Geophysical inverse theory and regularization problems

GEOPHYSICAL INVERSE THEORY AND RECULARIZATION PROBLEMS Methods in Geochemistryand Geophysics (Volumes 1-28 are out of...

Author: Michael S. Zhdanov

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GEOPHYSICAL INVERSE THEORY AND RECULARIZATION PROBLEMS

Methods in Geochemistryand Geophysics (Volumes 1-28 are out of" print) A.S. R I T C H I E - CHROMATOGRAPHY IN GEOLOGY R. BOWEN n PALEOTEMPERATURE ANALYSIS D.S. PARASNIS m M I N I N G GEOPHYSICS 3. I. ADLER n X-RAY EMISSION SPECTROGRAPHY IN GEOLOGY 4. THE LORD ENERGLYN A N D L. BREADLY m ANALYTICAL GEOCHEMISTRY 5. A.J. EASTON m CHEMICAL ANALYSIS OF SILICATE ROCKS 6. E.E. A N G I N O A N D G.K. BILLINGS - - ATOMIC ABSORPTION SPECTROMETRY IN GEOLOGY 7. A. VOLBORTH m ELEMENTAL ANALYSIS 1N GEOCHEMISTRY, A: MAJOR ELEMENTS 8. P.K. BHATTACHARYA A N D H.P. PATRA ~ DIRECT CURRENT GEOELECTRIC SOUNDING 9. J.A.S. ADAMS A N D P. GASPARINI ~ GAMMA-RAY SPECTROMETRY OF ROCKS 1o. W. ERNST ~ GEOCHEMICAL FACIES ANALYSIS 11. P.V. SHARMA ~ GEOPHYSICAL METHODS IN GEOLOGY 12. C.H. CHEN (Editor) ~ COMPUTER-AIDED SEISMIC ANALYSIS AND DISCRIMINATION 13. 14A. O. KOEFOED n GEOSOUNDING PRINCIPLES, 1. RESISTIVITY SOUNDING MEASUREMENTS 14B. H.P. PATRA A N D K. MALLICK m GEOSOUNDING PRINCIPLES, 2. TIME-VARYING GEOELECTRIC SOUNDINGS A.A. KAUFMAN A N D G.V. KELLER ~ THE MAGNETOTELLURIC SOUNDING METHOD 15. A.A. KAUFMAN A N D G.V. KELLER m FREQUENCY A N D TRANSIENT SOUNDINGS 16. C.H. CHEN (Editor) ~ SEISMIC SIGNAL ANALYSIS A N D DISCRIMINATION 17. J.E. WHITE ~ UNDERGROUND SOUND APPLICATION OF SEISMIC WAVES 18. M.N. BERDICHEVSKY A N D M.S. Z H D A N O V ~ ADVANCED THEORY OF DEEP 19. GEOMAGNETIC SOUNDINGS 2oA. A.A. KAUFMAN A N D G.V. KELLER ~ INDUCTIVE M I N I N G PROSPECTING, PART I: THEORY A.W. WYLIE ~ NUCLEAR ASSAYING OF M I N I N G BOREHOLES- AN INTRODUCTION 21. C.H. CHEN (Editor) n SEISMIC SIGNAL ANALYSIS A N D DISCRIMINATION III 22. R.P. PHILP ~ FOSSIL FUEL BIOMARKERS 23 . R.B. JOHNS (Editor) ~ BIOLOGICAL MARKERS IN THE SEDIMENTARY RECORD 24. J.C. D' ARNAUD GERKENS ~ FOUNDATION OF EXPLORATION GEOPHYSICS 25. P. TYGEL A N D P. H U B R A L - - T R A N S I E N T WAVES IN LAYERED MEDIA 26. A.A. KAUFMAN A N D GV. KELLER m INDUCTION LOGGING 27. 28. J.G. NEGI A N D P.D. SAI~AF ~ ANISOTROPY IN GEOELECTROMAGNETISM 29. V.P. DIMRI ~ DECONVOLUTION A N D INVERSE THEORY- APPLICATION TO GEOPHYSICAL PROBLEMS K.-M. STRACK ~ EXPLORATION WITH DEEP TRANSIENT ELECTROMAGNETICS 30. M.S. Z H D A N O V A N D G.V. KELLER ~ THE GEOELECTRICAL METHODS IN GEOPHYSICAL 31. EXPLORATION A.A. KAUFMAN A N D AoL. LEVSHIN n ACOUSTIC A N D ELASTIC WAVE FIELDS IN 32. GEOPHYSICS, I A.A. KAUFMAN A N D PoA. EATON ~ THE THEORY OF INDUCTIVE PROSPECTING 33. A.A. KAUFMAN A N D P. HOEKSTRA m ELECTROMAGNETIC SOUNDINGS 34. M.S. Z H D A N O V A N D P.E. WANNAMAKER m THREE-DIMENSIONAL ELECTROMAGNETICS 35. M.S. Z H D A N O V ~ GEOPHYSICAL INVERSE THEORY A N D REGULARIZATION PROBLEMS 36. 1o

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Methods in Geochemistry and Geophysics, 36

GEOPHYSICAL INVERSE TH EORY AND RECULARIZATION PROBLEMS

Michael S. ZH DANOV University of Utah

Salt Lake City UTAH, U.S.A.

2002

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Contents Preface

I 1

I n t r o d u c t i o n to Inversion T h e o r y F o r w a r d and inverse p r o b l e m s in g e o p h y s i c s 1.1 F o r m u l a t i o n of forward a n d inverse p r o b l e m s for different geophysical fields . . . . . . . . . . . . . . . . . . . . . . . . •. . . . . . . . . . . . 1.1.1 G r a v i t y field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 M a g n e t i c field . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 E l e c t r o m a g n e t i c field . . . . . . . . . . . . . . . . . . . . . . . 1.1.4 Seismic wavefield . . . . . . . . . . . . . . . . . . . . . . . . 1.2 E x i s t e n c e a n d uniqueness of t h e inverse p r o b l e m solutions . . . . .

1.3 2

xIx

1.2.1 E x i s t e n c e of t h e solution . 1.2.2 U n i q u e n e s s of t h e solution 1.2.3 P r a c t i c a l uniqueness . . . I n s t a b i l i t y of t h e inverse p r o b l e m

. . . . . . . . . . . . . . . solution

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. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

3 3 6 7 9 14 16 16 17 23 24

I l l - p o s e d p r o b l e m s a n d t h e m e t h o d s of t h e i r s o l u t i o n 2.1 Sensitivity a n d resolution of geophysical m e t h o d s . . . . . . . . . . . 2.1.1 F o r m u l a t i o n of t h e inverse p r o b l e m in general m a t h e m a t i c a l spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . .

29 29

2.2

F o r m u l a t i o n of well-posed a n d ill-posed p r o b l e m s . . . . . . . . . 2.2.1 Well-posed p r o b l e m s . . . . . . . . . . . . . . . . . . . . . 2.2.2 C o n d i t i o n a l l y well-posed p r o b l e m s . . . . . . . . . . . . . 2.2.3 Quasi-solution of the ill-posed p r o b l e m . . . . . . . . . . .

32 32 33 34

2.3

F o u n d a t i o n s of regularization m e t h o d s of inverse p r o b l e m solution . 2.3.1 Regularizing o p e r a t o r s . . . . . . . . . . . . . . . . . . . . . 2.3.2 Stabilizing functionals . . . . . . . . . . . . . . . . . . . . .

36 36 39

2.3.3

42

T i k h o n o v p a r a m e t r i c functional

................

29 30 31

VIII 2.4

2.5

II 3

4

CONTENTS Family of stabilizing functionals . . . . . . . . . . . . . . . . . . . . . 2.4.1 Stabilizing functionals revisited . . . . . . . . . . . . . . . . . 2.4.2 Representation of a stabilizing functional in the form of a pseudoquadratic functional . . . . . . . . . . . . . . . . . . . . . . . Definition of the regularization parameter . . . . . . . . . . . . . . . 2.5.1 Optimal regularization parameter selection . . . . . . . . . . . 2.5.2 L-curve m e t h o d of regularization p a r a m e t e r selection . . . . .

M e t h o d s of t h e S o l u t i o n of Inverse P r o b l e m s

45 45 50 52 52 55

59

L i n e a r discrete inverse problems 3.1 Linear least-squares inversion . . . . . . . . . . . . . . . . . . . . . . 3.1.1 T h e linear discrete inverse problem . . . . . . . . . . . . . . . 3.1.2 Systems of linear equations and their general solutions . . . . 3.1.3 The d a t a resolution matrix . . . . . . . . . . . . . . . . . . . . 3.2 Solution of the purely underdetermined problem . . . . . . . . . . . 3.2.1 Underdetermined system of linear equations . . . . . . . . . . 3.2.2 The model resolution matrix . . . . . . . . . . . . . . . . . . . 3.3 Weighted least-squares m e t h o d . . . . . . . . . . . . . . . . . . . . . 3.4 Applying the principles of probability theory to a linear inverse problem 3.4.1 Some formulae and notations from probability theory . . . . . 3.4.2 M a x i m u m likelihood m e t h o d . . . . . . . . . . . . . . . . . . . 3.4.3 Chi-square fitting ........................ 3.5 Regularization methods . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 The Tikhonov regularization m e t h o d . . . . . . . . . . . . . . 3.5.2 Application of SLDM m e t h o d in regularized linear inverse problem solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.3 Definition of the weighting matrices for the model p a r a m e t e r s and d a t a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.4 Approximate regularized solution of the linear inverse problem 3.5.5 The L e v e n b e r g - Marquardt m e t h o d . . . . . . . . . . . . . . 3.5.6 The m a x i m u m a posteriori estimation m e t h o d (the Bayes estimation) . . . . . . . . . . . . . . . . .............. 3.6 The Backus-Gilbert Method . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 The d a t a resolution function . . . . . . . . . . . . . . . . . . . 3.6.2 The spread function . . . . . . . . . . . . . . . . . . . . . . . 3.6.3 Regularized solution in the Backus-Gilbert m e t h o d . . . . . .

61 61 61 62 64 66 66 67 68 69 69 71 73 74 74

Iterative solutions of the linear inverse problem

91 91 91 93

4.1

Linear operator equations and their solution by iterative m e t h o d s 4.1.1 Linear inverse problems and the Euler equation . . . . . . . . 4.1.2 The minimal residual m e t h o d . . . . . . . . . . . . . . . . . .

. .

75 77 79 81 82 84 84 86 88

IX

CONTENTS

4.2

4.3

5

4.1.3 Linear inverse problem solution using M R M . . . . . . . . . . A generalized minimal residual m e t h o d . . . . . . . . . . . . . . . . . 4.2.1 T h e Krylov-subspace m e t h o d . . . . . . . . . . . . . . . . . . 4.2.2 T h e Lanczos minimal residual m e t h o d . . . . . . . . . . . . . 4.2.3 T h e generalized minimal residual m e t h o d . . . . . . . . . . . . 4.2.4 A linear inverse problem solution using generalized M R M . . . T h e regularization m e t h o d in a linear inverse problem solution . . . . 4.3.1 T h e Euler equation for the Tikhonov p a r a m e t r i c functional . . 4.3.2 M R M solution of the Euler equation . . . . . . . . . . . . . . 4.3.3 Generalized M R M solutions of the Euler equation for the parametric functional . . . . . . . . . . . . . . . . . . . . . . . . .

N o n l i n e a r inversion t e c h n i q u e 5.1

5.2

5.3

5.4

Gradient-type methods . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 M e t h o d of steepest descent . . . . . . . . . . . . . . . . . . . .

99 101 101 103 108 112 113 113 115 117 121 121 121

5.1.2 T h e Newton m e t h o d . . . . . . . . . . . . . . . . . . . . . . . 131 5.1.3 T h e conjugate gradient m e t h o d . . . . . . . . . . . . . . . . . 137 Regularized gradient-type m e t h o d s in the solution of nonlinear inverse problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 5.2.1 Regularized steepest descent . . . . . . . . . . . . . . . . . . . 143 5.2.2 T h e regularized Newton m e t h o d . . . . . . . . . . . . . . . . . 145 5.2.3 A p p r o x i m a t e regularized solution of the nonlinear inverse problem ................................ 147 5.2.4 T h e regularized preconditioned steepest descent m e t h o d . . . 147 5.2.5 T h e regularized conjugate gradient m e t h o d . . . . . . . . . . . 148 Regularized solution of a nonlinear discrete inverse problem . . . . . . 149 5.3.1 Nonlinear least-squares inversion . . . . . . . . . . . . . . . . 149 5.3.2 T h e steepest descent m e t h o d for nonlinear regularized leastsquares inversion . . . . . . . . . . . . . . . . . . . . . . . . . 150 5.3.3 T h e Newton m e t h o d for nonlinear regularized least-squares inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 5.3.4 Numerical schemes of the Newton m e t h o d for nonlinear regularized least-squares inversion . . . . . . . . . . . . . . . . . . 152 5.3.5 Nonlinear least-squares inversion by the conjugate gradient m e t h o d 1 5 3 5.3.6 T h e numerical scheme of the regularized conjugate gradient m e t h o d for nonlinear least-squares inversion . . . . . . . . . . 153 C o n j u g a t e gradient re-weighted optimization . . . . . . . . . . . . . . 155 5.4.1 5.4.2

T h e Tikhonov parametric functional with a pseudo-quadratic stabilizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Re-weighted conjugate gradient m e t h o d . . . . . . . . . . . . .

155 157

5.4.3

Minimization in the space of weighted p a r a m e t e r s . . . . . . .

160

x 5.4.4

III 6

CONTENTS The re-weighted regularized conjugate gradient (RCG) method in the space of weighted parameters . . . . . . . . . . . . . . .

Geopotential Field Inversion

167

Integral representations in forward modeling of gravity and m a g n e t i c fields 6.1 Basic equations for gravity and magnetic fields . . . . . . . . . . . . . 6.1.1 Gravity and magnetic fields in three dimensions . . . . . . . . 6.1.2 Two-dimensional models of gravity and magnetic fields . . . . 6.2 Integral representations of potential fields based on the theory of functions of a complex variable . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Complex intensity of a plane potential field . . . . . . . . . . . 6.2.2 Complex intensity of a gravity field . . . . . . . . . . . . . . . 6.2.3 Complex intensity and potential of a magnetic field . . . . . .

7

7.2

7.3

7.4

8

169 169 169 170 171 171 174 175

Integral representations in inversion of gravity and magnetic data 177 7.1

IV

161

Gradient methods of gravity inversion . . . . . . . . . . . . . . . . . . 7.1.1 Steepest ascent direction of the misfit functional for the gravity inverse problem . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.2 Application of the re-weighted conjugate gradient method . . Gravity field migration . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Physical interpretation of the adjoint gravity operator . . . . . 7.2.2 Gravity field migration in the solution of the inverse problem . 7.2.3 Iterative gravity migration . . . . . . . . . . . . . . . . . . . . Gradient methods of magnetic anomaly inversion . . . . . . . . . . 7.3.1 Magnetic potential inversion . . . . . . . . . . . . . . . . . . . 7.3.2 Magnetic potential migration . . . . . . . . . . . . . . . . . . Numerical methods in forward and inverse modeling . . . . . . . . . . 7.4.1 Discrete forms of 3-D gravity and magnetic forward modeling operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 Discrete form of 2-D forward modeling operator . . . . . . . . 7.4.3 Regularized inversion of gravity data . . . . . . . . . . . . . .

177 179 181 181 184 186 188 188 189 190 190 193 193

199

Electromagnetic Inversion F o u n d a t i o n s of electromagnetic theory 8.1 Electromagnetic field equations . . . . . . . . . . . . . . . . . . . . . 8.1.1 Maxwell's equations . . . . . . . . . . . . . . . . . . . . . . . 8.1.2 Field in homogeneous domains of a medium 8.1.3 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . .

177

..........

201 201 201 202 203

CONTENTS

8.1.4 8.1.5 8.1.6 8.1.7 8.1.8 8.1.9 8.2

8.3

8.4

9

XI

Field equations in the frequency domain . . . . . . . . . . . . Quasi-static (quasi-stationary) electromagnetic field . . . . . . Field wave equations . . . . . . . . . . . . . . . . . . . . . . . Field equations allowing for magnetic currents and charges . . Stationary electromagnetic field . . . . . . . . . . . . . . . . . Fields in two-dimensional inhomogeneous media and the concepts of E- and H-polarization . . . . . . . . . . . . . . . . . E l e c t r o m a g n e t i c energy flow . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Radiation conditions . . . . . . . . . . • ............. 8.2.2 Poynting's t h e o r e m in the time domain . . . . . . . . . . . . . 8.2.3 Energy inequality in the time domain . . . . . . . . . . . . . . 8.2.4 Poynting's t h e o r e m in the frequency domain . . . . . . . . . . Uniqueness of the solution of electromagnetic field equations . . . . . 8.3.1 Boundary-value problem . . . . . . . . . . . . . . . . . . . . . 8.3.2 Uniqueness t h e o r e m for the u n b o u n d e d domain . . . . . . . . E l e c t r o m a g n e t i c Green's tensors . . . . . . . . . . . . . . . . . . . . . 8.4.1 Green's tensors in the frequency domain . . . . . . . . . . . . 8.4.2 Lorentz l e m m a and reciprocity relations . . . . . . . . . . . . 8.4.3 Green's tensors in the time domain . . . . . . . . . . . . . . .

204 209 210 211 212 213 215 216 216 218 220 222 222 223 224 224 225 227

I n t e g r a l r e p r e s e n t a t i o n s in e l e c t r o m a g n e t i c f o r w a r d m o d e l i n g 231 9.1 Integral equation m e t h o d . . . . . . . . . . . . . . . . . . . . . . . . . 231 9.1.1 Background (normal) and anomalous parts of the electromagnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 9.1.2 Poynting's t h e o r e m and energy inequality for an anomalous field233 9.1.3 Integral equation m e t h o d in two dimensions . . . . . . . . . . 234 9.1.4 Calculation of the first variation (Fr6chet derivative) of the elect r o m a g n e t i c field for 2-D models . . . . . . . . . . . . . . . . . 237 9.1.5 Integral equation m e t h o d in three dimensions . . . . . . . . . 239 9.1.6 Calculation of the first variation (Fr6chet derivative) of the elect r o m a g n e t i c field for 3-D models . . . . . . . . . . . . . . . . . 240 9.1.7 Fr6chet derivative calculation using the differential m e t h o d . . 243 9.2 Family of linear and nonlinear integral approximations of the electromagnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 9.2.1 Born and extended Born approximations . . . . . . . . . . . . 246 9.2.2 Quasi-linear approximation and tensor quasi-linear equation . 247 9.2.3 Quasi-analytical solutions for a 3-D electromagnetic field . . . 248 9.2.4 Quasi-analytical solutions for 2-D electromagnetic field . . . . 251 9.2.5 Localized nonlinear approximation . . . . . . . . . . . . . . . 252 9.2.6 Localized quasi-linear approximation . . . . . . . . . . . . . . 253 9.3 Linear and non-linear approximations of higher orders . . . . . . . . . 256 9.3.1 Born series . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256

XII 9.3.2 9.3.3 9.3.4 9.3.5 9.3.6

9.4

CONTENTS

Modified Green's o p e r a t o r . . . . . . . . . . . . . . . . . . . . Modified Born series . . . . . . . . . . . . . . . . . . . . . . . Quasi-linear a p p r o x i m a t i o n of the modified Green's o p e r a t o r . QL series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Accuracy e s t i m a t i o n of the QL a p p r o x i m a t i o n of the first a n d higher orders . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.7 QA series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Integral r e p r e s e n t a t i o n s in numerical dressing . . . . . . . . . . . . . 9.4.1 Discretization of the model p a r a m e t e r s . . . . . . . . . . . . . 9.4.2 Galerkin m e t h o d for e l e c t r o m a g n e t i c field discretization . . . . 9.4.3 Discrete form of e l e c t r o m a g n e t i c integral equations based on boxcar basis functions . . . . . . . . . . . . . . . . . . . . . . 9.4.4 C o n t r a c t i o n integral equation m e t h o d . . . . . . . . . . . . . . 9.4.5 C o n t r a c t i o n integral equation as the p r e c o n d i t i o n e d conventional integral equation . . . . . . . . . . . . . . . . . . . . . . 9.4.6 M a t r i x form of B o r n a p p r o x i m a t i o n . . . . . . . . . . . . . . . 9.4.7 M a t r i x form of quasi-linear a p p r o x i m a t i o n . . . . . . . . . . . 9.4.8 M a t r i x form of quasi-analytical a p p r o x i m a t i o n . . . . . . . . . 9.4.9 T h e diagonalized quasi-analytical (DQA) a p p r o x i m a t i o n . . .

257 259 261 263 263 266 267 267 269 271 275 276 278 278 280 281

10 I n t e g r a l r e p r e s e n t a t i o n s in e l e c t r o m a g n e t i c i n v e r s i o n 287 10.1 Linear inversion m e t h o d s . . . . . . . . . . . . . . . . . . . . . . . . . 288 10.1.1 Excess (anomalous) current inversion . . . . . . . . . . . . . . 288 10.1.2 Born inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 10.1.3 C o n d u c t i v i t y imaging by the Born a p p r o x i m a t i o n . . . . . . . 292 10.1.4 I t e r a t i v e Born inversions . . . . . . . . . . . . . . . . . . . . . 296 10.2 Nonlinear inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 10.2..1 F o r m u l a t i o n of the nonlinear inverse p r o b l e m . . . . . . . . . . 297 10.2.2 Fr6chet derivative calculation . . . . . . . . . . . . . . . - . . . 298 10.3 Quasi-linear inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 10.3.1 Principles of quasi-linear inversion . . . . . . . . . . . . . . . . 300 10.3.2 Quasi-linear inversion in m a t r i x n o t a t i o n s . . . . . . . . . . . 301 10.3.3 Localized quasi-linear inversion . . . . . . . . . . . . . . . . . 306 10.4 Quasi-analytical inversion . . . . . . . . . . . . . . . . . . . . . . . . 311 10.4.1 Fr6chet derivative calculation . . . . . . . . . . . . . . . . . . 311 10.4.2 Inversion based on the quasi-analytical m e t h o d . . . . . . . . 312 10.5 M a g n e t o t e l l u r i c (MT) d a t a inversion . . . . . . . . . . . . . . . . . . 314 10.5.1 I t e r a t i v e Born inversion of m a g n e t o t e l l u r i c d a t a . . . . . . . . 315 10.5.2 D Q A a p p r o x i m a t i o n in m a g n e t o t e l l u r i c inverse p r o b l e m . . . . 317 10.5.3 Fr6chet derivative m a t r i x with respect to the l o g a r i t h m of the t o t a l conductivity . . . . . . . . . . . . . . . . . . . . . . . . . 319 10.5.4 Regularized s m o o t h and focusing inversion of M T d a t a . . . . 320

XIII

CONTENTS

10.5.5 Example of synthetic 3-D M T d a t a inversion . . . . . . . . . .

321

10.5.6 Case study: inversion of the Minamikayabe area d a t a . . . . .

324

11 Electromagnetic migration imaging

331

11.1 Electromagnetic migration in the frequency domain

..........

332

11.1.1 Formulation of the electromagnetic inverse problem as a minimization of the energy flow functional . . . . . . . . . . . . . .

332

11.1.2 Integral representations for electromagnetic migration field . .

335

11.1.3 Gradient direction of the energy flow functional

336

11.1.4 Migration imaging in the frequency domain 11.1.5 Iterative migration

........

..........

338

........................

343

11.2 Electromagnetic migration in the time domain . . . . . . . . . . . . .

344

11.2.1 Time domain electromagnetic migration as the solution of the b o u n d a r y value problem . . . . . . . . . . . . . . . . . . . . .

345

11.2.2 Minimization of the residual electromagnetic field energy flow

351

11.2.3 Gradient direction of the energy flow functional in the time domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

353

11.2.4 Migration imaging in the time domain

.............

354

11.2.5 Iterative migration in the time domain

.............

357

12 Differential methods in electromagnetic modeling and inversion

361

12.1 Electromagnetic modeling as a boundary-value problem . . . . . . . .

361

12.1.1 Field equations and boundary conditions . . . . . . . . . . . .

361

12.1.2 Electromagnetic potential equations and b o u n d a r y conditions

365

12.2 Finite difference approximation of the boundary-value problem . . . . 12.2.1 Discretization of Maxwell's equations using a staggered grid

366 .

367

12.2.2 Discretization of the second order differential equations using the balance m e t h o d . . . . . . . . . . . . . . . . . . . . . . . .

371

12.2.3 Discretization of the electromagnetic potential differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

376

12.2.4 Application of the spectral Lanczos decomposition m e t h o d (SLDM) for solving the linear system of equations for discrete electromagnetic fields . . . . . . . . . . . . . . . . . . . . . . . . . . 379 12.3 Finite element solution of boundary-value problems 12.3.1 Galerkin m e t h o d

..........

380

.........................

12.3.2 Exact element m e t h o d

380

......................

384

12.4 Inversion based on differential methods . . . . . . . . . . . . . . . . .

385

12.4.1 Formulation of the inverse problem on the discrete grid . . . . 12.4.2 Fr6chet derivative calculation using finite difference methods

385 .

386

xIv

V

CONTENTS

Seismic Inversion

393

13 W a v e f i e l d equations 395 13.1 Basic equations of elastic waves . . . . . . . . . . . . . . . . . . . . . 395 13.1.1 Deformation of an elastic body; deformation and stress tensors 395 13.1.2 Hooke's law . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 13.1.3 Dynamic equations of elasticity theory for a homogeneous isotropic medium .............................. 400 13.1.4 Compressional and shear waves . . . . . . . . . . . . . . . . . 402 13.1.5 Acoustic waves and scalar wave equation . . . . . . . . . . . . 405 13.1.6 High frequency approximations in the solution of an acoustic wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 13.2 Green's functions for wavefield equations . . . . . . . . . . . . . . . . 407 13.2.1 Green's functions for the scalar wave equation and for the corresponding Helmholtz equation . . . . . . . . . . . . . . . . . 407 13.2.2 High frequency ( W K B J ) approximation for the Green's function 410 13.2.3 Green's tensor for vector wave equation . . . . . . . . . . . . . 411 13.2.4 Green's tensor for the Lam6 equation . . . . . . . . . . . . . . 413 13.3 Kirchhoff integral formula and its analogs . . . . . . . . . . . . . . . . 414 13.3.1 Kirchhoff integral formula . . . . . . . . . . . . . . . . . . . . 415 13.3.2 Generalized Kirchhoff integral formulae for the Lam6 equation and the vector wave equation . . . . . . . . . . . . . . . . . . 417 13.4 Uniqueness of the solution of the wavefield equations . . . . . . . . . 420 13.4.1 Initial-value problems . . . . . . . . . . . . . . . . . . . . . . . 420 13.4.2 Energy conservation law . . . . . . . . . . . . . . . . . . . . . 422 13.4.3 Uniqueness of the solution of initial-value problems . . . . . . 425 13.4.4 Sommerfeld radiation conditions . . . . . . . . . . . . . . . . . 426 13.4.5 Uniqueness of the solution of the wave p r o p a g a t i o n problem based on radiation conditions . . . . . . . . . . . . . . . . . . 429 13.4.6 Kirchhoff formula for an u n b o u n d e d domain . . . . . . . . . . 434 13.4.7 Radiation conditions for elastic waves . . . . . . . . . . . . . . 437 14 I n t e g r a l representations in wavefield theory 14.1 Integral equation m e t h o d in acoustic wavefield analysis . . . . . . . . 14.1.1 Separation of the acoustic wavefield into incident and scattered (background and anomalous) parts . . . . . . . . . . . . . . . 14.1.2 Integral equation for the acoustic wavefield . . . . . . . . . . . 14.1.3 Reciprocity t h e o r e m . . . . . . . . . . . . . . . . . . . . . . . 14.1.4 Calculation of the first variation (Fr6chet derivative) of the acoustic wavefield . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Integral approximations of the acoustic wavefield . . . . . . . . . . . . 14.2.1 Born approximation . . . . . . . . . . . . . . . . . . . . . . .

443 443 443 445 447 448 449 449

CONTENTS

14.2.2 Quasi-linear a p p r o x i m a t i o n . . . . . . . . . . . . . . . . . . . 14.2.3 Quasi-analytical a p p r o x i m a t i o n . . . . . . . . . . . . . . . . . 14.2.4 Localized quasi-linear a p p r o x i m a t i o n . . . . . . . . . . . . . . 14.2.5 Kirchhoff a p p r o x i m a t i o n . . . . . . . . . . . . . . . . . . . . . 14.3 M e t h o d of integral equations in vector wavefield analysis . . . . . . . 14.3.1 Vector wavefield s e p a r a t i o n . . . . . . . . . . . . . . . . . . . 14.3.2 Integral e q u a t i o n m e t h o d for the vector wavefield . . . . . . . 14.3.3 Calculation of the first variation (Fr6chet derivative) of the vector wavefield . . . . . . ...................... 14.4 Integral a p p r o x i m a t i o n s of the vector wavefield . . . . . . . . . . . . . 14.4.1 Born t y p e a p p r o x i m a t i o n s . . . . . . . . . . . . . . . . . . . . 14.4.2 Quasi-linear a p p r o x i m a t i o n . . . . . . . . . . . . . . . . . . . 14.4.3 Quasi-analytical solutions for the vector wavefield . . . . . . . 14.4.4 Localized quasi-linear a p p r o x i m a t i o n . . . . . . . . . . . . . .

15 Integral representations in wavefield inversion 15.1 Linear inversion m e t h o d s . . . . . . . . . . . . . . . . . . . . . . . . . 15.1.1 Born inversion of acoustic and vector wavefields . . . . . . . . 15.1.2 Wavefield imaging by the Born a p p r o x i m a t i o n s . . . . . . . . 15.1.3 I t e r a t i v e Born inversions of the wavefield . . . . . . . . . . . . 15.1.4 Bleistein inversion . . . . . . . . . . . . . . . . . . . . . . . . . 15.1.5 Inversion based on the Kirchhoff a p p r o x i m a t i o n . . . . . . . . 15.1.6 Traveltime inverse p r o b l e m . . . . . . . . . . . . . . . . . . . . 15.2 Quasi-linear inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2.1 Quasi-linear inversion of the acoustic wavefield . . . . . . . . . 15.2.2 Localized quasi-linear inversion based on the Bleistein m e t h o d 15.3 Nonlinear inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3.1 F o r m u l a t i o n of the nonlinear wavefield inverse p r o b l e m . . . . 15.3.2 Fr6chet derivative o p e r a t o r s for wavefield problems . . . . . . 15.4 Principles of wavefield m i g r a t i o n . . . . . . . . . . . . . . . . . . . . . 15.4.1 G e o m e t r i c a l model of m i g r a t i o n t r a n s f o r m a t i o n . . . . . . . . 15.4.2 Kirchhoff integral formula for reverse-time wave e q u a t i o n migration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4.3 Rayleigh integral . . . . . . . . . . . . . . . . . . . . . . . . . 15.4.4 Migration in the spectral d o m a i n (Stolt's m e t h o d ) . . . . . . . 15.4.5 Equivalence of the spectral and integral m i g r a t i o n a l g o r i t h m s . 15.4.6 Inversion versus m i g r a t i o n . . . . . . . . . . . . . . . . . . . . 15.5 Elastic field inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5.1 F o r m u l a t i o n of the elastic field inverse p r o b l e m . . . . . . . . 15.5.2 Fr~chet derivative for the elastic forward m o d e l i n g o p e r a t o r . . 15.5.3 Adjoint Fr~chet derivative o p e r a t o r and b a c k - p r o p a g a t i n g elastic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

XV 450 451 452 453 456 456 457 458 460 460 461 461 462

467 467 468 470 475 475 491 494 496 496 497 499 499 500 503 503 507 510 514 516 517 518 518 520 522

xvI A

Functional A.1

A.2

A.3

A.4

A.5

A.6 B

C

D

CONTENTS spaces of geophysical models and data

531

Euclidean Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

531

A.I.1

Vector operations in Euclidean space

531

A.1.2

Linear t r a n s f o r m a t i o n s (operators) in Euclidean space . . . . .

534

..............

A.1.3

N o r m of the o p e r a t o r . . . . . . . . . . . . . . . . . . . . . . .

534

A.1.4

Linear functionals . . . . . . . . . . . . . . . . . . . . . . . . .

536

A.1.5

N o r m of the functional . . . . . . . . . . . . . . . . . . . . . .

536

Metric space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2.1

Definition of metric space

A.2.2

Convergence, Cauchy sequences and completeness . . . . . . .

Linear vector spaces

....................

537 537 538

. . . . . . . . . . . . . . . . . . . . . . . . . . .

539

A.3.1

Vector operations . . . . . . . . . . . . . . . . . . . . . . . . .

539

A.3.2

N o r m e d linear spaces . . . . . . . . . . . . . . . . . . . . . . .

Hilbert spaces

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

A.4.1

Inner p r o d u c t

. . . . . . . . . . . . . . . . . . . . . . . . . . .

A.4.2

A p p r o x i m a t i o n problem in Hilbert space

C o m p l e x Euclidean and Hilbert spaces

............

C o m p l e x Euclidean space

A.5.2

C o m p l e x Hilbert space . . . . . . . . . . . . . . . . . . . . . .

546

....................

546 547

E x a m p l e s of linear vector spaces . . . . . . . . . . . . . . . . . . . . .

Operators

541 544

.................

A.5.1

540 541

547

in t h e s p a c e s of m o d e l s a n d d a t a

553

B.1

O p e r a t o r s in functional spaces . . . . . . . . . . . . . . . . . . .

B.2

Linear operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

555

B.3

Inverse operators

556

B.4

Some a p p r o x i m a t i o n problems in the Hilbert spaces of geophysical d a t a 557

B.5

G r a m - Schmidt orthogonalization process

Functionals

. . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...............

in t h e s p a c e s of g e o p h y s i c a l m o d e l s

553

559

563

C.1

Functionals and their norms

. . . . . . . . . . . . . . . . . . . . . . .

563

C.2

Riesz r e p r e s e n t a t i o n t h e o r e m . . . . . . . . . . . . . . . . . . . . . . .

564

C.3

Functional r e p r e s e n t a t i o n of geophysical d a t a and an inverse p r o b l e m

565

Linear operators

and functionals revisited

569

D.1

Adjoint operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

569

D.2

Differentiation of operators and functionals . . . . . . . . . . . . . . .

571

D.3

Concepts from variational calculus . . . . . . . . . . . . . . . . . . . .

573

D.3.1

Variational o p e r a t o r

573

D.3.2

E x t r e m u m functional problems

. . . . . . . . . . . . . . . . . . . . . . . .................

574

CONTENTS

XVII

E

Some formulae and rules from matrix algebra E.1 Some formulae and rules of operation on matrices . . . . . . . . . . . E.2 Eigenvalues and eigenvectors . . . . . . . . . . . . . . . . . . . . . . . E.3 Spectral decomposition of a symmetric m a t r i x . . . . . . . . . . . . . E.4 Singular value decomposition (SVD) . . . . . . . . . . . . . . . . . . E.5 The spectral Lanczos decomposition m e t h o d . . . . . . . . . . . . . . E.5.1 Functions of matrices . . . . . . . . . . . . . . . . . . . . . . . E.5.2 The Lanczos m e t h o d . . . . . . . . . . . . . . . . . . . . . . .

577 577 578 579 580 582 582 583

F

Some formulae and rules from tensor calculus F.1 Some formulae and rules of operation on tensor functions . . . . . . . F.2 Tensor s t a t e m e n t s of the Gauss and Green's formulae . . . . . . . . . F.3 Green's tensor and vector formulae for Lam~ and Laplace operators

589 589 590 . 591

Bibliography

593

Index

604


Preface Inverse solutions are key problems in many natural sciences. They form the basis of our understanding of the world surrounding us. Whenever we try to learn something about physical laws, the internal structure of the earth or the nature of the Universe, we collect data and try to extract the required information from these data. This is the actual solution of the inverse problem. In fact the observed data are predetermined by physical laws, and by the structure of the earth or Universe. The method of predicting observed data for given sources within given media is usually referred to as the forward problem solution. The method of reconstructing the sources of some physical, geophysical, or other phenomenon, as well as the parameters of the corresponding media, from the observed data is referred to as the inverse problem solution. In geophysics, the observed data are usually physical fields generated by natural or artificial sources and propagated through the earth. Geophysicists try to use these data to reconstruct the internal structure of the earth. This is a typical inverse problem solution. Inversion of geophysical data is complicated by the fact that geophysical data are invariably contaminated by noise and are acquired at a limited number of observation points. Moreover, mathematical models are usually complicated, and yet at the same time are also simplifications of the true geophysical phenomena. As a result, the solutions are ambiguous and error-prone. The principal questions arising in geophysical inverse problems are about the existence, uniqueness, and stability of the solution. Methods of solution can be based on linearized and nonlinear inversion techniques and include different approaches, such as least-squares, gradient-type methods (including steepest-descent and conjugate-gradient) and others. A central point of this book is the application of so-called "regularizing" algorithms for the solution of ill-posed inverse geophysical problems. These algorithms can use a prion geological and geophysical information about the earth's subsurface to reduce the ambiguity and increase the stability of the solution. In mathematics we have a classical definition of the ill-posed problem: a problem is ill-posed, according to Hadamard (1902), if the solution is not unique or if it is not a continuous function of the data (i.e., if to a small perturbation of data there corresponds an arbitrarily large perturbation of the solution). Unfortunately, from the point of view of classical theory, all geophysical inverse problems are ill-posed.

XX

PREFACE

because their solutions are either non-unique or unstable. However, geophysicists solve this problem and obtain geologically reasonable results in one of two ways. The first is based on intuitive estimation of the possible solutions and selection of a geologically adequate model by the interpreter. The second is based on the application of different types of regularization algorithms, which allow automatic selection of the proper solution by the computer using a priori geological and geophysical information about the earth's structure. The most consistent approach to the construction of regularization algorithms has been developed in the works of Tikhonov (1977, 1987) (see also Strakhov, 1968, 1969; Lavrent'ev et al., 1986; Dmitriev et al., 1990). This approach gives a solid basis for the construction of effective inversion algorithms for different applications. In the usual way, we describe the geophysical inverse problem by the operator equation: ^ m = d, m G M, d G Z^, where D is the space of geophysical data and M is the space of the parameters of geological models; A is the operator of the forward problem that calculates the proper data d G D for a given model m G M. The main idea of the regularization method consists of approximating the ill-posed problem with a family of well-posed problems Aa depending on a scalar regularization parameter a. The regularization must be such that as a vanishes, the procedures in the family A^ should approach the accurate procedure A. It is important to emphasize that regularization does not necessary mean "smoothing" of the solution. Regularization may include "smoothing," but the critical element of this approach is in selecting the appropriate solution from a class of models with the given properties. The main basis for regularization is an implementation of a priori information in the inversion procedure. The more information we have about the geological model, the more stable is the inversion. This information is used for the construction of the "regularized family" of well-posed problems A^The main goal of this book is to present a detailed exposition of the methods of regularized solution of inverse problems based on the ideas of Tikhonov regularization, and to show different forms of their applications in both linear and nonlinear geophysical inversion techniques. The book is arranged in five parts. The first part is an introduction to inversion theory. In this part, I formulate the typical geophysical forward and inverse problems and introduce the basic ideas of regularization. The foundations of regularization theory described here include: 1) definition of the sensitivity and resolution of geophysical methods, 2) formulation of well-posed and ill-posed problems, 3) development of regularizing operators and stabilizing functional, 4) introduction of the Tikhonov parametric functional, and 5) elaboration of principles for determining the regularization parameter. In the second part, I describe basic methods of solution of the linear inverse problem using regularization, paying special attention to iterative inversion methods.

PREFACE

XXI

In particular, Chapter 4 deals with the classical minimal residual method and its generalizations based on different modifications of the Lanczos method. The important result of this chapter is that all iterative schemes, based on regularized minimal residual methods, always converge for any linear inverse problem. In Part II, I discuss the major techniques for regularized solution of nonlinear inverse problems using gradient-type methods of optimization. Thus, the first two parts outline the general ideas and methods of regularized inversion. In the following parts, I describe the principles of the application of regularization methods in gravity and magnetic (Part III), electromagnetic (Part IV), and seismic (Part V) inverse problems. The key connecting idea of these applied parts of the book is the analogy between the solutions of the forward and inverse problems in different geophysical fields. The material included in these parts emphasizes the mathematical similarity in constructing forward modeling operators, sensitivity matrices, and inversion algorithms for different physical fields. This similarity is based on the analogous structure of integral representations used in the solution of the forward and inverse problems. In the case of potential fields, integral representations provide a precise tool for linear modeling and inversion. In electromagnetic or seismic cases, these representations lead to rigorous integral equations, or to approximate but fast and accurate solutions, which help in constructing effective inversion methods. The book also includes chapters related to the modern technology of geophysical imaging, based on seismic and electromagnetic migration. Geophysical field migration is treated as the first iteration in the iterative solution of the general inverse problem. It is also demonstrated that any inversion algorithm can be treated as an iterative migration of the residual fields obtained on each iteration. From this point of view, the difference between these two separate approaches to the interpretation of geophysical data - inversion and migration - becomes negligible. In summary, this text is designed to demonstrate the close linkage between forward modeling and inversion methods for different geophysical fields. The mathematical tool of regularized inversion is the same for any geophysical data, even though the physical interpretation of the inversion procedure may be different. Thus, another primary goal of this book is to provide a unified approach to reconstructing the parameters of the media under examination from observed geophysical data of a different physical nature. It is impossible, of course, to cover in one book all the variety of modern methods of geophysical inversion. The selection of the material included in this book was governed by the primary goals outlined above. Note that each chapter in the book concludes with a list of references. A master bibliography is given at the end of the text, for convenience. Portions of this book are based on the author's monograph "Integral Transforms in Geophysics" (1988), where the general idea of a unified approach to the mathematical theory of transformation and imaging of different geophysical fields was originally introduced. The corresponding sections of the book have been written using research results originated by the author in the Institute of Terrestrial

XXII

PREFACE

Magnetism, Ionosphere and Radio-Wave Propagation (IZMIRAN) and later in the Geoelectromagnetic Research Institute of the Russian Academy of Sciences in 19801992. However, this text actually began as a set of lecture notes created for the course "Geophysical Inverse Theory," which I taught during the fall semester, 1992, and the spring semester, 1993, at the Colorado School of Mines. These notes resulted in a tutorial "Regularization in Inversion Theory," published in 1993 as Report #136 of the Center for Wave Phenomena (CWP), Colorado School of Mines. Over the years of teaching the "Inversion Theory and Applications" class at the University of Utah this set of notes was significantly expanded and improved. In this book, I also present research results created by the author and his graduate students at the Consortium for Electromagnetic Modeling and Inversion (CEMI). CEMI is a research and educational program in apphed geophysics based at the Department of Geology and Geophysics, University of Utah. It is supported by an industry consortium formed by many major petroleum and mining exploration companies. The general objectives of the Consortium are to develop forward and inverse solutions for gravity, magnetic, and electromagnetic methods of geophysics, and to provide interpretive insight through model studies (for additional information, please, see the CEMI web site at h t t p : / / w w w . m i n e s . u t a h . e d u / ~ w m c e m i ) . The research goal is to improve the effectiveness of geophysical techniques in mining, petroleum, geothermal, and engineering applications. Progress in these fields requires the development of mathematically sophisticated methods which are aimed at the solution of practical geophysical problems. This philosophy is reflected in the current book which contains a mixture of basic mathematical material and advanced geophysical techniques, which, I hope, will fill a gap in the presently available literature on geophysical applications of mathematical inverse theory. Some of the results contained in the book are based on research projects, which have been supported by grants and contracts from the National Science Foundation, the Department of Energy, the United States Geological Survey, and the Office of Naval Research. I am very grateful for the funding support provided by all these organizations. It is a great pleasure for me to acknowledge those many people who have influenced my thinking and contributed to my learning of mathematics and geophysics. Amongst many, I must single out the unforgettable influence and encouragement given to me by Academician Andrei N. Tikhonov. During all his life he demonstrated in his research how the synthesis of advanced mathematical theory and practically oriented applications can generate exciting progress in science and technology. His ideas lie at the foundation of this book. I am also indebted to Academician Vladimir N. Strakhov of the Institute of the Physics of the Earth, Russian Academy of Sciences, whose contributions to mathematical geophysics are unsurpassed. The inspiring discussions with Professor Vladimir I. Dmitriev of Moscow State University on the geophysical aspects of regularization theory and integral equation methods were very helpful and important to me as well.

XXIII

PREFACE

I also wish to thank Professor Frank Brown and other members of the University of Utah for providing stimulating support during the work on the book. I am thankful to all my past and present graduate students and research associates, who took this course in 1992-2001 and provided me with invaluable feedback and many constructive discussions and suggestions, which helped me to improve the text. While preparing the book, I received much assistance from Professor John Weaver of the University of Victoria, British Columbia, who touched every chapter and made the final version of the book much more readable and understandable. Thanks also go to Professor Robert Smith of the University of Utah, who read parts of the manuscript and made a number of useful suggestions and corrections. Last, but foremost, I wish to dedicate this book to my wife, Olga Zhdanov, whose continuous patience, support and unfailing love made this book a reality. Salt Lake City, Utah December, 2001

Michael S. Zhdanov


Part I I n t r o d u c t i o n to I n v e r s i o n T h e o r y


Chapter 1 FORWARD AND INVERSE PROBLEMS IN GEOPHYSICS Geophysical methods are based on the study of different physical fields being propagated through the earth's interior. The most important geophysical fields are gravity, magnetic, electromagnetic, and seismic wave fields. The observed values of these fields depend, first, on the physical properties of rocks. The conventional approach to geophysical data analysis consists of constructing different geological models and comparing the theoretical geophysical data computed for these models with the observed data. Numerical modeling of geophysical data for given model parameters is usually called a forward problem. The forward problem solution makes it possible to predict geophysical data for specific geological structures. The final goal of geophysical observation is determining the geological structures from the geophysical data. It is a very difficult problem due to the complex structure of the earth's interior. Usually we approximate real geology by a more or less simple model and try to determine the model parameters from the data. We call this problem an inverse problem,. The success of geophysical interpretation depends on our ability to approximate real geological structures by reasonable models, and to solve the corresponding inverse problems effectively. 1.1

Formulation of forward and inverse problems for different geophysical fields

In this introductory section, I will give a mathematical formulation of several forward and inverse problems typical for geophysical methods. The definition of general forward and inverse problems can be described schematically by the following chart: FORWARD PROBLEM: model {model parameters m } —^ data d. INVERSE PROBLEM: data d -^ model {model parameters m } . In studying the geophysical methods, we should also take into account that the field can be generated by some source. So we have to correct our chart accordingly. FORWARD PROBLEM:

4

Forward and inverse problems in geophysics

model {model parameters m, sources s} -^ data d : d =^(m),

(1.1)

where As is the forward problem operator depending on a source s. INVERSE PROBLEM: {data d, sources s} -^ model {model parameters m } : m = A;\d),

(1.2)

or {data d } —> model and sources {model parameters m, sources s}: {m,s)=A-'{d),

(1.3)

where Aj^ and A~^ are inverse problem operators. We will call the problem (1.2) an inverse model problem. Note that the problem (1.2), as applied to electromagnetic field or acoustic field propagation, is usually called an inverse scattering problem. In some geophysical applications the inverse problem is formulated with respect to the sources of the observed field only: {data d} —> { sources s} s = A-\d).

(1.4)

The problem (1.4) is called an inverse source problem. In this case an assumption is made that the model parameters (the physical properties of the medium) are known. Typical examples of this problem are the inverse gravity problem and the inverse seismological problem. In the first case, the density distribution of the rock formation is the source of the gravity field. In the second case, the goal is tofindthe location and type of the earthquake sources from the observed seismic field. In the solution of any inverse problem three important questions arise: 1) Does the solution exist? 2) Is it unique? 3) Is it stable? The question of the solution's existence is associated with the mathematical formulation of the inverse problem. From the physical point of view, there should be some certain solution, since we study real geological structures of the earth's interior. However, from the mathematical point of view, there could be no adequate numerical model from the given model set which would fit our observed data. The question of the uniqueness of the solution can be illustrated by the following formulae. Assume that we have two different models, m i and 1212, and two diff'erent sources, si and 52, which generate the same data do : ^ ( m i , 5i) = do, v4(m2, ^2) = do.


5

In this case, it is impossible to distinguish these two models from the given data. That is why the question of uniqueness is so important in inversion. The last question of solution stability is a critical one in inversion theory as well. In fact, geophysical data are always contaminated by some noise ^d. The question is whether the difference in the responses for different models is larger than the noise level. For example, let two different models, mi and m2, and two different sources, Si and 52, generate two different data sets, di and d2, which can be expressed schematically as follows: yl(mi,5i) = di, and A(m2, 52) = d2. Assume also that these two models and sources are very different, while the data difference is within the noise level e: \\6m\\ = | | m i - m 2 | | > C, \\8s\\ = \\si - S2\\ > C, ||(5d||-||di-d2|| < £ ,

C»6,

where symbol ||...|| denotes some norm, or measure of difference between two models, sources and data sets (see Appendix A for a rigorous definition). In this situation it is also impossible to distinguish these two models from the observed data. Considering the importance of these three questions for inverse problem solution, the famous French mathematician Hadamard expressed the opinion that a certain mathematical problem was formulated correctly if all three questions posed above had a positive answer. In other words, the mathematical problem was said to be well-posed, if its solution did exist, was unique and was stable. A problem was ill-posed according to Hadamard (1902) if the solution did not exist, or was not unique, or if it was not a continuous function of the data (i.e. if to a small perturbation of data there corresponded an arbitrarily large perturbation of the solution). Hadamard considered that an ill-posed mathematical problem was not physically and/or mathematically meaningful (that was why one could call it an " ill" problem). However, it turned out that the majority of the problems of mathematical physics and geophysics (actually the majority of the natural sciences problems) are ill-posed. Fortunately, it was subsequently found that Hadamard's opinion was wrong: ill-posed problems are physically and mathematically meaningful and can be solved. In the middle of the XX Century the Russian mathematician Andrei N. Tikhonov developed the foundations of the theory of ill-posed problem solutions. He introduced a regularization method in the solution of an inverse problem which was based on an approximation of an ill-posed problem by a number of well-posed problems. In this book we will systematically study the principles of ill-posed inverse problem solution. Now, let us turn to the analysis of typical formulations of the forward and inverse problems for major geophysical fields: gravity, magnetic, electromagnetic and seismic wave fields.

6

For^vard and inverse problems in geophysics

1.1.1 Gravity field According to Newton's law, the gravity field g (r) of a three-dimensional distribution of masses with a density p ( r ) , satisfies the equations (Zhdanov, 1988) V • g = -47r7p, V X g = 0,

(1.5)

where 7 is the universal gravitational constant. It is expressed in terms of the gravity potential U : g = V[/, (1.6) satisfying the Poisson equation V^f/ = -47r7p.

(1.7)

It follows from formula (1.7) that, outside masses, the gravity field is of Laplace form, i.e. it satisfies the Laplace equation: V ' g = 0,

(1.8)

where V^ is Laplace operator, or Laplacian. It is well known that the solution of the Poisson equation can be presented in the form of a volume integral (Zhdanov, 1988):

where D is a domain of mass concentration. Taking into account the identity: V'

1 -

=

r - r' _ r-rM"

(where prime at the gradient operator means differentiation with respect to the variable r'), we can obtain the known expression for the gravity field of a volume mass distribution:

g(r') = A^{p)

=jlJJ^p{r)V'j^^^dv

-ÎIL

^WfTT^-^'^^-

(1-10)

Here A^ is an operator of the forward gravity problem. In particular, for the vertical component of a gravity field, we write:

5^(o = VZ/p^'^'^F^'^''-

^^-^^^


7

Gravity prospecting is based on studying anomalous gravity fields generated by the anomalous density distribution in the earth, Ap. The notion of the "anomalous density" arises if we consider some homogeneous background model of the earth with the density p^, and calculate anomalous density as the difference between the real p(r) and background p^ densities: Ap(r) = p(r) — p^. The anomalous density distribution generates a corresponding gravity anomaly, Ag, that is related to this density by an equation of type (1.10),

Ag(r') = A•/

> Figure 1-9

One-dimensional electromagnetic induction inverse problem.

Weidelt's theorem of uniqueness can be treated as a generalization of Tikhonov's theorem for a two-dimensional model with an electrical conductivity described by an analytic function a{x,z). Assume that we observe simultaneously electric and magnetic field components of the frequency domain electromagnetic field over some interval of observation {a < x < b) along a profile above the two-dimensional conductivity model of the earth (a = a (x, z)), as shown in Figure 1-10. The theorem states that, if the observations are given over an entire frequency range 0 < c^ < oo, then one can reconstruct uniquely the conductivity distribution a {x, z) from the observed data. Gusarov's theorem is a further generalization of this result for a 2-D geoelectrical model with a piecewise analytic distribution of electrical conductivity. The last two theorems make it feasible to apply the electromagnetic method for examining inhomogeneous geological structures of the earth. Note that all these theorems can be easily understood by relying on intuition. We can assume that the frequency dependence of the field (electromagnetic skineffect) provides information about the vertical variations of conductivity, while the spatial dependence of the data on the surface allow us to reconstruct the horizontal changes in conductivity. Thus, one can expect that these theorems can be extended to 3-D cases as well. Note, also, that the proof of these theorems, including the 3-D case, can be obtained as a special case of a more general mathematical uniqueness theorem of inverse problems for general partial differential equations. We will outline this more

ForAvard a n d inverse p r o b l e m s in g e o p h y s i c s

22

u u u u nu £yfx, coj, Ey,(x, (oj, //v-fx, coj, H^ (x, m)

X

a

(y(x, z)

Figure 1-10

Two-dimensional electromagnetic induction inverse problem

general approach in the next section, considering the inverse problem for a seismic wavefield as an example. C. Seismic wavefield and general uniqueness theorems The mathematical formulation of seismic forward and inverse problems in the simplest case of an acoustic model in the frequency domain is given by equation (1.25)^ which we will repeat here for convenience:

c^(r)

(1.41)

This equation can be treated as a special case of a general differential equation: ^7 {P{^^ ^)) - ^ 7 (r) • Vp(r, u) -h a;^p(r, uj) = - 7 ( r ) / ( r , a ; ) ,

(1.42)

where L^ is an elliptic differential operator: ^7 (P) = V • ( 7 V p ) ,

(1.43)

and the coefficient 7 (r) is equal to the square of the velocity: 7(r) =c^(r). ^Note t h a t Hemholtz equation (1.41) describes not only acoustic waves, but, in some models, electromagnetic fields in the frequency domain as well (Zhdanov, 1988).

Existence

and uniqueness of the inverse problem solutions

23

The inverse problem in this case is formulated as recovery of the unknown coefficient 7 of the elliptic operator L^ from the known values of the field p(r, u) in some domain or in the boundary of observations. In a number of brilliant mathematical papers the corresponding uniqueness theorems for this mathematical inverse problem have been formulated and proved ^. The key result is that the unknown coefficient 7 (r) of an elliptic differential operator can be determined uniquely from the boundary measurements of the field, if 7 (r) is a real-analytical function, or a piecewise realanalytical function. In other words, from the physical point of view we assume that 7 (r) is a smooth function in an entire domain, or a piecewise smooth function. Note that this result corresponds well to Weidelt's and Gusarov's uniqueness theorems for the magnetotelluric inverse problem. I would refer the readers for more details to the papers by Calderon (1980), Kohn and Vogelius (1984, 1985), Sylvester and Uhlmann, (1987), and Isakov (1993). Returning to the seismic wavefield inverse problem, we can assume that, based on general mathematical uniqueness theorems, the seismic inverse problem,

c={AT\p), (where a pressure field p(r, LU) is given on the earth's surface) should have a unique solution for the models with a piecewise smooth distribution of seismic field velocity. This result, based on intuition, becomes even stronger in the case of seismic field observations with multiple source positions, because the different sources can illuminate the target from different angles. We will discuss the practical aspects of the uniqueness problem in the next section. Note, in conclusion, that multisourcc and multireciever observation systems arc typical for the modern seismic exploration technique, thus providing a basis for the unique inversion result. 1.2.3 Practical uniqueness Unfortunately, the number of uniqueness theorems for geophysical inverse problems is relatively small. These theorems cover very specific geophysical models. In practical situations we should rely on a more simple but important property of inverse problem solution. Following Hjelt (1992), we call this property practical uniqueness. It can be described using the following simple considerations. Consider a case where we would like to determine some function of n variables from the observed data. To obtain the solution of the inverse problem in this case, we obviously have to measure a function of the same or greater number of variables. For example, if the model parameters are given in 3-D space m = m(x, ?/, z) then the data §The most remarkable results were formulated for a special case of static field (u = 0) which corresponds, in particular, to the solution of an electric potential problem: i,(it(r)) = V-(a(r)Vu(r)) = - / ( r ) , where u (r) is the electric potential, a (r) is the conductivity distribution, and / (r) is the source of the electric field.

24

Forward £ind inverse problems in geophysics

should also be a function of three variables: d = d(ri,r2,r3). The important point is that it is not necessary that variables (ri,r2,r3) have the same physical meaning, or coincide with the coordinates (x, y, z). For example, we can change the position of the transmitter and receiver, r^ and r"": r^ = (x^2/^0), r ^ - ( a : ^ y ^ O ) . Then the observed data will be the function of four parameters: d = d(a;^2/â:^2/'-)• We can expect in this situation that the practical inverse problem would have a unique solution, because the space of the four-dimensional data parameters is bigger than the space of the three-dimensional model parameters. Another way to increase the dimension of the data parameter space is to take into account the data dependence on time or frequency: d = d(x'',y'',t), or d = d{x'^,y'',Lj). In this situation, again, we can expect that the inverse problem may have a unique solution. I would like to emphasize, at the end of this section, that practical uniqueness has nothing in common with theoretical uniqueness. The requirement that the observations should be a function of the same or greater number of parameters than the model provides a qualitative background for the uniqueness only. Theoretically, even in this situation, the inverse problem could have several equivalent solutions. However, this requirement serves as a useful practical guide to ways of obtaining a unique solution for a practical problem. 1.3

Instability of the inverse problem solution

Another critical problem of inversion theory is instability. This problem reflects the practical fact that two observed data sets could differ only within the noise level, while the corresponding model parameter distributions could be completely different. We will discuss below a typical example of instability in solving a downward analytical continuation problem for the gravity field potential. According to equation (1.7), the gravity potential outside masses satisfies the equation: VÛ = 0.

(1.44)

Let us introduce the Cartesian system of coordinates (x^y.z) with the vertical axis directed downwards. We assume that we observe the gravity potential at some level z = 0: [/(x,y,0) = /(x,2/),

25


where f{x,y) is some known function. The problem is to recalculate the potential from the level z = 0 to any other level z = h in the lower half-space: U{x,y,h)=7,

0 0 is a small number. In other words, we suggest that the function /i(x, y) describes precise observations, while the function f2{x,y) describes observations with

26


some noise. Solving the problem of downward analytical continuation for these two observations, we obtain two solutions at a depth h: Ui{x,y,h)

= ae'^^'^ sin {kxx) sin (kyy),

(1-51)

and C/2(x, y, h) = {a-\- eje''^^ sin (A:^x) sin (kyy).

(1.52)

Let us analyze the difference between these two solutions: I Ui{x,y,h)

- U2{x,y,h) H ee^"'' I sin (A:,x) sin (/c^y) | .

(1.53)

Note that, at the points TT

TT

Xn = ± (2n + 1) —-, 2/m = ± (2m -h 1) —-; n, m = 0,1, 2, 3,... this difference is equal to: I f/i(xn, ym, h) - U2{xn^ ym. h) \= ec^'K

(1.54)

The remarkable fact is that the expression ee^°^ grows exponentially with the increase of depth /i, and/or frequencies kx and ky, and it can be made bigger than any large number C for any small value of e. At the same time, the difference between two observations, /i(x, y) and /2(x, y), is equal or smaller than e : I hi^^y)

- f2{x,y) \< 6.

Figures 1-11 and 1-12 provide an illustration of these theoretical results. Figure 1-11 shows the plot of gravity potential at a distance 25 meters from the source (a material ball), and Figure 1-12 shows the same potential field analytically continued closer to the source. One can see that a small, practically invisible noise in the original data results in dramatic oscillations for downward analytically continued data. This is a classical example of an unstable inverse problem. Similar demonstrations of instability can be provided for practically all inverse geophysical problems (see, for example, Lavrent'ev et al., 1986). That is why any reasonable algorithm for an inverse problem solution must be able to take this effect into account. In the next Chapter we will study a regularization method of the stable solution of the ill-posed inverse problems.


27

Potential U at 25m

4—! U

0

10 20 30 40 50 60 70 80 90 Distance

Figure 1-11

Plot of gravity potential at a distance 25 m above the material ball.

J 0

mjxrr]

10 20 30 40 50 60 70 80 90 Distance

Figure 1-12 Dennonstration of instability in inverse problem solution. Smooth solid line shows the true distribution of the gravity potential at a distance 20 m above the material ball. Oscillating line presents a plot of gravity potential analytically continued downward at a distance 20 m above the material ball. One can see that a small, practically invisible noise in the original data results in dramatic oscillations for downward analytically continued data.

28


References to Chapter 1 Berdichevsky, M. N., and M. S. Zhdanov, 1988, Advanced theory of deep geomagnetic sounding: Elsevier, Amsterdam, London, New York, Tokyo, 408 pp. Bleistein, N., and J. Cohen, 1976, Non-uniqueness in the inverse source problem in acoustics and electromagnetics: J. Math. Physics, 18, 194-201. Blok, H., and M. Oristaglio, 1995, Wavefield imaging and inversion in electromagnetics and acoustics: Delft University of Technology, Report number: E t / E M 1995-21, 132 pp. Calderon, A., 1980, On an inverse boundary value problem: Seminar on numerical analysis and its application to continuum physics, Rio de Janeiro. Gusarov, A. L., 1981, On uniqueness of solution of inverse magnetotelluric problem for two-dimensional media (in Russian): Mathematical Models in Geophysics, Moscow State University, 31-61. Hadamard, J., 1902, Sur les problemes aux derivees partielles et leur signification physique, Princeton Univ. Bull. 13, 49-52: reprinted in his Oeuvres, Vol. Ill, Centre Nat. Recherche Sci., Paris, 1968, 1099-1105. Hjelt, S.-E., 1992, Pragmatic inversion of geophysical data: Springer-Verlag, Berlin, Heidelberg, New York, 262 pp. Isakov, v., 1993, Uniqueness and stability in multi-dimensional inverse problem: Inverse Problems, 6, 389-414. Kohn, R., and M. Vogelius, 1984, Determining conductivity by boundary measurements: Commun. Pure Appl. Math., 37, 281-298. Kohn, R., and M. Vogelius, 1985, Determining conductivity by boundary measurements, interior results II: Commun. Pure Appl. Math., 38, 643-667. Lavrent'ev, M. M., Romanov, V. G., and S. P. Shishatskii, 1986, Ill-posed problems of mathematical physics and analysis: Translations of Mathematical Monographs, 64, American Mathematical Society, Providence, Rhode Island, 290 pp. Novikov, P. S., 1938, Sur le probleme inverse du potential: Dokl. Acad. Sci. URSS, 18, 165-168. Stratton, J. A., 1941, Electromagnetic theory: McGraw-Hill Book Company, New York and London, 615 pp. Sylvester, J., and G. Uhlmann, 1987, Global uniqueness theorem for an inverse boundary value problem: Ann. Math., 125, 153-169. Tikhonov, A. N., 1965, Mathematical basis of electromagnetic sounding (in Russian): Zh. Vichisl. Mat. Mat. Fiz., 5, 207-211. Weidelt, P., 1978, Entwicklung und Erprobung eines Verfahrens zur Inversion Zweidimensionaler Leitfahigkeitsstrukturen in E>Polarisation: Dissertation, Gottingen Universitat, Gottingen. Zhdanov, M. S., 1988, Integral transforms in geophysics: Springer-Verlag, New York, Berlin, London, Tokyo, 367 pp. Zhdanov, M. S., and G. Keller, 1994, The geoelectrical methods in geophysical exploration: Elsevier, Amsterdam, London, New York, Tokyo, 873 pp.

Chapter 2 ILL-POSED PROBLEMS AND T H E METHODS OF THEIR SOLUTION The formal solution of the ill-posed inverse problem could result in unstable, unrealistic models. The regularization theory provides a guidance how one can overcome this difficulty. The foundations of the regularization theory were developed in numerous publications by Andrei N. Tikhonov, which were reprinted in 1999 as a special book, published by Moscow State University (Tikhonov, 1999). In this Chapter, I will present a short overview of the basic principles of the Tikhonov regularization theory, following his original monograph (Tikhonov and Arsenin, 1977). 2.1

Sensitivity and resolution of geophysical methods

We begin with the formulation of the notions of geophysical sensitivity and resolution, which are important in understanding the regularization principles. 2.1.1 Formulation of the inverse problem m general mathematieal spaces In the first Chapter we have introduced an inverse problem as the solution of the operator equation d = A{m), (2.1) where m is some function (or vector) describing the model parameters, and d is a data set, which can also be characterized as a function of the observation point (in the case of continuous observations), or as a vector (in the case of discrete observations). The solution of the inverse problem consists in determining such a model nipr (predicted model) that generates predicted data dp^, that fit well the observed data d. We have discussed already that we do not want to fit the observed data exactly, because they always contain some noise that we should not fit. Therefore, we are looking for some predicted data that will be close enough to the observed data (usually, within the accuracy of our observations). But what does "close enough" mean? How can we measure the closeness of two data sets? This is exactly the moment when we have to introduce some kind of "distance" between two data sets that will help us to evaluate the accuracy of the inverse problem solution. In other words, we need to introduce a geometry to measure the distance between the actual and predicted data. The mathematical theory of function spaces provides us with guidance to the solution of this problem. The basic principles of this

30

Ill-posed problems and the methods of their solution

theory are outlined in Appendix A. The simplest and, at the same time, the most important mathematical space which contains a geometry (in the sense that there is a distance between any two elements of this space) is a metric space. More complicated but still very useful spaces are the linear normed space and the Banach space. An extremely important example of a mathematical space used in inverse theory, is the Hilbert space. Using the basic ideas of the mathematical theory of function spaces and operators acting in these spaces, we can present a more rigorous formulation of mathematical inverse problems. Let us assume that we are given two Banach (complete normed) spaces, M and D, and an operator A that acts from the space M to the space D : ^ (m) = d, m G M, AeD.

(2.2)

We will call D a space of data sets and M a space of the model parameters. Operator ^ is a forward modeling operator that transforms any model m into the corresponding data d. The inverse problem is formulated as the solution of the operator equation (2.2). We examine some general properties of the forward and inverse problem (2.2) that can be treated as the sensitivity and resolution of the corresponding geophysical methods. 2.1.2 Sensitivity Any forward geophysical problem can be described by the operator equation (2.2). Let us consider some given model mo and corresponding data dg. We assume, for the sake of simplicity, that in some vicinity of the point mo, the operator A = Am^ is a linear operator. Then we have: ^ ^ ^ ( m - mo) = Am,m - Amoô = d - do, or yl^,(Am)=Ad,

(2.3)

where A m = m — mo and

Ad = d — do

are the perturbations of the model parameters and of the data. Following work by Dmitriev (1990), we can now give a corresponding definition of the sensitivity. Definition 1 The sensitivity Smo of the geophysical method is determined by the ratio of the norm of the perturbation of the data to the norm of the perturbation of the model parameters. The maximum sensitivity is given by bm.

- sup I ii^îi j - sup I

ii^îi

j . - ||^„J| ,

(2.4)

Sensitivity

31

and resolution of geophysical methods

where a symbol sup (/; denotes the least upper bound or supremum of the variable (/?, i.e. S^^^ is equal to the norm of the operator AmoIf we know 5^^^, according to (2.3) and (A.25), we can determine the variations of the model that can produce the variations of the data greater then the errors of observations S:

||m-mo||> S/SZT-

(2-5)

Therefore, the geophysical method is sensitive only to those perturbations of the model parameters that exceed the level S/S^^^. Any other variations of the model cannot be distinguished from the data. 2.1.3 Resolution Let us assume now that in some vicinity of the point mo the following inequality is satisfied \\Am^{Am)\\ > k\\Am\\ for any Am, where /c > 0 is some constant. Then, according to Theorem 64 from Appendix B, there exists a linear and bounded inverse operator A^^. It means that the solution of the inverse problem in the vicinity of the point mo can be written as m = mo + / l - i ( d - d o ) .

(2.6)

The same expression can be written for data d^, observed with some noise d^= d4-(5d: m , = mo + A;,^^(d,-do).

(2.7)

ms-mÂ;;,l{ds-d).

(2.8)

From (2.6) and (2.7) we have

Now we can determine the maximum possible errors in the solution of the inverse problem for the given level of the errors in the observed data, equal to S = \\Sd\\: ^max=

sup | | m , - m | | -

I|d5-d|h6

sup \\A;^l{ds-d)\\

I|ci,-dlh6

= \\A;^l\\6,

(2.9)

where by Theorem 64 in Appendix B,

ll^^lll < I-

(2.10)

Based on the last formulae, we can determine the resolution of the geophysical method. Two models m i and m2 in the vicinity of the point mio can be resolved if the following condition is satisfied: c

| | m i - m 2 | | > Ama. = \\A^l\\6 = -^—.

(2.11)

32


The value ^0 = ^ ^ (2.12) II^mo II is the measure of resolution of the given geophysical method. It follows from (2.10) and (2.12) that Rmo>k. (2.13) The smaller the norm of the inverse operator, the bigger the resolution, Rmo, and the closer to each other are models that can be resolved. If the inverse operator A^ is not bounded, i.e. its norm goes to infinity, the resolution goes to zero, Rmo — 0, and the maximum possible errors in the determination of m are infinitely large. We have exactly this case for the ill-posed problem. 2.2

Formulation of well-posed and ill-posed problems

We formulate an inverse problem as the solution of an operator equation d = ^(m),

(2.14)

where m G M, is some function (or vector) from a metric space M of the model parameters, and d G D is an element from a metric space D of the data sets. There are two important classes of inverse problems: well-posed and ill-posed problems. We will give detailed descriptions of these problems in this section. 2.2.1 Well-posed problems Following classical principles of regularization theory (Tikhonov and Arsenin, 1977; Lavrent'ev et al., 1986) we can give the following definition of the well-posed problem. Definition 2 The problem (2.14) is correctly (or well) posed if the following conditions are satisfied: (i) the solution m of equation (2.14) exists, (ii) the solution m of equation (2.14) is unique, (Hi) the solution m depends continuously on the left-hand side of equation (2.14) d. In other words, the inverse operator A~^ is defined throughout the space D and is continuous. Note that the well-posed inverse problem possesses all the properties of the "good" solution discussed in the previous chapter: the solution exists, is unique, and is stable. Definition 3 The problem (2.14) is ill-posed if at least one of the conditions, listed above, fails. We have demonstrated in the first chapter that the majority of geophysical inverse problems is ill-posed, because at least one of the conditions listed above fails. However, it may happen that if we narrow the class of models which are used in

Formulation

of well-posed and ill-posed problems

33

inversion, the originally ill-posed inverse problem may become well-posed. Mathematically it means that instead of considering m from the entire model space M, we can select m from some subspace of M, consisting of simpler and/or more suitable models for the given inverse problem. Thus, we arrive at the idea of the correctness set and conditionally well-posed inverse problems. 2.2.2 Conditionally well-posed problems Suppose we know a priori that the exact solution belongs to a set, C, of the solutions with the property that the inverse operator A~^^ defined on the image* AC^ is continuous. Definition 4 The problem (2.14) is conditionally well-posed (Tikhonov^s well-posed) if the following conditions are met: (i) we know a priori that a solution of (2.14) exists and belongs to a specified set C C M , (ii) the operator A is a one-to-one mapping of C onto AC C D, (iii) the operator A~^ is continuous on AC C D. We call set C the correctness set. In contrast to the standard well-posed problem, a conditionally well-posed problem does not require solvability over the entire space. Also the requirement of the continuity of A~^ over the entire space M is substituted by the requirement of continuity over the image of C in M. Thus, introducing a correctness set makes even an ill-posed problem well-posed. Tikhonov and Arscnin (1977) introduced the mathematical principles for selecting the correctness set C. For example, if the models arc described by a finite number of bounded parameters, they form a correctness set C in the Euclidean space of the model parameters. This result can be generalized for any metric space. First, we introduce a definition. Definition 5 The subset K of a metric space M is called compact if any sequence mi e K of elements in K contains a convergent subsequence m/^. G K, which converges to an element m in K. For example, it is known that any subset R of Euclidean space En is compact if and only if it is bounded: ||x|| < c, c > 0, for any x e R. We will demonstrate now that any compact subset of the metric space M can be used as a correctness set for an ill-posed inverse problem (2.14). Theorem 6 (Tikhonov): Let the solution of equation (2.14) be unique, let the set C be compact in M, and let A be a continuous one-to-one mapping of C to AC C D. Then A~^ is continuous on AC C D. *The domain AC C D formed by all vectors obtained as a result of operator A applied to all vectors m from the set C, m G C, is called an image of the set C in the space D.

34


Proof: Suppose that the theorem is not correct. Then there exists eo > 0 such that for any (5 > 0 there are m, m' G C with the properties / i ^ ( m , m') > 60, /xz)(^x, Ax!) < 6, where fXj^, /i^ are the metrics of the spaces M and D respectively. Let {6k} be some sequence of Â; ^ 0 for A: —^ cx) and let {rrifc}, {mifc} be some sequences of the elements in C such that /i^(mfc,m'fc) > eo, /x^(>lmfc,>lm'^) < 4 • (2.15) According to the fact that C is compact, the sequences {rrifc}, {m^} contain convergent subsequences. We assume, for the sake of simplicity, that these subsequences are identical to the original sequences. Let m = lim nifc, m' = lim m'^. Then, according to (2.15) and the continuity of the operator A^ we have / i ^ ( m , m ' ) > eo, /j.j^{Ain,Am')

= 0.

From the last result it follows that Am = Am\

(2.16)

Equation (2.16) contradicts the uniqueness of the solution of the inverse problem (2.14). Thus, according to Tikhonov and Arsenin (1977), any compact subset of M can be used as a correctness set for equation (2.14). 2.2.3 Quasi-solution of the ill-posed problem We assume now that the problem (2.14) is conditionally well-posed (Tikhonov's wellposed). Let us assume, also, that the right-hand side of (2.14) is given with some error: d^= d-h5d, (2.17) where fij,{ds,d) 0 is called a regularization parameter. We require also that nia—> mt, if Q -^ 0,

where nia = â^ (d) is the solution of the inverse problem (2.24), and m^ is the true solution of the original problem (2.23). Thus, we replace the solution of one ill-posed inverse problem by the solutions of the family of well-posed problems, assuming that these solutions, m^, tend asymptotically to the true solution, as a tends to zero. In other words, any regularization algorithm is based on the approximation of the noncontinuous inverse operator A~^ by the family of continuous inverse operators A~'^ (d) that depend on the regularization parameter a. The regularization must be such that, as a vanishes, the operators in the family should approach the exact inverse operator A''^. Let us now give a more accurate definition.

Foundations of regulaxization methods of inverse problem solution

37

Definition 8 Operator i?(d, a) (dependent on a scalar parameter a) is called the regularizing operator in some neighborhood of the element dt = A{nit), if there is a function a{6) such that, for any e > 0, there can be found a positive number (5(e) with the property, and / i ^ ( m a , m j < e, where nia =

R{d,a{S)).

In other words, m^ is a continuous function of the data and m« = i ? ( d , a ( ( 5 ) ) ^ m , ,

(2.25)

when a —^ 0. Definition 9 Operator i?(d, a) is called the regularizing operator for equation (2.23) m some subset Di C D, if it is a regularizing operator in the neighborhood of any point d G ZîFigure 2-2 illustrates the basic properties of the regularizing operator. Let m^, be the exact solution for exact data d^ = ^(m^). However, we can observe only the noisy data d^ = dt + 6d. We obtain a result m'^, which might lie far away from the true solution, if we apply some rigorous inverse operator to the noisy data d,^. One could obtain quite another result, m'^, also completely different from the true solution, with slightly different noisy data d^. The main advantage of the regularizing operator R is that it provides a stable solution in any situation. If we apply an operator R to the noisy data d^, we will get a solution, m^ = /?(d^,a), which is very close to the true model: ||ni^—m^H < e. Application of R to noisy data d^ will result in another solution m^ = R{ds,a), which is still close to m^. The accuracy of the true solution approximation by the regularized one depends on the regularization parameter a. The smaller the a, the more accurate the approximation. We can see that the regularizing operators can be constructed by approximating the ill-posed equation (2.23) by the system of well-posed equations (2.24), where the corresponding inverse operators A~^ are continuous. These inverse operators can be treated as the regularizing operators: A-'{d)

=

R{d,a).

The only problem now is how to find the family of regularizing operators. Tikhonov and Arsenin (1977) suggested the following scheme for constructing regularizing operators. It is based on introducing special stabilizing and parametric functionals.


38

m^^=A-^ (d^)

M m

'•

m•

m = R(d. ,a)

d^ = d( + db

m. = R(d. ,a) Figure 2-2 The scheme illustrating the construction of the regularizing operators. The bold point nit denotes the true solution, and the point dt e D denotes the true data. The noisy data are shown by a point d^ = d^ + 6d. Application of a formal inverse operator A~^ to the noisy data generates formal solutions, m'^ and m^, which are unstable with respect to a small perturbation in the data, d^ or d^. However, application of the regularizing operators, R{d,a) to any of the observed data, d^ or d 0 } , IS compact in the Hilbert space. as follows:

Therefore,

we can introduce

a stabilizing

5(m)=/i(m,mo),

functional (2.27)

where nio is any given model from M = L2. Obviously, m G M for which s{m) < c ,

the subset Mc of the

s{m) = / i ( m , m o ) < c,

elements (2.28)

IS compact. E x a m p l e 12 Let us consider a Sobolev space (which is at the same time a Hilbert space) W2 formed by the functions continuously differentiable to the order n in the interval [a, b] (see Appendix A). The metric in the space VK^ is determined according to the formula / / ^ p ( m i , m 2 ) = 0. Therefore, we can select the sequence of models {m^^} with the property lim„ôoP° = Po, (2.45) where P„" = P " ( m ^ , d ) .

(2.46)

Evidently, we can select {m^} in such a way that, for any n, P:+I < P: < P^Then, for any n and for any fixed o; > 0, we have as(m°) < ^lUA{ml,

d) + as(m^) - P„" < Pf.

Let us assume that a > 0 : s{inl) < - P f < c. a

(2.47)

44


Therefore, the sequence of the models {m^} belongs to the subset Mc C M, for which s{m) < c. According to the definition of the stabihzing functional, the subset Mc is a compact. Therefore, we can select from the sequence {m^} a subsequence {m^/^)} which converges to some model rria G M. Inasmuch as the operator A is SL continuous operator, we obtain infêMP"(m,d) = lim,ôoP"(m^,d) = lim^ôcP"(m^^),d) = l™A:ôo[M^(^(m^Ô,d) + a5(m^^^)] = fil{A{mc,),d)

+ a5(ma).

(2.48)

In the case where a = 0, the parametric functional is equal to the stabilizing functional for which there exists a model minimizing its value. From the last result, formula (2.48), the statement of Theorem 14 follows at once. Thus, for any positive number, a > 0, and for any data, ds G D, we have determined an operator, R{ds, a), with the values in M, such that the model ina = R{ds,a)

(2.49)

gives the minimum of the Tikhonov parametric functional P"(in,d^). The fundamental result of the regularization theory is that this operator, i?(d^,a), is a regularizing operator for the problem (2.14:). I do not present here the proof of this result, which requires an extensive mathematical derivation, referring interested readers to the original monograph by Tikhonov and Arsenin (1977). Thus, as an approximate solution of the inverse problem (2.14), we take the solution of another problem (2.41) (problem of minimization of the Tikhonov parametric functional P*^(m,d^)), close to the initial problem for the small values of the data errors 6. It is important to underline the fact that, in the case where A is a linear operator, where D and M are Hilbert spaces, and where s{m) is a quadratic functional, the solution of the minimization problem (2.44) is unique. Note that the quadratic functional is a functional g(ni) with the property

q{Pm)=(3\{m). Under these conditions, a parametric functional can be rewritten as P " ( m , d , ) = | | y l ( m ) - d , | | l , + ag(m). Let us assume that there are two different models mL and riia , satisfying (2.44), i.e., realizing the minimum of P°'(in, d^)). Let us consider the elements of the model space M, located in the interval of the "straight Hne," connecting the elements rrva and nia : m = mli)+/?(mi2)-mi^)).

45

Family of stabilizing functionals

The functional P"(m,d^) on the elements of this "line" is a nonnegative quadratic function of /? (because a misfit functional ||^(m) — d^||^ is a quadratic functional as well). Therefore, it cannot reach its minimum for two different values of p. For example, a quadratic functional Syj{m) can be introduced as s^{m) = {Wm,Wm)j^

= \\Wni\\\

(2.50)

and VF is a positively determined linear continuous operator in M. Note that in general cases for a nonlinear A, the solution nia can be nonunique. 2.4


2.4-i Stabilizing functionals revisited The main role of the stabilizing functional (a stabilizer) is to select the appropriate class of models for inverse problem solution. The examples listed above show that there are several common choices for a stabilizer. One is based on the least squares criterion, or, in other words, on the L2 norm for functions describing model parameters: SL2{m) = ||m||^^^ = (m,m)^^ = / \m {r)f dv =^ min . (2.51) Jv In the last formula wc assume that the function m (r), describing model parameters, is given within a three-dimensional domain V, and r is a radius-vector of an observation point. The conventional argument in support of the norm (2.51) comes from statistics and is based on an assumption that the least squares image is the best over the entire ensemble of all possible images (see, for example. Chapter 3). We can use, also, a quadratic functional 5^^ : = [ \w {T) m {T)\'dv = min, (2.52) Jv where w (r) is an arbitrary weighting function, and VK is a linear operator of multiplication of the function m (r) by the weighting function w (r) . Another stabilizer uses a minimum norm of difference between a selected model and some a priori model m^pr • s^ = \\Win\\% = {WmWm)^^

SL2apr{m)

=

11^1 -

niaprWl^

= Uliu .

(2.53)

The minimum norm criterion (2.51), as applied to the gradient of the model parameters Vm, brings us to a maximum smoothness stabilizing functional: 5max5m(m) = ||Vm||^^ =- (Vm,Vm)^^ -=

V

\Vm{r)fdv

= min .

(2.54)

46


In some cases, one can use the minimum norm of the Laplacian of model parameters 5maxsm (m) = || V^m|| = (V^m, V^m) = min .

(2.55)

It has been successfully used in many inversion schemes developed for geophysical data interpretation (see, for example, Constable et al., 1987; Smith and Booker, 1991; Zhdanov, 1993; Zhdanov and Fang, 1996). This stabilizer produces smooth models, which in many practical situations fail to describe properly the real blocky geological structures. It also can result in spurious oscillations when m is discontinuous. In Chapter 3 we will demonstrate that some of the stabilizers, introduced above, can be treated in terms of probability theory. We mention here, as an example only, a minimum entropy stabilizer, which selects the simplest possible solution required to fit the data: / Sminentroim)

f 1^1 ^

\ = -

1^1 7

~Q~ ^^^'Q'

^^ ^^\ '

^

^ ^

where Q =

Jv

\m\ dv.

Note that this stabilizer works similarly to the maximum entropy regularization principles, considered, for example, in Smith et al. (1991), and Wernecke and D'Addario (1977). However, in the framework of the Tikhonov regularization, the goal is to minimize a stabilizing functional, which justifies the "minimum entropy" name for this stabilizer. In the paper by Rudin et al., 1992, an approach based on total variation (TV) method for reconstruction of noisy, blurred images has been introduced. It uses a total variation stabilizing functional, which is essentially the Li norm of the gradient: STV (m) = ||Vm|j

/ \Vm{r)\dv.

Jv

(2.57)

This criterion requires that the distribution of model parameters in some domain V be of bounded variation (for definition and background see Giusti, 1984). However, this functional is not differentiable at zero. To avoid this difficulty, Acar and Vogel (1994) introduced a modified TV stabilizing functional: s^TV (m) = J yj\\/m{r)\^

+ p^dv,

(2.58)

where /? is a small number. The advantage of this functional is that it does not require the function m to be continuous, only piecewise smooth (Vogel and Oman, 1998). Since the TV norm does not penalize discontinuity in the model parameters, we can remove oscillations while preserving sharp conductivity contrasts. At the same time, it imposes a limit

Family of stabilizing

47

functionals

on the total variation of m and on the combined arc length of the curves along which m is discontinuous. That is why this functional produces a much better result than maximum smoothness functionals when the blocky structures are imaged. TV functionals STV (m) and Sf3Tv (j^), however, tend to decrease the bounds of variation of the model parameters, as can be seen from (2.57) and (2.58), and in this sense they still try to "smooth" the real image. However, this "smoothness" is much weaker than in the case of traditional stabilizers (2.54) and (2.53). One can diminish this "smoothness" effect by introducing another stabilizing functional which minimizes the area where significant variations of the model parameters and/or discontinuity occur (Portniaguine and Zhdanov, 1999). This stabilizer is called a minimum gradient support (MGS) functional. For the sake of simplicity we will discuss first a minimum support (MS) functional., which provides a model with a minimum area of the distribution of anomalous parameters. The minimum support functional was considered first by Last and Kubik (1983), where the authors suggested seeking a source distribution with the minimum volume (compactness) to explain the anomaly. We introduce a support of m (denoted spt m) as the combined closed subdomains of V where m ^ 0. We call spt m a model parameter support. Consider the following functional of the model parameters:

p' V m^ (r) + P ^^P'^'^-^^

./spt m / /spt

m

w? (r) + /?'

1 -U^^âi^'^^ 2 ( r ) -f 0-

dv

(2.59)

From the last expression we can see that s^ (m) -^ spt m, if/? -> 0.

(2.60)

Thus, s^ (VOL) can be treated as a functional, proportional (for a small /?) to the model parameter support. We can use this functional to introduce a minimum support stabilizing functional SMS (ni) as SMS (m) = 5^ (m - niapr) == / ^f^—-i^dv. Jv [m - rriapr) -f P

(2.61)

To justify this choice we should prove that SMS (n^) can actually be considered as a stabilizer according to regularization theory. According to the definition given above, a nonnegative functional s (m) in some Hilbert space M is called a stabilizing functional if, for any real c > 0 from the domain of the functional s (m) values, the subset Mc of elements m G M, for which s (m) < c, is compact. Let us consider the subset Mc of the elements from M, satisfying the condition

48

Ill-posed problems and t h e methods of their solution

SMS ( m ) < c,

(2.62)

where SMS (J^) is a minimum support stabiUzing functional determined by equation (2.61). It can be proved that SMS is a monotonically increasing function of II _ 11*^ SMS (mi) < SMS {^2),

if ||mi-mapr|li^2 < ||m2-maprlL2 •

(2-63)

To prove this, let us consider the first variation (see Appendix D) of the minimum support functional: 2

6 f 6SMS •{m) (m) = 61 ^—-^dv Jv (m - rriapr) + /? / —^6(rn-majyrfdv= IV i(m {{m - m.r.y rriaprf + -{-py Jv B^\

I 0?6 {m - ruaprf dv, Jv

where a =

((m - ruaprf + P^)

Using a mean value theorem, we obtain 6SMS (m) = a^ / 6{m-

Jv

rriapr)^ dv

= 0^6 / (m - rriapr)^ dv = 0^6 ||m - mapr||^2 = 2a^ ||m - maprWL^ ^ W^ " m^prlL^'

(2.64)

where a^ is an average value of a^ in the volume V. Taking into account that a^ > 0 and IIm — rriapT^H^^ > 0, we obtain (2.63) from (2.64). Thus, from condition (2.62) and (2.63), we see that ||m - maprllz.2 < ^' m G Mc,

(2.65)

where g > 0 is some constant, i.e. Mc forms a ball in the space M with a center a,t the point niapr' It is well known that the ball is compact in a Hilbert space. Therefore, the functional SMS (J^) is a stabilizing functional. This functional has an important property: it minimizes the total area with nonzero departure of the model parameters from the given a priori model. Thus, a dispersed and smooth distribution of the parameters with all values different from the a priori model niapr results in a big penalty function, while a well-focused distribution with a small departure from niapr will have a small penalty function.


49

Figure 2-4 Illustration of the principle of the nriinimum gradient support inversion. A smooth inversion produces a smooth image of a true rectangular model, while the inversion with the minimum gradient support stabilizer generates a sharp image, close to the true model.

We can use this property of the minimum support functional to increase the resohition of blocky structures. To do so, we modify SMS (J^) and introduce a minimum gradient support functional as SMGS (m) = sp

[Vm]

L

Vm • Vm zdv. Vm • Vm 4- /^^

(2.66)

We denote by spt Vm the combined closed subdomains of V where V m j^ 0. We call sptVm a gradient support. Then, expression (2.66) can be modified: SMGS (m) -= spt Vm -

p^

1 ::dv. spt Vm Vm • Vm + /3^

(2.67)

From the last expression we can see that SMGS (m) - ^ spt Vm,

if P

0.

(2.68)

Thus, the functional SMGS (i^) can really be treated as a functional proportional (for a small /3) to the gradient support. This functional helps to generate a sharp and focused image of the inverse model. Figure 2-4 illustrates this property of the minimum gradient support functional. Repeating the considerations described above for SMS (ni), one can demonstrate that the minimum gradient support functional satisfies the Tikhonov criterion for a stabilizer.

50


Another approach to selecting inverse models with sharp boundaries was considered by Ramos et al. (1999), who introduced a minimum first-order entropy method based on the following stabilizer: /• | V m | + / ? ^ _ | V m | - f / ?

/.

^minentrl \J^)

Geophysical inverse theory and regularization problems

Regularization of inverse problems

Introductory geophysical inverse theory