

3 Construction of LSVF After classifying SVPs, we can draw out representative SVPs from all SVPs observed at surveying water area according to their types and distributions. These representative SVPs will be used for building local sound velocity field (LSVF) by a cluster of polynomial functions. Due to the depth differences at different SVPs’ stations, it is difficult to directly use these representative SVPs in building LSVF. Thus, all of representative SVPs should be normalized at first before the construction of LSVF. The normalization means sound velocity at each defined depth layer is calculated by a linear interpolation with the sound velocities of adjacent two sampling depth in a SVP. After the normalization, we can get sound velocity at each defined depth layer of each representative SVP. The normalization is very useful to build LSVF. LSVF, which we will build, should reflect that sound velocity varies with location and depth. In order to present the spatial relationship, we designed a cluster of functions at different defined depth layers. At each defined depth layer, a polynomial function, which reflects that sound velocity varies with location, can be built by equation (1). C di = f di (B, L) = a0 + a1 ΔB + a2 ΔL + a3 ΔBΔL +a4 ΔB 2 + a5 ΔL 2

(1)

where C di is sound velocity at depth layer di ; f di is a polynomial function regarding to location (B, L), and reflects sound velocity variations with location. If there are m SVPs, we can draw out m sound velocities C j (1 ≤ j ≤ m) at defined depth layer di . Integrated with the locations (B j , L j ) (1 ≤ j ≤ m) of these representative SVPs’ stations, the equation (1) can be expressed as: ⎡ ⎢ ⎢ ⎢ ⎢ ⎣

C1 C2 . . . Cm

⎤

⎡

⎥ ⎥ ⎥ ⎥ ⎦

⎢ ⎢ ⎢ = ⎢ ⎢ ⎣

di

1

ΔB1

ΔL 1

ΔB1 ΔL 1

ΔB12

ΔL 21 ΔL 22 . . . ΔL 2m

1

ΔB2

ΔL 2

ΔB2 ΔL 2

ΔB22

. . . 1 ⎡

. . . ΔBm ⎤

. . . ΔL m

. . . ΔBm ΔL m

. . . 2 ΔBm

a0 ⎢ a2 ⎢ ×⎢ ⎢ .. ⎣ . a5

⎥ ⎥ ⎥ ⎥ ⎦

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(2)

where ΔBi = Bi –B0 , ΔL i = L i –L 0 , (B0 , L 0 ) is the center of surveying water area, ak (0≤ k ≤5) is the coefficient of polynomial f di .

Research on Precise Monitoring Method of Riverbed Evolution

The matrix expression of equation (2) is C = PX

(3)

where C = [C1 C2 . . .Cm ]T , P is the coefficient matrix and X = [a0 a1 . . .a5 ]T . In virtue of least-squares theory, we can estimate X and acquire the sound velocity function f di at depth layer di . X = (P T P)−1 P T C

(4)

If defined n depth layers, similarly, we can construct n f di (1≤ i ≤ n) at corresponding depth layers di (1 ≤ i ≤ n). The n f di (1≤ i ≤ n) form a cluster of sound velocity functions, which is named F. F is also a local sound velocity field (LSVF) model, which is correlated with location (B, L) and depth D and can be expressed by equation (5). C = F(B, L, D)

(5)

According to equation (5), we can calculate sound velocity C at any position and depth. Model of LSVF is very convenient for sound ray tracing in the sounding data processing of MBS. Besides, LSVF accurately reflects the actual variations of sound velocity in space rather than discrete SVPs at different observation stations, therefore can improve the accuracy of sound ray tracing in MBS survey.

4 Bathymetric Survey and Riverbed Evolution MBS can acquire dense sounding points by swath surveying mode. For each of sounding points, the point depth D can only be acquired by sound ray tracing in which sound velocity plays a very important role. LSVF provides an ideal spatial sound velocity structure, therefore can guarantee the sounding accuracy. If tidal level and squat parameter of the surveying vessel are also observed in MBS survey, the height of sounding point can be calculated as follows. H p = HT − ΔHsquat − D

305

building a digital terrain model (DTM) of surveyed riverbed. By comparing DTMs built at different periods, we can accurately calculate the height difference ΔH p at a point p by equation (7). ΔH p12 = H p2 − H p1

where H p1 and H p2 are the heights of point p acquired at 1st and 2nd observations; ΔH p12 is the difference between H p1 and H p2. Using these height differences (H ) at these discrete points (B, L), we can construct a DTM again which reflects the magnitude and distribution of riverbed scouring and depositing.

5 Experiments and Analysis An experiment was implemented for monitoring riverbed evolution with MBS near Chongming Island in Changjiang Estuary. In the experiment, 22 SVPs were observed in surveying water area (Figure 1). In Figure 1, the blue points are the observation stations of SVPs. Firstly, the 22 SVPs are classified with SOFM neural network. After the network training, the 22 SVPs are classified and shown in Figure 2. The explanation is shown in Table 2. According to Table 1, we draw out SVP 1, 2, 4 5, 7, 8, 11, 13, 14, 15, 16, 17, 19, 20 and 21 for constructing local sound velocity field (LSVF) by equatuon (1–5), and check the LSVF with other SVPs. Comparing the sound velocity from observations with that from LSVF at different depth and location, we acquire the statistic result about the accuracy of LSVF in Table 3. Table 2 shows that the accuracy of LSVF constructed by above method is better than that of sound velocity profiler HY1200 (0.2 m/s), and reaches 0.11 m/s. In other words, the LSVF accurately presents

(6)

where H P is the height of the sounding point; HT is tidal level acquired by tidal readings at tidal gauges; ΔHsquat is the squat of surveying vessel; D is the depth of the sounding point. Integrated with the location of sounding point p(B p , L p ), the 3-dimension position (B p , L p , H p ) of each sounding point can be acquired and used for

(7)

Fig. 1. Locations of MBS experiment and 22 SVPs.

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J.H. Zhao, H.M. Zhang Table 3. Accuracy of LSVF constructed with 22 SVPs

Fig. 2. Classifications of 22 SVPs.

the actual structure of sound velocity in the surveying water area. In the experiment, we designed five surveying lines and measured these lines with MBS. In MBS data processing, we adopt two kinds of data processing methods. One is the traditional method. In sound ray tracing, we directly use discrete SVPs which are close to the locations of beam observations. Correspondingly, we acquire depth at each sounding point. Using these sounding points, we construct DEMs of each surveying line. Comparing these DEMs with known riverbed DEM, the accuracy of traditional method in each surveying line are calculated and shown in Table 4. The other one is the new method presented in this paper. LSVF is used in sound ray tracing. Similarly, we construct DEMs of each surveying line and compare these DEMs with known riverbed DEM. Statistic accuracies of each surveying line are also shown in Table 4. The statistic results in Table 4 show that the new MBS data processing method expressed in the paper

Table 2. Explanation of Classifying 22 SVPs SVP

Explanation

2, 3, 4, 8, 9, 10, 12, 20 1,11,15, 17, 18

Small positive gradient Obviously negative gradient above 10 m, small positive gradient under 10 m Obviously negative gradient above 4 m and small positive gradient under 4m Obviously negative gradient above 2 m, small positive gradient under 2 m Obviously negative gradient above 6 m , small positive gradient from 6m to 16 m, and small negative gradient under 16 m Very small negative gradient above 9 m, near constant sound velocity under 9 m

7, 13, 16

5, 6

14, 22

19, 21

Maximum Bias (m/s)

Minimum Bias (m/s)

Mean Bias (m/s)

Standard Deviations (m/s)

0.20

0.04

0.13

0.11

Table 4. Accuracies of the new method and the traditional method used in each of surveying lines

Line 1 2 3 4 5

Traditional method

New method

Mean bias (cm)

Mean bias (cm)

3.7 3.8 3.0 5.8 –6.8

STD (cm) 14.1 15.2 14.2 14.3 13.2

0.5 0.3 0.6 0.4 0.6

STD (cm) 6.1 6.2 5.9 5.5 5.8

has zero mean and about 6 ∼ 7 cm of standard deviation. This means the new method is only influenced by random error in MBS surveying and has higher accuracy. For the traditional method the mean is not near zero, and the standard deviation is about 13 ∼ 15 cm. This shows that the new method significantly improves the accuracy of the traditional method in sound ray tracing of MBS data processing.

6 Conclusions and Advices The new method based on LSVF in sound ray tracing provides a good way to get precise sounding results for reflecting riverbed evolution by comparing two-period DEMs. In this study, a local sound velocity field (LSVF) is built with the SVPs observed at one surveying period. Thus, the LSVF only reflects the spatial change of sound velocity at that time, but no temporal change. If we observe SVPs at some designed SVP stations according to defined time interval in a studied water area, we can acquire a time-series SVPs at different SVPs’ stations and construct a spatio-temporal LSVF in the studied water area. Spatio-temporal LSVF is very useful for improving the efficiency and accuracy of riverbed evolution monitoring with MBS.

References Clarke J. E. (1996), Vertical Position Requirements and Method, Coastal Multibeam Training Course, ST Andrews, New Brunswick 15–25. Demartines P., Herault J. (1997) Curvilinear Component Analysis: A Self-organizing Neural Network for Nonlinear


Research on Precise Monitoring Method of Riverbed Evolution Mapping of DataSets. IEEE Transaction on Neural Networks, 8(1): 148–154. Dunham S. J., Handal J. T., Peterson T., O’Brien M. (2004) High Resolution Bathymetric System. IEEE 1139–1146. Jensen J. B., Kuperman W. A. and Porter M. B. (1994) Computation Ocean Acoustic. New York: AIP Press. Kohonen T. (1997) Self-Organizing Maps, Berlin: Springer Verlag, Second Edition. Nasrabadi N. M., Feng Y. (1988) Vector quantization of images based upon the Kohonen self-organizing feature

307 maps, International Joint Conference on Neural Networks. San Diego, 101–108. Zhang L. M. (1993) Models and Applications of Neural Network (in Chinese). Shanghai, Published by Fudan University. Zhou F. N., Zhao J. H., Zhou C. Y. (2001) The Determination of Classic Experimental Sound Speed Formulae in Multibeam Echo Sounding System (in Chinese). Journal of Oceanography in Taiwan Strait, 2001(4): 1–9.


Part IV

Geodetic Boundary Value Problems and Inverse Problem Theory

On the Universal Solvability of Classical Boundary-Value Problems of Potential Theory: A Contribution from Geodesy F. Sansò, F. Sacerdote DIIAR, Politecnico di Milano – Polo Regionale di Como, Via Valleggio, 22100 Como, Italy Dipartimento di Ingegneria Civile, Università di Firenze, Via S. Marta 3, 50139 Firenze, Italy

Abstract. The theory of elliptic boundary-value problems in Hilbert spaces has been extensively illustrated some decades ago, e.g., by J.L.Lions and E.Magenes for very general differential operators, with coefficients, right-hand sides of the equations and boundary conditions belonging to irregular function or distribution spaces; consequently solutions too are defined in some generalized sense and belong in general to distribution spaces. In Laplace equation, on the contrary, with constant coefficients and zero right-hand side, maximal regularity properties are met inside the domain of harmonicity. It is therefore interesting, along with the usual regularization procedures of the general theory, to develop an autonomous scheme that, making use of these regularity properties, allows to define a general topological structure for the space of the solutions of the Laplace equation in an open set. It is shown that they can therefore be classified according to the regularity of their boundary conditions, formulating suitable trace theorems. New Hilbert spaces of harmonic functions are then defined, which are different and in a sense complementary to the spaces described by Lions and Magenes. The results can be very simply proved and illustrated in the case of a spherical boundary, for which it is possible to use explicit spherical harmonic representations, and can be generalized to the case of an arbitrary regular boundary. As a matter of fact one can see that, endowing with a general topological structure the space of all the harmonic functions in an open simply connected smooth set, all these results are quite naturally generalized in a comprehensive theory of Dirichlet and Neumann problems for the Laplace operator.

Keywords. Harmonic spaces, topological montel spaces, boundary value problems

1 Introduction It is a long time that geodesists study boundary value problems (BVP) for harmonic functions in several formulations; let us mention Sansò and Sacerdote (1991) for a comprehensive list of works, from the most classical of S.M. Molodensky, through the seminal paper by L. Hormander, to more recent results published in geodetic literature in the last years. The push to enter into this branch of mathematics has always been the desire of understanding whether limit problems, modeling a situation in which the earth surface is covered by various kinds of observations on the gravity field (cf. Sansò (1995)), would provide a firm basis to the usual approximation procedures applied in geodesy to estimate the gravity potential. In particular, when we came to a linearized version of BVP’s we were interested in understanding how bad could be the data, still providing a unique regular harmonic solution. This problem translates in mathematical terms into the problem of properly characterizing the space of all the traces on the boundary of specific boundary operators applied to harmonic functions with some specific regularity property. This paper provides an answer to this question when these functions have no more regularity than being themselves harmonic. The study of solution of BVP’s for Laplace equation can indeed be seen as a particular case of the more general theory of BVP for elliptic problems, like those described in classical textbooks as Neˇcas (1967); Lions and Magenes (1968); and Taylor (1996). Yet a different classification of harmonic solution spaces may be given making use of the fact that harmonic functions belonging to the Sobolev spaces W s (Ω) form a closed subspace of this space, which can be equipped with the Hilbert space structure induced by W s (Ω) itself, and will be denoted by Hs (Ω) ; furthermore, harmonic functions are

311

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F. Sansò, F. Sacerdote

all infinitely differentiable inside the harmonicity domain, and what distinguishes the different Hs (Ω) is essentially their behaviour approaching the boundary of Ω. For example, a discussion on Sobolev spaces of harmonic functions and their duality relations is presented in Axler et al (2001), Ligocka (1986), where a coupling using integrals with weight functions vanishing on the boundary is introduced; this formulation, however is not fit to define traces on the boundary and, consequently, to obtain results on boundary-value problems. The investigation of the properties of the spaces Hs (Ω) is particularly simple when Ω is a sphere centered at the origin (or, equivalently, taking advantage of the properties of the Kelvin transform, its exterior); in this case, in fact, spherical harmonic expansions can be considered, and the different spaces Hs (Ω) are characterized by the convergence properties of the sequences of coefficients. The results are illustrated in detail in Sansò and Venuti (2005). In particular, it turns out that each Hs is densely included in Ht for whatever t < s (with completely continuous embedding), and that it is possible to represent the dual with a coupling induced by the scalar product in a pivot space Hs¯ suitably chosen. The choice of H1 as pivot space leads to the definition of a particular coupling, the socalled Krarup coupling, that proves to be extremely useful in defining the topological structure of the space H(Ω) of all the functions harmonic in Ω (and vanishing at ∞, as Ω is chosen to be unbounded) and of its dual. On this item one can consult as well Shlapunov and Tarkhanov (2003). The final result is a very general trace theorem for the functions in H(Ω), which leads to the solution of the classical boundary-value problems for the Laplace equation in their most general form. The aim of this paper is to obtain similar results for an arbitrary open set Ω with simply connected C ∞ boundary; in the present case, as usual in geodesy, it is assumed that Ω is the complement of a bounded set, denoted by B. It will be useful in the sequel to introduce in H1 (Ω) the norm

2

u =

2

Ω

|∇u| dx = −

⭸Ω

u

⭸u dS ⭸ν

(1)

where ν is the normal direction external to the closed surface ⭸Ω, i.e., in this case, pointing into the interior of Ω. The proof that this is a true norm, equivalent to the one induced by W 1 (Ω), will be given in Appendix 1.

This result, which is already known in geodetic literature (see Holota (1997)), is reproduced here for the ease of the reader. As starting point, consider a space H1+s (Ω) , s > 0 , for which classical results for trace operators hold. It is easy to see that any harmonic function v ∈ ¯ ( Ω¯ = closure of Ω ), with the coupling C ∞ (Ω) ∇u · ∇vdx = < v, u > = (v, u)H1 (Ω) = Ω ⭸u =− (2) v dS ⭸Ω ⭸ν represents a continuous linear functional on H1+s (Ω) , with norm vH1+s (Ω)∗ =

|(v, u)H1 (Ω) | . u∈H1+s (Ω) uH1+s (Ω) sup

(3)

Furthermore, the space of the functions ¯ is certainly dense in H1+s (Ω)∗ , as v ∈ C ∞ (Ω) ¯ ⇒ u =0. < v, u >= 0 ∀v ∈ C ∞ (Ω)

(4)

¯ Indeed, choosing a sequence {vn } in C ∞ (Ω) converging to u in H1 (Ω) , from < vn , u >= 0 it follows < u, u >= u2H1 (Ω) = 0 ⇒ u = 0 . Con¯ with sequently H1+s (Ω)∗ is the closure of C ∞ (Ω) respect to the norm (3). From now on H1+s (Ω)∗ will be denoted as H1−s (Ω) . Theorem. v ∈ H1−s (Ω) is a harmonic function in Ω . Proof. Consider a sequence of harmonic functions ¯ , vn → v in H1−s (Ω) . Let {vn } ⊂ C ∞ (Ω) G ν (x, y) be the Green function for the domain Ω , so that the identity vn (x) = vn (y)G ν (x, y)dSy (5) ⭸Ω

holds for any n and any fixed x ∈ Ω . Furthermore, let M(x, y) be the solution of the Neumann problem for the Laplace equation in Ω with boundary condition G ν (x, y) :

∇ y2 M = 0 ⭸M ⭸ν y = −G ν

in Ω on ⭸Ω

(6)

where ν y , consistently with the conventions previously introduced, points into the interior of Ω.

On the Universal Solvability of Classical Boundary-Value Problems of Potential Theory

Then, from (5) it follows that vn (x) = ∇vn (y) · ∇ M(x, y)dy = =

Ω (vn , M(x, ·))H1 (Ω)

Taking the sup with respect to u one obtains (7)

and consequently |vn (x)| ≤ M(x, ·)H1+s (Ω) vn H1−s (Ω)

313

(8)

is a convergent numerTherefore vn (x) ical sequence and one can define pointwise v(x) = limn→∞ vn (x). Furthermore, as M(x, ·)H1+s (Ω) is bounded for x ∈ K , where K is any compact subset of Ω, vn → v uniformly in K. Consequently, using classical results for harmonic functions (Harnack’s theorem, see for example Mikhlin (1970)), it follows that v is harmonic in Ω.

2 The Spaces Hs (Ω) : Trace Theorems It will be shown that it is possible to define in H1−s (Ω), for any s > 0, a trace operator, which is one-to-one, and, with respect to a suitable topology, continuous together with its inverse. Let v be an element of H1−s (Ω), with norm defined by (3). Let {vn } be a sequence of harmonic C ∞ functions converging to v (that certainly exists, as the space of harmonic C ∞ functions is dense in H1−s (Ω)). Then the limit of ⭸u (9) ∇vn · ∇udx = − vn dS ⭸ν Ω ⭸Ω is a linear functional of ⭸u/⭸ν localized on the boundary of Ω; this limit will be denoted by φv . On the other hand, from the classical theory of Neumann problem it follows that any function w ∈ W s−(1/2)(⭸Ω) can be taken as Neumann boundary value (⭸u/⭸ν)|⭸Ω of a function u ∈ H1+s (Ω); furthermore, positive constants α , β exist, such that αuH1+s (Ω) ≤ wW s−(1/2) (⭸Ω) ≤ βuH1+s (Ω) (10) Consequently, φv turns out to be a continuous functional on W s−(1/2)(⭸Ω). In fact, from (10) and from the equality φv (⭸u/⭸ν) = (v, u)H1 (Ω) , it follows |φv ( ⭸u |(v, u)H1 (Ω) | |(v, u)H1 (Ω) | ⭸ν )| ≤ ⭸u ≤ . βuH1+s (Ω) αuH1+s (Ω) ⭸ν W s−(1/2) (⭸Ω) (11)

1 1 vH1−s (Ω) ≤ φv ≤ vH1−s (Ω) , β α

(12)

from which, in addition, it follows that the mapping v → φv is one-to-one and continuous from H1−s (Ω) to the space W (1/2)−s (⭸Ω) of the continuous linear functionals on W s−(1/2) (⭸Ω). Furthermore, taking the limit in (9), it is clear that, if v is a regular function, φv is the classical trace; consequently it can be interpreted as a generalized trace, and (12) proves the existence and uniqueness of the solution of the Dirichlet problem for the Laplace equation, with the usual −(1/2) rule.

3 The Topology of the Space H(Ω) Recall that S is a closed regular surface in R3 , B its interior and Ω its exterior domain. In H(Ω) the following topology can be defined: let Ωm ⊂ Ω be defined as the exterior of the closed surface Sm = {x + h m ν|x ∈ ⭸Ω}, where ν is the unit vector orthogonal to ⭸Ω at x, pointing inside Ω; {h m } is a decreasing sequence, with ¯ where h¯ is so small that lim h m = 0 , h m < h, ¯ is a regular surface. It has been Sh¯ = {x + hν} proved in Sansò and Venuti (2005) (and, by the way, it is obvious) that, for u ∈ H(Ω), the sequence pm (u) = uH1 (Ωm ) is a sequence of seminorms. Indeed pm (u) are true norms, as pm (u) = 0 for every positive integer m implies that ∇u = 0 everywhere in Ω, and consequently u ≡ 0 in Ω. Yet H(Ω) is certainly not closed under pm . Lemma. H(Ω) is a Fréchet space, i.e. it is complete with respect to the topology induced by the seminorms pm (u). Proof. Let {u m } be a Cauchy sequence. Then limm→∞ pn (u m − u m+ ) = 0 for any fixed n, uniformly with respect to . Hence there exists u¯ (n) ∈ H1 (Ωn ) such that u m − u¯ (n) H1 (Ωn ) → 0. On the other hand u m − u¯ (n+1) H1 (Ωn ) ≤ u m + −u¯ (n+1) H1 (Ωn+1 ) , and consequently u m → u¯ (n+1) in H1 (Ωn ); hence u¯ (n) is the restriction of u¯ (n+1) to Ωn . Therefore there exists a unique u¯ in H(Ω) whose restriction to each Ωn is u¯ (n) , and lim u m = u¯ in H(Ω). Lemma. H(Ω) is a Montel space, i.e. for any sequence bounded in H(Ω) there exists a subsequence converging in H(Ω).

314

Proof. If {u } is bounded, then, for any fixed n, there exists a constant cn such that pn (u ) < cn ∀, that implies u H1 (Ωn ) < cn . As H1 (Ωn ) is compactly embedded in H1 (Ωn−1 ), there exists a subsequence of {u } converging in H1 (Ωn−1 ) to a certain element u. ¯ As this procedure can be repeated for all n, it is possible to extract a subsequence of {u } converging in H(Ω). It follows from a well-known theorem on Montel spaces (Miranda (1978), Treves (1967)) that H(Ω) is reflexive, and that its dual is Montel too. Furthermore, in both H(Ω) and H(Ω)∗ , strong and weak topologies are identical. For the representation of the dual H(Ω)∗ of H(Ω) an obvious suggestion would be to try to proceed similarly as for Hs (Ω), using some kind of H1 coupling. Indeed such a procedure can be easily exploited in the case of a spherical boundary, and leads to the conclusion that H(Ω)∗ can be represented by the space H+ (Ω) of the functions harmonic in an open set containing Ω¯ = Ω ∪ S. Furthermore, making use of the Kelvin transform, H+ (Ω) can be identified with the space H+ (B) of the functions harmonic in an open set containing the closed ball B complementary of Ω, and the coupling can be represented, for any v ∈ H+ (B), by an integral ˜ where B˜ is over a closed surface contained in Ω ∩ B, the harmonicity domain of v (see Sansò and Venuti (2005)). In the general case the application of the Kelvin transform is not so straightforward, and it is preferable to start with the representation of H(Ω)∗ with functions in H+ (B), with the coupling (introduced in geodetic literature by Krarup (1975)) ⭸v ⭸u 1 u− v dS , (13) < v, u >= 4π Sm ⭸ν ⭸ν v ∈ H(B+ ) , u ∈ H(Ω) Indeed, for m sufficiently large, Sm is enclosed in the harmonicity domain of v, and it is easy to see that < v, u > is independent of m. Theorem. Any continuous functional on H(Ω) can be represented by a function in H+ (B) via the Krarup coupling. Proof. That (13) actually represents, for fixed v ∈ H+ (B), a continuous functional on H(Ω) it can be easily proved considering that, when u tends to 0 in H(Ω), certainly it tends to 0 in H1 (Ωm ) for any m, and consequently, owing to well-known


properties of harmonic functions, both u and ⭸u/⭸ν converge uniformly to 0 on Sm , once m is fixed; hence, if Sm is in both the harmonicity domains of u (i.e. Ω) and v, also the integral in (13) tends to 0. Conversely, let be a continuous functional in ¯ | (u)| ≤ const · H(Ω)∗ . Then, for a suitable n, pn¯ (u). But pn¯ (·) is the norm in the Hilbert space H1 (Ωn¯ ), and the space of the restrictions of H(Ω) in H1 (Ωn¯ ) is dense in H1 (Ωn¯ ) (indeed, it contains all the finite combinations of spherical harmonics). Therefore, (u) can be extended as a continuous linear functional in H1 (Ωn¯ ), and consequently, by Riesz theorem, it can be represented by a unique element w in H1 (Ωn¯ ) itself: (u) = (w, u)H1 (Ωn¯ ) = (∇w, ∇u) L 2 (Ωn¯ ) ⭸w udS . (14) =− Sn¯ ⭸ν Now, for any x ∈ Bn¯ , the function Nx y = 1/|x − y|, as function of the variable y ∈ Ωn¯ , is an element of H1 (Ωn¯ ). Define v(x) = (Nx y ) = ∇w · ∇ y Nx y dy Ωn¯ ⭸w =− Nx y dSy . Sn¯ ⭸ν

(15)

It is clear that v(x) is harmonic in Bn¯ , i.e. v ∈ H+ (B). Furthermore, take n > n, ¯ so that Ωn¯ is exterior to S n (with positive distance). Therefore, for any y ∈ Ωn¯ ,

⭸u(x) ⭸ Nx y + u(x) Nx y dSx − ⭸ν ⭸ν Sn˜ (16) Consequently (cf. (14))

u(y) =

1 4π

⭸w (u) = − udS (17) Sn˜ ⭸ν ⭸u(x) 1 ⭸w = (y) Nx y + dSy dSx ⭸ν 4π ⭸ν Sn˜ Sn˜ ⭸ −u(x) Nx y ⭸ν ⭸u(x) ⭸w 1 (y)Nx y dSy + dSx = 4π Sn˜ ⭸ν Sn¯ ⭸ν ⭸Nx y ⭸w (y) dSy −u(x) ⭸ν ⭸νx Sn¯ ⭸v 1 ⭸u = u− v dS (18) 4π Sn˜ ⭸ν ⭸ν


Furthermore, the function v representing a functional via the Krarup coupling is unique, or, equivalently, the only function representing the functional (u) = 0 ∀u is v(x) ≡ 0, as it can be directly seen from (13), introducing u(y) = Nx y for any x ∈ B.

4 Trace Theorems and Boundary-Value Ω) Problems in H (Ω The possibility of defining a trace operator in H(Ω) and of establishing existence and uniqueness results for the main boundary-value problems for the Laplace equation is investigated. As for the spaces Hs (Ω) the procedure is based on the introduction of suitable functionals localized on the boundary; yet, in this case, owing to the irregular behaviour of the functions in H(Ω) near the boundary, it is necessary to start from integrals on surfaces inside Ω and to check carefully their behaviour when these surfaces approach S. 4.1 Dirichlet Problem The starting point is (13), that defines Krarup coupling and, as previously illustrated, is invariant with respect to the change of the surface Sm , provided it is contained in both the harmonicity domains of u and v and encloses S (from now on these surfaces will be called admissible). For any v ∈ H+ (B), the surface S is internal to its harmonicity domain; consequently v| S is in C ∞ (S), (ext) and it is possible to define v D ∈ H(Ω) with bound(ext) ¯ ary value v on S; clearly v D is in C ∞ (Ω). Now define the space ⭸ (ext) (−v D + v)| S , v ∈ H+ (B)} ⭸ν (19) Clearly D (S) ⊆ C ∞ (S); furthermore there exists a linear operator TD on H+ (B) that maps v to σ , i.e.

D (S) ≡ {σ =

TD v = σ

v ∈ H+ (B) , σ ∈ D (S) (20)

Remark. As it will be shown in Appendix 2, D (S) is strictly contained in C ∞ (S). Lemma. TD is an isomorphism. Proof. As a matter of fact, D (S) is built as the image of H+ (B) through the operator TD , so that, given any σ ∈ D (S), there is at least one v ∈ H+ (B) such that TD v = σ . Therefore it has only to be proved that TD is injective. Assume that, for a certain v ∈ H+ (B), TD v = (ext) 0, i.e. (⭸/⭸ν)(−v D + v)| S ≡ 0. It will be shown

315

that this is possible only if v ≡ 0. Obviously (ext) (⭸/⭸ν)(−v + v)(x)Nx y dSx = 0 for any y ∈ D S S. Making use of the equality

⭸v (ext) D (x) S

⭸ν

Nx y − v (ext) D (x)

⭸ Nx y dSx = 0 (21) ⭸ν

valid for any y ∈ B, one obtains ⭸v(x) 1 ⭸ (ext) 0= Nx y − v D (x) Nx y dSx = 4π S ⭸ν ⭸ν ⭸v(x) 1 ⭸ Nx y − v(x) Nx y dSx ≡ v(y) = 4π S ⭸ν ⭸ν (22) Now let Sm be an admissible surface, defined as ¯ above, and let v (ext) D,m be the function in H(Ωm ) whose trace on Sm coincides with vm ≡ v| Sm . Define v˜m on S setting v˜ (s) = v(s + h ν) , v˜ (s) = and v˜m m m m = (⭸/⭸ν)v(s + h m ν) , s ∈ S. Similarly define (ext) v˜ (ext) D,m (s) = (⭸/⭸ν)v D,m (s + h m ν), s ∈ S, where h m is defined at the beginning of sec. 3. As S and Sm for m sufficiently large are inside the harmonicity domain of v, it is easy to see that, when m → ∞, v˜m → (⭸/⭸ν)v| S and (ext) (ext) v˜ D,m → (⭸/⭸ν)v D | S both in C ∞ (S). Then σm = (ext) (⭸/⭸ν)(−v˜ D,m + v˜m )| Sm tends to σ in C ∞ (S). Furthermore, the definition itself of Krarup coupling implies 1 ⭸v ⭸u (ext) udS − v D dS = < v, u >= 4π Sm ⭸ν Sm ⭸ν (ext) ⭸v D 1 ⭸v udS − udS = = 4π Sm ⭸ν ⭸ν Sm 1 = σm udS (23) 4π Sm This expression is constant with respect to m, and consequently it has a finite limit when m → ∞. Clearly, if u is a regular function up to the boundary S, this limit is exactly 1/(4π) S σ udS; this expression can be viewed as a linear functional on D (S), represented, via a L 2 coupling on S, by the boundary value of u on S. Now define the following linear operator D from H(Ω) to D (S)∗ : D u is the linear functional on

D (S) defined by the coupling D u, σ =< TD−1 σ, u >

(24)

where is the Krarup coupling. From the expression obtained above in the case of a function u

316


regular up to the boundary, it is clear that the operator D can be viewed as a generalization of the classical trace operator. Theorem. D is a continuous one-to-one operator with continuous inverse from H(Ω) to D (S)∗ . Proof. From u n → u in H(Ω), it follows, for any σ ∈ D (S), that < TD−1 σ, u n > → < TD−1 σ, u > and, consequently, owing to (24), D u n , σ → D u, σ , that proves continuity. Similarly, D u n → D u in D (S)∗ means D u n , σ → D u, σ for any σ ∈ D (S), that implies < TD−1 σ, u n > → < TD−1 σ, u >; hence u n → u weakly in H(Ω), as any v ∈ H+ (B) can be expressed as v = TD−1 σ , σ ∈ D (S). Furthermore, if D u = 0, D u, σ = 0 for arbitrary σ ∈ D (S), and consequently < TD−1 σ, u >= 0. Hence u = 0. As a conclusion, it can be asserted that, for any τ D ∈ D (S)∗ , the Dirichlet problem for the Laplace equation with boundary value τ D has a unique solution in H(Ω), continuously dependent on τ D . More precisely, the correspondence τ → σ is given by the equality τ D , σ =< TD−1 σ, u >, i.e. u is defined as an element of H+ (B)∗ ≡ H(Ω)∗∗ ≡ H(Ω), as H(Ω), being a Montel space, is reflexive. 4.2 Neumann Problem In the present section it will be shown that, with a procedure similar to the one used for the Dirichlet problem, it is possible to define a generalized normal derivative on the boundary for the functions in H(Ω) and to establish an existence and uniqueness theorem for the Neumann problem. (ext) For an arbitrary element v of H+ (B), define v N the function of H(Ω) whose normal derivative on (ext) S coincides with (⭸/⭸ν)v| S , v N,m the function of H(Ωm ) whose normal derivative on S coincides with (⭸/⭸ν)v| Sm , where the index m is chosen so large that Sm is inside the harmonicity domain of v. Now define (ext) (ext) γ = (v N − v)| S , γm = (v N,m − v)| Sm . As both S and Sm are inside the harmonicity domain of v, it can be easily seen that, for m → ∞, γm → γ in C ∞ . Furthermore, similarly to (23), for u ∈ H(Ω) the Krarup coupling can be expressed as < v, u >=

1 4π

γm Sm

⭸u dS ⭸ν

(25)

independently of m, for m large enough. It is clear that, if u is so regular that ⭸u/⭸ν is defined on S in the classical sense, then, taking the limit for m → ∞, it is possible to write 1 < v, u >= 4π

γ S

⭸u dS ⭸ν

(26)

In addition, similarly as for the operator TD defined in the previous section, it is easy to see that the mapping v → γ defines a linear one-to-one operator TN , whose image is a linear space, denoted by

N (S). Then, as for the Dirichlet problem, one can introduce a linear operator N : H(Ω) → N (S)∗ , defined by N u, γ =< TN−1 γ , u >

(27)

and prove that it is one-to-one and continuous, as well as its inverse. From (26) it can be stated that the operator N is a generalization of the normal derivative on the boundary. Therefore, it can be concluded that, for any τ¯N ∈ N (S)∗ , the Neumann problem for the Laplace equation with boundary condition τ N has a unique solution in H(Ω), continuously dependent on τ N . 4.3 Oblique Derivative Problem Consider a vector field t = b + ν on a regular closed surface S, where ν is the unit normal vector pointing inside Ω and b is tangential; let Dt = (⭸/⭸ν) + b · ∇τ , where ∇τ is the tangential gradient, be the corresponding directional derivative. If u and v are regular scalar functions in a domain containing S, owing to the identity 0= S ∇τ · (uvb)dS = = S ∇τ · (vb)u + b · (∇τ u) v dS the following equality holds too: ⭸v ∗ ⭸u u− v dS = (Dt v)u − (Dt u)v dS ⭸ν S ⭸ν S (28) where ⭸v Dt∗ v = − ∇τ (bv) (29) ⭸ν Now, let Sm be as in the previous sections, u ∈ H(Ω), v ∈ H+ (B). Then the Krarup coupling can be expressed as < v, u >=

1 4π

Sm

∗ (Dt v)u − (Dt u)v dS

(30)


Define now v (ext) in Ω such that O ∗ v| and, similarly, v (ext) in Ω such Dt∗ v (ext) | = D S S m t O O,m (ext) ∗ ∗ that Dt v O,m | Sm = Dt v| Sm ; using the latter equality in (30), and taking into account that, owing to (28), the equality ∗ (ext) (Dt v O,m )u − (Dt u)v (ext) (31) O,m dS = 0 Sm

holds, the Krarup coupling can be expressed in the form 1 (ext) < v, u >= (Dt u)(v O,m − v)dS (32) 4π Sm The integral in (32) is independent of m; hence it has a limit for m → ∞ that can be viewed as a linear functional on the space O (S) of the functions of the (ext) form ρ = (v O − v)| S , v ∈ H+ (B). Now the procedure goes on as for Dirichlet and Neumann problems. The mapping v → ρm = (v (ext) O,m − v) from H+ (B) to a linear subspace O (Sm ) of C ∞ (S) defines a linear one-to-one operator TO . In fact, ρm = 0 implies 1 (ext) Dt Nx y (v O,m − v)dSx = 0= 4π Sm ∗ (ext) 1 = (Dt v O,m )Nx y − (Dt Nx y )v dSx = 4π Sm ∗ 1 (Dt v)Nx y − (Dt Nx y )v dSx = = 4π Sm ⭸v(x) 1 ⭸ Nx y − v(x) Nx y dSx ≡ = 4π Sm ⭸ν ⭸ν ≡ v(y) (33) Then it is possible to introduce a linear operator O : H(Ω) → O (S)∗ , defined by O u, ρ =< TO−1 ρ, u >

317

their normal derivatives and of their oblique derivatives along a vector field t = b + ν, together with their duals. The unconditional solvability of such problems, in the exterior domain Ω, represents a very satisfactory result, which however should be generalized in several senses. In particular, the whole machinery used in the proofs still works when the surface S is much less regular than C ∞ . Naturally the space of the traces of v ∈ H+ (B) on S will also be much less regular than C ∞ and therefore its dual will be accordingly constituted by less irregular generalized functions. Another interesting point could be to assess the impact of the present theorems on the full linear problem with known non-vanishing right-hand side, i.e. for the solutions of the classical boundary-value problems for the Poisson equation. All in all, summarizing, from the geodetic point of view we could say that the question of “existence” of solutions of linear geodetic BVP’s needs not to concern us any more, at least if we are in a condition of uniqueness. Rather a question needing a more precise answers is: what are the geometric and functional conditions that guarantee the uniqueness of the solution which is often claimed for the oblique derivative problem advocating a perturbative argument? This will be subject of further investigations in the future.

Acknowledgement The authors are thankful to F. Tomarelli for his stimulating suggestions and for the example reported in Appendix 2.

(34)

Appendix 1 and prove that it is one-to-one and continuous, as well as its inverse. Finally, it is easy to see that O , for regular u, reduces to the classical operator of oblique derivative on the boundary S.

5 Concluding Remarks The classical Dirichlet, Neumann and obliquederivative boundary-value problems for the Laplace equation in a regular domain have been reviewed. By using a coupling, known in geodesy as Krarup coupling, one is able to characterize the spaces of suitably defined traces of all the functions u ∈ H(Ω), of

It is proved that in the space H1 (Ω) of the harmonic functions belonging to W 1 (Ω) the functional u =

Ω

|∇u|2 dx

1/2

(35)

is a norm equivalent to the one induced by W 1 (Ω), that, for unbounded Ω, can be expressed as u1 =

u2 Ω

r2

1/2 + |∇u|2 dx

(36)

318


To this purpose it is sufficient to prove an inequality of the form

u2 dx ≤ C r2

Ω

|∇u|2 dx,

Ω

(37)

2

u2

|∇u| dx ≤ 2 Ω r ≤ (1 + C) |∇u|2 dx Ω

(38)

The result can be easily obtained if Ω is the exterior of a spherical surface, expanding u into spherical harmonics, and directly carrying out the computations. Hence, if S R is a spherical surface with radius R enclosing the boundary S of Ω and Ω R is its exterior, it is sufficient to prove Ω

u2 dx ≤ C1 r2

Ω

|∇u|2 dx

(39)

= Ω \ ΩR . where Ω Assume for simplicity that S can be expressed as r = r (σ ), i.e. it is the boundary of a star-shaped domain (but it is possible to modify the proof to meet less restrictive conditions). A harmonic function u in is regular inside Ω, and can be expressed as Ω u(r, σ ) = u(R, σ ) − r

R

⭸u (r , σ )dr . ⭸r

(40)

Hence

R ⭸u 2 u (r, σ ) ≤ 2u (R, σ ) + 2 (r , σ )dr ≤ ⭸r r R 2 ⭸u (r , σ ) dr . ≤ 2u 2 (R, σ ) + 2R ⭸r r(σ ) 2

2

Consequently R 2 u2 u (r, σ ) 2 dx = dσ r dr ≤ 2 r2 r Ω S1 r(σ ) R ⭸u 2 (r , σ ) dr + ≤ dσ 2R 2 r(σ ) ⭸r S1 2 +2Ru (R, σ ) ≤ (41) ⭸u 2 ≤ 2R 2 dx + 2R u 2 (R, σ )dσ ⭸r S1 Ω

u 2 (R, σ )dσ ≤ ≤ C2 Ω R |∇u|2 dx ≤ C2 Ω |∇u|2 dx .

(42)

Appendix 2

+ |∇u| dx ≤ 2

Ω

S1

that implies

It is obvious that Ω (⭸u/⭸r )2dx ≤ Ω |∇u|2 dx; in order to complete the proof of (39), using spherical harmonic expansions it is easy to show that

It is shown by a counterexample that the space (S) introduced in Section 4.1 does not coincide with the whole C ∞ (S). Let S be the unit sphere S1 . Then any

v ∈ H(B+ ) has the spherical harmonic expansion vnm r n Ynm , which converges in a ball with radius larger than 1, and can be differentiated term by term with respect to r inside this ball; therefore the series

2 R 2n converges for some R > 1, as well as | |v nm

|vnm |2 n 2k R 2n for any k. corresponding v (ext) has the expansion

The −(n+1) ⭸ (ext) + v)| vnm r Ynm ; hence S

σ = ⭸ν (−v has the expansion (2n + 1)v Y . As nm nm

|vnm |2 (2n + 1)2 R 2n certainly converges for some R > 1, it can be asserted that all the functions in (S) have spherical harmonic expansions wnm Ynm whose coefficients fulfil

|wnm |2 R 2n < ∞ for some

R > 1.√On the other hand, the function f = exp(− n)Ynm is in C ∞ (S), but does not fulfil this condition; therefore it does not belong to (S).

References Axler S, Bourdon P and Ramey W (2001) Harmonic function theory, 2nd ed. Springer Verlag, Berlin. Holota P (1997) Coerciveness of the linear fgravimetric boundary value problem and a geometrical interpretation. Journal of Geodesy 71:640–651. Krarup T (1975) On potential theory. Methoden und Verfahren der mathematischen Physik 12 (B. Brosowski and E. Martensed, eds.), Bibliographisches Institut Mannheim/Wien/Zürich, pp 79–160. Ligocka E (1986) The Sobolev spaces of harmonic functions. Studia Mathematica 84:79–87. Lions JL, Magenes E (1968) Problèmes aux limites non homogénes et applications, volume 1, Dunod, Paris. Mikhlin SG (1970) Mathematical physics: an advanced course. North Holland series in applied mathematics and mechanics, 11. North Holland, Amsterdam. Miranda C (1978) Istituzioni di analisi funzionale lineare, volume I. Unione Matematica Italiana. Neˇcas J (1967) Les methodes directes en theorie des equations elliptiques. Masson, Paris. Sansò F, Sacerdote F (1991) The boundary-value problems of geodesy. Proceedings of the geodetic day in honor

On the Universal Solvability of Classical Boundary-Value Problems of Potential Theory of Antonio Marussi (M. Caputo and F. Sansò, eds.), Accademia Nazionale dei Lincei, Roma, pp 125–136. Sansò F (1995) The long road from measurements to boundary value problems in physical geodesy. Manuscripta Geodaetica 20:326–344. Sansò F, Venuti G (2005) Topological vector spaces of harmonic functions and the trace operator. Journal of Geodesy 79:203–221, Springer.

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Shlapunov A, Tarkhanov N. (2003) Duality by reproducing kernels. International Journal of Mathematics and Science 6:327–395. Taylor ME (1996) Partial differential equations I – basic theory. Applied Mathematical Sciences 115, Springer, Berlin, Heidelberg, New York. Treves F (1967) Topological vector spaces, distributions and kernels. Academic Press, New York.

Model Refinements and Numerical Solutions of Weakly Formulated Boundary-Value Problems in Physical Geodesy P. Holota Research Institute of Geodesy, Topography and Cartography, 25066 Zdiby 98, Praha-východ, Czech Republic, e-mail: [email protected] O. Nesvadba Land Survey Office, Pod S´ıdliˇstˇem 9, 182 11 Praha 8, Czech Republic, e-mail: [email protected] Abstract. In solving boundary-value problems the weak formulation offers a considerable degree of flexibility. It has a natural tie to variational methods. In this paper the weak formulation was applied to the linear gravimetric boundary-value problem and also to the linear Molodensky problem. The numerical solution is interpreted in terms of function bases constructed in the respective Sobolev weight space of functions. Especially, the reciprocal distance and the reproducing kernel were used to produce the function basis. Subsequently, Galerkin’s matrix was constructed for an unbounded solution domain. The obliqueness of the derivative in the boundary conditions is taken into account through an iteration procedure. In the case of Molodensky’s problem, in addition, some modifications were done due to a nontrivial kernel of the problem. All these studies are added extensive numerical simulations using gravity data derived from EGM96. Keywords. Earth’s gravity field modelling, gravimetric boundary-value problem, Molodensky’s problem, disturbing potential, Galerkin’s method

1 Introduction In gravity field studies the approach to boundaryvalue problems (BVP’s) often represents the classical solution. We look for a smooth function satisfying Laplace’s (or Poisson’s) equation and the boundary condition “pointwise”. Alternatively, we can look for a function satisfying an integral identity connected with the BVP in question. This is the weak solution. We will follow this concept in the determination of the external gravity field of the Earth. Therefore, the solution domain Ω is an unbounded domain. Let ⭸Ω be its boundary. We will suppose that Ω = R3 − Ω ∪ ⭸Ω is a domain with Lipschitz’ boundary ⭸Ω . Clearly, ⭸Ω = ⭸Ω . In Ω we will 320

work with functions from Sobolev’s weight space W2(1) (Ω) endowed with inner product (u, v)1 =

Ω

uv dx + |x|2

Ω

grad u, grad v dx (1)

where . , . is the inner product of two vectors in R3 . L 2 (⭸Ω) will stand for the space of square integrable functions on ⭸Ω and we also put A1 (u, v) =

Ω

grad u, grad v dx

(2)

for u, v ∈ W2(1) (Ω). In some cases it is possible to replace the BVP by a minimization problem. This corresponds to variational methods. An example is the quadratic functional Φ(u) = A1 (u, u) − 2 ⭸Ω u f dS on W2(1) (Ω), where f ∈ L 2 (⭸Ω). It attains its minimum at a point (1) u ∈ W2 (Ω), defined by A 1 (u, v) =

⭸Ω

v f dS

(3) (1)

which is assumed to hold for all v ∈ W2 (Ω), see Neˇcas and Hlavácˇ ek (1981) and Holota (2000). This integral identity represents Euler’s condition and has also a classical interpretation. Under some regularity assumptions one can deduce that u has to be a solution of Neumann’s problem (NP) Δu = 0 in Ω, ⭸u/⭸n = − f on ⭸Ω

(4)

where Δ is Laplace’s operator and ⭸/⭸n means the derivative in the direction of the unit outer normal n of ⭸Ω. On the basis of the Lax-Milgram theorem the identity even defines the weak solution of the NP, see Neˇcas (1967) and Rektorys (1977).

Model Refinements and Numerical Solutions in Physical Geodesy

2 BVP’s in Gravity Field Studies We will discuss the linear gravimetric boundaryvalue problem and the linear Molodensky problem. W and U will stand for the gravity and the normal gravity potential of the Earth. Hence T = W − U is the disturbing potential, g = grad W , γ = grad U , g = |g| means the measured gravity and γ = |γγ | is the normal gravity. Let further δg = g − γ be the gravity disturbance and Δg = g(x) − γ (x ) the (vectorial) gravity anomaly (both corrected for the gravitational interaction with the Moon, the Sun and the planets, for the precession and nutation and so on). Finally, x and x are the corresponding points of the Earth’s surface and the telluroid, respectively.

321

The problem may be solved by iterations. We consider a sequence Tm , m = 0, 1, 2, . . . , defined by A 1 (Tm+1 , v) =

⭸Ω

v f dS + A 2 (Tm , v)

(10)

which is assumed to hold for all v ∈ W2(1) (Ω) and m = 0, 1, 2, . . . . In Holota (2000) it has been shown that [ Tm ] ∞ m=0 is a Cauchy sequence, so that it converges to a function T ∈ W2(1) (Ω), which solves the LGBVP. Under certain regularity assumptions the iterations may be interpreted as follows: v fm dS (11) A 1 (Tm+1 , v) = ⭸Ω

Linear Gravimetric BVP (LGBVP)

(1)

The problem is to find T such that ΔT = 0 in

Ω

s, grad T = ⭸T /⭸s = − δg

(5) on ⭸Ω

(6)

where s = − ( 1 /γ ) grad U , Ω is the exterior of the Earth and T is assumed regular at infinity. Following the weak formulation, we suppose that we have a bilinear form A(u, v) such that the problem may be written as an integral identity

is valid for all v ∈ W2 (Ω) and m = 0, 1, 2, . . . while f m = f − ( ⭸Tm /⭸t ) tan (s,n) and ⭸/⭸t is the derivative in the direction of t = (σ − n)/|σ − n|. Clearly, t is tangential to ⭸Ω (since σ − n = a × n). Note that formally equation (11) has the structure of Euler’s condition, so that every iteration step may be interpreted as a minimization problem. Linear Molodensky’s Problem (LMP) Within the classical setting the problem is to find T such that ΔT = 0 in Ω (12)

A(T, v) =

⭸Ω

v f dS

(7)

(1)

valid for all v ∈ W2 (Ω). The solution T is sought (1) in W2 (Ω), provided f ∈ L 2 (⭸Ω). The LGBVP, however, is an oblique derivative problem, so that A(u, v) has a more complex structure than in the case of the NP. Referring to Holota (1997), we have A(u, v) = A 1 (u, v) − A 2 (u, v)

(8)

A 2 (u, v) =

Ω

+

grad v, a × grad u dx Ω

v curl a, grad u dx

(9)

and a is a vector field with ai and |x|(curl a)i in L ∞ (Ω) (space of Lebesgue measurable functions defined and bounded almost everywhere on Ω). Moreover, for σ = n + a × n on ⭸Ω we suppose that σ = s s, n . Note also that the tie to the classical formulation requires f = γ (⭸U/⭸n) −1 δg.

h, grad T + T = F

on ⭸Ω

(13)

and (asymptotically) T (x) = c/|x| + O( |x| −3 ) as x → ∞. Recall that now Ω is the outer space of the telluroid, F = Δ W + h, Δg, Δ W and Δg are the potential and the (vectorial) gravity anomaly, c is a constant and O stands for Landau’s symbol. The vector h = − [ ⭸2U/⭸x i ⭸x j ] −1 grad U , for details see Krarup (1973), Hörmander (1975) or Moritz (1980). Recall that h(x) is close to x/2 for x ∈ ⭸Ω and that Δ W = 0 for Marussi’s telluroid. The LMP may be expressed by an integral identity as in equation (7). The structure of A(u, v), however, is different and also the function f on the R.H.S. differs. Following Holota (1996, 1999), A(u, v) = A 1 (u, v) − A 2 (u, v) + a(u, v)

(14)

where A 1 (u,v) and A 2(u, v) are as above, but with a such that h h, n = n + a × n on ⭸Ω. a(u, v) =

⭸Ω

χuv dS

(15)

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is a boundary bilinear form with χ ∈ L ∞ (⭸Ω) and u, v ∈ W2(1) (Ω). The tie to the classical formulation requires χ = − 1 h, n and also f = − F h, n . The solvability of the LMP has a somewhat more complicated nature. The problem has a non-trivial kernel spanned by vi = ⭸U/⭸x i , i = 1, 2, 3 (obvious solutions of the homogeneous LMP), see Hörmander (1975). For this reason, taking for H ∗ (1) those v ∈ W2 (Ω) which are harmonic in Ω and such that asymptotically v(x) = O( |x| −3 ) as x → ∞, we split W2(1) (Ω) into two parts K and Q. In this decomposition K is a supplementary space spanned by v0 = 1 |x| and vi , i = 1, 2, 3, as above, while (1) Q = H ∗ ⊕ W0,2 (Ω), where ⊕ is the direct sum (1) and W0,2 (Ω) = { z ∈ W2(1) (Ω); z = 0 on ⭸Ω }. (1) Thus W2 (Ω) = K ⊕ Q, see Holota (1996). In consequence, if T is the weak solution of the 3 c v ,T ∈ Q LMP, then T = TK + TQ , TK = i=0 i i Q and ci are some constants. Hence the LMP actually means to find c0 and TQ ∈ Q such that v f dS (16) A(v0 , v) c0 + A(TQ , v) = ⭸Ω

for all v ∈ W2(1) (Ω) and m = 0, 1, 2, . . . . Its convergence may be shown similarly as for the LGBVP.

3 Linear System and the Software LGBVP In our solution it is enough to consider just the space H2(1)(Ω) of those functions from W2(1) (Ω) which are harmonic in Ω and to reformulate our problem, i.e. (1) to look for Tm+1 ∈ H2 (Ω) such that A 1 (Tm+1 , v) =

⭸Ω

and note that h, n ≈ R/2, where R is a radius of a sphere that approximates the topography of the telluroid. Now we construct a sequence of functions TQ(m) and constants c0(m) , m = 0, 1, 2, . . . , defined by ((v0 , v)) 1 c0(m+1) + ((TQ(m+1) , v)) 1 − 3i=1 αi(m+1) ⭸Ω vi v dS = = ⭸Ω v f dS + ((v0 , v))2 c0(m) + ((TQ(m) , v))2 (19)

(20)

holds for all v ∈ H2(1)(Ω) and m = 0, 1, 2, . . . . Subsequently, we approximate Tm by means of n

T (n,m) =

(n,m)

i=0

ci

vi

(21)

where v j are members of a function basis of H2(1)(Ω). (n,m+1) can then be obtained from The coefficients ci Galerkin’s system n

(1)

holds for all v ∈ W2 (Ω), since A(v, vi ) = 0 for i = 1, 2, 3. As regards the coefficients ci , i = 1, 2, 3, they are rule out by the asymptotic condition for T at infinity. Recall in particular that the solution of this problem implies three conditions which have to be met by f , see Holota, 1996). Therefore, we introduce three extra unknown parameters αi , i = 1, 2, 3, and replace f by f ∗ = f + 3i=1 αi vi in order to equalize the balance between the unknowns and conditions, cf. a trick by Hörmander (1975). As in the case of the LGBVP we will solve the LMP by iterations. Put first β = h, n−1 − 2R −1 , ((u, v)) 1 = A 1(u, v) − 2R −1 uv dS (17) ⭸Ω β uv dS (18) ((u, v)) 2 = A 2(u, v) +

v fm dS

⭸Ω

i=0

ci(n,m+1) A 1 (vi , v j ) =

⭸Ω

v j f m dS (22)

where j = 0, . . . , n. LMP For the same reasons as above, here it is enough to work with the space H2(1)(Ω) too. Nevertheless, note (1) that H2 (Ω) = K ⊕ H ∗ . In addition we suppose ∗ that [ vi ]∞ i=4 is a Schauder basis of H and that vi , (m) i = 0, 1, 2, 3, are as above. Thus for TQ we have an approximation (n,m)

TQ

n

=

(n,m)

i=4

ci

vi

(23)

The coefficient cin,m+1 , i = 0, 4, . . . , n, can then again be obtained from Galerkin’s system n

(n,m+1)

((vi , v j )) 1 ci

i=0,i =1,2,3 3 − αi(m+1) ⭸Ω vi v j i=1

=

⭸Ω

v j f dS +

dS =

n

i=0,i =1,2,3

(n,m)

((vi , v j ))2 ci

(24)

where j = 0, 1, . . . , n. In both cases the matrix of the system is symmetric and positive definite.


In the development of the respective software an object oriented design was followed, so that the modularity, encapsulation and genericity are its main features. The software consists of basic functional elements called objects, in particular FUNCTION object and its descendants [which enable to represent N c v (x)], any function u in terms of u(x) = i=1 i i BOUNDARY (responsible for the boundary representation and the surface integration), MATRIX and VECTOR (offer a number of linear equations solvers), GBVP (responsible for Galerkin’s system setup in cases of any BVP considered) and GTASK (that controls the computation process and distributes load over parallel computers). Remark 1. In our opinion a number of these objects can be also used in Boundary Element Methods, which appeared in gravity field studies too, ˇ e.g. Klees (1997), Lehmann (1997) or Cunderl´ ık et al. (2007).

4 Basis Functions Within the software outlined above one can work with nearly any system of basis functions, since every function is inherited from the generic class FUNCTION ARG, RVAL, a template of argument and return value. We prefer basis functions, which are smooth up to the boundary. Clearly, for BVP’s governed by Laplace’s equation harmonic basis functions are a natural choice. An immediate example is the system of functions vi (x) =

1 |x − yi |

,

i = 0, 1, 2, . . .

(25)

with x ∈ Ω and yi ∈ Ω . Another system of basis function, we used, is vi (x) = K (x, yi )

,

i = 0, 1, 2, . . . ,

(26)

x, yi ∈ Ω, produced by a reproducing kernel K (x, y). We distinguish two cases in dependence of whether we solve the LGBVP or the LMP, where vi has to be freed from harmonics of degree 0 and 1. Solution of the LGBVP K (x, y) is considered in H2(1)(Ω) in this case, but with respect to the inner product (u, v) = A1 (u, v). [Note. (u, v) generates a norm which is equivalent 1/2 to ||.||1 = (., .)1 .] Thus (K (x, y), v(x)) = v(y) holds for all x, y ∈ Ω and v ∈ H2(1)(Ω). [Note also that there is no reproducing kernel in W2(1) (Ω), see Holota (2004).]

323

In particular for the exterior of the sphere of radius R we have K (x, yi ) =

∞ 1 2n + 1 n+1 Pn (cos ψi ) (27) z 4π R n+1 i n=0

2

where z i = R |x||yi | and ψi is the angle between yi and x. One can also find that 2z i L + z i − cos ψi 1 − ln K (x, yi ) = 4π R L 1 − cos ψi (28) where L = ( 1 − 2z i cos ψi + z i2 )1/2 , see Tscherning (1975) and Neyman (1979). Solution of the LMP In this case K (x, y) is considered in H ∗ endowed with inner product (u, v) = ((u, v))1 . Thus (K (x, y), v(x)) = v(y) holds for all x, y ∈ Ω and v ∈ H ∗. For the exterior of the sphere of radius R we have K (x, yi ) =

∞ 1 2n + 1 n+1 z Pn (cos ψi ) (29) 4π R n−1 i n=2

Clearly, for z = 1 we have the famous Stokes’ function. Moreover, it is not difficult to find that 1 2z i K (x, yi ) = + z i − 3Lz i 4π R L

1 − z i cos ψi + L 2 − z i cos ψi 5 + 3 ln 2 (30) In both cases the reproducing property of K (x, y) is very useful in constructing Galerkin’s matrix. In our work the points yi are located on a sphere, for simplicity reasons. Their position has an effect on parameters of the basis functions above. It is given by the refinement of an initial platonic solid in R3 , e.g. for vertices yi and y j on a sphere of radius ρ the refining vertex is yi j = ρ (yi + y j ) |yi + y j |. In our computations the 6th level of the refinement of the icosahedron produced a parking grid of 40962 points and thus also an approximation space of the same dimension, i.e. in H40962(Ω) in case of the LGBVP ∗ (Ω) in case of the LMP, see Figure 1. and in H40962

5 Boundary and the Simulation of Data The BOUNDARY class is able to represent the boundary of any star-shaped domain numerically (by a function defined on the sphere). For the surface numerical integration again a triangulation of the faces of the icosahedron was used to generate a subdivision of the boundary surface. The advantage

324

P. Holota, O. Nesvadba

90

90

1000 GPU 800 GPU

60

600 GPU

200 GPU 0

0 GPU – 200 GPU

–30

– 400 GPU – 600 GPU

–60

latitude [deg]

latitude [deg]

60

400 GPU

30

30 0 –30

– 800 GPU –90 –180

–1000 GPU –120

–60

0

60

120

–60

180

longitude [deg]

–90 –180

Fig. 1. Disturbing potential T on the spherical boundary [ 1 GPU ≡ 1 m 2 s −2 ].

of hierarchically created grids is exploited within Romberg’s integration method with Richardson’s extrapolation. This increases the accuracy and provides a feedback in the control of the integration error. [The method was used as a two-dimensional analogue to, e.g., Press et al. (1992).] In our numerical experiments we used: (i ) a spherical boundary of radius R = 6371 km, and (ii) the ETOPO5 boundary. On these boundaries the input data, i.e. δg for the LGBVP and Δg = h/|h| , Δg for the LMP, were simulated by means of the potential U of the Somigliana-Pizzetti normal gravity field with parameters given in the GRS80 (Moritz, 2000) and by the potential W of the gravity field model EGM96 (Lemoine et al., 1998). Clearly, the disturbing potential T = W − U should be then reproduced by the solution of our problems based on the simulated data, see Figure 1. 2560

2240

–120

–60

0

60

120

180

longitude [deg]

Fig. 3. LMP: Parking grid and the magnitude of the coeffi∗ approximation space. cients ci for the H40962

6 Numerical Solution Within each step of successive approximations one has to solve Galerkin’s system given by equations (22) in case of the LGBVP or equations (24) in case of the LMP. Due to the harmonicity and smoothness of the basis functions A 1 (vi , v j ) = − ⭸Ω vi (⭸v j /⭸n) dS and for the spherical boundary we even have an analytic expression for A 1 (vi , v j ) and for ((vi , v j ))1 as well, see Figure 2. From our computations we can conclude that the numerical stability of the system needs a special care in both cases, i.e. for the reciprocal distance (RD) and the reproducing kernel (RK) used in constructing the basis functions. Stress that the matrix of the system is a full and positive definite matrix. We tested Cholesky’s method as well as the SVD method. In particular the latter, though rather slow, offers an estimate of the numerical stability in terms of the spectral norm. Nevertheless, according to our experience, the use of a CG (Conjugate Gradients)

1920

1280

960

640 320

0

90

50 GPU 40 GPU

60

latitude [deg]

2.2E-007 2.1E-007 2E-007 1.9E-007 1.8E-007 1.7E-007 1.6E-007 1.5E-007 1.4E-007 1.3E-007 1.2E-007 1.1E-007 1E-007 9E-008 8E-008 7E-008 6E-008 5E-008 4E-008 3E-008 2E-008 1E-008 0 –1E-008 –2E-008

1600

30 GPU 20 GPU

30

10 GPU 0

0 GPU –10 GPU

–30

–20 GPU –30 GPU

–60

–40 GPU –90 –180

–50 GPU –120

–60

0

60

120

180

longitude [deg]

0

320

640

960

1280

1600

1920

2240

2560

∗ Fig. 2. LMP: Structure of Galerkin’s matrix for the H2562 approximation space – RK basis and the spherical boundary.

Fig. 4. LGBVP: Differences between the exact solution and the obtained solution (final iteration) – depicted on the boundary – max.: 67 GPU, rms: 3.3 GPU, mean: 0.05 GPU.


325

50 GPU

90

40 GPU

latitude [deg]

60

30 GPU 20 GPU

30

10 GPU 0 GPU

0

–10 GPU –30

–20 GPU –30 GPU

–60

–40 GPU –90 –180

–50 GPU –120

–60

0

60

120

180

Fig. 7. LMP: Difference between the first and the final iteration on the boundary – maximum: 7 GPU.

longitude [deg]

Fig. 5. LMP: Differences between the exact solution and the obtained solution (final iteration) – depicted on the boundary – max.: 72 GPU, rms: 3.7 GPU, mean: 0.00 GPU.

iterative solver, as in Ditmar and Klees (2002), proved to be the best choice. An important advantage of the CG method is its easy MPI implementation. In our case, a simple diagonal preconditioning was applied to CG. It offered an efficiency of the iterative process of about 94% (2330 iterations lead to a residual error of ca 5 · 10− 6) that can be further improved by applying a full matrix preconditioner. As to the radius ρ, we followed Nesvadba et al. (2007), putting ρ = 0.9856 R for the RD (i.e. yi are in the depth of ca 92 km) and ρ = 1.0146 R for the RK (i.e. yi are in the height of ca 93 km). Figures 3–7 illustrate results concerning the solution of the LGBVP and the LMP mainly for the spherical boundary and the RK basis. For the LGBVP they extend and improve results obtained in Nesvadba et al. (2007). Remark 2. In the solution of the LGBVP the stability of the results was reached after 7 iterations. In the solution of the LMP in addition a subsequent refinement of the dimension of the approximation space was used in the iteration process. In this case the results stabilized already after 3 iteration steps.

Fig. 6. LGBVP: Difference between the first and the final iteration (oblique derivative effect) on the boundary – max.: 0.8 GPU.

As regards the ETOPO5 boundary, the results are not reproduced here, mainly for the given page limit. Nevertheless for the LGBVP they can be also found in Nesvadba et al. (2007), but for a less detailed approximation space.

7 Conclusions The numerical experiments performed offer a relatively good and instructive insight into the computations connected with the use of Galerkin’s method in the solution of the LGBVP and the LMP. Also some specific features in the recovery of T can be identified in these two cases. The computations were done in approximation spaces of dimension that is comparable with the dimension of gravity field models today and can be even higher. The numerical implementation of our approach offers solutions of an accuracy limited only by the hardware used. Most of algorithms (Galerkin’s matrix, Right hand side, CG) are MPI parallelized, providing almost linear scalability with the dimension of the approximation space and the number of CPU available. Already now it enables to solve the problems under study with an accuracy and resolution comparable with present geopotential models. Among positive features of the software design especially its genericity has to be mentioned. It guarantees a considerable degree of flexibility concerning the choice of basis functions, the choice of the boundary and also the kind of the particular boundary-value problem solved. Future research will be aimed to problems associated with new basis functions under the particular emphasis on their spectral properties. The BOUNDARY and DOMAIN objects will be further developed, so as to better adapt the existing software to more complicated boundaries, to mixed boundaryvalue problems and also to combinations of satellite and terrestrial data.

326

Acknowledgements The work on this paper was supported by the Grant Agency of the Czech Republic through Grant No. 205/06/1330. Computing facilities (SGI Altix system) were kindly provided by Stichting Nationale Computer-faciliteiten (NCF), The Netherlands through Grant SG-027 of the Institute of Earth Observations and Space System (DEOS), the Delft Univ. of Technology and subsequently also by the Supercomputing Centre of the Czech Technical University, Prague. All this support is gratefully acknowledged.

References ˇ Cunderl´ ık R, Mikula K and Mojzeˇs M (2007) Numerical solution of the linearized fixed gravimetric boundary value problem. Journal of Geodesy DOI 10.007/s00190-0070154-0/. Ditmar P and Klees R (2002) A method to compute the Earth’s gravity field from SGG/SST data to be acquired by the GOCE satellite. DUP Science. Holota P (1996) Variational methods for quasigeoid determination. In: Tziavos IN and Vermeer M (eds) Techniques for Local Geoid Determination. Reports of the Finnish Geodetic Inst No 96:2, Masala: 9–21. Holota P (1997) Coerciveness of the linear gravimetric boundary-value problem and a geometrical interpretation. Journal of Geodesy, 71: 640–651. Holota P (1999) Variational methods in geoid determination and function bases. Physics and Chemistry of the Earth, Part A: Solid Earth and Geodesy, Vol 24, No 1: 3–14. Holota P (2000) Direct method in physical geodesy. In: Schwarz KP (ed.) Geodesy Beyond 2000. IAG Symposia, Vol 121, Springer: 163–170. Holota P (2004) Some topics related to the solution of boundary-value problems in geodesy. In: Sansò F (ed) V Hotine-Marussi symposium on mathematical geodesy. IAG Symposia, Vol 127, Springer: 189–200. Hörmander L (1975) The boundary problems of physical geodesy. The Royal Inst of Technology, Division of

P. Holota, O. Nesvadba Geodesy, Stockholm, 1975; also in: Archive for Rational Mechanics and Analysis 62(1976): 1–52. Klees R (1997) Topics on boundary element methods. In: Sansò F and Rummel R (eds) GBVPs in View of the 1cm Geoid. Lect Notes in Earth Sci, Vol 65, Springer: 482–531. Krarup T (1973) Letters on Molodensky’s problem III: Mathematical foundation of Molodensky’s problem. Unpublished manuscript communicated to the members of IAG special study group 4.31. Lehmann R (1997) Solving geodetic boundary value problems with parallel computers. In: Sansò F and Rummel R (eds) GBVPs in View of the 1cm Geoid. Lect Notes in Earth Sci, Vol 65, Springer: 532–541. Lemoine FG et al. (1998) The development of the Joint NASA GSFC and National Imagery and Mapping Agency (NIMA Geopotential Model EGM96. NASA/TP-1998206861, NASA, GSFC, Greenbelt, Maryland. Moritz H (1980) Advanced physical geodesy. H Wichmann Karlsruhe and Abacus Press Tunbridge Wells Kent. Moritz H (2000) Geodetic reference system 1980. The Geodesist’s handbook. Journal of Geodesy, 74: 128–133. Neˇcas J (1967) Les méthodes directes en théorie des e´ quations elliptiques. Academia, Prague. Neˇcas J and Hlavácˇ ek I (1981) Mathematical theory of elastic and elasto-plastic bodies: An introduction. Elsevier Sci Publ Company, Amsterdam-Oxford-New York. Nesvadba O, Holota P and Klees R (2007) A direct method and its numerical interpretation in the determination of the Earth’s gravity field from terrestrial data. In: Tregoning P and Rizos C (eds) Dynamic Planet. IAG Symposia, Vol 130, Chap 54, Springer: 370–376. Neyman YuM (1979) A variational method of physical geodesy. Nedra Publishers, Moscow (in Russian). Press WH, Teukolsky SA and Vetterling WT (1992) Numerical recipes in C: The art of scientific computing. Cambridge University Press, New York. Rektorys K (1977) Variational methods. Reidel Co, Dordrecht-Boston. Tscherning CC (1975) Application of collocation. Determination of a local approximation to the anomalous potential of the Earth using “exact” astro-gravimetric collocation. In: Bosowski B and Martensen E (eds) Methoden und Verfaren der Mathematischen Physik, Vol 14: 83–110.

On an Ellipsoidal Approach to the Singularity-Free Gravity Space Theory G. Austen, W. Keller Stuttgart University, Geodetic Institute, Geschwister-Scholl-Street 24/D, 70174 Stuttgart, Germany

Abstract. In 1977 F. Sansò found an elegant approach to solve the geodetic boundary value problem by transforming it into gravity space. Ten years later W. Keller proposed a revised theory to overcome some shortcomings of F. Sansò’s gravity space transformation. Considering the numerical implementation of the geodetic boundary value problem, linearisation of the problem with respect to a spherical normal potential is common to both methods so far. Indeed, Keller’s concept can benefit from the fact that in its linearised version the resulting boundary value problem is analogous to the one of the simple Molodensky problem. Yet, leadoff numerical experiments indicate that linearisation as applied so far results in computational deficiencies. Therefore, the aim of this contribution is to examine the suitability of using an ellipsoidal normal potential as linearisation point. The key questions to be answered are, whether the mathematical structure of a simple Molodensky problem can be preserved and whether introducing an ellipsoidal normal potential is numerically advantageous. Keywords. Gravity space theory, ellipsoidal approach, geodetic boundary value problem, gravimetric telluroid, adjoint potential

normal potential, the same mathematical structure as the simple Molodensky problem. Thus, all well known and commonly used algorithms and procedures developed for the solution of the traditional Molodensky problem can be used further on. Despite conceptual advantages of treating the geodetic boundary value problem in gravity space, the numerical evaluation of the linearised problem implies certain difficulties originating mainly from linearization at a spherical normal potential. Remedy is sought-after in adopting an ellipsoidal normal potential as linearisation point. The outline of this contribution is as follows. Next, the original concept of Sansò to solve the GBVP is reviewed followed by an overview of Keller’s refined theory in Section 3. In Section 4 the focus is on the new ellipsoidal transformation and closed-loop simulation studies are presented Section 5. Thereafter, summary and outline conclude the paper.

2 Review of Sansò’s Approach F. Sansò succeeded in Sansò (1977) by introducing new independent coordinates ξ = [ξ1 ξ2 ξ3 ] ξ := ∇V (x) ,

x = [x 1 x 2 x 3 ]

(1)

and the so-called adjoint potential ψ = ψ (ξξ )

1 Introduction Almost 30 years ago F. Sansò has pioneered a concept for solving the geodetic boundary value problem (GBVP) which has become known as the gravity space approach. Sansò proposed a solution of the GBVP by mapping it into the so-called gravity space using Legendre transformation, the simplest form of a contact transformation. Prompted by the fact that Sansò’s transformed problem suffered from a singularity at the origin W. Keller suggested later on a modified contact transformation. His singularity-free gravity space approach holds, after linearisation with respect to a spherical

ψ := x ∇V (x) − V (x) ,

x = x(ξξ ) ,

(2)

to transform the classical GBVP V (x) = 0 , V ⭸G = v ∇V = g ,

x ∈ ext ⭸G

⭸G

(3) (4) (5)

into the following boundary value problem tr 2 − (tr )2 = 0 , ξ ∈ int ⭸ , (ξξ ∇ξ ψ − ψ)ξ =g = v , ξ ∈ ⭸

(6) (7) 327

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G. Austen, W. Keller

subject to the Hessian matrix of second order derivatives of the adjoint potential = ∇ξ (∇ξ ψ) . Moreover, in order to express e.g. the gradient of the scalar potential function V , the operator ∇ stands

for the symbolic vector ⭸x⭸1 ⭸x⭸2 ⭸x⭸3 and accord ingly ∇ξ for ⭸ξ⭸1 ⭸ξ⭸2 ⭸ξ⭸3 . Naturally, represents the Laplacian. From its mathematical structure (3)–(5) is a linear free exterior boundary value problem in geometry space with high complexity both in the theory of existence and uniqueness as well as in its numerical implementation. ⭸G denotes the boundary surface, i.e. the topography of the Earth. In contrast, the boundary value problem (6), (7) constitutes a nonlinear fixed interior boundary value problem in gravity space. The second order differential equation of Laplace type (3) with the corresponding boundary condition (4) is replaced by a Monge-Ampère type differential equation (6) with a linear boundary condition (7). The gravity space boundary surface ⭸⌫ is the image of the Earth’s surface ⭸G under the mapping (1). As indicated before, the asymptotic behaviour of the gravitational potential V ∼

1 → 0, x

with x =

x → ∞

x 12 + x 22 + x 32

3 A Singularity-Free Gravity Space Transformation In order to eliminate the addressed obstacle, a similar gravity space transformation is defined in Keller (1987), which leaves the point at infinity as a fixed point. Similarly to equations (1) and (2) the modified transformation, this time generating new gravity space coordinates ξ given in [m] instead of [m/s 2 ] as in (1), is described by p = [ p1 p2 p3] := ∇V (x) √ ξ := − G M

p p3/2

(13)

ψ := x p − V (x) .

(14)

In (13), G M denotes the gravitational constant times Earth’s mass. The resulting exterior boundary value problem to be solved for reads as

and ξξ =

ξξ → 0

1 − ξ ∇ξ ψ − ψ 2

+ ξ22

+ ξ32 .

√

ξ =− G M

g g3/2

=v,

(15)

ξ ∈ ⭸⌫ , (16)

(8)

where denotes the following matrix of first and second order derivatives of the adjoint potential

⭸2 ψ ⭸ψ = (ik ) := γim γkj + βimk (17) ⭸ξm ⭸ξ j ⭸ξm

(9)

with the coefficients γkj and βimk defined by αik γkj = δi j βimk :=

(10) and

ξ12

ξ ∈ ext ⭸⌫

tr 2 − (tr )2 = 0 ,

results in an unfavourable asymptotic behaviour for the adjoint potential ψ ∼ − ξξ → 0,

(12)

(11)

Equation (10) demonstrates a singularity of the gravity space approach at the origin. The coordinate transformation (1) maps a point in infinity to the origin, thus generating an interior problem. Since the concept of differentiability has no meaning at the point in infinity, it is unsurprising that in the new coordinates ξ differentiability of the adjoint potential, a prerequisite in order to solve the differential equation of second order (6), is lost at the origin. Hence, the question to be settled in the following is whether this singularity can be avoided.

−G M αik := ξξ 3

⭸γim ⭸ pk

ξi ξk δik − 3 . ξξ 2

Equations (15) and (16) again establish a nonlinear boundary value problem with a fixed boundary in gravity space and are in accordance with equations (3)–(5) of geometry space. But this time for the adjoint potential ψ the asymptotic relation ψ ∼−

1 → 0, ξξ

ξξ → ∞

(18)

holds, which means that the modified gravity space approach (12)–(14) is free from any singularity.

On an Ellipsoidal Approach to the Singularity-Free Gravity Space Theory

3.1 Linearised Problem Nevertheless, the singularity-free boundary value problem (15) and (16) is still vastly complicated. Since with the spherical normal potential U :=

GM x

ξ ∈ ext ⭸⌫

(20)

(21)

1 − ξ ∇ξ τ − τ = (V − U )|⭸G = δV . (22) 2 ⭸⌫ The boundary surface ⭸⌫, also referred to as gravimetric telluroid, is defined by the condition g = ∇V (x) = ∇U (ξξ ) ,

Since the gravity space mapping (12), (13) is the identical transformation for a pure spherical potential M V = Gx GM

is a first approximation of the actual adjoint potential ψ. The linearisation of (15), (16) at ψ0 leads to the following boundary value problem for the adjoint disturbing potential τ := ψ − ψ0 τ (ξξ ) = 0 ,

4 An Ellipsoidal Regular Gravity Space Transformation

(19)

we have a first approximation of the actual gravitational potential U ≈ V , we can expect that the adjoint normal potential ψ0 , given by GM ψ0 := −2 ξξ

329

x ∈ ⭸G, ξ ∈ ⭸⌫ , (23)

which balances the gradient of the true gravitational potential V at the physical surface of the Earth to the gradient of a spherical normal potential U derived at the gravimetric telluroid. In Keller (1987) it is shown that this implicit definition of the boundary surface ⭸⌫ is equivalent to the definition of the gravimetric telluroid ⭸⌫ as the image of the Earth’s surface ⭸G under the mapping (13). It is remarkable that the linearised problem (21), (22) is mathematically of the same structure as the linearised Molodensky problem. Only potential and gravity have changed their places, which as discussed in Austen and Keller (2007) gives reason to the assumption that consideration of the linearised gravity space approach is superior to solving the Molodensky problem by classical methods. However, despite apparent conceptual advantages implementation of the gravity space approach proves numerically demanding. The prevailing reason seems to be the insufficient spherical linearization point. On this account the next section is devoted to a new gravity space transformation based on an ellipsoidal normal potential as linearisation point.

ξ (∇x

x √ GM x3 ) = GM 3/2 = x , x GM

(24)

x2

and since the true potential differs from the spherical potential by about 10−3 , also the gravimetric telluroid differs from the Earth’s surface by about the same factor. This leads to a maximum separation between ⭸⌫ and ⭸G of about 12 km (cf. Figure 2). One way to reduce this large separation, is to find a new gravity space transformation, which is the identical transformation for an ellipsoidal normal potential

GM x3 R 2 V (x) = ) . (25) P2 ( 1 − J2 x x x x3 In (25), P2 ( x ) specifies the Legendre Polynomial P2 (sin φ) expressed in cartesian coordinates, i.e.

P2 (sin φ) = P2 (

3 x3 )= x 2

x3 x

2 −

1 . 2

(26)

The corresponding gravity space transformation can easily be given in an implicit form: Let ξ (p) be the solution of p = ∇ξ

GM R 2 ξ3 1 − J2 ( ) P2 ( ) ξξ ξξ ξξ

(27)

then for ξ (∇x

R 2 x3 GM ) P2 ( ) ) = x. 1 − J2 ( x x x (28)

holds. 4.1 Series Expansion of Transformation Since no closed solution ξ (p) can be found, a series expansion of the solution with respect to the small parameter J2 will be given here. The equation to be solved is of the type p = F0 (ξξ ) + J2 F1 (ξξ ) .

(29)

330


For J2 = 0 the solution of the resulting equation

η = p2 (5 p14 + 5 p24 − 4 p22 p32 − 60 p34 + +2 p12 (5 p22 − 2 p32 ))

p = F0 (ξξ 0 )

ϑ = p3 (−51 p14 − 51 p24 + 20 p22 p32 + 20 p34 + + p12 (−102 p22 + 20 p32 )) . (34)

is obviously √ ξ0 = − GM

p p3/2

Introducing new coefficients γ¯i j and β¯im j

.

The implicit function theorem guarantees that for J2 small enough (29) has a unique solution ξ (p, J2 ), which is analytical with respect to J2 . Thus, the new gravity space transformation reads as ξ = ξ 0 + J2ξ 1 + J22ξ 2 + . . .

ψ = x p − V (x) .

(30) (31)

Inserting (30) into (29) and expanding it into a series with respect to J2 results in p = F0 (ξξ 0 + J2ξ 1 + J22ξ 2 + . . .) + J2 F1 (ξξ 0 + J2ξ 1 + J22ξ 2 + . . .) = F0 (ξξ 0 ) + ∇F0 (ξξ 0 )(J2ξ 1 + J22ξ 2 + . . .) + 1 (J2ξ 1 + J22ξ 2 + . . .) ∇ 2 F0 (ξξ 0 )(J2ξ 1 + 2 J22ξ 2 + . . .) + J2 F1 (ξξ 0 ) + ∇F1 (ξξ 0 )(J22ξ 1 + . . .) = F0 (ξξ 0 ) + J2 ∇F0 (ξξ 0 )ξξ 1 + F1 (ξξ 0 ) + 1 J22 ∇F0 (ξξ 0 )ξξ 2 + ξ 1 ∇ 2 F0 (ξξ 0 )ξξ 1 + 2 (32) ∇F1 (ξξ 0 )ξξ 1 + . . . .

The comparison of the coefficients of identical powers of J2 on both sides of (32) yields the sequence of equations p = F0 (ξξ 0 ) 0 = ∇F0 (ξξ 0 )ξξ 1 + F1 (ξξ 0 ) 1 0 = ∇F0 (ξξ 0 )ξξ 2 + ξ 1 ∇ 2 F0 (ξξ 0 )ξξ 1 + ∇F1 (ξξ 0 )ξξ 1 2 0 = ... ,

leading to the following expressions for ξ 1 and ξ 2

(γ¯i j ) =

ζ = p1 (5 p14 + 5 p24 − 4 p22 p32 − 60 p34 + +2 p12 (5 p22 − 2 p32 ))

,

¯ 2 − tr ⌽ ¯ 2=0, tr ⌽

⭸γ¯im , β¯im j = ⭸pj

(35)

␰ ∈ ext ⭸⌫ ,

(36)

¯ is given by where the functional matrix ⌽

⭸2 ψ ⭸ψ ¯ ¯ i j ) = γ¯im γ¯ j l ⌽ = ( + β¯im j . (37) ⭸ξm ⭸ξl ⭸ξm Comparing (36)–(15) the resulting boundary value problem is of identical structure only that the matrices (17) and (37) differ. 4.2 Linearised Problem Assume that the reference potential V0 is given by expression (25), then the corresponding adjoint reference potential ψ¯ 0 is deduced from (31) together with x = ξ since p = ∇V0 holds and we arrive at the following equation for ψ¯ 0 ψ¯ 0 = ξ ∇V0 − V0

2G M ξ3 R 2 =− P2 ( 1 − 2 J2 ) . ξξ ξξ ξξ Here, P2 ( ξ3 ) is used in analogy to (26). Splitting ξ the unknown adjoint potential ψ into its normal part ψ¯ 0 and its disturbing part δψ ψ = ψ¯ 0 + δψ ,

−3R 2

subject to

then the adjoint potential ψ satisfies the following field equation

⎡

⎤ p1 ( p12 + p22 + 2 p32 ) ⎢ ⎥ ξ1 = √ p2 ( p12 + p22 + 2 p32 ) ⎦ ⎣ 4 G Mp5/2 2 2 2 − p3 (3 p1 + 3 p2 + 2 p3 ) ⎡ ⎤ ζ 9R 4 ⎣ η ⎦ ξ2 = (33) 32G M 3/2 p7/2 ϑ

⭸ξξ ⭸p

and denoting the corresponding functional matrices ¯ 0 and δ ⌽ ¯ by ⌽

2ψ ¯0 ¯0 ⭸ ⭸ ψ ¯ 0 = γ¯mi γ¯kl ⌽ + β¯iml ⭸ξm ⭸ξk ⭸ξm

2 δψ ⭸ ⭸δψ ¯ = γ¯mi γ¯kl + β¯iml δ⌽ ⭸ξm ⭸ξk ⭸ξm

On an Ellipsoidal Approach to the Singularity-Free Gravity Space Theory

leads to the following field equation ¯ 2 − [tr ⌽ ¯ 0 ]2 = tr ⌽ 0 ¯ − tr [⌽ ¯ 0 δ ⌽] ¯ + O(δψ 2 ) ¯ 0 tr δ ⌽ 2 tr ⌽

(38)

for the linearised boundary value problem. In contrast to (21) equation (38) is an inhomogeneous partial differential equation. 4.3 Spherical Approximation In order to achieve a homogeneous problem spherical approximation of (38), i.e. omitting terms of order J2 and higher on the level of the coefficients α¯ ik , β¯imk and γ¯kj and the reference potential ψ¯ 0 , is conducted α¯ ik = αik + O(J2 ) , γ¯ik = γik + O(J2 ) ,

β¯imk = βimk + O(J2 ) , ψ¯ 0 = ψ0 + O(J2 ) , (39)

which reduces (38) into ¯ 2 ] − [tr ⌽ ¯ 0 ]2 = O(J2 ) tr [⌽ 0

(40)

¯ 0 holds since for ⌽ ¯ 0 = ( ¯ 0 ik ) = γik + O(J2 ) . ⌽

(41)

Spherical approximation according to (39) represents only a minor modification of (38), therefore (40) is an adequate approximation. Finally, the resulting linearised boundary value problem in spherical approximation reads as δψ(ξξ ) = 0 ,

ξ ∈ ext ⭸⌫

1 − ξ ∇ξ δψ − δψ = δV . 2 ⭸⌫

(42) (43)

Equations (42) and (43) show a remarkable symmetry to the simple Molodensky’s problem in the classical theory.

Remark. At this point it is worthwhile to comment on the different meanings with regard to spherical approximation and to summarize the particular assumptions involved. Within the scope of the classical treatment of Molodensky’s problem, a spherical reference potential together with a spherical coordinate representation has been applied to obtain the simple Molodensky’s problem. See, e.g., Moritz (1980). Whereas, in the context of the

331

presented gravity space approach spherical approximation has to be understood in the sense of assuming J2 = 0 for all involved quantities. That is, for the coefficients α¯ i j , β¯im j and γ¯i j as well as for the adjoint normal potential ψ¯ 0 . Furthermore, in contrast to the classical Molodensky’s approach, spherical approximation within the scope of the ellipsoidal gravity space approach is not only utilized to simplify the underlying boundary condition, but also to approximate the corresponding field equation, cf. (40) and (42). In both cases the boundary values remain unchanged.

5 Closed-Loop Numerical Simulation The setup of the error-free closed-loop simulation and the applied numerical methods is given in detail in Austen and Keller (2007) and is therefore only briefly reviewed here, cf. Figure 1. Based on given models for the Earth’s topography and gravitational potential the gravity space boundary surface, i.e. the gravimetric telluroid, and the corresponding boundary data are synthesized. The effect of using ellipsoidal linearisation instead of spherical is demonstrated for a selected meridian passing through the Himalaya region in Figure 2 and 3. The advantages of an ellipsoidal linearisation point are obvious. Whereas in the spherical variant the separation of the gravimetric telluroid and the topography amounts up to 12 km, the separation is well below 1.5 km for the ellipsoidal approach. Furthermore, this decrease in telluroid-topography separation is also reflected in the magnitude of the boundary data. The boundary values, i.e. potential disturbances, diminish from ≈70000 m2 s−2 to below 800 m2 s−2 . Since the boundary values have to be iteratively upward continued in the next step from the gravimetric telluroid to an enclosing Brillouin sphere using collocation in a remove-compute-restore mode, scaled down boundary data are advantageous from

Fig. 1. Flowchart of the numerical study.

332


The coefficients δψnm of the spherical harmonic expansion of the adjoint disturbing potential are related to the vnm in a simple way δψnm =

Fig. 2. Profile of telluroid and boundary data in the Himalayas in case of a spherical normal potential for linearization.

2 vnm . n−1

(44)

Thus, adding up the adjoint disturbing potential δψ and the ellipsoidal normal potential ψ¯ 0 , adjoint potential values ψˆ i j can be computed pointwise for the gravimetric telluroid and compared to the adjoint potential values ψi j derived directly by transforming potential values taken at the Earth’s surface from geometry space to gravity space. The resulting accuracy in terms of a geoid undulation commission error amounts to ≈1.5 cm rms.

6 Summary and Outline a computational point of view. Next, with a combination of FFT and Gaussian quadrature, spherical harmonic coefficients vnm of the upward continued potential disturbances are computed. The weights and the nodes of the grid for the Gaussian quadrature formula are designed in such a way, that (besides rounding errors) the spherical harmonic analysis is exact up to a certain degree and order, dependent upon the largest available set of nodes and weights for the Gaussian quadrature. In the underlying simulation study the only error contained in the spherical harmonic coefficients vnm stems therefore from the upward continuation error.

Equations (42) and (43) suggest that the adjoint disturbing potential satisfies Laplace equation and is therefore harmonic in the outside domain. Thus, the mathematical structure of the simple Molodensky problem is preserved. Furthermore, introducing an ellipsoidal reference potential as linearisation point reduces the telluroid-topography separation by a factor of about 10 and the magnitude of the corresponding boundary data by a factor of about 100. Finally, numerical simulation studies carried out in a closed-loop fashion so far, suggest a resulting accuracy of about 1.5 cm rms in terms of geoid undulation commission error. This result is comparable to the former spherical result. However, for the erroneous simulation study, which is still open, the ellipsoidal approach is expected to be superior.

References

Fig. 3. Profile of telluroid and boundary data in the Himalayas in case of an ellipsoidal normal potential for linearization.

Austen G., Keller W. (2007): Numerical implementation of the gravity space approach – Proof of Concept. In Rizos C., Tregoning P. (Eds.): Dynamic Planet – Monitoring and Understanding a Dynamic Planet with Geodetic and Oceanographic Tools; IAG Symposia, Cairns, Australia, August 22–26, 2005, Vol. 130, pp. 296–302, Springer, Berlin-Heidelberg. Keller W. (1987): On the treatment of the geodetic boundary value problem by contact- transformations. Gerlands Beitr. z. Geophysik, 96(3/4), pp. 186–196. Moritz H. (1980): Advanced Physical Geodesy. Herbert Wichmann Publishing, Karlsruhe. Sansò F. (1977): The Geodetic Boundary Value Problem in Gravity Space. Mem. Akad. Naz. Lincei, 14, pp. 39–97.

Local Geoid Modelling From Vertical Deflections W. Freeden University of Kaiserslautern, Geomathematics Group, 67653 Kaiserslautern, P.O. Box 3049, Germany, e-mail: [email protected] S. Gramsch University of Kaiserslautern, Geomathematics Group, 67653 Kaiserslautern, P.O. Box 3049, Germany, e-mail: [email protected] M. Schreiner University of Buchs, Laboratory for Industrial Mathematics, Werdenbergstrasse 4, CH–9471 Buchs, Switzerland, e-mail: [email protected] Abstract. This paper deals with the problem of improving geoidal undulations from discretely given vertical deflections on a local area (i.e., a regular region on the sphere) by virtue of a “zooming-inprocedure”. Keywords. Deflections of the vertical, regularization of Green’s function, multiscale modelling, geoidal undulations, local approximation

1 Mathematical Background Roughly spoken, the determination of the geoidal undulations N from vertical deflections v on a regular region, i.e., a simply connected open subset of the unit sphere ⊂ R3 , amounts to the inversion of the equation ∇ ∗ N = v, where ∇ ∗ denotes the surface gradient on the sphere (see Freeden and Schreiner 2006). This paper presents a three step solution of the inversion process, namely (i) the continuous approach to N based on the fundamental theorem for ∇ ∗ on , (ii) a multiscale approximation of N caused by an appropriate regularization process, and (iii) a realistic application under geodetically relevant assumptions.

2 Inversion and Regularization The point of departure for our considerations is a known integral formula involving the Green function with respect to ∗ (see Freeden and Schreiner 2006; Gramsch 2006). Theorem 1. (Fundamental theorem for ∇ ∗ on ) Let ⊂ be a regular region with boundary ⭸. Suppose that F is a continuously differentiable function on . Then, for every point ξ ∈ , we have

F(ξ ) = −

1 4π

F(η) d␻(η)

∇η∗ F(η) · ∇η∗ G(∗ ; ξ, η) d␻(η)

+

⭸

F(η)νη · ∇η∗ G(∗ ; ξ, η) d␴(η),

where ν is the unit normal field pointing into \ and the Green function with respect to ∗ = ∇ ∗ · ∇ ∗ is given by (see Freeden et al. 1998) G(∗ ; ξ, η) =

1 1 1 ln(1 − ξ · η) + − ln 2. 4π 4π 4π

In other words, knowing ∇ ∗ F (continuously over ) we are able to calculate F. As a matter of fact, F is determined on up to an additional constant. Before we come back to this problem under geodetic aspects, we first establish an approximation of F based on the regularization of the Green function (see Freeden and Schreiner 2006; Gramsch 2006). For real values ρ > 0, we consider – as an auxiliary function – the so-called ρ–regularized Green kernel function with respect to ∗

G ρ (∗ ; ξ, η) =

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

1 4π

ln(1 − ξ · η) 1 + 4π (1 − ln 2) for 1 − ξ · η > ρ,

1−ξ ·η 4πρ

+

1 4π (ln ρ

− ln 2)

for 1 − ξ · η ≤ ρ.

The kernel function (ξ, η) → G ρ (∗ ; ξ, η) only depends on the inner product of ξ and η, hence, G ρ (∗ ; ξ, η) is (as in the case of G(∗ ; ξ, η)) a zonal function. In addition, we can easily deduce that G ρ (∗ ; ·, η) is a continuously differentiable function on for every (fixed) η ∈ , G ρ (∗ ; ξ, ·) is a continuously differentiable function on for every 333

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(fixed) ξ ∈ . For F ∈ C(1) (), we let

where

P ρ (F)(ξ )= − ∇η∗ G ρ (∗ ; ξ, η) · ∇η∗ F(η) d␻(η) F(η)νη · ∇η∗ G ρ (∗ ; ξ, η) d␴(η) +

ψρ (ξ, η) =

⎧ ⎪ ⎪ ⎪ ⎨

0

⎪ ⎪ ⎪ ⎩

1 4πρ (ξ

for 1 − ξ · η > ρ, − (ξ · η)η)

for 1 − ξ · η ≤ ρ.

⭸

Furthermore, we use the abbreviation

as regularized counterpart to P(F) (ξ ) = − ∇η∗ G(∗ ; ξ, η) · ∇η∗ F(η) d␻(η) + F(η)νη · ∇η∗ G(∗ ; ξ, η) d␴(η).

(ST ) (F) (ρ;ξ ) = − ∇η∗ F(η) · gρ (ξ, η) d␻(η) + F(η)νη · gρ (ξ, η) d␴(η).

Then we are able to formulate a regularization of the inversion by virtue of the following integral formula (Freeden and Schreiner 2006; Gramsch 2006).

It should be pointed out that the kernels constituting the wavelet vector field possess a local support. This is of great significance for computational purposes, since approximate integration procedures have to observe only the contributions inside the local support of ψρ . It is obvious that the wavelets behave like O(ρ −1 ), hence, the convergence of the integrals in the following reconstruction theorem is guaranteed.

⭸

Theorem 2. Let ⊂ be a regular region with boundary ⭸. Suppose that F is a continuously differentiable function on . Then, for all ξ ∈ , we have 1 F(η) d␻(η) + lim P ρ (F)(ξ ). F(ξ ) = ρ→0 4π

3 Reconstruction Formulae and Wavelet Transform For ρ > 0 the family {gρ }ρ>0 of kernels gρ : × → R3 defined by gρ (ξ, η) = ∇η∗ G ρ (∗ ; ξ, η),

ξ, η ∈ ,

(1)

is called scaling vector function. Moreover, the scaling function g1 corresponding to the parameter ρ = 1 is called the mother kernel of the scaling vector function. Furthermore, the family {ψρ }ρ>0 of kernels ψρ : × → R3 defined by ψρ (ξ, η) = −ρ

d gρ (ξ, η), dρ

ξ, η ∈ ,

(2)

is called wavelet vector function. Equation (2) is called the scale–continuous scaling equation. For the scaling function {gρ }ρ>0 , the associated wavelet transform is defined by (W T ) (F)(ρ, ξ) = − ∇η∗ F(η) · ψρ (ξ, η) d␻(η) F(η)νη · ψρ (ξ, η) d␴(η), + ⭸

⭸

Theorem 3. Let ⊂ be a regular region. We denote with {gρ }ρ>0 the scaling vector function. Then, for ξ ∈ and F ∈ C(1) (),

2 0

dρ ρ 1 = F(ξ ) − F(η) d␻(η) 4π 1 F(η)P1 (ξ · η) d␻(η). − 4π

(W T ) (F)(ρ, ξ )

4 Multiscale Modelling Until now, we have been concerned with a scale– continuous approach to wavelets. In what follows, scale discrete scaling vector fields and wavelets will be introduced to establish a numerical solution process. We start with the choice of a sequence that divides the continuous scale interval (0, 2] into discrete pieces. More explicitly, (ρ j ) j ∈N0 denotes a monotonically decreasing sequence of real numbers satisfying the equations ρ0 = 2 and lim j →∞ ρ j = 0 (for example, ρ j = 2− j +1 ). Given a scaling vector field {gρ }, then we clearly define the (scale) discretized scaling vector field {g Dj } j ∈N0 as g D j = gρ j , j ∈ N0 . An illustration of |g Dj | can be found in Figure 1.

Local Geoid Modelling From Vertical Deflections

335

D Fig. 1. Illustration of the norm of the scaling function |g D j (cos ϑ)| (left) and of the wavelets |ψ j (cos ϑ)| (right).

In discretizing the scaling function we are naturally led to the following type of scale discretized wavelets.

(ST )D (F) (J + K ; ξ ) = (ST )D (F)(J ; ξ )

+ Definition 1. Let {g D j } j ∈N0 be the scaling vector function. Then the scale discretized wavelet function is defined by ψ jD (·, ·)

=

ρj ρ j +1

ψρ (·, ·)

dρ , ρ

j ∈ N0 .

(3)

A graphical impression of the scale-discretized wavelets can be found in Figure 1. It follows from equation (3) that ψ jD (·, ·) = ∞

dρ − ψρ (·, ·) ρ ρ j +1

∞

dρ = ψρ (·, ·) ρ

ρj D g j +1 (·, ·) − g D j (·, ·).

(4)

The last formulation is called the (scale) discretized scaling equation. It is not difficult to see that, for ξ, η ∈

ψ jD (ξ, η) =

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

j =J

for ρ j > 1 − ξ · η > ρ j +1 , 1 1 1 4π ρ j − ρ j +1 (ξ − (η · ξ )η) for 1 − ξ · η ≤ ρ j +1 .

Assume now that F is a function of class C(1) (). Observing the discretized scaling equation (equation 4), we get for J ∈ N0 , K ∈ N, and ξ ∈

ξ ∈ ,

(ST )D (F)( j ; ξ ) = (ST ) (F)(ρ j ; ξ ), j = J, . . . , J + K − 1, and (W T )D (F)( j ; ξ ) = (W T ) (F)(ρ j ; ξ ), j = J, . . . , J + K − 1. Therefore we arrive at the following corollary. Corollary 1. Let {g JD } j ∈N0 be a (scale) discretized scaling vector field. Then the multiscale representation of a function F ∈ C(1) (), for ξ ∈ , is given by ∞ j =0

for 1 − ξ · η > ρ j , 1 1 1 − 4π ρ j 1−ξ ·η (ξ − (η · ξ )η)

(W T )D (F)( j ; ξ ),

with

F(ξ ) = 0

J +K −1

(W T )D (F)( j ; ξ ) 1 F(η)dω(η) 4π 1 + F(η)P1 (ξ · η) dω(η) . 4π

+

The aforementioned corollary admits the following reformulation. Corollary 2. Under the assumptions of Corollary 1 we have for J ∈ N0

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(ST )D (F) (J ; ·) +

∞ j =J

(W T )D (F)( j ; ·)

1 F(η) dω(η) =F− 4π 1 F(η)P1 (· · η) dω(η). − 4π As in the classical theory of wavelets (see, e.g., Freeden et al. 1998), the integrals (ST )D (F)( j ; ·), (W T )D (F)( j ; ·) may be understood as low–pass and band–pass filters, respectively. Obviously, (ST )D (F)( j + 1; ·) = D (ST )D (F)( j ; ·) + (W T ) (F)( j ; ·). (5)

zero- and first – order moments on the whole sphere . In particular, we have D(η)Yn,k (η) dω(η) = 0;

n = 0, 1;

k = 1, . . . , 2n + 1.

A canonical discretization of the integrals occuring in Corollary 5 by (suitable) cubature formulas (involving the weights wl and the nodes ηl ) leads us to the following local improvement of the geoidal undulations based on deflections of the vertical N(ξ ) ≈ −

L

wl g JD (ξ, ηl ) · dl

l=1

Equation (5) may be interpreted in the following way: The ( j +1)-scale low–pass filtered version of F is obtained by the j -scale low–pass filtered version of F added by the j -scale band–pass filtered version of F.

5 Geodetic Application The geodetic problem to be addressed now can be formulated as follows (see Freeden and Schreiner 2006): According to the usual PizettiSomigliana concept, the zero – and first – order moments of the geoidal undulations N in terms of spherical harmonics {Yn, j } are known to be zero for the whole sphere , i.e., on the Earth understood to be spherical. Furthermore, for simplicity, we assume that there is a known approximation N˜ on the spherical Earth such that the error function D = N − N˜ restricted to = ∪ ⭸ (i.e., the local area under consideration) satisfies the following conditions: (i) D(η) = 0 on \ , ∇η∗l D(ηl )

= d(ηl ) = dl ∈ on a (dense) discrete set {η1 , . . . , η L } ⊂ ,

(iii) D(η)Yn,k (η) dω(η) = 0; n = 0, 1; k = 1, . . . , 2n + 1. (ii)

R3

In other words, we assume the approximation to be (ideally) known on \ and to have vanishing

−

L ∞

wl ψlD (ξ, ηl ) · dl .

j =J l=1

Finally, a suitable truncation of the infinite sum yields N(ξ ) ≈ −

L

wl g JD (ξ, ηl ) · dl

l=1

−

L T

wl ψlD (ξ, ηl ) · dl .

j =J l=1

In other words, finer and finer detail information about the geoid is obtained by wavelets with smaller and smaller local support. The PhD–thesis Gramsch 2006 shows that the calculation can be obtained by virtue of a tree algorithm thus providing fast wavelet transform (FWT).

References W. Freeden, T. Gervens, and M. Schreiner. Constructive Approximation on the Sphere (With Applications to Geomathematics). Oxford Science Publication, Clarendon, 1998. W. Freeden and M. Schreiner. Local Multiscale Modelling of Geoid Undulations from Deflections of the Vertical. J. Geod., 79:641–651, 2006. S. Gramsch. Integral Formulae and Wavelets on Regular Regions of the Sphere. PhD thesis, Geomathematics Group, University of Kaiserslautern, 2006.

Monte Carlo Integration for Quasi–linear Models B. Gundlich Central Institute for Electronics, Forschungszentrum Jülich GmbH, 52425 Jülich, Germany Now at: GFZ Potsdam, Telegrafenberg, 14473 Potsdam, Germany J. Kusche Delft Institute of Earth Observation and Space System (DEOS), TU Delft, Kluyverweg 1, PO box 5058, 2600 GB Delft, The Netherlands Now at: GFZ Potsdam, Telegrafenberg, 14473 Potsdam, Germany Abstract. In this contribution we consider the inversion of quasi-linear models by means of Monte-Carlo methods. Quasi-linear models are a special class of non-linear models, which can be formally written in matrix-vector formulation but whose design matrix depends on a subset of the unknown parameters. A large class of geodetic problems can be recast as quasi-linear models. As there exist no general analytical solutions for the quasi-linear model, Monte Carlo optimization techniques in the context of a Bayesian approach are investigated here. In order to accelerate the Monte Carlo method we utilize the analytical solution of the linear model under the condition that the unknown parameters in the design matrix are considered as constant. Thereby the sampling dimension in the Monte Carlo approach can be reduced. The estimators for expectation and covariance of the parameters that we derive turn out as weighted means of the individual sample leastsquares solutions. We develop an efficient set of algorithms for the solution of quasi-linear models using Monte Carlo techniques and demonstrate the efficiency of the method in a numerical example which is taken from satellite geodesy and gravity field recovery. Two groups of unknown parameters are relevant in this example: the spherical harmonic coefficients of a gravity field model, and the state vectors of the satellite(s) which affect the observation model through the design matrix. Keywords. Monte-Carlo methods, quasi-linear models, Bayesian inversion, satellite geodesy

1 Introduction Monte Carlo methods are sampling techniques that make use of generating random variables. By this means they offer a very general tool to solve inverse problems in geodesy numerically, even if no analytical expression for the ‘best’ estimator according to some optimization principle is available; e.g. due to a non-linear functional model. Monte Carlo methods

are known to be computationally demanding, but due to increasing computational power they become more and more interesting for various applications in geodesy and geophysics. Nevertheless, for largedimensional problems it is still desirable to develop and apply fast and efficient Monte Carlo techniques. For example, there is a growing interest in the use of Monte-Carlo methods for approximating large error covariance matrices in gravity field modelling, and facilitate the related error propagation computations (see Gundlich et al., 2003, Alkhatib and Schuh, 2007). In this contribution we consider so-called quasilinear models. A large variety of geodetic non-linear problems can in fact be written as quasi-linear models; including those involving state-vector integration, multiple model formulations, or models where the design matrix is affected by some mismodelling parameters. Quasi-linear models are a special class of non-linear models, which can be written in the usual matrix-vector formulation but whose design matrix depends on a subset of the unknown parameters. If this subset of parameters is kept fixed, a linear model with its associated analytical solution under assumption of normality is given. As there exists no general analytical solution for the total quasilinear model Monte Carlo techniques shall be applied here. But in order to accelerate the Monte Carlo method we utilize the analytical solution of the linear model under the condition that the unknown parameters in the design matrix are considered as constant. Thereby the sampling complexity (dimensionality) in the Monte Carlo approach can be reduced, accelerating the procedure. Here, the Monte Carlo solution for quasi-linear models is presented in a Bayesian framework. As in Bayesian statistics all kinds of estimates are derived from probability density functions it is the predestined approach for Monte Carlo methods that require the sampling of random variables. We develop an efficient set of algorithms for the solution of quasi-linear models using Monte Carlo techniques and demonstrate the efficiency of the method in a numerical example which is taken from 337

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B. Gundlich, J. Kusche

satellite geodesy and gravity field recovery. Two groups of unknown parameters are relevant in this example: the spherical harmonic coefficients of a gravity field model, and the state vectors of the satellite(s) which affect the observation model through the design matrix. This paper is organized as follows: In Section 2, we will review Monte Carlo integration in the context of the Bayesian approach, and we will in particular discuss conditional Monte Carlo integration. Then, in Section 3, quasi-linear models will be introduced and we will apply the methodology discussed previously to this important class of models, leading to estimators for expectation and covariance. In Section 4, we will apply our method to an example from satellite geodesy and gravity field determination, where the design matrix depends on the initial state vector of a satellite pair through assumed pure gravitational motion. Results obtained in a realistic setting will be discussed. Conclusions will be provided in Section 5.

In the Bayesian approach the information about the unknown parameters and the observations is formulated in probability density functions (pdf). The socalled prior density of the unknown parameters p(β) describes the a priori knowledge about the unknown parameters before observations y are obtained in the actual measurement. The Bayes theorem modifies the prior density p(β) by the likelihood function p(y|β), that describes the information gained by the measurement, and leads to the posterior density (see for example Koch, 1990) (1)

wherein p(y) acts as a normalizing constant. As the posterior density p(β|y) contains all informations, it forms the basis for all kind of estimations and hypothesis tests. The Bayes estimator minimizes the expectation of a loss function. In case of a quadratic loss function the Bayes estimator is the posterior expectation E(β|y) βˆ B = E(β|y) =

β p(β|y)dβ

(3)

B

In special cases, for example linear models and Gauss distribution for the observations, the equations (2), (3) can be solved analytically, but in more general cases a numerical solution is necessary. Simulation based techniques like Monte Carlo integration are then a helpful tool. In the following we consider the problem of solving f (x) p(x)d(x) (4) I = X

over the domain X of x. Herein p(x) is a pdf, so that (4) can be interpreted as expectation of f (x). If we are able to draw samples x(i) , i = 1, . . . , N, we approximate (4) by N 1 f (x(i) ) . Iˆ = N

(5)

i=1

2 Bayesian Approach and Monte Carlo Integration

p(β) p(y|β) p(β|y) = p(y)

D(β|y) = (β − E(β|y)) (β − E(β|y)) p(β|y)dβ .

(2)

B

with the domain B of β. The covariance matrix of the unknown parameters is given per definition by

Assume we are not able to draw samples from p(x) but from the pdf u(x) that approximates p(x). With w(x) = p(x)/u(x) we can describe I in (4) as f (x)w(x)u(x)dx (6) I = X X w(x)u(x)dx and approximate the integral by means of N generated samples x(i) from u(x) by Iˆ =

1 N

N

f (x(i) )w(x(i) ) N (i) i=1 w(x )

i=1 1 N

(7)

= with the importance weights w(x(i) ) p(x(i) )/u(x(i) ). The pdf u(x) is therefore often called importance sampling distribution, importance function or proposal distribution. The denominator of (7) is also an approximation of the integral p(x)dx . (8) 1= X

That means that p(x) can also be replaced by a function that is only proportional to a pdf. The normalizing constant is then estimated by the denominator in (7). In the Bayesian approach we have to solve integrals of the type f (β) p(β|y)dβ , (9) I = B

Monte Carlo Integration for Quasi–linear Models

339

see equations (2) and (3). In any case the posterior density p(β|y) is known from the Bayes theorem (1) up to a normalizing constant p(y). In place of (9) we find 1 f (β) p(β) p(y|β)dβ (10) I = p(y) B and want to solve the integral via Monte Carlo simulation but can not sample from the posterior distribution itself but have to use a proposal distribution, that means we can apply an approximation of the type of (7) and do not have to take care of the normalizing constant p(y). The proposal distribution that shall approximate p(β|y) can be defined for given observations, thus u(β|y), but more general it only depends on the parameters β, u = u(β). It is very common and obvious to use the prior density p(β) as proposal distribution u(β). For this special case, u(β) = p(β), the integral (10) is approximated by 1 N

Iˆ =

N

(i) (i) i=1 f (β ) p(y|β ) 1 N (i) i=1 p(y|β ) N

,

(11)

if β (i) , i = 1, . . . , N, are samples drawn from p(β).

Instead of solving the integral (9) completely by the Monte Carlo simulation we try to solve a part of it analytically. We subdivide the parameter vector into β = (β1 , β2 ) and express the common posterior density as (12)

The Bayes theorem (1) delivers the marginal density for β1 p(β1 ) p(y|β1 ) (13) p(β1 |y) = p(y) and in analogy the conditional posterior density for β2 given β1 p(β2 |β1 , y) =

p(β2 |β1 ) p(y|β1 , β2 ) . p(y|β1 )

(14)

We transform the integral (9) into I =

B1

B2

f (β1 , β2 ) p(β1 , β2 |y)dβ1 dβ2

=

B1

p(β1 |y)h(β1 , y)dβ1

h(β1 , y) =

B2

f (β1 , β2 ) p(β2 |β1 , y)dβ2

(16)

and with the domains B1 , B2 for the subvectors β1 and β2 . If we manage to solve the inner integral (16) for given observations and a given vector β1 analytically, and compute the normalizing factor p(y|β1 ) from the posterior density of β2 for a given β1 from (14), then we only have to solve I = h(β1 , y) p(β1 |y)dβ1 = B1

1 p(y)

h(β1 , y) p(β1 ) p(y|β1 )dβ1

(17)

B1

numerically. Drawing samples β1(i) , i = 1, . . . , N from p(β1 ) enables us to approximate (17) by Iˆ =

1 N

N

(i) (i) i=1 h(β1 , y) p(y|β1 ) (i) N 1 i=1 p(y|β1 ) N

.

(18)

3 Quasi–Linear Models We consider the model

2.1 Conditional Monte Carlo Integration

p(β1 , β2 |y) = p(β1 |y) p(β2 |β1 , y) .

with

(15)

y + e = Aβ1 β2 ,

D(y|β) = Q y

(19)

with observations y and their covariance matrix D(y|β) = Q y and errors e and with the unknown parameters β = (β1 , β2 ) . The design matrix Aβ1 has full rank but is not (completely) known. At least some of its elements depend on functions of β1 . Considering the complete parameter vector β, the above defined model is a non-linear model. But for a given β1 the model is linear with respect to β2 , thus we interprete the model as a quasi–linear model and in the following make use of the linearity in case of a given β1 . Similar approaches to solve nonlinear models by considering a number of single linear models are ensemble Kalman filters or the multiple model approach (see Bar-Shalom et al., 2001). Here the uncertainty of the design matrice is taken into account by considering a set of design matrices depending on various β1 . Allowing for non-linear dependences of the design matrix on β1 , the quasilinear model (19) is more general than the total-least squares approach (see Schaffrin and Felus, 2008; Golub and van Loan, 1980), where perturbations in the design matrix are modelled by addition of an error matrix. Linearizing quasi-linear model (19)

340


makes it comparable to the Gauss-Helmert model (see Koch, 1999, p. 212).

D(β1 |y) = E((β1 − βˆ 1 )(β1 − βˆ 1 ) |y) = B1 (β1 − βˆ 1 )(β1 − βˆ 1 ) p(β1 |y)dβ1

3.1 Conditional Monte Carlo Integration for Quasi–linear Gaussian Models A priori the unknown parameters are independent from each other, p(β1 , β2 ) = p(β1 ) p(β2 ), with Gaussian prior densities β1 ∼ N μ1 , Q0β1 and β2 ∼ N μ2 , Q0β2 (20) with expectations E(βi ) = μi and covariance matrices D(βi ) = Q0βi , i = 1, 2. The pdf of the observations y for given parameters β = (β1 , β2 ) is also Gaussian y|β1 , β2 ∼ N Aβ1 β2 , Q y .

(21)

The conditional posterior density for β2 for a given β1 following from the Bayes theorem (14) can be solved analytically, as the fixing of β1 leads to a linear Gaussian model, see Koch (2000 p. 104f), with Gaussian posterior density of β2 given β1 β2 |β1 , y ∼ N βˆ2 |β1 , Qβ2 |β1 (22) with covariance matrix −1 −1 Qβ2 |β1 = Aβ1 Q−1 y Aβ1 + Q0β2

(23)

and with expectation −1 βˆ2 |β1 = Qβ2 |β1 Aβ1 Q−1 y y + Q0β2 μ2 .

(24)

ˆ 1 |y) = D(β N (i) (i) ˆ MC (i) ˆ MC i=1 (β1 − β1 )(β1 − β1 ) p(y|β1 ) (.29) N (i) i=1 p(y|β1 ) In case of the estimation of the expectation for β2 E(β |y) = βˆ2 =

2 p(β1 |y) β2 p(β2 |β1 , y)dβ2 dβ1 (30) B1 B

2 ˆ β2 |β1 we use the conditional posterior density p(β2 |β1 , y) and find N

(i) ˆ (i) i=1 β2 |β1 p(y|β1 ) N (i) i=1 p(y|β1 )

(31)

The posterior covariance matrix for β2 is

B1

(β2 − βˆ 2 )(β2 − βˆ 2 )

D(β2 |y) = B1

Now we are able to solve integrals of the type (15) by Monte Carlo integration according to (18) if we gen(i) erate N samples β1 from p(β1 ) in (20). The Bayes estimation (2) for β1 E(β1 |y) = βˆ1 = β1 p(β1 |y)dβ1 (26)

B2

× p(β1 |y) p(β2 |β1 , y) dβ2 dβ1 .(32)

A part of the equation can be solved analytically,

(β2 − βˆ 2 )(β2 − βˆ 2 ) p(β2 |β1 , y)dβ2 B2

= Qβ2 |β1 + (βˆ2 |β1 − βˆ 2 )(βˆ2 |β1 − βˆ 2 ) ,

(33)

so that we can estimate the posterior covariance matrix D(β2 |y) by

is approximated by N

(i) (i) i=1 β1 p(y|β1 ) N (i) i=1 p(y|β1 )

(28)

is estimated by

ˆ 2 |y) = βˆ2 MC = E(β

The normalizing constant p(y|β1 ), see (14), respectively the likelihood function in (13) can also be computed analytically, y|β1 ∼ N Aβ1 μ2 , (Q y + Aβ1 Q0β2 Aβ1 ) . (25)

ˆ 1 |y) = βˆ MC = E(β 1

The covariance matrix for β1

.

(27)

ˆ 2 |y) = D(β

N i=1

N

f (i) p(y|β1(i) )

i=1

(i)

p(y|β1 )

(34)


341

with (i)

(i)

(i)

f (i) = Qβ2 |β1 +(βˆ2 |β1 − βˆ 2MC )(βˆ2 |β1 − βˆ 2MC ) . (35) In analogy the covariance matrix C(β1 , β2 |y) C(β1 , β2 |y) = (β1 − βˆ 1 )(β2 − βˆ 2 ) B1

B2

× p(β1 |y) p(β2 |β1 , y) dβ2 dβ1

(36)

is approximated by ˆ 1 , β2 |y) = C(β

N i=1

N

g (i) p(y|β1(i) )

i=1

p(y|β1(i) )

(37)

with (i) (i) g (i) = (β1 − βˆ 1MC )(βˆ2 |β1 − βˆ 2MC )

(38)

because of (β2 − βˆ 2 ) p(β2 |β1 , y)dβ2 = (βˆ2 |β1 − βˆ 2 ) . (39) B2

independent step of the process, based upon continuous GNSS tracking of the satellite. It is recognized that high-quality orbits are very important for the following gravity recovery step. The orbits enter equation (40) through the data preprocessing and through the computation of the design matrix elements. They are typically computed using the reduced-dynamic approach. In particular for GRACE, the quality of the orbits has been considered as a limiting factor for the recovery of the gravity field. Some groups have tried to mitigate orbit errors through explicit inclusion of orbit error parametrization schemes in equation (40). It is possible that estimated empirical parameters absorb orbit errors to some extent. In the context of this study, we formulate this problem as (41) y + e = Aβ1 β2 where β2 contains the spherical harmonic coefficients plus instrumental parameters as discussed above, β1 contains a set of parameters that is intended to model initial (e.g. at the begin of an orbital arc) orbit errors, and Aβ1 = A(β1 ) explicitly depends on β1 . For simplicity, we choose D(y|β) = σ 2 I. 4.1 Set-up

4 An Example from Orbital Mechanics With the dedicated gravity missions CHAMP, GRACE, GOCE, new concepts have been developed for satellite gravity field recovery. One important step was the implementation of functional models based on in-situ data y (residual potential or intersatellite residual potential differences from energy conservation considerations, satellite accelerations derived from kinematically determined orbits, satellite gravity gradiometry) that could be formulated as leastsquares problems y + e = Aβ2

(40)

We chose a simple simulation setup to test the developed estimators. Using a gravity field model, two GRACE-type satellite orbits (A and B) are numerically integrated. As observation vector y, we assume the intersatellite potential difference yi = V (riB , λiB , θiB ) − V (riA , λiA , θiA )

(42)

along the two orbits is given. Jekeli (1999) showed how to obtain this pseudo-observation from the original range-rate measurements. The vector β2 contains the spherical harmonic coefficients, β2 = c20 , c21 , . . . , slmax lmax .

(43)

Each row of the design matrix contains, because of without employing variational equations Ilk et al. (2008). These (possibly preprocessed) observations can be directly, in a linear model, related to the spherical harmonic coefficients of the Earth’s gravity. In β2 , the spherical harmonic coefficients are estimated, together with instrumental (e.g. accelerometer drift or scaling) and empirical (e.g. once/twice per revolution) parameters. The precise determination (POD) of the satellite position at the measurement epoch is completed in a prior almost

lmax l a l+1 GM V (r, λ, θ ) = a r

(44)

l=0 m=0

× (clm cos mλ + slm sin mλ) Plm (cos θ ) the difference in the corresponding spherical harmonic functions evaluated in the two orbital positions. In equation (44), G M is the universal gravitational constant times Earth’s mass, a a reference

342


length, and the Plm (cos θ ) are the fully normalized associated Legendre polynomials. Nongravitational forces are disregarded in this simulation. Finally, the initial state vectors x = A

r0A

x = B

r˙0A

r0B

r˙0B

(45)

– position and velocity in the celestial reference frame – for the two satellites are collected in

β1 = (x A , x B ) .

Two sets of each N = 100 samples have been generated for the state vector parameter, drawn from a Gaussian pdf p(β1 ), equation (20), and centered at (but excluding) the “true” state vector. In case 1, the standard deviations of position (3 cm) and velocity (1 mm/s) were chosen at realistic level. As performance indicators, we show the “sample weights” (Figure 1 and Figure 3). p(y|β1(i) ) α (i) = N (i) i=1 p(y|β1 )

(46)

The orbital positions r A , r B in equation (42) depend on the state vectors, equation (45), by d2 A r = ∇V (r A ) dt 2 r˙ A (0) = r˙0A r A (0) = r0A

4.2 Results

(47)

and for r B correspondingly. For any sample or (i) “guess” β1 , the orbits can be numerically integrated to provide the design matrix Aβ (i) . To summarize the 1 main conditions on the set-up: 1. The used gravity field model was EGM96, complete to degree lmax = 20 2. The observation vector y contains 1 arc of 24 h of intersatellite potential differences, with a sampling rate of 5 sec. 3. Initial orbital elements for GRACE A were (a = 6849208.325 m, e = 0.003443, I = 88.995◦, = 288.134 ◦, ω = 100.528 ◦, M = 141.816 at 31/7/2003 12:00:05) 4. A white noise realization ei has been added to the intersatellite potential differences, generated with standard deviation σ = σV AB = 0.01 m2/s2 , which is roughly consistent with the GRACE Kband range-rate accuracy. 5. The state vector samples are integrated using the Adams-Moulton-Cowell method and the EGM96 model. 6. For the state vector a-priori distribution and the generation of samples, a Gaussian with equal variances for the positions and the velocities each is used. No correlations are assumed here. We mention that the observable for estimating the gravity fields is y; the orbits themselves are not used directly here for estimating the gravity field.

(48)

ˆ 2 |y) and the ˆ 1 |y), E(β that govern the estimates E(β covariance matrices derived in Section 3, and we show a histogram of the sample weighted residual square sum (Figure 2 and Figure 4) (i) WRSS 1 (i) (i) (i) (i) (y − Aβ1 βˆ2 |β1 ) (y − Aβ1 βˆ2 |β1 ) = n

(49)

that represents an empirical sample for σ in D(y|β) = σ 2 I. Figure 1 shows the weights (48) for the first set of 100 samples. It is obvious that one single sample – the “best one”— gets a weight of nearly one. The consequence is that in this case, where the Bayes estimate is numerically equal to the MAP (maximum a posteriori) estimate, it is very easy to resolve the ‘correct’ initial state vector from comparing the gravity recovery residuals provided that a sufficient number of samples has been generated. The covariance matrices for the gravity field parameters β2 are numerically the same as for the “best” sample, the covariˆ 1 |y) are practically zero. A histogram ances in D(β

weights [–] vs sample [#] 1.0 0.8 0.6 0.4 0.2 0.0 10 20 30 40 50 60 70 80 90 100

Fig. 1. Sample weights according to equation (48), case 1.


343

samples [%] vs WRSS [m**2/s**2] 0 10 20 30 40 15 15

10

samples [%] vs WRSS [m**2/s**2] 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 20 20

15

15

10

10

5

5

10

5

5

0 0

10

20

30

0 40

0 0 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040

Fig. 2. Histogram of weighted RSS according to equation (49), case 1.

Fig. 4. Histogram of weighted RSS according to equation (49), case 2.

derived from the sample WRSS (49) of the first set of samples, Figure 2, tells that for only 10% of the samples the “empirical σ ” is below 1 m2 /s2 (it would be 0.01 m2 /s2 for the “true” state vector) whereas for 90% it is above. And, for example, for 5% it is amplified even to factor between 25 and 35 m2 /s2 . This is due to the state vector “noise” propagating into the design matrix. The implication is that in the common processing with fixed βˆ2 , a large WRSS points at a bad initial state vector. Although the figure does not resolve in detail the distribution of samples with a low WRSS, it is clear that the state vector noise has a much larger influence than the data errors. In the second set of samples, we have thus artificially decreased the noise in the state vector pdf by a factor of 1 · 10−3 . This means that we simulate a case where we know β1 much better relative to β2 and the data noise in y. The α (i) are shown in Figure 3, even now there are only two samples that contribute to the Bayesian estimates. The distribution of the sample

WRSS (the empirical σ ) only slightly “smearing out” from the reference value of 0.01 m2 /s2 due to the low noise in the state vector, can be seen from Figure 4. Having in mind that this is a hypothetical case in the gravity example, it clearly exhibits the characteristics of the Bayesian estimators as a weighted sample means.

weights [–] vs sample [#] 1.0 0.8 0.6 0.4 0.2 0.0 10 20 30 40 50 60 70 80 90 100

Fig. 3. Sample weights according to equation (48), case 2.

5 Conclusions We have investigated the inversion of quasi-linear models by means of Monte-Carlo methods. Quasilinear models, where the forward operator or design matrix depends on certain unknown parameters, play an important role in geodetic modelling. To accelerate the Monte-Carlo solution, we utilize the analytical solution of the linear model under the condition that these unknown parameters are constant. The Bayesian estimators turn out as weighted means of the sample solutions. We have applied these estimators in a synthetic example from satellite geodesy and gravity field recovery, where orbit errors play a role. For the case considered here, which involved a one-day arc of simulated in-situ potential difference data, collected along a dynamical orbit with uncertain (taken as stochastic) initial state vector, we found the MonteCarlo method suitable in solving simultaneously for the gravity parameters and the state vector corrections. In fact, with realistic data errors the method points unambiguously to the “correct” state vector in the simulation. However, this would have to be validated in a more realistic setting where the orbit error is modelled by much more parameters than just 6/day as considered here, e.g. in a reduced-dynamic

344

scheme. Our results are encouraging, but more detailed simulations have to be carried out before finally applying the algorithms to real data.

References Alkhatib H, Schuh WD (2007), Integration of the Monte Carlo covariance estimation strategy into tailored solution procedures for large-scale least squares problems, J Geodesy, 81: 53–66 doi: 10.1007/s00190-006-0034-z. Bar-Shalom Y, Li XR, Kirubarajan T (2001), Estimation with Applications to Tracking and Navigation. John Wiley & Sons, New York. Golub GH, van Loan CF(1980), An analysis of the total least squares problem, SIAM J Numer Anal, 17(6):883–893.

B. Gundlich, J. Kusche Gundlich B, Koch KR, Kusche J (2003), Gibbs sampler for computing and propagating large covariance matrices, J Geodesy, 77:514–528. Ilk KH, Löcher A, Mayer-Gürr T (2008), Do we need new gravity field recovery techniques for the new gravity field satellites. Jekeli C (1999), The determination of gravitational potential differences from satellite-to-satellite tracking, Cel Mech Dyn Astr 75:85–100. Koch KR (1990) Bayesian inference with geodetic applications. Springer, Berlin. Koch KR (1999), Parameter estimation and hypothesis testing in linear models. Springer, Berlin. Koch KR (2000), Einführung in die Bayes-Statistik. Springer, Berlin. Schaffrin B, Felus YA, (2008), Multivariate Total LeastSquares Adjustment for Empirical Affine Transformations.

Wavelet Evaluation of Inverse Geodetic Problems M. El-Habiby, M. G. Sideris, C. Xu Department of Geomatics Engineering, The University of Calgary, 2500 University Drive N.W., Calgary, Alberta, Canada T2N 1N4

Abstract. A computational scheme using the wavelet transform is employed for the numerical evaluation of the integrals involved in geodetic inverse problems. The integrals are approximated in finite multiresolution analysis subspaces. The wavelet algorithm is built on using an orthogonal wavelet base function. A set of equations is formed and solved using preconditioned conjugate gradient method. The full solution with all equations requires large computer memory, therefore, multiresolution properties of the wavelet transform are used to divide the full solution into parts. Each part represents a level of wavelet detail coefficients or the approximation coefficients. Wavelet hard thresholding technique is used for the compression of the kernel. Numerical examples are given to illustrate the use of this procedure for the numerical evaluation of inverse Stokes and Vening Meinesz integrals. Conclusions and recommendations are given with respect to the suitability, accuracy and efficiency of this method. Keywords. Wavelet multiresolution analysis, hard thresholding, inverse stokes, inverse vening meinesz

1 Introduction Wavelet analysis is a comparatively young branch in signal processing. Wavelet expansions allow better local description and decomposition of signal characteristics (Burrus et al., 1998). For many years the evaluation of geodetic integrals in physical geodesy has been completely governed by the theory of the Fourier transformation. The classical approach used for the efficient evaluation of geodetic integrals is based on the Fast Fourier Transform (FFT) (Schwarz et al., 1990). This approach is well established and is now a standard tool in the geodetic arsenal. Rauhut (1992) tested different regularization methods for the solution of the inverse Stokes problem. The solution was tested using simulated and observational data.

Hwang (1998) derived the inverse Vening Meinesz formula, which converts the deflection of the vertical to gravity anomalies. The formula was implemented by two-dimensional (2D) FFT method. Sandwell and Smith (1997) computed gravity anomalies from a dense network of satellite altimetry profiles of geoid heights and a grid of the two components of the deflection of the vertical, using 2D FFT. All these approaches are mainly depending on a stationary environment. The main advantage of wavelet is its localization properties in both space and frequency domains (El Habiby and Sideris, 2006). The wavelet transform is a very powerful tool for evaluating singular geodetic integrals, because of its localization power in the space and frequency domain (Gilbert and Keller, 2000). The wavelet transform of singular kernels will lead to a significant number of small value coefficients. Therefore, high compression levels of the kernels can be achieved. In this paper, two inverse geodetic problems are evaluated. The first problem is the inverse Stokes problem, which is solved using a combination of orthogonal wavelet transform and preconditioned conjugate gradient algorithm. The second problem is the inverse Vening Meinesz that is treated as a direct convolution problem using the formulae Hwang (1998) derived. A two-dimensional wavelet transform algorithm is used for the evaluation of this formula. The main objective of this research is the verification of the wavelet approach as an alternative to the FFT in order to solve the inverse geodetic problem under the stationarity condition.

2 Wavelets as a Filtering Tool The discrete wavelet transform (DWT) coefficients ω j,k of a signal or a function f (x) are computed by the following inner product (Chui et al., 1994) ω j,k = f (x), ψ j,k (x)

(1) 345

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where ψ j,k is the wavelet expansion function and both j and k are integer indices, for the scale and translation of the wavelet function, respectively. The inverse wavelet transform is used for the reconstruction of the signal from the wavelet coefficients ω j,k by (Mallat, 1997) f (x) =

j

ω j,k (x)ψ j,k (x)

(2)

k

Equations (1) and (2) are named as analysis and synthesis, respectively. In this research, the dyadic wavelets developed by Daubechies, are used (Daubechies, 1992). Shifted (k) and dilated ( j ) versions of the wavelet function ψ0,0 and the scaling function ϕ0,0 are used to create V spaces (approximation) and W spaces (detail) (Keller, 2004), respectively. These subspaces are depending on the concept of multiresolution analysis (MRA) The concept of multiresolution analysis (MRA) is introduced for the fast decomposition of a signal into independent frequency bands through a nested sequence as follows (Keller, 2004): {0} ⊂ . . . ⊂ V1 ⊂ V0 ⊂ V−1 ⊂ . . . ⊂ L 2 ()

(3)

V j = L 2 (),

j ∈Z

ψ(x) =

V j = {0}

(4)

j ∈Z

f (•) ∈ V j ⇔ f (2 j •) ∈ V0

(8)

k∈Z

ψ is the wavelet function and the coefficients gk are defined as: gk = (−1)k h 1−k (9) For the 2D wavelet transform, the Mallat algorithm will go through a tensor product of two different directional one-dimensional wavelet transforms (Chui et al., 1994; and Mallat, 1997): ϕ(x 1 , x 2 ) = ϕ(x 1 ).ϕ(x 2 ) ψ H (x 1 , x 2 ) = ψ(x 1 ).ϕ(x 2)

(10) (11)

ψ V (x 1 , x 2 ) = ϕ(x 1 ).ψ(x 2 ) ψ D (x 1 , x 2 ) = ψ(x 1 ).ψ(x 2 )

(12) (13)

Daubechies (db4) wavelets have been used in this research (Figure 1). Daubechies wavelets have energy concentrated in time, are continuous, have null moments, and decrease quickly towards zero when the input tends to infinity. Daubechies (db) wavelets have no explicit expression except for db1 (Haar wavelet).

(5)

Wavelet thresholding is a technique used to compress the wavelet coefficients of the geodetic integral kernel. Wavelet coefficients (absolute) larger than certain specified threshold δ are the ones that should be included in reconstruction. The reconstructed function can be show as, Ogden (1997):

and the scaling function ϕ j,k ∈ L 2 () with V0 = span[ϕ(• − k) |k ∈ Z ]

fˆ(t) =

√ 2 h k ϕ(2x − k)

j

(6)

Equation (5) illustrates that all spaces V j of the MRA are scaled versions of the original space V0 , which is according to equation (6) spanned by the shifted versions ϕ0,k of the scaling function ϕ0,0 (x). A scaling (approximation) space V j is decomposed into the scaling space V j +1 and a detail space W j +1 . A number of scaling coefficients h k represent the scaling function, which is the base of space V0 ϕ(x) =

√ 2 gk ϕ(2x − k)

2.1 Wavelet Thresholding

With the following properties:

where

I{|ω j,k |>δ } ω j,k ψ j,k (t)

where I{|ω j,k |>δ } is the indicator function of this set. The problem is always in making the decision about the thresholding value. The thresholding value is as follows: (15) δ = median(ω1,k )

1

1 0.5

0.5

(7)

k∈Z

where k is the translation parameter. The base of W0 is represented by the detail function ψ j,k (Keller, 2004)

(14)

k

0 0

–0.5 0

2

4

6

0

2

4

6

Fig. 1. Daubechies with 4 vanishing moments, scaling function to the left and wavelet function to the right.

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347

If it is equal to zero, then δ=

1 max(ω1,k ) 20

(16)

For more details see (Donoho and Johnstone, 1994). Hard thresholding is used in this paper, as follows: ω j,k if ω j,k ≥ δ (17) ωˆ j,k = 0 otherwise 2.2 Inverse Stokes and Vening Meinesz in the Wavelet Frame The two inverse geodetic integrals evaluated using the 2D wavelet technique are the inversion of the Stokes integral and the inverse Vening Meinesz integral. The equation for the Stokes integral in planar approximation is (Schwarz et al., 1990): 1 N(x 2 , y2 ) = Δg(x 1, y1 ) (18) 2πγ E

×K N (x 2 , y 2 , x 1 , y1 )dx 1dy1 where 1 [(x 2 − x 1 )2 + (y2 − y1 )2 ]1/2 (19) E is the planar approximation. (x 1 , y1 , x 2 , y2 ) are local Cartesian coordinates of the data points (x 1 , y1 ) and the computational points (x 2 , y2 ), respectively (m). Δg are the gravity anomalies (mGal) γ is the normal gravity (mGal) The Inverse Vening Meinesz integral is: ⎧ −γ ⎨ ξ(x 1 , y1 )K ξ dx 1 dy1 Δg(x 2, y2 ) = 2π ⎩ E ⎫ ⎬ + η(x 1 , y1 )K η dx 1 dy1 (20) ⎭ K N (x 2 , y 2 , x 1 , y1 ) =

E

The procedure for the implementation of these inverse problems in the wavelet frame is described in the following steps: The Stokes integral will be introduced as an example for this implementation. The first step is the 2D wavelet transform of the gravity anomalies, which are unknown in case of inverse Stokes: α appr. = αH = α =

Δg ϕ(x 1 ) ϕ(y1 ) dx 1 dy1

Δg ϕ(x 1 ) ψ(y1 ) dx 1 dy1 (23) Δg ψ(x 1 ) ϕ(y1) dx 1 dy1

V

αD =

Δg ψ(x 1 ) ψ(y1 ) dx 1 dy1

Second step is the wavelet transform of the kernel, which is known in case of inverse Stokes kernel: β appr. = βH = βV =

βD =

K N ϕ(x 1 ) ϕ(y1 ) dx 1 dy1 K N ϕ(x 1 ) ψ(y1 ) dx 1 dy1 (24) K N ψ(x 1 ) ϕ(y1 ) dx 1 dy1 K N ψ(x 1 ) ψ(y1 ) dx 1 dy1

Using Beylkin’s (1993) algorithm (which mainly relies on using orthogonal wavelets), the kernel and the gravity anomalies are represented on wavelet basis using the wavelet decomposition coefficients, h for the approximation decomposition and gfor the detail decomposition. Equation (25) gives an example of this representation in the case of the kernel, as follows: K =

k

+

S

k

+

Where Kξ = Kη =

y2 − y1 (x 2 − x 1 )2 + (y2 − y1 )2 x2 − x1 (x 2 − x 1 )2 + (y2 − y1 )2

−3/2 , (21) −3/2 , (22)

and ξ and η are the two components of the deflection of the vertical.

β V gk (x 1 )h s (y1 )

s

k

β H h k (x 1 )gs (y1 )

s

k

+

β appr. h k (x 1 )h s (y1 )

β D gk (x 1 )gs (y1 )

(25)

s

Similar representation is introduced for the gravity anomalies. The summation uses dyadic intervals to avoid redundancy and decrease the computational effort (Mallat, 1997). This reconstructed kernel formula (25) is substituted in the Stokes integral (18), to

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have the following equation: appr. 1 y(x 2, y2 ) = 2πγ [ β h k (x 1 )h s (y1 ) k s H V + β h k (x 1 )gs (y1 ) + β gk (x 1 )h s (y1 ) s k k s + β D gk (x 1 )gs (y1 )]Δg(x 1, y1 )dx 1 dy1 k

s

(26) y(x 2 , y2 ) is the output from the integration at one computational point (Geoid undulation, or gravity anomalies) The wavelet transform representation of the gravity anomalies are also substituted in (26). By interchanging the order of integration and summation and subsequently integrating, the solution will be as follows: appr. appr. y2 )= α β y(x 2 , D D + αH β H + αV β V + α β (27) Due to the orthogonal properties of the wavelet used, the wavelet coefficients multiplications in (27) will lead directly to the solution (the multiplication of the wavelets decomposition coefficients (h and g) will be either 1 or zero). This can be summarized as the element-byelement multiplication of wavelet transform coefficients of the kernel and the data. Then the product output is summed up in order to have directly the signal at the computational point. It should be mentioned that the step of inverse wavelet transform is done implicitly (because of the orthogonal properties of the wavelet function (db4) used, in addition to Beylkin’s algorithm). Consequently, using this algorithm decreases the computational effort compared to standard algorithms, where the inverse wavelet transform is required for the output (Keller, 2004). The same implementation is used twice in the case of the inverse Vening Meinesz integral (because of the two double integrals (20)). The problem will be formulated in the form of the following equation (linear system of equations): Ai j x j = y j

(28)

where Ai j is the design matrix containing the wavelet coefficients of the wavelet transform of the kernels, each line is corresponding to one kernel. i is the number of kernels (equal to the number of computational points) j number of the wavelet coefficients of the kernel. x j is a vector containing the wavelet coefficients of the unknown data (gravity anomalies in Stokes case)

or known data, which is one of the two components of the deflection of the vertical. y j is a vector containing the known signal (geoid undulations in case of inverse Stokes) and unknown signal (gravity anomalies in case of inverse Vening Meinesz). In the case of the Stokes integral, the wavelet coefficients of the Stokes kernel corresponding to each computational point is in the form of a matrix, which is turned into a row vector. These row vectors (wavelet coefficients) are used to build the design matrix Ai j during the evaluation of inverse Stokes formula. For gridded data with equal spacing, which is the case in this research, the elements of the matrix (of the kernel corresponding to each computational point) in the spatial domain, are as follows:

1 ΔxΔy 2πγ [(x 2 − x 1 )2 + (y2 − y1 )2 ]1/2

(29)

for (x 2 = x 1 ) or (y2 = y1 ), and 1 ΔxΔy γ π

(30)

for (x 2 = x 1 ) and (y2 = y1 ), where Δx, Δy are the grid spacing in the x and y directions. The second values (30) compensate for the singularity at the computational point (Heiskanen and Mortiz, 1967). The procedure used in solving inverse Stokes (inverting (18)) is a combination of Beylkin’s (1993) non-standard algorithm for fast wavelet computations of linear operators, Mallat’s algorithm (Mallat, 1997), and preconditioned conjugate gradient method, in order to invert a system of linear equations in the form of (28) (Barrett et al., 1994). In case of inverse Vening Meinesz integral (20), x is known and the yis unknown and direct multiplication is done for determining the geoid undulation using (28).

3 Data Used and Results 3.1 Data Used The data used here are grids of geoid undulations, the vertical and horizontal component of the deflection of the vertical. These are on a 3’ by 3’ grid between the geographic bounds (18E-21.2E, 38.8N42N). This data is synthetic over Greece. 3.2 Inverse Stokes’ Solution The solution is done using full matrices ( A matrix without thresholding). The difference between the

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349

100

relative residual

Full solution Global thresholding(78%) 10–2

10–4

10–6

10–8

0

20

40

60 80 iteration number

100

120

Fig. 2. Precondition conjugate gradient iterations versus relative residuals for inverse stokes integral.

solution (gravity anomalies) and the reference data has a root mean square error (RMSE) of around 4 mGal. Equations (15)–(17) are applied. A 78% compression level is achieved with an RMSE of around 0.01 mGal for the difference between the full matrix solution and the 78% threshold wavelet coefficients solution. The two solutions are achieved using a conjugate gradient iterative method for solving the linear equations in the form of equation (28). The solutions are achieved in both cases after the same number of iterations (102 iterations) and at relative residual equal to 1 × 10−6 , as shown in Figure 2. 3.3 Inverse Vening Meinesz The A matrix is built for each of the two terms of the integral shown in (20) (one for the wavelet coefficients of the kernel of the horizontal component of the deflection of the vertical and one for the vertical component kernel) and then multiplied with the wavelet coefficients of vertical and horizontal components of deflection of the vertical, respectively (28). The wavelet solution of inverse Vening Meinesz integral gives identical results to the FFT solution (RMSE 0.00 mGal). Almost 96% compression level is achieved with less than 1.5 mGal loss in accuracy in comparison to the FFT solution. Higher compression levels are achieved in the Inverse Vening Meinesz case because it has faster dropping kernels than the Stokes kernel case.

4 Conclusions Wavelet representation of inverse geodetic integrals is promising. Orthogonal wavelets are essential for the usage of this algorithm. The number of multiplications and the matrix storage can be reduced

by compression, through wavelet hard thresholding technique. In comparison with the FFT approach, it gives almost identical results using full matrices without any thresholding. The compression of the inverse Stokes kernel can reach 80% of the matrices elements with less than 0.01 mGal loss in accuracy. The compression of the Vening Meinesz kernel can reach 96% of the matrices elements with less than 1.5 mGal in comparison to FFT solution.

References Barrett R., M. Berry, and T. F. Chan (1994). Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, SIAM. Beylkin G. (1993),Wavelets and Fast Numerical Algorithms, Lecture Notes for short course, AMS-93, Proceedings of Symposia in Applied Mathematics, Vol. 47, pp. 89–117, 1993. Burrus C. S., R. A. Gopinath, and H. Guo (1998). Introduction to Wavelets and Wavelet Transforms: A Primer. Prentice Hall, New Jersey. Chui C. K., L. Monefusco, and L. Puccio (1994). Wavelets: theory, algorithms, and applications. Wavelet analysis and its applications, Vol. 5, Academic Press, INC. Daubechies I. (1992). Ten lectures on wavelets. Society for Industrial and Applied Mathematics. Donoho D. L., and I. Johnstone (1994). Ideal Spatial Adaptation via Wavelet Shrinkage. Biometrika 81, pp. 425–455. El Habiby M., and M.G. Sideris (2006). On the Potential of Wavelets for Filtering and Thresholding Airborne Gravity Data. Newton’s Bulletin of the International Gravity Field Service, No. 3, section 1, Reviewed papers. Gilbert A., and W. Keller (2000), Deconvolution with wavelets and vaguelettes, J Geod 74, pp. 306–320. Heiskanen W. A., and H. Mortiz (1967). Physical Geodesy. W. H. Freeman and Company, San Fransisco. Hwang C. (1998). Inverse Vening Meinesz formula and deflection-geoid formula: application to prediction of gravity and geoid determination over South China Sea. J Geod 72, pp.113–130. Keller W. (2004). Wavelets in geodesy and geodynamics. Walter de Gruyter. Mallat, S. G. (1997). A wavelet tour of signal processing, Academic Press, san Diego. Ogden R. T. (1997). Essential Wavelets for Statistical Applications and Data Analysis. Birkhäuser, Boston, USA. Rauhut A.C. (1992). Regularization methods for the solution of the inverse Stokes problem. UCGE reports, No. 20045, Department of Geomatics Engineering, University of Calgar. Sandwell D.T., and W.H.F. Smith (1997). Marine gravity anomaly from Geosat and ERS 1 satellite altimetry. J Geop R, 102: B5, pp. 10,039–10,054. Schwarz K. P., M. G. Sideris, and R. Forsberg (1990). The use of FFT techniques in physical geodesy. Geop J Int, 100, pp. 485–514.

Correcting the Smoothing Effect of Least-Squares Collocation with a Covariance-Adaptive Optimal Transformation C. Kotsakis Department of Geodesy and Surveying, Aristotle University of Thessaloniki, University Box 440, GR-54124, Thessaloniki, Greece, e-mail: [email protected]

Abstract. An optimal modification of the classical LSC prediction method is presented, which removes its inherent smoothing effect while sustaining most of its local prediction accuracy at each computation point. Our ‘de-smoothing’ approach is based on a covariance-matching constraint that is imposed on a linear modification of the usual LSC solution so that the final predicted field reproduces the spatial variations patterns implied by an adopted covariance (CV) function model. In addition, an optimal criterion is enforced which minimizes the loss in local prediction accuracy (in the mean squared sense) that occurs during the transformation of the original LSC solution to its CV-matching counterpart. The merit and the main theoretical principles of this signal CV-adaptive technique are analytically explained, and a comparative example with the classical LSC prediction method is given. Keywords. Least-squares collocation, spatial random field, smoothing, covariance matching

1 Introduction The prediction of the functional values of a continuous spatial random field (SRF), using a set of observed values of the same and/or other SRFs, is a fundamental inverse problem in geosciences. The mathematical model describing such a problem is commonly formulated in terms of the observation equation yi = L i (u) + vi ,

i = 1, 2, . . . , n

(1)

where u(P) denotes the primary random field of interest (P ∈ D, with D being a bounded or unbounded spatial domain) that needs to be determined, at one or more points, using n discrete measurements {yi } which are taken on the same and/or other locations. The symbols L i (·) correspond to bounded linear or linearized functionals of the unknown field, depending on the physical model that relates the observable 350

quantities with the underlying SRF itself, while the terms {vi } contain the effect of measurement random noise. Typical examples that fall within the realm of the aforementioned SRF prediction scheme include the determination of the disturbing gravity potential on or outside a spherical Earth model using various types of gravity field functionals, the prediction of stationary or non-stationary ocean circulation patterns from satellite altimetry data, the prediction of atmospheric fields (tropospheric, ionospheric) from the tomographic inversion of GPS data, the prediction of crustal deformation fields from geodetic data, etc. The predominant approach that is generally followed in geodesy for solving such problems is leastsquares collocation (LSC) which was introduced by Krarup (1969) in a deterministic context as a rigorous approximation method in separable Hilbert spaces with reproducing kernels, and formulated in parallel by Moritz (1970) in a probabilistic setting as an optimal prediction technique for spatially correlated random variables and stochastic processes; see also Sanso (1986) and Dermanis (1976). A critical aspect in LSC is the smoothing effect on the predicted signal values u(P), ˆ which typically exhibit less spatial variability than the actual field u(P). Consequently, small field values are overestimated and large values are underestimated, thus introducing a likely conditional bias in the final results and possibly creating artifact structures in SRF maps generated through the LSC process. Note that smoothing is an important characteristic which is not solely associated with the LSC method, but it is shared by most interpolation techniques aiming at the unique approximation of a continuous function from a finite number of observed functionals. Its merit is that it guarantees that the recovered field does not produce artificial details not inherent or proven by the actual data, which is certainly a desirable characteristic for an optimal signal interpolator. However, the use of smoothed SRF images or maps generated by techniques such as LSC provides a shortfall for

Correcting the Smoothing Effect of Least-Squares Collocation with a Covariance-Adaptive Optimal Transformation

applications sensitive to the presence of extreme signal values, patterns of field continuity and spatial correlation structure. While founded on local optimality criteria that minimize the mean squared error (MSE) at each prediction point, the LSC approach overlooks to some extent a feature of reality that is often important to capture, namely spatial variability. The latter can be considered a global field attribute, since it only has meaning in the context of the relationship of all predicted values to one another in space. As a result of the smoothing effect, the ordinary LSC solution does not reproduce either the histogram of the underlying SRF, or the spatial correlation structure as implied by the adopted model of its covariance (CV) function. In this paper we present an ad-hoc approach that enhances LSC-based field predictions by eliminating their inherent smoothing effect, while preserving most of their local prediction accuracy. Our approach consists of correcting a-posteriori the optimal result obtained from LSC, in a way that the corrected field matches the spatial correlation structure implied by the signal CV function that was used to construct the initial LSC solution. Similar predictors have also appeared in the geostatistical literature by constraining the usual unbiased kriging-type solution through a covariance-matching condition, thus yielding new linear SRF predictors that match not only the first moment but the second moment of the primary SRF as well (Aldworth and Cressie 2003, Cressie 1993). In contrast to stochastic simulation schemes which provide multiple equiprobable signal realizations according to some CV model of spatial variability (e.g. Christakos, 1992), the methodology presented herein gives a unique field estimate that is statistically consistent with a prior model of its spatial CV function. The uniqueness is imposed though an optimal criterion that minimizes the loss in local prediction accuracy (in the MSE sense) which occurs when we transform the LSC solution to match the spatial correlation structure of the underlying SRF.

2 Ordinary Least-Squares Collocation Denoting by si = L i (u) the signal part in the available data, the system of observation equations in (1) can be written in vector form as y = s+v

(2)

where y, s and v are random vectors containing the known measurements, and the unknown signal and noise values, respectively, at all observation points

351

{Pi }. The signal and noise components in (2) are considered uncorrelated with each other (a crucial simplification that is regularly applied in practice), and of known statistical properties in terms of their given expectations and co-variances. Assuming that the spatial variability of the primary SRF u is described by a known CV function model Cu (P, Q), the elements of the CV matrix of the signal vector s are determined according to the CV propagation law (Moritz 1980) Cs (i, j ) = L i L j Cu (Pi , P j )

(3)

where L i and L j correspond to the functionals associated with the i th and j th observation. In the same way, the cross-CV matrix between the primary field values (at the selected prediction points {Pi }) and the observed signal values is obtained as Cus (i, j ) = L j Cu (Pi , P j )

(4)

The CV matrix Cv of the data noise is also considered known, based on the availability of an appropriate stochastic model describing the statistical behavior of the zero-mean measurement errors. An additional postulate on the spatial trend of the primary SRF is often employed as an auxiliary hypothesis for the LSC inversion of (1) or (2). In fact, various LSC prediction algorithms may arise in practice, depending on how we treat the signal detrending problem. For the purpose of this paper and without any essential loss of generality, it will be assumed that we deal only with zero-mean SRFs and signals (E{u} = 0, E{s} = 0). Based on the previous assumptions, the LSC predictor of the primary SRF, at all selected prediction points, is given by the well known matrix formula uˆ = Cus (Cs + Cv )−1 y

(5)

which corresponds to a linear unbiased solution with minimum mean squared prediction error (Moritz 1980, Sanso 1986). The inherent smoothing effect in LSC can be identified from the CV structure of its optimal result. Applying CV propagation to the predicted field uˆ in (5), we obtain the result Cuˆ = Cus (Cs + Cv )−1 CTus

(6)

which generally differs from the CV matrix Cu of the original SRF at the same set of prediction points, i.e.

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Cu (i, j ) = Cu (Pi , P j ) = Cuˆ (i, j )

(7)

Moreover, if we consider the vector of the prediction errors e = uˆ − u, it holds that Cuˆ = Cu − Ce

(8)

new predictor uˆ , including its CV-matching property given in (10). The predicted field obtained from (11) should reproduce the CV structure of the primary SRF, in the sense that Cuˆ = Cu for the given spatial distribution of all prediction points {Pi }. Hence, the filtering matrix R has to satisfy the constraint

where the error CV matrix is given by the equation Ce = Cu − Cus (Cs + Cv )−1 CTus

(9)

The fundamental relationship in (8) conveys the meaning of the smoothing effect in LSC, which essentially acts as an optimal low-pass filter to the input data. The spatial variability of the LSC prediction errors, in terms of their variances and covariances, is exactly equal to the deficit in spatial variability of the LSC predictor uˆ with respect to the original SRF u.

3 Optimal “De-Smoothing” of the LSC solution

(10)

where Cu is the CV matrix formed through the CV function Cu (P, Q) of the primary SRF; see (7). In addition, the prediction errors e = uˆ − u associated with the field predictor uˆ should remain small in some sense, so that the new solution can provide not only a CV-adaptive representation for the SRF variability, but also locally accurate predicted values on the basis of the given data. For this purpose, the formulation of the operator (·) should additionally incorporate some kind of optimality principle by minimizing, for example, the trace of the new error CV matrix Ce . Let us now introduce a straightforward linear ˆ through approach to modify the LSC predictor u,

uˆ = Ruˆ

(12)

where Cu and Cuˆ correspond to the CV matrices of the primary and the LSC-predicted SRFs. The assessment of the prediction accuracy of the new solution uˆ can be made through its error CV matrix Ce = E{(uˆ − u)(uˆ − u)T }

(13)

which, taking (11) into account, yields ˆ T Ce = RCuˆ RT + Cu − RCuu ˆ − Cu uR

Our objective is to develop a correction algorithm that can be applied to the optimal field prediction obtained from LSC for the purpose of removing its inherent smoothing effect, while sustaining most of its local prediction accuracy. In general terms, we seek a “de-smoothing” transformation to act upon the ˆ such that the CV structure LSC predictor, uˆ = (u), of the primary SRF is recovered. This means that the transformation (·) should guarantee that Cuˆ = Cu

RCuˆ RT = Cu

(14)

Using (8) and the following relation that is always valid for the LSC predictor uˆ (assuming that there is zero correlation between the observed signals s and the measurement noise v) Cuu ˆ = Cuˆ

(15)

the new error CV matrix can be finally expressed as Ce = Ce + (I − R)Cuˆ (I − R)T

(16)

where Ce is the error CV matrix of the LSC solution. Evidently, the prediction accuracy of the modified solution uˆ will always be worse than the predicˆ regardtion accuracy of the original LSC solution u, less of the form of the filtering matrix R. This is expected since LSC provides the best (in the MSE sense) unbiased linear predictor from the available measurements, which cannot be further improved by additional linear operations. Nevertheless, our aim is to determine an optimal filtering matrix that satisfies the CV-matching constraint in (12), while minimizing the loss of the MSE prediction accuracy in the recovered SRF, in the sense that trace (Ce − Ce ) = trace(δCe ) = minimum (17)

(11)

where R is a square filtering matrix that needs to be determined according to some optimal criteria for the

where δCe = (I − R)Cuˆ (I − R)T represents the part of the error CV matrix of the new predictor uˆ which depends on the choice of the filtering matrix.

Correcting the Smoothing Effect of Least-Squares Collocation with a Covariance-Adaptive Optimal Transformation

The determination of the filtering matrix R which (i) satisfies the CV-matching constraint (12), and (ii) minimizes the loss in the MSE prediction accuracy of the predictor uˆ according to (17), is analytically described in Kotsakis (2007). Due to space limits, we will only present the final result herein, without going into any technical details regarding its mathematical proof. The optimal filtering matrix is R=

1/2 1/2 1/2 1/2 Cu (Cu Cuˆ Cu )−1/2 Cu

(18)

or equivalently (see Appendix) −1/2

R = Cuˆ

(Cuˆ Cu Cuˆ )−1/2 Cuˆ 1/2

1/2

1/2

(19)

Note that the above result was originally derived in Eldar (2001) under a completely different context than the one discussed in this paper, focusing on applications such as matched-filter detection, quantum signal processing and signal whitening.

4 Numerical Test A numerical example is presented in this section to demonstrate the performance of the CV-adaptive predictor uˆ , in comparison with the classical LSC preˆ The particular test refers to a standard noise dictor u. filtering problem for a set of simulated gridded gravity anomaly data. The image shown in Figure 1(a) is the actual realization of a free-air gravity anomaly field which has been simulated within an 50×50 km2 area and with a uniform sampling resolution of 2 km,

353

according to the following model of the spatial CV function

Cu (P, Q) =

Co 1 + (r P Q /a)2

(20)

where Co = 220 mgal2 , r P Q is the planar distance between points P and Q, and the parameter a is selected such that the correlation length of the gravity anomaly field is equal to 7 km. The noisy data grid is shown in Figure 1(b), with the underlying noise level being equal to ±15 mgals. Note that the additive random errors have been simulated as a set of uncorrelated random variables, thus enforcing a white noise assumption for the gridded data. In Figure 1(d) we see the filtered signal as obtained from the classical LSC algorithm (i.e. Wiener filtering), whereas in Figure 1(c) is shown the result obtained from the CV-adaptive solution uˆ . It is seen that, although LSC provides in principle the most accurate (in the MSE sense) filtered signal, the result obtained from the CV-adaptive predictor clearly looks more similar to the original SRF that is depicted in Figure 1(a). The emulation of the spatial variability of the primary SRF by the CV-adaptive solution uˆ , in contrast to the smoothed representaˆ can also be tion obtained by the LSC predictor u, seen in the histograms plotted in Figure 2, as well as in the signal statistics listed in Table 1.

(a)

(b)

(c)

(d)

Fig. 1. Plots of the actual (simulated) gravity anomaly signal (a), the noisy observed signal (b), the CV-matching filtered signal (c), and the LSC-filtered signal (d).

354

C. Kotsakis

(a)

(b)

(c)

Fig. 2. Histograms of the actual (simulated) gravity anomaly signal (a), the LSC-filtered signal (b), and the CV-matching filtered signal (c).

Table 1. Statistics of the actual (simulated) signal u, the LSCfiltered signal uˆ and the CV-matching filtered signal uˆ (all values in mgals)

Actual grid values LSC solution CV-matching solution

Max

Min

Mean

σ

45.33 27.42 41.47

–42.88 –29.10 –42.48

–0.04 0.74 0.73

±14.96 ±10.29 ±14.64

inbuilt in the theoretical development of the classical LSC method (Moritz, 1980). In practice, an empirical signal CV function is often first estimated from a given and possibly noisy data record, and then used in the implementation of the LSC procedure for the (sub-optimal) recovery of the primary SRF at a set of prediction points. For such cases, it is reasonable to question whether it would be meaningful to let the spatial variability of the LSC-predicted field to be adapted to an empirical CV function by following the CV-matching approach presented in this paper. A more reasonable methodology would be to additionally incorporate a variance component estimation approach, in a way that the final predicted field becomes adapted to an “improved” model of spatial variability (i.e. with respect to the one imposed by the empirical CV function). In contrast to the standard CV-matching constraint introduced in (10), we can impose in this case the alternative CV-tuning constraint (21) Cuˆ = σ 2 Qu where the known CV matrix Qu is formed through the empirically determined signal CV function, and σ 2 is an unknown variance factor which controls the consistency between the empirical and the true CV function for the underlying unknown signal. Tackling the above problem along with the study of one-step CV-matching linear predictors (see Schaffrin 1997, 2002), instead of the two-step constructive approach that was presented herein, may be an interesting subject for future investigation.

Acknowledgements The author would like to acknowledge the valuable suggestions and comments by the two reviewers, as well as the constructive criticism provided by the responsible editor Prof. Burkhard Schaffrin.

References 5 Conclusions Due to its inherent smoothing effect, the LSC prediction algorithm does not reproduce the spatial correlation structure implied by the CV function of the primary SRF that needs to be recovered from its observed functionals. The method presented in this paper offers an alternative approach for optimal SRF prediction, which preserves the signal’s spatial variability as dictated by its known CV function. Evidently, the rationale of the proposed technique relies on the knowledge of the true CV function of the underlying SRF, an assumption which is also

Aldworth J, Cressie N (2003) Prediction of nonlinear spatial functionals. J Stat Plan Inf, 112: 3–41. Christakos G (1992) Random Field Models for Earth Sciences. Academic Press, New York, NY. Cressie N (1993) Aggregation in geostatistical problems. In A. Soares (ed) Geostatistics Troia 1992, vol 1, pp. 25–36, Kluwer, Dordrecht. Dermanis A (1976) Probabilistic and deterministic aspects of linear estimation in geodesy. Rep No. 244, Dept. of Geodetic Science, Ohio State University, Columbus, Ohio. Eldar YC (2001) Quantum signal processing. PhD thesis, Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA.

Correcting the Smoothing Effect of Least-Squares Collocation with a Covariance-Adaptive Optimal Transformation Krarup T (1969) A contribution to the mathematical foundation of physical geodesy. Danish Geodetic Institute, Report no. 47, Copenhagen. Kotsakis C (2007) Least-squares collocation with covariancematching constraints. J Geod, 81(10): 661–677. Moritz H (1970) Least-squares estimation in physical geodesy. Deutsche Geodätische Kommission, Reihe A, Heft 69, Muenchen. Moritz H (1980) Advanced Physical Geodesy. Herbert Wichmann Verlag, Karlsruhe. Sanso F (1986) Statistical Methods in Physical Geodesy. In H. Suenkel, (ed): Mathematical and Numerical Techniques in Physical Geodesy, Lecture Notes in Earth Sciences, vol. 7, pp. 49–156, Springer Verlag, Berlin. Schaffrin B (1997) On suboptimal geodetic network fusion. Poster presented at the IAG Scientific Assembly, September 3–9, Rio de Janeiro, Brazil. Schaffrin B (2002) Reproducing estimates via least-squares: An optimal alternative to Helmert transformation. In: E.W. Grafarend, F.W. Krumm, V.S. Schwarze (eds) Geodesy – The Challenge of the 3rd Millennium, pp. 387–392, Springer, Berlin, Heidelberg, New York.

Appendix In this appendix, we will establish the equivalency between the two forms of the optimal filtering matrix R that were given in (18) and (19), respectively. Starting from (18) and using the following matrix identity (which is easy to verify for all invertible matrices S and T)

355

(ST)−1/2 = S(TS)−1/2 S−1

(1)

we have R = Cu (Cu Cuˆ Cu )−1/2 Cu 1/2

1/2

1/2

1/2

= Cu (Cu Cuˆ Cuˆ Cu )−1/2 Cu 1/2

1/2

1/2

1/2

1/2

T 1/2

S 1/2 1/2

1/2

= Cu (Cu Cuˆ )(Cuˆ Cu Cu Cuˆ )−1/2 × 1/2

S

× (Cu Cuˆ )−1 Cu 1/2

1/2

S 1/2

1/2

1/2

1/2

T

S

−1/2

= Cu Cuˆ (Cuˆ Cu Cuˆ )−1/2 Cuˆ 1/2

1/2

1/2

−1/2

Cu

1/2

Cu

−1/2

Cuˆ Cu Cuˆ (Cuˆ Cu Cuˆ )−1/2 Cuˆ

−1/2

(Cuˆ Cu Cuˆ )(Cuˆ Cu Cuˆ )−1/2 Cuˆ

−1/2

(Cuˆ Cu Cuˆ )1/2 Cuˆ

= Cuˆ = Cuˆ = Cuˆ

1/2

1/2

1/2

1/2

1/2

1/2

1/2

1/2

1/2

1/2

−1/2 −1/2

−1/2

(2) The last expression in the above equation is identical to the matrix form given in (19), and thus the equivalency between (18) and (19) has been established.

Analytical Downward and Upward Continuation Based on the Method of Domain Decomposition and Local Functions Y.M. Wang, D.R. Roman, J. Saleh National Geodetic Survey, Silver Spring, MD 20910, USA

Abstract. Upward and downward continuation of gravity anomalies are usually based on the Poisson integral and its iteration. By using the method of domain decomposition, a local function can be used for upward and downward continuation of gravity data. This approach decomposes the total area into small domains, and uses local functions to model the disturbing potential within each of these domains. One appropriate set of local functions, used in this paper, is the 3-D Fourier series. The results of the domain decomposition approach are compared to the Poisson integral for upward continuation in a flat area (to avoid large topographic effect) and downward continuation in a rough area in the Rocky Mountains. The data consist of 1 × 1 grid of NGS’ surface gravity over the conterminous US (CONUS). In addition, INTERMAP’s airborne gravity tracks in the Baltimore area are used to validate the upward continuation results of the two methods. The advantages and disadvantages of the local functions versus the Poisson integral approach are discussed. Keywords. Poisson integral, harmonic continuation, local functions, 3-D Fourier series and domain decomposition

1 Introduction The Poisson integral links a harmonic function above a sphere with its value on the sphere and can be used for upward continuation (Heiskanen and Moritz, 1967, p. 37). After a slight manipulation of the Poisson integral, it can also be used for downward continuation (Heiskanen and Moritz, 1967, p. 317). Although upward and downward continuations produce harmonic functions, they are quite different in stabilities. The upward continuation is a stable process, while downward continuation is an ill-posed problem. Alternatively, upward and downward continuation can be performed by using global functions, such 356

as a spherical harmonic series. The estimation of global functions that represent the fine structure of the gravity field is very difficult for two reasons: (1) The downward continuation of NGS’ 1 × 1 gravity grid requires global gravity with the same data density. (2) Such data density implies an expansion degree of 10800 for spherical harmonic series. The estimation of the coefficients of such a series entails solving linear systems with about 100 million unknowns. Even if the spherical harmonic series is used as a set of approximating base functions in local or regional analyses, capturing the fine structure of the data requires that the series must extend to a high degree and order. Therefore, more localized functions may be numerically more efficient for local and regional analysis. In this paper, another alternative for upward and downward continuation of gravity data is used: the domain decomposition and local functions approach. This method has several advantages: (1) the local functions easily capture the high frequencies of dense local or regional data; (2) there is no need for global data; (3) estimating the coefficients of local functions is a simple matter.

2 Analytical Continuation Based on Poisson’s Integral 2.1 Upward Continuation The gravity anomaly at any point, P, above the Earth’s surface can be computed by: 2 2 ) g g P = t (1−t dσ 4π D3 σ t = R r , D = 1 − 2t cos ψ + t 2

(1)

where r is the radial coordinate of the computation point P, D and ψ are its chord and spherical distance to the infinitesimal element dσ , respectively, and R is the radius of a sphere that approximates the geoid. Upward continuation using the Poisson integral has several disadvantages: (1) it is based on the

Analytical Downward and Upward Continuation Based on the Method of Domain Decomposition and Local Functions

assumptions that the gravity data is given on the geoid and that the geoid is a sphere of radius R. In reality, gravity data are measured on the Earth’s physical surface, not on the geoid, and the geoid is not a sphere; (2) the numerical integration of Poisson’s integral is time consuming. Also, the fast Fourier method can not be directly applied, if the computation points are not at the same altitude. 2.2 Analytical Downward Continuation (ADC) ADC is computed by Poisson’s integral in an iterative fashion (Heiskanen and Moritz, 1967) p. 318 g1∗ = g ······ ∗ gn+1

k 1 [(ψ Pk ) − f Pk ]2 2

n

I =

k

σ

gn∗ − (gn∗) P dσ D3 (2)

where the anomalies, g, are given on or above the Earth’s surface and the downward continued gravity, g∗, is usually computed on the geoid. The index stands for the iteration number. As can be seen from the equations above, initially, the downward continued data is taken as the given anomalies. The result of the previous iteration is then inserted in the integral, resulting in the downward continued data of the current iteration. The procedure has to be stopped before it diverges. The iterative ADC using Poisson’s integral has several disadvantages: (1) Its computation is timeconsuming; (2) Its convergence depends on the roughness and density (grid size) of the data. It has been reported that the grid size must be limited in order to guarantee convergence of the iterative process (e.g., Martinec, 1996).

3 Analytical Continuation Based on Local Functions Let I be a functional of the disturbing potential given by: I =

n 1 [(ψ P ) − f P ]2 2

(3)

P=1

where ψ P is a function approximating the disturbing potential at point P, is a functional, f P is an observation of (ψ P ) and n is the total number of observations. The domain decomposition approach divides the total area into local domains or cells, k=1, 2, 3, · · · , thus the functional I becomes:

(4)

P=1

where each ψ Pk is defined only for cell k and is trivial elsewhere, f Pk represents the corresponding observation and n k is the total number of observations used to estimate a solution for cell k. As a demonstration, we choose the local functions, ψk , to be the 3-D Fourier series (Wang, 2001):

ψk (x, y), H =

t 2 (1 − t 2 ) = g − 4π

357

N M

λmn [amn cos u m x cos vn y+ m=0 n=0 cos vn y + cmn cos u m x sin vn y+

+bmn sin u m x +dmn sin u m x sin vn y] exp(−ωmn H ) (−L ≤ x ≤ L; −W ≤ y ≤ W )

(5)

with

um =

mπ nπ , vn = , ωmn = L W

u 2m + vn2

(6)

where M and N are the cutoff frequencies of the series, x,y are the local coordinates which refer to a coordinate system with its x axis pointing to the north and y axis pointing to the east with origin at the cell center ; amn , bmn , cmn and dmn are coefficients of the Fourier expansion; L and W are the half width and half length of the cell. The λmn is the normalization factor given by:

λmn

⎧ (m = 0, n = 0) ⎨ 0.25 0.5 m > 0, n = 0; m = 0, n > 0 = ⎩ 1 (m > 0, n > 0)

(7)

The coefficients amn , bmn , cmn and dmn for cell k constitute the parameters of the local solution ψk, to be estimated from the data in and around cell k. The (least squares) solution is the one that minimizes the functional I. To ensure continuity and smoothness of the solution ψk across the borders of cell k, all data located in margins of width L/2 on all sides of cell k are included in estimating the solution ψk. Alternatively, the values of the disturbing potential and its first and second height-derivatives on the four corners of each cell may be constrained to be equal for all neighboring cells. This alternative requires the solution of a block tri-diagonal linear system rather than a block diagonal one.

358

Y.M. Wang et al.

4 Numerical Tests in the Baltimore Area for Upward Continuation The local functions approach is compared to the traditional Poisson integral and the results of both methods are validated by comparisons to INTERMAP’s airborne gravity tracks (6 km above the ground). Figure 1 presents the airborne tracks on the background of NGS point (surface) gravity data in the Baltimore area. Poisson’s integral for upward continuation requires the gravity data to be given on a sphere. To minimize the topographic effects on our comparison resulting from the deviation from this assumption, the upward continuation is tested in the flat Baltimore area, where the topography is lower than 400 m. The NGS 1 × 1 surface gravity data is upward continued to the flight altitude using the Poisson integral. The integration is done over the whole CONUS (34◦ × 76◦ ) area. The local functions are chosen as the 3-D Fourier series, which in this test, represent the surface gravity anomalies. The CONUS area is decomposed into 1◦ × 1◦ cells. The coefficients of the Fourier series in each cell are estimated by least squares. Table 1 shows statistics of the airborne and upward continued gravity anomalies and Table 2 shows the statistics of the differences between them. The differences between the two upward continuation methods

mGal

40' 00'

50

Table 1. Statistics of the airborne gravity and upward continued gravity anomalies (mGal)

No. of Pts Mean RMS Min. Value Max. Value

Airborne

F-Series

Poisson

6252 20.4 23.8 −27.2 42.0

6252 20.5 23.7 −19.8 37.8

6252 20.3 23.4 −19.3 38.4

Table 2. Statistics of the differences (mGal)

No. of Pts Mean RMS Min. Value Max. Value

Airborne F-Series

AirbornePoisson

F-SeriesPoisson

6252 −0.2 2.6 −11.4 8.3

6252 0.1 2.6 −11.1 8.9

6252 −0.2 0.4 −1.0 0.9

agree on the sub-mGal level. The fact that Poisson’s integral assumes the gravity anomaly data to be on the mean Earth sphere is one of the sources for these differences. The mean difference between each method and the airborne data is a small fraction of a mGal. The RMS of the differences is 2.6 mGal, mainly due to tilts in several tracks. This number could have been substantially better had there been more crossovers in the airborne survey (see Figure 1).

5 Numerical Tests in the Rocky Mountains for Downward Continuation

45 40 35 39' 30'

30 25 20 15

39' 00'

10 5 0 –5

38' 30'

–10 –15 –20 –25

38' 00' 282' 00'

%

20

0 –40

–30 282' 30'

283' 00'

283' 30'

N = 14098 Mean = 16.62 SD = 21.54

–20

284' 00'

Min = –55.40 Max = 59.90

0

20

40

60

(mGal)

Fig. 1. Airborne and surface gravity in Baltimore.

We next test downward continuation of gravity anomaly data in a rough area in the Rocky Mountains using both methods, Poisson’s integral and the local functions (3-D Fourier series) with the domain decomposition approach. The test area extends over the geographic window 39◦ ≤ ϕ ≤ 40◦, 250◦ ≤ λ ≤ 251◦ where the elevations range from 1300 to 2800 m (Figure 2). The surface gravity data consist of point free air anomalies on the Earth surface from the NGS gravity database, gridded into a 1 × 1 grid. The ADC effect is the change in gravity anomalies due to the downward continuation, i.e., the downward continued anomalies on the geoid minus the surface gravity anomalies. The ADC effect computed by each iteration of the Poisson integral was compared to that obtained using the Fourier series. Figure 3 shows this effect after 4 iterations of the Poisson integral (top) and by the 3-D Fourier series (bottom). Clearly, both plots are quite similar. The

Analytical Downward and Upward Continuation Based on the Method of Domain Decomposition and Local Functions m

40' 00'

39' 50'

39' 40'

39' 30'

39' 20'

39' 10'

39' 00' 250' 00' 250' 10' 250' 20' 250' 30' 250' 40' 250' 50' 251' 00' Min = 1318.11 Max = 2787.22

0 1200

1600

2000 (m)

2400

mGal

39' 50'

39' 40'

39' 30'

39' 20'

39' 10'

39' 00' 250' 00' 250' 10' 250' 20' 250' 30' 250' 40' 250' 50' 251' 00' 10 Min = – 67.21 N = 3600 %

Mean = 2.54 SD = 31.29

0 –80

–60

–40

12 39' 50'

8 4 2

40

60

0

39' 30'

–2 –4 39' 20'

–6 –8 –10

39' 10'

–12 –14

39' 00' 250' 00' 250' 10' 250' 20' 250' 30' 250' 40' 250' 50' 251' 00' 10 N = 3600 Min = –17.93 Mean = 1.65 SD = 7.42

00 70 60 50 40 30 20 10 0 –10 –20 –30 –40 –50 –60 –70 –80

80

Fig. 2. Elevations (top) and surface anomalies (bottom) in a test area in the Rocky Mountains.

RMS and range of their differences is 0.8 and 2.79 to 5.75 mGals, respectively. However, the first Poisson iteration (the g1 term), differs considerably from the Fourier series. Their RMS difference and range is 2.8 and −7.4 to 12.9 mGal, respectively. As expected, the differences between the downward continued anomalies obtained by the Fourier series and those from the Poisson iterations become smaller as the number of iterations increases. However, after three iterations, the process starts a slow divergence characterized by increasing range of the extreme values. Table 3 shows the statistics of the differences between the Fourier series downward continued

–12

–16

Max = 29.63

–8

–4

0 (mGal)

4

8

12

16 mGal

40' 00'

16 14 12

39' 50'

10 8 6

39' 40'

4 2 0

39' 30'

–2 –4 39' 20'

–6 –8 –10

39' 10'

–12 –14

39' 00' 250' 00' 250' 10' 250' 20' 250' 30' 250' 40' 250' 50' 251' 00' 10

0 20 (mGal)

6

0 –16

Max = 71.61

–20

10

39' 40'

2800

40' 00'

16 14

%

N = 3600 Mean = 1928.81 SD = 327.71

40' 00'

%

%

10

2000 2700 2600 2500 2400 2300 2200 2100 2000 1900 1800 1700 1600 1500 1400 1300 1200

359

N = 3600 Mean = 2.12 SD = 7.75

0 –16

–12

–16

Min = –16.73 Max = 31.63

–8

–4

0 (mGal)

4

8

12

16

Fig. 3. ADC effect using the Poisson integral (top) and the 3-D FS (Bottom).

gravity anomalies and those obtained by 1, 2, 3 and 4 iterations of the Poisson integral. Clearly, the first three iterations bring the results increasingly closer to the Fourier series results. The larger extreme values in iteration 4 over iteration 3, and the lack of improvement in the mean with only a very small improvement in the RMS differences from iteration 3 to 4, imply that the divergence has started after the third iteration. Since the Poisson iterations are essentially equivalent to the traditional g1, g2, g3, . . . correction terms, Table 3 shows that the g2 and higher terms still contain significant power, which is lost if only the g1 term is taken into account.

360

Y.M. Wang et al.

Table 3. Statistics of the differences between the ADC effect by Fourier series and those obtained by 1, 2, 3 and 4 iterations of the Poisson integral, in mGal Iteration#

1st

2nd

3rd

4th

Mean RMS Min. Max.

−0.72 2.81 −12.92 7.40

−0.50 1.36 −7.16 3.90

−0.47 0.90 −5.52 2.48

−0.47 0.80 −5.75 2.79

For downward continuation of gravity anomalies in a rugged test area in the Rocky Mountains, the 3D Fourier series shows significant advantage in computational efficiency. The change in gravity anomalies due to the downward continuation using the Poisson iterations agree with those of the 3D Fourier series to a 0.8 mGal RMS after 4 iterations. However, the Poisson integral iterations start divergence after the third iteration whereas the Fourier series does not suffer from any numerical instability.

Conclusions

References

The use of the Poisson integral for upward and downward continuation of gravity data is time consuming and its iteration diverge for dense data. The local functions and domain decomposition approach demonstrate efficiency and accuracy. For upward continuation in a flat area, the results of the traditional Poisson integral and the 3D Fourier series are almost identical (RMS difference of 0.4 mGal with maximum differences below ±1 mGal).

Heiskanen W. A. and H. Moritz (1967). Physical Geodesy, W. H. Freeman, San Francisco. Martinec Z. (1996). Stability investigations of a discrete downward continuation problem for geoid determination in the Canadian Rocky Mountains. Journal of Geodesy, Vol. 70, pp 805–828. Wang Y. M. (2001). GSFC00 mean sea surface, gravity anomaly, and vertical gravity gradient from satellite altimetry data. Journal of Geophysical Research, Vol. 106, C11, 31, pp.167–175.

Author Index

Altamimi Z. 101 Amiri-Simkooei A.R. 200, 273, 277 Austen G. 327

Kulkarni M. N. 88 Kusche J. 337 Kutterer H. 232

Bao J. 258, 297

El-Habiby M. 345

Le A.Q. 166 Li B. 195 Li C. 155 Li J. 35 Li M. 258, 297 Li S. 293 Li W. 293 Li Z. 287 Liu C. 189 Liu J. 133, 211 Liu J. 189 Liu X. 17 Liu Yan-chun 258, 297 Liu Y. 248 Löcher A. 3 Loon J.P. van 43 Luo N. 207

Fan P. 248 Felus Y. A. 238 Freeden W. 333

Ma C. 243 Mayer-Gürr T. 3 Monico J.F.G. 179

Gao B. 293 Ge M.R. 133 Geng J.H. 133 Grafarend E. W. 79, 216 Gramsch S. 333 Grund V. 128 Gui Q. 243 Gundlich B. 337 Guo J. 207

Nesvadba O. 320 Neumann I. 232 Nicolas J. 73

Cai J. 79, 217 Cai Z. 252 Chao B. F. 56 Chao B. 207 Chao D. 35 Chen J. 138 Chen Y. 62 Collilieux X. 101 Dai W. 200 Dermanis A. 111 Ditmar P. 17 Durand S. 73

Han S. 243 Heck B. 49 Holota P. 320 Hu C. 216 Huang D. 155, 185 Huang J. 185

Odijk D. 200 Pagamisse A. 179 Peters T. 67 Petrovich D. 173 Piché R. 173 Polezel W.G. C. 179 Poutanen M. 79 Reguzzoni M. 263 Rizos C. 126 Roggero M. 160 Roman D. R. 356

Ilk K.H. 3 Keller W. 10, 29, 327 Klotz J. 128 Koivula H. 79 Kotsakis C. 350

Sacerdote F. 311 Saleh J. 356 Sánchez L. 119 Sansò F. 311 Schaffrin B. 62, 238 361

362

Schreiner M. 333 Sharifi M.A. 29 Shen C. 56 Shen Y. 195 Shi C. 133 Shum C.K. 62 Sideris M.G. 23, 345 Sneeuw N. 23, 29 Souza E.M. 179 Sui L. 248 Sun S. 56 Teunissen P.J.G. 143, 149, 166, 200, 227, 273, 280 Tian H. 189 Tiberius C.C.J.M. 280 Tselfes N. 263 Venuti G. 263 Verhagen S. 143, 149 Wang J. 138, 216 Wang W. 207 Wang W. 248 Wang X 269, 293 Wang Y. 335

Author Index

Wang Z. 35 Wild F. 49 Xiang A. 56 Xiong Y. 185 Xu C. 23, 345 Xu C.Q. 269 Xu R. 155 Yin H. 185 Yuan L. 63 Zhang C. 252 Zhang H. 303 Zhang Q. 94 Zhang X. 211 Zhao C. 94 Zhao D. 252 Zhao J. 303 Zhao Q. 133 Zhou L. 155 Zhou X. 287 Zhu J. 287 Zhu P. 56 Zou Z. 287

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