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
2.
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
ELSEVIER Amsterdam- Boston- London - New York-Oxford- Paris-Tokyo San Diego- San Francisco- Singapore- Sydney
E L S E V I E R SCIENCE B.V. Sara Burgerhartstraat 25 P.O. Box 211, 1000 A E Amsterdam, The Netherlands
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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
This Page Intentionally Left Blank
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
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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
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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.
Formulation of forward and inverse problems for different geophysical fields
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
-^IIL
^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^'^''-
^^-^^^
Formulation of forward and inverse problems for different geophysical fields
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<^{Ap) = 7 JJJ^ Ap{v)j^^^dv.
(1.12)
Thus, the forward problem for an anomalous gravity field consists in computing the integral (1.12) over the domain D filled with anomalous mass, i.e. Ag =
A^Ap).
The main goal of gravity prospecting is to reconstruct the anomalous density distribution Ap(r) from the data, Ag, given on the surface of the earth: Ap=(A')-'(A.<7), where (A^)~^ is an operator of the inverse gravity problem (inverse gravity operator). The density distribution provides unique information about deep geological structures. It is typically used in the stage of a regional geological study of the earth's crust. However, highly precise gravity surveys can be used for detailed oil, gas, and mineral prospecting to contour the location of mineral deposits. Figure 1-1 presents an example of anomalous gravity field distribution over a typical massive ore deposit. 1.1.2 Magnetic field Another major geophysical method, magnetic prospecting, is based on measuring an anomalous magnetic field which is related to the anomalous magnetization distribution within the earth's crust. Magnetic properties of rocks vary over a wide range for different rock formations. Magnetized materials are characterized by the magnetization vector I, which is proportional to the applied magnetic field Hy^: I =XmH^,
(1.13)
where Xm is magnetic susceptibility. Note that the magnetic susceptibility of purely diamagnetic materials is negative, and in paramagnetic materials Xm i^ positive. In other words, in diamagnetic materials the induced magnetic field tends to reduce the applied field, while in paramagnetic materials the induced field tends to increase the applied field.
Forward a n d inverse p r o b l e m s in g e o p h y s i c s
U 100 200 Q
\-300
Figure 1-1
i^^^p^^y^j^^
Anomalous gravity field profile over a massive ore body.
Most minerals behave as diamagnetic or paramagnetic materials. This property depends on the behavior of the rotating electrons within the atoms of different minerals, which align themselves with or against an applied magnetic field. This field may be the earth's ambient magnetic field. A few minerals exhibit ferromagnetism, an intense magnetic polarization which occurs in some materials when large groups of atoms, termed magnetic domains, orient themselves in unison, causing a large enhancement of the magnetic field. The most common ferromagnetic mineral is magnetite, and so, in most cases, the magnetic susceptibility of a rock is determined by its magnetite content. The susceptibility of ferromagnetic minerals can be as large as 10^, so, in ferromagnetic materials the applied field can increase dramatically. These minerals are often associated with ore deposits. That is why magnetic anomalies are often good indicators of mineral deposits. Assume that the volume D is filled now with magnetic masses with the intensity of magnetization I(r). It is well known that the corresponding magnetic field H(r') can be represented as follows (Zhdanov, 1988):
H(r') = ^^(I) = V Iff I(r) • V'—^Av,
J J JD I ^ ~ '^ I where A^ is an operator of the forward magnetic problem. The inverse problem in this case can be described as the determination of the intensity of magnetization I(r) from the observed magnetic data:
Formulation
of forward and inverse problems for different geopfiysical
Ai
LB
Ml
fields
N
^k^^^JliW^
Figure 1-2 Current AB and receiver MN electrodes configuration in the typical DC geophysical survey.
The important common feature of both gravity and magnetic forward problems is that operators A^{p) and ^ ^ ( I ) are hnear operators. Therefore the inverse problem operators {A^)''^{g) and (74^^)~^(H) are also linear. We will study carefully the properties of linear operators in Appendix B. We will see below that linear inverse problems have many important features that simplify their solution. 1.1.3
Electromagnetic field
Electromagnetic geophysical methods are based on the study of the propagation of electric currents and electromagnetic fields in the earth. There are two major modifications of electromagnetic methods: one is based on direct current (DC), and another employs a transient electromagnetic field. DC methods, or resistivity methods, involve injecting electric current in the earth by a system of current electrodes and measuring the electrical potential with receiver electrodes. In practice, it is more convenient technically to use a low frequency (below 10 Hz) current, which propagates inside the earth practically like a direct current. DC surveys are used to determine the resistivity of rock formations. Resistivity is a very important physical parameter that provides information about the mineral content and physical structure of rocks, and also about fluids in the rocks. In a DC survey the current and receiver electrodes are placed on the surface of the earth (Figure 1-2) or in a borehole (Figure 1-3). In both cases the electric potential distribution recorded by the receiver electrodes is used to map the spatial resistivity distribution of the rock formation. The main limitation of the resistivity method is that direct current cannot penetrate through resistive formations. Electromagnetic induction methods, based on transient electromagnetic fields, overcome this difficulty because a transient field can easily propagate through resistors like a radiowave propagates in the air. At the same time.
Forward and inverse problems in geophysics
10
^y777
A
A M
N
{y Figure 1-3 Current AB and receiver MN electrodes configuration in the resistivity well-logging.
a transient electromagnetic field provides information not only about the resistivity of rocks, but also about two other important electromagnetic parameters: magnetic permeability, //, and the dielectric constant, e. Electromagnetic induction methods can be used for ground, airborne, seabottom and borehole observations. There are many different electromagnetic survey configurations. The advantage of the electromagnetic technique is that one can generate a transient field using different types of sources (transmitters). For example, one can use different systems of electric bipoles or loops of wire as transmitters. We can also measure electric or magnetic components of induced electromagnetic fields using receiver electrodes or receiver loops. Figure 1-4 presents a typical electromagnetic survey formed by a paired horizontal transmitter loop and receiver loop, moving together along a profile on the surface. Figure 1-5 shows another typical survey configuration with a fixed transmitter loop and multiple receivers. The transmitter loop in these surveys generates a transient electromagnetic field penetrating the earth. The receivers measure the total field formed by the primary signal in the transmitter and a scattered signal from the internal structures of the earth. The main goal of these observations is to map the spatial resistivity distribution inside the earth plus magnetic permeability, /i, and dielectric constant, e. This problem can be solved by inverting the observed data into electromagnetic parameters of the examined media. Another typical electromagnetic survey is used in an airborne system. In this case the transmitter loop is located on the plane while the receiver loop is positioned
Formulation
of forward and inverse problems for different geophysical
Tx
Figure 1-4
J
Rx
Tx
T
Rx
Tx
T
fields
RX
Moving transmitter loop (Tx) - receiver loop (Rx) EM induction survey.
Rx J
RX
2 Rx ^ Rx^
Rx ^
/ - y -EJ-EB- EM3 -£3-1
Figure 1-5
Fixed transmitter (Tx) EM induction survey.
11
12
Forward and inverse problems in geophysics
Tx
Figure 1-6
Airborne EM induction survey.
on the "bird" flying a few hundred meters behind the plane (Figure 1-6). The airborne observations can cover huge prospecting areas. The transmitter loop generates a monochromatic signal or sends repetitive electromagnetic pulses. Interpretation of electromagnetic data collected by the receiver is another typical inverse geophysical problem. The basis of the theory of electromagnetic fields studied by geophysicists is provided by Maxwell's equations (Stratton, 1941; Zhdanov and Keller, 1994): V xH=j-Hr-K
(1.15)
V XE V - B = 0, V-'D^q
(1.14)
+ q",
(1.16) (1.17)
where E and D are the vectors of the electric field; H and B are the vectors of the magnetic field; j is the conduction current density; q is the spatial density of free
Formulation of forward and inverse problems for different geophysical fields
13
electric charges; j ^ and q^ are the densities of extraneous electric currents and charges (in the transmitter), interrelated by the continuity equations:
-•y-%^
(MS,
Equations (1.14) through (1.17) should be supplemented by constraint equations reflecting the electromagnetic properties of a medium, which are, for linear and isotropic media, as follows: D =eE, B =/iH,
(1.19)
where e and /i are the dielectric constant (permittivity) and the magnetic permeability, respectively. Note that magnetic permeability is related to magnetic susceptibility Xm by a simple formula: M = Mo(l + Xm),
(1-20)
where /iQ = 47r X 10"'' Henry/meter is the free space magnetic permeability. The conduction current density is expressed in terms of the electric field by means of Ohm's law (in a differential form): J ='TE,
(1.21)
where a is the electric conductivity of a medium which is the reciprocal of its resistivity p , (7 - 1/p. We can sec from Maxwell's equation that the electromagnetic properties of the media are characterized by three electromagnetic parameters: conductivity a, dielectric permittivity £:, and magnetic permeability p,. Therefore, the forward problem can be described by the following operator equation:
{E,n} =
A'"'{a,e,fi},
where A^"^ is an operator of the forward electromagnetic problem. Note that this operator is, in general cases, a nonlinear one. An inverse problem in an electromagnetic case involves determining electromagnetic parameters of the media cr,£:,/x from the observed electromagnetic field E,H: {a,5,/i} = ( y l ^ - ) - ^ { E , H } . This inverse problem is a nonlinear one that makes the inversion of electromagnetic data a challenging problem in geophysics.
Forward and inverse problems in geophysics
14
Seismic wave source
Seismic receivers i^eophones) A 4 A A AA
Figure 1-7
A typical reflection seismic survey.
1.1.4 Seismic wavefield Seismic methods are based on the study of elastic wave propagation inside the earth. These waves are generated by explosions on the earth's surface (in seismic exploration), or by seismic sources in the earth's crust (in seismology). The seismic receivers - geophones - record on the surface of the earth the "echo" signal reflected from deep geological structures (Figure 1-7). The spatial structure of a seismic signal depends on the velocity of elastic wave propagation, which is a function of the physical parameters of rock formations. In the case of a layered model of the earth, one can use a simple technique of geometrical seismics, which is based on studying the geometry of rays of seismic wave propagation. In more complicated geological structures, comprehensive imaging and inversion methods must be used to analyze seismic data. In order to develop these methods, one should study carefully the physics of seismic waves. The simplest model of these waves is one based on acoustic principles. Assume that the earth can be treated as an acoustic medium and the influence of variations in density can be ignored. In this case the propagation of seismic waves in the earth can be described by the acoustic wave equation: V'P(r,t)-
1 9^ P(r,t) = -F(r,t), c2(r)9t2
(1.22)
where P ( r , t) is a pressure field, F{r,t) is the strength of a source of energy, and c(r) is the velocity of wave propagation. The forward problem in this case is formulated as the solution of the differential equation (1.22) with respect to P ( r , t ) for the given velocity c(r) :
A'ic),
(1.23)
Formulation
of forward and inverse problems for different geophysical fields
15
where A^ is the acoustic forward modeUng operator. The inverse problem consists in reconstructing the velocity distribution from the observed pressure field: c=(A'')-\P). (1.24) Both operators of forward A^ and inverse (A^)"^ problems are nonlinear operators. The analysis of a seismic wavefield can be significantly simplified in the frequency domain: 1
r+oo
27r7_oo
Equation (1.22) in this case is reduced to the Helmholtz equation: V^Kr,^) + ^P{r.uj)
= -/(r,c^),
(1.25)
where p{r,uj) and f{r,cu) are the Fourier spectra of P{r,t) and F{r,t),
OO
f
+ 00 iu)t ^
Numerical solution of the Helmholtz equation for a given velocity distribution describes the forward problem, while the inverse problem is actually aimed at determining the coefficients (velocity c(r)) for the given pressure field p(r,cj). Both of these problems are nonlinear. Note that often inverse seismic problems are formulated not for the velocity but for the slowness, which is the inverse velocity: s(r) = l / c ( r ) . In general cases of elastic media, the seismic field equations will be much more complicated than acoustic wave equations. We will study these equations in Chapter 13. Note in conclusion that due to the high resolution of the seismic method, it is one of the leading geophysical methods, especially in oil and gas prospecting. We can now summarize the operator equations for different geophysical forward and inverse problems: Gravity field Forward problem: g = A^ip)-
(1.26)
P={A^r'{g).
(1.27)
Inverse problem:
Forward and inverse problems in geophysics
16
Magnetic field Forward problem: H = A^(I).
(1.28)
I=(^^)-i(H).
(1.29)
Inverse problem:
Electromagnetic field. Forward problem: {E,li}
=
A^^{a,e,^i}.
(1.30)
Inverse problem: {a,e,M} = ( A n ~ ' { E , H } .
(1.31)
Seismic wave field Forward problem: P = ^"(c).
(1.32)
Inverse problem: c=
{A'^)-\P).
(1.33)
To develop an efficient method of solving geophysical inverse problems, it is important to better understand the properties of these operators and to study the general principles of inverse problem solution. We will discuss this issue in the next chapter. Before moving to the theoretical foundations of inversion, let us take a quick look at the questions of the existence, uniqueness, and stability of the geophysical inverse problems. 1.2
Existence and uniqueness of the inverse problem solutions
1.2.1 Existence of the solution We can write the operator equations of forward modeling (1.26), (1.28), (1.30), and (1.32) in a general form: d = A{m),
(1.34)
where d denotes different geophysical data (gravity, magnetic, electromagnetic, and seismic), and m denotes corresponding model parameters (density p, magnetization I, conductivity cr, dielectric permittivity e, magnetic permeability /i, and velocity c distributions).
Existence and uniqueness of the inverse problem solutions
17
The question of the existence of the inverse problem solution has two aspects. One is the physical existence of some geophysical parameter distribution that generates observed data, and another one is the existence of the mathematical solution of the operator equation (1.34). There is no doubt about the physical existence of the inverse problem solution; however, the mathematical existence can be questionable. To understand this phenomenon better, note that measured geophysical data d^ always contain certain errors ^d : d^ = d + 6d. The question is whether it is possible to find such a model m,^ that generates exactly the observed noisy data d^: ds = A{ins). (1.35) The answer is that sometimes we can't find such a model, and it is easy to understand why. Really, one should remember that the noise has no relation to the real model parameters; it can be generated for reasons that have no common ground with the geophysical field equations described above. Therefore, noise cannot even be described by the same operator equation (1.34) as the theoretical geophysical data. For this very reason we should not expect that we can always find a physically meaningful model that would fit exactly the observed data. But why should we even try to find such a model? Usually an inverse problem solution is sought in the class of simplified models. Hence the question must be about the quasi-solution of the inverse problem, i.e. about such a solution from a chosen model class that fits the observation results best. Thus, we arrive at an idea of practical existence: solution of the inverse problem exists if there exists m^ such that \\ds-A{ms) \\<S, where S is the measurement error, and ||...|| denotes some measure of diff"erence between the theoretical (predicted) data ^(m^^) and observed noisy data d^. In Appendix A we study the measure ||...|| in detail. However, now it is important to understand that we should not even try to get an exact solution of the inverse problem. It has no practical meaning for noisy data (and the data are always noisy!). So we should think about some approximate approach to the inversion, based on searching for a model that would fit the observations within the given accuracy 6. This simple idea is the keystone of regularization theory. 1.2.2 Uniqueness of the solution Another important question, which we outlined above, is the uniqueness of the solution. Again, it seems that mother nature dislikes simple solutions: as a rule, the solutions of geophysical inverse problems are not unique! In analysing the non-uniqueness problem, we should distinguish between the two classes of inverse problems we introduced above: the inverse model (or inverse scattering) problem and the inverse source problem. The advantage of the inverse
18
Forward and inverse problems in geophysics
scattering problem is that one can choose many different positions of the source to illuminate the medium under examination. Each source position will generate a different data set that, as a rule, reduces the ambiguity of the inverse problem solution (Blok and Oristaglio, 1995). The inverse source problem is more ambiguous because there usually exists a source distribution that generates a zero external field. This type of source is called a nonradiating source. The detailed analysis of the analytical properties of the nonradiating source was given in a classical paper by Bleistein and Cohen (1976). Following this paper, we can easily demonstrate this fact as applied to inverse source problems for an acoustic field. Let us assume that we are seeking a nonradiating source for an acoustic wave equation: V2p(r, t) - -Jr-r^P{r, c2(r)dt^
t) = - F ( r , i ) .
(1.36)
Let W{T, t) be a function which is zero outside some finite domain DQ: W{r,t)=0, iir^Do.
(1.37)
We apply the wave operator to W{r, t) : V'W^(r, t) - - i - ^ ^ H ^ ( r , t) = -iV(r, t).
(1.38)
Let this new function N[r, t) be a source of some wavefield. The solution of the wave equation with this new source will be W{Y,t) itself, which is zero outside a domain DQ. Therefore, the source, A^(r, t)^ is nonradiating. In a similar way we can construct an infinite number of nonradiating sources. Adding any of these sources to the given source, F(r,^), will not change the observed pressure field. Thus, the inverse source problem can have an infinite number of equivalent solutions. Fortunately, the situation is different for inverse model (or inverse scattering) problems. There are many favorable situations when inverse geophysical problems happen to be unique. These situations are outlined by corresponding uniqueness theorems. For example, I list below some important uniqueness theorems of geophysics. Theorems of uniqueness A. Gravity field The inverse gravity problem is actually an inverse source problem, because the masses are the sources of the gravity field. Thus, it may have many equivalent solutions. For example, all spherical material balls with the same center and the same total mass (but with different radii) produce the same gravity field (outside these balls). Obviously, the densities Pi and P2 of ^^e balls Bi and B2 with the different radii Ri and i?2 must be different to produce the same total mass value M: 4.77-
M = P,—R\
4-7r
= P2-jRl.
(1.39)
Existence
and uniqueness of the inverse problem solutions
Figure 1-8
19
Non-uniqueness in a gravity inverse problem.
If one considers the material spherical body obtained by subtracting the smaller ball Bi from the larger ball B2, this spherical body will generate a zero external gravity field (Figure 1-8). Note that the distribution of the density Ap within this spherical body will be:
Therefore, the body will consist of two spherical layers: the inner layer (ball) with a negative density, and the outer layer with a positive density*. Thus we come to an idea of the existence of a density distribution that generates a zero external gravity field. This idea is the centerpiece of the non-uniqueness principle for gravity field inverse problems. One can add this kind of density distribution to any given density model and obtain another model generating the same gravity field. Unfortunately, examples of the density distribution generating a zero gravity field are not limited to the combinations of the spherical material layers. There are much more complicated distributions of density that do not generate a gravity field. These distributions are "invisible" to observations of anomalous gravity fields. The general mathematical theorem states that any density distribution p(r) *Note that, in reality, the density is always positive. However, in geophysical applications we usually consider an anomalous density distribution which can be both positive and negative. In this case, the model of two spherical layers described above can generate a zero gravity anomaly.
20
Forward and inverse problems in geophysics
that satisfies the equation
L'
h(r)p(T)dv = 0,
(1.40)
for any harmonic function^ /i(r) in D generates a zero external gravity field. It is assumed in formula (1.40) that the support of mass £) is a star-type local domain. In other words, the boundary S oi D can be uniquely described in some spherical system of coordinates (r, (f, 9) by the equation: S : r = r {(p^9). Novikov (1938) proved the uniqueness theorem for a star-type body with the given homogeneous density distribution p{r) = p^. The theorem states that if it is known that the gravity field is generated by a star-type body with the a given constant density, the gravity inverse problem has a unique solution. In the case of spherical balls, this result is obvious: according to formula (1.39), if Pi = P2y then Ri = i?2B. Electromagnetic field There are three famous uniqueness theorems for the electromagnetic inverse problem (Berdichevsky and Zhdanov, 1984): 1) Tikhonov's theorem of uniqueness for a one-dimensional (1-D) model (Tikhonov 1965); 2) Weidelt's theorem of uniqueness for a two-dimensional model with an electrical conductivity described by an analytic function (Weidelt, 1978); 3) Gusarov's theorem of uniqueness for a two-dimensional model with a piecewise analytic function of electrical conductivity (Gusarov, 1981). These three theorems rest on the condition that the field is known over an entire frequency range 0 < u; < CXD. I present here a brief overview of these theorems without going into the mathematical details of their formulation and proof. Tikhonov's theorem considers a 1-D (horizontally layered) geoelectrical model with plane electromagnetic wave excitation (Figure 1-9). Assume that in the frequency domain we observe simultaneously the electric and magnetic field components of a plane electromagnetic wave propagating vertically in a 1-D conductivity model of the earth {a = cr{z)). The observed field components are known over an entire frequency range 0 < a; < oo. In this case one can reconstruct uniquely the conductivity distribution a (z) from the observed data. This theorem forms the basis of one of the most important electromagnetic geophysical methods - magnetotelluric sounding (MTS) (Zhdanov and Keller, 1994). ^The function h (r) is harmonic in D if it satisfies the Laplace equation in D : V^/i (r) = 0.
Existence and uniqueness of the inverse problem solutions
21
u u u u uu E^(m), Ey(m), H^ (m), Hy(m) ^
X
•
^ >•/
> 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/^a:^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: V^U = 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
Instability of the inverse problem solution
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
where H is the distance from the observation level 2; = 0 to the sources of the gravity field. This problem is called a downward analytical continuation of the gravity potential. We can write an operator equation which connects the potential at a level z = h with the potential at the observation line, z = 0, U{x,y,0)
= A[U{x,y,h)].
(1.45)
The application of operator A is usually called an upward analytical continuation of the gravity potential. We can treat this problem as the forward problem, while the downward analytical continuation (toward the gravity masses) is an inverse problem. Let us demonstrate that small noise added to the observed values of the gravity potential could result in unpredictably big variations of the potential at a level h (inverse problem (1.45) solution). Let us assume that the known function f{x,y) is given by the formula: f{x,y)
= a s'm (kj^x) sin {kyy) ,
(1-46)
where a, k^, ky are some coefficients. It is easy to find that the solution of the Laplace equation (L44) with the boundary conditions (L46) can be described by the formula U{x, y, z) = ae^^^^ sin (k^x) sin {kyy) ,
(1-47)
where no = ^/k^ -\- k'^. In fact, direct substitution of the expression (1.47) into (1.44) shows that this formula satisfies the Laplace equation, and U{x,y,z
= 0) = a sin {kj,x) sin {kyy) = f{x,y).
(1-48)
Now, we assume that we have two different observations (two different boundary conditions) at a level z = 0: fi{x,y)
= a sin (kj^x) sin {kyy),
(1-49)
and /2(x, y) = (a -h e) sin {k^^x) sin {kyy),
(1.50)
where e > 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
Forward and inverse problems in geophysics
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.
Instability of the inverse problem solution
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
Forward and inverse problems in geophysics
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 - Amo^o = 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^^ii j - sup I
ii^^ii
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-m^A;;,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
Ill-posed problems and the methods of their solution
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
Ill-posed problems and the methods of their solution
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 ^A; ^ 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)<6.
(2.18)
Definition 7 A quasi-solution of the problem (2.14) in the correctness set C is an element ms G C which minimizes the distance iij){Am^ds), i.e.: lij^{Ame,ds)
= inf / i ^ ( ^ m , d ^ ) , meC
where inf (/? denotes the greatest lower bound of the variable ^.
(2.19)
Formulation
of well-posed and ill-posed
35
problems
m=A-'(d)
.s
d d^d+M
m^ =A^
(d^)
Figure 2-1 A quasi-solution, ms, is selected from the correctness set C under the condition that its image, A (m^), is the closest element to the observed noisy data, d^, from the subset AC : /ij^(Ani<5, d^) = inf^^c/i^(v4m, d^) Obviously, we can reach the minimum of the /.i^{Am, ds) in C, if the correctness set is a compact. In this case the quasi-solution exists for any data ds. Figure 2-1 illustrates the definition of a quasi-solution. The element m € M is an exact solution of the inverse problem d = A{
(2.20)
Subset AC of the data space D is an image of the correctness set C obtained as a result of the application of operator A. A quasi-solution, ni<5, is selected from the correctness set C under the condition that its image, A (ms), is the closest element in the subset AC to the observed noisy data, d,^. It can be proved also that the quasi-solution is a continuous function of d^. Indeed, let us consider the triangle inequality /j.j^{Ains, d) < iJ.jj{Ams, ds) + /J^oi^s^ d)-
(2.21)
According to the definition of the quasi-solution and condition (2.18), it follows from inequality (2.21) that fijj{Ams,d) < 28. (2.22)
36
Ill-posed problems and the methods of their solution
Based on Tikhonov's theorem, we know that the operator A~^ is a continuous one on the correctness set C. Therefore, we conclude from (2.22) that the quasi-solution is a continuous function of d^. Note that this property holds only in the correctness set C. If one selects a solution, m^, from outside the correctness set, it may be no longer a continuous function of the data (see Figure 2-1). The idea of the quasi-solution makes it possible to substitute the inverse problem solution by minimization of the distance / i ^ ( ^ m , d^) in some appropriate class of suitable models. The standard method of functional minimization can be used to solve this problem and, therefore, to find the quasi-solution. In this way, we significantly simplify the inverse problem solution.. However, this approach is effective only if we know a priori the corresponding class of the models (the correctness set) where we are searching for the solution. In many situations, it is difficult to describe this class completely, or we prefer not to restrict ourselves to some specific class. In this case we have to use a more general approach to a stable solution of the inverse problem. 2.3
Foundations of regularization methods of inverse problem solution
2.3.1 Regularizing operators Let us consider the inverse geophysical problem described by the operator equation d = A(m),
(2.23)
where m represents model parameters, and d is observed geophysical data. In general cases, the inverse operator A~^ is not continuous and, therefore, the inverse problem (2.23) is ill-posed. The main idea of any regularization algorithm is to consider, instead of one ill-posed inverse problem (2.23), a family of well-posed problems, d = ^a(m),
(2.24)
which approximate the original inverse problem in some sense (Strakhov, 1968, 1969a,b). The scalar parameter a > 0 is called a regularization parameter. We require also that nia—> mt, if Q -^ 0,
where nia = ^a^ (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 Z^iFigure 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.
Ill-posed problems and the methods of their solution
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<5, produces stable results: the inverse models m^ and m^ are close to each other and to the true solution, if the observed data, ds and d<5, are close to each other and to the true data d^.
39
Foundations of regularization methods of inverse problem solution 2.3.2
Stabilizing
functionals
A stabilizing functional (or a stabilizer) is used to select from t h e space M of all possible models t h e subset Mc, which is a correctness set. D e f i n i t i o n 1 0 A nonnegative functional s{ni) in some metric space M is called a stabilizing functional if, for any real number c > 0 from the domain of functional values, the subset Mc of the elements m E M , for which s{m) < c, is compact. We will give now several examples of stabilizing functionals. E x a m p l e 11 Let us consider a real Hilbert space L2 formed by functions m the interval [a, h] (see Appendix A). The metric in the space L2 is according to the formula /i(nii,ni2) = < /
[mi (x) — 7712 (x)] dx\
.
integrable determined
(2.26)
It can be proved that any ball, b (nio,c) = { m : /i(m, nig) < c, c > 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 ) = <^ /
Vgfc(x)
d^Am{x) dx^
dx
where Am{x) = mi{x) — m2{x); and qo{x), qi{x), ....,qp{x) are the given real functions (qp{x) 7^ 0^. We can introduce a stabilizing functional in the space Wf as follows: 5(m) = /iv^p(m,mo), where mo is any given model from M — W^, and the sphere 5(m) = / / ^ p ( m , m o ) < c IS compact.
40
Ill-posed 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
Let us analyze now more carefully how one can use a stabilizer to select an appropriate class of the models^ Assume that the data d^ are observed with some noise d^= d^-f (5d, where dt is the true solution of the problem. In other words, we assume that the misfit (distance) between the observed data and true data is less than the given level of the errors, 5, in the observed data, /i^(d,,d,)<(5,
(2.29)
where 6 = ||<5d||. In this situation, it is natural to search for an approximate solution in the set Q^ of the models m such that /i^(A(m),d,)<(5.
(2.30)
Thus, (3<5 C M is a set of possible solutions. The main application of a stabilizer is to select from the set of possible solutions Qs the solutions that continuously depend on the data and which possesses a specific property depending on the choice of a stabilizer. Such solutions can be selected by the condition of the minimum of the stabilizing functional: 5(m; m G Q<5) = min .
(2.31)
We have introduced a stabilizing functional under the condition that it selects a compact subset Mc from a metric space of the model parameters. Thus, we can say that a stabilizer from a set of possible solutions, Qs^ selects a solution, which at the same time belongs to the correctness set Mc- Figure 2-3 helps to explain this role of the stabilizing functional. The existence of the model, minimizing (2.31), was demonstrated by Tikhonov and Arsenin (1977). We will denote this model as m^: ^(m^; m^ G Qs) = min .
(2.32)
One can consider a model m<5 as the result of an application of the operator R{ds,6) to the observed data d^, depending on the parameter 8\ m, =
fi(d,,(5)
(2.33)
We will prove the following theorem: Theorem 13 The operator R{ds, 6), introduced by formula (2.33), is the regularizing operator for the equation (2.23), and m^ can he used as an approximate solution of the inverse problem (note that in this case a = 6, while in general cases a = a{6)). "^Here we follow mainly the exposition given by Tikhonov and Arsenin (1977) and Lavrent'ev et al. (1986).
Foundations of regularization methods of inverse problem solution
41
d = A (m)
m ^=R(d g,6)
eg={|Li^(Am,jp<5} s (m ^\ m^€
Jo = d -\- 8d
2g ) = ynin
Figure 2-3 The stabilizing functional selects from a set of the possible solutions, Qs, a solution, in^, which at the same time belongs to the correctness set Mc-
42
Ill-posed problems and the methods of their solution
Proof: First, it is shown above that m^ is a continuous function of d^; therefore, R{ds^5) is a continuous operator. Let us assume now that condition (2.25) for the regularizing operator is not satisfied. This assumption requires that for some £o > 0 there exist sequences d^ and a^ such that fij^{ma^,mt)
= /J.M{R{di,a^),mt)
> So, when a , - ^ 0,
(2.34)
where nia- = R{di,ai). On the other hand, based on the definition (2.32) of 111^. = R{di, a^), we can write 5 ( m , J < 5 (m,), (2.35) whence we see that the sequence 111^. belongs to a compact set. It follows that we can find the limit of the sequence 111^^ when a^ —> 0 : ma, -^ m.
(2.36)
At the same time, according to the definition of 111^^ and inequality (2.29) one can write the following inequality
//^ (^ (m^J , mO
(2.37)
Therefore, in the limit a^ —> 0, we have fij^{A{m),mt)
=0,
which contradicts assumption (2.34). Thus, in the framework of the approach we have developed, the problem of the solution of equation (2.23) with approximate left hand part d^ can be reduced to the problem of minimization of the stabilizing functional on the set Qs: s{m; m G Qs) = min, where Qs = {m; /i^(A(m),d,) < 6}.
(2.38)
2.3.3 Tikhonov parametric functional It has been proved by Tikhonov and Arsenin (1977) that, for a wide class of stabilizing functionals, their minimum is reached on the model ms such that /i^(yl(m^), d^^) = S. Thus, we can solve the problem of minimization (2.31) under the condition that lijj{A{ms).ds)=6.
(2.39)
In other words, one should consider the problem of minimization of the stabilizing functional (2.31), when the model m is subject to the constraint (2.39). A common
43
Foundations of regularization methods of inverse problem solution
way to solve this problem is to introduce an unconstrained parametric P " ( m , d^), m G M, given by P"(m, d,) = fil{A{m),ds)
functional
+ as{m),
(2.40)
and to solve the problem of minimization of this functional: P^(m,d^) = min.
(2.41)
Functional /j,jj{A{in), ds) is often called a misfit functional. Thus, the parametric functional P^{in, d^) is a hnear combination of the misfit and stabihzing functionals, and the unknown real parameter a is similar to the Lagrangian multiplier. It is determined under the condition HoiA{m„),d6) = 6,
(2.42)
where 111^ is the element on which F"(m,d^) reaches its minimum. The functional P " ( m , d^) is called the Tikhonov parametric functional. Theorem 14 Let operator A he a continuous operator from M to D. For any d ^ D and any parameter a > 0, there is a model nic G M, on which the functional P^(m, d) - fil{A{in),d)
+ as{m)
(2.43)
readies its lower hound, mf^^MP''(m^d)
= P"(m,„d).
(2.44)
Proof: There is an exact lower bound of the parametric functional inf P " = PQ\ because for any m, P " > 0. Therefore, we can select the sequence of models {m^^} with the property lim„^ooP° = 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
Ill-posed problems and the methods of their solution
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 inf^eMP"(m,d) = lim,^ooP"(m^,d) = lim^^ocP"(m^^),d) = l™A:^oo[M^(^(m^^O,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
Family of stabilizing functionals
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
Ill-posed problems and the methods of their solution
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^^^ai^'^^ 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.
Family of stabilizing functionals
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
Ill-posed problems and the methods of their solution
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^)
Jv
where
"•^^-^r^"'.
(2«»)
Q' = [ {\Vm\^P)dv. Jv Note, that, in principle, we can construct the stabilizers using different monotonic functions of the model parameters or their gradients, for example, exponential functions: = / exp {\mf) dv, (2.70) JV
or 5,expVm
(m) = / exp (Vm • Vm) dv. JV
(2-71)
We just have to check every time that the corresponding functional satisfies all necessary conditions for a stabilizer. 2.4-2 Representation of a stabilizing functional in the form of a pseudo-quadratic functional Note that all stabilizing functionals introduced above can be expressed as pseudoquadratic functionals of the model parameters: 5(m) = {We (m - rriapr), VK^ (m - mapr))^^ \we (r) (m (r) - m^p, (r))|' dv,
(2.72)
IV
where We is a linear operator of multiplication of the model parameters function m (r) by the function We (r),which may depend on m. If the operator We is independent of m (r) , we obtain a quadratic functional, like the minimum norm (2.53) or the maximum smoothness (2.54) stabilizing functionals. In general cases, the function We may even be a nonlinear function of m, like the minimum support (2.61) or minimum gradient support (2.66) functionals. In these cases, the functional 5(m), determined by formula (2.72), is not quadratic. That is why we call it a "pseudo-quadratic" functional. However, in Chapter 5, we will show that presenting a stabilizing functional in a pseudo-quadratic form simplifies the solution of the regularization problem, and makes it possible to develop a unified approach to regularization with different stabilizers. For example, the maximum smoothness stabilizer is expressed by formula (2.72) if rUapr = 0 Q^nd ^;e (r) =: w , — ( r ) =
/ ' [im? (r) + e^J '
(2.73)
51
Family of stabilizing functionals
where, ultimately, we shall let e —> 0. In the case of the TV stabilizing functional, 5^Ty(^), we assume rriapr = 0, and the function We (r) in (2.72) is:
(m"= (r) 4- e^) ' In the case of the minimum support functional, SMS ( ^ ) , we have w, (r) = < - (r) =
— ^ —. [(m(r) - m a p r (r)) + e^J
(2.75)
For the minimum gradient support functional SMGS ('^) , we assume ma^^ = 0, and find w, (r) = 7i;f ^^ (r) =
Vm(r) ^ ,, -. [Vm (r) . Vm (r) + /?2]'/' [m2 (r) + e2] V2
(2.76)
And finally, for the exponential stabihzers (2.70) and (2.71), we have oxp {-K | m ( r ) | , w, (r) = « ; « - (r) = ^ ^, \ ' \ ' , [m2(r)+621^/2 or .;. (r) = « ; r ^ - (r) -
exp [^ Vm (r) • Vm (r)l ^ % , , ^ ^ ,./2 '
(2.77)
(2.78)
Similar expressions for w^ (r) can be easily derived for other types of stabilizing functionals. Using the pseudo-quadratic form (2.72) of stabilizing functionals, we can present the corresponding parametric functional (2.43) as P " ( m , d) = (^(m) - d, yl(m) - d ) ^ + a {W, (m - m^p.) , 14^^ (m - m,^,))^^ . (2.79) Therefore, the problem of the minimization of the parametric functional introduced by equation (2.79) can be treated in a similar way to the minimization of the conventional Tikhonov parametric functional. The only diff'erence is that now we introduce some variable weighting operator Wg, which depends on the model parameters. We will discuss in Chapter 5 the different practical techniques of minimizing the parametric functional (2.79). Thus, we can see that minimization of the stabilizing functionals can impose diff'erent conditions on the class of model parameters. One case (e.g., maximum smoothness or minimum entropy stabilizers) requires a smooth distribution of the model parameters. Another case (e.g., a minimum gradient support stabilizer, or
52
Ill-posed problems and the methods of their solution
minimum first-order entropy stabilizer) imposes sharpening conditions on the model parameter distribution. As a result, by choosing one or another type of stabilizer, we can select different classes of inverse problem solutions. In other words, stabilizing functionals help to use a priori information about the desired properties of inverse problem solutions. This is the main role of stabilizing functionals in regularization theory. 2.5
Definition of t h e regularization p a r a m e t e r
2.5.1 Optimal regularization parameter selection The regularization parameter a describes the trade-off between the best fitting and most reasonable stabihzation. In a case where a is selected to be too small, the minimization of the parametric functional P"(ni) is equivalent to the minimization of the misfit functional; therefore we have no regularization, which can result in an unstable incorrect solution. When a is too large, the minimization of the parametric functional P^{m) is equivalent to the minimization of the stabilizing functional 5(ni), which will force the solution to be closer to the a priori model. Ultimately, we would expect the final model to be exactly like the a priori model, while the observed data are totally ignored in the inversion. Thus, the critical question in the regularized solution of the inverse problem is the selection of the optimal regularization parameter a. The basic principles used for determining the regularization parameter a are discussed in Tikhonov and Arsenin (1977). The solution of this problem can be based on the following consideration. Let us assume that data d^ are observed with some noise, d^ = d^ -h 5d, where dt is the true solution of the problem and the level of the errors in the observed data is equal to 6\ /i^(d,,dj<(5. (2.80) Then the regularization parameter can be determined by the misfit condition (2.49) /i^(>l(m,),d,)=(5.
(2.81)
To justify this approach we will examine more carefully the properties of all three functionals involved in the regularization method: the Tikhonov parametric functional and the stabilizing and misfit functionals. Let us introduce the following notations: p{a) = P'^(mc^,d^), parametric functional, s{a) = s{ma), stabilizing functional, i{a) = /i|,(yl(ma),d^), misfit functional. We examine some properties of the functions p(ce), z(a), 5(a). Property 1
(2.82)
53
Definition of the regularization parameter
Functions p{a),i{a)^s{a) are monotonic functions: p{a) and i{a) are not decreasing and s{a) is not increasing. Proof Let a i < 0^2 and ik = i[oLk) = / i ^ ( ^ ( m c , J , d J , 5fc = s{ak) = 5(mc,J. The following inequality holds: P2 = i2+ <^252 > ^2 + <^l52,
(2.83)
P " H m , , , d , ) =22 + c^i52 > 2 i + c.i5i = p i = P"Umai,d^),
(2.84)
because ai < 0^2On the other hand
because rriQ,^ realizes the minimum pi of the functional P"^(m, d^). Thus from (2.83) and (2.84) we have (2.85)
P2 > Pi for a2 > au which means that p{a) is a monotonic function of a. Furthermore, F"^ ( m , , , d^) = 21 + ^2^1 > Z2 + a2S2 = P"^ ( m , , , d , ) ,
(2.86)
because Yn^^ realizes the minimum p2 of the functional P^2(m, d^). Subtracting the left-hand side of inequality (2.86) from the right-hand side of inequality (2.84) and the right-hand side of inequality (2.86) from the left-hand side of inequality (2.84), we obtain (ai - a2)52 > {ai - a2)si.
(2.87)
5i > 52.
(2.88)
Since ai < 0^2, From inequalities (2.84) and (2.88) it follows that i2-ii>
ai{si
and hence Z2 >
ii.
- S2)
Ill-posed problems and the methods of their solution
54
ao Figure 2-5
a;
Illustration of the principle of optimal regularization parameter selection.
Property 2 It can be proved that the functions p{a)^i{a),s{a) (if the element rUa is unique). Note, also, that p{a) —> 0 for a -^ 0,
are continuous functions
and p(0) = 0.
(2.89)
From the fact that i{a) + as{a) — p{a) -^ 0, for a —» 0, it follows that z(0) = 0.
(2.90)
Thus we have proved the following theorem.
Theorem 15 / / i{a) is a one-to-one function, then, for any positive number S < SQ — iijj{A{mo),dg) (where UIQ is some a priori model), there exists a{6) such that /i^(A(mc,(6)),d^)
=6.
Note that i{a) is a one-to-one function when element m^ is unique. It happens, for example, when ^ is a linear operator, D is a Hilbert space and 5(in) is a quadratic functional. Figure 2-5 helps in understanding of the principle of optimal regularization parameter selection. One can see that because of the monotonic character of the function z(a), there is only one point, QQ, where i{ao) = fi'j^{A{Ynao),d^) = 5^.
55
Definition of tfie regularization parameter
A
^ ^
OC
quasi-optimal
^
log i(a) Figure 2-6 L-curve represents a simple curve for all possible a of the misfit functional, z(a), versus stabilizing functional, s{a), plotted in log-log scale. The distinct corner, separating the vertical and the horizontal branches of this curve, corresponds to the quasi-optimal value of the regularization parameter a. Let us consider one simple numerical m e t h o d for determining t h e p a r a m e t e r a. Consider, for example, a progression of numbers: ak = aiq
k-i
/ c - 1,2,,
0 < g < 1.
(2.91)
For any number ak we can find the element ma^ minimizing P"'^(ITI, d^) and calculate the misfit i.iij{A{mc^),d^). T h e optimal value of the p a r a m e t e r a is t h e n u m b e r <^o = <^fcO) for which, with the necessary accuracy, we have the equality (2.92)
/i^(^(m«,J,d^) T h e equality (2.92) is called the misfit
condition.
2.5.2
parameter
L-curve
method of regularization
selection
L-curve analysis (Hansen, 1998) represents a simple graphical tool for qualitative selection of t h e quasi-optimal regularization parameter. It is based on plotting for all possible a t h e curve of t h e misfit functional, i(a),versus t h e stabilizing functional, s{a) (where we use notations (2.82)). T h e Lcurve illustrates t h e trade-off between t h e best fitting (minimizing a misfit) a n d most reasonable stabilization (minimizing a stabilizer). In a case where a is selected t o be too small, t h e minimization of the parametric functional P " ( m ) is equivalent t o t h e minimization of t h e misfit functional; therefore i{a) decreases, while s{a) increases.
56
Ill-posed problems and the methods of their solution
When a is too large, the minimization of the parametric functional P*^(m) is equivalent to the minimization of the stabilizing functional; therefore 5(0;) decreases, while i{a) increases. As a result, it turns out that the L-curve, when it is plotted in log-log scale, very often has the characteristic L-shape appearance (Figure 2-6) that justifies its name (Hansen, 1998). The distinct corner, separating the vertical and the horizontal branches of this curve, corresponds to the quasi-optimal value of the regularization parameter a.
References to Chapter 2 Acar, R., and C. R. Vogel, 1994, Analysis of total variation penalty methods: Inverse Problems, 10, 1217-1229. Constable, S. C , Parker, R. C , and G. G. Constable, 1987, Occam's inversion: a practical algorithm for generating smooth models from EM sounding data: Geophysics, 52, 289-300. Dmitriev, V. I., Editor in chief, 1990, Computational mathematics and techniques in exploration geophysics (in Russian): Nedra, Moscow, 498 pp. Giusti, E., 1984, Minimal surfaces and functions of bounded variations: BirkhauserVerlag, 240 pp. Hansen, C , 1998, Rank-deficient and discrete ill-posed problems. Numerical aspects of linear inversion: Department of mathematical modeling. Technical University of Denmark, Lyngby, 247 pp. Last, B. J., and K. Kubik, 1983, Compact gravity inversion: Geophysics, 48, 713-721. 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. Portniaguine, O , and M. S. Zhdanov, 1999, Focusing geophysical inversion images: Geophysics, 64, No. 3, 874-887. Ramos, F. M., Campos Velho, H. F., Carvalho, J. C , and N. J. Ferreira, 1999, Novel approaches to entropic regularization: Inverse Problems, 15, 1139-1148. Rudin, L. L, Osher, S. and E. Fatemi, 1992, Nonlinear total variation based noise removal algorithms: Physica D, 60, 259 - 268. Smith, J. T., and J. R. Booker, 1991, Rapid inversion of two- and threedimensional magnetotelluric data: J. Geophys. Res., 96, 3905-3922. Smith, R. T., Zoltani, C. K., Klem, G. J., and M. W. Coleman, 1991, Reconstruction of the tomographic images from sparse data sets by a new finite element maximum entropy approach: Appl. Opt., 30, 573-582. Strakhov, V. N, 1968, Numerical solution of incorrect problems represent able by integral equations of convolution type (in Russian): DAN SSSR, 178 (2), 299. Strakhov, V. N., 1969a, Theory of approximate solution of the linear ill-posed problems in a Hilbert space and its application in applied geophysics, Part I (in Russian): Izvestia AN SSSR, Fizika Zemh, No. 8, 30-53.
Definition of tlie regularization parameter
57
Strakhov, V. N., 1969b, Theory of approximate solution of the Hnear ill-posed problems in a Hilbert space and its application in applied geophysics, Part II (in Russian): Izvestia AN SSSR, Fizika Zemh, No. 9, 64-96. Tikhonov, A. N., 1999, Mathematical geophysics (in Russian): Moscow State University, 476 pp. Tikhonov, A. N. and V. Y. Arsenin, 1977, Solution of ill-posed problems: V. H. Winston and Sons, 258 pp. Vogel, C. R., and M. E. Oman, 1998, Fast total variation based reconstruction of noisy, blurred images: IEEE Trans. Image Processing, 7, 813-824. Wernecke, S. J., and L. R. D'Addario, 1977, Maximum entropy image reconstruction: IEEE Trans. Comput. 26, 351-364. Zhdanov, M. S. and S. Fang, 1996, 3-D quasi-linear electromagnetic inversion: Radio Science, 3, No. 4, 741-754. Zhdanov, M. S., 1993, Tutorial: regularization in inversion theory: CWP-136, Colorado School of Mines, 47 pp.
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P a r t II M e t h o d s of t h e S o l u t i o n of I n v e r s e Problems
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Chapter 3 LINEAR DISCRETE INVERSE PROBLEMS We will study, first, the most important and, yet, relatively simple type of inverse problem, which arises in the case of a linear forward modeling operator and discrete data and model parameters. This situation is typical, for example, in gravity and magnetic inverse problems. The same approach can be used in special cases for electromagnetic and seismic inverse problems, when one can approximate the nonlinear forward modeling operator by a linear one. Discrete data are typically used in most applications as well. We will begin our study with the traditional least-squares method. After that, I will demonstrate how the same problem can be solved based on the principles of probability theory. Finally, I will discuss different aspects of implementing regularization principles in the solution of the linear discrete inverse problem. 3.1
Linear least-squares inversion
3.1.1 The linear discrete inverse problem Let us consider a general inverse geophysical problem, described by the operator equation: d = ^(m), (3.1) where m represents the model parameters, and d are observed geophysical data. We assume that A^ measurements are performed in some geophysical experiment. Then we can treat these values as the components of the A^-dimensional vector d. Similarly, some model parameters can be represented as the components of a vector m of order L: m = [mi,m2,m3,...,mL]^, where the superscript T denotes the transpose of the two vectors. In a case where A is a linear operator, equation (3.1) can be rewritten in matrix notation: d - Am, (3.2) where A is the N x L matrix of the linear operator A.
62
Linear d i s c r e t e inverse p r o b l e m s
Expression (3.2) describes a system of A^ linear equations with respect to L unknown parameters, mi, m2, ma,..., rriL'. L
d^^^AijrUj,
z = 1,2,3,...,A^.
(3.3)
3= 1
Thus, solving the inverse problem (3.1) means solving the system of linear equations (3.3) with respect to parameters mi,m2,m3, ...,mL. The system (3.3) is called underdetermined UN < L. The system (3.3) is called overdetermined ii N > L. Very often in geophysical applications, we work with an overdetermined system wherein the number of observations exceeds the number of model parameters. At the same time, in many situations it may be necessary to work with an underdetermined system. We will examine both types of linear equation systems below. Thus, we can see that in the case of a linear discrete inverse problem the operator equation (3.1) is reduced to the matrix equation (3.2). To solve this equation we have to use some formulae and rules from matrix algebra, described in Appendix E. 3.1.2 Systems of linear equations and their general solutions Let us consider again a system of linear equations determining the relationship between the observed data {di, ^2, (is,..., G^AT} and parameters of the model {mi, m2, ma,... L
dr = ^Aijmj,
z = l,2,3,...,A^,
(3.4)
3= 1
where N > L. In other words we assume now that equation (3.4) describes an overdetermined system. We know the column vector d = [di,d2,ds, ....d^]^ of the observed data and matrix A of the linear operator of the forward problem. Our goal is to determine the column vector m = [mi, m2, ma,..., TUL]^ of the model parameters. Note that in this chapter we assume, for simplicity, that all model parameters and data are represented by real numbers. Let us denote by d^ = [df, ^2, c^a, •••, c?^]^ a vector of the predicted data: < = ^A,,m^-,
2 = 1,2,3,...,A^.
3= 1
We can now write r = d^ - d, where r = ['^'i, ^2, ra,..., ryv]^ is the column vector of the residuals (errors) between the observed, d, and predicted, d^, data . However we have more data {di, (i2,c?a, ...,G?Ar} than model parameters {mi, m2, ma, ..., m^}, so we cannot fit all the data. The best that we can do is to minimize
63
Linear least squares inversion
the misfit between the observed and predicted data, which in the Euchdean metric of data space can be calculated as a sum of the squares of the errors: TV
/(mi,m2,m3,...,mL) = | | r f = ^ ^ ^ = min.
(3.5)
Functional / is called a misfit functional. It can be written in the form /(m) = | | d ' ' - d | r = ||Am-d||2 = = (Am — d, A m — d) = min .
(3.6)
Note that, using matrix notations, one can rewrite the last equation in the form / ( m ) = (Am - d)'^(Am - d) = min .
(3.7)
The column vector mo = [^oi,'^02,'^03, •••,'^OL]"^, in which the misfit functional reaches its minimum, is called a pseudo-solution of the system (3.4). If /(mo) = 0, then mo is the conventional solution of the system (3.4). Minimization of the misfit functional It is well known that the best way to solve an optimization problem for conventional functions is based on differentiating the functions and equating the resulting derivatives to zero. A similar approach can be applied in principle to functionals. However, we have to use an analog of calculus for functionals and operators, which is called variational calculus. This generalization has been discussed in Appendix D. The problem of minimization of the misfit functional (3.7) can be solved using variational calculus. Let us calculate the first variation of / (m) : ^7 (m) = 2(A^m)'^(Am - d). The necessary condition for the minimum of the functional / (m) has the form Sf (m) - 2(A(5m)^(Am - d) = 2{6m)^A^\Ani
- d) = 0,
for any 6m. From the last formula we have the following system of equations: A'^Am = A^d.
(3.8)
System (3.8) is called the normal system for (3.4). The matrix A^A is an L X L square matrix. Thus the solution of the normal system can be given in the form mo = (A^A)-^A^d. (3.9)
64
Linear discrete inverse problems
Note that the normal system can be obtained formally by multiplication of the original system (3.2) by the transposed matrix A-^. However, in general cases, the pseudosolution mo is not equivalent to the solution of the original system, because the new system described by equation (3.8) is not equivalent to the original system (3.2) if matrix A is not square. The main characteristic of the pseudo-solution is that it provides the minimum of the misfit functional. To find a pseudo-solution numerically, we can apply the method of singular value decomposition (E.13) and obtain A^A = (UQV^)'^UQV^ = VQU^UQV^ = VQ^V^,
(3.10)
where Q is diagonal, and U and V are orthogonal matrices. Thus system (3.8) can be rewritten as VQ^V^mo = VQU^d.
(3.11)
Let us apply the inverse matrix (VQ^V-^)"^ to the right-hand and left-hand parts of (3.11): mo = ( V Q V ^ ) - i V Q U ^ d = V Q - ' U ^ d = V[diag(-^)]U^d. Expression (3.12) gives directly the pseudo-solution of the system (3.4). The matrix A+ = V [ d i a g ( i - ) ] U ^
(3.12)
(3.13)
is called the pseudo inverse matrix for A. The pseudo inverse matrix is equal to the inverse matrix of a square matrix A:
Equation (3.13) is a natural generalization to rectangular matrices of formula (E.14) from Appendix E for square matrices. Thus, minimization of the misfit functional opens a way to construct a generalized inverse matrix for any matrix, rectangular or square, with the only limitation being that the elements of the diagonal matrix Q are not equal to zero: Q^ 7^ 0, z = 1, 2, ....L. 3.1.3 The data resolution matrix Since matrix A^ solves the inverse problem A m = d, it is often called the generalized inverse matrix: A~-^ = A+ (Menke, 1989). Figure 3-1 provides a schematic illustration of a generalized inverse matrix. According to equation (3.9), we have A-^ = (A^A)-^A^,
(3.14)
65
Linear least squares inversion
Generalized Inverse (Pseudo Inverse) Matrix A = UQV ^
A
u
L
L
A-^ = V
A-^
Figure 3-1
Q
V L
diag
(^)>'
ft-\
Generalized inverse matrix computation.
and the pscudo-sohition of the inverse problem is given by mo = A~-M.
(3.15)
We can now retrospectively ask how well this estimate IHQ fits the observed d a t a d: d^^ = A m o = A A " M = N d ,
(3.16)
where N is t h e (A^ x TV) square data resolution matrix. If N = I is an identical matrix, then d^ = d, and the prediction errors are zero. In general cases r = d^-d = (N-I)d. (3.17) Let us say, for example, t h a t the 2—th row of the m a t r i x N contains t h e elements [.... 0 0 0 0.1 0.9 0.1 0 0 0 ....] T h e n t h e i—th d a t u m is given by
From the last formula one can see t h a t the predicted value d^ is the weighted average of the observed d a t a d^^i, d^, and d^J^l. So, the rows of the resolution m a t r i x N
66
Linecir discrete inverse problems
show how well the neighboring data can be independently predicted, or resolved. The diagonal elements of the data resolution matrix indicate how much weight a datum has in its own prediction. That is why these diagonal elements are called the importance n of the data (Menke, 1989): n = diag[N]. Note also that the data resolution matrix is not a function of the data but only of the operator of the forward problem. It can therefore be studied without actually performing the geophysical observations and can therefore be used for planning the field experiment. 3.2
Solution of the purely underdetermined problem
3.2.1 Underdetermined system of linear equations Assume that the inverse problem d = Am
(3.18)
is purely underdetermined. This means that N < L and there are no inconsistencies in these equations. It is, therefore, possible to find more than one set of model parameters which precisely fit the observed data. Let us try to select from all possible solutions the one which is the simplest in some sense, for example, it has the smallest Euclidean norm: l{m) = ||m||^ = min, (3.19) where \\mf
= (m, m) = m ^ m .
(3.20)
Thus we have the following problem: find the m^*^ that minimizes /(m) subject to the constraint that d-Am = 0. (3.21) We can solve this problem of the conditioned minimum by using the method of Lagrange multipliers: 0(m) = m ^ m + A (d - A m ) ^ = min,
(3.22)
where A is a diagonal matrix of the Lagrange multipliers. Let us calculate the first variation of the functional (f){m): S(t){m) = 2Sm^m - ^m^A^A = 6m^{2m - A^A).
(3.23)
The necessary minimum condition for the functional gives 6(P{m) = ^m^(2m - A^A) = 0, for any 6in^.
(3.24)
Solution of the purely underdetermined problem
67
Thus we have the following expression for the solution m^^^ of our problem:
On the other hand, this solution must satisfy equation (3.18): d = - A A ^ A. 2 T h e matrix A A ^ is a square N x N matrix and, if it is not singular, can be inverted: A = 2(AA^^yM.
(3.26)
By substituting the last equation into (3.25) we have ^e,s/^ = A^^^(AA^^')~M.
(3.27)
Formula (3.27) provides a minimum norm solution of the underdetermined problem. ^t2.2
The viodd
rcsohittofi
inatrix
'V\\v ni()d(^l li^solut ion inatrix prcxHcts whether model ])aram(ieis can \)v prc^dictiMJ w(^n or resoh^ed. Assume t h a t there is a true but unknown set of p a r a m e t e r s rn'''"' t h a t solves the e(|uation d = Am''^'"".
(3.28)
Let us substitute equation (3.28) into (3.27):
T h e L X L square matrix, R = A^^'(AA^^VA,
is called the model resolution
matrix.
(3.30)
From (3.29) and (3.30) we have
m'"' = Rm'""^
(3.31)
T h e estimated model parameters are the weighted averages of the true model p a r a m e ters, where t h e weights are determined by the rows of the model resolution matrix. In the case where R = I, the model parameters are exactly determined. Like t h e d a t a resolution matrix, the model resolution matrix is completely determined by t h e matrix of the operator of the forward problem.
68
Linear discrete inverse problems
3.3
Weighted least-squares method
Let us introduce some weighting factors w1 for estimation of the residuals r^. The reason for the weighing is that in practice some observations are made with more accuracy than others. In this case one would like the prediction errors r^ of the more accurate observations to have a greater weight than the inaccurate observations. To accomplish this weighting we define the weighted misfit functional f^j as follows /^(mi,m2,m3,...,mL) -= ||r||^ = ^{wjrjf
= min.
(3.32)
i=l
We can introduce the weighting operator VK, which is a linear operator, acting in the space of data D and having the diagonal matrix W : W = [diag(i^,)].
(3.33)
Then equation (3.32) can be rewritten in the form / ^ ( m ) = llWd^ - W d f - ||WAm - W d f = (WAm - W d ) ^ ( W A m - W d ) = min .
(3.34)
The problem of minimization of the weighted misfit functional can be solved by calculating the first variation of this functional and setting it equal to zero: 8U{m) = 2 (^WAm)^ (WAm - W d ) = 2{drnf (WA)^ (WAm - W d ) = 0. Thus, we obtain the following system of equations: (WA)^ W A m = (WA)^Wd or A^W^Am = A^W^d.
(3.35)
Assuming that the matrix A-^W^A is non-singular, we can write mo = (A^W2A)-^A^W2d.
(3.36)
The solution of the weighted least-squares problem is obtained from equation (3.36). The problem of the selection of the proper weights, Wi^ will be discussed in the next sections. The matrix A-^ = ( A ^ W 2 A ) - ^ A ^ W 2 (3.37) is called the weighted generalized inverse matrix.
Applying
the principles of probability
theory to a linear inverse
d
Ad
problem
69
d + Ad
measurement d Figure 3-2 The shaded area of the probability distribution gives the probability that the datum will fall between d and (i + A d . 3.4
A p p l y i n g t h e p r i n c i p l e s of p r o b a b i l i t y t h e o r y t o a l i n e a r i n v e r s e p r o b lem
S.4.1
Some formulae
and notations
from probability
theory
111 this section wo cxaiiiino a diffcreiit approach to the least sc|uares m e t h o d s , ])ase(l on i(l(>as h'oin |)r()ha])ihty^ th(H)ry. We can trcuit tlic^ ()l)serv(xl data d as a randoin x'ariabk^ because ineasurenients always contain some random noise. In this connection thc^ observed d a t a d.^.^^ can be considered as the realization of the r a n d o m varial)le d. T h e true properties of a random variable d, can be specified by the probability density function (or probability distribution) P{d). T h e probability t h a t t h e measurement is between d, and c/+ Ad is determined by the value of P{d,)Ad, (see Figure 3-2). In particular, we have for a discrete random variable t h e following equality:
J2
P{d)Ad=].
(3.38)
(i== —00
T h e mean value of the discrete random variable d can be determined by the expression {d) =
^
dP{d)Ad.
(3.39)
d= — oo
A variance a'^ of t h e discrete r a n d o m variable d describes the deviation of d from its mean value. It is calculated by the formula {{d-{d)f)=
^(d-(d))2p(d)Ad
(3.40)
Linear discrete inverse problems
70
d,,
measurement d
Figure 3-3 The maximum likelihood point (IML 'S a point having the maximum value of the probability distribution P {d). This point will be different from the mean value {d) in general.
The value a is sometimes called the standard deviation of data d. The most likely measurement dML is the one with the highest probability: P(dML) = max {P(d)}.
(3.41)
The value df^L is called the maximum likelihood point (see Figure 3-3). Let us assume that we know the column vector d = [(ii, (i2, c/a,..-, <^N]^ of the observed data. The joint probability density P ( d ) determines the probability that the first datum is between di and (ii+Adi, the second datum between d2 and d2-\-/S.d2-i the third datum between d-^ and ^3 + Ada etc. For example, if the data are independent, then the joint distribution is just the product of the individual distributions: P(d) = P{d^)P{d2)P{d3)
P{dN).
(3.42)
If the data are correlated, it is more difficult to estimate P ( d ) . The mean value of the datum di is determined by the formula + 00
Y^ ... J2 diP{d)AdiAd2...Ad!,
(dr)
(3.43)
djsf =
For correlated data we can introduce some measure of their correlation which is called the covariance: coY{d,,dj) = ((d, - {d^)){dj - {dj))) +00
+00
E E
di— — oo d2 =—o<.
+00
. ^ d// = —00
{d,-{d,)){d,-{dj))P{d)MiM2...M,
(3.44)
Applying
the principles of probability
theory to a linear inverse
_
0.50 _
problem
71
^
0.25
d Figure 3-4
Typical Gaussian distributions w i t h zero mean and a i = 1 for curve A, and
a2 = 2 for curve B. The variance a'^ corresponds t o the sharp probability distribution, while the variance ai describes the smooth distribution.
Notc^ t h a i the covariaiico of a clatuni with itsch" is just the variance: (X)v(c/,, d,) — aj.
(3.15)
l l i u s , ior a cohiiiiii vector d — [(l'l^(^''2l (hi •••I^'N]' ^^i ili<^' obscrvcxl d a t a we can nit-ro(hice a covariancc matrix a- = [a,j] = [cov{(k,(lj)]. (3.46) This matrix is symmctricaL S.4-2
Max'irriinji likdiJiood
rnetfwd
As wo have mentioned above, in general cases, the probabihty distribution can be described by a very comphcatcd function. However, according to the central lirnit theorem, a large sample of a random variable tends to a very simple distribution, the so-called Gaussian (or normal) distribution, as the size of r a n d o m sample increases:
P{d) =
1 (27r)2cr
exp[
{d-{d)) 2^2
(3.47)
This distribution has a mean {d) and variance a^. Figure 3-4 shows two typical Gaussian distributions with zero mean, a = \ for curve A, and a = 2 for curve B. One can sec t h a t the smaller variance corresponds to the narrower and sharper probability distribution, while the bigger variance describes the wider and smoother distribution. T h e joint distribution for two independent Gaussian variables is just t h e product of two univariate distributions. W h e n the d a t a forming a vector d are correlated
72
Linear discrete inverse problems
(with mean (d) and covariance a = [cTy]) , the appropriate distribution turns out to be ^ ( d ) = ^ ^ e x p [ - i ( d - ( d » V - i ( d - (d))].
(3.48)
The idea that the model and data are related by an explicit relationship Am = d
(3.49)
can now be reinterpreted in the sense that this relationship holds only for the mean data: A m - (d). (3.50) Substituting (3.50) into (3.48), we can rewrite the distribution for the data as ^ ( d ) = 7 f S e x p [ - i ( d - A m ) V - H d - Am)]. (ZTTJ 2
Z
(3.51)
Under this assumption we can say that the optimum values for the model parameters are those that maximize the probability that the observed data are in fact observed. Thus the method of maximum likelihood is based on maximization of the probability function (3.51) P(d) = max. (3.52) Clearly the maximum of P(d) occurs when the argument of the exponential function is maximum or when /^(m) = (d - A m ) V - ^ ( d - Am) = min.
(3.53)
Let us calculate the first variation of /^j: 5/^(m) - - ( 5 A m ) V - ^ ( d - Am) - (d - Am)^o--^(^Am). It can be shown that for the symmetrical matrix CT~^ a^cr-^b = b V - ^ a , where a and b are two arbitrary column vectors. Therefore, we can write the necessary condition for the functional ja to have a minimum as follows: 5/^(m) = - 2 ( A 5 m ) V - ^ ( d - Am) = -2(^m)^A^o--^(d - A m ) - 0. From equation (3.54) we obtain at once A V - ^ ( d - A m ) = 0,
(3.54)
Applying the principles of probability theory to a linear inverse problem
73
or the following normal system of equations for the "pseudo-solution" of the minimization problem (3.53):
If the m a t r i x A-^cr"^ A is non-singular, then mo = ( A ^ V ~ ^ A ) - ^ A V ^ M .
(3.55)
Comparing t h e last formula with the corresponding equation for the weighted leastsquares m e t h o d (3.36), we see t h a t we have obtained exactly t h e same result, if we s u b s t i t u t e t h e m a t r i x W ^ for a~^: (3.56)
W'^ = (7--\
S.4• 3
CAii-square
fitting
Notice tliat if d a t a ha})pcn to be uncorrelatcd, then the covariancc m a t r i x l^ecomes diagonal: a = [diag(af)],
(;?.57)
and tlic^ eknnents of the; main diagonal arc Ihc variances of thc^ data. In this ca*sc the weights ar(^ given by: (3.58)
Wt = ~ . T h e functional TV
N
,
,p
is called a '"chi-sciuare." In cases where the measurement errors are normally distributed, t h e qTiantity X^ is a sum of TV squares of normally distributed variables, each normalized to unit variance. T h u s , by applying the weighted least-squares m e t h o d we can select t h e smaller weights for d a t a with bigger s t a n d a r d deviations (less accurate d a t a ) and t h e bigger weights for d a t a with smaller s t a n d a r d deviations (more certain d a t a ) . If t h e ,2 d a t a have equal variances, CTQ, 07 t h a n t h e weighting m a t r i x becomes scalar
w^ = CT~' = - I T (J.0
and the chi-square functional becomes equal to the ordinary misfit functional.
74
3.5
Linear discrete inverse problems
Regularization methods
3.5.1 The Tikhonov regularization method DifTerent modifications of least square solutions of linear inverse problems have resulted from the straightforward minimization of the corresponding misfit functional. However, all these solutions have many limitations and are very sensitive to small variations of the observed data. An obvious limitation occurs when the inverse matrices (A^A)""^ or (A'^W^A)"^ do not exist. However, even when the inverse matrices exist, they can still be ill-conditioned (become nearly singular). In this case our solution would be extremely unstable and unrealistic. To overcome these difficulties we have to apply regularization methods. Let us consider first the general approach based on the Tikhonov regularization technique (Tikhonov and Arsenin, 1977). The corresponding parametric functional can be introduced in the following form: P " ( m , d) = IIW^Am - W ^ d f -F a | | W ^ m - W ^ m , p , f , where W^ and W ^ are some weighting matrices of data and model (not necessary diagonal); niapr is some a priori model and ||...|| denotes the Euclidean norm in the spaces of data and models. In a majority of practical applications we assume that W ^ = I, but it also can be chosen arbitrarily (for example as a matrix of first or second order finite-diff"erence differentiation to obtain a smooth solution). We will discuss some specific choices of Wm later. We can also rewrite the functional P°'(m, d) in the matrix notation: P«(m, d) = (WdAm - Wdd)^(WdAm - W^d) ^ m^^apr ) •
According to the basic principles of the regularization method, we have to find a quasi-solution of the inverse problem as the model ma that minimizes the parametric functional P ^ ( m a , d ) = min. The regularization parameter a is determined from the misfit condition: WWdAnia - Wdd\\ = 5, where 6 is some a priori estimation of the level of "weighted" noise of the data: \\Wd6d\\ = 6.
(3.60)
To solve this problem let us calculate the first variation of P^(m, d): 5 P " ( m , d ) = 2{WdA6mf{WdAm
- W^d)
75
Regularization methods
+2a{W^6mf{Wmm
- W„map,) = 0.
The last equation can be rewritten as 6m^ [ ( A ^ W j W d A + a W ^ W „ ) m
from which we obtain at once a regularized normal equation for the original inverse problem (3.2) (A^^ W j W . A + a W ; ^ W ^ ) m « = A^ W j W . d + aWjNmma^r.
(3.61)
and its regularized solution m„ = (A''W;';W,A + a W ; ' „ W „ , r ' (A'^'WjW.d + a W r , W „ m „ , „ ) .
(3.62)
Usually the weighting matrices W,/ and W^„ are selected to be synnnetric (or even diagonal), so equation (3.62) can be rewritten: m,, - (A'^ W^^A + rvW;J-^(A^^ W;id + rvW;^„rrv,).
(3.63)
The last (\x])i"essi()n giv(\s the regularized solution of the generalized least-sc|uares |)rol)lem. ^L5.2 Ap'plicatton of SLUM nieUiod in regularized, linear inverse jrrobleni solution One of th(^ key ])roblems in regTilarized inversion is the selection of the optimal regularization parameter rv. The traditional methods of determining the optimum value of this parameter, outlined in Chapter 2, are based on nuiltiple inverse problem solutions with the different values of (y. This procedure, if implemented explicitly, could be extremely comi)uter resource and time consmning. One way to overcome this difhculty is to apply the Spectral Lanczos Decomx)osition Method (SLDM), presented in Ap])endix E. Let- us address again ec^uation (3.62), which provides a regularized solution of the generalized least-squares problem. Introducing notations c = A^^Wj^Wdd,
B = A^WjWrfA,
(3.64)
and assuming for the sake of simplicity that niapr = 0, we can rewrite this equation in the form m , = (B + c . W ^ W ^ ) - ^ c . (3.65) Note that matrix B is symmetric because B'^' = ( A ' ^ X i W d A ) ' ' = A'^'WjWdA = B.
76
Linear discrete inverse problems
Denoting by /^ the matrix function /„ (B) = (B + a W ^ W J - i ,
(3.66)
m , = / , (B)c.
(3.67)
we finally obtain Thus, we arrive at the problem of computing a function of matrix B . This problem can be most effectively solved by the Spectral Lanczos Decomposition Method (SLDM), which involves partial tridiagonalization of the given matrix (Druskin and Knizhnerman, 1994; Golub, and Van Loan, 1996). We will introduce the following definition. Definition 16 The finite dimensional subspace KL of the Euclidean space EN, spanned by the vectors c, Be, ...., B^~^c, is called a Krylov space:
KL = spein{c, Be, . . . . , B ^ " ^ e } . The Lanczos method is based on generating the orthonormal basis in Krylov space KM =span{e, Be, ....,B^~^e} by applying the Gram-Schmidt orthogonalization process, described in Appendix A. In matrix notation this approach is associated with the reduction of the symmetric matrix B to a tri-diagonal matrix Tyv and an orthogonal matrix QAT, and also with the special properties of TA^ and Q^^. The reduction formula, according to (E.21) and (E.23), can be written as B = Q;vT^Q?;.
(3.68)
This reduction (called also the QT decomposition) can be accomplished by the Lanczos algorithm (E.29): PQ = 1, qo = 0, qi = e / | | e | | ,
(a)
while Pj ^ 0, q^+i = r^//?^-, a^I = qjBq^-,
{b)
Vj = (B-a,l7v) q^- - /?^_iq^-i, /3j = ||r^|| ,
(c)
(3.69)
j = l, 2,....,
N-1;
where I^ is N x N identity matrix. As the result, we find an orthogonal matrix Q^r, formed by the Lanczos vectors q^ (see Appendix E), Qiv = [qi,q2, ,qiv],
Regularization methods
77
and the tri-diagonal symmetric matrix Tyv formed by the coefficients aj and Pji ai
/?!
...
Pi
(^2
02
0
PN~I
0
O^N
In A p p e n d i x E we show t h a t we can reduce t h e original problem of calculating the function / « of m a t r i x B to a much smaller problem of calculating t h e s a m e function of t h e tri-diagonal matrix Tyv (expression (E.39)) (TV)
(3.70)
m« = | | c | | Q y v / a ( T ^ ) e i
/her -1 is the TV-dimensional vector with the first component equal to one, a n d all other components equal to zero: AN)
(1,0,...,0,...0)'\
Taking into account t h e known analytical expression for the m a t r i x function (3.66), wc arrive at the following formula for the regularized solution: m,, - | | c | | Q A . ( T ^ + a W ; ( ; W .
-iJ^)
(3.71)
T h e main advantage of representation (3.71) over original inverse formula (3.62) is t h a t we havc^ to run the Lanczos algorithm only once for all different values of the regularization p a r a m e t e r cv. After t h a t wc have to invert only a tri-diagonal m a t r i x for different a, which is much more simple operation t h a n the {Tpsj 4- (yWIJ,^m) inversion of full m a t r i x (B + aWj^^Wrn)After application of SLDM, we can use any m e t h o d of optin^al a selection, for example, wc can use Tikhonov algorithm based on misfit condition (2.92), or L-curve m e t h o d (see C h a p t e r 2). In particular, the misfit condition (2.92) can be rewritten now in t h e form: | W , A m „ - Wrfdj! = ilWrfA llcjl Q;V(TA, + aW;',;W, 3.5.3
Defimtion
of the weighting
Wrfdjl = 6. (3.72)
matrices for the model paramieters
and
data
Let us analyze the sensitivity of the d a t a to t h e p e r t u r b a t i o n of one specific p a r a m e t e r m/e. To solve this problem, we apply t h e variational operator to b o t h sides of equation (3.3): 6d, = AkSruk. (3.73) In the last formula, A^k are the elements of t h e m a t r i x A of t h e forward modeling operator, and there is no summation over index k. Therefore, the norm of t h e p e r t u r b e d vector of the d a t a can be calculated as
78
Linear discrete inverse problems
\\6d\\ = / ^ ( M ) ' = Ij^iAikfSm,.
(3.74)
We determine the integrated sensitivity of the data to the parameter ruk as the ratio Sk =
\M\ Snik
(3.75)
= JTM^'-
One can see that the integrated sensitivity depends on the parameter k. In other words, the sensitivity of the data to the different parameters varies, because the contributions of the different parameters to the observation are also variable. Definition 17 The diagonal matrix with the diagonal elements equal to Sk = \\6d\\ /SrUk is called an integrated sensitivity matrix:
S = diag [ JY^ {A,f j = diag (A^A) '^' .
(3.76)
In other words, it is formed by the norms of the columns of the matrix A. The basic idea of introducing a weighting matrix, W ^ , for the model parameters is as follows. We identify this matrix as the diagonal integrated sensitivity matrix W™ = [Wj] = [Sj] = S. (3.77) Thus, the weights are selected to be equal to the sensitivities: Wj = Sj.
(3.78)
We can now introduce the weighted model parameters: m- - W^m.
(3.79)
Using these notations, we can rewrite the inverse problem equation (3.2) as follows d = AW^^W^m = A ^ m - ,
(3.80)
where A^ is a weighted forward modeling operator, A- - A W - ^
(3.81)
Now we perturb the data with respect to one specific weighted parameter m^ :
6di = Ar,5m^
79
Regularization methods
and calculate a new integrated sensitivity 5 ^ of t h e d a t a to t h e weighted p a r a m e t e r m ^ as the ratio
J E
(^*W/,-^)'
= W,'\Y^ {A,,f = W^'S, = 1.
(3.82)
Formula (3.82) shows t h a t the new m a t r i x of the integrated sensitivity S'^ is a unit matrix: S^ = 1 . Therefore, d a t a are uniformly sensitive to the new weighted model parameters! Note t h a t the corresponding weighted stabilizing functional takes the form ,s,,; (m) = ( m - m„p,,)^^ W,^,(m - m„,p,) -:: ( m - m„p^)'^'S'^(m - m„,p.,).
(3.83)
It imy)oscs a stronger penalty on departure from the a prion model for those paramc^ters that contri])utc more significantly to the data. Thus. th(^ model weighting ix^sults in pra(*tically ecjual iTsolut ion of the iuATnsion with respect to different parameters of the model. In a similar way, we can define the diagonal d a t a weighting matrix, formed by the norms of the rows of the matrix A :
W, = diag
Y. (^'*)'
= diag (AA''')'''.
(3.84)
These weights make normalized d a t a less dependent on the specific p a r a m e t e r s of observations (for example, frequency and distance from the anomalous domain), which improves the resolution of the inverse method. 3.5.4 ApproxtmMe regularized solution of the linear inverse problem In practical geophysical applications the size of the linear inverse problem (3.2) can be so large, t h a t direct matrix inversion can be extremely time consuming. This situation is typical for seismic or electromagnetic d a t a inversion. In this case we can use a simple approach to approximate the solution of the inverse problem. T h e basic ideas of this technique as applied to the Born inversion in electromagnetics and acoustics were formulated by Block and Oristaglio (1995). We rewrite the regularized normal equation (3.61) in the form (A^W^A + a W ^ ) m „ = A^W^d + aW^m„p,,
(3.85)
80
Linear discrete inverse problems
where we use a tilde sign above W ^ , because we will introduce the different model parameters weights in the case of approximate solution. Let us assume that the regularization parameter a is so big that we can neglect the term A ^ W ^ A compared with the term aW'^ on the left-hand side of (3.85): a W ^ m , ^ A ^ W ^ d + aW^niap..
(3.86)
From the last formula we obtain at once an approximate regularized solution xn^ ^ a - ^ W - ^ A ^ W ^ d + m.^,.
(3.87)
We can see that computing the approximate solution nic does not require an inversion of the matrix A and can be found by simple matrix multiplication. The coefficient a~^ can now be treated as just a scaling factor, which can be found by minimizing the misfit between the observed and predicted data II A m , - d|| = | | a - ^ A W - ' A ^ W ^ d + Aniapr - d|| = min.
(3.88)
In the following chapters we will demonstrate that approximation (3.87) results in effective imaging schemes for different geophysical data interpretations, including gravity, electromagnetic and seismic migration. Note, however, that in the case of the approximate solution, the weighting matrix for the model parameters should be selected as the square root of the integrated sensitivity matrix:
w„
H^nx,
V ^ = ^ = diag
JYl (^''^)' = d^^g (A^A)'/' . (3.89)
This choice of the weighting matrix can be justified by the following consideration. Let us denote by m^ the approximate regularized solution (3.87) with weights, m f = W^m,.
(3.90)
According to (3.86), the weighted solution is expressed by the formula: m f = a - ^ A ^ W ^ d + mf;„
(3.91)
where m^^ = W^m^prIntroducing the notation of (3.90), we rewrite the inverse problem equation (3.2) as follows d = AW^2 W ^ m = A^' m^2, (3.92) where A^^ jg ^ j^g^ weighted forward modeling operator, A^2 = AW;;,^
(3.93)
Regularization
81
methods
P e r t u r b i n g t h e d a t a with respect to one specific weighted p a r a m e t e r m ^ , 6d, A^j^ 6m^ A^f^ 6rri^ ,, we we calculate calculate a new integrated sensitivity 5^^ of t h e d a t a to one new weighted p a r a m e t e r rn.]^
E(^^ Y.{Akf-w,^'S,
= 1.
Formula (3.94) shows t h a t the new matrix of the integrated sensitivity S^ matrix: S ^ ' = I.
(3.94) is a unit
As a result, the weighted approximate regularized solution, given by formula (3.91), provides a uniform estimate of the different model parameters. However, at t h e fmal stage of inversion we should return from the weighted parameters, m^' , to the real model parameters, m^^, using the formula m,, = W , , f m f
= a:~^ W - ; ' A ' ' ' W , ^ d + m,,,,.
(3.95)
Note in conclusion t h a t diifercnt selections of model paramc^ter weights are chctatcxl by the chlic^rent methods of solution of the inverse i)roblem. if we apply tht^ dir(x:t invi^rsion accorchng to formula (3.62), we should use weights equal to sensitivities. If we use the ai)proxiniate inversion described by formula (3.95), we select the weights as square roots of the sensitivities. In both cases the weights are selected under the condition t h a t they provide imiform sensitivity of the solution to the observed data. S.5.5
Tfie Levenberg - Marquardt
method
Let us consider a special case when W^i = I and W^^^ = I. T h e n equation (3.63) turns out to be m^. = (A^^'A + aiy^ (A^^'d + (y.map (3.96) Assume now t h a t lUapr = 0 m„ = (A''A + aI)~'A'^d.
(3.97)
T h e solution (3.97) describes the classical Levenberg - M a r q u a r d t m e t h o d of d a m p e d least squares, where a plays the role of a "damping factor" (Levenberg, 1944; M a r q u a r d t , 1963). For a b e t t e r understanding of how the regularization p a r a m e t e r or "damping factor" a works, let us apply a singular value decomposition m e t h o d to m a t r i x A : UQV'
82
Linear discrete inverse problems
Then we have A ^ A = VQ^V'^.
(3.98)
From the last equation we obtain at once (A^A + a l ) - ' = (VQ^V^ + a V I V ^ ) - i = (V[diag(a + QDIV^)-^ = V [ d i a g ( — ^ ) ] V ^ .
(3.99)
We can clearly see from equation (3.99) how regularization makes the nearly singular matrix well-conditioned; even ii Qi —^ 0 division by zero does not occur. 3.5.6 The maximum a posteriori estimation method (the Bayes estimation) Let us consider the regularization technique from the point of view of probability theory (Tarantola, 1987). First of all, we introduce the following (normally distributed) densities of probability: 1) P ( d / m ) is a conditional density of probability for the data d, given the model m. It means that it is the probability density of theoretical data d to be expected from a given model m. 2) P ( n i / d ) is a conditional density of probability for a model m, given the data d. According to the Bayes theorem, the following equation holds: ^ , , ,, P(d/m)F(m) ^ ( m / d ) = ^ ^^^^^^ ^
,^ _ , (3.100)
where P ( d ) and P(ni) are unconditional probability densities for data and model parameters respectively. It is assumed that (m) = niapr, where rUapr is an a priori constrained expectation of the model, and [coy{mi,mj)] = dmThus, considering normally distributed parameters, we have ^ ( " ^ ) ""
^r ^k (27r)#
^ X P [ - o ( ^ ^ - ^aprf(T^{m "'^' 2^
Analogously, it is assumed that [cow{di,dj)\ = ad
- rriapr)]-
(3.101)
83
Regularization methods and P(d/m) = ^
^
e x p [ - i ( d - Ani)^V,7i(d - A m ) ]
(3.102)
Z
2
[ZTT)
T h e m a x i m u m likelihood method can now be used to find the model mo which m a x imizes the conditional probability P ( m / d ) : P ( m / d ) = i ^ ^ ^ e x p [ - i ( d ^ Am) V 7 i ( d - Am)] [ZTT) 2
x'"^"^'
Z
' e x p [ - ^ ( m - inaprfa;^\m
[27T)'2
- m,^,)]p-^(d).
(3.103)
Z
It is evident t h a t to maximize the P ( m / d ) we have to minimize the s u m of t h e expressions in the exponential factors in equation (3.103): fnayes = (d - AuiY^a'^d
~ A m ) + ( m - mapr)^^V-/(m - lUapr)-
(3.104)
Note t h a t t h e minimization of the first t e r m gives the classical m a x i m u m likelihood or weighted least-squares method. Let us calculate the first variation of fnayes'^fiuuirs - ~2{A(Sm)'cT-,\d
- A m ) + 2(/)m)^^'o-,;;/(m - m,,,,) ™ 0.
From the last eciuation we have (^m)'^'[A^V, ^(d - A m ) - ^ ; / ( m -~ m,,,,)] = 0. T h u s the normal system of equations for minimization of fuaycs i^ equal to A^^'cr^7^(d - A m ) - cr^7/(m - m„p^) - 0, from which we have at once (A^cr^^ A + a;^^)m
= A^^V~M + cr,7/ma^,,,
and mo = ( A ' V . - ' A + < , ^ ) - 1 ( A V , ~ M + cT-^'ma^).
(3.105)
By comparing equation (3.105) and (3.63) we see t h a t cT'„l=aWl,
(3.106)
SO a^ is playing the role of the regularization p a r a m e t e r and the model p a r a m e t e r weights simultaneously. Let us assume now t h a t we have uncorrelated d a t a with t h e equal variances,
84
Linear discrete inverse problems
and similarly for the a priori covariance of the model,
Then equation (3.105) turns out to be mo = (A^A + H)-^(A'^d + kmaj^), where
2
a k = -4-=a
(3.107)
(3.108)
plays the role of the regularization parameter. We can see from formula (3.108) that large values of the variance cr^ of the model parameters correspond to a small regularization parameter a, and vice versa, large values of a correspond to a small variance cr^. This means that, without regularization (a close to zero), the uncertainty in determining the inverse model is great, while with regularization it becomes smaller. The last formula illustrates once again the close connection between the probabilistic (Tarantola, 1987) and deterministic (Tikhonov and Arsenin, 1977) approaches to regularization. 3.6
The Backus-Gilbert M e t h o d
3.6.1 The data resolution function Backus and Gilbert (1967) have developed a solution of the linear inverse problem that estimates the resolution of the model parameters. In this section, we will outline the basic ideas of this approach and will show how the regularization can be introduced in the Backus-Gilbert method. Suppose that we have observed the data in a fixed number of observation points: j = 1, 2, ...AT; dj e Eu me
M,
where M is a Hilbert space L2 of model parameters, function m =m (r) is determined within some 3-D domain V of the Earth interior, and fj{ni) represents a linear functional defined on M. According to the Riesz representation theorem (see Appendix C) there exist functions \j=lj{r) (elements of the space M), called the data kernels, such that d, = ( m , l , ) = ( m ( r ) , / ^ ( r ) ) , (3.109) j = l,2,...Ar;,
IjEM,
where the scalar product is calculated in the L2 metric over domain V : ( m , 1,) = (m (r),/,(r)) = ffj
m(r)lj{r)dv.
(3.110)
The Backus-Gilbert
85
Method
In the majority of geophysical problems we know the data kernels 1^. It is shown in Appendix C that the solution of the inverse problem which has the minimum norm in M, can be presented in the form: (3.111)
m'-''iTo) = Y.f^Mro), 1=1
where coefficients P^ satisfy the equation N
N
d: = ^ A a . ( r ) , / ; ( r ) ) = ^ r , . ^ ,
(3.112)
d = r/3.
(3.113)
1=1
1=1
or in matrix form Hero, /3 = [/?,,/?2,
PN]''
and [r,,:]==r = [(l,:,l,)] = [(/,(r),/,(r))]
(3.114)
is a synnnctric square N x N Gram matrix. From (3.112) we have at once
whore
7 = (7,:,,] = r - \ or in matrix form (3.116)
(3 = T 'd. Substituting equation (3.115) into (3.111), we obtain N
m-'(r„) = E 1=1
N
E
N
^rj^M^o)
= E
'^^•('•o)'^:"
(^^-ll"^)
j= l
3= 1
where TV
1=1
or using matrix notation, nf'\vo) = L''(ro)/3 = L-^'(ro)r-M = b^(ro)d, whore b^'(ro) = [6i(ro),62(ro), ,6yv(ro)] = L ^ ( r o ) ^ - ^
(3.118) (3.119)
86
Linear discrete inverse problems
and L is a vector formed by the data kernels, L^ = [/i(ro),/2(ro),
,^iv(ro)].
Alternatively, data d can be expressed through the true model by equation (3.109). Thus, substituting (3.109) into (3.117) we find the relationship between the estimated model and true model N
m''\ro)
= ^6,(ro)(m(r),/,(r)) =
fi(ro,m(r)).
(3.120)
The linear operator R is called the resolution operator. Using the definition of the scalar product (3.110), it can be written /?(ro,m(r)) = ^ 6 ^ ( r o ) / / / m{r)lj{r)dvr
= ///
p{ro,r)m{r)dv,
where p(ro,r) = ^ 6 , ( r o ) / , ( r )
(3.121)
is the resolving kernel of the resolution operator or the data resolution function. The "sharper" this function, the better the estimated model fits to the true model. The best estimation can be obtained if p(ro,r)
^6{ro-r),
where ^(r) is a delta function (Fig. 3-5). Note that equations (3.120) can be rewritten in matrix form m^'\ro)
= (m(r),b^(ro)L(r)) = (m(r), p(ro,r)),
(3.122)
where p(ro, r ) = b^(ro)L(r) = L ^ ( r o ) r - i L ( r ) . (3.123) 3.6.2 The spread function Let us introduce a spread function as a measure of difference between the resolving kernel and the delta function: 5(ro) = III
[p(ro, r) - ,5(ro - r)]^ dv.
(3.124)
We will prove that the resolving kernel given by (3.121) delivers us the minimum of the spread function.
87
The Backus-Gilbert Method
f5(r-ro)
p(ro, r)
Figure 3-5
T h e plots of the resolving kernel of the resolution operator. The "sharper"
this function, the better the estimated model fits t o the true model. T h e best estimation can be obtained if the resolving kernel is approximated by the delta function.
Let lis siil)stitiitc (3.121) into (3.124) and find the constants bj{r{)) (for given To) from the conchtion t h a t the spread function ^(ro) is miniimini: 5^/;,(ro)/,(r)-^(ro-r;
^S'(rn
(In ~ mill.
j - i
Wo then calculate tlic derivative of the spread function with respect to //.(ro ): OS{T,
N
rr, 9
14v)
|^/;,(ro)/,(r)~^'(ro-r)|r/r;-0. j=i
From the last equation we liavc ^6,(ro)(/.(r),/,(r))=/.(ro) Using (3.114), we write ^6^-(ro)r^-,; = /^(ro or in matrix form, b^^(ro)r = L^^\ro). Inverting the last equation, we obtain (3.119). Thus, we have proved t h a t t h e minimum norm solution (3.111), which we have found using the Riesz representation theorem, is at the same time the minimum spread function solution, obtained in the Backus-Gilbert method.
88
Linear discrete inverse problems
3.6.3 Regularized solution in the Backus-Gilbert method According to equations (3.116) and (3.123), the key formulae in the Backus-Gilbert method include the inversion of the Gram matrix P. In general, this matrix can be ill-conditioned so that the solution of the inverse problem (3.111) and the estimation of the resolving kernel p(ro,r) turn out to be ill-posed problems. To overcome this difficulty, we can use the Tikhonov regularization method to obtain the regularized Backus-Gilbert method (Zhdanov, 1993). We suppose that the regularized approximation of the model parameters m^^^(ro) can also be presented in the same form (see equation (3.111)) we have used in the classical Backus-Gilbert method, but with different coefficients /?^^^: N
m'-'=«(ro) = Y. PT'liiro) = L^(ro)/3'-^^
(3.125)
i=l
The matrix column of the unknown coefficients /3^^^ can now be found based on the minimization of the Tikhonov parametric functional: P^(/3""^d) = (d-d"^^)^(d-d"^^)+a(m""^(r),m'""^(r)) = min.
(3.126)
Here, the column vector of theoretical data d^^^ for the given model m^^^(ro ) can be determined, according to expression (3.113), by the formula:
Substituting (3.127) and (3.125) into (3.126), we have P"(/3^^^ d) = (d - Tf3'''f{d
- r/3^^^) + a (L^ (r) /3^^^ L^ (r) (3'^') = min.
Now we should calculate the first variation of the parametric functional with respect to the perturbation of the parameters /3^^^: Spc^^f^reg^ d) - -2{T6f3''')^{d
- r/3""^) -^ 2a (L^ (r) 5^"^^ L^ (r) ^''^) .
(3.128)
The second term in (3.128) can be calculated as: 2a(L^(r)(5/3'"^^L^(r)^^^^) = 2a {Sf3'''f
(L ( r ) , L^ (r)) /3""^ = 2a {6f3'''f
^/3""^
where we use an obvious identity; r = (L(r),L^(r)). Substituting (3.129) into (3.128) and setting SP" equal to zero, we obtain -2(6/3'"'f
[ r ^ ( d - rf3''s) - ar/3""s] = 0,
(3.129)
89
The Backus-Gilbert Method from which we have Since T^ = F , we can solve the last equation preg ^ ( r + « I ) ^ M .
(3.130)
Comparing (3.130) with (3.116), we can see now t h a t we have to invert t h e well-conditioned m a t r i x T + al rather t h a n the possibly ill-conditioned m a t r i x T. Substituting (3.130) into (3.125), we write m--'(ro) = L ^ ( r o ) ( r + aiy'd
= b^'(ro)ci,
where hlivo)
=L'{r
(3.131)
o){T + aiy\
By comparing (3.131) and (3.119) we see t h a t , in the case of the regularized solution we invert the well-conditioned m a t r i x (F + al) instead of ill-conditioned F , which makes our result stable. Also, we can rewrite equation (3.122) for the regularized m e t h o d as: m - " ( r „ ) = ( m ( r ) , b ; : ( r „ ) L ( r ) ) = ( m ( r ) , ft,(rn, r ) ) , where we have a new ex|)ression for the regularized resolving kernc^l /;„(ro, r ) =. b:i'(ro)L(r ) ^- L''{r,){T
+ aiy
' L ( r ).
T h e o])timal value of the regularization parameter a can be found from the misfit condition (d - Tf3'''^)'\d - F/3""0 = ^\ (3.132) where 6'^ = \\6d\(^ = ((^^d)'^X^^d) is the level of noise in the observed data. S u b s t i t u t i n g equation (3.130) into (3.132) we obtain a clearer form of the misfit condition (d - F ( F + a:I)-M)^^' (d - F ( F -f- aiy^d)
= 6.
T h e value of the regularization parameter should therefore be balanced with t h e level of t h e noise in the data. T h u s , we have shown t h a t the ideas of the regularization theory can be n a t u rally implemented in the Backus-Gilbert method, providing it with optimal stability and efficiency.
References to Chapter 3 Backus, G. E., and T. I. Gilbert, 1967, Numerical applications of a formalism for geophysical inverse problems: Geophys. J. R. Astr. S o c , 1 3 , 247-276.
90
Linear discrete inverse problems
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. Druskin, V., and L. Knizhnerman, 1994, Spectral approach to solving threedimensional Maxwell's diffusion equations in the time and frequency domains: Radio Science, 29, No. 4, 937-953. Golub, G. H., and C. F. Van Loan, 1996, Matrix computations. Third edition: The Johns Hopkins University Press, Baltimore and London, 694 pp. Levenberg, K., 1944, A method for the solution of certain nonhnear problems in least squares: Quart. Appl. Math, 2, 164-168. Marquardt, D. W., 1963, An algorithm for least squares estimation of nonlinear parameters: SIAM Journal, 11, 431-441. Menke, W., 1989, Geophysical data analysis: Discrete inverse theory: Academic Press, Inc., San Diego, 289 pp. Tarantola^ A., 1987, Inverse Problem Theory: Elsevier, Amsterdam, Oxford, New York, Tokyo, 613 pp. Tikhonov, A. N. and V. Y. Arsenin, 1977, Solution of ill-posed problems: John Wiley & Sons, 258 pp. Zhdanov, M. S., 1993, Tutorial: regularization in inversion theory: CWP-136, Colorado School of Mines, 47 pp.
Chapter 4 ITERATIVE SOLUTIONS OF THE LINEAR INVERSE PROBLEM In this C h a p t e r I wih examine a general approach to the solution of the linear inverse problem based on the iterative method. T h e practical problem is t h a t in m a n y applications the m a t r i x of the forward modeling operator is so huge t h a t it becomes very difficult to find the inverse matrix, or generalized inverse matrix, required by the least squares m e t h o d s described in Chapter 4. In this situation the problem can be solved by iterative methods. I consider, first, the general principles of the iterative teclmique applied to linear operator equations. I then apply these principles to the hnear inverse problem and implement the regularization. 4.1 4.1.1
Linear operator equations and their solution by iterative m e t h o d s Jyinrar inverse 'problems and the Elder
eq'iiaiion
Let us consider a general linc^ar inverse geophysical i)roblem, described by an operator e([uation: d = ylm, (4.1) where m G M represents the model parameters (an element from a real Hilbert space M of the model parameters), d G D is observed geophysical d a t a (an element from a real Hilbert space D of the d a t a sets), and A is a linear operator acting from M to D. One general way to solve the inverse problem (4.1) is the trial and error m e t h o d . It is based on minimizing a square norm of the residual, r, between the predicted d a t a , d^^ = ylm, for a given model, m , and observed data, d: / ( m ) = | | r f - \\(F - d\\' = II A m - d f = m m . Functional / is called a misfit functional.
(4.2)
It can be written in t h e form
/ ( m ) = {Am - d. Am - d)jj ^ min,
(4.3)
where t h e lower subscript D indicates the inner product operation in the space D. It is well known t h a t the best way to solve an optimization problem for a conventional function is based on differentiating t h e function and equating t h e derivatives to zero. A similar approach can be applied, in principle, to functionals. However, in
92
Iterative solutions of the linear inverse problem
doing so, we have to use analogous results from calculus for functionals and operators. This generalization has been discussed in Appendix D. The problem of minimization of the misfit functional (4.3) can be solved using variational calculus. Let us calculate the first variation of / (m) : 6f (m) = 2 {A6m, Am - d ) ^ . The necessary condition for the minimum of the functional / (m) has the form 6f (m) = 2 (^^m, ^ m - d ) ^ = 2 ((5m, ^* [Am - d ] ) ^ = 0, for any ^m, where A^ is an adjoint operator (see Appendix D), and the lower subscript M indicates the inner product operation in the space M. From the last formula we have the following operator equation: ylM = ^ M m .
(4.4)
Equation (4.4) is called the Euler equation. The element m at which a misfit functional achieves a minimum is a solution of the corresponding Euler equation. In the case of discrete data and model parameters, the Euler equation becomes a normal equation (3.8) for the corresponding system of linear equations (3.4). Note that the Euler equation can be obtained formally by applying the adjoint operator, A^, to both sides of equation (4.1). However, the Euler equation (4.4) is not in general cases equivalent to the original inverse problem (4.1). The main characteristic of the Euler equation is that it provides the minimum of the misfit functional. The Euler equation (4.4) is equivalent to the original equation (4.1) if each of these equations has a unique solution in M. Note that, in this case, the operator A*A is always a positive self-adjoint operator, because {A'^Af = >1M, ( ^ M x , x ) = (^x,Ax) > 0,
(4.5)
and (ylMx,x) -= (>lx,Ax) = 0 , if and only if x = 0. The linear operator A acts from the space M to the space D. It is easier to solve the inverse problem if the corresponding operator equation describes the transformation within the same vector space, for example, within a space of the model parameters M. By introducing the Euler equation, we have reduced the general linear inverse problem (4.1) to the linear operator equation (4.4) determined for the elements of the model parameter space. We will consider below several methods of solving this linear equation.
Linear operator equations and their solution by iterative methods 4.1.2
The minimal
93
residual m^ethod
Let us consider a general linear operator equation: f = Lm,
(4.6)
where f, m G M , and L is a linear continuous operator in M, We assume also, in t h e beginning, t h a t M is a real Hilbert space. T h e general iterative solution of equation (4.6) can be expressed by t h e formula m^+i = m ^ + A m , , = m ^ - knYn, n = 0 , 1 , 2,...
(4.7)
where mo is some initial approximation, A m , , is an iteration step, A m , , = -/cnrn,
(4.8)
r„, = Lm,, - f,
(4.9)
r„ is a residual on the n - t h step,
and A:„ is an iteration coefficient. Note t h a t the residual on the (n -f l)-tli step, r„)_i, is related to r„ by the formula r„_^i ^ Lm,/1 1 — f ~ L m „ -- A'„Lr„ — f ^^ r„ — k,JjY,,.
('I-IO)
An obvious choice^ of the iteration coefficient A:„ is to minimize the norm of the residual r,M 1 •
||r„+i II' = ||r„ - A:„Lr„||' = cl>2 (A-,,) = m m .
(4.11)
T h e function ^^ (A"„) can be expressed using the inner product operation in the Hilbert space M : ^
[r\'n) = (r„ — knLTnj r„ — A:,,Lr„j
= (r,,, r„) - 2kn (r„, Lr„J + kl (Lr„, Lr„,) = min .
(4.12)
Differentiating the function ^^ {kn) with respect to the iteration coefficient A:„, we can write an equation, realizing the minimmn condition (4.12): }
"^ = - 2 (r„, Lr„) + 2A':,, (Lr„, Lr,,) = 0.
ClKn
From the last formula we obtain immediately an expression for the value of A:,, t h a t minimizes ||r,^^^.i|| : ^^n = 7 ^ ^ ^ ^ . (Lr„,Lr„)
(4.13)
94
Iterative solutions of the linear inverse problem
The corresponding minimum of the function $ (kn), according to (4.12), is equal to ^{kr.) = l l r . ^ J = J K , r O - P z ^ ^ = Iknll JI - / ' ; : 2 t ' ^ ^ ' 2 > y (Lr^,Lrn) y Hr^H ||Lr^||
(4.14)
According to (4.14) the residual on the (n + l)-th iteration is always less than or equal to the residual on the previous iteration: Ikn+ill < ||rn||,
(4.15)
and equality is possible only if the elements Lr^ and r^ are mutually orthogonal: {Lvn, Fn) = 0. The iterative method of the linear operator equation solution, based on recursive formulae (4.7) and (4.13), is called the minimal residual method (MRM). Note that, according to the Schwarz inequality (A.38),
from which it follows that inequality (4.15) holds for any linear operator L in the Hilbert space. However, this inequahty does not guarantee the convergence of the iteration process, based on the recursive formulae (4.7) and (4.13). In order to obtain a converging method, we should impose an additional condition on the linear operator L. Let us assume that L is a positively determined (PD) linear continuous operator acting in a real Hilbert space M. According to the definition of the PD linear operator (Appendix D), the following inequality is valid for all r^GM : ( r n , i : r 0 > 7 ( r n , r , ) = 7 | | r n f , 7 > 0.
(4.16)
Also, any linear and continuous operator L is bounded: ||L|| < c , c > 0 .
(4.17)
Note that, according to inequality (4.17) and the Schwarz inequality (A.38),
||i^r„f
^ ' ^
Substituting (4.16) and (4.17) into (4.14), and taking into account (4.18), we obtain the inequality " ^ " ^ ^ I ' = P V< A / ^r^ < 1 , V
(-
(4.19)
95
Linear operator equations and their solution by iterative methods from which it follows t h a t l|rn+i||<||r„||,
(4.20)
l|r„+i||=p||r„!|.
(4.21)
and It is i m p o r t a n t to emphasize t h a t for a P D linear operator we always have p < 1 according to (4.19), because 7 > 0 and c > 0. Applying formula (4.21) to each subsequent residual, we find
l!r„+i||=p"+M|roil.
(4.22)
Therefore, t h e residuals tend to zero as n ^ 00 : ||r^+i|| ^ 0, i f n - ^ cx).
(4.23)
Based on formulae (4.22) and (4.23), we see t h a t the minimal residual m e t h o d converges if L is a positively determined (PD) linear continuous operator, acting in a real Hilbert space M. Actually, the following i m p o r t a n t theorem holds. T h e o r e m 18 .• Let L he a positively determined (PD) linear eontinuous operator, aeting in a real Hilbert spaee M. Then the solution, of the linear operator equation (4.6) exists a.nd, is iiniqiie in A/, and the niininud residual method, based on the reenrsiue formulae (4-''0 o^nd (4-13), exrnvercjes to tins sedation for any uiitial approximation niQ.
Proof: Let us evaluate the difference between the (n + l ) - t h and r/-th iterations. According to formulae (4.7), (4.13), and the Schwarz inequality II
M
[^TD P'^n)
I,
II
On the other hand, from the condition (4.16) and the Schwarz inequality we have \\LrJ
\\rj
> ( r „ , L r „ ) > 7 (r„, r j = 7 l l r „ f ,
and | | L r „ | l > 7 l | r „ | | , 7 > 0.
(4.25)
Substituting (4.25) back into (4.24), we obtain ||m„+i-m„(| < - | | r „ | j .
(4.26)
96
Iterative solutions of the linear inverse problem
From inequality (4.26) and taking into account (4.22), we find ||mn+i-mn|| < V l k o l l . 7
(4.27)
In particular, for any / > n, we obtain ||m/ - rrinll = ||(m/ - m/_i) -h (m/_i - m/_2) + .... (m^+i - m^)!! < < (p^-^-" +p^-2-" + ... + 1) V V
^7
llroll < - T ^ llroll . 7 1 —p
(4.28)
Thus, we conclude that the sequence of elements irin, generated by the minimal residual method, is a Cauchy sequence, because the distance between any two elements goes to zero, \\nii — mn\\ —> 0, as /, n —^ oo (see Appendix A, section A.2). Since the Hilbert space M is a complete linear space, the Cauchy sequence m^ converges to the element in e M : rrin —> m, if n —> oo. Now, we can consider the limit of formula (4.9) for the residual as n ^ cx) : f = Lmn-r, ^L(m),
(4.29)
because r^ -^ 0, according to (4.23), and L is a continuous operator. Formula (4.29) completes the proof that the minimal residual method converges to a solution of the linear operator equation (4.6). Note that this solution, m, is unique. For, let us assume that there is another solution, m^ Applying the linear operator L to the difference between these two solutions and taking into account (4.25), we obtain \\L (m - m')|| > 7 ||m - m'||, 7 > 0. On the other hand, ||L(m-m')|| = ||Lm-Lm'|| =0, therefore ||m — fn'll = 0, and m = m'. As a result, we conclude that there exists an inverse operator, L ~ \ and it is a bounded operator because of the condition (4.25) (see Appendix B): \\L~'\\<^.
(4.30)
This result was extended by Samokhin (1998) to the more general case of the linear operator equation in a complex Hilbert space M (see Appendix A), which is extremely important for many geophysical applications, for example, in electromagnetic forward and inverse problems. The main difference between the real and the
97
Linear operator equations and their solution by iterative methods
complex Hilbert spaces is t h a t in the complex space the s y m m e t r y property of the inner product operation is different: (f,g) = (g,f)%
(4.31)
where the asterisk * means complex conjugate. From the last formula it follows t h a t (f,A;g) = r ( f , g ) .
(4.32)
T h e b e a u t y of t h e minimal residual m e t h o d is t h a t it applies to this situation as well. T h e only difference is t h a t in the case of the complex Hilbert space, the o p t i m u m iteration coefficient k^, determined according to equation (4.13), may have a complex value, because an inner product (r„, Lr^J is a complex number. T h e corresponding formula (4.14) is modified accordingly: $ (kn) = | | r n + l | | = V (I'n - /•'..LF,,, F,, -
V
k,,LYn)
iirniniLrnir
In a comi)lex Hilbert s})ace the Schwarz inequality (A.38) takes the form |(r„,Lr„,)|"<||r„,||^||Lr„||^
(4.34)
from which it follows t h a t inequality (4.15) holds for any linear operator L in a complex Hilbert space as well. In a similar way, in order to obtain formula (4.22) for a complex operator L, we have to require t h a t L is an absolutely positively determined (APD) linear continuous operator in a complex Hilbert space. T h e last property, according to the definition of Appendix D, means t h a t operator L satisfies the condition | ( m , L m ) | > 7 ( m , m ) = 7 | | m | | ' , 7 > 0,
(4.35)
for all m G M . Based on this condition, one can repeat all calculations (4.16) -(4.21) and arrive at t h e same result, ||r^+i|| - ^ 0 , i f n ^ 00,
(4.36)
in a complex Hilbert space. Formula (4.36) shows t h a t the minimal residual m e t h o d converges if L is an A P D linear continuous operator, acting in a complex Hilbert space M. As a result, we arrive at the corresponding theorem, which was proved originally by Samokhin (1993, 1998).
98
Iterative solutions of the linear inverse problem
Theorem 19 ; Let L be an absolutely positively determined (APD) linear continuous operator, acting in a complex Hilbert space M. Then the solution of the linear operator equation (4-6) exists and is unique in M, and the minimal residual method, based on the recursive formulae H-l) and (4-^3), converges to this solution for any initial approximation UIQ The proof of Theorem 19 repeats practically all the steps of Theorem 18. For example, according to formulae (4.7), (4.13), and the Schwarz inequality (4.34), we obtain II
II
K^n, Lrn)\ ,,
,,
knll \\LYn\\ < " : . " ,2"" llrn|| = ^ ^ | | r „ | | . (4.37) \Lvr.\\' " ' " • ll^r.ll On the other hand, from the condition (4.35) and the Schwarz inequality, we have \\LYn\\ \\Tn\\ > {{Vn, LVn)] > 7 (fn, T^) = 7 I k n f ,
and ||i^r,||> 7l|rn||, 7 > 0 . Substituting (4.38) back into (4.37), we find
(4.38)
||mn+i-mn|| < - | | r n | | , (4.39) 7 which is similar to inequality (4.26) in the real Hilbert space. Note that, owing to condition (4.35) for the linear operator L, in the complex Hilbert space equation (4.22) also holds. From inequality (4.39) and taking into account (4.22), we find | | m , + i - m , | | < Vllroll(4.40) 7 The remaining steps in the proof repeat those in the proof of Theorem 18 Note, in conclusion, that one can estimate also the convergence rate of the minimal residual method based on formula (4.22) for the residuals: M=p"||ro||
llroll-
Observing that ||r^|| = WLnin - f II = ||Lm^ - Lm|| > 7 ||mn - m|| , we find
||m,-iii||<Myi-Jj
||Lmo-f||.
(4.41)
The last formula can be applied to the MRM solution of the linear inverse problem in both real and complex Hilbert spaces.
99
Linear operator equations and their solution by iterative methods 4.1.3 Linear inverse problem solution using MRM In this section we apply the minimal residual method (MRM) to the solution of the Euler equation (4.4). Writing L = A*A, and f = A^d, we can rewrite the iterative process (4.7) - (4.13) as follows m^+i =nin~
kJn^ n = 0,1,2,...
(4.42)
where 1^ and kn are calculated by the formulae h - Lnin -f
= A' [Amn - d] = A^r^,
(4.43)
(L,^ML) ^ p u f and the residual, r^, is calculated for the original inverse problem (3.1): r^ = AiUn — d. Thus, the MRM algorithm for the inverse problem (4.1) solution can be summarized as follows: r,, = Anin - d, 1^ = l(m^J =A^r^, (a) (4..15) kn-\\gnf/\\K\\\
(C)
The iterative process (4.45) is terminated when the misht reaches the given level ^o : l l r ^ f <£(,. It is also worthwhile finding the iteration coefficients k[^ corresponding to (4.4r)(c)), based on the residual minimum in the original inverse problem equation (4.1) rather than in the Euler equation (4.4). In this case we have to minimize the function ^' iK) = IK^if
= P m . + i - d||^ = WAmr, -~ A^Al,, - d||^
- ||r, - k'^Alnf
= (r^ - k'^Ah, r , - k'^Aln) = mm,
(4.46)
which is equal to the misfit functional value at the (n + l)-th iteration. Applying the variational operator to the last equation, we find the minimum condition <5fc;,$2 (ki) = 4 , (r„ - k'^Aln, r„ - fc^/ll,,,)
100
Iterative solutions of t h e linear inverse problem
= - 2 Re [8k'^ (^U, r , - k'^AXn)] = 0. One can verify this and the next formulae by comparing with example 84 from Appendix D. From the last formula we obtain at once 7 / _ (fn; A^n) _ (r^t, AA Tn) _ [A Fy^, A Tn) _
"~
||Al„f "
||^l„f
"
\\Al„f
||^n||
~ \\Al„f'
/ . .^x
^ •
and the corresponding minimum of the function ^ {kn) is equal to
Note that this choice of coefficients /c^ provides the orthogonality between vectors U and l^+i. In fact, according to (4.42) and (4.43), Tn+l = Amn+1 - d =Amn
- k'n^ln - d = T^ - k'n^ln,
(4.49)
and U i
= A^Tn^i = A'Vr, - k'^A'Aln
= U " k'^A'Air..
(4.50)
From the last formula and taking into account (4.47), it follows that ('n+l5 In) — v'n ~ k^A Aln, in) = ||ln||
~ ^n ( ^ Ain, In)
= ||l„f - k'„ \\AUf = ||l„f - J
^ Pl„||^ = 0.
(4.51)
This new modification of the minimal residual algorithm for the linear inverse problem can be summarized as follows: r^ = Anin - d, In = l(m^) =A'rn,
(a) grt = ^In,
(b)
(4.52)
^;=l|ln||Vl|gn|r,
(C)
Note that this algorithm is equivalent to the steepest descent method, which we will describe in Chapter 5. Accordingly, we refer to this technique as the steepest descent method for the linear inverse problem solution. A detailed explanation of the geometrical ideas behind this method will be given in Chapter 5. Substituting (4.43) into formula (4.48), we obtain the following estimate for the residual, r^+i, generated by the algorithm (4.52): Ir
II
l l r II / l
K..IIHMW1-
. .. . .
li^n.AA^TnW
,
,.
(4.53)
A generalized minimal residual method
101
Let us assume now that the operator AA*" is an absolutely positively determined (APD) operator, which satisfies the conditions > 7 (m, m ) = 7 | | m f , 7 > 0,
\{m,AA''m)\
(4.54)
and WAA-'W < c, c > 0. Then, using an inequality, similar to (4.18),
c2 -
\\AA*rnf\Kf
~
'
one can prove t h a t
fe^-p
'«*'
and l|rn+i||=p"+Mkoll-
(4.57)
Based on formulae (4.55) - (4.57), wc can prove a theorem, similar to T h e o r e m 19. T h e o r e m 20 .• Let A be a Iniear coniinii.oiLs openitor, acting from a complex Hilhcrt space AI to a complex Hilbcrt space D. If L ~ A/V ts an absolutely positively de~ termviicd linea,r conlinuxnis operator, actiny vn a complex Ililbert space M, tJien the solution of tlie linear inverse problem (4-V <^'3:z5Ls and is unique in M, and tJie nmiimal residuxil mxdhod, based on the recursive form/ulae (4-52), converges to this solution for any initial approximation mo. T h e proof of this theorem is similar to the proof of Theorem 19. T h e main problem with the practical application of this m e t h o d is t h a t , in the case of an ill-posed inverse problem, t h e required condition for t h e operator L = AA'^ does not hold. Therefore, one should use the regularization technique, which will be discussed later. 4.2 4.2.1
A generalized minimal residual m e t h o d The Krylov-sub space
method
An effective m e t h o d to accelerate the convergence of the M R M algorithm is based on the Krylov-subspace m e t h o d (Kleinman and van den Berg, 1993). We introduced t h e Krylov subspace in Chapter 2 as the finite dimensional subspace K^ of t h e Hilbert L^^^r^: space M , spanned by t h e vectors r^, L r „ , L^r^, Ks = span {r^, Lr^, L^r^,
L'^^hn}
.
Iterative solutions of the linear inverse problem
102
Note that the dimension, 5, of the Krylov subspace Kg is always less than or equal to g, {s < q), and 5 = g, if and only if the system {Lr^,L^rn, L^^n} is linearly independent. The Krylov-subspace method is based on approximating the iteration step, Anin, in the recursive formula (4.7) by an element of the Krylov subspace Ks (4.58)
mn+i = rUn + Arrin, where Arrin = - ^
(4.59)
KiL^ ^r^, n = 0,1,2,...; g > 1,
Fn is a residual vector (4.60) and (fcni, kn2, ^nq) is a set of iteration coefficients which may be complex, in general. The difference between the iteration procedure (4.58)-(4.59) and (4.7) is that now we move from one iteration to another not only in the direction of one residual vector Yn, but along a multidimensional Krylov subspace spanned by the vectors {r^jLr^i,
L
r^} .
It is obvious that the coefficients kni should be selected based on the minimum of the norm of the residual Yn+i ' \Yn+lf = \\Lmn+l -ff
=
Lnin
- ^
knlL^Yn
~ f
1=1
^n
/
^ f^nlL
Yji
knq) = m i u .
^^{knukn2,
(4.61)
l=\
The minimum condition for the function $^ (/c^i, kn2, knq) can be obtained, as in example 84 from Appendix D, by applying the variational operator 6 with respect to the coefficient knp and equating the result to zero: ^kr^p^^ {knukn2,
knq) = ^fc^p \ ^ r i ~ ^ knlL^Yn,
^n - ^
1=1
-6kn
^ ^n, '^n ~ / ^ knlL Yn I —
6kn
Z=l
- 2 Re 6k np
^1*72 5 ^n
knlL^Y^
1=1
^ I*n? ^n
/
^ l^nl^ I*n
/=1
/ ^ knlL/ r „ 1=1
0,p=l,2,....q.
A generalized minimal residual
method
103
From t h e last formula, we obtain at once a system of linear equations for t h e unknown coefficients kni 'J2kni{L'r^.
U'r^)
=(rn,
(4.62)
L^^n)
1=1
We will denote by K'^ the finite dimensional subspace spanned by t h e vectors {LYn.L'^n,
L'^Yn} • K[ = span [LY^, L^YU,
iv^r^} , s < q.
Note t h a t K'^ is a subspace of the Krylov space K[_^^ = span {r^, Lr^, L'^Yn, i^^r^} . Expressions (4.61) and (4.62) actually d e m o n s t r a t e t h a t we find t h e coefficients ^7?/ by solving an approximation problem in the Hilbert space (see A p p e n d i x A, section A.4.2): wc determine the vector r^ = X^Li KiiL^^n ^ ^ 5 , closest to r^^. Note t h a t the system of equations (4.62) may have a nonunique solution, if the vectors { L Y „ , IJY,,, L^^Yri} are linearly dependent, which happens if ,s < q. In order to select a unique solution, and also, to simplify the system of equations (4.62), wc can a|)|)ly the Laiiczos method. 4.2.2
The Lanczos
iiiintrnal residual rneUiod
The Lanczos m e t h o d is ])a,sed on generating an orthogonal basis <^ f/
, f,
,
fs" >
in the su])space A'^, using the Gram-Schmidt orthogonahzation process (see Ap])endix B), and h u d m g the cxx^ihcicnits k^a a.s ])r()jecti()ns of th(^ r(\si(hial Y,, 011 thc^ vcx'lors from this basis. In Ap])endix 13 (section 13.4.1) we have outlined the constructive procedure for An)
converting an arbitrary set of vectors into a corresponding orthogonal set < f|' r{n)
with the property t h a t the linear subspace of H spanned by < ^1
, f^'
r{n) ' ^2 )
^[n)
the same a.s t h a t spanned by {Lr^, /^^fn, L'^^^} • T h e first step of the orthogonahzation process is
Z.gl"' = Lr,„ where (77,)
gi
(4.63)
= rn.
T h e next step is constructing a vector ^ \ which is orthogonal to f|" L^Y„
A21 J^Yn
L LY,,
^21 g i
where L''rn,fi ^21
An)
(77)
and g2(n)
LY^
^21 g l
•
: — ^B2
5
Iterative solutions of the linear inverse problem
104
n(n)
(n)
On the third step we construct a vector fg , which is orthogonal to fj;
^
^n — ^ 3 2 ^ & 2
~ ^31 -^Bl
(^)rv(^)
= L^t\
\(^)«.(^)
A32 g 2
— I^ \J^ Tn
and pw. ^
A31 g ^
where ^31
2
n
-
(n)
' ^32
-
?{n)
2
'
and §3
— ^
^n — ^ 3 2 6 2
~ ^31 6 1
'
On the step number p p-i
p-i n)
f (^) = L-r. - J2 A?f/^^ - L^^n - J2 ^r^^g^^ = ^^^P 1=1
1=1
where
„)_K:^ p(n)
and
2
'
l-l
i^'Z -_ i ' r „ - ^ A M " \ / = l,2,3....,p.
(n) _
k=l
According to the construction, we have f^") = Lg("), p=
1,2,3....,s,
and any vector Lgp is orthogonal to every preceding vector ^gp_i, ^gp_2 5 ••••5 ^^nUsing the new set of orthogonal vectors, we can replace the minimization problem (4.61) by the following
I*n
/
(4.64)
kns) = miu .
'^^{knukn2,
^ rCnl^i
1=1
The corresponding minimum condition is given by the equation <5fcnp^^(^nl,fcn2,
—
OKrip I t p
5 I*n
/
^ knlti
(n)
kns)
^knp I ip
, r„
/
(n)
^ knii-i
/=1
A generalized minimal residual
= - 2 Re
method
I Ip
^rCnp
105
5 ^n
/
^ ^n/t/
(n)
= 0, p = l , 2 , . . . . 5 .
1=1 p(n)
From the last equation and taking into account t h a t aU vectors fp orthogonal, (n) r{n)
f.(n)
n[n{n)
are m u t u a h y
, ifp = /,
we find F(^)
^n, Ip
rH
I
I Tn, i^gp
(4.65)
p = 1,2, . . . . 5 .
(n) Ip
Lg^r'
Now t h e iteration coefficients knp can be treated as t h e coefficients of t h e projection of the residual r^^ on the orthogonal basis of the subspace /C^ s
s
r „ « ^ f c „ , f / " ' = L^fc,,g<"\
(4.66)
/=!
/= 1
where the sc^uare norm of the difference.
rn -
(1.67)
LY^knlgi'^' /-I
is minimal. T h e corresponding minimum e'^ of the function ^P^ is equal to *-^=
r„-^fc„,f<"),r„-^Av,f/"' /=1
/=1
|r,,,||-^ - ;^Av,,, ( f ( " ) , r „ ) -J2K, 1=1
Tn
E-
(f/"',r„,)* +
J2kn,Ki
1=1
^ n ? 1/
1=1 {n)
(n)
p(n)
.("•)
=
1-E1=1
r„
1*72 5 ^l
r„
r{n)
(4.68)
Note t h a t from the last formula it follows at once t h a t
T. (=1
An)
nj ^l
r„
F(")
< 1,
(4.69)
106
Iterative solutions of the linear inverse problem
because otherwise e^ will be equal to the negative number. On the other hand
E '=1
|(r„,f,'"')f r„
An)
>0.
(4.70)
= 0
(4.71)
Actually, the condition
E ^=1
|(r.,f«) (n)
r„
(n)
would mean that r^ is orthogonal to any vector f^^ ^ G i^^, and all the coefficients knp are equal to zero. Taking into account that ||rn-fi|| = WLuin+i - f II = WLuin "h I/Alii^ - f II = llr^ + LAmnll
Fn - L ^
knlgl'''^ + ^ X ^ ^nZg/""^ + I ^ A m ^
1=1
<
Tn-
L"^
(n)
knlg,
1=1
+ 11^11
Y^ knigt^ + Am„ ./=i
we find that the norm of the residual Hr^+iH reaches the minimum, equal to e, (4.72)
Ikn-flll = ^, if the iteration step is given by the formula Anin = - ^ A : n / g [ ' ' ^
(4.73)
1=1
Therefore, we arrive at the following formula for iterations: s
m„+i = m„ - ^
k„igl"\
1=1
which minimizes the residual r^+iNote that, if s = 1 ,
l^nA
—
(1*715
•^'^nj
iLVr,
(4.74)
A generalized minimal residual method
107
and t h e Lanczos m e t h o d reduces to the conventional M R M process (4.7) a n d (4.13). According to formulae (4.68) and (4.72), we have t h e following e s t i m a t e for the residual: f^in)
r
|rn+i|
1-E
1=1
\
(4.75)
= p < 1,
p(n)
rr>
because, due to (4.70) and (4.69), 6'
r
f in]
1=1 r„
(4.76)
< 1.
P\om equation (4.75) it follows t h a t (4.77)
Ikn+lll < IK and
Ikn+ill =p\K\\
n+1
=/>
(4.78)
rol
From (4.78), wc can prove the convergence theorem and estimate t h e convergence rate of tlic Lanczos minimal residual method, in a manner similar to t h a t used in the case of the original MRM. T h e o r e m 2 1 .• Let L he an absolutely positively determined (APD) linear eontinuoiis operator, aeting in a eomplex Hilhert spaee M. Then the solution of the linear operator equation (4.6) exists and is unique in M , and the Lanezos minimal residual method, based on the reeursive formulae (4.^^/4) ci-'^d (4-65), eonverges to this solution for any initial a,pproxima,tion mo . Proof: Let us evaluate, first, the difference between the (n + l ) - t h and n - t h iterations. According to formulae (4.74), (4.65), and the Schwartz inequality
^^nlgj.
An+l
<
E-
ig, ig,
n)
<
(n)
^g-
^
/=1
(n)
in) g/
(n)
^f- \K\\
(n)
g(
(n)
E
rn,^g/
llr„(n) ig.
(") g;'
(4.79)
On the other hand, from condition (4.25), ^g;
(«)
(")
> 7 g;
7 > 0.
(4.80)
Iterative solutions of the linear inverse problem
108 and we obtain
(4.81)
|mn+i - m n l l < - ||rn|| From the last inequality, taking into account (4.78), we find g
(4.82)
\\mn+i - n i n l l < -p^'llroll . 7
Thus, we conclude that the sequence of elements nin, generated by the minimal residual method, is a Cauchy sequence. Following the further steps of Theorem 19, we prove that the Lanczos minimal residual method converges to a unique solution, m, of the linear operator equation (4.6): rrin -^ m, n -^ oo. In a similar way, one can estimate also the convergence rate of the Lanczos minimal residual method based on the estimate (4.78) for the residuals: / m„
i-E
mil < \
i=i
(rn,Lg<"^) r„
ig,
2 \
\\Lmo - f II
(n)
(4.83)
/
In summary, the iteration process of the Lanczos MRM technique can be outlined as follows: r„ = Lm„ - f,
g^"' = r„,
gj") = L'-^r„ - E i = \ X^V:\
(a) 1 = 2,3, ...s,
(b) (4.84)
X^^ = (L'r„, Lgi"') / ||Lgi") f , fc„, = (r„, Lg|"^) / ||Lg("' m„+i = m„ - YlLi knigT^-
(c)
{d)
4.2.3 The generalized minimal residual method Saad and Schultz (1986) suggested a slightly different algorithm of the generalized minimal residual method (GMRM), which has been extended for the case of an operator equation in a complex Hilbert space by Samokhin (1998). It is based on the same idea of the Lanczos method of introducing an orthogonal basis < L^i
, L^2 5
•> Lgi
in the subspace K'^ spanned by the vectors {Lr^jl/^rn, L^Vn} • However, the recursive method of constructing this basis is now slightly different. We apply the same formula for iterations as (4.74): rrin+i = nin - ^ knigl""^.
(4.85)
\
A generalized minimal residual
method
109
However, now we treat the iteration step, A m ^ = — X]J=i ^niEi \ as a superposition of 5 substeps. In the first substep we move along a vector gi'^ = r^ : 7
{n)
T h e corresponding residual can be calculated as rn+i,(i) = ivm^+i,(i) - f = L m ^ - KiLg^^^
- f = r^ -
kniLg[''\
where the coefficient kni is selected from the minimum condition for the n o r m of the residual vector rn+i^(i) (||r^4_i^(i)|| = min): ^n^Lg\ knl
Lg\
n)
in)
2
(4.86)
•
Note t h a t the last formula is derived by analogy with the basic formula (4.13) of the minimal residual method. n) In the second substep we move along a vector g2 J") ^
m,,^^.l,(2) - - m , , , + i,(|) -
kn2g2
and the corresponding resichial is computed as 1*77,-1-1,(2) =
•^->n^7;.+ l,(2) "
f = L m , , + i^(i) — kn2Lg2
" f = r n - f l , ( l ) — kri2Lg2
,
wnerc g2
= I^n+l,(l) - P2I g l ^
and coefficient /i^2i i^ selected from the condition of orthogonality between Lg^
and
A;n)
Lr n + l , ( l ) .Lg\ P21
Lg\
(n)
in)
while the coefficient kni is selected from the condition t h a t the new residual vector rn4-i,(2) has t h e minimum norm: (n)
rn+l,(l)5^g2 kni ~" LE^^
In the third substep a+l,(3) — ^72+1,(2) -
n) ^"nSga
Iterative solutions of the linear inverse problem
no
the corresponding residual is computed as rn+l,(3) - ^n-f 1,(2) " ^ n S ^ g s
5 gs
" ^72+1,(2) " P32 g2
~ P3I g l
'
where the coefficients P^i ^^d /?3i are selected to secure the orthogonality of the vectors I/g^ , Lg2 \ and Lg^\
0:" "
llr„(")||^ ig^^
' ^ " ~
llr„(«)|l^
igi
^
and the coefficient /c^a is selected from the minimum norm condition for the residual vector rn+i,(3) : (rn+l,(2),^g3''^) ^nS
^g^' etc.
In substep number p we have nin+l,(p) = n i n + l , ( p - l ) -
knpgp\
and the corresponding residual is equal to p-i
r.+i,(,) = r.+i,(p-i) - /cnpLg(-\ g^"^ = rn-M,(p-i) - ^
^P/^S^^"^'
(^.87)
z=i o(n) where all coefficients /?^ ^ and knp are determined by the orthogonalization and minimal residual principles:
M
_
(^Lrn+i,(p_i),Lg^'' j ^n)
(^rn+l,(p-l),^gp
j
^np
(4.88)
ig^"' p = 2,3, ....s; Z = 1,2, ...,s — 1. In this case, obviously,
(Lg["\Lgi"))=0, fc
(4.89)
A generalized minimal residual method
111
Note t h a t this technique is equivalent to t h e Lanczos method, described above. Certainly, after making s substeps, we arrive at a new residual on t h e (n-f 1) iteration step,
s
(4-90)
Vn~Y.^:nMp'\
where according to the Lanczos method the coefficients k^p have to be determined, using minimal residual principles, from formulae (4.65): (n)
I-n, L^p K,jip
, p = l,2,....s.
(4.91)
Lg^"' Note t h a t expressions (4.91) and (4.88) arc equivalent, because from (4.89) and (4.90)
We should also make IIK^ obvious o])scrvation t h a t all the basic r(\sults obtained in IIK^ ])r(nn()us scH'tion coiiccniiing the convergence rate of the Lanczos method, and Ihc^ corresi)onding Theorems 19 and 21, hold for the G M H M techni(|ue as well. T h e algorithm of the G M R M described above can be summarized as follows: T
r
^n+l,(p) -^ r n + l , ( p - l ) ^
gp
7
(^^)
T (n)
(a)
(n)
knp-L^p
p-1 Ppi r^i^)„in) - rn+i,(p-i) - l^i^i g/ ,
p - z, 6, ....'
(c)
(4.92)
P^;^ = ( L r . , , . , ( , _ , ) , LgJ-)) / ||Lg{^^)||', k^, = ( r . „ Lg^^^) / | | L g ^ - | ' , (d) lYin+i = m ^ - ^^'^^ knigi\
n = 1, 2, 3
(e)
T h e iterative process (4.92) is terminated when t h e misfit reaches t h e given level ^o '• ||ryv||'<eo. T h e advantage of the generalized M R M and Lanczos m e t h o d s over t h e original minimal residual m e t h o d is t h a t now the iteration coefficients are determined a s
112
Iterative solutions of the linear inverse problem
projections of the residual vector on the orthogonal basis in the Krylov subspace, which decreases tremendously the effect of numerical errors on the inversion result. The advantage of the GMRM algorithm (4.92) over the Lanczos method is that the former does not require computation of the linear operator L raised to some power (as one can see by comparing formulae (4.84b,c) and (4.92c,d)), which could lead to a significant increase in numerical error. 4-2.4
A linear inverse problem solution using generalized MRM
Now we can apply the Lanczos and generalized MRM algorithms to the Euler equation (4.4). Denoting L = A*A^ and f = A*d, we can rewrite the Lanczos algorithm (4.84) as follows iiin+i = mn - ^
knil['^\ n == 0,1, 2,...
(4.93)
where i\\H ' and kni are calculated by the formulae /-I
1„ = t^ = A*v^, l[") = {A*A)'-' 1„ - ^ AL")li"\ / = 2,3, ...s,
(4.94)
k=l
(in.A^At^) _ ( A 1 . , 4 ^ ) ) "^nl
A^A\\ in)
A^A\\
(n)
(4.95)
the residual, r^, is determined for the original inverse problem (4.1) A{mn) and \n = t^ = A*rn =A^ [^(mO - d]
(4.96)
is the residual for the Euler equation. The coefficients A}^^ are determined, as usual, from the condition of the orthogonality of the vectors A^'Ali, / = 1, 2, ....5 :
^ik -
u
77772
AMll"^
•
Thus, the Lanczos MRM algorithm for solution of the inverse problem (4.1) can be
The regularization method in a Unear inverse problem solution
113
summarized as follows: r„ = Anin - f,
1„ =
(a)
l{mJ=A*r„,
ib)
^Mli"^ (d) (4.97) In a similar way we can construct a generalized MRM algorithm for solution of t h e linear inverse problem: r„ = yl(m„) - f, ".+ 1,(1)
1„ = l ( m j = / l * r „ ,
iH l^,"-AVn/l^v41|
,
(n)_
1^^= 1„,
-^n+l,(p) ~" l n + l , ( p - l ) ~
(a) in)
knp^'^^\
V - 1 ,c;(''Oi(''0
A'Mn
r /
i(ri)
VAliW
' ^'•"' "
• l l , n . l l,(") i"
.l*.ir(")
, ('/)
(e) (4.98) T h e iterative proeesses (4.97) and (4.98) are terminated when t h e misfit reaehes t h e given level ^o : niy/.f
mn -
Xl/':^! ^ ' n / V
,
n =
1, 2 , 3 ,
T h e advantage of the G M R M algorithm over the Lanczos m e t h o d is again in t h e absenee of t h e power (A'^A)' calculations, which decreases the effect of numerical errors in the inverse problem solution. 4.3 4.3.1
T h e r e g u l a r i z a t i o n m e t h o d in a linear i n v e r s e p r o b l e m s o l u t i o n The Euler equation for the Tikhonov
parametric
functional
In general cases, the inverse problem (4.1) is an ill-posed problem. Therefore, the linear operator L = A'^A on the left-hand side of the Euler equation (4.4) m a y not satisfy condition (4.35) required for the convergence of the iterative m e t h o d s . In this case, one should use the principles of regularization theory, outlined in C h a p t e r 3. Note t h a t we assume again t h a t both M and D are some real or complex Hilbert spaces with corresponding inner product operations, (..., ...)^^ and (...,...)/;.
114
Iterative solutions of the linear inverse problem
The regularized solution of the ill-posed inverse problem is provided by minimization of the corresponding parametric functional: P"(m) = \\Am - df
-f- as(m) = min,
(4.99)
where a > 0, 5(ni) is any quadratic functional, for example 5(m) = {Wm,Wm)j^
=
\\Wmf,
and W is di positively determined linear continuous operator in M. We have established in Chapter 2 that, in the case where A is a linear operator, D and M are Hilbert spaces, and s(in) is a quadratic functional, the solution of the minimization problem (4.99) is unique. Let us find the equation for the minimum of the functional P " ( m ) . We will use the same technique for solving this problem that we considered above for the misfit functional minimization. Applying the variational operator 6 to P " ( m ) , and taking into account that both operators A and W are linear operators, we find the necessary condition for the minimum of the functional P^" (m) : (5P^ (m) = 2 Re {A6m, Am-d)^-\-2a
Re {WSm, Wm)^^ =
2 Re (5m, A'' [Am - d] -h aWWm)j^
= 0, for any 6m.
From the last formula we have the following Euler equation: A^d = ( ^ M -h aWW)
m.
(4.100)
We denote as nia the solution of this equation. Let us introduce a linear operator L" according to the formula L" = AM-h aH^*iy.
(4.101)
This operator is always an absolutely positively determined (APD) operator, because |(L"m, m ) | = \{[A*A + aW*W] m, m ) ^ | = |(AMm, m)^ + a {W*Wm, m ) ^ | = {Am,Am)^ + a {Wm,Wm)j^ > a {Wm,Wm)^ > a^l | | m f , 7«, > 0, (4.102) for all m G M . The last formula is based on a property (4.25) of the positively determined linear operator W : {Wm,Wm)^
= \\Wm\f>jl\\m\\\
7 . > 0.
Thus, the Euler equation has a unique solution, rria, which can be obtained by the minimal residual method, MRM, or by the generalized MRM. We noted in the beginning of this section that the solution of the minimization problem (4.99) is also unique. Thus, we can conclude that it is equal to rria. In other words, we have proved that minimization of the Tikhonov parametric functional (4.99) is equivalent to the solution of the corresponding Euler equation (4.100).
The regularization
115
method in a Unear inverse problem solution
4.3.2 MRM solution of the Euler equation We outline now the computational scheme for finding the solution of the Euler equation based on the MRM technique. Using notation (4.101) and assuming that f = A^d, we can rewrite the iterative process (4.7) - (4.13) as follows m,+i = m , - O n , n = 0,1, 2,...
(4.103)
where 1" and /c" are calculated by the formulae 1^ = {A'A -f aW^W) xnn-A'd ( I n , Lnn)M
k^ = (L-l«, L"l«),,
_
= yl" (y4m^ - d) -f aW'^Wuin, ( I n , ^^^l^n)M
+ ^ K .
(4.104)
^ ^ W ^ ) , ,
||L«l«f
\Arj' + a\\W\-f
(4.105)
Thus, the regularized MRM algorithm for the ill-posed inverse problem (4.1) solution can be summarized as follows: r,, = Amn ~ d, 1^ = l"(m,,J =/lVn+aI4/^Vl/m,„ (a) gn = ^^In,
K = L'%1 ^Z. = "^'K
^^n=[l|gnll' + « l | P n l l 1 / l | h n f
(^^)
,
(4.106)
(c)
mn+1 -= mn - k'^X^
(d)
where L" is determined by expression (4.101). The iterative process (4.106) is terminated when the misfit reaches the given level So '• \\rM\'<eo. We can also apply the steepest descent algorithm (4.52) to the regularized linear inverse problem solution. In this case we have to minimize the parametric functional $2 {k':) = P"(m„+i) = ||/lm„ - C ^ l n ~ d f + a \\W (m„ - k':iZ)f (r„ - CAln,
r„ - k':A\^)^
+ a{W (m„ - ^1^),
=
H/ (m„ - C 1 " ) ) M = ^ i n .
Applying the variational operator to the last equation, we find the minimum condition (see example 84 from Appendix D for comparison): 6K.^'
ik'n) = - 2 Re [5K {Al^, r„ - C ^ l " ,
)D\
116
Iterative solutions of the linear inverse problem
- 2 a Re [8k'^ {Wr^, W (m„ - Cln))M] = 0From the last formula we obtain at once ,,„ _ (rn,Alg)^ + a ( V F m „ , m g ) ^ _
a
a. i a \
lllc>;||'^
|2'
(4.107)
This new modification of the regularized minimal residual algorithm for the linear inverse problem can be summarized as follows: r^ = Artin - d,
(a)
1^ = r ( m j - A * r n + a W ^ W m , ,
(6) (4.108)
C = 111^11 V[ll>ii^ll' + ^l|H^i^ll']
W
Note that this algorithm is equivalent to the regularized steepest descent method, which we will describe in Chapter 5. According to formulae (4.102), we know that operator L^ is always the APD operator for any linear inverse problem. Because of this, we can immediately formulate the following important theorem. T h e o r e m 22 .• Let A be an arbitrary linear continuous operator, acting from a complex Hilbert space M to a complex Hilbert space D, and W be a positively determined linear continuous operator in M. Then the Tikhonov parametric functional P^(m) = \\Am - df
+ a
\\Wmf
has a unique minimum, m^ G M, and the regularized minimal residual method, based on the recursive formulae (4-52), converges to this minimum for any initial approximation UIQ: m^ —> m^^ n —> cx). The proof of this theorem is similar to the proof of Theorem 19. A critical point of the regularization method is the definition of the regularization parameter a. Usually for the solution of this problem it is convenient to select a subsequence of the numbers QQ > ai > a2 > > O^L, where QQ and a^ are sufficiently large and small enough respectively. For example, we can use a progression of numbers (2.91): ak = aog^, /c = 1, 2,..., n; q > 0.
The regularization
method in a Unear inverse problem solution
117
To determine the regularized solution nia of the minimization problem, the regularized MRM algorithm is used: n—>oo
For example, let 111^^^ be a solution, obtained for the given QQ- T h e n we can use this solution as t h e initial approximation HIQ for the regularized M R M in t h e next cycle, when we use t h e next value of regularization p a r a m e t e r a i , etc. T h e calculations arc repeated until we find the quasi-optimal value of the regularization p a r a m e t e r m^^^^^^ based on the misfit condition (2.81): p m „ „ „ , , , - d||?5 = ,5.
(4.109)
In particular, one can use an approach based on u p d a t i n g the regularization parameter a during the iterative process. This approach leads to the so-called adaptive regularized M R M algorithm. I will discuss the basic idea of the adaptive m e t h o d later in section 5.3, considering the general iterative m e t h o d s of the regularized solution of the inverse problem. 4.S.3 Generalized MRM solutions of the Euler equation for the paranietrie funetional We now apply the I^anczos and generalized M R M algorithms to finding the solution of the Eulcr ecjuation for the Tikhonov parametric functional. Writing IJ^ ~~ A*/\ + nW*VV' and f — /1*d. wc can rewrite the Lanczos algorithm (1.84) as follows: m,,+ i = m,, - ^
A^^yl"^", n = 0, 1, 2 , . .
(4.110)
/=i
where I^
' and kH; are calculated by the formulae /-I
l(n)a ^ ( L « ) ' - l l ^ - ^ A < : ' " l l " ' " , I = 2 , 3 , . . . . S ,
(4.111)
k=l
Kii = \
— ^ ,
the residual, r^, is determined for the original inverse problem (4.1) r^ = A{inn) and is the residual for the Euler equation (4.100).
- d,
(4.112)
Iterative solutions of the linear inverse problem
118
The coefficients X]^^^ are determined, as usual, from the condition of the orthogonality of the vectors L^lJ"^^"", / == 1,2, ....s : An) _ V^ ^Ik
~
> ^r..^ K , ,
n
Ualin)a
)
||2
Thus, the regularized Lanczos MRM algorithm for the ill-posed inverse problem (4.1) can be summarized as follows: {n)a
Tr, = A m , - f, l(n)a
[^ )
1^ = 1 ^ " = A^Vn + >/—1 x ( n ) a | ( n ) a
^n~2^k=l'^lk
h
(a)
aW^Wnin,
ib)
, t — Z,Ci, ...5,
|2
a,(n)a AL")"=((L")'l„,L-4"))/||L"li")"|| , A:J^,= ( l ^ , L " t ) " ) / L"l
m„+i = m„ -
(c)
(^)
X ; L I A;^!/"^",
(4.113) where L^ is determined by expression (4.101). In a similar way we can construct a generalized MRM algorithm for the linear inverse problem solution: r„ = Aran - f,
{n)a
1^ = li
W l , ( l ) ~ ^n f^nl^ h l(n)a
n{n)a Ppl
_ ~
1? n+l,(p-l)
'
A*rn +
Wl,(p) — Wl,(p-1)
^p-l^(„)aj(n)a^
( r ot^oc T OL\{n)oc\ , \ ^ A n + l , ( p - l ) ' ^ ^l ) '
aW*Wm„,
L^l
a
Ya}{n)oi
P '^np^ ^:
(b)
5
P = 2,3,...S, {n)a
/c;j,= ( l ^ , L " l | " ) " ) / L"l
(c) {n)a
id)
n)a
(e) (4.114) The iterative processes (4.113) and (4.114) are terminated when the misfit reaches the given level EQ • m^+i = riin - E L i ^nJz ^^ n = 1, 2, 3
\\rNf<eo. The definition of the regularization parameter a is based on the same principles, outlined above for the case of the conventional regularized MRM algorithm. The important result of this chapter is that all iterative schemes, based on regularized minimal residual methods, always converge for any linear inverse problem! The proofs of the corresponding convergence theorems are similar to Theorem 21.
The regularization method in a linear inverse problem solution
119
References to Chapter 4 Kleinman, R. E., and P. van den Berg, 1993, Iterative methods for radiowave problems: The Review of Radio Science, Oxford University Press, 57-74. Samokhin, A. B., 1993, Integral equations of electrodynamics for three-dimensional structures and iteration method of solving them (Review): J. of communications, technology, and electronics, 38, No. 15, 15-34. Samokhin, A. B., 1998, Integral equations and iterative methods in electromagnetic scattering (in Russian): Radio i sviaz, Moscow, 160 pp. Saad, Y., and M. N. Schultz, 1986, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems: SI AM J. Sci. Stat. Comput., 7, No. 3, 856-869.
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Chapter 5 NONLINEAR INVERSION T E C H N I Q U E In this chapter I wih describe the most general case of geophysical inverse problems, when operator A of the forward problem is nonlinear. This case arises in electromagnetic and seismic methods, and in m a n y other applications. In t h e framework of general regularization theory, the solution of the inverse problem can be reduced to minimization of the Tikhonov parametric functional. T h e last problem can be solved by different optimization methods (see, for example, Tarantola, 1987; Press et al., 1987; Fletcher, 1995; Gih et al., 1995; Nocedal and Wright, 1999). G r a d i e n t - t y p e m e t h o d s belong to a family representing the most powerful optimization technique. I begin this chapter with a presentation of the general principles of gradient-type methods. 5.1 5.1.1
Gradient-type methods MeiJiod of steepest
deseent
Gonsider again the inverse problem d = A(m),
(5.1)
where m G M is some function (or vector) from a real Hilbert space M of t h e model parameters, and d G D is an element from a real Hilbert space D of d a t a sets, and /I is a nonlinear operator. Previously, we found t h a t one of the central problems in inversion theory was the solution of t h e minimization problem for different functionals. This problem can be solved directly in the case of the linear operator A of a forward modeling. However, in the general case of a nonlinear operator A, t h e solution can only be found iteratively. There are many different approaches to t h e construction of the iteration process for functional minimization. One of the most widely used techniques for optimization is based on gradient-type methods. We start our discussion with the most i m p o r t a n t and clearly u n d e r s t a n d a b l e m e t h o d of steepest descent. T h e idea of this m e t h o d can be explained using t h e example of misfit functional minimization: 0 ( m ) = / i ^ ( A ( m ) , d) = min . In t h e last formula, iijj is some given metric in a Hilbert space D\ / i ? , ( A ( m ) , d ) =. \\A{m) - d f = (yl(m) - d, A ( m ) - d ) .
(5.2)
122
Nonlinear inversion technique
where (...,...) means a scalar (inner) product of the elements from the space D. Local and global minima of the misfit functional Note that, in general cases, the misfit functional (5.2) may have several minima. We will distinguish three types of minimums: strong local minimum, weak local minima, and global minimum. Definition 23 The vector msL 'is a strong local minimum of the functional 0(m), if there exists 5 > 0 such that 1. 0(m) is defined in a neighborhood of m5L, 0{msL,^) 2. (l){msL) < 0(m) for all m G O (m^^, 6), m^
= {m; ||m — m^^ll < 5} ;
m^^.
Definition 24 The vector UVWL ^5 d weak local minimum of the functional 0(m)^ if there exists 8 > 0 such that 1. (/)(m) is defined in a neighborhood of m ^ i , , O {mwL^^) — {i^; ||i^ —
J^VKLH
< 8} ;
2. 0(mv^L) < 4>{^ for all m G O (mwL, S), and 3. mwL is not a strong local minimum. Figure 5-1 illustrates the difference between the strong and weak local minima. Actually, a weak local minimum area corresponds to some "plateau" on the misfit curve (or surface, in general cases). Definition 25 A vector mc at which the misfit assumes its least value, is called the global minimum: 0(mGf) < 0(m) for all m G M, m j^ m c (5.3) Note that there is no global minimum of the misfit functional if the solution of the original inverse problem (5.1) is nonunique. We know also that in this case we have to apply regularization theory to solve the ill-posed inverse problem. In this section, however, we will assume that misfit functional (5.2) has a global minimum, so there is only one point at which (j){m) assumes its least value. Our main goal will be to find this point. We should also notice that the gradient-type methods seek, as a rule, only a local minimum, a point at which the functional 0(m) is smaller than at all other points in its neighborhood. They do not always find the smallest of all such minima, that is, the global minimum. At the same time, many successful global minimization algorithms are based on solving a sequence of local minimization problems (Nocedal and Wright, 1999; Floudas and Pardalos, 1992), to which the gradient-type methods can be applied.
Gradient-type methods
123
Strong local minimum
Weak local minima
Global minimum
Figure 5-1 Three types of functional nninima: strong local mininnum, weak local minimunn, and global minimum.
Descent m e t h o d Wo consider an iterative algoritliin for misfit functional minimization. It is rca.sonable to bnikl this algorithm on the idea that, at every iteration nin, a misht decreases. In other words we impose the descent condition that (/)(rQ,,^i) < (y6(m„,) for all n > 0.
(5.4)
A method that imposes tliis condition is called a descent method. The question is how to find iterations {m„} which satisfy the descent condition. To solve this problem we calculate the first variation of the misfit functional at the point m: (50(m) = (^"/4(v4(m), d) = S{A{m) ~ d, A{m) - d) = 2 ( M ( m ) , A{m) ~ d).
(5.5)
We assume also that the operator A is nonlinear (in general cases), but that it is a difierentiable operator, so that 6A{m) = FmSm.
(5.6)
Here Fm is a linear operator which is called the Frechet derivative of A (see Appendix D). By substituting equation (5.6) into (5.5), we obtain 5(/.(m) = 6^Jl (A(m), d) = 2 {F^Sm, A(m) - d ) ,
(5.7)
124
Nonlinear inversion technique
Equation (5.7) can be simplified using the properties of the adjoint of the Frechet derivative operator, considered in Appendix D. The notion of the adjoint operator makes it possible to move a Hnear operator of a Frechet derivative, F ^ , from the left to the right-hand side of the inner product in equation (5.7): ,5<^(m) = 2 ( , 5 m , F ^ ( A ( m ) - d ) ) , (5.8) where F ^ is the adjoint operator of the Frechet derivative of A. In order to satisfy the descent condition (5.4), we select (5.9)
Sm = -kl{m),
where k is some positive real number and l(in) is a direction determined by the formula l(m) = F A ( A ( m ) - d ) . (5.10) Certainly, by substituting equations (5.9) and (5.10) into (5.8) we have 6(f){m) = -2/c(l(m), l(m)) < 0,
(5.11)
so l(m) describes the "direction" of increasing (ascent) of the functional (?^(m), because it is opposite to the descent direction, ^m. Remark 1 It can be proved that the direction, l{ni), determined above, is the steepest ascent direction. This means that, along this direction, the misfit functional increases most rapidly local to m. Proof: The absolute value of the first variation of the misfit functional can be estimated using equation (5.8) and the Schwarz inequality (A.38): \S{m)\ = 2 \{6nx, F4(yl(m) - d))| < 2 ||,5m|| ||l(m)||. From the last formula we obtain an estimate for the relative speed of the functional increase in the direction 6m :
M E M < 2 ||F* (A(m) - d)|| = 2 ||l(m)||.
(5.12)
On the other hand, the relative speed of the functional increase in the direction ^m = l(ni), according to (5.10) and (5.8), is equal to: \6{m)\_^{l{m),l{m))
IJTRF "
l|l(m)||
^ "^^™^" •
^^-^^^
Comparing (5.13) and (5.12) we deduce that l(m) really is the direction of steepest ascent, mm)\ ^ mm)\ \\6m\\ - ||l(m)
125
Gradient-type methods and t h e opposite direction, 6in = —/cl(m), is t h e steepest descent direction. T h u s , we can construct the iteration process as An-fl
m ^ + Snin = mn-
(5-14)
knl{mj,
where kn is a step length on the n - t h iteration. Iterations (5.14) wih satisfy t h e descent condition (5.4) if the corresponding step lengths, A:^, are appropriately selected. T h e traditional way of selecting k^ is based on the condition (j) (m^+i) := (j) (m^ - / c j ( m ^ ) ) = ^{kn)
= min,
(5.15)
where t h e m i n i m u m is determined v/ith respect to kn- T h e last condition allows us to define a step length along the direction of the steepest descent. T h e iteration process (5.14) together with the condition (5.15) gives us a numerical scheme for the steepest descent m e t h o d applied to misfit functional minimization. Figure 5-2 shows schematically the plot of the misfit fimctional value as a function of model parameters m . T h e vector of the steepest ascent, l(ni^J , shows the direction of "climbing on the hill" along the misfit functional surface. If we are at the point on this surface which corresponds to the r?,th iteration, we will be moving "downhilF' using formula (5.14) along the steepest descent direction to reach the (//. + l ) t h iteration. Line search m e t h o d We will now discuss the problem of determining a length of the step, A:„. Over the years numerous line search algorithms have been proposed based on optimization of the functional (Av,) : c[)(A':,,) - (l){mn+i) = 0 ( m , , - A v J ( m J ) (5.16) = (yl(m,, - kJimJ)
- d, A{mn - kJimJ)
- d) = (r,,+i, r^+i) = min,
where Tn+i = ^ ( m , , - kj{mj)
~ d,
is a residual vector at the (n + l ) t h iteration. We now consider some approaches to finding the minimum of the last functional with respect to kn- Let us calculate the first variation of $(A:„) and e q u a t e it to zero at a m i n i m u m point: 6^{kn)
= -26kn{Fr,J{inJ,
A{mn - kj{mj)
- d) = 0.
(5.17)
Using an adjoint operator, we move a linear operator of t h e Frechet derivative, F^^, from t h e left to the right-hand side of the inner product in equation (5.17): 6^kn)
= -26kn{l{niJ,
F ^ ^ [A{mn ~ U ( m J ) - d]) = 0.
126
Nonlinear inversion technique
SPACE M OF MODEL PARAMETERS Figure 5-2 The plot of the misfit functional value as a function of model parameters m . The vector of the steepest ascent, l ( m ^ ) , shows the direction of "climbing on the hill" along the misfit functional surface. The intersection between the vertical plane P drawn through the direction of the steepest descent at point vein and the misfit functional surface is shown by a solid parabola-type curve. The steepest descent step begins at a point (j){mn) and ends at a point ^ ( n i n + i ) at the minimum of this curve. The second parabola-type curve (on the left) is drawn for one of the subsequent iteration points. Repeating the steepest descent iteration, we move along the set of mutually orthogonal segments, as shown by the solid arrows in the space M of the model parameters.
127
Gradient-type methods
From the last formula we obtain immediately an equation for the optimum length of the step F4^ [A{mn - ^ n l ( m j ) - d] = 0. (5.18) The simplest technique of the line search arises if one assumes that in the last equation /c^l(m^) is small enough that one can use a linearized representation of the operator A(m^ — knl{m^) : A ( m , - M ( m J ) ^ A{mn) - KFmJimJ.
(5.19)
Substituting (5.19) into (5.17), we have Scl>{kn) = -2SK{FmJ{mJ,
A ( m J - A : , F ^ J ( m J - d) - 0.
(5.20)
Solving the last equation, we finally obtain Av^ —
[F^MmJ, A{m„) - d) _ ( l ( m j , F;,„ (/l(m„) - d)) (F„,„l(mJ, F„,„l(mJ) ||F,„„l(mjf ^ (l(m,J, l ( m j ) _
lll(mjf
(5.21)
Foniiula (5.21) works well if A is close to being a hnear opc^rator, so that rcprc^sentation (5.20) holds for every iteration step. That is why this technique may ])c referred to as a linaar line secvrcli. However, for a general nonlinear operator it is more cfiicient to use tlic quadratic approximation (D.27) of Appendix D: A{m,, - kjimj)
« A{m„:) ^ fc„F„„l(mJ + \kiFJ:^l{\{mJ),
(5.22)
where Fmi is the operator of the second variation of the operator A. We can use formula (5.22) to describe the quadratic behavior of the residuals Tn+i {kn) at the point m,,: Tn+i {kn) = A {rUn - kJimJ)
- d = r,, - g^.A:^ + h,,/c^^,
(5.23)
where gn is a gradient of the operator A along the step direction, gn =
FmJimJ,
and h^ is proportional to the second variation: h„ = ^ F ( f ) l ( m J .
(5.24)
128
Nonlinear inversion technique
In the case where /c^ = 1 we can compute the residual as Fn+l ( 1 ) = Tn - g n + h n ,
and knowing the gradient along the step direction gn and the residual r^ at the current point, we can estimate the vector h^: hn = r^+i (1) - Tn 4- gn.
(5.25)
Thus, equation (5.16) can be replaced by a 4-th order polynomial with respect tofcn,if we know the residual r^+i (1): ( r n - gnkn + K^l,
T^ - g n ^ n + ^71^1) = m i u .
(5.26)
Using the rules of inner product calculations, we can rewrite the equation (5.26) in the form of a scalar fourth-order polynomial minimization problem with respect to parameter k — kn'Po + Pik + P2k'^ 4- Psk^ + P4^^ = min,
(5.27)
where the polynomial coefficients are Po = (rn,rn), Pl = - 2 ( r n , g n ) , P2 = 2 ( h n , r ^ ) + ( g n , g n ) ,
Ps = - 2 (hn, gn) ,
PA
= (hn, K) •
One can solve equation (5.27) numerically using the secant root finding method and select the smallest positive root as an optimal step length, because we have to be conservative and not go too far from the previous iteration. It can be demonstrated that, if we apply the steepest descent method with the line search, the subsequent gradient directions are mutually orthogonal (l(ni„+i),l(m„))=0.
(5.28)
This result comes from the simple fact that, if we minimize a functional along some direction, described by a parametric line, the direction of the steepest ascent must be perpendicular to this line at the minimum point on the line (see Figure 5-3); otherwise we would still not be reaching the minimum along this line. A formal proof of this result was presented in Chapter 4 for the linear operator A, when we discussed the minimal residual method for the linear inverse problem solution (formula (4.51)).
Gradient-type
Figure 5-3
methods
129
T h e t o p part of the figure shows the isolines of the misfit functional map and
the steepest descent path of the iterative solutions in the space of model parameters. T h e b o t t o m part presents a magnified element of this map w i t h just one iteration step shown, from iteration ( n — 1) t o iteration number n .
According t o the line search principle,
the direction of the steepest ascent at iteration number n must be perpendicular t o the misfit isoline at the minimum point along the previous direction of the steepest descent. Therefore, many steps may be required t o reach the global m i n i m u m , because every subsequent steepest descent direction is perpendicular t o the previous one, similar t o the path of experienced slalom skiers.
130
Nonlinear inversion technique
Figure 5-2 also illustrates this property of the steepest descent method. The intersection between the vertical plane P drawn through the direction of the steepest descent at point nin and the misfit functional surface is shown by a solid parabolatype curve. The steepest descent step begins at a point (f) (rrin) and ends at a point (j) (m„+i) at the minimum of this curve. The second parabola-type curve (on the left) is drawn for one of the subsequent iteration points. With repeated steepest descent iterations, we move along the set of mutually orthogonal segments, as shown by the solid arrows in the horizontal plane, representing the space M of model parameters. Numerical schemes for the steepest descent method The iteration process (5.14) together with the linear line search described by formula (5.21) gives us a numerical scheme for the steepest descent method for misfit functional minimization. Thus, this algorithm for the steepest descent method can be summarized as follows: Tn = A{mn) - d,
(a)
In = l ( m ^ ) = F^Jn^
gn = F ^ J n
(&)
(5.29) 7
111 II ^ / II
II "^
( \
f^n — ||-^n|| / llSnII 5 m^+i =mn-
\^)
kJri-
{d)
The iterative process (5.29) is terminated at n = A/", when the misfit reaches the given level €o : (/)(mAr) = ||rAr||^ < ^oNote that the steepest descent method can be applied to the solution of the linear inverse problem as well. In this case the Frechet derivative of the linear operator A is equal to the operator itself: Fmr, = A, for any n.
(5.30)
Taking the last formula into account, we can simplify the algorithm (5.29): r^ = A{mn) - d,
(a)
In = l ( m ^ ) = A-'Tn,
gn = ^ U
{b) (5.31)
k - 111 11^ / llff 11^ f^n
||*n||
/ ||&n||
{c\ 5
m^+i = m^ - knln-
V^/
{d)
This algorithm is equivalent to the minimal residual algorithm (4.52) for solution of the linear inverse problem, introduced in Chapter 4. Thus, Theorem 18 and
131
Gradient-type methods
T h e o r e m 20 can be applied to this algorithm as well. In other words, t h e steepest descent m e t h o d converges to the solution of the linear inverse problem for any initial approximation mo, if L = AA*" is a positively determined linear continuous operator, acting in a real Hilbert space M, or if AA'^ is an absolutely positively determined ( A P D ) linear continuous operator, acting in a complex Hilbert space M. In the case of a nonlinear operator A, it is preferable to use an algorithm of the steepest descent m e t h o d with the quadratic line search. It can be summarized as follows: Tn = ^ ( m ^ ) - d,
(a)
U - l ( m j = F^^r.,
(6)
rn+i(l) = A ( m n - l ( m J ) - d
(c)
g n = Fmjn^
{d)
h ^ = T^+l ( l ) " I*n + g n
po = (rn, r^) ,
Pi = -2 {vn, gn) ,
P2 = 2 ( h , , , r , J -f ( g n , g n ) , p, - - 2 (h,„ g,,) , p4 = (hn, K)
{e) (/) (5.32)
{(j)
/;() + Pik-n + P'zk'i + p:ik;\ + p^ky, = m m (h) m,,+i = m,, - kJn-
U)
T h e iterative process (5.32) is terminated at n = N, when the misfit reaches the given level £Q :
5.1.2
The Newton
method
T h e main idea of t h e Newton method is to try to solve the problem of minimization in one step: m i = mo + A m .
(5.33)
T h u s , instead of moving downhill along a long p a t h formed by mutually orthogonal directions of t h e steepest descent (see Figure 5-3), one can try to reach the m i n i m u m of t h e misfit functional along one direction, as shown in Figure 5-4. To determine this specific direction, A m , let us calculate t h e misfit functional for this first iteration
= {A{m,)
- d, A{m,)
- d) (5.34)
= ( ^ ( m o + A m ) - d, A{mo + A m ) - d ) .
Nonlinear inversion technique
132
Ufmj
St St
(|)fmo + Amj
'•-> J")\
1 mb
mrj=^n\oA Am/
SPACE M OF MODEL PARAMETERS Figure 5-4 The plot of the misfit functional value as a function of model parameters m . In the framework of the Ne\A/ton method one tries to solve the problem of minimization in one step. The direction of this step is shown by the arrows in the space M of model parameters and at the misfit surface.
Gradient-type
133
methods
The first variation of the misfit functional with respect to A m is equal to 6(p{mi) = 2{6A{mo + Am), A{nio + Am) - d) (5.35) = 2{FmoSAin, A{mo + Am) - d). Using an adjoint operator for the Frechet derivative, we find (5(/)(mi) = 2(^Am, F^Jyl(mo + Am) - d]). Note that, according to Theorem 81 of Appendix D, the first variation of the misfit functional at the minimum must be equal to zero: ^(/)(mi) = 2 ( M m , F4Jyl(mo + Am) - d]) = 0,
(5.36)
and equation (5.36) must hold for any variation 6Am. Therefore, the second multiplier in (5.36) is equal to zero as well: F ; j / l ( m o + Am) - d] - 0.
(5.37)
It is difficult to find the exact solution of the operator equation (5.37). However, one can simplify this problem by linearization of the operator /I(mo -f Am), using a Frechet derivative operator: A{ino + Am) ^ y4(mo) + i^,,,,Am.
(5.38)
Substituting (5.38) into (5.37), we obtain F;j/l(mo) + F , , „ A m - d ] = 0 . From the last equation we find immediately F,;„F„,, A m = - F ; , „ [A{mo) - d].
(5.39)
According to (5.10) F ; j / l ( m o ) - d ] = l(mo), SO, from equation (5.39) we have the normal equation for the optimum step ^H^^Am
= -l(mo),
(5.40)
where H„„=2F4„F„„.
(5.41)
The operator Hmo is called a quasi-Hessian operator, because it does not take into account the second variation Frnl of the operator A (see expression (D.31)
134
Nonlinear inversion technique
of Appendix D). This operator is Unear. If there exists an inverse quasi-Hessian operator H:^^, one can solve equation (5.40): A m = - 2//;;^4(mo).
(5.42)
Substituting (5.42) into (5.33), we finally define mi = mo - 2i/-Jl(mo).
(5.43)
Note that expression (5.43) produces the final solution of the inverse problem only in the case of a linear forward operator A. In this case expression (5.43) takes the form (5.44) m i = mo - ( F ^ , F ^ o ) " ' I K ) - mo - (^M)"^ \{m^). Of course, it is usually not enough to use only one iteration for the solution of a nonlinear inverse problem in the framework of the Newton method (because we used the linearized approximation (5.38)). However, we can construct an iterative process based on the relationship (5.43): mn+i=mn-2i/-il(m^). Thus, the algorithm of the Newton method can be summarized as follows: Tn = A{mn) - d,
(a)
1. = l ( m j = F^^r,,
(b)
nin+i = m^ - 2//-^In.
(5.45)
(c)
The iterative process (5.45) is terminated at n — N when the misfit reaches the given level eo ' 0(mA^) = llfArf < So. The N e w t o n method with the line search Note, that in general cases of an arbitrary nonlinear operator A, algorithm (5.45) may not converge (see for details Fletcher, 1995), and, in fact, (/>(mn) may not even decrease with the iteration number n. This undesirable possibility can be eliminated by introducing a line search on every step of the Newton method: mn+i = m^ - A:n/f~^ln.
(5.46)
As in the steepest descent method, the length of the Newton step, /c^, is determined from the condition that ^{kn) = 0 (m^+i) = (t> (mn - /cni/~iln) = min .
(5.47)
135
Gradient-type methods
Applying a linear line search to the last problem, we obtain the optimum length of the step equal to .„ = ' " " - • ' " • • • ' • " " • • ! ? .
(5.48,
\\F,.,H-n(mJ\\' The algorithm of the Newton method with the linear line search can be summarized as follows: Yn = A^nin) - d,
(a)
^^.^(//-il(mj,l(mj)/||g,.f,
(c)
(5.49)
m„,+i = m,, - knH-^lln.
{d)
The iterative ])rocess (5.49) is terminated at n = N when the misfit reaches the given level 6() : 0(mAf) = llrA^II" < £oOne can also use a, formula similar to (5.22) to describe the ciuadratic behavior of the Newton's residuals, r,,H (A-,,) = yl(m,, ^ A - J / , ; 4 ( m J ) ^ d, at a point m,,:
where
gn = i^...(/C:KmJ), K = ^F(?)(//-;j(mJ). 2
As a result, we arrive at a similar algorithm to the one described by formula (5.32): r,, = A(m^) - d,
(a)
In = l(m^) = F^^jn.
{b)
rn+i(l)=A(m„-//^,il(mJ)-d g; = ^m.//-lln,
h; = r ^ + i ( l ) - r n + g;
(c) {d)
136
Nonlinear inversion technique
Po = (rn,rn),
Pi = - 2 ( r n , g ; , ) ,
P2 = 2 ( h ; , r . ) - f - ( g ; , g ; ) ,
(e) (/)
P3 = - 2 ( K , g;) , p4 = ( K , h^)
(^)
nin+i = nin - /cni/^^^ln-
(j)
(5.50)
The last algorithm converges much more rapidly than the steepest descent method (5.29). The main difficulty is that it is a rather complicated problem to calculate the inverse quasi-Hessian operator. To overcome this difficulty, we can use some approximation Gn for the inverse quasi-Hessian H^]^ and search for the solution by using the algorithm mn+i= nin - knGJ{mJ,
(5.51)
where kn is, again, defined by the corresponding line search. In the case where On is equal to the identity operator / , Gn = ^, (5.51) reduces exactly to the method of the steepest descent (5.14). By a different choice of Gn^ we select different methods of optimization, which we will discuss below. The preconditioned steepest descent method The simplest choice for Gn is to use some good approximation for the initial inverse Hessian: Gn = Go^H-l. (5.52) By substituting (5.52) into (5.51) we have mn+i= nin -
knGol{mJ,
where the coefficients kn are selected from the condition that (/)(mn+i) = (/)(mn - knGo\{mJ)
= ^{kn) = min.
(5.53)
For example, for the linear line search, we have from (5.53) f^n
—
(Gol(mJ, l(mj) |F„„Gol(mJ||2
The modification of the gradient method, described above, is called the preconditioned steepest descent method. Note that there are many different types of preconditioned algorithms depending on the choice of the approximation in (5.52).
137
Gradient-type methods
5.1.3 The conjugate gradient method The conjugate gradient method is based on the same ideas as the steepest descent, and the iteration process is very similar to the last one: mn+i== m^ + Anin = m^ - W ( m ^ ) , where However, the "directions" of ascent l(m^) are selected in a different way. On the first step we use the "direction" of the steepest ascent: I(mo) = l(mo). On the next step the "direction" of ascent is a linear combination of the steepest ascent on this step and the "direction" of ascent l(mQ) on the previous step:
I(mJ = l(mJ+/^il(mJ. On the n-th step l(m„+i)=»(m„+i)+/ViT(m,J. The sto])s A"„ arc^ selected, as usual, by a line search to minimize the misfit fimctional: 0(m,,+ i) = (p{mn - kj{nij)
= ^^(AvJ = min .
(5.54)
A linear line search in t h e conjugate gradient m e t h o d To solve the problem (5.54) we consider more carefully the functional ^{kn) : ^K^n) = 0 (mn+i) = 0 (nin - A;J(m^) j = (5.55) (A(mn - /cj(m^)) - d, /l(m^ - M ( m ^ ) ) - d). Let us find the minimum of the last functional with respect to kn- We calculate now the first variation of $(/Cn) : 6$(fc„) = -2,5fc„(F„,„I(mJ, /l(m„ - U ( m J ) - d).
(5.56)
In the last equation we assume that /Cnl(ni^) is small enough that we can use a linearized representation for the operator A(nin — /c^l(ni^) : Aixxxn - U ( m ^ ) ) ^ yl(m,) - Iz^F^XivoL^Substituting (5.57) into (5.56) , we have
(5-57)
Nonlinear inversion technique
138 S^kn)
= -26kniFmj{mJ,A{mn)
- knF^JimJ
- d) = 0.
(5.58)
Solving the last equation, we finally find f^n
( F ^ J ( m J , A{mn) - d) _ ( F „ J ( m J , A(m„) - d) {FmJimJ,
FmAmJ)
( l ( m j , F ^ J A ( m . ) - d])
FmJimJ (T(mJ, l ( m . ) )
FmJimJ
(5.59)
FmAmJ
Note that one can consider a quadratic line search similar to one outlined for the steepest descent method. C o n j u g a t e directions There are different ways of defining the coefficients /3^. The basic idea is to make directions l(ni^_^J and l(ni^) "conjugate" in some geometrical sense, which would guarantee much faster convergence of the iterations. We will introduce a definition of the conjugate directions below. Consider first a linear inverse problem. Suppose that we have moved from the point nin to the point m^i+i in the space of models. The change in the gradient directions can be described by the formula 7n=l(m^+l)-l("^n) = ^*[^(m,+i) - d] - A'[A{mn) - d] = A M A m , = //n^.Am,,
(5.60)
where Hmr^ = ^ * ^ is the quasi-Hessian operator. Note that the last formula holds approximately for a nonlinear operator as well:
= F^^^J/l(m.^x) - d] - F* JA(mO - d] ^ F ^ ^ F ^ ^ A m , = Hm^Anin^
(5.61)
where Hmr, = F^^Fm^ is a quasi-Hessian operator. It has been demonstrated that if we apply the steepest descent method with the line search, the subsequent gradient directions are mutually orthogonal
(l(m„^j,l(mj)=0.
(5.62)
This result comes from the simple fact that if we minimize a functional along some direction, described by a parametric line, the direction of steepest ascent must
Gradient-type
139
methods
be perpendicular to this line at the m i n i m u m point on t h e line; otherwise we would still not be reaching the minimum along this line. Therefore, m a n y steps m a y be required to reach t h e global minimum, because every subsequent steepest descent direction is perpendicular to the previous one, as shown in Figure 5-5. T h e t o p p a r t of the figure shows the isolines of the misfit functional m a p and the steepest descent p a t h of the iterative solutions in the space of model parameters. T h e circled area at the b o t t o m part presents a magnified element of this m a p with just one initial iteration step presented. According to the line search principle, the direction of t h e steepest ascent in the first iteration, li , must be perpendicular to t h e direction of t h e steepest ascent, IQ , in the initial iteration. It is b e t t e r if the direction of gradient does not vary so dramatically from step to step. For example, we can select t h e vector —li, directed straightforward at the minimum of the misfit functional, as shown by t h e long dashed arrow in Figure 5-5 (the top p a r t ) . This direction can be constructed as a vector sum of the (negative) direction of the steepest ascent — li in the first iteration and the (negative) direction of the steepest ascent, — IQ , in the initial iteration with some coefficient P^, as shown schematically within the circled area in Figure 5-5:
Suppose t h a t we have moved along a direction l,, i = l(ni,, i), and now i)lan to move along a new direction 1„ ^= l(ni,,). T h e condition t h a t minimization along 1,, will not spoil oui' minimization along 1„ i is just t h a t the graxlient stays po]-])cn(li('ular to 1„ 1 (see Figure 5-6). This will h a p p e n if the variation of the gradient itself, 7 , , ^ l(m^^i ]) — l(ni^J, at the point ni„ is perpendicular to the previous line search (Urcction l(?r?.„_i): 7,,-Ll(m,_J, (5.63) because l(m,^JJ_ l(m^,^^J. Condition (5.63) holds if
= -kn
( l ( m , , _ i ) , / / , , J ( m O ) - 0.
(5.64)
T h e last condition is equivalent to (l(m^_i),i^^J(m,))
=0
(l{mn),
=0,
or HrnJinin^i))
because Hm,js a self-adjoint operator. But l(ni^_i) is parallel to Anin-i Am^-i = -^n-il(mn-i).
:
140
Nonlinear inversion technique
Figure 5-5 The top part of the figure shows the isolines of the misfit functional nnap and the steepest descent path of the iterative solutions in the space of model parameters. The bottom part presents a magnified element of this map with just one initial iteration step shown. According to the line search principle, the direction of the steepest ascent at the first iteration must be perpendicular to the directiori^of the steepest descent in the initial iteration. The circled area shows that direction —li at the minimum of the misfit functional can be constructed as a vector sum of the (negative) direction of the steepest ascent, — l i , in the first iteration and the (negative) direction of the steepest ascent, —IQ, in the initial iteration with some coefficient P^.
Gradient-type methods
141
-In
Figure 5-6
\Am„_, 111,.
Construction of conjugate gradient directions.
mization along 1,, does not spoil the minimization along l ^ - i
The condition t h a t miniis t h a t the gradient stays
perpendicular t o 1„ i . This will happen if the variation of the gradient, 7,^= In+i—l,?, at the point m „ is perpendicular t o the previous line search direction In-i'
7n~L l r , - i -
so wc luivc^ l(ni.,J, //^„,^^ Arrin^'n-1
-1
l(m„),7n-i
=0
^'7^-1
Definition 26 The 'vectors u and v are said to be conjugate if they satisfy the condition ( u , / / v ) = 0, (5.65) wJierc II IS a linear operator. Tims, we require that the vectors l(ni^^) introduced above,
I(m^) = l(m^) + /^ij(m^^_ J, to be mutuany conjugate in this sense. In other words we require that (l(m,,),i/^J(m,_i)) - 0 ,
(5.66)
which is equivalent to the condition W),7„_i)=0.
(5.67)
142
Nonlinear inversion technique
To simplify our analysis we will use the notations l(m„) = l„, l ( m j = l „ .
(5.68)
To satisfy condition (5.67), let us calculate f l n , 7 n - l ) = Un + /?nln-l,ln —ln-1 J
= {\nX) - (ln,ln-l) +/?n (jn-lX)
' Pn
(jn-lX-l)
=0,
= XX)-f3n(}n-lX-l) because (ln,l„-i)=0
(5.69)
(l„_i,l„)=0.
(5.70)
and
Equation (5.69) follows from (5.62), and equation (5.70) holds because in the previous step we moved along the search line in the direction l„_i to the minimum, so the steepest descent direction 1„ at the minimum point will be perpendicular to In-iAlso, it can be shown that ( ^ l n - l , l n - l j = yn-1
= ( l n - 1 , I n - l ) + Pn-1
-\-Pn-X-2An-lJ
( l n - 2 , K-l)
= (ln-1, In-l) •
(5-71)
Therefore XX)-Pn(jn-lX-l)
=(ln,ln)-/?n(ln-l,ln-l)
=0,
and we finally determine /3^ :
^"
(1„,1„)
(l(mj,l(mj)
||l(mj||^
(l„-i,l„_0
(l(m„_J,l(m„_0)
||l(m„_0|f
^ ' '
Note that there are several other popular techniques for determining the coefficients /3^, which I do not describe here. I refer the interested readers to the books by Tarantola (1987) and Fletcher (1995).
Regularized gradient-type methods in the solution of nonlinear inverse problems
143
N u m e r i c a l s c h e m e s for t h e c o n j u g a t e g r a d i e n t m e t h o d T h e algorithm for the conjugate gradient m e t h o d introduced above, can be s u m m a rized as follows: r^ = ^ ( m ^ ) - d,
In = l ( m ^ ) =F^j^,
Pn = I l l n l l V l | l n - l f ,
l n = I n + ^ J n - l , IQ = IQ,
kn = I In, In ) / K'mnln
7 J ^ n + l = ^Un ~ r^n^n-
(a) (b)
(5.73)
[C)
T h e iterative process (5.73) is terminated when the misfit reaches t h e given level SQ : (/)(m7v) = ||ryvf < £()• T h e algorithm of the conjugate gradient m e t h o d described above has been substantiated for a linear inverse problem. For example, in the simplest cavSe when operator A is a linear operator with a square matrix of the order TV, it can be proved t h a t the conjugate gradient algorithm will give an exact solution of the inverse ])rob1cm in A^ iterations (Fletcher, 1995). This algorithm can be also used, similarly to the Newton method, for solution of the nonlinear inverse j)rol)lcm. In general nonlinear ca.scs 1 he number of iterations is not fixed, but still the m e t h o d converges very rci])i(lly. Another advantage of the conjugate gradient m e t h o d is t h a t we go downhill not along a line, but on various planes. In this case we can overcome small local minima of the misfit functional and go faster directly to its global miniminn. 5.2 5.2.1
R e g u l a r i z e d g r a d i e n t - t y p e m e t h o d s in t h e s o l u t i o n of n o n l i n e a r i n verse problems ReguloTized steepest
descent
In a previous section, we considered the problem of minimization of the misfit functional. However, we know this problem is ill-posed and unstable. To find t h e stable solution for t h e minimization problem we have to consider the regularized minimization problem, P ' ' ( m ) = /i^(v4(m), d) + as{in) = min, (5.74) where d are observed data, m is the unknown model, v4(m) is the operator of forward modeling and s ( m ) is some stabilizing functional. If we assume t h a t the space of d a t a D is a real Hilbert space with some given metric, the space of models M is also a real Hilbert space with perhaps another metric, and 5(m) is any quadratic functional, for example:
s{m) = \\Wm\
M5
144
Nonlinear inversion technique
where W is a positively determined linear continuous operator in M, then P " ( m ) = (A(m) - d, A{m) - d ) ^ + a {W (m - niapr) ,W{ni-
inapr))M ^
where niapr is some a priori given model. To solve the problem of minimization (5.74) we have, as usual, to calculate the first variation: ^F"(m) = 6 (A(m) - d, A{m) - d ) ^ + a6 {W (m - niapr), W {m - mapr))j^ = 2 ( M ( m ) , ^ ( m ) - d)D + 2a{6Wm,
W{m -
^apr))^-
Taking into consideration that operator A is differentiable, with 6A{in) = FmSm, where Fm is a linear operator of the Frechet derivative of A, we have (5P"(m) = 2 (^FmSm, A{m) - d)D + 2a{6Wm, = 2 {6m, F ^ [A{m) - d] + aW'W{m
W{in - m , ^ ) ) ^ - niapr))M .
where F* and VK* are the adjoint operators. Let us select 5m = - A : " r ( m ) ,
(5-75)
(5.76)
where /c" is some positive real number and P ( i n ) is the direction of the steepest ascent of the functional P^{in): P ( m ) = F4 [A{m) - d)] + aW'W{m
- niapr)-
(5.77)
Certainly, by substituting (5.76) and (5.77) into (5.75), we have (5P^(m) - - 2 f c ( r ( m ) , r ( m ) ) < 0, so P ( m ) describes the "direction" of increasing (ascent) of the functional P " ( m ) , in other words, the direction of "climbing on the hill." Thus, we can construct an iteration process for the regularized steepest descent as mn+i= m^ -f 6nin= nin - /c^P(m^), where the coefficient k^ is found as described previously, by using the minimization of the parametric functional with respect to /c^: P"(m„+i) = F«(m„ - fc:P(m„)) = ^^{K)
= min.
(5.78)
In particular, applying the hnear line search, we find that the minimum of the parametric functional is reached if k^ is determined by the following formula: k'i =
l|l"(mjf l^m„l"(m„)r + a | i m - ( m „ )
Regularized gradient-type methods in the solution of nonUnear inverse problems
145
Numerical schemes for the regularized steepest descent miethod The algorithm for the regularized steepest descent method can be summarized as follows: r^ = A{mri) - d, (a) 1::= l " ( m j = F ^ ^ r n + a l ^ W ( m - m^^J,
(6) (5.79)
m,,+i = mn - k^l^.
(d)
The iterative process (5.79) is terminated at n = A^ when the parametric functional reaches the given level ^o : P"(m^) < ^0. In the special case when A is a linear operator, all Frechet derivatives Fm,, = A, and the regularized steepest descent method takes the form r,, = A{mn) - d, i;:- r{inj
(a)
=A^T,,+aW'W{in^~in,,,r).
{b) (5.80)
m,,„i - m , , - A : - l - .
{d)
The la.st algorithm is equivalent to the regularized minimal residual method (4.108), introduced in Chapter 4. Therefore, we can extend this method to the solution of the inverse problem in the complex Hilbert spaces M and D, and apply Theorem 22 (section 4.3.2) to the regularized steepest descent algorithm (5.80). As a result, we conclude that algorithm (5.80) converges to the unique minimum of the Tikhonov parametric functional, P"(m) = \\Am - d f 4- a \\Wmf
,
for any initial approximation mo: nin ~^ i^a, n ^ 00, if v4 is an arbitrary linear continuous operator, acting from a complex Hilbert space M to a complex Hilbert space D, and VK is a positively determined linear continuous operator in M. This is a fundamental result of rcgularization theory which opens the way to the unique and stable solution of ill-posed inverse problems. 5.2.2 The regularized Newton method The regularized Newton method can be obtained if we consider again only one iteration, m i = m o + Am, (5.81)
146
Nonlinear inversion technique
and calculate the parametric functional for the first iteration: F " = (A(mi) - d, A{im)-d)^
+ a{W{im-majrr),
H^(mi-m«p,))^
(5.82)
We linearize the operator ^ ( m i ) : ^ ( m i ) = ^ (mo + Am) ^ A (mo) + Fmo^ni.
(5.83)
We then have F " ^ {A (mo) + FmoAm - d, A (mo) + FmA^
- d)^ (5.84)
-ha (W (mo + A m - m^pr), W (mo 4- A m - mapr))^ • The first variation of the parametric functional with respect to A m is equal to 6P^ ^ 2{FmoSAm, A (mo) + F^o A m - d)D -{•2a{WSAm, W{mo + A m - mapr))M
(5.85)
- 2 {6Am, F ^ {A (mo) + F ^ , A m - d) + aW^W (mo + A m - m ^ ^ , ) ) ^ . Setting 6P^ = 0 in (5.85), we find at once the regularized normal equation for the optimum step, {F:,^Fm,+aW*W)Am
= -F;,[yl(mo) - d]-aW*Wimo-mapr)
= - l " ( m o ) , (5.86)
and the value of the step, A m = - ( F ^ „ F „ „ + aW*Wr'r{m,),
(5.87)
where P(mQ) is determined by equation (5.77). Substituting (5.87) into (5.81) we obtain m i = mo - (F4„F„„ +
aW^WrTim^).
The last equation allows us to construct the algorithm of the regularized Newton method: m„+i= m„ - (F*„F„„ + aW*W)-'r{mJ.
Regularized gradient-type methods in the solution of nonlinear inverse problems 5.2.3
Approximate
regularized solution
of the nonlinear
inverse
147
problem.
We can find an approximate solution of the regularized normal equation (5.86) for t h e o p t i m u m step, using the same idea which we applied for the approximate solution of t h e linear inverse problem in Chapter 3. Let us assume t h a t t h e regularization p a r a m e t e r a is big enough t o neglect t h e t e r m / / ^ Q A I H with respect t o t h e t e r m aW^WAm in equation (5.86). T h e n we obtain: aWWAm
^ ~F^[A{mo)
- d] -
alV^Wimo-mapr)-
Applying the inverse weighting operators (VK*VK)~ to b o t h side of t h e last equation, we find: A m ^ -a~' {W'Wy'
i \ ; [ ^ ( m o ) - d] - ( m o - m . p , ) .
(5.88)
Formula (5.88) plays an important role in fast inverse imaging. I t provides an approximate solution of the inverse problem, m i = mo + A m ^ ^ m^,r - cy-' {W'Wy'
F*JA(mo) - d)],
(5.89)
which requires only forward modeling. In the special case when t h e initial and a prioT'i models are equal t o zero, we arrive a t the extremely simple and i m p o r t a n t formula m,^a-^ {W'Wy' F;,d, (5.90) which serves a.s a ba.sis for geophysical migration imaging. In particularly, if we have a linear inverse problem, tlie Frechet derivative operator Fj^ is equal to t h e forward modeling operator A, and we can cast expression (5.90) in the form (5.91)
mi^a-'^ {W'Wy^ A'd.
Note t h a t the coefficient a~^ in the previous formulae can be treated as j u s t a scaling factor t h a t can be found by minimizing the misfit between the observed and predicted d a t a . 5.2.4
The regularized preconditioned
steepest descent
m,ethod
As we have mentioned above, the calculation of the Hessian operator is a r a t h e r complicated problem. For this reason we follow the recommendation given above a n d try to find some approximation for the inverse operator ( F ^ F^^, + cyW*Wy^. If we select this approximation in the form
we obtain the regularized steepest descent m e t h o d , m„+i=m„-fc^G'^r(mJ, where the coefficients /c^ are again selected from the condition (5.78). Note t h a t several other forms of the preconditioning operator are available (see, for example. Gill et a l , 1995).
N o n l i n e a r inversion t e c h n i q u e
148
5.2.5 The regularized conjugate gradient method This method uses the same ideas as the conventional conjugate gradient method. However, the iteration process is based on the calculation of the regularized steepest descent directions _ mn+i= nin + 5m -^ nin - fc^P(m^), where the "directions" of ascent P ( m ^ ) are selected according to the algorithm described above. In the first step, we use the "direction" of regularized steepest ascent: P(mo) = r ( m o ) . In the next step, the "direction" of ascent is a linear combination of the regularized steepest ascent on this step and the "direction" of ascent l"(ino) on the previous step: P ( m i ) = r ( m i ) + /3?r(mo). In the (n -f- l)th step r(m„+0=l"(m„^i)+/?^+iT"(m„). The steps k^ are selected based on the minimization of P"(m„+i) = P" ( m „ - k:r{mj)
= $"(fc°).
Solution of this minimization problem gives the following best estimation for the length of the step using a linear line search: (la k^ =
ia\
(l-, (F^^F^^+aiy^M/)!-) (la ia\
[a j a \ ^m ^n J (F^J-,
FmX)+a(w\-,
m-)
F 1"
+ a
Wl-
2'
One can use a parabolic line search also (Fletcher, 1985) to improve the convergence rate of the regularized conjugate gradient (RCG) method. The coefficients (3^ are determined by the formula (3n =
l|l°(mj||^
149
Regularized solution of a nonlinear discrete inverse problem A n u m e r i c a l s c h e m e for t h e r e g u l a r i z e d c o n j u g a t e g r a d i e n t m e t h o d T h e algorithm for the R C G m e t h o d can be summarized as follows:
r^ = /l(m^) - d, P:
1^= r ( m j =F:^j^^aW^W{in^-ina,r).
= lllnf / ||ln-l|r '
l n = K+P^^n^l.
IQ = I Q ,
«, i:^)/|||F^J^||Va II^T^fj, m^+1 = m ^ - A:^ I^.
(a) W
(c)
(5.92)
(d)
T h e critical point of rcgularization methods is t h e definition of the regularization p a r a m e t e r a. As we discussed in Chapter 4, for t h e solution of this problem it is usually convenient to select a subsequence of real numbers QQ > ai > a^ > > Q^L, where (IQ and cy.i are respectively sufficiently large and small enough numbers. To determine the regularized solution lUa of the minimization problem, t h e gradient (steepest descent, Newton, or conjugate gradient) methods are used: m ^ = lim lUn-^i. T h e calculations arc rej)eated tor different aj imtil we find the quasi-optimal value of the rcgularization parameter agopt based on the misfit condition (2.81):
/i„(/l(m„„„,J,d) = ,^-.
(5.93)
One can also use an adaptive rcgularization in the sense discussed in C h a p t e r 4. In this Ccxse the rcgularization parameter a is u p d a t e d on each iteration. T h e basic ideas of the adaptive m e t h o d will be discussed in section 5.3. 5.3 5.3.1
R e g u l a r i z e d s o l u t i o n of a n o n l i n e a r d i s c r e t e i n v e r s e p r o b l e m Nonlinear
least-squares
inversion
We now return to the discrete problem, which we have discussed in section 3.1. T h e forward geophysical problem is described, as usual, by the relationship d = A(m'"^^), where, in general, A is a nonlinear m a t r i x operator, and m^^^^ is a t r u e b u t u n k n o w n set of model parameters. For simplicity, we assume within this section t h a t all parameters are real numbers. However, the same technique, with t h e little modification, can b e applied to complex parameters as well. As usual, t h e misfit functional / ( m ) can be written in the form / ( m ) = ( A ( m ) - d ) ^ ( A ( m ) - d).
(5.94)
150
Nonlinear inversion technique
We can apply different types of minimization methods (steepest descent, Newton method, etc.) to the straightforward minimization of the corresponding misfit functional / ( m ) - However, all these solutions have many limitations and are very sensitive to small variations of the observed data due to the principal instability of the inverse problem. To overcome this difficulty, we have to apply a regularizing method. For a regularized solution of a nonlinear inverse problem, let us introduce a parametric functional. P " ( m , d) = IIWdA(m) - Wdd||2 + a|| W ^ m - W ^ m ,apr\\ |2 - (WdA(m) - Wdd)^(WdA(m) - W^d) - h a ( W ^ m - W ^ m a p r ) ^ ( W ^ m - W^rriapr),
(5.95)
where W^/ and W ^ are some weighting matrices of data and model parameters, niapr is some a priori model. Some possible methods of introducing W^ and Wm were discussed in Chapter 3. Numerical experiment demonstrates that for iterative methods of inversion of gradient type, the recommended choice of the model parameters weighting matrix W^^ is the square root of the integrated sensitivity matrix according to formula (3.89). This choice of matrix W ^ is based on the simple fact that the first iteration of a gradient method, as in the approximate solution (5.88), is usually proportional to the inverse square of the matrix W ^ . Therefore, application of the weights (3.89) helps to produce a suitable first iteration close to the true model. Under these circumstances, the convergence of the entire iteration process is improved. According to the basic principles of the regularization method, we have to find the model rric^, a quasi-solution of the inverse problem, which minimizes the parametric functional P ^ ( m , d ) = min. 5.3.2
The steepest descent method for nonlinear regularized least-squares inversion
To solve the problem of minimization of the parametric functional using the steepest descent method, let us calculate the first variation of P " ( m , d ) , assuming that the operator A(m) is differentiable, so that ^A(m) = FmSm,
(5.96)
where F ^ is the Frechet derivative matrix of A. Thus, we have 5 P ^ ( m , d ) = 2{WdFmSmf{WdA{ni) +2a{Wm6mf{Wmm
- W^d)
- W ^ m aprj)
151
Regularized solution of a nonlinear discrete inverse problem or after some algebra <5P"(m,d) = 2(^m)^F^W2(A(m) - d) + 2a(<5m)'^W2^(m - m^^),
where we assume that the matrices W^ and W ^ are diagonah Fohowing the general scheme of the steepest descent method, we can again select (5m = -/c"P(m), where k"^ is some positive real number (length of a step) and P(ni) is a column matrix defining the direction of the steepest ascent of the parametric functional: r(m) = FlWliAim) - d) + aWl{m - m.^J.
(5.97)
An iterative process of the method is constructed according to the formula m,,^_i = m,, + (5m = m,, - A:"P(mn), where the coefficient A:^ is defined by a line searcli according to the condition F'imn^-i) - P " (m,, - k^Y'inin)) = ^I>"(A<:) = nun . 5./y./y
(5.98)
The Nciuton vicUiod for 'nonlinear regularized least-squares iniH'rsion
he\ us consider-, again, the lirst iteration, mi =^ rrio+Am, and write the linearized parametric functional P"(mi,d) - P^^(mo+Am,d) = (W,A(m,) + W , F . , , A m - W,d)^'\W,A(mo) + W , F , , , A m - W , d ) +r^(W„,mo + W,, Am - W,,,m,,^,)^'lW,,mo + W,, Am - W,,m,,,,).
(5.99)
The first variation of the parametric functional is equal to <5AmP"(mi,d) = .^Am/'^lmo+Am.d) = {6^my'
[F^„W2 (A(mo) + F„„Am - d) + aWl{nvo + Am - m„p,)] .
It is evident that the necessary condition for the minimum of the parametric functional, (5P"(mi,d) = 0 , is satisfied only if F;^„W2 (A(mo) + F„„Am - d) + aW^(mo + Am - m„p,) = 0.
152
Nonlinear inversion technique
From the last equation, taking into consideration (5.97), we obtain
Am=-2H-Xl"(m„), where H„,„„=2(F^W2F„„ + aW^) is the regularized quasi-Hessian matrix. Thus, the Newton algorithm for the nonUnear regularized least-squares inversion can be expressed by the formula irin+i = rrin - 2H-^^J"(mn), where H„,„„ = 2 ( F ^ „ W ^ F „ „ + a W ^ ) . 5.3.4
Numerical schemes of the Newton method for nonlinear regularized least-squares inversion The algorithm of the Newton method for a discrete inverse problem can be summarized as follows: Tn = A(m„) - d,
(a)
1^ = F " ( m j = F ^ ^ W ^ r , 4- a n W ^ ( m , - m , ^ , ) ,
(6)
H«„„^^ = 2 (F^^ W^F^^ + a ^ W ^ ) ,
(5.100)
(c)
mn-fi = nin - 2H-1 J^"(mn),
{d)
where an are the subsequent values of the regularization parameter, updated on each iteration. This method is called the Newton method with adaptive regularization. In a similar way we can construct an algorithm of the regularized Newton method with a linear line search: Tn = A(mn) - d,
(a)
1-" = P ^ ( m J = F^^W^rn 4- a . W ^ ( m , - m . , , ) ,
(6)
H«„.,^„. = 2 (F^^ W ^ F ^ ^ + a n W ^ ) ,
(c)
hOtr,. =
^n
nin+l
\\U-1
P n | | 2 / | l p
\\^^an,mn'-n
=mn-
||
TT-1
lanll^
/ j|•*^ mn-"-an,mn*n
K-nzlmX''"
||
5
(5.IOI)
(J\
\^)
i^)
Both iterative processes (5.100) and (5.101) are terminated when the misfit reaches the given level SQ : 0(miv) = ||rivf < eo.
Regularized
solution of a nonlinear discrete inverse
problem
153
5.3.5 Nonlinear least-squares inversion by the conjugate gradient method This method uses the same ideas as the regularized conjugate gradient method: m^+i= m^ + ^m == m„ - ^ ^ r ( m ^ ) ,
(5.102)
where the "directions" of ascent P(ni^) are selected according to the algorithm described above: P ( m „ + 0 = r ( m „ + j ) + /?:?+iP(m„,). (5.103) However, the regularized steepest descent directions are now determined according to the formula for the least-squares method: r ( m „ ) = F^„W2(A(mJ ~ d) + a W ^ „ ( m „ - m apr
(5.104)
Determination of the length of iteration step, a coefficient /c", can be based on the linear or parabolic line search: P " ( m „ + 0 = P"(m„ - k:rim„))
= /(fc^) = min.
Solution of this ininimization problem gives the following best estimation for the length of the step using a linear line seareh; A:'! =
l"''Ym„)l"(m„ l"'-(m„) [(F^;„W2F„„. + r^W2,) l"(m„) ^•'•(m„,)P(m„)
WrfF„,J«(m„)
(5.105)
+ a WrnlH^n]
One can use a parabolic line search also (Fletcher, 1985) to improve the convergence rate of the RCG method. The CG method requires that the vectors P(ni^) introduced above will be mutually conjugate. This requirement is fulfilled if the coefficients P^^ are determined by the formula (Tarantola, 1987) Pn+l
|P(m,n-fly
WH^nW-
(5.106)
Using equations (5.102), (5.103), (5.105), and (5.106), we can obtain m iteratively. 5.3.6 The numerical scheme of the regularized conjugate gradient method for nonlinear least-squares inversion In this section we first consider a method of selecting the parameter a. The regularization parameter a describes a trade-off between the best fitting and most reasonable stabilization. In a case when a is selected to be too small, the minimization of the
154
Nonline£ir inversion technique
parametric functional P'^(m) is equivalent to the minimization of the misfit functional 0(m), and therefore we have no regularization, which can result in an unstable incorrect solution. When a is too large, the minimization of the parametric functional P"(ni) is equivalent to the minimization of the stabilizing functional 5(m), which will force the solution to be closer to the a priori model. Ultimately, we would expect the final model to be exactly like the a priori model, while the observed data are totally ignored in the inversion. Thus, the critical question in the regularized solution of an inverse problem is the selection of the optimal regularization parameter a. The basic principles used for determining the regularization parameter a were discussed in the previous sections of the book. We introduced in Chapter 2 a simple numerical method to determine the parameter a. Consider for example the progression of numbers ak = aiq^-^] A: = 1,2,3 ; g > 0. (5.107) The first iteration of the steepest descent or any other gradient method is run usually with Qo = 0- The initial value of the regularization parameter, a i , is determined after the first iteration, m i , as a ratio ||WrfA(mi)-W^df IW^mi In this way we have an approximate balance between the misfit and stabilizing functional. For any number ak we can find an element ma^, minimizing P"''(m), and calculate the misfit || A (nic^^) — d |p. The optimal value of the parameter a is the number a^o, for which we have II A ( m , , J - d f = (5,
(5.108)
where 6 is the level of noise in the observed data. The equality (5.108) is called the misfit condition. This algorithm, as well as the L-curve method (see Chapter 2), has clear practical limitations, because it requires a complete numerical solution of the inverse problem for each value of the regularization parameter ak. An alternative approach is based on the simple idea, which we discussed already above, that the regularization parameter a can be updated in the process of the iterative inversion. For example, one can use the following algorithm for the RCG method P"+^(m„+i) = P"+'(m„+i) + /3„+J«"(m„), where a^ are the subsequent values of the regularization parameter. This method is called the adaptive regularization method. In order to avoid divergence, we begin an iteration from a value of a i , which can be obtained as a ratio of the misfit functional and the stabilizer for an initial model, then reduce a^ according to formula (5.107) on each subsequent iteration and continuously iterate until the misfit condition (5.108) is reached.
Conjugate gradient re-weighted optimization
155
The algorithm of the regularized conjugate gradient method can be summarized as follows: r „ = A{mn)
-
(a)
d,
1°" = P " ( m J = Fj;;„W^r„ + a „ W ^ , ( m „ - m ,
(b) (5.109)
W.
a m„,+ i — nin — k^^"" 1^"-,
, id) (e)
where a;^ are the subsequent vahies of the regularization parameter. T h e iterative process (5.109) is t e r m i n a t e d when the misfit reaches the given level ^o • (j){m!sj) = | | r / v f < ^Q. This metliod is cahed the conjugate gradient m e t h o d with the adaptive regularization. 5.4 5.4.1
Conjugate gradient re-weighted optimization TJic TikJionov pa'rarnetric fiinctional
with a
'pseAido-quadratic
stabilizer
In C h a p t e r 2 we have demonstrated t h a t , in general cases, a stabilizing functional can be represented in the form of the pseudo-quadratic functional (2.72): s ( m ) = {W^m,
(5.110)
WeUi)
where operator W^ is a linear operator of multiplication of the model p a r a m e t e r s function 777. (r) by the function We (r) , which depends on m. For discrete model p a r a m e t e r s , using m a t r i x notations, operator We can be expressed as the m a t r i x multiplication Wein — W g n i ,
(5.111)
where diagonal m a t r i x W ^ computes differently for different stabilizers. For example, in the case of the m a x i m u m smoothness stabilizer
W"
d i a g [we] = d i a g
VTTI
(m2 + e2)^/2
(5.112)
where d i a g [we\ is a diagonal matrix formed by the values of the function We (r) determined by t h e discrete values of the function m (r) describing the model p a r a m e t e r s and where e is a small number.
156
Nonlineeu- inversion t e c h n i q u e
In the case of the minimum entropy functional -I 1/2 •y^minentrO __ (JJagp ,
,21 - 1 / 2
\m + e^]
Q
\m\
Q= ^
Im^l.
(5.113)
where
In the case of the TV stabilizing functional, sprvi'm), the matrix is 2\l/4'
Wr"^ = diag
(|Vm|^+/?^)
(5.114)
(m2 + e2)'/2
In the case of the minimum support functional, 5MS {'m), we have W.^
diag
1
Finally, for the minimum gradient support functional T^MGS = d i a g
(5.115)
( m 2 + e2)i/2 SMGS
{^
, we find
Vm [ [Vm • Vm + f3^] ^'^ [m? (r) + e^f'^
(5.116)
where /3 is a small number. The corresponding parametric functional (5.95) can be written as P " ( m , d) = v' (m) + a s (m) = (WdA(m) - Wdd)^(WdA(m) - W^d) + a ( W , W „ m - W e W „ m a p . ) ^ ( W e W „ m - W e W „ m'•aprj') ,
(5.117)
where (p (m) is a misfit, if (m) = (WdA(m) - Wdd)^(WrfA(m) - W^d);
(5.118)
matrix Wg is a variable matrix, which depends on m, and W^^ is the conventional fixed diagonal matrix for weighting the model parameters. Therefore, the problem of minimizing the parametric functional, given by equation (5.117), can be treated in a similar way to the minimization of the conventional Tikhonov functional. The only difference is that now we introduce some variable weighting matrix Wg for the model parameters. The minimization problem for the parametric functional introduced by equation (5.117) can be solved using the ideas of traditional gradient-type methods.
157
Conjugate gradient re-weighted optimization Let us find the Frechet derivative of P " ( m , d ) : 6P"{m,d)
= 2(WdF„6m)^(WdA(m) - W^d)
+2a,5(W,W,„m)'-'^(W,W„m - W , W „ m „ p , ) ,
(5.119)
We can calculate 6 (WeWmm) as 6 ( W e W „ m ) = [WeW„ + diag (m) f„] ,5m,
(5.120)
where diag (m) is a diagonal matrix formed by the components of the vector m, and im is the relevant Frechet derivative of the weighting operator WgW^^. For example, in the case of the minimum support functional, m C-^ = -diag (m2 + e^f^ W
V V TO
Substituting exprcKsion (5.120) into (5.119), wc find 5P'-'(in,d) = 2(W,F„,,5m)'^'(WrfA(m) - W^d) +2r>([W„W,„, + diag (m) f,„] ,5m)^'(W,W„,m - W,W,„m„p,,).
(5.121)
Following tlic^ general scheme of the stee])est descent method, we can again select (^m = -A:"P(m), where k^^ is some positive real number (length of a step) and F'(rQ) is a column matrix (lehning the direction of the steepest ascent of the parametric functional: ir.(m) - F:,W;i(A(m) - d) +
+ diag (m) f,J^^^ W,W,,,(m - m^^^J. (5.122) Based on expression (5.122) for the steepest ascent direction, we can derive all the formulae for the regularized steepest descent, Newton and conjugate gradient methods for minimization of the Tikhonov parametric functional with a pseudoquadratic stabilizer. 5.4-2 Re-weighted conjugate gradient method Q [W,W,,,
Portniaguine and Zhdanov (1999) have developed a simplified approach to minimizing the parametric functional (5.117), using the so-called re-weighted regularized conjugate gradient (RRCG) method. In the framework of this approach, the variable weighting matrix Wg is precomputed on each iteration, We = Wen = We (m„) based on the values nin, obtained on the previous iteration. As a result, it is treated as a fixed matrix on each iteration, and the expression (5.122) for the steepest ascent direction is simplified as r ( m „ ) = F;^„W2(A(mJ - d) + a W L w i ( m „ - m „ p , ) ,
(5.123)
158
Nonlinear inversion technique
where F ^ ^ is the Frechet derivative matrix computed on the n-th iteration. Thus, the RRCG method is based on the successive hne search in the conjugate gradient direction P(ni„): mn+i= mn-hSin = mn-A:^r(m^).
(5.124)
The conjugate gradient directions P(m„) are selected as follows. In the initial step, we use the "direction" of regularized steepest ascent for the initial model mo P(mo) = r ( m o ) = F^„W^(A(mo) - d) + a W , 2 o W ^ ( m o - m „ ^ ) , where F^^ is the Frechet derivative matrix for the initial model and W^Q = W^Q (mo). In the next step, the "direction" of ascent is a hnear combination of the regularized steepest ascent on this step and the "direction" of ascent P(mo) on the previous step: _ _ P(mi) = r ( m J + / ? ? P ( m o ) . In the {n -h l)th step P(m„+i) = P ( m „ ^ J + /3^+iP(m„),
(5.125)
where the regularized steepest ascent directions are determined now according to formula (5.123), The length of each iteration step, the coefficient /c^, can be determined with a linear or parabolic line search: P - ( m , + i ) = P ^ ( m , - fc^ P ( m , ) ) = f{k^) - min. Solution of this minimization problem gives the following best estimate for the length of the step using a linear line search: fc. = T"^(mn)P(m„) " l«^(m„) (F^„W2F„„ + aW^Wl) One can use a parabolic line search also (Fletcher, rate of the RRCG method. The CO method requires that the vectors mutually conjugate. This requirement is fulfilled if by the formula
^26) l«(m„)
1995) to improve the convergence _ P ( m ^ ) introduced above will be the coefficients /3^ are determined
^r(nwi)f
Using equations (5.124), (5.123) and (5.126), we can obtain m iteratively. We call this algorithm conjugate gradient re-weighted optimization because the weighting matrix W^^ is updated on every iteration (Portniaguine and Zhdanov,
Conjugate gradient re-weighted optimization
159
1999). One can find a formal proof of t h e convergence of this t y p e of optimization technique in Eckhart (1980). Note t h a t due to re-weighting, the stabihzing functional can change, a n d even increase from iteration to iteration, s (m^+i) = (m^+i - mapr) W ^ ( ^ ^ i ) W ^ ( m ^ + i - m^pr) = 7^^ i^n) where
,
(5.127)
5 (m^+i) _ (mn+i - m^pr)^ W ^ ( ^ ^ ^ ) W 4 ( m , , + i - m«p^
s (m^) (m^ - majyrVWl^W^inin - niapr) In order to insure the convergence of the misfit functional to the global minimum, we use adaptive regularization and decrease the a ^ + i , if 7 > 1 :
_ J a„, if 7 < 1, " " " ^ ' ^ 1 a J J, i f 7 > l . So, the product of the regularization parameter a^+i and the stabilizer 8(111^4.1) decreases or does not change: an-^is{mn.^i)
=
(inS (mn+i) = dnis (IriS ( m n + l ) h
(m,,) , if 7 < 1,
= ^n^^ i^ri)
, if 7 > 1 •
Wc also decrease the regularization parameter an+i,
if the misfit docs not decrease fast enough: | | W , A ( m , , ) - W , d f - ||W,A(m,,_H) - W^dH'-^ < 0.01 ||W,yA(m,,) - W , ^ d | | ' . Numerical experiment demonstrates t h a t the recommended choice of the empirical coefficient q is within an interval (0.5; 0.9) . T h e algorithm of the R R C G m e t h o d can be summarized as follows: v,, = A{mr,) - d,
s„ = W e „ W „ ( m „ - m„p^),
ia)
ib)
p^" = iic'ii7i|i"ri'ir. A-"= i ^ n ^ /
'n
m „ + i = m „ - /c^" 1^",
i-=c+/?:"C-r, c = c , V " i „ ' * d'- rn„
' " ** en * ' rn) *n
7 = ||s„+i|l / ||s„|| ,
a „ + i = Q-n, if 7 < 1, and «„+! = a „ / 7 , if 7 > f, a'„,+ i = (?a„+i, g < 1, if WWdTnf
- l l W ^ r ^ + i f < 0.01 WW^rnf
(c) (d) (e) (/) , {9)
(5.128)
160
Nonlinear inversion technique
where an are the subsequent values of the regularization parameter. The iterative process (5.128) is terminated when the misfit reaches the given level ^o : (/)(mAr) =
IIFA^II^
< 5o.
This method is called the RRCG method with adaptive regularization. d.J^.S Minimization in the space of weighted parameters The minimization problem (5.117) can be reformulated using a space of weighted parameters: m^ = W,W^m. (5.129) We introduce also the weighted data d^ = Wdd.
(5.130)
We can consider a forward operator, which relates the new weighted parameters m ^ to the weighted data: A^(m") = WrfA(W-^W;^m").
(5.131)
Using these notations, we can rewrite the parametric functional (5.117) as follows P " ( m - , d"') = (A^(m») - d'")^(A^(m"') -d^)
+ a (m^ - mrp,)^(m'" -
m:^), (5.132)
where d"" = A ^ [(m""'"=)"'] . As one can see, we keep the same misfit, as in equation (5.117) because V ( m - ) = >f^ (m) = ( A ^ ( m - ) - d - ) ^ ( A ^ ( m - ) - d^) (5.133) = (WdA(m) - Wdd)^(WdA(m) - W^d) = ||WdA(m) - W ^ d f , and the same stabilizer, as in equation (5.117), equal to the least square norm of difference (m — mapr) with the weights s ( m - ) = s^ (m) = ( m - - m ^ , , ) ^ ( m - - m - , ) = (m - mapr)^WlWl^{m
- m,^).
(5.134)
Therefore, the minimization problem (5.132) is equivalent to the minimization of the same parametric functional as in formula (5.117). Note, however, that the unknown parameters are now weighted model parameters, m^. In order to obtain the original model parameters we have to apply inverse weighting to the result of minimization of the parametric functional (5.132): m = W-^W;im^.
(5.135)
Numerical experiments show that, as a rule, the iterative process converges faster for (5.132) than for (5.117). The minimization method in the space of weighted parameters is similar to the RRCG method described above.
161
Conjugate gradient re-weighted optimization 5.4-4
The re-weighted regularized weighted parameters
conjugate
gradient
(RCG)
method
in the space of
T h e re-weighted regularized C G m e t h o d in the space of weighted p a r a m e t e r s is based on a successive Une search of the minimum of the parametric functional (5.132) in the R C G direction I;i(m;^"): ^Zli
= m-- + ^m-" = m - - ~ ^ v X ( m r ) -
(5-136)
T h e iteration step (coefficient /c") is determined from the linear line search: K =
" ^ " ^^^ " , l«/(m-")(FLF^n + aI)lS(m-")
(5.137)
where F,,;„ is t h e Frechet derivative m a t r i x of the op)erator W^iA (W~^^W^;^^m'''). We can also use a parabolic line search for A:^^. According to (5.135), the weighted model parameters, mj^'", are related to the original model parameters, m^^, by the expression m ^ = W , , W,,,m,„
(5.138)
where W , , , - W , (m,,) . Note t h a t from formula (5.131), a.ssuming t h a t the weighting m a t r i x W^7^/ is ah'c^ad}^ |)rccomputed ])ased on the values m,, and is fixed, we obtain a sim])le relationship between the Frechet derivative matrices, F,,; and F respectively, of the weighted A,,, (m'"), and the original A ( m ) forward modeling operators: l)A,„ ( m " ) = F,,(5m"' - W,/(3A (W-^W^Jm'^^) -
^N,i¥W;^^^N;2.^m'".
Therefore, F,,,=W,F,,W;/W;^
(5.139)
T h e R C G directions Y^^^{m^'') are selected according to t h e same rules as for the conventional R C G method presented above. In the first step we use the steepest ascent direction: I ^ ( m - ) = l ^ ( m 7 ) = ¥l,
( A „ ( m ^ ) - d^) + a ( m ; r - m - ) ,
(5.140)
where F^o is the weighted Frechet derivative m a t r i x (5.139) for t h e initial model, m ^ = W ^ W , o m o , rK^, = W ^ W e o m a ^ , , and W e ^ = W ^ (mo) . In the n - t h step, the "direction" of ascent, l^(mj^^), is a linear combination of the regularized steepest ascent l^(m'^'') of this step and t h e "direction" of ascent l^(m|^^i^) of t h e previous step: T^(m-) = i : ( m r ) + / 9 X ( m : : r ) -
(5.141)
162
Nonlinear inversion technique
The regularized steepest ascent directions for the re-weighted RCG method are determined according to the formula l^(mr) = F L (A. ( m - ) -d-) + a ( m - - m - ) .
(5.142)
where m;^- = WmWenmr,, m^^ = W^Wenrriapr. The coefficients /?^ are defined from the condition that the directions l^{m'!^'') and l^(m^r:7^) are conjugate: n
/r 1 /iq\
W'-wK'-'-'-n J\\
^""iiis(m-r)ir
^^
Note that at each step we recompute the real parameters of the model from the weighted parameters determined according to formula (5.136) at the n-th iteration m„+i=W-iW,->:;';i,
(5.144)
and then re-weight it again with the new weighting matrix W~A_^^x computed for the current model nin+i : ninlV = W^We(n+i)mn4-i. (5.145) We call this algorithm the re-weighted RCG method, because the weighting matrix Wen is updated on each iteration. As a result of re-weighting (5.145), the misfit does not change, <^ (m::;t') = <^ (m::';i) = (WdA(m„+i) - Wrfd)^(W,A(m„+i) - W^d), but the stabilizer changes,
= (m„+i - mapr)^W^(„+i)W^(m„+i - niapr) = 7S
{T^UII)
,
(5.146)
where ^ 5(m;;';|') ^ (mn+i - m„p,)^W^(„^i)W^(mn+i - niapr)
Therefore, on each iteration of the re-weighted RCG method we actually minimize the parametric functional with the different stabilizers, because the weighting matrix Wen is updated on each iteration. In order to insure the convergence of the misfit functional to the global minimum, we use adaptive regularization and decrease the Qn+i, if 7 > 1 : ^n+l
[ Qn/T, lf7 > 1-
Conjugate gradient re-weighted optimization
163
So, the product of the regularization parameter an+i and the stabihzer s (m^^^^) decreases or does not change: (5.149)
anS (m;^;t') / 7 = ^nS (m;^+i) , if 7 > 1We also decrease the regularization parameter a^^^
(5.150)
qan+u ^ < 1,
^n+l
if the misfit does not decrease fast enough: | | W , A ( m J - W , d f - ||W,A(m^+i) - W , d f < 0.01(/^«-(mO. The empirical coefficient q is selected, as in the case of the RRCG method considered above, from an interval (0.5; 0.9) . The algorithm of the re-weighted RCG method in the space of weighted parameters can be summarized as follows: («)
K"=-\\K:nr/
|2
_
'(i;(n—1)
(d) (e) (/) 7 = | | s r + t ' II / | | S n + l | |
) « n + l = « n , if 7 < 1, ^ u d »„,+ ! = ^n/l^
<+i = 9«n+i, g
if 7 > 1, {fj)
{h)
(5.151) The iterative process (5.151) is terminated when the misfit reaches the given level eo :
0(m7v) = llrTvll^ < £o. Finally, we present an algorithm for the re-weighted RCG method of solving the hnear inverse problem. It can be easily obtained from (5.151), assuming that all Frechet derivative matrices F^jn are equal to the matrix of the weighted linear
164
Nonlinear inversion technique
operator A^j'. C - A„ ( m ^ ) - d - = WdA ( W - i W , - > - " ) -Wad,
(a) (&)
|2
wn /.an _
/^TanT ^ a n ^ / \\cxnT ( \T
K
ii;(n—1)'
wn'^r-^n
ifO
. ^T^ Tan]
(c) (d) (e)
^ ^ n +l _
XX/-
^
/ ||Sn+l||
a;n + l
^ n +l _
fy^^n+l
_
j^ll^n + n
(/)
' ^ n + 1 = Oin, if 7 < 1, ^ n d Q^n+l = C^n/7, if 7 > 1,
ga.+i, g < l ,
(^)
if | | C f - ||r-^i|| < 0.01 ||r-|
(5.152) The advantage of the re-weighted RCG method in the space of the weighted parameters is that it converges even faster than the original re-weighted RCG method. In a practical implementation of the method, it is recommended that the weights, matrix W~J, and the regularization parameter, a^, not be updated on every iteration, but after performing a sequence of iterations (usually five or ten) with the fixed values of W~^^^ and Qno • This improves the convergence rate and robustness of the algorithm, while keeping the value of the regularization parameter OLU from being too small during the iteration process. For the final sequence of iterations the weighting matrix We is fixed until the required level of the misfit is reached. The number of "re-weighting" steps is controlled by the value of the stabilizing functional. Obviously, this is an interactive algorithm. It illustrates that the regularized inverse problem solution involves comprehensive numerical experiment. Some examples of practical application of this algorithm in geophysics will be given in the next chapters of the book. Note also that algorithms (5.151) and (5.152) involve computation of the inverse matrix W~'^. In this case we can set the small parameter e equal to zero without generating a singularity in formulae (5.112)-(5.116). For example, in the case of the minimum support functional, SMS (^^) -> we have W;^ = ( W f ^)-^ = diag (m^ H- e^)'
-^ diag [|m|], if e -> 0.
For the minimum gradient support functional
SMGS
( ^ ) , we find
(5.153)
165
Conjugate gradient re-weighted optimization il/2r
diag
2_L.2ll/2'
diag
Vm + ZJ^
ife-^0.
(5.154)
We have shown in this section how regularization theory can be a p p h e d to nonhnear inverse solutions. In the foUowing chapters of the book, we wih illustrate the general theory for some important geophysical inverse problems.
References to Chapter 5 Eckhart, U., 1980, Weber's problem and Weiszfeld's algorithm in general spaces: M a t h . Programming, 18, 186-196. Fletcher, R., 1995, Practical methods of optimization: J o h n Willey & Sons, Chichester - New York, 436 pp. Floudas, C , and P. Pardalos, eds., 1992, Recent advances in global optimization: Princeton University Press, Princeton, NJ, 633 pp. Gill, P. E., Murray, W., and M. H. Wright, 1995, Practical optimization: Academic Press, London, San Diego, 401 pp. Nocedal, J., and S. J. Wright, 1999, Numerical optimization: Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo, 636 pp. Portniaguine O., and M. S. Zhdanov, 1999, Focusing geophysical inversion images: Geophysics, 6 4 , 3, 874-887. Press, W. li., Flannery, B. P., Tcukolsky, S. A., and W. T. Vcttermg, 1987, Numerical recipes, T h e art of scientific computing: Cambridge University Press, C a m bridge, 818 pp. Tarantola, A., 1987, Inverse problem theory: Elsevier, A m s t e r d a m , Oxford, New York, Tokyo, 613 pp.
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P a r t III G e o p o t e n t i a l Field Inversion
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Chapter 6 INTEGRAL REPRESENTATIONS IN FORWARD MODELING OF GRAVITY AND MAGNETIC FIELDS 6.1
Basic equations for gravity and magnetic fields
6.1.1 Gravity and magnetic fields in three dimensions The basic formulae for gravity and magnetic fields were introduced in Chapter 1. According to these formulae the gravity field, g, satisfies the equations V - g = -47r7p, V X g = 0,
(6.1)
where 7 is the universal gravitational constant, and p is the density distribution within a domain D. The solution of these equations is given by the formula
g(r") = A»(p) = y^'
lll/'''\r-r'\'''
In particular, the commonly measured vertical component of the gravity field, g^^ can be found from the expression
In a similar way, we can represent a magnetic field. Note that in geophysical applications we usually consider the magnetic anomaly caused by the induced magnetization in the rock formations only. Assume that the volume D is filled by magnetic masses with the intensity of magnetization I(r), which is given as a product of magnetic susceptibility, x (r), and the inducing geomagnetic field H^: I(r) = x(r)HO.
(6.4)
It is well known that the corresponding magnetic field H(r'), generated by the magnetic masses within a domain D, can be represented as follows (Zhdanov, 1988): H(rO = V ' / / / I(r) • V'-^-dv.
(6.5)
Integral representations in forward modeling of gravity and magnetic fields
170
Figure 6-1 sources.
Ilustration of the concept of the plane field. F is a domain occupied by the
Substituting (6.4) into (6.5), we obtain (6.6)
where H^ = |H°| is the magnitude of the inducing field, and d/dl' denotes a directional derivative in the direction of magnetization H^. Formulae (6.2) and (6.6) provide the basis for the solution of forward and inverse geopotential (gravity and magnetic) field problems. 6.1.2 Two-dimensional models of gravity and magnetic fields In some practical situations we can assume that the density, p(r), or magnetization, I(r), of the rocks does not change in one horizontal direction, say in the y direction. In this case p(r) and I(r) are the functions of two variables, x and z: p(r) = p{x,z),
I(r) = I ( x , z ) .
(6.7a)
We assume, also, that the domain D is a cylindrical volume extended infinitely in the y direction with cross-section F in the vertical plane xz (Figure 6-1). Under these assumptions, obviously, gravity and magnetic fields vary only in the x and z directions, and formulae (6.2) and (6.5) take the form
l{x\z')=^
J JT
J-OO
H(a;', z') = V where ds is an element of area of F, i.e. ds = dxdz.
v' |3
dyds,
v' |3
dyds,
(6.8) (6.9)
Integral representations
of potential fields based on the theory of functions
of a complex
variablel71
Note that the y - components of gravity and magnetic fields vanish, and the X- and z- components are determined by calculating an elementary integral along the y axis: 1
,
1
r -
[{x-xf
+ {y-yf
^^ + {z-z'ff'
'^
2 {x — x/) 4- (2: — z')
(6.10)
Substituting expression (6.10) back into formulae (6.8) and (6.9), we find
JJv
(.r - .X-')'+ ( s " 2')
where V (kniotc^s the oi)crator of 2-D differentiation: V ~ {D/Ox'A)/dz') . a n d d,,. and d , are tlie unit ])asis vectors of t h e Cartesian system of coorcUnates. In particular, t h e vertical component of t h e gravity field is ecjual t o <)A-
r/,s.
'x ^ x'Y +
fn
Note in conclusion t h a t two-dimensional mock^ls of geoi)otential fields ai'c wick^ly used in geophysics, because they rechice dramatically the volume of numc^rical calculations r(X|uircd for forward and inverse problems. Another advantage of 2-D models is t h a t one can use t h e |)owerful tool of complex analysis t o s t u d y 2-D ])otential fields (Zhdanov, 1988). We will consider some aspects of this approach in t h e next sections. 6.2
I n t e g r a l r e p r e s e n t a t i o n s of p o t e n t i a l functions of a c o m p l e x variable
fields
b a s e d o n t h e t h e o r y of
Most acliicvcmcmts attained in tlie theory of tw^o-dimensional potential fields involve the theory of functions of a complex variable (Strakliov, 1970a,b; Tsirulsky, 1963). This theory was exposcxl in detail in t h e book by Zhdanov (1988). We shall present here those results developed in these publications most relevant t o t h e inverse problem solution. 6.2.1
Complex
intensity
of a plane potential
field
Let us consider a 2-D potential field F satisfying t h e equations V - F = g, V x F = 0,
(6.14)
172
Integral representations in forwetrd modeling of gravity and magnetic fields
where V denotes the operator of 2-D differentiation in the vertical plane xz, and q is a source of the potential field F , concentrated within a local domain V. According to (6.14), we can write the equations for the scalar components of the vector field F : OF, dF, — + — = , ( . , z), (6.15)
In particular, outside the sources we have dx
dz
We will introduce a complex variable C, = x ^ iz and designate by F(C) the following complex function: F{0 = - F , ( x , z) + iF,{x, z).
(6.18)
This function, F{(), is called a complex intensity of a plane field. It is an analytical function outside the sources, which vanishes at infinity because, according to equation (6.17), its real and imaginary parts, R e F = —F^, I m F = F^, satisfy the Cauchy-Riemann conditions: dReF dx
dlmF dz
dReF dlmF dz dx Let us proceed from the real variables x, 2 to the complex ones C = x + iz, C =x-iz,
(6.19)
(6.20)
and employ differentiation operators: d__l,d__ dC ~ 2^dx
.d_ ^dz''
where the asterisk * means complex conjugate. Multiplying equation (6.15) by (—1) and equation (6.16) by (—z) and summing them up, we obtain — ( - F , + zF,) + i-^i-F.
+ iF,) = - g ( x , z),
Integral representations of potential fields based on the theory of functions of a complex variablel73
Figure 6-2
Solution of the equation for the complex intensity of the potential field using
the Pompei formula. F is the domain occupied by the field sources.
-|.F,o^-i,(»
(6.22)
It is iiotcHvorthy t h a t for an arbitrary function vHC) fli^if '^^ analytical in a cloniain 6'. t-h(^ Cauchy-Hicniaini relations imply t h a t
^ V ' ( C ) - 0.
(G.23)
T h e solution of the differential equation (6.22) can be obtained using the Pornpei formula (Zhdanov, 1988):
FiO = ~ I y^FiOdC ~- ff 2711 Jos C-C
T^ JJsC
-(,
^^FiOds, oC
(G.24)
where S is a domain in tlic complex plane, dS is its boundary, and ("' is a fixed point within 5'. Note t h a t for any function ip t h a t is analytical in S, property (6.23) reduces (6.24) to the well known Cauchy integral formula:
^iC)
1
1
7V^(C)^C-
(6.25)
We will use the Pompei formula to examine the complex intensity of a plane field. In accordance with the definition, tlic complex intensity satisfies equation (6.22) everywdiere in a complex plane and vanishes in infinity. Let us take an a r b i t r a r y point
174
Integral representations in forward modeling of gravity and magnetic fields
(^ and draw therefrom a circle LR (Figure 6-2). The domain bounded by Lji will be SR. We assume that the radius R is big enough that the domain F is completely inside 5 ^ . Applying Pompei formula (6.24), we write
The integral taken over the circle LR can be written, upon substitution of the variable C = C -^R-e'^, in the form
Being analytical outside F, the function F(C' + /^ • e^^) tends uniformly over 6 to zero at infinity, R —^ oo. Hence the limit of the integral (6.27) as i? ^ oo is zero. Thus we write finally:
where we have substituted the domain of the source concentration F for the domain of integration SR, assuming that q{(^) = 0, if C ^ F. 6.2.2 Complex intensity of a gravity field The gravity field g of a two-dimensional distribution of masses concentrated with a density p(x, z) within the domain F satisfies the equations V • g = -ATTJP,
V X g - 0,
(6.29)
where 7 is the universal constant of gravitation. Let us define a complex intensity: ^(C) = -9x{x, z) + ig,{x, z).
(6.30)
In accordance with (6.22), the function g{C,) satisfies the equation ^ g ( C ) = 27r7p,
(6.31)
whose solution is governed by a formula following from (6.28),
ff(C') = -27 jj^ ^p{Ods,
(6.32)
where p{Q = p{x,z). In particular, the complex intensity of the gravity field generated by masses with a constant density p{() = PQ is defined by ge(C') = - 2 7 P o / ( ^ -
(6.33)
Integral representations
6.2.3
Complex
of potential fields based on the theory of functions of a complex
intensity
and potential
of a magnetic
variablel75
field
Let domain T be filled with magnetized masses with an intensity of magnetization I(x, z) = {Ix{x, z),Iz{x, z)). Let us define the complex intensity of mugnetization as I{0 T h e complex intensity (6.12), as
(6.34)
= L{x,z)+ih{x,z).
of a plane magnetic
field
can be defined, according to
H{C) = - i / . ( x ^ z') + ^7/,(x^ z') = - 2 JJ^ ^-±-^^I{(;)ds.
(6.35)
We can introduce a complex magnetic potential ^(C) as H{C) = ^U{C).
(6.36)
From (6.35) and (6.36), we have at once
f/(C') = - 2 11^ ^ / ( C ) r f . s .
(6.37)
It is useful to compare cc|uation (6.37) with the corresponding formula for a gravity iicld:
fjiO = - 2 7 11 ^^piOd-^.
(6.38)
One can see t h a t these expressions are similar, with one very i m p o r t a n t dificrence: the density p{Q is a real function, while the complex intensity of magnetization, /(C), is a complex function. T h e complex magnetic potential generated by a uniformly magnetized body, /(C) = /(), is defined by
U^{C) = ^2IoJI^^.
(6.39)
Comparing formulae (6.39) and (6.33), we see
IPo Substituting the last expression into (6.36), we arrive at the famous Poisson
theorem
which provides a simple connection between the magnetic field, H^ and t h e gravity field, gc, of t h e same b o d y with uniform distribution of the density and magnetization.
176
Integral representations in forward modeling of gravity and magnetic fields
The analytical representations derived above for anomalous gravity and magnetic fields provide a useful tool for the solution of the inverse problems.
References to Chapter 6 Strakhov, V. N., 1970a, Some aspects of the plane inverse problem of magnetic potential (in Russian): Izvestia AN SSSR, Fizika Zemli, No. 9, 31-41. Strakhov, V. N., 1970b, Some aspects of the plane gravitational problem (in Russian): Izvestia AN SSSR, Fizika Zemh, No. 12, 32-44. Tsirulsky, A. V., 1963, Some properties of the complex logarithmic potential of a homogeneous domain (in Russian): Izvestia AN SSSR, Fizika Zemli, No. 7, 1072-1075. Zhdanov, M. S., 1988, Integral transforms in geophysics: Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo, 367 pp.
Chapter 7 INTEGRAL REPRESENTATIONS IN INVERSION OF GRAVITY AND MAGNETIC DATA 7.1
G r a d i e n t m e t h o d s of g r a v i t y i n v e r s i o n
7.1.1 Steepest ascent direction of the misfit functional for the gravity inverse problem Let us consider the problem of inversion of the gravity field using gradient-type m e t h ods. We will use a two-dimensional case to illustrate this approach. According to formula (6.32), we have the following expression for t h e complex intensity of the gravity field:
fiiC) - .4" {(>) = -27 I J ^^piOds,
c' i r,
(7.1)
wherc^ A-^ {p) denotes the forward modeling operator. Let us assume t h a t we have observed some gravity field (jr{C) o^i ^^^c liorizontal axis .7: {z -^ 0), and the domain F is located in the lower half-plane. T h e problem is to determine the density p[C^). We introduce a complex Hilbert space D of d a t a (gravity field) with t h e metric U.9)u=
/
J —00
f{^')9*{^)d^\
(7.2)
f,geD,
and a real Hilbert space M of models (functions p(C)) with the metric (^. P)M = / / ^^iOpiOds;
cr,pe
(7.3)
M.
T h e gravity inverse problem can be formulated as the mininnzation of t h e misfit functional: HP)
= 11,9 - grWl = ( A ' (p) - gr, A" (p) - g^)^ = min .
(7.4)
To solve the minimization problem (7.4) we calculate, as usual, the first variation of the misfit functional: 64,{p) = 6 (A" (p) - gr, A« (p) - gr)^
= 2 Re (A-' {6p), A-' (p) -
gr)^
178
Integral representations in inversion of gravity and magnetic d a t a = 2 {8p, Re \A^* [A^ (p) - 9T)])M
= 2 {6p, I {p))^
(7.5)
,
where / {p) is a direction of the steepest ascent at the point p of the space M of the model parameters: I (p) = Re [A^* {A^ (p) - 5r)] = Re [A^* {g - gr)] •
(7.6)
Let us find an exphcit form of the adjoint operator A^* for the gravity problem: {A^p)J)^
(7.7)
= {p,A^*{f))^.
Using the definitions (7.2) and (7.3) of inner products, and expression (7.1) for forward operator, we can rewrite formula (7.7) as {A" (p), f)j, = r
A^ ip) fdx'
= -27 r
If
^dsf* oo
{x') dx'
f^/•* /
/
(7.8) M
where, as usual, the asterisk * means complex conjugate. By comparing (7.7) and (7.8) we can see that OO
A>>*{f) + 2jJ
/•* /
/
= 0,
C-x'
(7.9)
M
where the star • means adjoint operator. Equation (7.9) holds for any p, for example, for
J-OO C -
(7.10)
^
Substituting (7.10) into (7.9), we obtain
^•(^''i,'
0.
From the last formula we find that adjoint operator, A^*, applied to some function f {(,'), is given by the formula
A^^f) = 2j r i^dx'.
(7.11)
Therefore, according to (7.6), the direction of the steepest ascent is equal to Z(p)=27Re
r J —C
{g{x')-gr{x')y x' -C
dx'
(7.12)
Gradient methods
179
of gravity inversion
where g (x') is the predicted gravity field on the observation hne z = 0 : g{x')
=
AHp).
Note t h a t , according to equation (7.6), the direction of t h e steepest ascent at t h e initial model with zero density is equal to / ( p = 0) = - R e A ^ V -
(7.13)
We will examine the physical meaning of expression (7.13) later in t h e section on gravity field migration. 7.1.2
Application
of the re-weighted
conjugate gradient
method
We have established in the previous chapters of this book t h a t t h e gravity inverse problem is an ill-posed problem. We have to use rcgularization to generate a stable solution. Following the general principles of the rcgularization theory, we introduce a stabilizing fimctional, say, minimum norm stabilizer: •''• (P) = \\P^ PaprWlf ^
J^^"' ^Vr| ^^'''"
In (his ca.s(^ \\\c regularized direction of the stee])cst ascent is given ])y the formula r{p)^-Rc[A"*{yV'ip)-fjr)]+a{p-p„„,)
- 27 Rx; / . / - OO
dx + a [p - p,,,,,,) .
y-— -^
(7.14)
S
We can apply now the general scheme (5.151) of the re-weighted regularized conjugate gradient m e t h o d in the space of the weighted parameters, to form an iterative process for gravity field inversion. According to this scheme, we introduce a space of weighted densities, P'' = W,p^
(7.1 0
where the weighting operator Wp is the linear operator of multiplication of t h e density p by a function iii^ equal to the square root of the integrated sensitivity (see formula (3.89) in C h a p t e r 3 and discussion in section 5.3.1): Wp = ^/S.
(7.16)
T h e integrated sensitivity, in accordance with t h e definition (3.75), is calculated by the formula 5 = % ^ , 8p
(7.17)
180
Integral representations in inversion of gravity and magnetic d a t a
where 6gr is the perturbation of the gravity field resulting from a local perturbation of the density, 6p{Q = p{()ds^ within a differential element of area ds, located at the point ^ = x -\- iz oi the lower half-plane (z < 0): (7.18)
S9r = 6gr{C) = - 2 j ^ ^ . Substituting expression (7.18) into (7.17), we find
•'^M^siF^^^^^^'
'Ik^'
(7.19)
'^^\T-—^dC',
where b is some line of observations of the gravity field. In particular, if the profile of observations coincides with the horizontal axes x, z' = 0, the definite integral in (7.19) can be calculated as
•{i-
Z ..2 . J^ X — X') + Zy2
S^^^lxl I I 1
/*°° 1
= 27W— /
/""TT"
T—-2dri = ^iJ-n^
y-^j-ooi + T
V 1^1
' < 0-
(^-20)
Formula (7.20) should be treated as the integrated sensitivity of the gravity data to the local density anomaly located at the depth \z\ in the lower half-plane {z < 0). We can see that the sensitivity is inversely proportional to the square root of the depth of the density anomaly. Thus, expression (7.15) takes the form
p^{0 = w,{z)p{0,
(7.21)
where Wp{z) = J2-fJ—
V
V 1^1
z<0.
(7.22)
We can consider a forward operator, which relates the new weighted parameters p^ to the data: g{C)=Ai{p^) = Ao{W~'p-). (7.23) Therefore Ai=A^Wr\
(7.24)
Gravity field
migration
181
and the weighted regularized direction of the steepest ascent is equal to IZ ip) = Re [Ai* (Ai [pn - gr)] + a (p^ - p^,) .
(7.25)
Finally, the iteration procedure is described by the following formulae: Pn+l
= Pn^^P
= Pn -
KL\Pn)
(7.26)
^
v^here
(7.27) Pn ~
\rwn\\M
I
I/" M 5
\rw{n-l)\ M
|^t/;(n-l)|
"-^
and W7ii
'wri
M
+a IZnCnd^
7
dC + (y
^ (C - C)
Note that at every iteration step we have to recalculate the real model density from the weighted parameter: (7.28)
pAC) = ^^^p{z)Pr;AO. 7.2
Gravity field migration
7.2.1 Physical interpretaUon of the adjoint gravitij operator According to iterative formulae (7.26), (7.27), the first iteration of the inverse problem solution is given by the expression (7.29)
Pi - PO + (^P ~ Po - ^'O^'wO^
where ('u>Oi
'•wO/M
P^COIID + "
IICoil M
and Co = C iPo) = Re [Ar [Ai (p-) - gr)] + a {p^" Assuming that p^ = p^^^ = 0, we arrive at the formulae 7a
ReA^V,
p^^)
182
Integral representations in inversion of gravity and magnetic d a t a
p^ - k^ Re A^:gr = k^ Re {A^W;^ and
g^ - W;'k^
Re A^^gr.
2
^oc ^ II^S'^riiM ' \\AiAi^gr\\l + a \\Ai^gr\\V
(7.30) /y 3-|^\
We recall now that, according to (7.28), the real model of the density distribution pi (C) is related to the weighted model p^ {() by the expression Pi (C) = %-' {z) Pr (C) = k^wf
(z) Re Ao*gr.
(7.32)
Let us analyze the result of applying the adjoint gravity operator to the observed gravity field, A^'^gr. According to (7.11), A'^*gr = 27 r
(7.33)
^rzydx'.
We analyze more carefully the physical meaning of this formula. First of all, let us examine the expression for g'^ {('). According to (7.1)
= -2711^^ ^^PiOds
= gr- {x'),
(7.34)
where gr*{x^) can be treated as the gravity field of the masses located in domain F*, which is a mirror image of domain F with respect to the real axis x (see Figure 7-1). We will call this field, ^r*, an adjoint gravity field. The density distribution p(C) within F* is a mirror image of the density distribution p{Q in F : p((^) = p{C)- Obviously, the field ^r* of the sources located above the line of observation b (coinciding with the real axis x) is an analytical function everywhere in the lower half-plane. It can be expressed in equivalent form as ^r*(C) == - 2 7 / /
- ^ ^ ( 0 ^ 5 - , C ^ r*, z < 0,
(7.35)
where ( = x -{- il ^ F* is a variable of integration, and d^ = dxd^. Now we examine the expression A^^gr in formula (7.33). Let P^ stand for the upper half-plane of a complex plane (, bounded by the real axis x, and P~ for the lower half-plane. We consider an arbitrary point C ^ ^ and draw therefrom a circle of radius R. That part of the real axis X that happens to lie inside the circle will be represented by 6^ , while that part of the circle found inside P~ will be denoted by CR (see Figure 7-1). Evidently, by virtue of the Cauchy integral formula (6.25), 5r-(C)^7 ,
1
/ 5r-(C),7 dC, Cn C-C
(7.36)
Gravity field migration
183
Figure 7-1 Definition of the adjoint gravity field gr*. The field gr* is generated by the sources located in F*. The density distribution J){C) within F* is a mirror image of the density distribution /;(() in F : J){() = piC)\vho!"o the ill!ogral ion over \hc closcxl coiitonr f)j{UCj{ is tak(Mi in the conntc^i'-clockwiscv In ])articular, the integration over the segment bj{ of the real axis is from right to left. Let us now proceed to the hmit as R ^ oo. T h e integral taken over the p a r t of the circle Cji is written, upon substitution of the variable C = C + R(''^^ in t h e form 1 27TI
!]r.{C + Bx:'")d0. C-C
(7.37)
2^
Being analytical, the function gr*{C) tends uniformly over 0 to zero at infinity, hence the limit of t h e integral (7.37) is zero. T h e integral along bfi as /?. —^ oo tends to t h e integral taken over the entire real axis x. Thus, in the limit, formula (7.36) takes the form giAO
27rz J_^
x-C
-dx, C e P~
(7.38)
where tfic minus sign arises because we have clianged the direction of integration: it is now conducted from the left (-oo) to the right ( + o o ) . Note t h a t from formula (7.34) it follows t h a t , on the real axis .T, (7.39) Substituting (7.39) into (7.38), we find .'?r.(C) =
- ^ ZTTZ
dx, C e P~
(7.40)
184
Integral representations in inversion of gravity and magnetic data
Taking into account (7.40), we can rewrite formula (7.33) as follows: A^^gr = 27gj | _ ^ f ^ d x '
= -^^^ligT-iO-
(7.41)
Thus, we see that application of the adjoint operator to the observed gravity field ^r is equivalent to analytical continuation of the adjoint gravity field ^r* iii the lower half-plane. Taking into account that, on the real axis x, the complex conjugate of the observed field is equal to the adjoint field (formula (7.39)), we conclude that the adjoint gravity operator is equivalent to analytical continuation of the complex conjugate of the observed field in the lower half-plane. We will call this transformation a gravity field migration, and will use the notation (7.42)
gT{0 = 9r'{0 = ^A^'9T.
According to formulae (7.13) and (7.41), the steepest ascent direction at the point p = 0 is equal to / (p = 0) = - Re A^'gr - 4^7 Re [ig^{0].
(7.43)
Note that there is significant difference between downward analytic continuation of the observed field and its migration, g'^^Q. The original gravity field has singular points in the lower half-plane, associated with the source of the field, so that its analytic continuation can be extended downward only to these singularities, and it is an ill-posed, unstable procedure (Strakhov, 1970b; Zhdanov, 1988). On the contrary, migration results in the field being analytical everywhere in the lower halfplane, and it is a well-posed, stable transformation. From the physical point of view, the migration field is obtained, according to formulae (7.42) and (7.34), by moving the sources of the observed gravity field above the observational profile. Nevertheless, as we will see below, the migration field contains some remnant information about the original sources of the gravity anomaly. That is why it can be used in imaging the sources of the gravity field. 7.2.2 Gravity field migration in the solution of the inverse problem Using gravity migration, we can find the first iteration for the density of the gravity field sources. Substituting (7.41) and (7.42) into (7.32) we find Px (C) = K^f
{z) Re A<^*gr = -^i^lk^w^^
{z) Re [ig^^iQ].
(7.44)
We call expression (7.44) a migration density p^ {(): p „ ( 0 = -47r7fc>;2 (^) j^g j.^m(^)]
(745)
It is proportional to the magnitude of the weighted migration field ^ p . Thus, migration transformation provides a stable algorithm for calculating migration density.
Gravity field migration
185
Let us examine t h e basic properties of t h e migration density distribution Pm (C) 7 obtained as the result of gravity migration. Using representation (7.42) of t h e migration field in the form of the field generated by the sources shifted to t h e upper half-plane, a n d taking into account formula (7.35), we compute a real part of ig^ : 1
-27 Re
Re [ig^
(x—x) — i(z — z)
= - 2 7 Re
r* {x - x)
-h ( ? - z)
z — z
= -27
IT' {X - x)
(7.46)
p{Q)ds, ^ < 0.
+ {z - z)
Substituting (7.46) into (7.45), and substituting expression (7.22) for the weighting function, wc obtain p„Ax,z)
=47r7A;^-
v {x - x)
+{z-
(7.47)
z)
For example, if the real density distribution is given by a delta function, p[Q
=
f\)^ (C - Co) , Co ^ -^"o - iZi) e r , then the density p(C) -= Po^ [^ - C(* j ^ C(* = --^^o + ?-o ^ r*. This means t h a t th(^ soince of the observed field is a material point located at a depth of —Z() , while the source of the migration field is a material point located at a height of +zn (Figure 7-2).
In this case we have
jjv{x^xy 2
~Z 1- {x -
.T)
z
fpiOds zo 2
2P0^ ( C ~ Co ) '^^'^ ~
~
+ (z -
+ ii^z)
z)
V
y
2/^0-
+ (zo -
{XQ-X)
(7.48)
z)
Substituting (7.48) into (7.47), we find p ^ ( x , z ) = 4^7A^o"-^'^' ^
Zo •
{x,~xf~^{zo-~z)'^''
< 0.
(7.49)
In particular, on the vertical axis, passing through the point x = XQ p ^ ( x o , z ) =47r7Po/co-
1 ^M /TT (ZO
-
(7.50)
z)'
Differentiating the last expression, we obtain dp^{xo,z) dz
z Un - z
(^0 ~ ^ )
Integral representations in inversion of gravity and magnetic d a t a
186
z
Zo I Xo X
-Zo
C,o=xo-izo Figure 7-2 The source of the observed field is a material point, located at the depth — Zo , while the source of the migration field is a material point, located at height +^o• -{ZQ
47r7Po^o-7=
-
Z)
-\-2 \z\
Z\[Zr,
y/TT
where we use a simple equality \z\ Thus, at extremum point z
= 47r7PoA:o
(^0 + z)
2vM^(^o - ^Y
- 0 , ifz (7.51)
-z, because z < 0. -ZQ expression (7.50) is equal
Pm(^o,^o) = 47r7Pofco
1 2./7rzo
(7.52)
Figure 7-3 presents a map of the migration density distribution given by formula (7.49). One can see that this distribution has a local maximum at the position of the point source (xo,>2^o) • Thus, migration density does not solve directly the inverse problem but provides an image of the source associated with the maximum of the function p^ {x^z). 7.2.3
Iterative gravity migration
Note that we can give the same physical interpretation to every subsequent iteration in the iterative scheme (7.26). According to formulae (7.12) and (7.43), the direction of the steepest ascent / (/O^) on each iteration can be computed using migration of the residual field [g^ (C) — ^'r (C)]? which is the difference between the predicted field on the n-th iteration, g^, and the observed gravity field ^r/ ( p J = 27Re
f
J —c
dx' = -47r7Re{z [C(C) " 9^0]}
•
(7.53)
Gravity field
0
Figure 7-3
187
migration
0.5
A map of the migration density distribution. This distribution has a local
maximum at the position of the point source (2, —2) .
A^'^-'Hdiiu^:
iz) Re {> [^-(C) - fh'iC)]} + « {p'"(0
- f^UO)
•
(7.54)
conchide t h a t calculation of Z^, (p) can be based on the migration Now we can use this technique in the general conjugate gradient al)ove. Using formula (7.27), we calculate the conjugate gradient , iteratively and define the weighted density step, 6p^, on each iterSp: = -fcnCXftT).
(7.55)
Midtiplying 6p'" by tlie inverse weighting function, we finally find the corresponding density step 6p^ : This approach to the solution of the inverse problemi is called an iterative migration. It is practically equivalent to the basic conjugate gradient inversion algorithm with one very i m p o r t a n t diff"erence - the density p e r t u r b a t i o n on each iteration is determined by migration of the corresponding residual field [gn{0 — .^r(C)] • This procedure, as we have demonstrated above, is essentially a downward analytical continuation of tlie complex conjugate field, [gn{C) — griOT , under an assumption t h a t all the sources
188
Integral representations in inversion of gravity and magnetic d a t a
of this field are located above the observation line. It means that migration is equivalent, from the computational point of view, to upward analytical continuation of the potential field generated by the sources located in the lower half-plane. The last transformation is very well developed in the theory of the potential field interpretation. In other words, the iterative migration makes it possible to use the powerful and stable technique of upward continuation for the solution of the inverse problem. In the later sections of this book we will demonstrate that the same technique can be applied to electromagnetic and seismic wave field inversion. In these areas of geophysics, migration serves as a useful practical tool for imaging geophysical data, because of the relative numerical simplicity and transparent physical interpretation of the results of electromagnetic and seismic migration. 7.3
Gradient methods of magnetic anomaly inversion
7.3.1
Magnetic potential inversion
Let us consider a problem of inversion of the magnetic potential. According to formula (6.37), we have f/(C') = A" (/) = - 2 jj^ ^I{Qds.
(7.56)
Let us assume that we have observed some magnetic potential t/r(CO ^^ ^^^ surface of the Earth. The problem is to determine the function /((") of complex magnetization distribution. We introduce a complex Hilbert space D of data (magnetic potential) with the metric given by formula (7.2), and a complex Hilbert space M of models (functions /(C)) with the metric
IL
(J, I)^ = JJ JiOriOds; J, I EM.
(7.57)
The only difference between the inverse problem formulation for gravity and magnetic fields is that now we use a complex Hilbert space of magnetization models /(C)The magnetic potential inverse problem can be solved by minimization of the misfit functional: (/>(/) = \\U - UrWl = (^" (^) - Ur. A^ (/) - C/r)^ = min,
(7.58)
where L^r = Ur{C) is the observed magnetic potential. We apply the same technique to solve the minimization problem (7.58) that we used in gravity field inversion. The direction of the steepest ascent of the functional (/){I) is calculated by a formula similar to (7.6), only now it is a complex function: / (/) = A"* (/I" (/) - C/r),
(7.59)
189
Gradient methods of magnetic anomaly inversion
where adjoint operator A^* is given by the expression A^^ {U)=2
(7.60)
^-^dx'.
Therefore 7-00
^
S
where U{C,') is the predicted magnetic potential for the given magnetization / : U{C) =
A-{I).
In particular, the steepest ascent at the initial model with zero magnetization is equal to / ( / = 0) = -A"*t/r. (7.62) Thus we can construct an iteration process of the re-weighted conjugate gradient method in a similar way as for the gravity problem, using recursions (7.26)-(7.27)
C+i = C + 61^ = C -
fcX(C),
(7.63)
where the regularized direction of the steepest ascent is given by the expression C {p) = Ai: {Al i n - Ur) + a { r - / - J ,
(7.64)
and the weighted operator and magnetization, AJ^, P"^, are related to the original operator and magnetization, A^, /, by formulae similar to (7.24) and (7.21): Al=A^Wr\
r iO = w;{z) I iO,
where Wf^ is a linear operator of multiplication by the weighting function, wj (z) , which is the same as the weighting function Wp (z) for the gravity case with the multiplier ^ 7 set equal to unity. wi{z)
= W 2 W T ^ , Z<0.
(7.65)
7.3.2 Magnetic potential migration We can introduce a migration magnetic potential as a field generated by the magnetic masses concentrated in the mirror image F* of the domain P. The magnetization distribution /((") within F* is a mirror image of the magnetization distribution I{() in F : I{() = /*(C*)- The migration potential is connected to the adjoint operator by the formula U?{0 = Ur'{0 = ^A^*Ur,
(7.66)
190
Integral r e p r e s e n t a t i o n s in inversion of gravity a n d m a g n e t i c d a t a
and therefore, the direction of the steepest ascent I (In) on each iteration can be computed using migration of the residual field [Un {x') — UY {X')]^ which is the difference between the predicted field on the n-th iteration [/„, and the observed magnetic potential t/pl{Pn)
2 / i ^ ^ M _ J ^ i ^ d : , ' = _4^^ [ [ / - (c) - [/- ( 0 ] . Jb
(7.67)
^ ~ S
Thus, as in the gravity case, the calculation of the each iteration in the conjugate gradient method for magnetic potential inversion can be based on the migration transformation. Note that a similar approach can be outhned for magnetic field inversion as well. The corresponding formulae are obtained from the basic formula for magnetic potential inversion by simple differentiation according to expression (6.36). I leave this derivation as an exercise for interested readers. In conclusion of this section I should note that all the techniques of integral representations in gravity and magnetic inverse problems, formulated above for twodimensional fields, can be generalized to three dimensions. Based on the theory of 3-D analogs of Cauchy-type integrals, introduced by Zhdanov (1988), one can demonstrate that the steepest descent and conjugate gradient iterations can be treated as results of the migration of observed gravity or magnetic fields. However, in a practical solution of 3-D gravity and magnetic inverse problems, where the corresponding forward modeling operators are linear, it is more common to discretize the original integral representations of the gravity and magnetic fields and to apply the numerical methods of solution of linear inverse problems directly to the corresponding systems of linear equations. We shall discuss this technique in the next section. Nevertheless, the approach based on integral representations in inverse modeling and migration transformation is useful in order to understand the similarity between the interpretation of different geophysical fields. For example, as I mentioned above, migration transformation is a powerful tool for imaging seismic and electromagnetic data. We shall address this problem in the chapters of this book that deal with electromagnetic and seismic inversion. 7.4
Numerical methods in forw^ard and inverse modeling
7.4.. 1 Discrete forms of 3-D gravity and magnetic forward modeling operators We return now to the full 3-D gravity inverse problem. We divide the domain D, filled with the masses of a density p ( r ) , into A^^ small rectangular cells, Dk-, D — U^^^Dk, and assume that the density is constant within each cell, p (r) = p^., r ED^ : 9z(r') = 7 E P. / / /
r ^ ^ ^ ^ -
(7-68)
Numerical methods in forward and inverse modeling
191
Assume that we use small rectangular cells D^. We denote the coordinates of the cell center as rk= {xk, yk,Zk), k = 1, ...Nm, and the cell sides as dx, dy, dz. Also, we have a discrete number of observation points r'^ = (x^,y^, 0), n = 1, ...Nd- Using discrete model parameters and discrete data, we can present the forward modeling operator for the gravity field, (7.68), as 9Z{T'J
~ Y^KkPk^^=
(7.69)
1,---A^d,
/c=l
where the gravity field kernel yl^^ according to (7.68) is equal to Kk-i
Zkdxdydz
(7.70)
'nfc
and
Tnk = y{xk - O ^ + {yk - y'nf + 4In a similar way, we can develop a discrete magnetic forward modeling operator. Note that we consider only the magnetic anomaly caused by the induced magnetization in the rock formations. Assume that the volume D is filled by magnetic masses with intensity of magnetization I(r), which is given by the product of magnetic susceptibility, x {^) ^ and the inducing geomagnetic field H*^: I(r) = x (i*) H*^. We apply formula (6.5) and again divide the domain D into Nm small rectangvilar cells, Dk-, D = U^^^D^, assuming that magnetic susceptibility is constant within each cell, x{^) ^ Xk^ ^ ^^k '•
HW^VT.X.[///V^
(7.71)
dv
Using Poisson's theorem, we rewrite the last formula in the equivalent form: d
H(r') A:=l
D,
dV
r ./|3
(7.72)
dv.
where H^ is the magnitude of the inducing field, and d/dV denotes a directional derivative in the direction of magnetization 1, where 1 is a unit vector: H^ = H^\. Note that d
dV
r
-1
3(l.(r-O)(r-r0
./|3
(7.73)
./|2
Substituting equation (7.73) into (7.72), we obtain
HM^-«»|x.//X^^
3(l-(r-r'))(r-r')
dv.
(7.74)
Integral representations in inversion of gravity and magnetic d a t a
192
For example, in the case of vertical magnetization, 1 = (0,0,1), and the vertical magnetic field, Z, is equal to
m-H'tx.jJij^
?>{z-z'f
1-
dv.
./|2
(7.75)
r —r
Measurements in airborne magnetic surveys are usually taken along the total magnetic field, and can be computed approximately as
T(r0^1(rO-H(rO
-«"gx.///-^ k=i
1-
3(l.(r-r'))^ r - Y'\
^ ^ -^^fc
dv.
(7.76)
Using discrete model parameters, introduced above, and discrete magnetic data, T(r^), we can represent the forward modeling operator for the total field, (7.76), as (7.77)
TK)^Y.fnkXk. k=l
where the magnetic field kernel /^^ according to (7.76), is fT
_
ZTO
Jnk ~ ^
3 [k {xk - x'J + ly [vk - Vn) + hzk]^ ' nk
-1
dxdydz
(7.78)
' nk
For example, for vertical magnetization, 1 = (0,0,1)
Z{<)^T.fnkXk. k=i
where the magnetic field kernel /^^ is expressed as
/4 = H°
dxdydz 2
Tnfc
(7.79)
' nk
Thus, the discrete forward modeling operators for gravity and/or magnetic anomalies can be expressed in general matrix notations as Am.
(7.80)
Here m is a vector of model parameters (densities, p^, or magnetic susceptibilities, Xk) of order Nm, d is a vector of observed data, g^, T, or Z, of order A^^, and A is a rectangular matrix of a size A^^ x ^m, formed by either gravity or magnetic field kernels.
193
Numerical methods in forward and inverse modeling
7.4-2 Discrete form of 2-D forward modeling operator The basic formula for the vertical component of the 2-D gravity field, (6.13), was derived at the beginning of Chapter 6: gz{x,z')
= 27 / / p{x,z)JJT
-J—^ [X
-
X')
-^ds.
-i-[Z - Z')
As in the 3-D case, we now divide the domain F into the Nm small rectangular cells, Ffc, r = U^^^Tk, and assume that the density is constant within each cell.
k=l
We denote the coordinates of the cell center as {x^, Zk), k = 1, .••Nm, and the cell sides as dx, dz. Also, we have a discrete number of observation points (x^,0) , n = 1, -.•Nd- Using discrete model parameters and discrete data, we can represent the forward modeling operator for the gravity field, (7.81), as g.K,0)«^a>„
(7.82)
k=l
where, according to (7.81), the gravity field kernel a^^ Zkdxdz
(7-83)
and rnk = \J{xk - x'^f -^ zl. Using matrix notation, equations (7.82) are written as d = A^m,
(7.84)
where d stands for the vector of observed gravity data: d = {^^(^n' 0), n = 1, ...A^^}, in is a vector of the densities m = {p^, /c — 1, ...Nm} , and A^ is a rectangular Nd x Nm matrix representing the forward gravity operator A^ = {a^^, n = 1, ...Nd] A; = l, ...Nm] In a similar way, we can develop a discrete matrix operator for 2-D magnetic forward modeling, based on the integral representation (6.12). 7.4-3 Regularized inversion of gravity data We have demonstrated in the previous sections that the solution of a gravity or magnetic field inverse problem in discrete form is reduced to a linear matrix equation. Therefore, the full arsenal of solutions to the linear inverse problem, developed in Part
Integral representations in inversion of gravity and magnetic data
194
True gravity data
-800
-600
-400
-200
0
200 meters
400
600
800
1000
True model, g/cm3 1
\J
0.8
2 200
-10.6
0
E xf400
0.4
Q.
600
0.2
-800
Figure 7-4
-600
-400
-200
0 meters
200
400
600
800
g/cm^
0
T h e observed gravity field ( t o p panel) and the vertical cross-section o f the
true model of the density distribution ( b o t t o m panel).
II of this book, can be applied to gravity and magnetic inversion. As an illustration, we will show in this section how the conjugate gradient method, outlined in Chapter 5, works in this case. At the same time, we will use gravity field inversion to demonstrate the importance of the regularization technique in the solution of the inverse problems. A vertical cross-section of the true model of the density distribution is shown in Figure 7-4, bottom panel. The vertical and horizontal lines show the model discretization. The size of each rectangular cell is 50 x 50 m^. Altogether we have 33 x 15 cells with unknown density. The gravity field, gz, is observed along a profile z = 0 at 33 points with 50 m separation (Figure 7-4, top panel). This gravity inverse problem is described by matrix equation (7.84). It represents an underdetermined problem, because the number of unknown parameters, 315, far exceeds the number of observations, 33. So the solution of the inverse problem is nonunique and, of course, is also unstable. We will apply inversion to the theoretical data, shown in Figure 7-4, with 3% random noise added. The re-weighted and regularized conjugate gradient method in the space of weighted model parameters, (5.152), discussed in Chapter 5 will be used. This method includes weighting of the model parameters, which forms a very important part of
Numerical
methods
in forward and inverse
modeling
195
Misfit functional
15 20 iteration number
Inverse model, g/cm3
0) 0)
0.04
200
0.03
E
r4oo
0.02
600
0.01
QCD •o
-800
Figure 7-5
-600
-400
-200
0 meters
200
400
600
800
g/cm^
T h e result of traditional minimum norm inversion w i t h o u t model parameter
weights. The b o t t o m panel shows the density distribution obtained after 32 iterations. T h e t o p panel presents the normalized misfit functional versus iteration number
the inversion, especially for an underdetermined problem. In order to illustrate the importance of weighting, we apply, first, the traditional minimum norm inversion without model parameter weights. Figure 7-5, bottom panel, presents the inversion result: a density distribution, which generates a predicted gravity field fitting the observed field with an accuracy of 10"^%! Figure 7-5, top panel, shows the behavior of the misfit functional as a function of the iteration number. One can see that the method converges very rapidly, but to the wrong solution! This result has a very simple physical explanation. Due to nonuniqueness of the gravity inverse problem, the observed data can be fitted by an equivalent material layer located just below the observation profile. The smooth unweighted inversion tries to find the smooth density distribution approximating this layer. Now we introduce model parameter weights, using the approach outlined in Chapter 3, sections 3.5.4: W ^ = [Wj] =
Vs,
(7.85)
196
Integral representations in inversion of gravity and magnetic d a t a Misfit functional
15 20 iteration number
Inverse model, g/cm3 0.05 V
;i£
avif
^:"" . 0) -^ 1$; E - :<^ir<;r400 * .^lm ;^i S ft" ^ •D ^ -; jwi 600 "". m '-ik ^ 1200
.s;-
'il
>
kh
0.04
;;t "^V •-''I
' 'i\
CD
"'S.7f • fix
'.''^^
-600
-400
\, i %
•
0.03
'
%', '-:;; 'KX \ ir. %i 'mit-: tU '<'.'.v ; ^1^ ';; ^ mC?tt,',. ^S :>r^ ^ 200 400 •
•
^
'ii
"
V'
0.02
k^^
i'.-
,v^
-800
Figure 7-6
^ Mi:^^4 '¥< Mi ;-;„
-200
0 meters
0.01 ;
i^
600
800
g/cm^
0
Results of the minimum norm inversion w i t h the model parameter weights.
where
ll^dll Sruk
\}
^nkJ n=l
and coefficients a^^. are determined by expression (7.83). Note that weighting matrix (7.85) is a numerical analog of the weighting function (7.22) introduced in the previous sections. Figure 7-6 presents the results of the minimum norm inversion with the model parameter weights. One can see that now the depth of the density anomaly corresponds better to the true model (7-4); however, the image is unfocused and dispersed due to the smoothing nature of the minimum norm stabilizing functional. The density values are significantly underestimated. We can see only one deep dense body in this image, while the true model consists of two bodies. The top panel in Figure 7-6 shows the behavior of the misfit functional as a function of the iteration number for this inversion. We still observe a very fast convergence. The density model in Figure 7-6 corresponds to the 32-nd iteration. It generates the predicted gravity field fitting the observed field with the a accuracy of 10~^%. However, we observe a drastic difference between the model shown in Figures 7-6 and 7-5, which confirms once again
197
Numerical methods in forward and inverse modeling Misfit functional
20
25 30 iteration number
Inverse model, g/cm3 1 0.8
I 200
IJIIIIIIIIIIII
CD
E ^"400 Q) •D
600 -800
Figure 7-7
-600
-400
-200
0 meters
200
0.6 0.4 0.2
400
600
800
0 g/cm^
Results of the focusing inversion obtained by the re-weighted regularized
conjugate gradient method.
the nonuniqueness of the gravity inversion. The next step is an illustration of the importance of selecting an appropriate stabilizing functional. We use now the minimum support stabilizing functional (2.61) (Portniaguine and Zhdanov, 1999), introduced in Chapter 2 : SMS (P)
IL
P' + P'
ds.
(7.86)
In numerical implementation of the re-weighted conjugate gradient scheme (5.152) we apply the following inverse weighting matrix according to (5.153): W ; ^ = diag[|p,|
(7.87)
Figure 7-7 illustrates the focusing inversion result obtained by the re-weighted regularized conjugate gradient method. The plots of the misfit and parametric functionals are shown in the top panel of Figure 7-7. In this case the data fitting after 50 iterations is within 4%; nevertheless the inverse image adequately reconstructs the true model. We can clearly recognize two bodies in this image, and the densities correspond well to the true model.
198
Integral representations in inversion of gravity and magnetic data
The numerical examples shown in this section clearly demonstrate the importance of using regularization in the solution of inverse problems.
References to Chapter 7 Portniaguine O. and M. S. Zhdanov, 1999, Focusing geophysical inversion images: Geophysics, 64, No. 3, 874-887. Strakhov, V. N., 1970b, Some aspects of the plane gravitational problem: Izvestia AN SSSR, Fizika Zemh, No. 12, 32-44. Zhdanov, M. S., 1988, Integral transforms in geophysics: Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo, 367 pp.
P a r t IV E l e c t r o m a g n e t i c Inversion
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Chapter 8 FOUNDATIONS OF ELECTROMAGNETIC THEORY In this Chapter I will introduce the basic equations governing the electromagnetic field in inhomogeneous conductive media, and review the basic physical laws important in developing electromagnetic inverse theory. 8.1
Electromagnetic field equations
8.1.1 Maxwell's equations The basis of the theory of electromagnetic fields studied by geophysicists is provided by the macro-electrodynamic equations, i.e., the Maxwell's equations: VxH=j+r+—,
(8.1)
V x E = - —,
(8.2)
V • B = 0,
(8.3)
V - D = g-f g^
(8.4)
where E and D are the vectors of the electric field; H and B are the vectors of the magnetic field; j is the conduction current density; q is the spatial density of free electric charges; j ^ and q^ are the densities of extraneous electric currents and charges. Equation (8.1) is the generalized Ampere's law, which states that both conduction currents and displacement currents, dT)/dt, generate the magnetic field. Equation (8.2) is Faraday's law of electromagnetic induction. Equation (8.4) is Coulomb's law, which states that electric charges generate the electric field. Equation (8.3) is Gauss' law of non-existence of magnetic charges. The electric currents and charges are interrelated by the continuity equation
V.(j+n = - ^ ( ^ ,
(8.5)
which expresses the fact that the divergence of electric currents from an infinitesimal volume is equal to the rate of decrease of electric charge density with time.
202
Foundations of electromagnetic theory
Equations (8.1) through (8.4) should be supplemented by the constraint equations reflecting the electromagnetic properties of a medium, which are for linear and isotropic media as follows: D =eE, B =/xH,
(8.6)
where e and fi are the dielectric constant (permittivity) and the magnetic permeability, respectively. The free space dielectric constant, ^o, and magnetic permeability, /ig, have the values £o ~ 8.85 X 10~^^ farad/meter, /JLQ = An x 10"^ henry/meter. The conduction current density is expressed in terms of the electric field by means of Ohm's law (in differential form): J =^E,
(8.7)
where a is the electric conductivity of the medium, and is the reciprocal of its resistivity p , i.e. a = 1/p. 8.1.2 Field in homogeneous domains of a medium In domains where the values of e, /x and a are constant, equations (8.1) - (8.4) take the form V x H = (7E+r+£^,
(8.8)
V X E = -fi—,
(8.9)
V • H = 0,
(8.10)
V-E = ^
.
(8.11)
In many problems, it is useful to separate Maxwell's equations (8.8) - (8.11), i.e., to write the equations for the electric and magnetic fields separately. To do this, we will apply the curl operator V x to the left- and right-hand sides of (8.9) and substitute the result into (8.8): V . V . E
+ ,e—
+ , a - = -,J^.
(8.12)
Similarly, using the curl operator V x on (8.8) and allowing for (8.9) we arrive at ^2TT
VxV
xn
+ ^le-^
OTT
+ |MJ— = V xf.
(8.13)
203
Electromagnetic field equations In view of the vector identity V XV XE - W
•E - V^E
(8.14)
and of Maxwell's equation (8.11), the relation (8.12) can be cast in t h e form
It is common knowledge (Stratton, 1941; Kong, 2000) t h a t in t h e absence of extraneous electric currents and charges, free electric charges disappear (relax) very rapidly in a homogeneous medium; hence, we can write 9^E (9E V ^ E - z x e — - / x a — = 0.
(8.16)
Similarly, equation (8.13) can be brought into t h e form i^2TT
V'H-f,e^^
OTT
- ^la—
= -V
x f,
(8.17)
or for j ^ = 0 ^2TT
QTT
^ ' H - M £ ^ ^ - na—
= 0.
(8.18)
Equations (8.16) and (8.18), satisfied by the vectors of the electric and magnetic fields in a homogeneous domain of a medium, in the absence of extraneous currents and charges, are called homogeneous telegraph equations. 8.1.S Boundary conditions At the interface of diff"crent homogeneous domains, the values of the p a r a m e t e r s £, /i and a may undergo step-like variations. In this case, according to formulae (8.6) and (8.7), some field vectors are also b o u n d t o change abruptly. To solve problems in electrodynamics, it is necessary, therefore, to formulate t h e b o u n d a r y conditions - t h a t is the relations between the vectors of the field at two adjacent points on the diff"erent sides of the interface of media with different electromagnetic properties. We will formulate the b o u n d a r y conditions for practical applications of critical importance, namely those in which a smooth surface separates two media, 1 a n d 2, whose p a r a m e t e r s are either constant or vary from point to point very slowly, so t h a t in a small neighborhood of any point on the interface, the interface can be regarded as plane and t h e m e d i u m parameters can be constant. T h e derivation of these b o u n d a r y conditions can be found in the textbooks on electromagnetic theory (see, for example, S t r a t t o n , 1941; Zhdanov and Keller, 1994; Kong, 2000). Let n denote t h e unit vector normal to t h e surface 5* at a given point. We also assume t h a t no extraneous current or charges are present at t h e interface S. T h e n t h e following relations hold good:
Foundations of electromagnetic theory
204
1) For normal components of the field:
where 7 is the surface density of the electric charge on S. Hence, the normal component of the magnetic field B is continuous in transition through the media interface, while the normal component of the electric field D experiences a discontinuity equal to the surface density of the electric charges. 2) For the tangential components of the field
where js is the density of the surface electric current, n x ( E @ ) - E ( ~= ) )0.
(8.21)
The surface current density js is nonzero only at the surface of a perfect conductor; hence for real media, (8.20) and (8.21) can be cast in the form
H ~ ) - H ? )= o , E:)-E~)
=o,
(8.22)
where the subscript T refers to the field component tangential to S . Thus, tangential components of the vector of the magnetic field H are continuous at interfaces of real media, while those of the vector of the electric field E are continuous at interfaces of any media. 8.1.4
Field equations in the frequency domain
By virtue of the linearity of the Maxwell’s equations, a field varying arbitrarily in time can be represented as a sum of harmonic fields whose time dependence is expressed by the factor exp(-iwt). For a monochromatic field, equations (8.1) through (8.4) take the form
V
x H = ZE +je,
V
x E = ZWB,
V.B=O,
(8.23)
where 5 = c - ZWE stands for the complex electric conductivity of the medium. The first simplification of Maxwell’s system of equations is based on writing the separate equations for the electric and magnetic fields.
Electromagnetic
205
field equations
Separation of Maxwell's equations The system of Maxwell's equations (8.23) can be modified so that the first two equations involve only the electric or the magnetic fields. Expressing the magnetic field H from the second Maxwell's equation, H = — V X E,
(8.24)
and substituting it into the first equation, and taking into account Ohm's law (8.7) and the constraint equations (8.6), we find: /xVx I —V X E j - {iLJiicF+u?iie) E =iuj^'f
.25)
In a similar way, determining the electric field E from the first Maxwell's equation, E=^VxH-:^j^ (J
(8.26)
a
and substituting it into the second equation, we obtain: aVx
X H j - iLUfiaH = ? V x flA
flv
.
(8.27)
Introducing the wave number /c of a medium as k"^ =iuji.i(j = iiUfia -f
(8.28)
LJ'^/IS^
we can cast the last two equations in the form /iVx ( - V X E j - P E = z a ; / i j ^ , aVx
(Iv
XH j - P H =aVx ( l A
.
(8.29)
In a homogeneous domain of a medium, where the parameters e, /i, and a are constant, equations (8.23) take the form V x H = 5E-f-r, V X E = iuj/jJi, (8.30) V - H = 0,
V • E = g7?.
206
Foundations of electromagnetic theory
where ! = £-{- ia/cj = —a/iuj stands for the complex permittivity of a medium. Note that the fourth equation in (8.30) can be derived by applying the divergence operator V- to the left- and right-hand sides of the first Maxwell's equation, V • V X H = 5 V • E + V - f = 5 V • E-hia;g" = 0, where we take into account the continuity equation for extraneous electric current and charges. Accordingly, using the vector identity (8.14), we can write equations (8.29) in the form: _ V ^ E + A : 2 E = -icj/xj^ -h (Vg^) / ? ,
(8.31) V2H-h/c2H =
-Vxf.
Equations (8.31) are known as the Helmholtz equations. The value of k is chosen to have a positive real part: Re^>0.
(8.32)
If the homogeneous domain in question is devoid of extraneous currents or extraneous charges, then we have V X H = 5E, V X E = zcj/iH, (8.33) V • H = 0, V - E = 0. In this case, equations (8.31) are reduced to the homogeneous Helmholtz equations V ^ E - h P E - 0, (8.34) V ^ H - h P H = 0. E l e c t r o m a g n e t i c potentials Another approach which can be used to simplify the set of equations (8.23) is based on electromagnetic potentials. In view of the third Maxwell's equation, the magnetic field B is solenoidal (the magnetic flux lines flow along closed paths) and can be written as
Electromagnetic
207
Held equations
B = V X A,
(8.35)
where the auxihary field quantity A bears the name electromagnetic vector potential. Substituting equation (8.35) in the second Maxwell's equation (8.23), we obtain: V X E = iiuV X A and Vx(E-za;A) =0. Because the term in parentheses above has a curl of zero, it can be written as the gradient of some scalar function, t/, which is named the scalar electromagnetic potential, E-ZCJA =
-V^,
and so, E = ZUJA - V/7.
(8.36)
Next, we rewrite the first of Maxwell's equations (8.23) in terms of these electromagnetic potentials: Vx [ - V XA j =
iidcjA-aVU+y^
or, multiplying both sides of the last equation by the magnetic permeability, /i, /iVx f - V x A j - zcj//aA+/iaV[/=/ij^
(8.37)
To obtain an additional equation for electromagnetic potentials, we can take divergence of the first Maxwell's equation (8.23) V-5E + V • j ' = V- (V X H) = 0.
(8.38)
Substituting expression (8.36) into (8.38), we find: V- [a {iLuA - VU)] = - V • j " .
(8.39)
Thus, we have one vector equation (8.37) and one scalar equation (8.39), which, together with the radiation conditions in infinity (see below), uniquely determine the vector. A, and scalar, U, electromagnetic potentials. In the majority of geophysical applications we assume that fi = fx^ = const, where /J,Q is the free space magnetic permeability. Using also the vector identity given earlier in equation (8.14), we can write:
208
Foundations of electromagnetic theory
V^A -f- iujfiQaA- ( V V • A+/io5Vf/) = -Moj^
(8.40)
We note that while we have determined the curl of the vector potential in equation (8.35), the divergence of this potential remains undetermined. We may arbitrarily specify that V • A = 0.
(8.41)
Other convenient specifications for the divergence can be conceived (see below); the gauge defined by equation (8.41) is known as the Coulomb gauge. With use of this gauge condition, equation (8.40) can be rewritten in the form: V^A + fiQ^
{IUJA
- VU) = -/ioj".
(8.42)
Thus, with the choice of the Coulomb gauge, the electrical field, according to (8.36), is decomposed into a curl-free part, —Vf/, and a divergence free part, iouA. In conclusion of this section I present the equations for electromagnetic potentials in a homogeneous domain of a medium, where a = const and e = const. In this case, equation (8.40) takes the form V^A -h iujfiQaA - V (V • A-hfio'^U) = - / i o j ^
(8.43)
We can use now a Lorentz gauge condition V • A = -/io5t/,
(8.44)
which results in a Helmholtz equation for a vector potential: V2A + p A = - / i o j ^
(8.45)
Substituting expression (8.36) for the electric field in the fourth Maxwell's equation (8.30),
V'{iL0A-VU)
= lr,
and taking into account the Lorentz gauge condition, we finally arrive at the Helmholtz equation for a scalar potential: V'^U + k^U=-^.
(8.46)
In summary, we have transformed a problem requiring the solution for six unknown variables (the six scalar components of the vectors E and H) to a problem requiring solution for four unknown variables (the three scalar components of the vector potential A and the single variable, the scalar potential U). As we will see below, this reduction in the number of variables allows us to simplify markedly the solution of electromagnetic forward and inverse problems in many cases.
Electromagnetic
8.1.5
209
field equations
Quasi-static (quasi-stationary) electromagnetic field
In many problems, geophysicists deal with electromagnetic fields varying in time very slowly. In the equations for this field the second derivative with respect to time d'^/dt'^ can be discarded. This situation occurs, in particular, if we neglect the displacement currents, d'D/dt, in the first Maxwell's equation. This model is called quasi-static (or quasi-stationary according to Berdichevsky and Zhdanov, 1988/ A quasi-static electromagnetic field is described by the following equations:
VxH-aE+f, V XE =
-fidU/dt, (8.47)
V-/iH = 0, V - E = -(E-Vcr + V-j')/a. Note that the fourth equation in (8.47) can be derived by applying the divergence operator V- to the left- and right-hand sides of the first equation in (8.47), V • V X H = V- (o-E) -hV • j ' - V a • E + a V • E + V • j " = 0. According to equations (8.12) and (8.13), we have in domains of constant electric conductivity and magnetic permeability V X V X E -h ^adE/dt
=
-^id'f/dt, (8.48)
V X V X H -f- fidd'ti/dt
= V X j^
As a consequence, the telegraph equations (8.17) become the equations of diffusion V^H - iia—
= - V X f,
(8.49)
and in a homogeneous domain equation (8.15) takes the form <9E B\^ B\^ 1 V^E - iicj— = ; , ^ + V V • E = M - ^ - - V V • j ^
ot
at
at
a
(8.50)
Just as in a general case, the quasi-stationary field can be represented by Fourier transforms as a sum of harmonic fields whose time dependence is reflected by the factor exp(—icjt). The quasi-stationary field harmonics are described by the equations
210
Foundations of electromagnetic theory
V x H = (7E+f, V XE =
iuj/jH,
(8.51) V-/iH = 0, V - E = - ( E - V a + V-j")/a. Separating the electric and magnetic fields in (8.51) for a = const, /i = const, we write V^E+fc^E = -icufi}^ - - V V -f, a V^H+Zc^H - - V X f.
(8.52)
Here the wave number for a quasi-static model of the field in a medium is defined by the equation k^ = iujfiG. (8.53) The k values are also chosen to have a positive real part:
where A is the quasi-static electromagnetic wave length calculated by the formula A = 2lX yj 21UJ [IG.
8.1.6 Field wave equations In an insulator (for (7 = 0), equation (8.1) takes the form V x H = r + — .
(8.54)
If here e = const and fi = const, field equations (8.15) and (8.17) go over into the wave equations:
V^H - M £ ^ = - V X j ^
(8.55)
According to (8.55), the velocity v of the electromagnetic wave propagation is 1 For example, in free space this equation gives us the velocity of the light c : c = — = = 3 X 10^ m/s. \/Mo^o
Electromagnetic
211
field equations
<^.i.7 Fze/(i equations allowing for magnetic currents and charges Refer again to equations (8.8) through (8.11) for an arbitrary electromagnetic field in a homogeneous and isotropic domain of a medium. The first striking feature is the asymmetry of the right-hand sides of the equations for the electric and magnetic fields. This asymmetry arises from the absence of extraneous magnetic currents and charges in the equations concerned. This situation can be rectified by introducing the required additional terms
V x ;H =
(8.56)
9H -^^-
V X E ==
r,
(8.57) (8.58)
V I
(8.59)
Here j " ^ and q^ are the densities of extraneous "magnetic currents" and "magnetic charges," respectively, which are interrelated by a continuity equation similar to (8.5): V - r = — ^ .
(8.60)
Evidently, the quantities j ^ ^ and q'^' are fictitious, since to the best of our knowledge, no magnetic charges exist in nature. Yet these fictitious magnetic currents and charges may prove extremely useful for describing electromagnetic field discontinuities in certain theoretical problems. Employing the curl operator with the left- and right-hand sides of equations (8.57) and (8.56), we obtain ^
^
^
d^E
dE
dr
V X V X H + M £ ^ ^ + ^l(T— = -ar
- e ^
^
.m
+ Vxf.
(8.61)
Equations (8.56) - (8.59) become particularly elegant and symmetric when transformed to the frequency domain V x H = ^E+j^ V X E = iLonn - j " , (8.62) V • H = g-/M, V •E =
q^s.
212
Foundations of electromagnetic theory
The forth equation in (8.62) can be obtained by applying the divergence operator V- to both sides of the first equation and allowing for the continuity equation for the extraneous currents and charges. In particularly, the Helmholtz equations (8.31) can be written as follows V ^ E - h P E = -icj/ij" + Ivq^ V ^ H + P H = ar
+ V Xj^,
+ - V^^ - V X j ^
(8.63) (8.64)
where a and ? stand for the complex electric conductivity and the complex permittivity of a medium, respectively, a = a — icue, ? = £ -h ia/uj, and the wave number k is given by k = \/iuj\iG,
Re/c > 0.
8.1.8 Stationary electromagnetic field A stationary field is known to be independent of time. The equations it satisfies in a homogeneous and isotropic domain of a medium are derived from (8.56)- (8.59) by setting all the time derivatives equal to zero: V x H - a E + f , V x E = -j^, V . H = -q"^, V • E = ^ - ^ . /i
e
(8.65)
Here the continuity equations for extraneous electric and magnetic currents take the form V - f = 0, V -r = 0. (8.66) Accordingly, equations (8.61) are expressed as follows: V x V x E = - V xj^, V X V X H = -ap + V X f. (8.67) Note that the divergence of the stationary electric field in a homogeneous conductive medium (cr 7^ 0) is equal to zero: V-E-0. The last equation can be derived by applying the divergence operator V- to the leftand right-hand sides of the first equation in (8.47), V - V x H = a V - E - ^ V - f = crV-E-=0.
213
Electromagnetic Held equations
As a result, stationary electric and magnetic fields satisfy the Poisson equations:
V^H = - V^™ + (Tj™ - V X f.
(8.68)
8.1.9
Fields in two-dimensional inhomogeneous media and the concepts of E- and Hpolarization The two-dimensional model is characterized by electromagnetic parameters, 5, /i, cr, which vary only in two directions (say, in the vertical plane), and are constant along one specific direction (perpendicular to this plane). In order to simplify our discussion of two-dimensional models, we will always use coordinate systems in which this specific direction of constant parameters lies along the y axis. We will also assume that the extraneous current density will not vary in the y direction. Therefore, the electromagnetic field in a given model will also be two-dimensional; that is, it will not vary along the y axis. For such two-dimensional models, we consider a quasi-stationary electromagnetic field and assume that the magnetic permeability is that of a free space /IQ. We can write the first two Maxwell's equations (8.51) for this model: V x H = crE-f j ^ V x E = zcj/igH. Considering that all derivatives with respect to y are zero, the scalar expressions for the Maxwell's equations can be written as dHjdz
- dHjdx
= aEy -h j ^ ,
dHy/dx^GE,-\-jl, -dHy/dz -dEy/dz dEjdz
(8.69) (8.70)
= c7E,^jl,
(8.71)
- iujfiQH^,
(8.72)
- dEjdx
= luii^Hy,
dEy/dx = iujii^H,,
(8.73) (8.74)
where cr = cr(x, z) is the conductivity as a function of location on the xz plane. We find that only the field components E^, Hy and Ez are present in equations (8.70), (8.71), and (8.73), while only the components H^^ Ey, and H^ are present in equations (8.69), (8.72) and (8.74). The advantage of using two-dimensional models
214
Foundations of electromagnetic theory
lies largely in the fact that Maxwell's' equations separate naturally into two independent systems: -dHy/dz = aE,^jl,
(8.75)
dHy/dx = aE,+Jl,
(8.76)
dE:,/dz - dEjdx
= iuoyL^Hy,
(8.77)
and -dEy/dz
= iujfiQH:,,
dEy/dx = iujfiQH,, dHJdz - dHJdx = aEy + j^. At this point, we introduce a standard notation:
(8.78) (8.79) (8.80)
E^ = (s,,o,£;,), H^ = (0,i/„0),
(8.81)
E^ = (0,E„0), H^ = (//,,0,//,).
(8.82)
The field E ^ , H ^ is called an H-polarized field {transverse magnetic - T M mode), because the magnetic field has only a single component, Hy, which is nonzero (that is, the magnetic field is polarized to lie always on the long axis of the structure). Likewise, the field E ^ , H ^ is called an E-polarized field {transverse electric - T E mode), because only the single component Ey of the electric field is nonzero (the electric field is polarized to lie always on the long axis of the structure). Equations (8.75) to (8.77) characterize the behavior of the //-polarized field, and equations (8.78) to (8.80) characterize the behavior of the £^-polarized field. Because these two sets of equations are independent of one another, we conclude that when the conductivity model is structured in only two dimensions, the total electromagnetic field reduces to the sum of //-and £^-polarized fields which propagate in the model independently of one another: E = E^-hE^,
215
Electromagnetic energy How
H = H^+H^. The problem of studying the interaction of an electromagnetic field with a two-dimensionally structured model reduces to the solution of two far simpler problems, the solution of problems for £^-polarized and /f-polarized fields separately. The simplicity lies in the fact that when £^-polarized or //-polarized fields are considered, Maxwell's vector equations reduce to scalar diff"erential equations. Initially, let us examine a case with E polarization (equations (8.78) through (8.80)). Substituting equations (8.78) and (8.79) into (8.80), we obtain (8.83)
V'Ey^k^Ey^-iu^^fy,
where V^ is a two-dimensional Laplace operator, (9^/9x^4-5^/5?/^, and /c^ = k'^{x^ z) = iijfiQa{x^ z). Thus, the y component of the electric field for E polarization satisfies a twodimensional Helmholtz equation with a variable wave number, k{x, z). The magnetic components of the field in this case are found simply by difi'erentiating Ey according to equations (8.78) and (8.79). In the case of i/-polarization, by substituting equations (8.75) and (8.76) into equation (8.77), we find d ( 1 dUy\
d ( 1 dHy\
, _
d f I .:\
d ( \
In summary, two-dimensional models permit a great reduction in the amount of computation necessary for the solution of forward or inverse geoelectric problems. Because of this, such models are widely used in most problems of electromagnetic modeling and inversion. 8.2
Electromagnetic energy flow^
In this section we consider an electromagnetic field generated by the sources (extraneous currents j^), concentrated within some domain Q and propagated in a medium with a given distribution of electromagnetic parameters e, /x, and a. For the sake of simplicity, domain Q is considered to be local (bounded). This means that there is a sphere, CR, with its center at the origin of the Cartesian coordinate system and of a radius R (Figure 8-1), big enough that domain Q belongs completely to the ball OR bounded by the sphere CR : Q C OR. Let us surround the domain Q by another domain, Vi, which completely contains the domain Q (Figure 8-1). It is evident that the change of field energy within the volume Vi must be associated with afiowof energy through the surface 5*1 of Vi from or toward the sources. A relation between the rate of change of the energy stored in the electromagnetic field and the energy flow is given by Poynting's theorem (Stratton, 1941; Kong, 2000).
216
F o u n d a t i o n s of e l e c t r o m a g n e t i c t h e o r y
8.2.1 Radiation conditions Before considering Poynting's theorems, we should analyze the field behavior on the sphere CR when its radius R tends to infinity. Based on physical intuition, one should expect that the field generated by an arbitrary source, located within the bounded domain Q of space, can be approximated accurately enough by a spherical wave at a large distance from the source and should comprise only a divergent (or so-called "outgoing") electromagnetic wave*. The transverse field in this spherically divergent wave should decay like 1/r at large distances r = |r| from the origin of coordinates, so that the product of the field / ( r ) and the distance r is bounded, r / ( r ) is bounded,
(8.85)
where / stands for any component of electromagnetic field transverse to the radiusvector r. Another property of the divergent spherical wave is that it behaves locally like a plane wave travelling to infinity, with each field component transverse to r behaving as exp {ikr). The last requirement can be mathematically expressed by the following formula (Felsen and Marcuvitz, 1994): lim r | / ( r ) - ^ > )
0.
(8.86)
These conditions, introduced by Sommerfeld (1912), are called "Sommerfeld radiation conditions.'' We will see below that radiation conditions ensure the obvious physical requirement that the energy of the electromagnetic field travels away from the source domain, i.e. the electromagnetic sources radiate energy outwards from these sources. 8.2.2 Poynting^s theorem in the time domain Poynting's theorem can be derived based on the first two Maxwell's equations (8.1) and (8.2): VxH = a E + r + ^ ,
(8.87)
V X E = ——. (8.88) ot Note that the dot product E • j ^ has the dimensions of energy per unit volume per unit time. Scalar multiplying the first Maxwell's equation (8.87) by E, we find E - V x H = aE.E4-E- j^-f-E~.
(8.89)
In a similar way we can obtain from the second Maxwell's equation (8.88) an expression containing the terms which have dimensions of energy per unit volume per unit *For a rigorous definition of the divergent waves see the section on radiation conditions in Chapter 13.
Electromagnetic energy Bow
Figure 8-1
217
Derivation of Poynting's theorem and the energy inequality.
time, by scalar multiplication of each term of this equation by H H V X E = -H
dB dt
(8.90)
Subtracting equation (8.89) from (8.90) and using the vector identity (8.91)
H-VxE-E-VxH-V-(ExH), we obtain V- (E X H) =
(8.92)
E-r + a E - E ^ - E - ^ + H . ^ at at
Expression (8.92) represents a differential form of Poynting's theorem. The energy flow of the electromagnetic field can be calculated using the Poynting vector P , introduced by the following formula: .93)
ExH.
The Poynting vector P may be interpreted as the intensity of EM energy flow at a given point, that is the energy per second crossing a unit area whose normal is oriented in the direction of the vector E x H . Integrating expression (8.92) over domain Vi and applying the Gauss formula, we obtain an integral form of Poynting's theorem: Ft - / / ( E x H) 'iiids = ff P- mds J Jsi J Js^
f E-^dv^ [ff aE-Edv-h / / / Vi
J J Jvi
J J Jvi
dB
dB
dv
(8.94)
218
Foundations of electromagnetic theory
where Ft is the total energy flow of the electromagnetic field through the surface Si of the domain Vi, and rii is a unit vector of normal to the surface ^ i , directed outwards from the domain Vi. The first term on the right-hand side of equation (8.94) represents the power introduced through the extraneous currents 'f. The second term represents the power dissipated in Joule heat within the domain V\. The third term represents the rate of decrease of electric and magnetic energy stored in the volume Poynting's theorem states that the diminishing of electromagnetic energy stored within the volume Vi (the third term on the right-hand side of equation (8.94)) is partly accounted for by the energy flow Ft outside domain Fi, by the energy loss accounted for by the Joule heat, and is partly compensated by the power introduced through the extraneous currents. In linear isotropic media the last formula takes the form Ft= E-rdv^
III
I
J Jsi
P'Uids
a\Efdv+~JJJ
[e\Ef + ^\H\']dv\.
(8.95)
8.2.3 Energy inequality in the time domain An important property of the energy flow Ft out of domain Vi, containing sources j ^ , is that its integral over a time period, f_^ Ftdt, is always non-negative independently of the type of the extraneous sources. To prove this we introduce a domain Ofi\Vi formed by the ball On without domain Vi (Figure 8-1). We can apply now Poynting's theorem (8.95) to the volume OR\VI, taking into account that extraneous currents j ^ = 0 in OR\VI
:
-
//
P -nids-i-
//
P -njids
where n^^ is a unit vector of a normal to the surface CR, directed outside domain O^, and we have changed the sign of the integral over 5i, because now the outer normal to OB\VI is directed opposite to the vector rii. Setting i? —> oo in (8.96) and taking into account that, according to the radiation conditions (Harrington, 1961; Kong, 2000), the surface integral over CR tends to zero at infinity.
IL
P ' njids —> 0 for i? —> oo,
CR
we arrive at the following formula:
J Jsi
P • liids
Electromagnetic
219
energy Bow
where domain CVi is a complement of the domain Vi to the entire space. From the last formula we conclude that Ft=
V-n^ds
J Jsi
.98) Now we integrate both sides of equation (8.98) over a time period: rt
I Ftdt = f J—CO
= / J-oo
P • rvidsdt
J—C
/ / / a\Efdvdt^-fff J J JcVi
'^
[s\Ef^fi\nf]dv,
(8.99)
J J JcVi
Taking into account that all integrals on the right-hand side of expression (8.99) are non-negative, we arrive at the important energy inequality, t
Ftdt > 0,
(8.100)
— oo
and Ftdt = 0, if and only if E —0, and H =0 in CVi, which may happen, as we will see below, only if there is no source in the entire space j*^ = 0. Formula (8.100) has a clear physical interpretation: the total energy radiated out of the domain with sources of electromagnetic field, over the time period from the initial field generation until any time t, is always non-negative. Thus, the electromagnetic sources continuously radiate energy outside these sources! Substituting expression (8.95) for Ft in (8.100), we finally obtain:
a\Efdvdt^^ J —c
Vi
[[[ [£\Ef + /j.\li\^]dv '^ J J Jvi
<- [ Iff E-'fdvdt. 7-00 J J Jvi
(8.101)
The last inequality means that the energy introduced through the extraneous currents j ^ always exceeds the sum of the energy dissipated in Joule heat within the domain Vi and the electric and magnetic energy stored in the volume Vi. In other words, there is always a net radiation of electromagnetic energy from any source.
220
Foundations of electromagnetic theory
If j ^ = 0, then /
///
G |E|^ dvdt^\
J-oo J J Jvi
[[[
[e |E|^ + /i |H|^] dv = 0.
^ J J Jvi
An obvious result follows immediately from this equation, namely that in this case the electromagnetic field is equal to zero as well: E = 0, H = 0. 8.2.4 Poynting^s theorem in the frequency domain In the frequency domain equations (8.87) and (8.88) take the form V X H = aE-\-j^-iLueE
=5E + j ^
(8.102)
(8.103)
V xE = icj/jH,
where a = a — iuoe is a complex electric conductivity. The energy flow of the electromagnetic field in this case can be calculated using the complex Poynting vector P (Stratton, 1941), introduced by the following formula: P = ^ E X H*,
(8.104)
where asterisk "*", as usual, indicates a complex conjugate value. The real part of the complex Poynting vector P may also be interpreted as the intensity of EM energy flow at a given point, that is the energy per second crossing a unit area whose normal is oriented in the direction of the vector E x H*. For example, the total energy flow of electromagnetic field through the surface Si of the domain Vi, containing domain Q (Figure 8-1), is equal to F^ = Re ff
P • nids = ^ Re / / " ( E x H*) -uids,
(8.105)
where rii is a unit vector normal to the surface 5i, directed outwards from the domain Vi. In view of the Gauss formula, expression (8.105) can be cast in the form F^ = Re fff
V-Pdv
= ^Re
fff
V • (E x H*) dv.
(8.106)
We can obtain Poynting's theorem by taking the scalar product of the second equation for the total field (8.103) with H* and the complex conjugate of the first equation (8.102) with E, and subtracting one from the other:
221
Electromagnetic energy Row
2V • P - V • (E X H*) = H* • V X E - E • V X H* (8.107) = - 5 * | E | ^ - E - j ^ * + ia;/i|H|^ Substituting equation (8.107) into (8.106), we obtain F^ = ^Re
ff
(ExH*)-nid5 = - i
/ " / / {a\Ef-\-Re{E
-T)}
dv.
(8.108)
This is Poynting's theorem in the frequency domain (Stratton, 1941). It says that the energy flow outside the domain Vi is formed by the balance between the energy dissipated in heat per unit volume per second, and the energy introduced through the extraneous currents j ^ . We can prove, also, that in the frequency domain the energy flow F^ is always non-negative. Following the technique outlined above for the time domain case, we apply Poynting's theorem to the volume Oi^\Vi, taking into account that extraneous currents j ^ = 0 in OR\VI :
-1 / / /
I.IE'^ {a\Ef}dv OR\VI
= --Re
(ExH*)-nid5-f-^Re / /
[f
(E x H*)-ri/^ds.
(8.109)
Setting /?. -^ cx) in (8.109) and noting that, according to the radiation conditions, the integral over Cn goes to zero (Harrington, 1961), we find ^ Re ff
(E X H*) -uids = ^ fff
- Re ff
(E X H*) -111^5 = - fff
a |E|^ dv > 0,
(8.110)
and 2
a |E|^ dv = 0,
if and only if cr = 0. In other words, we have found that the total energy flow F^ of the electromagnetic field through the surface ^i of the domain Vi containing the extraneous sources of the field, is always positive, and is equal to zero if and only if the conductivity of the medium is zero (the case of a lossless medium): F^ > 0, and F^ = 0, if and only if a = 0. Substituting equation (8.108) into (8.110), we find {cr|E|^-hRe(E*-j^)}dt;<0, Vi
(8.111)
222
Foundations of electromagnetic theory
or / / / a\Efdv< - [[[ Re{E*-j')dv, (8.112) J J Jvi J J Jvi for any lossy medium (a ^ 0). Energy inequality (8.111) means that we have steadily positive energy flow out of a domain containing a source within a lossy medium. 8.3
U n i q u e n e s s of the solution of electromagnetic field equations
8.3.1 Boundary-value problem Let us consider an electromagnetic field given within a local domain V ^ bounded by the surface S. In the frequency domain, this field satisfies equations (8.102) and (8.103) VxH=?E-hf,
(8.113)
V x E = zu;//H,
(8.114)
where a = cr — iujs is the complex electric conductivity. We assume that the source of electromagnetic field (extraneous electric currents j^) and the boundary values of the field components on the surface S are known. In this case one can be assured that only one (unique) solution of Maxwell's equation is possible. Actually, it is sufficient to specify either the tangential electric field or the tangential magnetic field on the various parts of the boundary S. Indeed, let us assume that this boundary-value problem admits two solutions. Then we can express the difference of those solutions as the electromagnetic field { E ' , H ' } . This field satisfies the equations V X H'=5E',
(8.115)
V X E' = ia;/iH',
(8.116)
and the following boundary conditions (E'xH'*)-n = 0,
(8.117)
where n is a unit vector of the outward normal to S. Equation (8.117) holds because either the tangential electric field E' or the tangential magnetic field H ' is equal to zero on the various parts of the boundary S. We now can write Poynting's theorem (8.108) for the field { E ' , H ' } given within domain V : Re / / (E'xH'*) -Jiids = - fff
a \E'fdv,
(8.118)
Uniqueness of the solution of electromagnetic
field equations
223
where we take into account that this field, according to equation (8.115), has no extraneous electric currents in V. Substituting boundary conditions (8.117) into the last equation, we find
/ / / fv.
a\E'\^dv
= 0.
(8.119)
Condition (8.119) in any lossy medium {a > 0) undoubtedly leads to |E'| ^ 0, i.e. E', = E'y^Ei^O
(8.120)
everywhere within domain V. Substituting (8.120) in the second Maxwell's equation, we conclude that the magnetic field H^ is also identically equal to zero (assuming cc^O). Thus we have proved the uniqueness theorem for the boundary-value electromagnetic problem. 8.3.2 Uniqueness theorem for the unbounded domain We now examine the solution of Maxwell's equation in an unbounded domain with the given distribution of electromagnetic parameters e, //, and a. The electromagnetic field is generated by the sources (extraneous currents j""), concentrated within some local domain Q . Using the energy inequality (8.112), we can prove that there is only one (unique) solution of this problem. As in the case of the boundary-value problem considered above, we assume first that there may be two different solutions of the Maxwell's equations in the unbounded domain with the same source j"". Introducing the difference of those solutions, the electromagnetic field {E',H'} , we notice immediately that the source of this field is zero j ^ ' = 0. Let us draw a sphere, CR, with the center at the origin of the Cartesian coordinate system and of a radius R, big enough that domain Q belongs completely to the ball OR bounded by the sphere CR : Q C OR (Figure 8-1). Applying inequality (8.112) to the difference field inside this ball. OR, we have a | E f dv<-
[[[ Re (E* • j ^ ^ dv = 0. (8.121) J J JOR From this equation it follows immediately that in any lossy medium (a > 0), the electric field is equal to zero inside OR as well: E ' ( r ) = 0, reOR.
(8.122)
Formula (8.122) holds for any point r of the space, because for any r we can always select a radius R big enough that r ^OR. Substituting (8.122) in the second Maxwell's equation, we conclude that magnetic field H' is also identically equal to zero (assuming This completes the proof of the uniqueness theorem for the unbounded domain.
224
8.4
Foundations of electromagnetic theory
Electromagnetic Green's tensors
8.4'^ Greenes tensors in the frequency domain The electromagnetic Green's tensors G^; , G// are introduced as the fields of an elementary electric source (Zhdanov, 1988; Felsen and Marcuvitz, 1994). They satisfy the Maxwell's equations
+ T5(r-rO,
VXGH=^GE
(8.123) V X G^; = iLo^Gn, where G// (r | r') and G^; (r | r') are the magnetic and electric Green's tensors, I is the identity tensor, and 6{Y) is three-dimensional Dirac delta-function ^(r) = 6{x)6{y)6{z), satisfying the condition
where V = V -\- S is domain V with its boundary S. Equations (8.123) suggest that in a homogeneous domain G^; and G// satisfy the Helmholtz equations = -iujfjIS (r - r ' ) ,
(8.125)
V^G/z+fc^G// - -VxT(5(r - r ' ) .
(8.126)
V^GE+H^GE
The importance of Green's functions in electromagnetic theory is dictated by the following considerations. According to the basic property of ^-function (8.124), any source, j^(r'), can be represented as a sum of elementary current sources:
J J Jvoc
j%r)-l6{T'-r)dv,
(8.127)
where the volume integral is calculated over the entire infinite space Voo and I is the identity tensor. Equation (8.127) indicates that the density of the distribution of elementary current is described by the tensor expression 16{r'—r). Taking into account the linear character of Maxwell's equations and using the superposition principle, we find that the sum of electromagnetic fields produced by elementary currents must be equal to the field generated by the original current source j^(r'). In other words, the electromagnetic field { E , H } generated by the source described as a linear combination of some other elementary sources is equal to the linear combination (with the same coefficients) of the fields < GE.GH \ generated by the individual elementary current sources. Therefore, we can conclude that with the knowledge of electromagnetic Green's tensors G^; and GH one can also find
Electromagnetic
225
Greenes tensors
t h e solution of Maxwell's equations for an arbitrary allocation of extraneous electric currents j ^ ( r ) , as the convolution of t h e corresponding Green's functions with t h e function j ^ ( r ) , i.e., E(r,)
=
JJj^
GE{vj\v)-r{r)dv^GEir), (8.128)
H(r,)
=
/ / /
G^(r, |r)-r(r)dt; = G H ( n ,
where G^; a n d G// are t h e electric and magnetic Green's operators, and r^ is t h e position of t h e observation point. 8.4-2
Lorentz
lemma and reciprocity
relations
T h e Green's electromagnetic tensors for an isotropic m e d i u m satisfy t h e reciprocity relations which reflect the symmetry of t h e electromagnetic field in an inhomogeneous medium. T h e traditional way of deriving these relations is based on t h e Lorentz lemma (Zhdanov, 1988; Felsen and Marcuvitz, 1994), which is formulated for t h e case of electromagnetic fields excited by electric (j^) and magnetic (j^) sources in an inhomogeneous isotropic medium. We assume t h a t two sets of sources, {Ja, j ^ } and {j^, j ^ } are located within a domain Q and t h a t b o t h generate electromagnetic fields of the same frequency UJ. We denote by {E",H'^} and { E ^ , H ^ } the fields produced by a-type and 6-type sources, respectively. These fields satisfy the corresponding Maxwell's equations: V X H ^ = 5E^-fJa,
(8.129)
V X E^ = z c j / i H ^ - j ^ ,
(8.130)
V x H ^ = 5E^4-J5,
(8.131)
V X E^ = z c j / i H ^ - j ^ .
(8.132)
Calculating t h e dot products of equation (8.129) with E^, equation (8.132) w i t h H^, equation (8.130) with H^, and equation (8.131) with E", we obtain t h e expression V - { [ E ^ x H " ] - [E" x H ^ ] } - E ^ - j ^ + H ^ - j ^ - E ^ - J a - H ^ - J r -
(8-133)
Integrating (8.133) over a ball OR, containing domain Q (see Figure 8-1) a n d applying t h e Gauss theorem, we find: / /
J JCR
{ [ E ' X H"]
= [[[ [E'^-il J J JOR
-
[E" X H ' ] } • URCIS
+ H " • j - - E" • J : - H " • j ^ ] dv.
Foundations of electromagnetic theory
226
Now let the radius R of the ball OR tend to infinity. The surface integral over a sphere CR vanishes due to radiation conditions (8.85) and (8.86), and we arrive at a mathematical formulation of the Lorentz lemma: [E^-il -f H^ • r
- E ' • J : - H^ • J r ] dv = 0,
(8.134)
where \4o, as usual, denotes the entire infinite space. We now can use the Lorentz lemma to derive the reciprocity relations for the Green's electromagnetic tensor. Let us assume that the electric dipoles with moments a and b are located at points with the radius-vectors r' and r'', j^ = a,5(r-r'), j^ = M ( r - r " ) ,
(8.135)
and it is assumed that there are no magnetic-type sources, fi
i"
0.
(8.136)
(r I r') • a, and E*" (r) = G ^ (r | r") • b .
(8.137)
According to formulae (8.128), we can write E" (r) =
GB
Substituting expressions (8.135) through (8.137) into the Lorentz lemma (8.134), we find fff
{
\GE
(r I r') • a] • b5 (r - r") -
\GE
(r | r") • b] • a5 (r - r ' ) } dv = 0.
Taking into account the basic property of the delta-function, we obtain from this formula b- GE (r" I r') • a = a- GE (r' I r") • b The last formula directly implies that G^(rnrO = e E ( r M O ,
(8.138)
where the large tilde denotes the operation of transposition. Reciprocity relation (8.138) shows that the electric field component excited at r'' in the direction of b by an electric current element a at r' is identical with the electric field component excited at r' in the direction of a by an electric current element b at r". Note that we can derive a similar reciprocity relation, G ^ (r" I r') = G ^ (r' | r " ) ,
(8.139)
Electromagnetic
227
Green's tensors
for the magnetic Green's tensor as well, assuming that the magnetic dipoles with the moments a and b are located at points with the radius-vectors r' and r'' respectively, j - = a < 5 ( r - r ' ) , j ^ = M (r - r " ) , and that there are no electric-type sources, J : = O,
j^ = o.
In the conclusion, we consider the reciprocity relations between the magnetic and electric fields generated by the electric and magnetic currents, respectively. We now assume that the electric dipole with moment a is located at a point with the radius-vectors r', and the magnetic dipole with moment b is located at a point with the radius-vector r'' respectively, j : = a ^ ( r - r ' ) , j ^ = W (r - r " ) ,
(8.140)
while the sources j^*^ and jl are equal to zero: r
= 0, j ^ = 0.
(8.141)
Substituting expressions (8.140) and (8.141) into the Lorentz lemma (8.134), we find
= - [[[
[E' (r) • a(5 (r - r^ -f H" (r) • b(^ (r - r")] dv = 0
Taking into account the basic property of the delta-function, we obtain from this formula H^ (r'O • b = - E ' ( r O • a. (8.142) Reciprocity relation (8.142) shows that the magnetic field component excited at r^' in the direction of b by an electric current element a at r' is equal to the electric field component (multiplied by the minus sign) excited at r' in the direction of a by a magnetic current element b at r". 8-4-3 Green's tensors in the time domain The time domain electromagnetic Green's tensors G^; , G^ are introduced, as in frequency domain, as the fields of an elementary electric source (Felsen and Marcuvitz, 1994). They follow the Maxwell's equations V X G ^ =aGE + sdGE/dt
+ T^ (r - r^ 6 {t - t'), (8.143)
V X G^; =
-^dGn/dt,
228
F o u n d a t i o n s of e l e c t r o m a g n e t i c t h e o r y
where G// = G// (r,t | r',t') and G^; = G^; (r,t | r'^f) are the magnetic and electric Green's tensors in the time domain^ They are causal: G^; (r, t I r\t') ~ 0, GH (r, t \ T',t') = 0,t<
t'.
Equation (8.143) suggests that in a homogeneous domain G^; also satisfies the equation V X V X G e + fxe^i^+^ia^
=-fxIS
{r - T')6'{t
- t'),
(8.144)
where the prime superscript over the delta function denotes its time derivative. The EM Green's tensors exhibit symmetry and can be shown, using the Lorentz lemma, to satisfy the following reciprocity relations (Felsen and Marcuvitz, 1994): G^(r,t|r',O = G^(r',-t'|r,-0, (8.145) G^(r,^|r',O = G H ( r ' , - t ' | r , - 0 , where a tilde denotes the operation of transposition. The last conditions show that by replacing source and receiver (i.e. the points r' and r) and by going simultaneously to the reverse time —t, (therefore, by retaining the causality, because the condition t < t' in ordinary time implies the condition —t > —t' in reverse time), we obtain the equivalent electromagnetic field, described by the Green's tensors G^; {r',t' \ Y,t) and G// [r',t' \ r,t). Following Morse and Feshbach (1953) and Felsen and Marcuvitz (1994), we can also introduce the adjoint Green's tensors: G+(r,t|r',0-G^(r',t'|r,t), (8.146) G + ( r , t | r ' , 0 = GH(r',t'|r,t). They satisfy the equations, obtained from (8.143) by reversing the sign of all space-time coordinates: V X G + = -abG+
+ dG+ldt -18{Y-
r') 6{t-
t'), (8.147)
V X G+ =
-ndG+fdt,
"^Note that we use the same symbols for Green's tensors in time and frequency domains to simplify the notations. One can easily recognize the corresponding tensor by checking for arguments t or uj in the corresponding equations.
Electromagnetic
229
Green's tensors
and equation (8.144) takes the form: V X V X G + + M £ ^ ^ - f^<^b-Qf
= M K (r - r') S' {t - t').
(8.148)
The adjoint Green's tensors are anticausal: G+ (r, t I r^t') = 0, G+ (r, t \ r'.t') = 0, t> t'. Electromagnetic Green's tensors represent an important tool in the solution of the forward and inverse electromagnetic problems and in migration imaging. We will illustrate Green's tensor applications in the next Chapter.
References to Chapter 8 Berdichevsky, M. N., and M. S. Zhdanov, 1988, Advanced theory of deep geomagnetic sounding: Elsevier, Amsterdam, London, New York, Tokyo, 408 pp. Kong, J. A., 2000, Electromagnetic wave theory: EMW Publishing, Cambridge, Massachusetts, USA, 1007 pp. Felsen, L., and N. Marcuvitz, 1994, Radiation and scattering of waves, IEEE Press, New York, 888 pp. Harrington, R. F., 1961, Time harmonic electromagnetic fields: McGraw-Hill Book Co., New York, 480 pp. Morse, P. M., and H. Feshbach, 1953, Methods of theoretical physics, McGraw Hill Book Co., New York, 1978 pp. Stratton, J. A., 1941, Electromagnetic theory: McGraw-Hill Book Co., NewYork and London, 615 pp. Sommerfeld, A., 1912, Die Greenschc Function der Schwingungsgleichung. Jahresher: D. D. Math. Vereinigung, 21, 309 pp. Zhdanov, M. S., 1988, Integral transforms in geophysics: Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo, 367 pp. Zhdanov, M. S., and G. Keller, 1994, The geoelectrical methods in geophysical exploration: Elsevier, Amsterdam, London, New York, Tokyo, 873 pp.
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Chapter 9 INTEGRAL REPRESENTATIONS IN ELECTROMAGNETIC FORWARD MODELING There are several techniques available for electromagnetic (EM) forward modeling. They are based on numerical implementation of the differential equation (DE) approach (finite diff"erence, FD, or finite element, FE, methods) or the integral equation (IE) approach. In this chapter I will describe a technique of forward electromagnetic modeling based on diff"erent types of integral representation of electromagnetic fields in inhomogeneous conductive media. The IE method is a powerful tool for forward EM modeling. The basic principles of constructing integral equations were outlined by Dmitriev (1969), Wcidelt (1975a) and Hohmann (1975). A comprehensive implementation of the IE method in the 3-D case was realized by Wannamaker (1991), Xiong (1992) and Xiong and Kirsch (1992). The main advantage of this method in comparison with the FD and FE methods is the fast and accurate simulation of the EM response in models with compact 2-D or 3-D bodies in a layered background. At the same time, the main area of application of FD and FE methods is the modeling of EM fields in complex structures with large-scale EM parameter variations. In principle, the IE method can handle these models; however, the demand on computer resources is tremendous, which places practical limits on its use. This happens because of the large dense matrix arising in the IE formulation. In this direction, several approximate methods have been developed recently. These are the extended Born (localized nonlinear) approximation (Habashy et al, 1993), the quasi-linear approximation (Zhdanov and Fang, 1996a;b), quasi-linear series (Zhdanov and Fang, 1997), quasi-analytic approximation and quasi-analytic series (Zhdanov et al., 2000), etc. In this Chapter we discuss the basic principles of the integral equation method and introduce a family of linear and nonlinear approximations for electromagnetic modeling in conductive inhomogeneous media. 9.1 9.1.1
Integral equation method Background (normal) and anomalous parts of the electromagnetic field
Consider a 3-D geoelectrical model with the normal (background) complex conductivity ^5 and local inhomogeneity D with an arbitrarily varying complex conductivity a = 3^5-fAa. The inhomogeneity is considered to be local, so there exists some radius
232
Integral representations in electromagnetic forward modeling
RQ such that for y x ^ + yM-z^ > RQ the medium has the background conductivity at. Within this chapter we will confine ourselves to consideration of nonmagnetic media and, hence, assume that ^ = fiQ = An x 10~'^H/m, where /XQ is the free-space magnetic permeability. The model is excited by an electromagnetic field generated by an arbitrary source with an extraneous current distribution j ^ concentrated within some local domain Q. This field is time harmonic as e~^'^^. Complex conductivity includes the eff"ect of displacement currents: a = a — iue, where a and e are electrical conductivity and dielectric permittivity. Our goal is to derive the integral equations for the electromagnetic field for this model. The solution of this problem is based on introducing normal (background) and anomalous electromagnetic fields. The electromagnetic field in the model described above can be presented as a sum of the background (normal) and anomalous fields: E = E^ + E ^ H = H^ -h H ^
(9.1)
where the background field is a field generated by the given sources in the model with the background (normal) distribution of conductivity a^, and the anomalous field is produced by the anomalous conductivity distribution A5. The total electromagnetic field in this model satisfies the Maxwell's equations:
VxH
=
V XE
=
5E+r, (9.2) za;/ioH,
which can be written separately for the background field E^, H^, V X H^ - SfeE^+j", V X E^ = ^ct;/ioH^ and for the anomalous field E^, H^, V X H^ = 56E^+j^ (9.3) V
X E" = ZCJ/XQH",
where j ^ is the density of extraneous electric currents, and j " (r) = Aa (r) E (r) = Aa (r)
[E"
(r) + E" (r)]
(9.4)
is the density of excess (anomalous) electric currents within the inhomogeneity D. Equations (9.3) follow from the previous equations by subtraction.
Integral equation method
Figure 9-1
233
Derivation of Poynting's theorem and energy inequality for an anonnalous
field. 9.1.2 Poynting^s theorem and energy inequality for an anomalous field We now apply Poynting's theorem to the anomalous field only. We can introduce a domain V2 with the boundary ^2 containing the inhomogeneity D (Figure 91). Applying Poynting's theorem (8.108) to the anomalous field E^, H" distributed within the domain V2, and taking into account Maxwell's equations (9.3) for the anomalous field, we obtain {cr6|E^|VRe(E"-r*)}^i; = ^Re / /
^ a
(E^xH^*)-nsds.
(9.5)
V2
The energy flow F^ of the anomalous field outside the domain V2 with the anomalous domain D, is also always non-negative, F^ > 0,
(9.6)
and it is equal to zero only for a lossless background medium: F^ = 0, if and only if a^ ~ 0.
(9.7)
The proof of this formula is similar to the proof of formula (8.111). Based on formulae (9.6) and (9.7), we can obtain, after some algebraic transformations, the following important energy inequality, derived originally by Singer (1995) and Pankratov et al. (1995):
J J JV2
234
Integral representations in electromagnetic forward modeling
CTb E ^ - f •
(9.8)
4^6 J
2at
From the last formula we have
J J Jv
E«
^a |22
2a,
ppp
dt
|.a|2
JJJv2 4^6
(9.9)
Energy inequality (9.9) holds for any lossy medium. 9.1.3 Integral equation method in two dimensions Let us consider a two-dimensional model in which there exists a localized cylindrical inhomogeneity in the region F with complex conductivity a(x, z) = a^ -{- Aa(x, z) embedded in a background (normal) horizontally stratified section characterized by a conductivity sequence, ab{ai, a2, ... , ?n) (Figure 9-2). The structure is illuminated with an ^-polarized field, which originates from linear currents distributed with a density j ^ = j^dy (where d^^ is a unit vector directed along the y-ax.is) flowing in a region Q in the upper half-space. This model is typical for geoelectrical methods. The upper half-space is usually associated with nonconductive air, while the lower half-space represents the structure of the conductive earth. For such geoelectrical models, the electric field has but a single component, Ey, which can be represented as the sum of a background (normal) part and an anomalous part: Ey = El + E^,
(9.10)
where the background field is generated by the currents j ^ in the horizontally stratified model without inhomogeneity, while the anomalous field is caused by the conductivity inhomogeneity. It should be clear that the total field, Ey^ satisfies the equation V^Eyir) + iujfi^iab + Aa{T)]Ey{r) = -iuj/^^fy,
(9.11)
where V^ is a two-dimensional Laplace operator, d'^/dx'^ -h d^/dy^. The background (normal) field is described by the equation V^E'yir) + iujfi^abE'yiv) = -iujfiojy^
(9.12)
Subtracting the last equation from the one before it, we find an equation for the anomalous field: V^E^iv) + iujfio^bE^ir) = -iujfioJlir),
(9.13)
where jy is viewed as the excess current flowing in the inhomogeneous region, F : j^{v) = Aa{r)Ey{v).
(9.14)
Integral equation method
235 CR
ao = 0
\
ai di (52 di
\
\ ^x
/
1
Air
Earth
\
oj *
jK
/^
T
/ ^ \L
/ ( rv^Gd (x,z) =^^ ] ah + Aa ix,z)/
yfGN-l dN-1 GN
^"-^
yz
Figure 9-2 The two-dimensional model of a localized cylindrical inhomogeneity T with the conductivity 5(.T, z) = ai) + /S.a{x,z) embedded in a background (normal) horizontally stratified section characterized by a conductivity sequence, 56(5i. 52. ••• . ^n)- The structure is illuminated with an E'-polarized field, which originates from linear currents distributed with a density j ^ = j^dy flowing in a region Q in the upper half-space.
An important role is played in the integral equation method by the Green's function for the background (normal) model of the earth. The Green's function, Gh{Y'\Y), depends on the positions of two points, M(r) and M'{Y'), and is determined by a solution of the equation: V'^Gb{v'\Y) -\- iu^^GbGb{Y'\Y) = -iuJiiQ8{Y' - r),
(9.15)
which tends to zero at infinity. In this last equation, 8 is the two dimensional Dirac delta-function which is defined as follows:
jjj{v)8{v'-v)dS=\^
fir');
0
r' ^ D.
(9.16)
where / ( r ) is an arbitrarily continuous function in some domain D, and D = D S is domain D with its boundary S. Using the Green's function G^, we find an expression for the background component of the electric field, Ey. To do so, we select an arbitrary point r' and around it draw a circle, CR, with a radius great enough that the region Q falls within the region OR bounded by the circle CR; that is, Q e OR (Figure 9-2). We now write the expression for Green's theorem within the region OR :
236
Integral representations in electromagnetic forward modeling
IJiE'yV^G, - G,V^El]ds =• £ j ^ ^ ^ - G>,^]dl
(9.17)
OR
where Ey = Ey{r), Gt = G6(r'|r), and d/dn denotes a directional derivative in the direction of the external unit vector n normal to the contour C^. Substituting V'^Ey and V^G^ from equations (9.12) and (9.15) into this last equation, and taking into account (9.16), we find f r iLUfi, J J Cfyds
f - £
f)r^ dE^ [El-^ ~ Gt-g^]dl
= iujii^Eliv') if r' G OR.
(9.18)
Q
According to radiation conditions (Stratton, 1941), the functions Ey and Gb decrease as l / \ / r as |r| —> oo. Consequently, if the radius, i?, is expanded without limit, the line integral along CR will tend to zero. Hence, we find (9.19)
El{v') = jJGb{v'\v)fy{v)ds. Q
This result has a clear physical meaning: the background electric field is merely the superposition of elementary electric field contributions excited in the background geoelectrical sequence, 5^, by the current filaments within the region Q. Using a similar approach, and starting with equation (9.13), it can be shown that the anomalous field is the superposition of elementary field contributions excited in the normal geoelectrical sequence by the excess currents, j ^ , flowing within the inhomogeneous region, F : E;{V')
= j j G,{v'\v)jl{v)ds. r
(9.20)
Substituting the expression in equation (9.14) into this last equation, and considering equation (9.10), we find K(^') + / / Gb{r'\v)^a{v)Ey{v)dS r
= Ey{v'),
(9.21)
For r' E F, this relationship leads to a Fredholm integral equation of the second type for the field Ey within the region F. The Green's function for the background sequence, G^, which is present in equation (9.21) can be calculated from a simple recursive formula. Details about this procedure can be found in a monograph by Berdichevsky and Zhdanov (1984).
237
Integral equation method
The derivative of the Green's function, which is needed in computing the magnetic field components, is determined similarly. In accord with equation (9.21) and Maxwell's second equation, we can write F.(rO ^ H'^ir') - - ^ [[ luj^Q J J r
dz'
^^^Eyds,
r 9.1.4
Calculation of the first variation (Frechet derivative) of the electromagnetic field for 2-D models
The problem of the Frechet derivative, or sensitivity matrix calculation for the electromagnetic field, is very important for electromagnetic inversion. It was examined in many publications (Weidelt, 1975b; Rodi, 1976; Madden and Mackie, 1989; Oldenburg, 1990; McGillivray and Oldenburg, 1990; Zhdanov and Keller, 1994; de Lugao and Wannamaker, 1996; and de Lugao et al., 1997). We will consider, first, the technique for calculating the first variation of electromagnetic field for 2-D models. Let us perturb the conductivity distribution, a{x,z), within the region F. The equation for the corresponding perturbed electric field, 6Ey, can be obtained by perturbing equation (9.11):
We now introduce the Green's function Ga of the geoelectrical model with conductivity a = 5 (x, z) . The Green's function depends on the position of the points (x, z) and {x\z') and is determined by an equation similar to (9.15): V ' G , (r^lr) + luj^^aG, ( r » - -lu^^S
{v' - r ) .
(9.24)
We now can apply Green's formula (9.17) to the perturbed electric field, 8Ey^ and Green's function G^. Repeating the derivations conducted in the previous subsection, we arrive at the formula analogous to (9.20): 8Ey (rO = 11 G, ( r » 8a (r) Ey (r) ds.
(9.25)
The first variation of the magnetic field 6Hx can be calculated from SEy using Maxwell's equations:
238
Integral representations in electromagnetic forward modeling
Formulae (9.25) and (9.26) for the first variations of the electromagnetic field provide an important tool for computing the Prechet derivative of the forward electromagnetic operator. Actually, according to the definition (see Appendix D, equation (D.21)), the Frechet diff'erential Fx{6a) of the electromagnetic forward modeling operator Af^^(a) is given by:
f,„(*.) = MS„,..*., = { Jl; '- . t ^
^
(.27)
Substituting formulae (9.25) and (9.26) into (9.27), we find the expressions for the corresponding Frechet diff'erentials: FE{6a) = 6Ey (rO -
FH{6^)
/ / G^ ( r » 6^ (r) Ey (r) ds,
= 6Hx (r') = — ^
/ / ^^^^^6a
(r) Ey (r) ds.
(9.28) (9.29)
Formulae (9.28) and (9.29) can be used to estimate the sensitivity of electromagnetic data to geoelectrical model parameters. For example, substituting 6a (r) = 6 {Y"—Y) 6aS {V") in equation (9.28), we find a perturbation of electric field, 6Ey (r'), corresponding to the local perturbation of the complex conductivity 6as {r") at a point Y" : 6Ey {Y') = G, (YY) Ey (r'O 6as (r''). (9.30) The diff'erential sensitivity s {Y'\Y") of the electric field in the point r' to the electrical parameter at the point r" is calculated by the formula
Substituting (9.30) into (9.31), we have SE{Y'\v') = GAr'W')Ey{Y').
(9.32)
The integrated sensitivity, S (r''), of the data, collected over some observation line b over a frequency interval 0, in accordance with the definition (3.75), is equal to ^ - ( ^ " ' = ^ ' where the L2 norm
||...||Q5
is determined by the formula
\\6Ey\\^, =
Jj^Jj6Ey{r',cv)\'dl'du,,
(^-'^
239
Integral equation method
where "prime" over dl denotes that the integration is conducted over the variable r'. Therefore, the integrated sensitivity of the electric field to the local perturbation of the conductivity in point r'' is equal to SE (r'O = \ J \Ey (r^cj)l' | | G , (r'|r'Ol'^^'^^-
(9.34)
In a similar way, we can find the differential and integrated sensitivities for magnetic field:
1 dGajr'y)>Ey (r"), SH (r'|r") = --±-^^^1^' iujfiQ
and Mo \ Jn
(9.35)
dz'
dG„ (r'|r") dz' Jh\
^^
dl'dio.
(9.36)
We shall use these formulae (9.34) and (9.36) in electromagnetic inversion. 9.1.5 Integral equation method in three dimensions We can apply an approach, similar to the one used in the 2-D case, to derive the electromagnetic integral equations in three dimensions. Electromagnetic Green's tensors, introduced in the previous chapter, make it possible to determine the electromagnetic field of an arbitrary current distribution j (r) within a medium with background conductivity (75 :
(9.37) H(r,)
=
| | | ^ G , , ( r , |r).j(r)dt; = G„(j),
where G^: and G// are the electric and magnetic Green's operators. The proof of these formulae is similar to the proof of formula (9.19) and is based on an application of Green's theorem to the electric or magnetic field and the corresponding Green's tensors (Zhdanov, 1988). Maxwell's equations (9.2) can be rewritten for the background medium with excess current j ^ = A a E : VxH
=:
5,E+r+f,
V XE
== iuiiQH.
(9.38) We can apply the integral representations (9.37) to these equations:
E(r,) = GEin
+ Gsiy:), (9.39)
H(r,.) = GniD
+ Gn
in,
240
Integral representations in electromagnetic fonvard modeling
where the first terms describe the anomalous fields generated by the excess currents E"(r,)
=
G B (j") = G B ( A 5 E ) ,
(9.40)
H"(r^)
=
G ^ ( r ) = GH(AaE),
(9.41)
and the second terms correspond to the background field of the extraneous current in the background media, E*(r,) = G ^ ( f ) , (9.42) H''(r^) = G ^ ( n .
(9.43)
Substituting the expression for the excess current (9.4) into (9.39), we finally obtain the well-known representation for electromagnetic field as an integral over the excess currents in the inhomogeneous domain D (Raiche, 1974; Hohmann, 1975; Weidelt, 1975a): E (r,) = G^ (AaE) + E^ ( r , ) , (9.44) H (r,) = GH (A5E) + H^ ( r , ) .
(9.45)
Using integral formulae (9.44) and (9.45), one can calculate the electromagnetic field in any point r^, if the electric field is known within the inhomogeneity. Expression (9.44) becomes the integral equation for the electric field E ( r ) , if TJ e D. Equations (9.44) and (9.45) form the basis for EM forward modeling. 9.1.6 Calculation of the first variation (Frechet derivative) of the electromagnetic field for 3-D models We can find the equations for the Frechet derivative in the 3-D case by differentiating the corresponding Maxwell's equations. Let us write the general field equations (8.56) and (8.57) in the frequency domain, allowing for magnetic currents: V x H = ?E4-f,
(9.46)
V x E = za;/ioH-j^,
(9.47)
where j ^ and j ^ are the densities of extraneous electric and magnetic currents. Let us perturb the conductivity distribution a ( r ) . Applying the perturbation operator to both sides of equations (9.46) and (9.47), we obtain the equations for corresponding variations of the electromagnetic field: Vx6H
= a6E-{-6aE, (9.48)
V x ^ E = zcj/ig^H, where 6a is the conductivity variation, and 5H, 6E are the corresponding variations of the magnetic and electric fields.
241
Integral equation method
Applying the integral equation method (formulae (9.40) and (9.41)) to the solution of equations (9.48), we find that the variations of the electromagnetic field are associated with the variation of the conductivity structure by the following integral operators: 6E{rj)
= GE{SaE),
(9.49)
m(r,)
=
(9.50)
GniSaE).
The last two equations provide a straightforward way to calculate the Frechet differential (for definition see Appendix D) of the electromagnetic operator for forward modeling of complex three-dimensional electrical structures: - ^E = / / / G^; (r' | r) • (5? (r) E (r) dv,
(9.51)
FH(5,(5a) = ^Ag^(5,^?) = (^H = / / / GH (r' | r) • (55 (r) E (r) dv.
(9.52)
FE{a,6a) = 6A^^{a,8a) and
Note that the arguments in the expressions for the Frechet diff"erentials, F2^;//(a,(5a), consist of two parts. The first part, 5, is a conductivity distribution, at which we calculate the forward modeling operator variation, 6A^j^] the Green's tensors are calculated for this conductivity. The second part, 8a, is the corresponding variation of the conductivity a. Vector E in expressions (9.51) and (9.52) represents the electric field for the given conductivity a. We can estimate the sensitivity of electromagnetic data to geoelectrical model parameters by means of formulae (9.51) and (9.52). Substituting 6a (r) = 8 (r''—r) 8'ay {r") in equations (9.51) and (9.52), we find the perturbations of the electric and magnetic fields, 8E {r'), 8YL [r'), corresponding to the local perturbation of the complex integrated conductivity 8a^ {Y") at a point Y":
8E {Y') = G ^ (r' I Y") • 8a, [Y") E {Y") , m (r^ = GH [Y' \ Y") • 8a, (Y") E (Y") . (9.53) The diff"erential sensitivities s^; (r'|r'') and s^/ (r'|r'') of the electric and magnetic fields at the point Y' to the integrated conductivity perturbation at the infinitesimally small neighborhood of the point Y^' are calculated by the formulae
Substituting expressions (9.53) into (9.54), we finally obtain: s s (r'|r") = GE (r' | r") • E (r"), s^ (r'|r") =. G ^ (r' | r") • E (r").
(9.55)
242
Integral representations in electromagnetic forward modeling
Note that the differential sensitivities are vector functions, because they characterize the sensitivity of the vector electric and magnetic fields to the conductivity variation. From the last formulae we see that Green's electromagnetic tensors provide the sensitivity estimation of the electromagnetic field to the model conductivity. We can give a simple, but important physical interpretation of the expressions for sensitivities, (9.55), based on the reciprocity principle. Note that, according to definition (see Chapter 8), the Green's tensors Gj^ (r' | r") and G// (r' | r " ) , are the electric and magnetic fields at the receiver point, r', due to a unit electric dipole source at the point Y" of the conductivity perturbation. Let us introduce a Cartesian system of coordinates {x, y, z}, and rewrite these tensors in matrix form: G ^ (r' I r") = [Gf^ (r' | r")] , G ^ (r' | r") = [G^, (r' | r")] , a, /3 = x, y, z. The tensor component Gf^ is essentially an a-directed electric field at the receiver point, r', due to a /^-directed unit electric dipole source at the point r". The tensor component G^^ is an a-directed magnetic field at the receiver point, r', due to a /3-directed unit electric dipole source at the point r". According to the reciprocity principle these tensors can be treated also as an electric field at T" due to electric or magnetic current dipole sources at r'. To be more specific, the tensor component Gf^ can be treated as a /3-directed electric field at the point r", due to an a-directed unit electric current dipole source at the receiver point, r'. Correspondingly, the tensor component G^^ is a /3-directed electric field (with a negative sign according to formula (8.142)) at the point r'', due to an a-directed unit magnetic current dipole source at the receiver point, r'. We introduce the following notations: Gf^ (r' I r") = W^ (r") • d^, G^^ (r' 1 r") = -E™ (r") • d^,
(9.56)
where E^ (r'') and E ^ (r'') are the respective electric fields due to the a-directed unit electric or magnetic current dipole sources at the receiver point, r'; d^, /? = x,y,z, are the orthonormal vectors of the Cartesian basis. The expressions for a-directed components of the sensitivities, (9.55), can be rewritten in the form ^
""-^^'^^"^"^W^)^
G f , (r'I r") £;, (r") =
(3=x,y,z
= EI{T")-
Y^ I3=x,y,z
and, in a similar way,
where a =
x,y^z.
Y.
(E^ ( O " d , ) £;, (r")
j3=x,y,z
{d,E0{T")) = El{r")-E{r"),
(9.57)
243
Integral equation method
The final result is that the sensitivities can be obtained from the scalar product of the original electric field and the auxiliary electric fields, generated by elementary electric or magnetic dipole current sources, located at the receiver position, r'. This result was originally demonstrated by Pellerin et al., 1993, and then a derivation based on Lorentz lemma was given by McGillivray et al., 1994. In accordance with definition (3.75), the integrated sensitivity SE (r'O ^f ^^^ data, collected over some surface D of observations over a frequency interval ^2, is equal to
where the L2 norm ||.-.||^5] is determined by the formula
\m\n,i, =
Jmj^'^i^'^^)\'ds'du.
Therefore, the integrated sensitivity of the electric field to the local perturbation of the conductivity at the point r'^ is 2
ds^duj.
(9.59)
In a similar way we can find the integrated sensitivities of the magnetic field: SH (r") = Jj
j j
\GH (r' | r") • E {r")\'ds'dw.
(9.60)
Formulae (9.59) and (9.60) play an important role in electromagnetic inversion. 9.1.7 Frechet derivative calculation using the differential method We now present another way of calculating the Frechet derivative using the differential method, proposed by McGillivray et al., 1994. Let us introduce auxiliary electric and magnetic fields, E' and H', satisfying the equations V X H' =. 5 E ' + r , (9.61) V X E' = iufiW - J ^ ,
(9.62)
where the auxiliary electric and magnetic currents J^ and J"^ will be defined below. Multiplying the first equation (9.48) by E' and equation (9.62) by 5H, subtracting one from another, and taking into account the vector identity E' • V x ^ H - ^ H . V X E' = V • [(5H X E ' ] ,
244
Integral representations in electromagnetic forward modeling
we obtain V • [(5H X E'] = 5 E ' • 5E+55E' • E-iu/iSH
• H' + 5H • J ^ .
(9.63)
In a similar way, multiplying the second equation (9.48) by H' and equation (9.61) by (5E and subtracting one from another, we find H'] = iuJi^Sn ' H ' - 5 E ' - 6E - 6E - J ^
V-[6Ex
(9.64)
Adding the last two equations, we arrive at the formula V • [(5H X E' + ^E X H'] - 6aE' • E -f 5H • J ^ - 5E • J ^
(9.65)
We introduce now a domain VR, bounded by a sphere OVR of a radius i?, with its center at the origin of some Cartesian coordinate system, {x, y, z } . Integrating both sides of equation (9.65) over the domain VR, and applying Gauss's theorem, we find: [Sn xE' + SEx H ] • nds
II
IdVR
J Jdv
[55E' • E + 5H • J ^ - ^E . J"] dv, (9.66) J J Jvn fVR where n is a unit vector of the outer normal to dVn. Note that both the electromagnetic field perturbations, ^E, (5H, and the auxihary fields, E', H', are caused by the local sources. Therefore, according to the radiation condition (Harrington, 1961), the surface integral goes to zero if the radius R tends to infinity, i? —> oo :
III
[6aE' •E + 6Ji'3'^-6E-r]dv-^
0.
(9.67)
VR-^OO
Formula (9.67) provides a simple way of computing the sensitivity of the electromagnetic field in the receiver location, r', to a perturbation of the complex conductivity Say within a differential element of the volume dv located at a point r''. In this case, we represent 6a (r) by the formula (55(r) = ( 5 ( r " - r ) ( 5 5 4 0 ,
(9.68)
where r denotes the variable of integration in formula (9.67). We introduce the auxiliary field E' = E^, H' = H^, generated by an elementary electric current only, located at the receiver position r', and polarized along the respective axis x,y, or z: a = X,y,z- J ^ = 0,
Jl = dj{v-v'),
(9.69)
where da, a = x,y, z] are the orthonormal vectors of the Cartesian basis. Substituting formula (9.69) into (9.67), and taking into account (9.68), we obtain SEa (r'|r") = ^ ^ ^ day
= El (r") • E (r").
(9.70)
245
Family of linear and nonlinear integral approximations of the electromagnetic field
In a similar way, assuming that the auxihary field E' = E ^ , H ' = H ^ , is generated by an elementary magnetic current only, located at the receiver position, and polarized along the respective axis x, 7/, or z, : J ^ = dj
(r - r ' ) , a = x, y, z; r = 0,
(9.71)
we find the following expression for the magnetic field sensitivity SHa (r'|r") = ^ % ^
= -E™ (r") • E (r").
(9.72)
Note that formulae (9.70) and (9.72) are identical to formulae (9.57) and (9.58). 9.2
Family of linear and nonlinear integral approximations of the electromagnetic field
The main difference between the integral representations in potential and electromagnetic field theory is that in the first case the solution of the forward modeling problem is given in a form of direct quadrature, while in the second case we have to solve a system of integral equations. There is, however, an important approach to the fast solution of the EM integral equations, based on neglecting the anomalous (scattered) EM fields under the Green's integral operators. This approach is usually called the Born approximMion. The Born approximation was developed originally to describe quantum mechanical scattering (Born, 1933; Born and Wolf, 1980). Since the basic idea behind this method has broad applications, it is possible to apply the Born approximation to different geophysical problems as well. For example, it has been used quite extensively and successfully in seismic geophysics (Bleistein, 1984; Bleistein and Gray, 1985; Tarantola, 1987). Let us formulate a general forward EM problem so that the anomalous conductivity can be treated as a perturbation from a known background (or "normal") conductivity distribution. The solution of the EM problem in this case contains two parts: 1) the linear part, which can be interpreted as a direct scattering of the source field by the inhomogeneity without taking into account coupling between scattering (excess) currents, and 2) the nonlinear part, which is composed of the combined effects of the anomalous conductivity and the unknown scattered field in the inhomogeneous structure. The Born approximation is based on the assumption that this last part, which represents the actual non-linearity of the physical problem, is negligible when compared to the first linear part. As a result, a linear expression is obtained for the solution of the EM induction problem in a form of simple quadrature over the inhomogeneous domain, similar to the forward modeling solution for potential fields. This makes the method especially attractive for geophysical applications. The Born approximation has been widely used in inverse schemes, because it provides a linearized approach to the solution of inverse problems (Oristagho, 1989;
246
Integral representations in electromagnetic forward modeling
Habashy et al., 1993). However, the Born approximation works reasonably well only for small conductivity contrasts, relatively small inhomogeneities, and low frequencies Lj (Born and Wolf, 1980; Habashy et al., 1993). It has limited application for solving general 3-D electromagnetic problems, because the Born approximation breaks down when the "anomalous induction number" of the anomalous region, that is u/iAaL'^ (where L is the upper bound of the distance between any two points belonging to the region D with anomalous conductivity Acr, and /i is the magnetic permeability) gets too large. It also has problems for high contrast bodies even in the case of the stationary field or direct current (DC) approximation. Habashy et al (1993) and Torres-Verdin and Habashy (1994) introduced an extended Born approximation for the EM field, based on the assumption that the internal electrical field has a small spatial gradient, which can be neglected inside an inhomogeneous structure. This new approximation for the internal field is given by the projection of the background or normal electric field (i.e. the electric field excited in the absence of the conductivity inhomogeneity) onto a scattering tensor. It is shown that the scattering tensor does not depend on the illuminating sources and is a nonlinear functional of the anomalous conductivity distribution. In addition, they demonstrated the efficiency of the new approximation which remains accurate within a wide frequency band for large anomalous structures and large conductivity contrasts. In a series of publications, Zhdanov and Fang (1996a, 1996b, 1997) and Zhdanov et al. (2000) presented an approach to the extension of the Born method, based on so-called quasi-linear and quasi-analytical approximations. In the following sections I will present an overview of a family of linear and nonlinear integral approximations of the electromagnetic field in inhomogeneous media, which can be used both for forward modeling and inversion. 9.2.1 Born and extended Born approximations We consider again the basic integral equation of 3-D EM forward modeling (9.44), written for the anomalous electric field: E" (r^) = IIJ^
GE {rj | r) • {A5 (r) [E" (r) + E» (r)] } dv.
(9.73)
The conventional Born approximation E ^ (r^) for the anomalous field can be obtained from (9.73) if the anomalous field is negligibly small inside D in comparison with the background field: E^ (vj) = 111^ GE (r, | r) • [A? (r) E " (r)] dv.
(9.74)
The approximation (9.74) works reasonably well only for small conductivity contrasts between background media, a relatively small inhomogeneity and low frequencies (Berdichevsky and Zhdanov, 1984).
Family of linear and nonlinear integral approximations
of the electromagnetic
field
247
Habashy et al., (1993) and Torres-Verdin and Habashy (1994) developed the extended Born approximation, which replaces the total field in the integral (9.73) not by the background field, like in the Born approximation, but by its projection onto a scattering tensor T (r) : E (r) = E ' (r) + E" (r) = T (r) •
E'
(r)
(9.75)
The expression for the scattering tensor is obtained from equation (9.73) by approximating E (r) in the integral by its value at the peak point r = Vj of the Green's tensor: E"(r,) = E ( r , ) - E ' ' ( r , )
IIL IIIM GE
or E(r,
I
(vj I r) A a (r) dv
Tj I r)
ACT (r)
E(r,).
dv
•E^r,),
(9.76)
where I is the identity tensor. Therefore, we have an analytical expression for the scattering tensor,
r(r,
I
QE
(r, I r) Aa (r) dv
(9.77)
which is independent of the illuminating sources. Note that approximation (9.76) is often referred to as a localized nonlinear (LN) approximation. 9.2.2 Quasi-linear approximation and tensor quasi-linear equation The quasi-linear (QL) approximation is based on the assumption that the anomalous field E"^ inside the inhomogeneous domain is linearly proportional to the background field E'' through some tensor A (Zhdanov and Fang, 1996a): E" (r) ^X{r)-
E^ ( r ) .
(9.78)
Substituting formula (9.78) into (9.73), we obtain the QL approximation Eg^ (r) for the anomalous field: E^QL(r,)=-G^[A5(l + A(r)) E'
(9.79)
The last formula gives us the tensor quasi-linear (TQL) equation with respect to the electrical reflectivity tensor A, A (r,) • E" (r,) = G ^ A5A (r) •
E"
+ E^ (r,),
(9.80)
Integral representations in electromagnetic forward modeling
248
where E ^ (r^) is the Born approximation (9.74), and GE AcrA (r) • E' is a linear operator of A (r) : G,
[A5A
(r) • E^] = ffj
GE
(r,- | r) •
[AC
(r) A (r) • E^ (r)]dv.
(9.81)
The original QL approximation, introduced by Zhdanov and Fang (1996a), is based on the numerical solution of a minimization problem arising from the TQL equation (9.80): A (r,) . E" (r,) - G ^ A5A (r) • E^l - E ^ (r,) = m m .
(9.82)
The advantage of this approach is that we can determine the electrical reflectivity tensor A by solving a minimization problem (9.82) on a coarse grid. The accuracy of the QL approximation depends only on the accuracy of this discretization of A , and, in principle, can be made arbitrarily good. The disadvantage, however, is that, similar to the full integral equation (IE) method, the QL approach still requires solution of the corresponding system of linear equations arising from the minimization problem (9.82). We shall analyze in the next sections the diff"erent techniques of the TQL equation solution which result in diff'erent analytical expressions for the electrical reflectivity tensor A (r^). In particular, one of these solutions gives rise to the extended Born, or localized nonlinear (LN) approximation, introduced above. Note that these approximations may be less accurate than the original QL approximation with a fine grid for the discretization of A (r^). 9,2.3 Quasi-analytical solutions for a 3-D electromagnetic field In this section we analyze different approximate solutions of the tensor quasi-linear (TQL) equation (9.80) following the paper by Zhdanov et al. (2000). Solution for a scalar reflectivity coefRcient In the framework of the quasi-linear approach, the electrical reflectivity tensor can be selected to be a scalar (Zhdanov and Fang, 1996a): A=A. In this case, integral equation (9.80) can be cast in the form A (r,) E^ (r,) = G ^ [AaXE'] + E ^ ( r , ) .
(9.83)
Following ideas of the extended Born approximation outlined above, we use the fact that the Green's tensor G^; (r^ | r) exhibits either singularity or a peak at the point where r^ = r. Therefore, one can expect that the dominant contribution to the integral GJE; [ A 5 A E ^ ] in equation (9.83) is from some vicinity of the point r^ = r. Assuming also that A (r) is slowly varying within domain J9, one can write A (r,) E" (r,.) « A (r,) G ^
[ A S E " ] + E«
(r,)
Family of linear and nonlinear integral approximations of the electromagnetic field
= A(r,)E^(r,)+E^(r,).
249
(9.84)
Taking into account that we are looking for a scalar reflectivity tensor, it is useful to introduce a scalar equation based on the vector equation (9.84). We can obtain a scalar equation by taking the scalar product of both sides of equation (9.84) with the complex conjugate background electric field: A(r,)E''(r,)-E''*(r,) = A (r,) E ^ (r,) • E"* (r,) + E ^ (r,) • E"* ( r , ) .
(9.85)
Dividing equation (9.85) by the square of the background field and assuming that E''(r,).E''*(r,)^0,
(9.86)
we obtain A(r,) = where
(9.87)
l-5(r,)'
E^(r,).E''*(r,) 5(ri) = '^ E''(r,)-E^*(r,)
(9.^
Note that in the points where the background field vanishes we can select A equal to (-1) A(r,) = - 1 , if E ' ' ( r , ) - E ^ * ( r , ) = 0 . Substituting (9.87) into (9.1), we find E (r) = E" (r) + E*" (r) a [A (r) + 1]
E''
(r)
E^r)
(9.89)
E ^ r ) dv,
(9.90)
E^(r) dv.
(9.91)
l-ff(r)
Therefore from (9.44) and (9.45) we finally determine
E^^(r,)=E(r,)-E''(r
'' = ///„GE(J,
A?(r)
1-5(0
and
H ^ ^ ( r , ) = H ( r , ) - H^(r,) = | | | ^ G ^ ( r ,
A5(r)
i-^W
Formulae (9.90) and (9.91) give quasi-analytical (QA) solutions for 3-D electromagnetic fields. Note that the only difference between the QA approximation and the Born approximation (9.74) is in the presence of the scalar function [1 — ^ (^^)]~^ • That is why the computational expense of generating the QA approximation and the Born approximation is practically the same. On the other hand, it was demonstrated by Zhdanov et al. (2000), that the accuracy of QA approximation is much higher than the accuracy of the Born approximation.
Integral representations in electromagnetic forward modeling
250
Solution for a reflectivity tensor The QA solutions developed in the previous section were based on the assumption that the electrical reflectivity tensor was a scalar. This assumption reduces the areas of practical appUcations of the QA approximations because in this case the anomalous (scattered) field is polarized in a direction parallel to the background field within the inhomogeneity However, in general cases, the anomalous field may be polarized in a direction different from that of the background field, which could generate additional errors in the scalar QA approximation. To overcome this difficulty we introduce in this section a tensor quasi-analytical (TQA) approximation. The TQA approximation uses a tensor A, which permits different polarizations for the background and anomalous (scattered) fields. In the general case of a tensor A (r^), we can assume that the product A (r) • E^ (r) is a smoothly varying function of the coordinates, and it can be taken outside the integral over the anomalous domain D. As a result, we obtain from the TQL equation (9.80) A(r,.).EVr,-)«G,
A5I
A(r,).E'>(r,.)
+E^(r,)
= g(r^)-[A(r,.).E''(r,)]+E^(r,), or
i-g(r,)].[A(r,).E'>(r,)]=E^(r,)
(9.92)
g(r,.) = GB[A5(r)T'
(9.93)
where Note that in the original definition of the Green's operators (8.128), the argument of Gfi was a vector. In expression (9.93) we extend this definition to include a tensor argument T: QE{T)=JJJ
GE{vj\v)-T:{v)dv,
(9.94)
where G^; ( T 1 now becomes a tensor of the second rank. This means that the Green's operator applied to a vector generates a vector field, while the Green's operator applied to a tensor, generates a tensor field. We shall use this extended definition throughout the book. Solving equation (9.92) gives A(r,).E''(r,)=fl-g(r,)l"'.E^(r,). Substituting (9.95) into (9.78) and using (9.1), we obtain E (r) = E" (r) + E*" (r) ^ \\ (r) + 1 ] • E* (r)
(9.95)
Family of linear and nonlinear integral approximations
of the electromagnetic
field
= [l-g(r)J~ .E^(r)+E^(r).
251
(9.96)
Therefore, from (9.44) and (9.45) we find
G B (r, I r) • A5 (r) <^ I - g ( r )
-1
• E« (r) + E" (r) U t ; ,
(9.97)
and H;TQA ( r , ) = H ( r , . ) - H ' ' ( r , ?(r)( I-g(r) G„ (r. I r) • .Ac
• E ^ ( r ) + E''(r >dv,
(9.98)
where, according to (9.93) and (9.94),
IIL
g ( r , ) = G s Aa (r) I
GE {TJ
I r) A ? (r) dv.
(9.99)
We call expressions (9.97) and (9.98) tensor quasi-analytical (TQA) approximations for an electromagnetic field. These approximations provide a more accurate solution for a forward problem than a scalar QA approximation (Zhdanov et al., 2000). However, it involves computing the tensor multiplier I - g ( r )
which is
more time consuming than calculation of the scalar coefficient I — g (r)] in the case of the scalar QA approximation. 9.2.4 Quasi-analytical solutions for 2-D electromagnetic field Assume now that both the electromagnetic field and the complex conductivity a in a geoelectrical model are two-dimensional, i.e., they vary only along the directions X and z in some Cartesian system of coordinates, and they are constant in the y direction. In this case, repeating the derivations described above for the 3-D case, we can obtain the following quasi-analytical expressions for a 2-D electromagnetic field: E^QAyir,) « J J G,(r, | r ) ^ ^ ^ E ^ ( r ) d s ,
(9.100)
and similarly for magnetic field components 'Mo lUJflQ
Ji
tiQAA^j) ^ - ^
[[ D
^" dz
!/ 11 -_ 9^c..^ (0
dx
I-
'
•Ey{r)ds, g [r]
(9.101)
(9.102)
Integral representations in electromagnetic forward modeling
252 where
^W = ^ , i f ^ » v ^ O .
(9.103)
These formulae can serve as a new effective quasi-analytical tool in the solution of both direct and inverse 2-D electromagnetic problems. Numerical tests demonstrate that these approximations produce a very accurate result for 2-D models (Dmitriev et al., 1999). 9.2.5 Localized nonlinear approximation We will demonstrate now that the TQA approximation can be treated as a modification of the extended Born (locahzed nonlinear - LN) approximation, introduced above. Let us rewrite equation (9.96) in the form
E(r) =
:(r)
I-g(r)r
E^(r)+[T-g(r)]"'.{[T-g(r)].E''(r)}
• [E^(r)-g(r).E*(r)] +
I-g(r)
• E^ fr)
(9.104)
Taking into account once again the fact that the Green's tensor G ^ (r^ | r) exhibits either a singularity or a peak at the point where r^ = r, one can calculate the Born approximation G ^ [Aa (r) E* (r)] using the formula E ^ (r,) =
GE
[Aa (r)
E"
(r)] « g (r,) •
E"
(r,).
In the framework of this approach E^(r)-g(r).E^(r);^0.
(9.105)
This approximation is particularly appropriate if the background field is a smoothly varying function of the coordinates; it forms the basis of localized nonlinear approximations (Habashy et al., 1993). Under this assumption E (r) = E" (r) + E*" (r)
I-g(r)
•E'-W,
(9.106)
Therefore, from (9.44) and (9.45) we find
JJI^
GE (r, I r) • I A 5 (r) [l - g ( r ) ] " ' • E* (r) } dv,
and H2;v(r.)=H(r,)-H''(r,)
(9.107)
Family of linear and nonlinear integral approximations of the electromagnetic field
= III
GH{TJ
I r) . | A 5 ( r ) [ T - g ( r ) ] ' . E ^ ( r ) | d ^ .
253
(9.108)
Formulae (9.107) and (9.108) give the localized nonlinear (LN) approximation introduced by Habashy et al. (1993), and the tensor
I-g(r)l"'=f(r) is the scattering tensor introduced above. Thus, we can see that the difference between the tensor quasi-analytical approximation and the localized nonlinear approximation is determined by a term: ^TQA i'^j) ~ GE
^1N
i^j)
(r, I r) . { Aa (r) f (r) • [E« (r) - g (r) •
E"
(r)] }dv.
(9.109)
Note that both TQA and LN approximations use the same scattering tensor r (r) , which is based on the principle that the effect of the electric field in the Green's integral operator is localized. The only difference is that in the case of LN approximation, we use this principle twice for computing both the scattering tensor and the expression for the Born approximation E ^ (r^) on the right-hand side of the tensor quasi-linear equation (9.80), while in the case of TQA approximation we use the exact formula for E^^ (r^) . That is why we can consider TQA as a partially localized approximation. At the same time, this difference results in different behaviors of these two approximations for different electrical models (Zhdanov et al., 2000). 9.2.6 Localized quasi-linear approximation Let us analyze again the equations of the QL approximation E°(r,)«A(r,).E''(r,),
.110)
and E^^(r,) = G s ( A a ( l + A ( r ) ) . E ' ' ( r ) )
(9.111)
Following the same principle that was used in deriving the localized nonlinear approximation, we assume that the dominant contribution to the integral G^; AaA • E^ in equation (9.111) is from some vicinity of the point r^ == r. Assuming also that E^ (r^) is slowly varying within the domain D, one can rewrite equation (9.111) as E"Q^ (r,) « GE ( A 5 ( T + A ( r ) ) ) • E^ ( r , ) ,
(9.112)
where the tensor Green's operator is given by the formula GE ( A 5 ( I + A ( r ) ) ) = fff
GE {VJ \ r) • { A 5 ( r ) [ T + A ( r ) ] } dv.
(9.113)
254
Integral representations in electromagnetic forward modeling
Comparing (9.110) and (9.113), we find that A(r,)E''(r,)«Gs(A5(T+A(r))).E''(r,), where the electrical reflectivity tensor can be determined, similar to the original QL approximation, from the solution of the minimization problem: A (r,) • E^ (r,) - G^
(A5 (I
+ A ( r ) ) ) • E^ (r,)
(9.114)
L2{D)
Noting that A (r,) . E" (r,) - G ^
( A ? (I
+ A (r)) ) • E^ (r,)
A(r,)-GB(Aa?(T+A(r)))
L2{D)
<
L2(D)
|E''(r 3n\L2{D)
'
we can substitute A(r,)-G£;(A5(T-f A(r)))
L2{D)
= mm,
(9.115)
for the minimization problem (9.114). The solution of equation (9.115) gives us a localized electrical reflectivity tensor XL (r), which is obviously source independent. Expression (9.111) with \L (r) determined according to (9.115), is called a localized quasi-linear (LQL) approximation (Zhdanov and Tartaras, 2001): KQL
(r,) « G^; ( A ? ( T + XL (r)) • E " (r)) .
(9.116)
In a similar way we can find a localized quasi-linear approximation for the magnetic field: H V (r,) « GH ( A 5 ( i + Ai (r)) • E " (r)) .
(9.117)
The LQL approximation has a similar background to the localized nonlinear (LN) approximation, but there are some important diff'erences. The LN approximation also replaces the total field inside inhomogeneity with a product of the background field and a scattering tensor T (r), E2^(r)^r(r).E^r).
(9.118)
However, the scattering tensor is defined explicitly through an integral of the anomalous conductivity. On contrast, the reflectivity tensor in the LQL approximation is determined as the solution of the minimization problem (9.115). Note also that the
Family of linear and nonlinear integral approximations of the electromagnetic field
255
reflectivity tensor is assumed to be slowly varying, and therefore the minimization problem (9.115) can be computed on a much coarser grid than the field itself. Another important diff'erence between the LQL approximation and LN approximation is that for the former one can choose difl'erent types of reflectivity tensors. For example, one can introduce a scalar or diagonal reflectivity tensor. In the case of a scalar reflectivity tensor XL = A^I, and E" (r) « Ai (r)
E'
(r).
(9.119)
Substituting formula (9.119) into (9.116), we obtain the scalar LQL approximation E J Q I : (r) for the anomalous field: E V ( r ) = G ^ ( A 5 ( l + A^(r))E'')
(9.120)
where the scalar reflectivity coefficient A^ (r) is determined by the solution of an optimization problem: A ^ ( r ) T - G ^ ( A 5 ( l + A^(r))l)
L2{D)
mm.
).121)
In the special case of a diagonal reflectivity tensor A^ = diagAjr, XL = diagA;
Ax 0 0 0 A,^ 0 ,0 0 A,
where XL is a reflectivity vector Ax Xr =
A7;
we have E"(r)«diagAar)-E''(r)
(9.122)
Substituting formula (9.122) into (9.116) we obtain a LQL approximation ^IQL (J") for the anomalous field: E V (r) = G^ ( A 5 ( I + diagA^ (r)) • E^)
(9.123)
where the reflectivity vector A^ (r) is determined as the solution of an optimization problem diagA^ (r) - G^ ( A 5 ( T + diagA^ ( r ) ) )
L2{D)
(9.124)
256
9.3
Integral representations in electromagnetic forward modeling
Linear and non-linear approximations of higher orders
It is well known that the conventional Born approximation can be applied iteratively, generating N-th order Born approximations. This approximation can be treated as the sum of A'' terms of the Born (or Neumann) series. However, the convergence of the Born series is questionable and depends on the norm of the integral equation (Green's) operator. It seems to be very attractive to construct similar series on the basis of the QL and QA approximations. In this section of the book I present a solution to this problem. It is based on a method of constructing a convergent Born series developed by Singer (1995), Singer and Fainberg (1995), Pankratov et al. (1995), Pankratov et al. (1997), and Zhdanov and Fang (1997). This method transforms the conventional EM Green's integral operator G^; of forward modeling in inhomogeneous media into a modified Green's operator G ^ with norm less than 1: || G^|| < 1. We begin with a short review of the method of constructing an always convergent Born series, following mainly Zhdanov and Fang (1997). In the subsequent sections we introduce QL and QA series based on QL and QA approximations. 9.3.1 Born series Expression (9.73) can be rewritten using operator notations: E« = C[E^] ,
(9.125)
where C [E"^] is an integral operator of the anomalous field E" C [E^] = G^ [AaE^] -f G ^ [A5E«] .
(9.126)
Equation (9.125) can be treated as an integral equation with respect to the anomalous field E". The solution of this integral equation has to be a fixed point of the operator C (See Appendix B). In other words, the application of the operator C to the anomalous field E^ should not change this field. This solution can be obtained using the method of successive iterations which is governed by the equations ^a{N) ^ ^ [E^(iv-i)j ^ iv - 1,2,3...
(9.127)
It is well known that successive iterations converge, if operator C is a contraction operator (see Appendix B, Banach theorem), that is ||C [E"(^) - E^(2)] II < k ||E"^^) - E"(2)|| ,
(9.128)
where ||...|| is an L2 norm, /c < 1, and E"^-^^ and E"^^) are any two different solutions. Substituting (9.126) into (9.128), we obtain: ||C [E"(i) - E^(2)] II = ||G^ [Aa {E<'^ - E^^^))] || < IIG^II ||Aa|| ||E"(i) - E"(2)||
^9 ;^29)
Linear and non-linear approximations of the higher orders
257
Therefore, condition (9.128) holds if IIGBII | | A 5 | |
< 1.
(9.130)
If operator C is a contraction, then the A^-th iteration approaches the actual anomalous field E«W - . E ^ (9.131) when N -^ oo. The Born approximation is simply the first iteration of this method, if the initial approximation E"^^^ (zero order iteration) is selected to be equal to zero ^B ^ j.a(i) ^ c [0] = G^ [AaE^] .
(9.132)
The second iteration is equal to Ea(2) ^ Q [E"(i)j ^ Q ^ [AaE^] 4- G^; [AaE^] = ( G ^ M A ^ ) [E^] 4- ( G ^ M A ^ ) ' [E^] ,
where we use an operator MASF of multiplication by the function Aa. The A^-th iteration is represented as the sum of A^ terms of the Born series: E^W = ( G , , M A ^ ) [E^] 4- ( G Z ^ M A ^ ) ' [E^] + ... 4- ( G ^ M A . ) ^ [E^] .
(9.133)
The Born series would be a powerful tool for EM modeling if they were convergent. However, in practice the condition (9.130) does not hold, because in a general case the L2 norm of the Green's operator is bigger than 1. That is why the Born series has not found a wide application in EM modeling. 9.3.2 Modified Green's operator In a general case the L2 norm of the Green's operator can be arbitrarily big. Following Pankratov et al. (1995) and Zhdanov and Fang (1997), we apply some linear transformations to the Green's operator to obtain a new modified Green's operator G ^ with norm less than 1. The specific form of this linear transformation is motivated by the energy inequality (9.9) for an anomalous field. Actually, we construct a new linear operator G ^ , which transforms the integrand from the right part of the energy inequality (9.9) into its left part: y ^ ( E - + JV2a,) = v ^ G ^ [ 2 v ^ ( j V 2 ^ / ^ ) ] (9.134) Operator G g can be applied to any vector function: GE (X) = V^bGE ( 2 V ^ x ) + X,
(9.135)
258
Integral representations in electromagnetic forward modeling
X G L2 (I^), where I/2 (i^) is the Hilbert space of the vector functions determined in domain D and integrable in D with the norm:
The remarkable property of this operator is that according to inequahty (9.9),
l|G^(x)||<||x|i for any x € L2 {D). In other words the L2 norm of the modified Green's operator is always less than or equal to one: IIGfill < 1-
(9.136)
By virtue of (9.4), equation (9.134) can be simplified: oE" + tE* = G ^ [6 (E" + E*)] , where
_ 2(76 + Aa
(9.137)
_ ACT
Equation (9.137) can be treated as an integral equation with respect to the product aE^ : aE'^ - C"^ (aE"),
(9.139)
where C ^ (aE") is an integral operator of the anomalous field C^ (aE") = G ^ (/?aE") + G ^ (/3aE^) - PaE^
(9.140)
and a The solution of integral equation (9.139) is similar to equation (9.125) and can also be obtained using the method of successive iterations, which is governed by the equations
These iterations are always convergent, because the operator C^ is a contraction operator! To prove this result, consider the following inequality: | | C ^ [aE"(i) _ aE^(2)j || _ ||Qm [-^ (^^^a{i) _
^E«(2))]
II
Linear and non-linear approximations of the higher orders
where
11/311 = max
259
\Aa\
6(r) a(r)
max rGD 12(76 + A a |
Taking into account (9.136), we can conclude that C^ is a contraction operator if:
l-la
II/3II,
(9.141)
< 1.
Simple calculations show that
A?|' .~|2
\2at 4- A(j|
1
Aaab \a — ab\
+4(jcr6
< 1
under the natural condition that 0 < CTjnin < Cr < (Jrnax < OO, 0 < Cr6min < O-b < CTftmax < OO.
Thus, we have proved that C"^ is a contraction operator for any lossy background media (where cr^ min > 0)- Therefore the N-th iteration approaches the actual anomalous field when N ^^ OO. 9.3.3 Modified Born series We obtain the first iteration in the solution of operator equation (9.139), assuming that the initial approximation E^^°^ (zero order iteration) is selected to be equal to zero (E^(o) = 0): ^j.a(i) ^ ^m jQj ^ G ^ M ^ (aE^) - /3aE^ (9.142) where we use an operator M^ of multiplication by the function j3. By analogy with the classical Born approximation, we will call expression (9.142) a modified Born approximation E^"^: j.a(i) ^ ^Bm ^ I G ^ M ^ (aE^) - /3E^ The second iteration is ^Ea(2) ^ ^m (^E^(i)) _ G ^ M ^ (aE^^) + aE^"^. The third iteration is
(G^M^)^ (aE^^) + G ^ M ^ (aE^^) + aE^"^.
(9.143)
260
Integral representations in electromagnetic forward modeling
Finally the N—th iteration can be treated as the sum of TV terms of the Born (or Neumann) series:
+ G ^ M ^ (aE^"^) + (G^M^)^ (aE^"^) + ... + ( G g M ^ ) ^ " ' (aE^^) .
(9.144)
The remarkable result is that these series are always convergent for any lossy background media! Let us analyze more carefully the modified Born approximation introduced by formula (9.143). Simple calculations show that ^Bm ^ I G ^ M / J (aE^) - f3E^ = - v ^ E ^ ,
(9.145)
a a where E ^ = Gb [A5E^] is a conventional Born approximation. Therefore, we obtain ^Bm ^ _J^±^^B
(g ;^4g)
From equation (9.146) we can see that the modified Born approximation is equal to a conventional Born approximation outside the inhomogeneous domain D: E ^ ^ = E^,
ifr^D.
However, inside D they are different. Actually, it is this difference that makes the new Born series always convergent. Taking into account formula (9.145), we can obtain the following expression for the modified A^-th order Born approximation as the sum of N terms of the Born (or Neumann) series, which helps to understand better the internal structure of the new series: aE°(^) = y ^ E ^ + G ^ M ^ ( V ^ ^ ' ' ) + ( G ^ M ^ ) ' ( ^ / ^ E ^ ) + ... + (G^M^f-i(V57E^). These series unconditionally converge to the anomalous field for any lossy background media, 1 "^
E« = i^(G-M^)^(y^E^), a k=0
because operator (G^M/5) is a contraction operator:
IIGl^M^II < IIG^II II/3IU < 1.
Linear and non-linear approximations of the higher orders
261
We can estimate the accuracy 6 of the A^—th iteration: | a E « - a E ^ W | | = EZN{GTM,)'{^,E^)
(y^E^)ll - \\{G^M^faE^
[G^M^fZZoiG^M^)' <
(G^M^r
(9.147)
| | a E " | | < | | / 3NC | | a E l | .
Since \\P\\^ < 1, we conclude that 6 progressively goes to zero with the number of iterations N. 9.3.4 Quasi-linear approximation of the modified Greenes operator We will obtain a more accurate approximation even on the first step if we assume that the anomalous field E" inside the inhomogeneous domain is not equal to zero, as it was supposed in the previous section, but is linearly related to the background field E^ by some tensor A : E" (r) ^ A (r) • E^ (r).
(9.148)
Subsequently, we use the expression (9.148) as the zero order approximation for the scattered field inside the inhomogeneity, E^(o) _ X • E^, and calculate the first approximation as follows: aE^^i)
aA-E^
G^/3 (aX • E^) -f G^/3 (aE^) - /3aE^ (9.149)
^p(a(l-{-X)
-E^) - PaE^.
We call this approximation a quasi linear approximation of the first order E Q ^ or a modified quasi-linear approximation (MQL) E ^ g ^ for an anomalous field: K'L
= EMQL = ^G^{^' (l + A) • E^) - /JE".
(9.150)
Note that conventional QL approximation is given by the formula E^^ = G B ( A ? ( I + A ) - E ' ' ) .
(9.151)
Obviously, outside inhomogeneities, where ACT = 0, the MQL approximation is identically equal to the conventional QL approximation: E W = EQL>if'-^^-
(9.152)
Integral representations in electromagnetic forward modeling
262
However, inside inhomogeneity (ACT 7^ 0) they are different. In particular, inside domain D we obtain the following approximate formula for E ^ g ^ : E^QL « A • E*- « i c r (h ( l + = -^bGb
(Aa (T+
A)
A)
• E*) - /JE"
• E*-) + 0X • E ^
We estimate an accuracy of the first order QL approximation, ^QI = E ^ g ^ , using the following formula which is taken from (9.168) derived in section 9.3.6: \\a (E"-E«^Q,)|| < e = j ^ q j ^ ^ ( A ) ||aE^||,
(9.153)
where ^ f\\
=
||G^
(b ( l +
A V E^)
- bE^ - aX ' EHI / ||aE^
The inequality (9.153) can be modified as follows: (9.154)
\a\l-\\Ph, where I Loo
b{r) max a{r) reO+
|A5| max \2ab + Acr|' reo+
and a is an average value of a : \a\ • < \a\ < \a\
I I mm — 1^1 — 1^1 max
Formula (9.154) shows that the minimum of (f Ix) determines the accuracy of the QL approximation. This criterion is used to find the electrical reflectivity tensor: ^E^QL - aA • E
=
(6 ( l + A ) • E^) - bE^ - aX • EHI = ip (x\ \\aE^\\ m m .
(9.155)
Note that minimization problem (9.155) is equivalent to the minimization problem which we have used in determining the electrical reflectivity tensor for the original QL approximation. This means that we can use exactly the same reflectivity tensor in both cases.
263
Linear and non-linear approximations of the higher orders
9.3.5 QL series We have noticed that the background of the modified Born approximation and the new first order QL approximation is the same. The main diff'erence is that in the case of the Born approximation the starting point (zero order approximation) for the iteration process is the zero anomalous field, while in the QL approach we start with the anomalous field proportional to the background field. In principle we can extend our approach to computing all iterations by (9.149). In this case we will obtain a complete analog of the Born series. For example, the second order QL approximation is equal to aE^^l^ = C"^
UEQ^IA
= {C^pf
(aX • E^) -f (G^/3) (aE^"^) + aE^"^.
The third order QL approximation is given by the formula
= (G^/?)^ (aX ' E^) + (G^/3)^ (aE^^) -f (G^/3) (aE^^) -h aE Bm Finally the A^—th order QL approximation can be treated as the sum of TV terms of the QL series: o'E^t^ = E t o ^ ( G r / ^ ) ' («E^^-) + {G^(3f
(aA • E^) (9.156)
= aE^(^) + {Gl^(3f
(aX • E'^ ,
where, according to (9.144), E""^^^ is an A^-th order modified Born approximation. Let us compare the N-th order Born approximation with the A^—th order QL approximation: g a w _ ^a(N) ^ 1 (G-/3)^ (a ( A • £ " ) ) .
(9.157)
From the last formula we can see that higher order QL approximations are more accurate than the modified Born approximation of the higher order because they take into account an additional term in the series. 9.3.6 Accuracy estimation of the QL approximation of the first and higher orders Let us estimate the accuracy of the QL approximation. According to (9.140) and (9.139), the actual anomalous field E^ can be determined by the equation aE" - G ^ {b (E" + E^)) - bE\
(9.158)
Comparing aE" with the corresponding formula for modified QL (MQL) approximation (9.150), we can obtain an accuracy criteria for an MQL solution :
Integral representations in electromagnetic forward modeling
264
| | a E " - a E ^ Q ^ | | = | | G r (/? ( a E " - aX • E " ) ) (9.159)
<mU\Gn\\aE'^-a\-E'^^ Using equation (9.150) we can express aE° — aA • E* as follows:
aE" - aX • EHI aE" - aElfQL + G^ (b ( l + A ) • E*) - bE* - aA • E '
(9.160)
< llaE^-aE^Q^II + ^ ( A ) WaE"]], where ifi (x) = IIG^ (b (l + X) • E*-) - bE'' - aX • EHI / \\aE''\ (9.161)
< IIG^ (b (l + x) • E^) - bE'' - aX • EHI / ||aE''||. From inequalities (9.159) and (9.160), we have WaE^-aEl.Q.W < |i/J|U ilG^H {||aE«-aE^,,|| +
(9.162)
According to inequalities (9.136) and (9.141) ii/3iLiiGrii
\\<e = I n l t ^
( ^ ) I'^^'H '
^^"^^^^
where a is an average value of a : I imin — I 1 — I I m a x
Finally we obtain (E«-E^Q^)
<, _ 1
ll/?lloo ^
-'' - N r^w::^(A) llaEll,
(9.164)
Linear and non-linear approximations of the higher orders
where
265
|A5| b{r) (9.165) max a{r) = max GO+ 12(76 4- Acrl r£0+ Now, let us estimate the accuracy of the QL approximation of the N-th order:
ll^llc
aiN)
Z^fcliV i^E p)'' (aE^™) - {G'SPf
aE^ - aKQL
(aX • E^)
(G^/3)^ (aA • E*-) iV-1
aE" - a E ^ " - G^/? (aX • E*)l
<
{G^f3)
N-l
Thus we have
aE^-aE^^P
(9.166)
C\\aE^-aEl,Q,\\<mCe^
<
where e is an accuracy of QL approximation, determined in equation (9.163). Therefore, we obtain ,a{N)
E^-E;QL
<
m
N
1
mi
N
^
(x\ 11 E^ a
(9.167)
Another possible estimation comes from the following:
||a(E"-E"^Q^)|| = lallKE-^-E;;,,)!! < l«ki Therefore aE°-aE^W
<mC\a\e,<
-f ( ^ )
1
aE"
(9.168)
Finally we obtain
aE^-aE^Jf laE^I
N
<
1
-(f
(X).
(9.169)
Comparison of the inequalities (9.147) and (9.169) clearly demonstrates that the accuracy of the Born approximation depends only on the order A^, while the accjaracy of QL approximation of the same order can be increased by a proper selection of A. This circumstance makes the QL approximation a more efficient tool for EM modeling than a conventional or modified Born series. At the same time we can use the inequality (9.169) to prove that QL series are always convergent. This result comes from the fact that \\P\\^ < 1 for any model with lossy background media. Therefore QL
0, when N —^ oo.
266
Integral representations in electromagnetic forward modeling
9.3.7 QA series Now we can start iterations by the modified Born series with a quasi-analytical approximation for the anomalous field: a(0)
9
E: '^^~l-9
(9.170)
-E^
In this case, the first order QA approximation is equal to ,a(l) aE°7 = C™(aE"J°')=GS
,a(0)
(3aE:QA
+ G ^ [(3aE''] -
(9.171)
paE\
We will call the first iteration determined by expression (9.171) a modified quasianalytical approximation (MQA): '^QA
-
'(ia^t2 -h - G ^
'^MQA
[/?aE^] - /3E^
Taking into account the definition of the modified Green's operator (9.135) and formula (9.170), we obtain 2
Ea
a(0)
^{'
' ^MDA « ^ -— 2<7, + A ? ^ ^ ^
2^' ^
2ab + Aa
A5E;QA
.GJ-^E^' ^ [1 - g~
+ QE [ASE*-] } 2<J6
E aA/r
2(75+ A 5 ^ ^ ^
(9.172)
Equation (9.172) shows that the modified QA approximation is equal to the original QA approximation outside inhomogeneity D: ^MQA (r,) = E ^ ^ (r,)
(9.173)
,rjiD,
while they are different inside the geoelectrical inhomogeneity. The second order QA approximation can be expressed as aEfJ
= C™ (aE^W) = (G^/?)^ ( a E ^ ) ) + (G^/3) (aE^™) + a E Bm
where E^"^ is a modified Born approximation determined by formula (9.143). The third order QA approximation is given by the formula
aEgl^ = C^ « ^ ) = (G™/?)^ («EQ^A ) + {GEP?
( a E ^ " ) + (G^/3) (aE^'") + a E ^ " .
Finally, the N-th order QA approximation can be treated as the sum of A'' terms of the QA series: « E " j r = Zto
(GS/?)' (^E^™) + {G'SPf
(aE»(°') ,
(9.174)
267
Integral representations in numerical dressing
where G ^ is the modified Green's operator: (G^/?) (aE"J°)) = x / ^ / / / ^ GE (r, | r) 2^,bE''^''Jdv •JIJ^ CTb
+ tE^
GE (r, I r) AaE^^'Jdv + ^ E ^ ( ° ) ( r , ) .
Note that outside inhomogeneity D, the modified Born approximation coincides with the conventional Born approximation: E^™ (r,) = E ^ ( r , ) , r,- ^ D. We should note that QA series can also be built on TQA and LN approximations for the first iterations. 9.4
Integral representations in numerical dressing
In practice we usually solve forward and inverse problems in the space of discrete data and model parameters. To apply the integral representations of an electromagnetic field, introduced in the previous sections of the book, we have to develop the corresponding numerical methods which will operate with discrete data. The conventional approach to discretization of integral operator equations is based on the Galerkin method, named after Russian scientist B. Galerkin, who introduced it originally in 1915 (Galerkin, 1915). We shall use this approach to develop a numerical technique for the integral equation method and in order to compute linear and nonlinear approximations of an electromagnetic field. We will begin our study with discretization of the model parameters. 9.4-i Discretization of the model parameters We assume that the model parameters, anomalous conductivity distribution, A a (r), belong to some complex Hilbert space MQ of the functions 7 ( r ) , 7 G MD, defined in domain D and integrable in D with the inner product: (7,^)MZ,
"= / / / 7(1*)^* W ^ ^ ,
l^veMo-
We can discretize the model, in this case the anomalous conductivity distribution, A a (r), by introducing a set of basis functions, {T/^^ ( r ) , 7/^2 (^)' ••••V^AT (^)} in the finite dimensional Hilbert space Mjv, which is a subspace of the complex Hilbert space Mjj : Mp^ C M]j- Let us approximate the anomalous conductivity by its projection over the basis functions: N
A5(r)^^Aa,V^,(r),
(9.175)
Integral representations in electromagnetic forward modeling
268
n- • — n• — rs • — p i — • — • — • — • — •
Figure 9-3
FL • — • —
The domain D of anomalous conductivity is divided into elementary cells, N
Dn, formed by some rectangular grid in the domain D = U Dn.
where Acr^ are the coefficients of the expansion. In particular, if the basis functions are mutually orthogonal, (I
f 0^ ^ 7 ^ P
I \
Mo ' " = P
then the coefficients Aa„ can be determined by scalar multiphcation of both sides of equation.(9.175) by tp^:
Aan =
(A^.V-JMO IIV':TIUMD
///pAg(r)C(r)c^^; m^^PAr)rn{r)dv^
= 1
N.
(9.176)
The choice of the basis functions, t/^^, is important in both forward and inverse electromagnetic problems. It depends on the specific geophysical problem and the required accuracy of discretization. Let us divide the domain D into elementary cells, D^, formed by some recN
tangular grid in the domain D = U Dn (Figure 9-3). In the simplest case we can use as the basis functions the so-called "boxcar" functions, which are constant over elementary cell Dn, and which are equal to zero outside this cell: ^ „ ( r ) = box„(r) = { j ; ^ ^ 2 ;
.
(9.177)
An example of the boxcar basis function in two dimensions is shown in Figure 9-4. These functions form an orthogonal basis:
Integral representations
in numerical
269
dressing
b0Xn(x,z)
Figure 9-4 An example of the "boxcar" basis function, which is constant over the elementary cell D^, and which is equal to zero outside this cell.
boxn (r) boxp (r) dv
(boXn,boXp) =
0, n y^ p \Dn\ , n = p, TnG
D
where \Dn\ denotes a volume of the elementary cell DnSubstituting (9.177) into (9.176), we find Aar>
fjjj^ ACT (r) boxn (r) dv Iboxnl
M
1 Wr.
Aa{r)dv,
n= 1,....7V.
(9.178)
Dn
Thus, we have the following representation for the anomalous conductivity distribution: ACT (r) ^ ^
Acr^box^ (r)
(9.179)
where the coefficients of the expansion over the boxcar basis functions represent the average values of the Aa (r) over an elementary cell. In particular, if Aa (r) has the constant value Aa^ within the cell, then the coefficient in the boxcar function expansion is equal to this constant. Note that the coefficients AJ^ can be represented as the components of a vector cr of a length N: o--[A5i,A52,..-,A5^]^. 9.4-2 Galerkin method for electromagnetic field discretization In order to develop the numerical analogs of the electromagnetic field integral representations, we have to discretize, also, the field components and the Green's tensors within the anomalous domain D of integration. We can treat all integral representations, considered in this chapter, as operators acting on the vector functions, x.
270
Integral representations in electromagnetic forward modeling
defined in domain D: x G L2 (D), where L2 (D) is a complex Hilbert space of the vector functions determined in domain D and integrable in D with the inner product:
(X' y)L2{D) = JJj
X i^) • y* (r) dv.
For example, we show here how to represent formula (9.44) written in operator form, E = G^; (A5E) + E ^
(9.180)
where the electric field E is determined in D. We introduce a finite dimensional Hilbert subspace, L2 , of the space L2 {D) : Z/2 C L2 {D). Let {vi (r), V2 (r), ....v^v (r)} be an orthogonal basis in L2 • We will be looking for an approximate solution of (9.180) in the form N
E(r)«^XnV,(r),
(9.181)
n=l
where x^ (n == 1,2,..., A/") are the scalar coefficients of the expansion, equal to (E.Vn)^,(P)
J/^E(r)-v;(r)dt;
l|vn|li,(z,)
If ID Vn (r) • V* (r) dv'
(9.182)
Substituting (9.181) into (9.180), we have
Y, ^r{Vn = Y^ X„GE ( A 5 V „ ) + E^
(9.183)
n=l
n—\
Now, following the conventional technique of the Galerkin method (Kress, 1999), we form the inner product of both sides of equation (9.183) with the basis function Vp : N
N
Y^ Xn (Vn, ^V)L2{D) n=l
= I ] ^ n ( G ^ ( A 5 v ^ ) , ^p)^^^^^ n=\
+ ( E ^ Vp)^^^^^ .
The last equation can be written as follows: N
Y, Xn ([v„ - GE (A5v„)], vp)^^(o) = ( E ^ vp)^^(^,. n=l
Using t h e notations Any = ([Vn " G^; (ASv^)] , ^V)L-,{D)
' ^ ^ ^ 6^ = ( E ^ V^)^^^^^ ,
(9.184)
271
Integral representations in numerical dressing
we cast equation (9.184) in the form (9.185)
Y^XnAnp = bp. n=l
The coefficients Xn of the expansion (9.181) are determined as the solution of this system of Hnear equations (9.185), which can be obtained by one of the numerical methods described in Chapters 3, 4, and 5. 9.4-3 Discrete form of electromagnetic integral equations based on boxcar basis functions We can select the basis functions in expansion (9.181) in the form of the boxcar functions multiplied by the values of the vector E (r^) at some internal point of the cell Tn e Dn '.
V, (r) = E (r^) box, (r), n = l,2,...A^.
(9.186)
These functions form an orthogonal basis, and ||v„|r = / / / E (r„) box„ (r) • E* (r„) box„ (r) dv = |E (r^)^ |D„| 2
/ / / T 7 > /
\ U
/ \ T 7 « * /
\ U
/
\
J
ITTI/
M 2 |
In this case, the scalar coefficients of the expansion (9.181) are equal to ^ ^ / / / , , E (r) • E* (rQ box, (r) dv _ JJJ^^ E (r) • E* (r.) dv
llvnf
|E(r,)|^|D,|
(9.187)
According to a mean value theorem for integrals of the continuous functions, there is a "mean value" point r , G / ) , , for which E (r) • E* (r,) dv = E (?,) • E* (r,) | D , | .
(9.188)
We assume now that all points r , coincide with the corresponding mean value points, Tn = ?n- In this case formula (9.187) for expansion coefficients takes the form Xn =
///D„EW-E*(rn)rft; \E{r„)f\D„\ ^ ^ ^ ^^ o = o-^ = 1, n = 1, 2, ....A'. |E(r„)p|D„| |E(r„)nD„|
Therefore, the vector field E (r) can be represented as N
E (r) ^ ^
E (r,) box, ( r ) .
(9.189)
n=l
The points r , were selected above as the averaged points for the corresponding functions. In the case of a slow varying field and/or small cells D , , one can assign these points to the centers of the cells.
Integral representations in electromagnetic forward modeling
272
Note that within the framework of the integral equation method, the vectors E (r^) are still unknown and have to be determined by the solution of the corresponding equations. Substituting expansion (9.189) into the right-hand side of formula (9.180), we arrive at the discrete form of the operator equation: N
E(rp) =
^
GE
(vp I r) A 5 (r) dv • E (r^) + E^ (r^),
(9.190)
Dn
n=l
Now we can use also expansion (9.179) of the anomalous conductivity distribution over the boxcar functions. Substituting (9.179) in the last formula, we finally find N
E (rp) = Yl III
^E (rp \v)dv A 5 „ E (r^) + E ' (r^).
(9.191)
Thus, inside the anomalous domain D, the discrete analog of equation (9.44) takes the form eo = Gz)S^ez)+e^,
(9.192)
where ? is a (3A^ x 3N) diagonal matrix* of anomalous conductivities: A5i
•••
0
0 0
• •• • •
Ao-AT 0
0
•••
0
0
•••
0
0
• ••
A?i
• ••
0 0
0 0
•• • •• •
0 0 (9.193)
0 0 0
• • •
0 0
0 0
0
0
• •• Aa^v • •• 0 0
0 ACTI
0
•• • •• •
0 0 ACTA
en and e^^ are the vectors of the total and background electric fields formed by the X, y and z components of these fields at the centers of the cells Dn of the anomalous domain D: en = \E^ E'^ E^ E^ E'^ E^ E^ E'^ E^V rj> — rZ7'^,l Vrb,2 r^b,N rpb,l 77.6,2 j^b^N 77-6,1 77'6,2 T^b,N^T 5 •••^z J • ^D — [^x 1-^x ^ ' ^ x i^y 5^2/ )"-^2/ '^z ^^z
These vectors have order 3A/". The ZN X 3A/' matrix G^ is formed by the volume integrals over the elementary cells Dn of the components of the corresponding electric Green's tensor G^;, acting *In this section we use a "hat" sign above the matrix to distinguish between the column vectors and matrices.
Integral representations
in numerical
273
dressing
inside the domain D : ^^xx rxNN XX
XX
filN xy
xy
xy
' '
fin
^yx
^yx
^yx
ryNN ^yx
^yy
^zx
^zy
^xy ^yy
^xz
' ' '
^xz ri\\ ^yz
' ' ' . "
^xz
r
(9.194)
G, ' Nl
L Gzx
G NN
^yz
^yy '
GnNl zy
'
^zy
^zy
'
'
^yz
Gil ' • Gl^ ^
Nl
G NN
where GEa(3 (rp I r) dv, a,P = x, y, z; p,n=
1, 2, ...A^.
Note that equation (9.192) can be rewritten in equivalent form, ).195) where matrix eo is a sparse tri-diagonal 3A^ x N matrix containing the x, y and z components of the total electric field at the centers of the cells of the anomalous domain D:
El E' ^
(9.196)
GD
<
El
E^N and vector a of order N is composed of the anomalous conductivities of the cells: tT = [A5i,A52,..-,A5iv]^Thus, forward electromagnetic modeling based on the IE method is reduced to the solution of the matrix equation (9.192) in the unknown vector e^ of electric field components inside domain D. The equation is a 3A^ x 3A^ linear system, Aen = e*
JD7
(9.197)
Integral representations in electromagnetic forward modeling
274
where A = I - GD^.
(9.198)
Matrix A is a 3A^ x 3N dense matrix. For a small number of unknowns, direct solvers are practical, especially in the case of multiple sources. One can use different types of iterative methods, discussed in Chapter 4, for the solution of this problem. However, if A^ is large, the storage of A is extremely memory consuming, not to mention the complexity of direct matrix inversion. Using the discretization methods introduced above, we can write the discrete analogs of the integral formulae (9.44) and (9.45) representing the observed electromagnetic field measured at L different receiver positions (ri,r2, ....ri,) • We introduce vectors e, h, and e^, h^ representing the discrete total and background electric and magnetic fields at the observation points: e = \E^
E'^
U _ flijl zj2 " — l^x^^x^ Jb _ ^
\pb,l [-^x
E^
E'^
E^
E^
E^
E'^
E^V
TjL Tjl TT2 TTL TT\ TT2 zjLiT "-^x ^^y^^y^ -"^y ^^z^^z^ "'^z\ '
77.6,2 rpb.L 77.6,1 E^6,2 rph.L "i^x •)-"^x 1-^y ' y "> " y
rpb,l 1 z
77.6,2 i^z
Trb,L]T 1''^z J '
V,b _ r r r 6 , l rr6,2 r76,L rrb,! rr6,2 rr6,L 7:r6,l rr6,2 Ljb,L]T ^ — i^x ) ^ x ^--^x 5 ^ y ' ^ y ^-'-^y i^z ^^z ^'-^z J •
Then integral formulae (9.44) and (9.45) can be cast in the following discrete forms: (9.199) h = GneDO- + h ^
(9.200)
where matrices G^; and G// are discrete analogs of the corresponding Green's tensors. These matrices consist of the volume integrals over elementary cell Dn of the components of the corresponding electric or magnetic Green's tensor acting from the anomalous domain to the receivers. L is the number of receivers, and N is the number of cells in the anomalous body. The number of rows in GE^H equals the length of the data vector, 3L. Matrix GE,H has 3N columns: ^11 ^-^xx
^xx
GE,H
^^xx
'
'
^xx
yx
yx
yx
yx
^11 ^xy
'
^xy
^xz
'
^xy
'
^xy
^xz
' •*
^xz
yy
^yz
'
^yz
'
^yy
^yz
' "
Gil •
'
^zy
^zy
'
r
^zz
'
yy
'
'
^xz
— ^zx
'
^zx
'
'
^zx
fiLl yy
'
r
Gil • •• Gl^ •• G,^f
Integral representations in numerical dressing
275
where
After finding CD from equation (9.192), we can substitute it in equations (9.199) and (9.200) written in the form e = GEO-en-^e ,
(9.201)
h =
(9.202)
GH^eD-^h\
to calculate the electromagnetic field in the receiver positions. 9.4-4 Contraction integral equation method In section 9.3 we have applied some linear transformations to the Green's operator to obtain a new modified Green's operator G ^ with norm less than one. The specific form of this linear transformation is motivated by the energy inequality for the anomalous electromagnetic field, which expresses a fundamental physical fact, that the energy flow of the anomalous field outside the domain of anomalous conductivity is always non-negative. As a result, the original integral equation (9.44) can be converted into a contraction operator-based equation: aE" + bE^ = G ^ [6(E" -h E^)] where a=
2(76 + ACT , —=—, b 2^/d^
ACT
2yfa~h
(9.203)
(9.204)
and operator G ^ (x) is defined as a linear transformation of the original electric Green's operator: G ^ (x) = V ^ G B ( 2 V ^ X ) + X.
(9.205)
Equation (9.203) can be rewritten with respect to the product of a and the total electric field E, using simple algebraic transformations: E + ( 6 ~ a ) E ^ = E - V ^ E ^ = G;^ p a ' ^ E
- G ^ |6a-^E| ,
(9.206)
where E is the scaled electric field E-^aE.
(9.207)
Finally, we can present equation (9.206) in the form E-C"
{-)
6a-^E + V ^ E ' '
(9.208)
276
Integral representations in electromagnetic forward modeling
In this equation operator C ^ ( E l is a contraction operator for any lossy medium (see section 9.3.2): (E(^)
~ E(2)^
E(I)
- E(2)
(9.209)
where ||...|| is L2 norm, /c < 1, and E^^^and E^^^ are any two different solutions. For this reason integral equation (9.208) is called a contraction integral equation (CIE), and we call C^ a contraction Greenes operator. Using the original Green's operator given by expression (9.37) and taking into account formula (9.205), one can rewrite equation (9.208) as follows: E = y/o^GE (2y/d~bba-^E^ 4- ba'^E + v ^ E ^ or, after some transformation (1 - ba-^) E - y/B~bGE (Aaa-^E)
= v^E^
Note that according to (9.204), 1 — ba~^ = (a — 6) a~^ =
y/ala~^.
Therefore, we finally obtain the following form of the contraction integral equation (CIE) with respect to the scaled electric field E : o~ba-'E - ^tGE
(A5a-^E) = > ^ E ^
(9.210)
This is the basic equation of the CIE method of electromagnetic modeling. 9.4-5
Contraction integral equation as the preconditioned conventional integral equation One can demonstrate that the contraction integral equation (9.210) can be treated as the preconditioned conventional integral equation (9.44). This fact becomes especially important in numerical implementation of the CIE method. In discrete form, equation (9.210) can be written as M l (T-GjT)?) M2eD = Mie^^,
(9.211)
where G ^ is determined by (9.194); a and M i are the 3A^ x 3N diagonal matrices of the anomalous conductivities and the square root of the background conductivities respectively, similar to matrix (9.193): a = d i a g (Acri, A a 2 , . . . , AdTv, Acri, Acr2,..., Aa^,
Acri, Acr2,..., AGN) ,
277
Integral representations in numerical dressing
Ml - diag Ty^, y^,..., ^Z^, y^, y^..., y < , y^, y^..., and where GD
=
(9.212)
M^^BD.
with M2 defined as the diagonal matrix M2 = diag {a^\a2\
. . . , a ^ \ a ^ \ a 2 \ . . . , a ^ \ a 5 " \ a ^ \ ...,a^^) ,
(9.213)
where, according to (9.204), a^ = ——-=r—, ^ = 1, 2,..., N. ^y/CTbi
Considering (9.197) and (9.198), we can easily see that equation (9.211) is just the preconditioned variant of (9.197), Aen = e'j,,
(9.214)
A = M1AM2, S^ = Mie^^.
(9.215)
where and e^ =
fe^,
(9.216)
Thus, we have transformed the original matrix equation (9.197) into the preconditioned equation M1AM2 ( ^ ^ e ^ ) = M l e ^
(9.217)
where M i is the left preconditioner, and M2 is the right preconditioner. Note that there are different mathematical methods for introducing preconditioners for a general matrix equation (Golub and Van Loan, 1996). The main goal is to construct a matrix M1AM2, which produces a significantly higher convergence rate of iteration than the original matrix A. Note that the act of finding a good preconditioner is that of making A = M i AM2 as close as possible to the identity matrix. In the case of the CIE method we reach this goal by using the contraction Green's operator with the norm always less than or equal to one. In other words, we take into account the physical nature of our system of equations and find the preconditioners based on the fundamental physical law that the energy flow of the anomalous electromagnetic field outside the domain containing the conductivity inhomogeneity is always positive.
278
Integral representations in electromagnetic forward modeling
9.4-6 Matrix form of Born approximation We will present, as an illustration, the discrete forms of several types of linear and nonlinear approximations, introduced above. We begin with the original Born approximation (9.74): e^ = GE^D<^,
(9.218)
where vector e^ represents the discrete Born approximation of the anomalous electric field at the observation points. Matrix e^ is a sparse tri-diagonal 3A^ x N matrix containing the x, y and z components of the background electric field at the centers of the cells of the anomalous domain D: Eb' E\
K^'
^
(9.219)
El^ Eb,N ^
Similar to equation (9.192), we can cast the Born approximation in the form e^ = G e ^ e ^ .
(9.220)
9.4. "7 Matrix form of quasi-linear approximation Expression (9.79) for the quasi-hnear approximation of the anomalous field can be written in discrete form as e^L = G s ( l + Ap)e^^c
(9.221)
where matrix AD is the 37V x iN block-diagonal matrix consisting of the elements of the electrical reflectivity tensor inside the anomalous domain: ^Al
0
Ai
0
AL
0
• •
0
AL
^xy
0
0
0 •
^xz
"
\ z
0 (9.222)
0 AL
^yx
0
•
0
^zx
•• 0
•
\ y
0
^zy
0
0
• •• • ••
\ z
0
A-
Integral representations
in numerical
279
dressing
and I is the 3N x 3N identity matrix. Using the notation of (9.193), we can rewrite equation (9.221) in equivalent form: e^^ = G^ (I^XD) ?e5,. (9.223) In a special case when we use a scalar electrical reflectivity coefficient, equation (9.223) takes the form e^^ = G ^ ( T + A ) ? e ^ ^ , (9.224) where A is a diagonal 3A^ x 3A^ matrix Ai •••
0
0
0 •• • 0 • •'
XN
0
0
Ai
0 • • 0 • • 0
0
0
• •• • •
0 0
0 0
•• • •• •
0 0
0 0
0 • • 0 • •
XN
0
0
Ai
•• • •• •
0 0
0
0
0
0
(9.225)
A =
A,
formed by the electrical reflectivity coefficients in different cells of the grid. Introducing the notation m
(t.x)
(9.226)
where m is a so-called modified conductivity matrix, we can cast formula (9.224) in the form (9.227) Note that matrix m is a 3A^ x 3A^ diagonal matrix, mi---
m =
0
0
•••
0
0
•••
0
0 0
0 0
-- •• -
0 0
rriN
0
0
mi
•• • • ••
0 0
0
0
0 0
• •• • -•
rriN
0
- •
0
mi
- •
0 0
• •• • •-
0 0
0 0
• • • •-
0
0
rriN
280
Integral representations in electromagnetic forward modeling
where mi - (1 -f Xi) Aai, i = l,2, ....N.
(9.228)
Finally, we can introduce a modified conductivity vector m, m== [mi,m2,...,m7v]^, and present the quasi-hnear approximation in the form ^QL =
(9.229)
GE^D^,
and in a similar way for the magnetic field hQL =
(9.230)
GH^D^-
The modified conductivity matrix and vector play an important role in the method of quasi-linear inversion, which we will discuss in the next chapter. 9.4-S Matrix form of quasi-analytical approximation Finally, the quasi-analytical approximations (9.90) and (9.91) of the anomalous electric and magnetic fields in discrete form can be written as e^QA = GE^D [diag (I - g(o-))]-^ o-,
(9.231)
and h^^ = GH^O [diag (I - g(o-))]-^ a.
(9.232)
We use the following notations in the last formulae. The vectors eg^ and h g ^ represent the discrete quasi-analytical approximations of the anomalous electric and magnetic fields at the observation points. Vector I is a A^ x 1 column vector whose elements are all unity. The N xl column vector g (
g('^)
1^6,2* , J^B,2
^b,u . E^i ' E^'2* • E^'2 '
•p6,Ar* ^ J^B,i
' E^'-^* • E^'^
(9.233)
where E^'-^ and E^'-^ {j = 1,2,...A^) denote the Born approximation and the background electric field in each cell within the anomalous domain. Direct calculations show that vector g(cr) can be expressed by matrix multiplication: g(*T) = (e^oe^^*)-'e5;eg, (9.234) where the vector of the Born approximation inside the anomalous domain, e £ , can be expressed by a formula similar to (9.218): eg = GD'^D^-
(9.235)
Integral representations in numerical dressing
281
Substituting (9.235) into (9.234), we obtain {^^•^*)-'^^GD^D
g ( ^ ) = (e^^e^^*)
where
/"^ '^*\ ~^ ^ C={^^^^)-'^^Gn^^,
(9.236) (9.237)
Thus, we can represent equations (9.218), (9.231) and (9.232) in the form = GE^D^ =^Ecr,
eQ^=AE Idiag (I - Co-j 1 a =A£;B(cr) or, h^^ - A^ [diag (l - C ( T ) ] ' a = A H B ( O - ) a,
(9.238)
where AE
= GE^E,, A// == G//e^,
(9.239)
and the diagonal matrix B(cr) = [diag ( l - C o - ) 1
(9.240)
Let us introduce a notation d for an electric or magnetic vector of the anomalous part of the observed data. This vector contains the components of the anomalous electric and/or magnetic fields at the receivers. The discrete forward modeling problem for the electromagnetic field based on the QA approximation can be expressed by the following matrix operations: I-Ccr d = A diag (^
<j =AB(cr) cr,
(9.241)
where A stands for the electric or magnetic matrices, A^; = GE^^E ^^ ^H = G^/e^^, respectively. The last formulae serve as a basis for the method of quasi-analytical electromagnetic inversion, which I will present in the next Chapter. 9.4-9
The diagonalized quasi-analytical (DQA) approximation
One can see that matrix C is responsible for the scattering inside the anomalous body, because it is determined by the domain scattering matrix GE (see equation (9.237)). We can further simplify the quasi-analytical approximation by assuming that the anomalous electric field in a specific cell is affected mostly by the scattering in the same cell. In other words, we can simplify C by retaining its diagonal elements only. This is the basic idea of the diagonalized quasi-analytical approximation (DQA). The expressions of the anomalous electric and magnetic fields are the same as (9.238) for
282
Integral representations in electromagnetic forward modeling
10 m 40 m 100 ohm-m
10 m 10 m 20 m
Figure 9-5 The geoelectrical model consisting of a rectangular conductive body of 10 Ohm-m resistivity embedded in a homogeneous half-space with a resistivity of 100 Ohm-m. It is excited by a current in a horizontal rectangular loop with the frequency of 1000 Hz. The receivers are located above the body along the y axis.
Integral representations in numerical dressing
-40
-20 0 20 y [meters]
10
-20 0 20 y [meters]
283
40
20 0 20 y [meters]
40
20 0 20 y [meters]
Figure 9-6 The real and imaginary parts of the horizontal electric and vertical magnetic components of the scattered field computed by solving the full integral equation (IE), the QA and the DQA approximations.
284
Integral representations in electromagnetic forward modeling
the QA approximation, except that formula (9.237) is replaced now by the following expression: C = diag [{^^^n)"'^DGD^D]
•
(9-242)
To illustrate the accuracy of the DQA approximation, consider the following numerical experiment. The geoelectrical model consists of a rectangular conductive body of 10 ohm-m resistivity embedded in a homogeneous half-space of resistivity 100 ohm-m (Figure 9-5). It is excited by a current in a horizontal rectangular loop with the frequency is 1000 Hz. The receivers are located above the body along the y axis. Figure 9-6 shows the real and imaginary parts of the horizontal electric and vertical magnetic components of the scattered field computed by solving the full integral equation and the QA and the DQA approximations. We can clearly see that the accuracy of the diagonalized QA^ approximation is about the same as the QA approximation using the full matrix C. Since the cost of computing the diagonal of C is 0{N) instead of 0{N'^), the calculation of the DQA approximation is practically the same as the Born approximation.
References to Chapter 9 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 S. H. Gray, 1985, An extension of the Born inversion method to a depth dependent reference profile: Geophys. Prosp., 33, 999-1022. Bleistein, N., 1984, Mathematical methods for wave phenomena: Academic Press Inc., (Harcourt Brace Jovanovich Publishers), New York, 341 pp. Born, M., 1933, Optik: Springer, Berlin, 591 pp. Born, M., and E. Wolf, 1980, Principles of optics: Pergamon Press, New York, 808 pp. de Lugao, P., and P. E. Wannamaker, 1996, Calculating the two-dimensional magnetotelluric Jacobian in finite elements using reciprocity: Geophys. J. Int., 127, 806-810. de Lugao, P., Portniaguine, O., and M. S. Zhdanov, 1997, Fast and stable twodimensional inversion of magnetotelluric data, J. Geomag. Geoelectr., 49, 1469-1497. Dmitriev, V. I., 1969, Electromagnetic fields in inhomogeneous media (in Russian): Proc. of Computational Center, Moscow State University, Moscow, 134 pp. Dmitriev, V. I., Pozdniakova, E., Zhdanov, M. S., and S. Fang, 1999, Quasianalytical solutions for EM field in inhomogeneous structures based on unified iterative quasi-linear method: 68th Annual Mtg., Soc. Expl. Geophys., Expanded Abstracts, 444-447. Galerkin, B.C., 1915, Expansions in stability problems for elastic rods and plates (In Russian): Vestnik inzhenerov, 19, 897-908.
Integral representations in numerical dressing
285
Golub, G. H., and C. F. Van Loan, 1996, Matrix computations. Third Edition: The Johns Hopkins University Press, Baltimore, MD, 694 pp. Habashy, T. M., Groom, R. W., and B. R. Spies, 1993, Beyond the Born and Rytov approximations: a nonhnear approach to electromagnetic scattering: J. Geophys. Res., 98, No. B2, 1759-1775. Harrington, R. F., 1961, Time harmonic electromagnetic fields: McGraw-Hill Book Co., New York, 480 pp. Hohmann, G. W., 1975, Three-dimensional induced polarization and EM modehng: Geophysics, 40, 309-324. Kress, R., 1999, Linear integral equations: Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo, 365 pp. Madden, T. R., and R. L. Mackie, 1989, Three-dimensional magnetotelluric modeling and inversion: Proc. IEEE, 77, 318-333. McGillivray, P. R., Oldenburg, D. W., Ellis, R. G., and T. M. Habashy, 1994, Calculation of sensitivities for the frequency-domain electromagnetic problem: Geophys. J. Int., 116, 1-4. McGillivray, P. R., and D. W. Oldenburg, 1990, Methods for calculating Frechet derivatives and sensitivities for the nonlinear inverse problem: a comparative study: Geophys. Prosp., 38, 499-524. Oldenburg, D. W., 1990, Inversion of electromagnetic data: an overview of new techniques: Surv. Geophys., 11, 231-270. Oristaglio, M. L., 1989, An inverse scattering formula that uses all the data: Inverse Problems, 5, 1097-1105. Pankratov, O. V., Avdeev, D. B., and A. V. Kuvshinov, 1995, Electromagnetic field scattering in heterogeneous earth: A solution to the forward problem: Physics of the Solid Earth, 3 1 , 201-209 (English edition). Pankratov, O. V., A. V. Kuvshinov, and D. B. Avdeev, 1997, High-performance three-dimensional electromagnetic modeling using modified Neumann series. Anisotropic case: J. Geomagn. Geoelectr., 49, 1541-1547. Pellerin, L., Johnston, J. M., and G. W. Hohmann, 1993, Three-dimensional inversion of electromagnetic data: Expanded Abstracts, 63rd SEG Annual International Meeting, Washington, D. C , 360-363. Raiche, A. P., 1974, An integral equation approach to three-dimensional modelling: Geophys. J. R. Astr. Soc, 36, 363-376. Rodi, W. L., 1976, A technique for improving the accuracy of finite element solutions for magnetotelluric data: Geophys. J. R. Astr. Soc, 44, 483-506. Singer, B. Sh., 1995, Method for solution of Maxwell's equation in non-uniform media: Geophys. J. Int., 120, 590-598. Singer, B. Sh., and E. B. Fainberg, 1995, Generalization of the iterative dissipative method for modeling electromagnetic fields in nonuniform media with displacement currents: Journal of Applied Geophysics, 34, 41-46.
286
Integral representations in electromagnetic forward modeling
Stratton, J. A., 1941, Electromagnetic theory: McGraw-Hill Book Company, New York and London, 615 pp. Tarantola, A., 1987, Inverse Problem Theory: Elsevier, Amsterdam, Oxford, New York, Tokyo, 613 pp. Tikhonov, A. N. and V. Y. Arsenin, 1977, Solution of ill-posed problems: John Wiley & Sons, 258 pp. Torres-Verdin C , and T. M. Habashy, 1994, Rapid 2.5-dimensional forward modeling and inversion via a new scattering approximation. Radio Science, 29, No. 4, 1051-1079. Wannamaker, P. E., 1991, Advances in three-dimensional magnetotelluric modeling using integral equations: Geophysics, 56, 1716-1728. Weidelt, P., 1975a, EM induction in three-dimensional structures: J. Geophysics, 4 1 , 85-109. Weidelt, P., 1975b, Inversion of two-dimensional conductivity structures: Phys. Earth planet. Inter., 10, 282-291. Xiong, Z., 1992, Electromagnetic modelling of three-dimensional structures by the method of system iterations using integral equations: Geophysics, 57, 1556-1561. Xiong, Z. and A. Kirsch, 1992, Three-dimensional earth conductivity inversion: Journal of Comp. and Appl. Math., 42, 109-121. Zhdanov, M. S., 1988, Integral transforms in geophysics: Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo, 367 pp. Zhdanov, M. S., and G. Keller, 1994, The geoelectrical methods in geophysical exploration: Elsevier, Amsterdam, London, New York, Tokyo, 873 pp. Zhdanov, M. S. and S. Fang, 1996a, Quasi-linear approximation in 3D EM modeling: Geophysics, 6 1 , 646-665. Zhdanov, M. S. and S. Fang, 1996b, 3-D quasi-linear electromagnetic inversion: Radio Science, 31 No. 4, 741-754. Zhdanov, M. S. and S. Fang, 1997, Quasi hnear series in 3D EM modeling: Radio Science, 32, No. 6, 2167-2188. Zhdanov, M. S., Dmitriev, V. L, Fang, S., and G. Hursan, 2000b, Quasianalytical approximations and series in electromagnetic modeling: Geophysics, 65, 1746-1757. Zhdanov, M. S., and E. Tartaras, 2002, Inversion of multi-transmitter 3-D electromagnetic data based on the localized quasi-linear approximation: Geophys. J. Int., 148, No 3.
Chapter 10 INTEGRAL REPRESENTATIONS IN ELECTROMAGNETIC INVERSION Electromagnetic (EM) inverse methods are widely used in different technical and engineering applications, including determining the internal defects in metal or concrete constructions, studying the underground geological structures in mineral, hydrocarbon and groundwater exploration, solving environmental clean-up problems, imaging the internal structure of the human body in medical applications, and remote sensing, etc. A feature common to all the above applications is that EM transmitters generate a transient EM field which propagates through the medium and illuminates the object. The scattered back EM response from the object is recorded in the receivers as values of the electric or magnetic field components. The goal is to generate an image of the object. The image reconstruction is based on the numerical solution of an inverse scattering EM problem. This problem requires the development of intelligent methods for EM forward modeling and inversion in inhomogeneous media. These problems are extremely difficult, especially in three-dimensional cases. During the last decade we have observed remarkable progress in the development of a multi-dimensional interpretation technique. Several papers have been published during the last years on 3-D inversion of EM geophysical data (Madden and Mackie, 1989; Eaton, 1989; Smith and Booker, 1991; Lee and Xie, 1993; Oristaglio et al., 1993; Nekut 1994; Zhdanov and Keller, 1994; Torres-Verdin and Habashy, 1994; Zhdanov and Fang, 1996b; Alumbaugh and Newman, 1997; Newman and Alumbaugh, 1997; Zhdanov and Fang, 1999; Zhdanov et al., 2000a; Zhdanov and Hursan, 2000; Zhdanov and Tartaras, 2002; etc.). Note that this reference list includes papers which are focused on inversion in the earth and are most relevant to geophysical inverse problems only. Difficulties arise even in forward modeling because of the huge size of the numerical problem to be solved for adequate representation of the complex 3-D distribution of EM parameters in the media. As a result, computer simulation time and memory requirements could be excessive even for practically realistic models. Additional difficulties are related to EM imaging which is based on EM inverse problems. These problems are nonlinear and ill-posed, because, in general cases, the solutions can be unstable and/or nonunique. In order to overcome these diflSculties one should
288
Integral representations in electromagnetic inversion
use the methods of regularization theory to obtain stable, unique solutions of the original ill-posed EM inverse problems. The methods for solving multi-dimensional EM inverse problems are usually based on optimization of the model parameters by applying different inversion schemes. The key problem in the optimization technique, as we demonstrated in Chapter 5, is the calculation of the Prechet derivative (sensitivity matrix), which usually requires much computational time. A powerful tool for EM modeling and inversion is the integral equation (IE) method and the corresponding linear and nonlinear approximations, introduced in the previous chapter. One important advantage which the IE method has over the finite difference (FD) and finite element (FE) methods is its greater suitability for inversion. Integral equation formulation readily contains a sensitivity matrix, which can be recomputed at each inversion iteration at little expense. With finite differences, however, this matrix has to be established anew on each iteration at a cost at least equal to the cost of the full forward simulation. In this chapter we will outline the basic principles of electromagnetic inversion and imaging based on integral representations of the electromagnetic field developed in Chapter 9. 10.1
Linear inversion methods
The most common approach to the solution of nonlinear electromagnetic inverse problems is based on linearization of the forward modeUng operator. This approach has found wide practical application because of the ease of its implementation, the accessibility of software for linear inversion, and the speed of numerical calculations. Linearization uses the family of linear and nonlinear approximations based on the Born method, described in Chapter 9. 10.1.1 Excess (anomalous) current inversion Consider a 3-D geoelectric model with a normal (background) complex conductivity Gb and a local inhomogeneity D with an arbitrarily varying complex conductivity 5 = ^5 H- Aa. The model is excited by an electromagnetic field generated by an arbitrary source which is time harmonic as e~*^*. The magnetic permeability of the media is equal to the free-space magnetic permeability /x = /Xo — ^ T T X I O " ^ henry/meter. Complex conductivity includes the effect of displacement currents: a — a — lUJE.
The electromagnetic fields in this model can be presented as the sum of the background and anomalous fields: E = E^-f-E^ H = H ^ ^ - H ^
(10.1)
where the background field is a field generated by the given sources in the model with the background distribution of conductivity 5^, and the anomalous field is produced by the anomalous conductivity distribution Aa.
289
Linear inversion methods
According to formulae (9.40) and (9.41), the anomalous field in the frequency domain can be represented as an integral over the excess (anomalous) currents, j " = A5E, in the inhomogeneous domain D : E« (r,)
= G^; (A5E) = fff
H^(r,) = GH{AaE) = III
GE {TJ \ r) A 5 (r) E (r) dv, GH{rj\T)Aa{r)E{r)dv,
(10.2) (10.3)
where GE,H {^j \ r) stands for the electric or magnetic Green's tensor defined for an unbounded conductive medium with normal conductivity a^. We assume that the anomalous field E" (r^) is given on some surface of observation S. The goal is to find the anomalous conductivity Aa ( r ) . One simple idea for linear inversion of the anomalous electric field is to solve the linear equation (10.2) with respect to excess (anomalous) currents j " = A a E : E»(r,) = G £ ( r ) ,
(10.4)
and then find Aa from Ohm's law:
assuming that E (r) • E* (r) > 0. The electric field E (r) inside the inhomogeneous domain D is determined using the same integral forward modeling operator, acting within the domain D, E" (r) = GE {D = jjj
GE (r | r') j " (r') dv\
(10.6)
where r and r' ^ D. However, equation (10.4) is ill-posed, because it is non-unique and unstable. We have already discussed in Chapter 1 that there are nonradiating sources of electromagnetic field. For example, consider Maxwell's equations (9.3) for an anomalous field: V X H^ - a^W = j ^ (10.7) V X E^ - icj/iH^ = 0. Let eo be a vector function which is zero outside domain D: eo(r)=0, i f r ^ D , and let ho (r) = — V X eo ( r ) .
(10.8)
290
Integral representations in electromagnetic inversion
Obviously, the function ho is also zero outside domain D: ho(r)=0,
iir^D.
We apply the operators on the left-hand side of Maxwell's equations (10.7) to ho (r) and eo (r) : V X ho (r) - atoo (r) = jo ( r ) , (10.9) V X eo (r) - iuj/jho (r) =: 0. Let this new function jo (r) be an extraneous current density for some electromagnetic field. The solutions of the Maxwell's equations with this source are the fields eo (r) and ho (r) themselves, which are zero outside domain D. Therefore, the currents jo (r) are nonradiating: GE (Jo) = 0 outside D.
(10.10)
In a similar way we can construct an infinite number of different nonradiating currents. We can add these nonradiating currents to the excess (anomalous) currents j ^ in equation (10.4) without changing the anomalous field: E°(r,.) = G s ( r + J o ) .
(10.11)
Therefore, the linear inverse problem (10.4) can have an infinite number of equivalent solutions. All these nonradiating currents form a so-called null-space of the linear inverse problem (10.4). In principle, we can overcome this difficulty using a regularization method, which restricts the class of the inverse excess currents j " in equation (10.4) to physically meaningful solutions only. We will discuss an approach to the solution of this problem below, considering a quasi-linear method. 10.1.2 Born inversion The most common approach to the solution of the electromagnetic inverse problem is based on linearization of the integral equations (10.2) and (10.3) using a Born approximation: E^ (r,)
^
G^; (A5E^) = / / / G^; (r,- | r) A 5 (r) E^ (r) dv,
(10.12)
H" (r^)
«
GH ( A 5 E ^ ) = jjj
(10.13)
GH (r,- | r) A 5 (r) E^ (r) dv.
Note that comparing formulae (10.12), (10.13) and (9.51), (9.52), we see that the expressions in the right hand sides of the Born approximations can be represented as the Frechet derivative operators calculated for the background conductivity a^ and the anomalous conductivity Aa *: G^; (A?E^) - F ^ ia^A^),
(10.14)
*We call these expressions the Frechet differentials as well (see Chapter 9 and Appendix D).
291
Linear inversion methods GH ( A ? E ^ ) = FH (5,A5^)
(10.15)
The argument in the expressions for the Prechet differentials, FE^ni^bi^^), consists of two parts. The first part, 5^, is the background conductivity distribution, for which we calculate the forward modeling operator variation; the second part, Aa, is the anomalous conductivity, which plays the role of the background conductivity variation. We will use below the following simplified notations for the Frechet differentials F^,H(5,,A5) = F^^^^(Aa), where the upper subscript "6" denotes that the differentials are calculated for the background conductivity. Substituting the Born approximation (10.14) into formula (10.2), we arrive at the linearized form of the forward modeling operator E" = F ^ ( A 5 ) .
(10.16)
In a solution of the inverse problem we assume that the anomalous field, E", and the background conductivity 5^, are given. The goal is to find the anomalous conductivity distribution, Aa. In this case, formula (10.16) has to be treated as a linear equation with respect to Aa. The linear inverse problem (10.16) is ill-posed. Therefore, we should solve this equation by a regularization method, imposing additional conditions on the class of the inverse models with corresponding stabilizing functionals. We introduce a complex Hilbert space D of data (anomalous electric fields d = E"^ given on the surface of observations S) with the metric (f, d ) ^ = ff
f (rO • d* (rO ds' f, d G D,
(10.17)
and a complex Hilbert space M of models (conductivities a) with the metric (?(!), 5(2))^ = fff
3^(1) (r) . 5(2)* ^r)dv; a^'\a^^^ G M.
(10.18)
According to the conventional Tikhonov regularization method, we substitute for the solution of the linear inverse problem (10.16) a minimization of the corresponding parametric functional with, for example, a minimum norm stabilizer: P-=(F^^(Aa)-d, +a {WE (Aa-Aaapr),
F^^(A5)-d)^ WE {Aa-Aaapr))M
.
(10-19)
where Aaapr is some a priori model, and WE is a weighting operator. The minimization problem (10.19) can be solved by the conjugate gradient or re-weighted conjugate gradient methods introduced in Chapter 5.
Integral representations in electromagnetic inversion
292
10.1.3
Conductivity imaging by the Born
approximation
Before moving to the iterative Born inversion technique, we introduce a fast imaging technique based on a Born approximation. Let us recall formula (5.91) for an approximate regularized solution of the linear inverse problem. In the case of equation (10.16), this formula takes the form Aa^kiW^WE)"
(10.20)
F'^{E^)
where F ^ is an adjoint Frechet derivative operator, WE is a weighting operator, and the scalar coefficient A; is a solution of the minimization problem: ||A:F^^ {{W^WE)-'
F^ ( E ^ ) ) - E^|f = min.
(10.21)
Let us find, first, an explicit form of the adjoint Frechet derivative operator (F^(A?),f)^ = (A5,F^(f))^.
(10.22)
Using the definitions (10.17) and (10.18) of inner products and the expressions (10.14) and (9.37) for a forward operator, we can rewrite formula (10.22) as
{F»E (A5) ,i)^ = JJGE
-IL J J JD
KJ
GE
Js
(A5E*) • f* (r') ds'
(r' I r) • A ? (r) E** (r) dv r (r') ds'
G £ ; ( r ' | r ) . E ^ ( r ) ] ' - f ( r ' ) d 5 ' | dv = {Aa,F'^ {{)) ^ . (10.23)
Prom the last formula we have ACT,
Jib*
'''-//.
Gfi (r'I r) • E^ (r)
•t{r')ds'
= 0.
(10.24)
M
Equation (10.24) should hold for any ACT, for example for A ? ' = F^e*
<''-//.
GE
(r' I r) • E" (r) •{(r')ds'.
Therefore, F'B* «
- / / .
G^ (r'I r) . E^ (r)
^ {{r') ds^
0. M
From the last formula we conclude that the adjoint Frechet derivative operator F ^ is given by the formula:
'Bii) = Jl[GE{T'\r).E'{v)
•UT')ds'
293
Linear inversion methods
(10.25)
= E^*(r).y|Gl(r'|r).f(rV^',
where a tilde denotes the operation of transposition. In a similar way we can obtain the formula for the adjoint Frechet derivative operator F ^ : F ^ (f) = E^* (r) • JJ^ G^ (r' | r) • f (r') ds'.
(10.26)
Substituting expression (10.25) into formula (10.20), we finally arrive at the following imaging formula: A 5 (r) ^k
{W^EWE)''
E^* (r) • ff
G^
{r' | r) • E^ (r')
ds'.
(10.27)
In a similar way we can derive an imaging formula based on anomalous magnetic field observations: A 5 (r) ^k {W^HWHy' E'* (r) • / / G ^ (r' | r) • H^ (r^ ds'.
(10.28)
The weighting operators WE and WH for electric and magnetic forward operators are linear operators of the multiplication of conductivity Aa by the functions WE (r) and Wjj (r) equal to the square root of the integrated sensitivity (see formula (3.78) in Chapter 3): "^E ~ y SE, and wjj = ySn(10.29) The integrated sensitivity, SE, in accordance with (9.59), is calculated by the formula SE{V)
'IL
GE
(r' I r) . E^ (r) ds'
GE
{r' I r) • E^ (r)
(10.30)
Formula (10.30) should be treated as the integrated sensitivity of the anomalous electric field to the local conductivity anomaly located at a point with the radius vector r. Similarly, we can find the integrated sensitivity of the anomalous magnetic field to the local conductivity anomaly, using formula (9.60):
SH
(r) = JJJ
\GH
(r' | r) • E" ( r ) | ' ds' =
\\GH
(r' | r) • E" (r
(10.31)
The weighting functions, according to (10.29), are equal to WE (r) = ^/ \\GE (r' I r) • E^ (r)||^, WH (r) = ^ | | G ^ (r' | r) • E" (r)||^.
(10.32)
294
Integral representations in electromagnetic inversion
Substituting formulae (10.32) into (10.27) and (10.28), we finally find: ACT (r)
E''{r)'!!sGAr'\r)-E^{r')ds^
^k-
GE{r'
(10.33)
|r)-E^(r)
or ACT (r)
«fc
G^(r'lr).EMr)
(10.34) D
Formulae (10.33) and (10.34) have a clear physical interpretation. Let us introduce a unit conductance EQ = 1 S. According to the reciprocity theorem (see Chapter 8), the integral term in equation (10.33) can be treated as the complex conjugate of the electric field E ^ (divided by the unit conductance EQ), E^(r)
//.
GE
(r' I r) • E" (r') ds' (1035)
= 1 1 GE (r I r') • EQE"* (r') ds'/Eo = J J GE (r | r') • j | (r') ds'/Eo,
generated at the point r by the fictitious electric current, distributed over the surface of observations S with the surface density jf, equal to the complex conjugate anomalous field multiplied by SQ:
j|(rO:=EoE-(0.
(10.36)
We have introduced the unit conductance EQ in both equations (10.35) and (10.36) to keep the correct dimensions of the fictitious electric current j f and the normalized electric field E ^ ( r ) . The approximate anomalous conductivity in formula (10.33) is obtained as a scalar product of the auxiliary field E^* (r) with the complex conjugate background field at the point r, normalized by the magnitudes of the background field and the norm of the corresponding Green's tensor at the same point: ACT (r)
^ kAa^ ( r ) , and Aa^ (r)
E^* (r) • E^* (r)
G^ (r' I r) . E^ (r)
(10.37)
where the scalar coefficient k is the solution of the minimization problem ||/cF5j(Aa'B(r))-E«||' = min. Formula (10.34) has a similar interpretation, only in this case we should introduce by reciprocity the magnetic field H ^ , H ^ (r) = J I ^H (r' I r) • H«* (r') ds' = Jf
GH
(r | r') • H"* (r') ds',
295
Linear inversion methods
generated at the point r by the fictitious electric current, distributed over the surface of observations S with the surface density j ; ^ , determined by the expression^ j^(r')=H-(r'). As a result, we arrive at the following formula for the anomalous conductivity Aa (r) « kAa'ff (r), and Aa'^ (r) = j p :
^-^
- ^ '
G^ (r'I r) • E" (r)
(10-38)
llD
where the scalar coefficient k is the solution of the minimization problem \\kF% {Aa^ (r)) - H"||^ = min. Note that in the definition of the auxihary fields E^* (r) = E^* (r,cj) and H^* (r) = H^* (r,6j) we use the complex conjugate Green's tensors. It can be demonstrated that the transition to the complex conjugate function in frequency domain corresponds to changing a sign in front of time variable t in the time domain. Thus, using the convolution theorem (Arfken and Weber, 1995, p. 863-865), we can write the expressions for the auxiliary electric and magnetic fields in time domain as follows:
//./ and
G£;(r, -t\ 1
r\ - t ^ E " (r^t) dtds' = E ^ (r,^^ ,
(10.39)
r+oo
— / H^*(r,u;)e-^"*'da; 27r 7_oo
ILL
+00
5.'
_
G//(r, -^1 r', -t')W
(r',t) dtds' = H ^ (r,tO .
(10.40)
The minus sign in front of time variable t in the expression for the Green's tensors indicates that the electromagnetic field propagation occurs backward in time. Therefore, the fields E ^ (r,t') and H ^ (r,^0 are referred to as the "back-propagated" or "back-scattering" fields. We will discuss the properties of these fields in more details in Chapter 11 on electromagnetic field migration. "•"Note that, according to the first Maxwell's equation, the surface density of the electric current has the dimensions of magnetic field.
296
Integral representations in electromagnetic inversion
10.1.4 Iterative Born inversions Other techniques available for linearization of the original electromagnetic integral equation are based on transforming this nonlinear problem to a sequence of linear problems. Two common iterative schemes are the Born iterative and distorted-Born iterative methods (Chew, 1990). In the Born iterative method the original integral equation for the electric field E is written twice: first as a domain equation (10.41) with respect to the electric field inside the anomalous domain D, and second, as a data equation (10.42) connecting the observed anomalous field (electric data d£;= E"= E — E^, or magnetic data d/f = H " = H — H^) with the electric field within the anomalous domain E = G^; (A?E) + E ^
(10.41)
d^; = G^; ( A 5 E ) , d ^ = G// ( A ? E ) ,
(10.42)
where G^; and GH are the electric and magnetic Green's operators determined by formula (9.37). Note that each equation, (10.41) and (10.42), is bi-linear because of the product of two unknowns, A a and E. However, if we specify one of the unknowns, the equations become linear. The iterative Born inversion is based on subsequently determining Aa from equation (10.42) for specified E, and then updating E from equation (10.41) for predetermined A5, etc. Within the framework of this method the Green's tensors G^; and G//, and the background field E^ stay unchanged. In the distorted-Born iterative method, the background field and the Green's tensors are updated after each iteration according to the updated conductivity Aa. However, this approach is difficult to implement in practice, because it requires calculation of the Green's tensors for inhomogeneous media, which is an extremely time consuming problem in itself. We can write the basic equations for the iterative Born inversion using the matrix notations introduced in Chapter 9. For example, the domain equation (10.41) takes the form eD = GD^eu + e^D. (10-43) and the data equations (10.42) are cast as e = GE^eu
+ e^ h - G^S^eD -h h ^
(10.44)
The critical problem of both the Born iterative and the distorted-Born iterative methods is the convergence of the algorithms. To ensure convergence we can use a modified Green's operator G"^, which has the contraction property ||G^(x)|| < ||x||.
(10.45)
The modified Green's operator is introduced by formula (9.135), which we reproduce here: G ^ (x) - y/^bGE (2 V ^ x ) + x. (10.46)
297
Nonlinear inversion
Using this operator, we can apply formula (9.137), aE" + 6E^ = a (E" + E^) + (6 - a) E^ = aE - v ^ E ^ = G^ [b (E" + E^)] , and cast the domain equation (10.41) in the form
where
aE = Gg[6E] + E^
(10.47)
_ _ 2cr6 + Acr , ACT a = ^ - = - , 6 = - — .
/ .ON (10.48)
The iterative Born inversion based on the modified Green's operator involves subsequently determining Aa from equation (10.42) for specified E, and then updating E from equation (10.47) for predetermined A?, etc. 10.2 10.2.1
Nonlinear inversion Formulation of the nonlinear inverse problem
Consider a general nonlinear inverse problem for anomalous electric or magnetic fields: dE = AE (Aa) = GE ( A 5 E ) ,
(10.49)
d// = AH (Aa) = G// ( A 5 E ) , (10.50) where AEJI {Aa) are nonlinear forward modeling operators, given by the formulae: AE,H
/ / / fv.
(Aa) =
GE,H
(A5E)
G^; (r' I r) • A 5 [E^ (r) -f E" (r)] dv.
(10.51)
The nonlinearity of the operators AE^H (r' | Aa) with respect to the anomalous conductivity Aa comes from the fact that the anomalous electric field E^ is also a function of A5. According to the conventional Tikhonov regularization method, we substitute for the solution of the inverse problems (10.49) or (10.50) a minimization of the corresponding parametric functional with, for example, minimum norm stabilizer: F " {Aa) = (A^,^(A5) - d, A ^ , H ( A 5 ) -
-fa {W {Aa-Aaapr),
W {Aa-Aaapr))M
d)^
'
(10-52)
where W is the corresponding weighting operator. The minimization problem (10.52) can be solved by the conjugate gradient or re-weighted conjugate gradient methods introduced in Chapter 5.
298
Integral representations in electromagnetic inversion
Let us describe, for example, the algorithm based on the regularized conjugate gradient method (5.92), which we reproduce here with small modifications: ^E,Hn = A£;,//(A5n) " d, l^=l-{Aan)
(a)
=Fl^^RE,Hn^aW'W{Aan-A^apr).
Aan+i=A^n-kt
{b)
id)
^^^'^^^
k^ is determined from the minimization problem P - (A5n+i) = ^ " (A5n - Kt)
= min,
(e)
where, as usual, we denote by Aa^ the corresponding anomalous conductivity distribution at the n-th iteration. We can see that practical implementation of this algorithm requires computing the adjoint of the Frechet derivative F ^ / / ^ of the corresponding forward modeling operators AE,H for each iteration n. 10.2.2
Frechet derivative calculation
In Chapter 9 we developed the expressions for the Frechet differentials of the forward modeling electromagnetic operator (9.51) and (9.52), which we reproduce here for convenience: Fs(a,.55) = F | (r' | 6d)
^ISL^'"'
r)-55(r)E(r)dt;,
(10.54)
and
/ / / .D
&H (r' I r) • 65 (r) E (r) dv.
(10.55)
In the last formulae a — a^-f A a is a conductivity distribution, for which we calculate the forward modeling operator variation; 55 is the corresponding variation of the conductivity 5, which is^obviously equal to the variation of the anomalous conductivity, (55 = 5A5. Tensors G^ j^ are electric and magnetic Green's tensors calculated for the given conductivity 5. Vector E in expressions (10.54) and (10.55) represents the total electric field, E = E^4-E^ for the given conductivity a.
299
Nonlinear inversion
Note that in the RCG algorithm (10.53) the expression F^ HrJ^E,Hn denotes the result of an application of the adjoint Frechet derivative operator to the corresponding electric or magnetic residual field IlE,Hn = A£;,//(A5n) — dE,H on the n-th iteration. The expressions for the adjoint Frechet derivative operators are given by formulae (10.25) and (10.26). Based on these formulae, we can write: F^^^R^
= F | - ^ (r I Rn) = J J [GE,H {T' | r) • E , (r)] * • R^;,^, (r') ds' =
E;
(r) . 11^ G ^ , ^ , (r' I r) • R ^ , ^ . (r') d5^
(10.56)
where E^ (r) = E^ (r) -f E^ (r) is the electric field, and GE,Hn are the electric or magnetic Green's tensors, computed on n-th iteration for the conductivity distribution, an (r) = at (r) -h Aa^ (r); a tilde denotes the operation of transposition. According to the reciprocity theorem (see Chapter 8), the integral term in the last formula can be treated as the complex conjugate of the electric field, E ^ (r) (in the case of the electric Green's tensor G^; and the electric residual field HEU), or the complex conjugate of the magnetic field H ^ (r) (in the case of the magnetic Green's tensor G// and the magnetic residual field R//n),
Ef (r)
G;,„
H f (r)=
G'^Ar\r')-RHn{r')ds'
(r I r') •
REU
(r') ds' =
IL
GEn(r|r')-Rk(r')rfs' {v\v')-W,,Jr')ds' (10.57)
Here, the fields E ^ (r) and H ^ (r), E«(r) = | ^ G s „ ( r | r ' ) - R k ( r ' ) r f 5 ' , H « (r) = jj^ GHU (r | r') • R ^ „ (r') ds\
(10.58)
are generated at the point r inside the volume V by the fictitious electric currents distributed along the surface of observation S with the surface density j f ^ (r') or '^^Hn (^0 ) determined by the complex conjugate residual fields:
Rk(0-jL(rO;Rk(0-jL(rr
(10.59)
Therefore, from (10.56) and (10.57) we find th^t the result of applying the adjoint Frechet operator to the residual field is just equal to the scalar product of the complex conjugate of the electric field E^, computed at the n-th iteration, with
300
Integral representations in electromagnetic inversion
the corresponding auxiliary electric E^* or magnetic H^* fields due to the reciprocal sources on the observation surface: F|"* (r 1
RB„)
=
E;
(r) • E«- ( r ) ,
F r ( r | R H „ ) = E ; ( r ) . H f (r).
(10.60)
Thus, we can conclude, that the calculation of the terms F^^Hn^E,Hn requires just one additional forward modeling solution on each iteration for the auxiliary electric field E ^ or magnetic field H ^ due to the reciprocal sources. In the practical application of the solution of the nonlinear inverse problem described above significant difficulties are experienced, however. The most critical problem can be described as follows. The numerical calculation of the reciprocal fields E ^ and H ^ on the n-th iteration requires, according to formula (10.58), the knowledge of the corresponding Green's tensors GE^HU for the conductivity a^, obtained on the previous iteration. In general cases, the function 5^ (r) cannot be represented by simple horizontally or cylindrically layered models, for which it is relatively easy to compute the Green's tensors. The computation of the Green's tensors for an arbitrary conductivity distribution becomes a difficult numerical problem itself, similar to distorted-Born iterative method. One practical way to overcome this difficulty is to abandon the integral equation approach for nonlinear inverse problems and to consider the finite difference or finite element methods of forward modeling. We will present this approach in Chapter 12. Another way is based on using approximate, but accurate enough, quasi-linear and quasi-analytical approximations for forward modeling, introduced in Chapter 8. We will discuss these techniques in the following sections of this chapter. 10.3
Quasi-linear inversion
10.3.1 Principles of quasi-linear inversion One approach, which can be used in 3-D inversion, is based on principles of quasilinear (QL) approximation. According to QL approximation, the anomalous electric field E" inside the inhomogeneous domain D is linearly proportional to the background field E^ through some electrical refiectivity tensor A: E" - A • E ^
(10.61)
Substituting (10.61) into (10.42), we obtain d^,H - GE,H ( A ? ( I + A ) . E^) = GE^H (m • E^),
(10.62)
where m is a material property tensor: m=Aa
(^I+A)
.
(10.63)
Quasi-linear
301
inversion
Note that equation (10.62) is linear with respect to m. The reflectivity tensor A can be determined from the following linear equation inside the inhomogeneous domain D, as long as we know m : A-E^ = G £ ; ( m - E ^ ) .
(10.64)
After determining m and A it is possible to evaluate the anomalous conductivity distribution A 5 from equation (10.63). This inversion scheme reduces the original nonlinear inverse problem to three linear inverse problems: the first one (the quasiBorn inversion) for the tensor m, the second one for the tensor A, and the third one (correction of the result of the quasi-Born inversion) for the conductivity Aa. This method is called a quasi-linear (QL) inversion^. 10.3.2 Quasi-linear inversion in matrix notations Note that, in the case of a scalar electrical reflectivity coefficient A, we can rewrite equation (10.62) according to (9.229) and (9.230), in matrix notation: (10.65) Here m is the modified conductivity vector m =^ [mi, 7712, ...,m7v]^, related to the electrical reflectivity coefficient and Aa by formula (9.228), which we reproduce here m, = Aa, (1 + A,), 1 = 1,2, ....A^. (10.66) The last formula can be written in matrix notations as (10.67)
I+A
m
where vector cr of the order N represents the anomalous conductivity distribution, Aa (r) , on some grid within domain D a = [Acri,Aa2,...,Aaiv]^, and A is a diagonal N x N matrix^ formed by electrical reflectivity coefficients: Ai •••
0
A =
(10.68) 0
AA
*The method of quasi-linear inversion has been developed in collaboration with S. Fang (Zhdanov and Fang, 1996b, 1999; Zhdanov et al., 2000). Hn this section we use a "hat" sign above the matrix to distinguish between the column vectors and matrices.
302
Integral representations in electromagnetic inversion
Equation (10.66) can be treated as a modification of Ohm's law, because it comes directly from the differential Ohm's law for the quasi-linear approximation of the anomalous field: j ^ - A a E =Aa (E^+E") = A a [l-f A (r)] E ^
(10.69)
Thus, the discrete forward modehng problem for the electromagnetic field based on the QL approximation can be expressed by the following matrix operations: d = A m,
(10.70)
where A stands for the electric or magnetic matrices, A^; = GE^^Z) or d stands for electric d^; or magnetic dn data, respectively. Similarly, we deduce from (10.64) e^A = Gi^e^m,
AH
= G^/eJ^, and
(10.71)
where A =[Ai, A2,..., A^v]^ is a vector formed by the electrical reflectivity coefficients in different cells of the grid; ej^ is a sparse tri-diagonal 3A^ x A^ matrix containing the X, y and z components of the background electric field at the centers of the cells of the anomalous domain D, according to formula (9.219); and the 3N x 3A^ matrix G£) is a discrete analog of the corresponding electric Green's tensor acting inside the domain D (formula (9.194)). Thus, the solution of the inverse problem is reduced to inversion of the linear system (10.65) with respect to m and then to solving (10.71) with respect to A : X={^^y'GD^D^-
(10.72)
After that we obtain a using (10.67): (T= I-fA
-1
m.
(10.73)
We apply the Tikhonov regularization theory to solve the linear system (10.65). The regularized conjugate gradient method for solving linear inverse problem equations Note that the solution of equation (10.62) and, correspondingly, equation (10.65), can be nonunique in a general case, since there may exist m (r) distributions that produce a zero external field, similar to the existence of the current distributions producing zero external field demonstrated in the beginning of this chapter. However, according to the uniqueness theorem by Gusarov (1981, see Chapter 1), the excess conductivity A 5 (r) can be found uniquely in the class of piecewise-analytic functions. Therefore,
Quasi-linear inversion
303
to make our solution unique, we have to develop a unified approach for simultaneously finding m, A, and Aa. We use the method of constrained inversion developed by Zhdanov and Chernyak (1987). A similar approach to 2-D inverse scattering problem was discussed also by Kleinman and van den Berg (1993). It is based on introducing the Tikhonov parametric functional P"(m) = 0(m) + a 5(m), (10.74) where the misfit functional is specified as (m)
W , ( A m - d ) ) \' + \ Iw
^* m
m — I+A cr )\
(10.75)
and the stabilizer is selected to be equal to S'(m) =11
Wm{m-nia
(10.76)
The a priori model niapr is some reference model, selected on the basis of all available geological and geophysical information about the area under investigation. The scalar multiplier a is a regularization parameter. As one can see from (10.75), the misfit functional takes care not only of the solution of the equation (10.65) but, simultaneously, of the fulfilling of the modified Ohm's law (10.69). W^ and W^^ are data and model weighting diagonal matrices respectively. Now the inversion is reduced to the solution of the minimization problem for the parametric functional: P^{m) = min.
(10.77)
The solution of this problem is similar to one outlined in Chapter 5. However, there are some important differences related to the construction of the specific misfit functional (10.75). That is why the derivation of the corresponding CG algorithm is presented here. One can calculate the first variation of the parametric functional under the assumption that cr and A are, temporarily, constants: ^P"(m) = 2 R e { « m * [ A * ( A m - d ) m
I+A
A +a\ m — mapr ']},
where asterisk * denotes transposed complex conjugate matrices. Let us select Sui as 6in = -kr{m.)
,
0 < A: < oo.
(10.78)
Integral representations in electromagnetic inversion
304
where P ( m ) = A * ( A m - d ) + f m - [ I + A ] (T) + a (m - niapr),
(10.79)
and k is the length of the iteration step. This selection makes 5P"(m) = -2fc" R e { P * ( m ) P ( m ) } < 0. That means that the parametric functional is reduced if we apply perturbation (10.78) to the model parameters. In order to accelerate the convergence, we apply the regularized conjugate gradient method described by the following iteration process: nin+i = irin + ^m^ = nin — A;nl"(mn), where the "directions" of ascent P(mn) are selected according to the algorithm described below. In the first step, we use the "direction" of regularized steepest ascent determined by equation (10.79): P(mo) = r ( m o ) . Therefore, the first iteration is computed by the formula mi = mo + Smo (10.80) = mo~ ko A*(Amapr - d ) ) where m,apr
T+A^apr
' apri
and (Tapr and Xapr are the parameters of the reference model. The second iteration is m2 = m i -f- (5mi = m i — ki P ( m i ) ,
(10.81)
where P(mi)=P(mJ+/3iP(mo) and l"(mi) (10.82) = A * ( A m i - d ) + (mi - fl+Ai
''apr] + a ( m i -
mapr).
In the (n + l ) t h step m„+i = m„ + ^m„ = m„ - fc„ P ( m „
(10.83)
305
Quasi-linear inversion and
_ _ P(mJ=r(mJ+/3„P(m„_i),
where r{mn)
= A*(Amn - d ) + frrin - [ I + A J cr„-ij + a (m„ - niapr),
(10.84)
and asterisk * denotes transposed complex conjugate matrices. The reflectivity vector A^^ is determined from the vector m^ using equation (10.72): \n={^Dy'Gne'j,inn.
(10.85)
The coefficient kn can be determined based on the corresponding hne search. The conjugate gradient method requires that the vectors P(in^) introduced above will be mutually conjugate (see Chapter 5). This requirement is fulfilled if the coefficients /3^ are determined by formula (5.106), which we reproduce here:
Pn
|l"(m,) |l"(mn_ I)
12-
Using equations (10.83) and (10.84), we can obtain m iteratively. One can use the adaptive regularized conjugate gradient method, introduced in Chapter 5, P-*-(m^+i) = P-^Hmn+i) +/^n+il""(mn), where an are the subsequent values of the regularization parameter. The anomalous conductivity on the n-th iteration can be found using equation (10.73). Note that anomalous conductivity at every iteration has to satisfy the condition A(j,(r) > -(76 ( r ) , (10.86) because the total electrical conductivity has to be positive. The conventional way to satisfy this condition in the inverse solution is to introduce a new model parameter, cr^ equal to the logarithm of total conductivity: a' (r) - In {at (r) -f A a (r)).
(10.87)
The anomalous conductivity can be found from the simple inversion of (10.87): ACT (r) - exp a' (r) - a^ ( r ) .
(10.88)
Obviously, the new model parameter a' (r) can never produce negative total conductivity. Thus, we express the anomalous conductivity at the n-th iteration by parameter cr^ (r), determined at the same iteration: AcTn (r) = exp a'^ (r) - a^ ( r ) .
(10.89)
306
Integral representations in electromagnetic inversion
This parameter can be found from equation (10.73), which should hold for any frequency. In reality, of course, it holds only approximately. Therefore parameter G'^ (r) can be found by using the method of least squares to solve equation (10.73) over a number of frequencies: Wrun (r,u;) - [expa^ (r) - a^ (r)] [1 + A, (r,c^)]||^,(^) - min .
(10.90)
We solve the minimization problem (10.90) for every point r within the cells of the grid, introduced in domain D, and calculate the corresponding anomalous conductivity using expression (10.89). The iterative process is terminated when the misfit reaches the given level of noise in the data. 10.3.3 Localized quasi-linear inversion The quasi-linear inversion, introduced above, cannot be used for interpretation of multi-transmitter data, because both the reflectivity tensor A and the material property tensor m depend on the illuminating background electromagnetic field. However, in many geophysical applications, for example, in airborne EM and in well-logging, the data are collected with moving transmitters. In this case one can build an effective inversion scheme based on the localized quasi-linear approximation, introduced in Chapter 9, which is source independent. Following Zhdanov and Tartaras (2002), we introduce a new tensor function: fhi(r) = A5(r)(l + Ai(r)),
(10.91)
which we call a localized material property tensor. Note that this tensor function is independent of the transmitter position, because A^ does not depend on the background field. This is the main diff"erence between the localized QL inversion and the original QL inversion (Zhdanov and Fang, 1999). In the case of a scalar reflectivity tensor, we introduce a scalar parameter rriL (r) :
JUL (r) = A 5 (r) (1 + A^ (r)).
(10.92)
We assume now that the anomalous parts of the electric, E" (r^), and/or magnetic, H^ (r^), flelds (generated by a transmitter with one or different positions) are measured in a number of observation^points, r^. Using LQL approximation for the observed fields, d, we arrive at the following equation, similar to (10.62): d-G4m^(r)E^(r)),
(10.93)
which is linear with respect to the localized material property function TUL ( r ) . In the last equation d stands for the electric or magnetic field, E or H, and G^ denotes the corresponding operator G ^ or G//. The main difference between equations (10.93) and (10.62) is that in the framework of the LQL approximation the material property function rriL (r) does not depend on the illuminating source.
307
Quasi-linear inversion
We can solve the linear equation (10.93) with respect to TUL ( r ) , which is source independent. A scalar reflectivity coefficient A^ (r) is determined, based on the condition: \\XL (r) I - G^; {rriL (r)) - min. (10.94) 11
\\L2{D)
Knowing X^ (r) and m^ (r), we can find ACT (r) from equation (10.92), as in the QL inversion scheme. Note that equation (10.92) should hold for any frequency, because the electrical reflectivity and the material property parameters are functions of frequency as well: A^, = AL (r,a;), mi = rriL (r,^) • In reality, of course, it holds only approximately. Therefore the conductivity ACT (r) can be found by using the least squares method to solve equation (10.92) with respect to the logarithm of the total conductivity a' (r), similar to equation (10.90): ||mz. (r,a;) - [exp a' (r) - a^ (r)] [1 -f A^ (r,c^)] H^^^^^ = min,
(10.95)
and ACT (r)
= exp a (r) — a^ (r).
This inversion scheme can be used for a multi-source technique, because A/, and TRL are source independent. It reduces the original nonlinear inverse problem to three linear inverse problems: the first (the quasi-Born inversion) for the parameter m/,, the second for the parameter A/,, and the third (correction of the result of the quasi-Born inversion) for the conductivity Aa. Note that in the discrete case we can rewrite equation (10.93) in matrix notation: d = GniL. (10.96) Here mi is the column vector of the localized material property function TTIL^ d is the column vector of the field data, and G is the matrix representation of the corresponding linear operator G^; or G//. The solution of the inverse problem is reduced to inversion of the linear system (10.96) with respect to m^ and then to computation of XL using condition (10.94). After that we find Aa as a least squares solution of the optimization problem (10.95). We can use the re-weighted regularized conjugate gradient method, introduced in Chapter 5, for solving the system of the linear equations (10.96). One of the most important possible applications of the LQL inversion technique is the interpretation of frequency-domain helicopter-borne data. This type of airborne survey is used extensively in mining exploration. We use the integral equation code SYSEM (Xiong, 1992) to simulate such a survey over a conductive (20 ohm-m) dipping dike located in a resistive (500 ohm-m) half-space. Five lines were flown over the target at an altitude of 30 m and at a distance of 25 m from each other. A schematic plan view of the survey is shown in Fig. (10-1). The moving transmitter-receiver system was a pair of vertical magnetic dipoles (simulating a coplanar pair of horizontal coils) and a pair of horizontal magnetic
Integral representations in electromagnetic inversion
308
flight lines
inverted area
Figure 10-1
A schematic plan view of the helicopter-born EM survey.
dipoles (simulating a coaxial pair of vertical coils) with 8 m horizontal separation. The yy (coaxial) and zz (coplanar) components of the anomalous magnetic field were measured every 15 m along the lines (70 observation points in each line). Two frequencies were used: 900 Hz and 7.2 KHz. We added 5% random noise to the anomalous magnetic field and then inverted it using the LQL inversion method with a scalar reflectivity tensor and focusing inverse imaging (Zhdanov and Tartaras, 2002; Portniaguine and Zhdanov, 1999a). The area of inversion, centered around the body, was 150 m x l 5 0 m x l 5 0 m, and was divided into 12 X 12 X 12 cells. Figure 10-2 shows the comparison of "observed" (synthetic) and predicted (for the resulting inverse model) data for the central profile passing above the centre of the dike. The predicted data fit the observed data very well (within the 5% noise level). Fig. 10-3 shows a 3-D view of the true model. Fig. 10-6 presents a 3-D image of the LQL inversion result. Only values of resistivity below a certain threshold have been plotted out of the inverted cube-shaped area. Both the location and the depth of the dike are determined very well. Its dip is also remarkably well defined. We can even distinguish some of the details of its shape. Its resistivity, however, is overestimated. These results demonstrate the stability of the method in the presence of noise (results of inversion of noise-free data, not shown here, are very similar).
Quasi-linear inversion
309
X10-"
X 10-'^
0
0 -5
0) DC
X ---10
-10
E
-15 on
-500
,
-300
— obs - - pred
v -100
0
100
300
Y — obs - - pred
\ -500
500
-300
-100
0
.
100
300
500
xlO
x10~
0
^ ^ S N-5 X
--.^ E
~ -10
-300
-100
0
100
Distance (m)
300
500
-500
1/ f/ -300
-100
0
— obs - - pred 100
Distance (m)
,
300
500
Figure 10-2 Comparison of "observed" data (solid lines) and predicted data for the resulting inverse model (dashed lines) for the central profile passing above the centre of the dike.
Integral representations in electromagnetic inversion
310
X(m)
Figure 10-3 3-D view of a true model of a conductive (20 ohm-m) dipping dike located in a resistive (500 ohm-m) half-space.
50 J .
100 J .
150 J -50 ^ s : ^ ^ ---"-"-/•'4. 0 ^X^'.'-"-".-"]50 ^ s L X(m)
Figure 10-4
Y(m)
3-D image of the LQL inversion result for a model of a dipping dike.
Quasi-analytical inversion
10.4
311
Quasi-analytical inversion
Quasi analytical approximations (9.90) and (9.91) provide another tool for fast and accurate electromagnetic inversion. This approach leads to a construction of the quasi-analytical (QA) expressions for the Prechet derivative operator of a forward problem, which simplifies dramatically the forward EM modeling and inversion for inhomogeneous geoelectrical structures^. 10.4-1 Frechet derivative calculation By introducing a perturbation of the anomalous conductivity 6/SS- (r) we can calculate the corresponding perturbation of the electric field ^E(rj) on the basis of equation (9.90): SA^ir)^, GE{V,\V)5E(r,) = E^(r) dv
1-^W
+
GE
(r, I r) •
where 69 (r) GE
^^MMl)E^(r)
dv,
(l-5(r))
(10.97)
^E^ (r) • E''* (r) Eb (r) . E*-* (r)
(r i r') • 6Aa (r') E" (r'
E*"* (r) Eft (r) • Eb* (r)
d,v'.
(10.98)
Substituting equation (10.98) into the second integral in (10.97) and changing the notation for the integration variables, r ^ r' and r ' ^ r, we obtain GE {VJ
I r)
(l-5(r))'
".
dv
.5A5 (r) K (r, | r) • E*" (r) dv,
(10.99)
where K(r,|r) Aa (rQ lD{l-9{r')f
GE{rjW){\GE{r'\r)-E'{v')
b* (r' tW\ E** Eft(r')-E''*(r')
dv'. (10.100)
Therefore 6E (vj) = JJJ^ 6Aa (r) F ^ (r,- | r) dv.
(10.101)
where the vector function F ^ (r^ | r) is the kernel of the integral Frechet derivative operator: FE
(r, I r)
l-5(r)
GB(r, | r ) + K ( r , |r)
F," (r).
(10.102)
^The sections on quasi-analytical (QA) inversion and magnetotelluric QA inversion were prepared in collaboration with G. Hursan.
Integral representations in electromagnetic inversion
312
In particular, considering the infinitely small domain of the conductivity perturbation, we arrive at the following formula for the Prechet derivative of the electric field: 5E(r,
aA5 (r)
= F^(r,|r)
The last formula provides an analytical expression for computing the Frechet derivative for the forward modeling operator. Note that, in this case, the amount of calculation for the forward modeling solution and for the Frechet derivative is equivalent to computing the Born approximation. 10.4-2 Inversion based on the quasi-analytical method In practice we usually solve forward and inverse problems in the space of discrete data and model parameters. For a numerical formulation of QA inversion we can use the matrix formula for QA approximation (9.241), reproduced here for convenience: d = A Q A (O-) = A [diag (l - Ca)]
a =AB(o-) cr,
(10.103)
where P^QA is a forward modeling operator, a stands for a set of the model parameters describing anomalous conductivity distribution, Aa, vector d is an electromagnetic data set, and A stands for electric or magnetic matrices", respectively: A^; = Q^E^D O^ ^E =
(10.104)
GE^D
Matrix C depends on the matrix of the background electric field ej^, introduced in Chapter 9 (formula (9.219)): C = (e^e'*)" e^*Gz^e^;
(10.105)
and the diagonal matrix, B(o-)= [ d i a g r i - C t r ) !
\
(10.106)
is determined by the vector of the model parameters cr, describing the anomalous conductivity distribution, Aa. Let us consider the derivation of the Frechet derivative matrix of the discrete forward operator (10.103). Noting that the model parameters are the anomalous conductivity values in the cells of the anomalous body, that matrix A is independent of the model parameters, and that B is a diagonal matrix, one can express the perturbation of the forward operator (10.103) with respect to the model parameters in the form 6d = A6 B(or) cri = A i%{(T)8(T + diag(o')5
[B(O-)1
}.
I'As in Chapter 9, we use a "hat" sign above the matrix to distinguish between the column vectors and matrices.
313
Quasi-analytical inversion
Since
6 [B(O-)1 = 6 [diag (l - Co-)] ~^ = [diag (l - Co-)] "^ 6 ^ 0 - = B2(cr) C Sa, we obtain M = A { B ( O - ) -f diag(o-)B2(o-) c } ^o- - F(o-) ^o-,
where F(o-) = A {B(O-) + diag(o-)B2((T) c }
(10.107)
is the Frechet derivative matrix. Expression (10.107) is a discrete analog of formula (10.102). Note that the terms depending on the model parameters are diagonal matrices. The remarkable fact is that the full matrices A and C in the discrete form of the QA approximation are independent of the anomalous conductivity. Therefore, formulae (10.103) and (10.107) are very efficient in iterative inversion, because these matrices have to be computed only once for the entire inverse process. Thus, after the initial computation of the full matrices A and C for the background model, the iterative updating of F(cr) is relatively inexpensive during the inversion process. Note that the Green's functions are computed using the Fast Hankel Transform (Xiong, 1992). The only term depending on the model parameters cr is the diagonal matrix diag ( I — Ccr) , which is easy to compute for a given C. These results make QA approximation a very powerful tool in inversion. The inverse problem (10.103) is usually ill-posed, i.e., the solution can be non-unique and unstable. The conventional way of solving ill-posed inverse problems, according to regularization theory (Chapter 2), is to minimize the Tikhonov parametric functional: P " (cr). = (/? (or) -h as (cr) , (10.108) where (p {cr) is a misfit functional between the theoretical values AQA (cr) and the observed data d, s {cr) is a stabilizing functional, and a is a regularization parameter. The optimal value of a is determined from the misfit condition ip{cT) = 6d,
(10.109)
where 6d is the noise level of the data. The minimization problem (10.108) can be solved using any gradient type technique, say, by the re-weighted conjugate gradient (RCG) method. The critical point of an inversion algorithm is the calculation of the Frechet derivative (sensitivity) operator F at every iteration of the CG method. The quasi-analytical solutions described above provide a very eff'ective and elegant way of computing the Frechet derivative (sensitivity matrix) directly, using formula (10.107).
314
Integral representations in electromagnetic inversion
Note again that numerical computations based on formula (10.107) are very fast and efficient, because the full matrices A and C are precomputed^ for the background model and are fixed; we update only the diagonal matrix B ( m ) on each iteration of the inverse process. 10.5
Magnetotelluric (MT) data inversion
In magnetotelluric (MT) and controlled source magnetotelluric (CSMT) methods the observed data are not usually recorded as pure electric or magnetic field components. In particular, the results of magnetotelluric measurements are the apparent resistivity and phase calculated from the mutually orthogonal electric and magnetic field components (Zhdanov and Keller, 1994). Let us denote an arbitrary horizontal electric field component by Ey and define H as the magnetic field component perpendicular to the direction of JB at a particular receiver site. Note that both E and H are complex numbers. The magnetotelluric impedance is defined as the ratio of E and H : Z = | .
(10.110)
Since the impedance does not depend on the actual strength of the source, it is a very practical parameter for natural electromagnetic field observations. The magnetotelluric apparent resistivity is calculated from Z, using (10.111)
p=—\Z\\
If the incident field is a plane-wave and the model is a homogeneous half-space, p is equal to the true resistivity, making it indicative of the resistivity structure of the subsurface. Another commonly measured parameter is the phase angle of the impedance:
0 = tan-If.
(10.112)
This quantity is less sensitive to the eff^ect of shallow subsurface anomalies, so it is particularly useful for the detection of deep structures. To obtain a simple relationship between the anomalous field components and the MT parameters, let us express the logarithm of Z by writing E = E^ -^ E^ and
Thus, I n Z = In (1 -F EyE^)
- In (l + HyH^)
where Z^ = E^/H^ is the background impedance.
-f l n Z ^
(10.114)
315
Magnetotelluric (MT) data inversion
From equation (10.111) we obtain \n\Z\
=\n (u;/ioP)'/'j = \ Hcjfio) -^l^^P-
(10.115)
The same relationship holds for the background impedance as well. Considering the identities In Z = In |Z| + z0 and In Z^ = In |Z^| -f i(l>t we find i Inp, + ia = In (1 + EyE')
- In (l + HyH')
,
(10.116)
where In p^ = In p — In p^ is the log anomalous apparent resistivity, and (pa — ^ ~ ^b is the anomalous phase. Knowing the background cross-section, these quantities can be easily obtained from the measured total apparent resistivities and phases. 10.5.1 Iterative Born inversion of magnetotelluric data Portniaguine and Zhdanov (1999b) suggested linearizing expressions \ii{l -{- E"^/E^) and ln(l -h H^/H^) by neglecting the higher order terms in the power series expansion of the logarithmic function: I n ( H - x ) ^x. (10.117) Thus, (10.116) becomes
1
-\np^
ipa
Tja
+ i
(10.118)
This is a linear expression between the observed MT parameters and the anomalous electric and magnetic field components. If we measure only the absolute value of apparent resistivity, but not phase, all we need to use is the real part of the logarithmic equation (10.118): In I P J = 2 ReiE'^/E^ - HyH^).
(10.119)
Note also that expression (10.119) for the log-anomalous apparent resistivity is an approximation. Experiments with synthetic and real data show that the maximum of log anomalous apparent resistivity is of the order of 0.2. For this value, linearization (10.118) holds with an accuracy of 0.01. The estimated noise in the 3-D inverse problem with respect to log-anomalous apparent resistivity is also on the order of 0.01. Thus, we can safely use the notion of log-anomalous apparent resistivity if its maximum value is less than 0.2. The inversion process consists of two major steps. First, we estimate normal apparent resistivity by fitting the log of observed apparent resistivity ln(p) for all stations and all frequencies with the log of apparent resistivity derived from the layered model In(p^). The residual of fitting is associated with log-anomalous apparent resistivity. As we can see from (10.118), log anomalous apparent resistivity In(p^) is a linear combination of anomalous and background fields. Further, we represent the anomalous field as a linear combination of responses from individual cells and solve the 3-D inverse problem using the method of focusing inversion.
Integral representations in electromagnetic inversion
316
We introduce a vector of the data d, which combines the values of loganomalous apparent resistivity at the receivers, for all frequencies. In matrix notation, formula (10.118) is cast as follows: iMT
\ l n ( p j + i 0 , = (e*)-ie« - {^y^\v\
(10.120)
where e^ and h^ are diagonal matrices containing the background electric and magnetic fields at the receivers: E '6,1 ^ e^ = diag(Eb) ^6,iV
ifM (10.121)
h^ = diag(Hb) = R
b,N
Matrix equations (10.44) connect the anomalous fields at the receivers to anomalous currents ae^ inside the domain: e^ = GEO-BJ^,
h"" =
Gno-ej).
(10.122)
We introduce sensitivity to the currents, matrix F^, as a linear combination of matrices Ge and G/^: Fj = ( e * ) - ' G e - ( h ^ ) - ' G h . (10.123) Using (10.120) and (10.123), we convert equations (10.122) into a single equation: d = Fj?eD.
(10.124)
We formulate the inverse problem with respect to a vector cr of order N composed of the anomalous conductivities of the cells: 0--[A5i,A52,...,A5iv]^. The following relationships hold: ? = diag(o-),
(10.125)
aeD= diag(e^)tT.
(10.126)
Using (10.126) and (10.124), we establish the equation of the forward problem: d - Fjdiag(e^)o-.
(10.127)
317
Magnetotelluric (MT) data inversion
Or, expressing BD via (10.43), as BD = (^T-Gz)?j
e^jj,
we may rewrite equation (10.127) as d = Fjdiag((T- G^diag(o-))-^e'^)o-.
(10.128)
Our goal is to find the parameters a given the data d, that is to solve the inverse problem. The inverse problem is ill-posed. To solve it, we use the Tikhonov regularization method and minimize the parametric functional with the appropriate stabilizer s (cr): ||d - F,diag((T - GDdiag(a-))~^e^)o-f + as (a) = min .
(10.129)
Note that problem (10.129) can be expressed in bilinear form, using (10.127): ||d - F^diag(e^)cr|p -f as (cr) = min, en = {i-
GDdmg{a))-'e'j,.
(10.130) (10.131)
The numerical solution of (10.130) and (10.131) follows the conventional logic of the iterative Born method. First, we solve (10.130) for a, using some approximation to multiplier e/j. Note that if e/) is known, problem (10.129) becomes linear. Second, we update eu, for given cr, using equation (10.131). It is very important to find a good initial approximation to the unknown multiplier ep. To derive this initial approximation, we can use, for example, a diagonalized form of expression (10.131) in which all off'-diagonal elements of the matrix are neglected, (10.132) eD= d i a g ( T - G^ay'e'^, which is easier to compute. If we consider an anomalous domain consisting of one cell, expression (10.132) gives the exact solution. It will also be exact if we consider multiple cells located far away from each other. Physically, this means that formula (10.132) provides the first approximation for the solution with no electromagnetic coupling between the currents in different cells. The subsequent iterations of the method properly take into account the EM coupling eflPect. 10.5.2 DQA approximation in magnetotelluric inverse problem The DQA approximation is particularly suitable for constructing massive 3-D magnetotelluric inversion schemes, because of the low cost and simplicity of the expressions for forward modeling. In this section I discuss the implementation of the DQA approximation in MT inversion, following the paper by Hursan and Zhdanov, 2001. The main advantage of the QA method over the iterative Born method is that now
Integral representations in electromagnetic inversion
318
we do not need to use the linearized approach to constructing the forward modeling operator for MT data. However, we still can organize the log anomalous apparent resistivities and anomalous phases into a data vector similar to (10.120). 1 d ^ ^ = ^ In(p^) + 20" = In [I + (e*)-^e"] - In 1 +
(h'Y'h^]
(10.133)
where vector I is a A^ x l column vector whose elements are all unity; e^ and h^ are diagonal matrices containing the background electric and magnetic fields at the receivers: W 6,1
^6,1
(10.134)
e^ Eb.N
H'b,N
and it is assumed that In is applied to each element of the corresponding vector. Substituting the QA approximations (9.238) into the anomalous field components in (10.133), we obtain the QA approximation for the data containing the log anomalous apparent resistivities and phases: iMT
In I + (e^)-^A£;B(m)m] - In [l + (h^)-^A//B(m)m] ,
(10.135)
where AE = Gj^e^, and AH = G/fe^, respectively and
B(m)= [diag(l-Cm)l~\
(10.136)
Thus, the MT inverse problem is reduced to the solution of the nonlinear matrix inverse problem (10.135). Now let us consider the derivation of the Prechet derivative matrix of the forward operator (10.135). Noting that the model parameters are the anomalous conductivity values in the cells of the anomalous body, and that matrices C, A^; and AH are independent of the model parameters, one can express the perturbation of the forward operator (10.135) with respect to the model parameters in the form Sd^T ^ | ( 9 * ) - i A s d i a g - i [ l + (e*)-iABB(m)m] ('h''V^AHdiag-i [l + (h''ViAffB(m)m] } 6 [B(m)m'
(10.137)
where diag [...] means a diagonal matrix formed by the elements of the vector [...]. Since 5 [B(m)l = S [diag ( l - C m ) ] diag ( l - C m ) ] ~ C 6m = B^{m) C 6m,
Magnetotelluric (MT) data inversion
319
we obtain 6 [B(m) ml = { B ( m ) + diag(m)B2(m) c \ 6m = D(m) 6m,
(10.138)
where D(m) = B(m) + diag(m)B2(m) C
(10.139)
is a diagonal matrix. Substituting expression (10.138) into (10.137), we find 6d^^ = {(e^)-iA^diag-i [ l + (e^)-iA^B(m)m] - fh^V^A/^diag-^ [iH- (h^V^A/^B(m)ml}D(m) 6m. From the last formula we find the MT Frechet derivative matrix F^^(m) = A^^(m)D(m),
(10.140)
where A^^(m) = (e^)-iA^diag-^ [ l + (e^)-^A^B(m)m
fh^V^A//diag-^ [l+ (h^V^AHB(m)ml
(10.141)
Note that the terms depending on the model parameters are diagonal matrices. The full matrices A^; and A// depend only on the background conductivity distribution. Therefore, after precomputing full matrices AE and AH for the background model, the iterative updating of F ^ ^ ( n i ) is relatively inexpensive during the inversion process. 10.5.3
Frechet derivative matrix with respect to the logarithm of the total conductivity
We have introduced above the formulation of a magnetotelluric inverse problem with respect to the anomalous conductivity vector, denoted by a. However, during the minimization we may obtain anomalous conductivity values such that the total conductivity becomes negative. Thus, we have to transform the anomalous conductivity into a new space of model parameters with the property that the total conductivity always remains positive. We have already introduced above the conventional way of solving this problem, which uses the logarithm of the total conductivity as a model parameter: 5^ = ln(cr6 4-cr). (10.142)
320
Integral representations in electromagnetic inversion
Obviously, this model parameter can never produce negative conductivity. The inverse transform is cr = exp{a) - at. (10.143) To calculate the Prechet derivative with respect to this new model parameter, let us rewrite the forward operator in terms of a: d{a) = d[{a{a)] = d[exp(5^) - cr^].
(10.144)
The variation of d with respect to m can be found using the chain rule: 6^d{a) = 6ad[{a{a)] = 6^d{a)6^a.
(10.145)
Since 6^d{a) = F(o-) = F[exp(5^) - 0-5],
(10.146)
where F is given by (10.107) and 6^(T = exp{a)6a,
(10.147)
6^d{a) = F\a)6a,
(10.148)
we obtain where F^(5^) = F[exp(5^) - at] exp(a)
(10.149)
is the Frechet derivative with respect to the logarithm of the total conductivity. 10.5.4 Regularized smooth and focusing inversion of MT data The MT inverse problem described by the matrix nonlinear equation (10.135) is an ill-posed problem. The solution of this problem requires the appHcation of the corresponding regularization methods (Tikhonov and Arsenin, 1977). We discussed in Chapter 2, that the traditional way to implement regularization in the solution of the inverse problem is based on a consideration of the class of inverse models with a smooth distribution of the model parameters. Within the framework of classical Tikhonov regularization, one can select a smooth solution by introducing the corresponding minimum norm, or "smoothing" stabilizing functiona l . This approach is widely used in geophysics and has proven to be a powerful tool for stable inversion of geophysical data. The traditional inversion algorithms providing smooth solutions for geoelectrical structures have difficulties, however, in describing the sharp geoelectrical boundaries between different geological formations. This problem arises, for example, in inversion for the local conductive target with sharp boundaries between the conductor and the resistive host rocks, which is a typical model in mining exploration. In these situations, it can be useful to search for the stable solution within the class of inverse models with sharp geoelectrical boundaries.
Magnetotelluric (MT) data inversion
321
The mathematical technique for solving this problem was described in Chapter 2. It is based on introducing a special type of stabilizing functional, the so-called minimum support or minimum gradient support functionals. We call this technique a focusing regularized inversion to distinguish it from the traditional smooth regularized inversion. We can use the re-weighted regularized conjugate gradient (RRCG) method to solve the nonlinear inverse problem (10.135). The basic principles of the RRCG method are outlined in Chapter 5. This method can incorporate both the smooth regularized inversion, which generates a smooth image of the inverted conductivity, and a focusing regularized inversion, producing a sharp focused image of the geoelectrical target.
10.5.5 Example of synthetic 3-D MT data inversion The synthetic MT data set is generated by the IE modehng code SYSEM (Xiong, 1992), based on the integral equation method, for a model of a dipping dike (Figure 10-5, left panel) of resistivity 10 ohm-m, submerged in a half-space of 100 ohm-m. The top of the dike is at depth 200 m, and its bottom is at depth 700 m beneath the surface. The dike consists of two separated parts. This model is excited by plane EM waves with four different frequencies: 1, 10, 100 and 1000 Hz. Following the traditional approach used in practical MT observations, we calculate the synthetic observed apparent resistivities and phases based on two off-diagonal elements of the magnetotelluric tensor at each observation point. The quantities Py^ (apparent resistivity) and (t)y^ (impedance phase) are assigned to the nominal TE mode, whereas p^y and (p^y are assigned to the nominal TM mode. Note, that this 2-D nomenclature is artificial and approximate in nature for 3-D structures. However, it is widely used in practical MT observations, and often only the quantities p^^, ^^j,, p^y and (f)^y are available for inversion. That is why we use the same approach in the model study. The apparent resistivity and phases of impedance were calculated at 195 receivers placed on the nodes of a square grid on the surface. The distance between the observation points is 100 m in the x and y directions. The volume of inversion is covered by a homogeneous mesh consisting of 15x13x8 cubic cells surrounding the anomalous structure to be inverted. Each cell has a dimension of 100 m in the x, y and z directions. The region of inversion is also shown in Figure 10-5 (left and right panels). The data vector consists of 780 simulated field components (195 sites for 4 frequencies). The synthetic data set is generated by a full integral equation method and it has been-contaminated by 3 percent random noise. The model parameters are the unknown anomalous conductivity values of each cell in the volume over which the inversion is carried out. It has been demonstrated by Wannamaker (1999) that the best results for elongated anomalies are obtained with TM mode. The anomalous body has a strike
322
Integral representations in electromagnetic inversion
True model
Anomalous body, XY, CSMT3D
Figure 10-5 A model of a clipping dike (left panel) and inversion results obtained by the iterative Born method with focusing (right panel).
True model
Anomalous body, XY, QAINVNL
Figure 10-6 A model of a dipping dike (left panel) and inversion results obtained by the iterative DQA method with focusing (right panel).
Magnetotelluric (MT) data inversion
323 KAYABE
41:57:29
•
«
« * . * • • * ******
41:57:10
*
'41:56:51
2
*
41:56:13
* • • * m* ** * • « * • * * . * • * • * *
•
* 140:53:17
500
Figure 10-7
*
*\**\*\*\* •»*.%». * V -• : * ' ,*
41:56:32
« 140:53:39
140:54:2
»
»
140:54:24
Longitude (East)
^ 140:54:47
1000
The location of the MT measurement sites in the Minamikayabe area.
along the y axis, so the TM mode corresponds to xy polarization {Ex and Hy). In the case of the TE mode, the primary electric field is polarized along the strike direction, and therefore, it is less sensitive to the dike boundaries parallel to the strike. In the case of the TM mode, the primary electric field is directed across the strike, and it becomes more sensitive to the dike boundaries. Figures 10-5 - 10-6 present the results of inversion for an xy polarized field using, respectively, the iterative Born and the diagonalized quasi-analytical (DQA) inversion methods, with focusing. Both methods produce good results. Note that inversion of theoretical MT data takes just few minutes on a personal computer. A remarkable fact is that it is possible for this polarization to separate the upper and lower parts of the dike. The position and shape of the dike are also reconstructed quite well. However, the resistivity of the dike is not recovered correctly, which is a common problem in nonlinear inversion. The first method (iterative Born inversion) underestimates the anomalous conductivity, while the second (DQA inversion) produces a more correct result. The reason for the underestimation of the conductivity by the iterative Born inversion is very simple. The linearized response, which is used in this technique, usually overestimates the anomalous field, which results in an underestimation of the conductivity in inversion. The quasi-analytical approximation produces a more accurate electromagnetic response, so the inversion works more accurately as well. Thus, the modeling study demonstrates that both methods can resolve a three-
324
Integral representations in electromagnetic inversion 0
200
400
meters
10 100 1000 Resistivity, ohm-m
Figure 10-8 Sketch of a typical cross-section below the center of the region of inversion, based on a 2-D interpretation of MT data (according to Takasugi, 1992)
dimensional conductive target well, using very modest computational resources (computer memory and CPU time). 10.5.6
Case study: inversion of the Minamikayabe area data
The New Energy and Industrial Technology Development Organization (NEDO) has been conducting a "Geothermal Development Promotion Survey" in the Minamikayabe area located in the southern part of Hokkaido, Japan (Takasugi et al., 1992). This area is particularly interesting because of its geothermal potential. It is characterized by a resistive magmatic intrusion surrounded by fractured host rocks with relatively high conductivity. In 1988, MT sounding and AMT sounding were conducted in this area. Figure 10-7 shows the location of the MT measurement sites. The site spacing is 100 m along and across observational profiles. The frequency range of MT data is from 1 Hz to 130 Hz. We use MT data for inversion at seven different frequencies between 1 Hz and 100 Hz (96 Hz, 48 Hz, 24 Hz, 12 Hz, 6 Hz, 3 Hz, and 1.5 Hz). The selected background model consists of a four layer model with resistivities of 480 ohm-m, 6 ohm-m, 15 ohm-m and 270 ohm-m, and thicknesses of 30 m, 450 m, and 200 m, respectively. We use both nominal TE mode and TM mode data for a total of 161 MT soundings in the inversion. A sketch of a typical cross-section below the center of the area of inversion, based on a 2-D interpretation of MT data (according to Takasugi, 1992) is shown in Figure 10-8. The subdivision of the region of inversion can be seen in Figure 10-9. It was divided into 26 x 26 cells horizontally (each cell has a
Magnetotelluric (MT) data inversion
325
MT stations;
Plane wave source Frequencies : 1.5.3.6, t2r24,48 andQS Hz Measured: 9u ^m Tt ^m
Figure 10-9 The region of inversion was divided into 26 x 26 cells horizontally (each cell has a 50 m x 50 m size in the x and y directions), and 11 layers in the vertical direction. The total number of cells is 7436.
326
Integral representations in electromagnetic inversion
Predicted resistivity after smooth inversion
•10000
1000
100 110 -500 500
500
x(m)
ohm-m
y(m)
Predicted resistivity after focusing Inversion
•10000 11000 100 10
.500 500
x(m)
500
y(m)
1
ohm—fit
Figure 10-10 Volume images of the underground fo'mations in the Minamikayabe area predicted by smooth (top panel) and focusing (bottom panel) inversion.
Magnetotelluric (MT) data inversion
327
Obaanwd ipparMfit rMiatfvlty
PnMHetad apfMnMtt rMkttlvity
i{m)
y(««)
Figure 10-11 Pseudosections of the observed (top panel) and predicted (bottom panel) apparent resistivities in the Minanfiikayabe area.
Figure 10-12 Pseudosections of the observed (top panel) and predicted (bottom panel) phases in the Minamikayabe area.
328
Integral representations in electromagnetic inversion
50 m X 50 m size in the x and y directions), and 11 layers in the vertical direction. The vertical cell sizes are 25 m for the first layer, 50 m for the next seven layers and 100 meters for the cells at the bottom three layers, i.e. the cell sizes increase with the depth. The total number of cells is 7436. It takes only few minutes on a PC to invert these data using the QA inversion method. Figure 10-10 shows the volume images of the models predicted by smooth (top panel) and focusing (bottom panel) inversion. The inversion results are consistent with the 2-D interpretation results of Takasugi et al., (1992), and with 3-D inversion results obtained by QL inversion (Zhdanov et al., 2000). Comparisons of observed and predicted TM mode apparent resistivities and phases along four profiles are presented in Figures 10-11 to 10-12. They show good agreement between the observed and predicted apparent resistivity and phase pseudosections.
References to Chapter 10 Alumbaugh, D. L., and G. A. Newman, 1997, Three-dimensional massively parallel inversion - II. Analysis of a cross-well electromagnetic experiment: Geophys. J. Int., 128, 355-63. Arfken, G., and H. J. Weber, 1995, Mathematical methods for physicists, Fourth Edition: Academic Press, San Diego, New York, Boston, London, Tokyo, Toronto, 1028 pp. Chew, W. C , 1990, Waves and fields in inhomogeneous media. Van Nostrand Reinhold, New York, 608 pp. Eaton, P. A., 1989, 3-D EM inversion using integral equations: Geophysical Prospecting, 37, 407-426. 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. Hursan, G., and M. S. Zhdanov, 2001, Rapid 3-D magnetotelluric and CSAMT inversion: 71st SEG Annual International Meeting, San Antonio, Texas, 1493-1496. Kleinman, R. E., and P. M. van den Berg, 1993, An extended range-modified gradient technique for profile inversion: Radio Sci., 28, 877-884. Lee, K. H., and G . Xie, 1993, A new approach to imaging with low frequency EM fields: Geophysics, 58, 780-796. Madden, T. R., and R. L. Mackie, 1989, 3-D magnetotelluric modelling and inversion: Proceedings of the IEEE, 77, 318-332. Nekut, A., 1994, EM ray-trace tomography: Geophysics, 59, 371-377. Newman, G. A., and D. L. Alumbaugh, 1997, Three-dimensional massively parallel inversion - I. Theory: Geophys. J. Int., 128, 355-363. Oristaglio, M., Wang, T., Hohmann, G. W., and A. Tripp, 1993, Resistivity imaging of transient EM data by conjugate-gradient method: Expanded Abstracts, 63rd SEG Annual International Meeting, Washington, D. C , 347-350.
Magnetotelluric (MT) data inversion
329
Portniaguine O., and M. S. Zhdanov, 1999a, Focusing geophysical inversion images: Geophysics, 64, No. 3, 874-887. Portniaguine O., and M. S. Zhdanov, 1999b, 3-D focusing inversion of CSMT data: Second International Symposium of Three-Dimensional Electromagnetics, Utah, 132-135. Smith, J. T., and J. R. Booker, 1991, Rapid inversion of two- and threedimensional magnetotelluric data: J. Geophys. Res. 96, 3905-3922. Takasugi S., Keisaku T., Noriaki K., and M. Shigeki, 1992, High spatial resolution of the resistivity structure revealed by a dense network MT measurement - a case study in the Minamikayabe area, Hokkaido, Japan: J. Geomag. Geoelectr., 44, 289-308. Torres-Verdin, C., and T. M. Habashy, 1994, Rapid 2.5-dimensional forward modeling and inversion via a new nonlinear scattering approximation: Radio Sci.; 29, 1051-1079. Wannamaker, P. E., 1999, Affordable magnetotellurics: interpretation in natural environments: Three Dimensional Electromagnetics, SEG Monograph, Tulsa, OK, 349-374. Xiong, Z., 1992, EM modeling of 3-D structures by the method of system iteration using integral equations: Geophysics, 57, 1556-1561. Zhdanov, M. S., and V. V. Chernyak, 1987, An automated method of solving the two-dimensional inverse problem of electromagnetic induction within the earth (in Russian): Transaction (Doklady) USSR Acad. Sci., Earth Sci., 296, 59-63. Zhdanov, M. S., and G. Keller, 1994, The geoelectrical methods in geophysical exploration: Elsevier, Amsterdam, London, New York, Tokyo, 873 pp. Zhdanov, M. S. and S. Fang, 1996a, Quasi-linear approximation in 3-D EM modeling: Geophysics, 6 1 , 646-65. Zhdanov, M. S. and S. Fang, 1996b, 3-D quasi-linear EM inversion: Radio Sci., 3 1 , 741-54. Zhdanov, M. S. and S. Fang, 1999, 3-D quasi-linear electromagnetic modeling and inversion: Three Dimensional Electromagnetics, SEG Monograph, Tulsa, OK, 233-255. Zhdanov, M. S., Fang, S., and G. Hursan, 2000, Electromagnetic inversion using quasi-linear approximation: Geophysics, 65, No. 5, 1501-1513. Zhdanov, M. S., and G. Hursan, 2000, 3-D electromagnetic inversion based on quasi-analytical approximation: Inverse Problems, 16, 1297-1322. Zhdanov, M. S. and E. Tartaras, 2002, Inversion of multi-transmitter 3-D electromagnetic data based on the localized quasi-linear approximation: Geophys. J. Int., 148, No 3.
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Chapter 11 ELECTROMAGNETIC MIGRATION IMAGING One of the most challenging problems in electromagnetic (EM) geophysical methods is developing fast and stable methods of imaging inhomogeneous underground structures using electromagnetic data. Solution of this problem is important for many practical applications ranging from mineral exploration to waste and building site characterization. In this chapter I present an approach to electromagnetic imaging based on the migration transformation of an electromagnetic field. The basic principles of electromagnetic migration have been formulated in Zhdanov and Frenkel (1983a,b); Zhdanov (1988); Zhdanov et al. (1988); Zhdanov and Keller (1994); Zhdanov et al. (1996), and Zhdanov (1999). Electromagnetic migration has important features in common with seismic migration (Schneider, 1978; Claerbout, 1985), but differs in that for geoelectric problems, electromagnetic migration is done on the basis of Maxwell's equations, while in the seismic case it is based on the wave equation. Electromagnetic migration, similar to geopotential field migration, is based on a special form of downward continuation of the observed field or one of its components. This downward continuation is obtained as the solution of the boundary value problem in the lower half-space for the adjoint Maxwell's equations, in which the boundary values of the migration field on the earth's surface are determined by the observed electromagnetic field. At the same time, electromagnetic migration can be treated as an approximate solution of the corresponding electromagnetic inverse problem (Zhdanov and Traynin, 1997; and Zhdanov and Portniaguine, 1997). We can introduce the residual electromagnetic field as the difference between the simulated electromagnetic field for some given (background) geoelectrical model and the actual electromagnetic field. The electromagnetic energy flow of the residual field through the surface of observations can be treated as a functional of the conductivity distribution in the model. Electromagnetic migration imaging is considered as the initial step in the general electromagnetic inversion procedure, based on the minimization of the residual field energy flow through the surface of observations. By applying migration iteratively, we arrive at iterative electromagnetic migration which provides a rigorous solution of electromagnetic inverse problems.
332
Electromagnetic migration imaging
Figure 11-1 2-D geoelectrical model with a background electrical conductivity a and a local inhomogeneity F with conductivity cr == cr^ -|- ACT, varying spatially
11.1
CTb
Electromagnetic migration in the frequency domain
For the sake of simplicity, I consider in this section only the 2-D geoelectrical problem (E-polarization case) in the frequency domain. However, all the results, developed below, can be generalized to a 3-D case. The solution of the 3-D migration problem in the time domain will be discussed in the next sections. 11.1.1
Formulation of the electromagnetic inverse problem as a minimization energy flow functional
of the
Electromagnetic migration can be treated as an approximate solution of an inverse problem. That is why we begin with the formulation of the corresponding 2-D inverse problem. Consider a 2-D geoelectrical model with a background electrical conductivity a = at and a local inhomogeneity F with conductivity a = aij -h ACT, varying spatially (Figure 11-1). Note that in general the background conductivity can be a function of the coordinates a^ = ab{x,z). However, it is assumed that it is known a priori. We assume that fi = /J^Q = in x 10~^H/m, where /ig is the freespace magnetic permeability. The model is excited by an E-polarized field generated by a linear current density j ^ = j^dy, which is distributed in a domain Q in the upper half-plane (z < 0) with a constant conductivity crb{x,z < 0) = const. Here {da:, d^;, d^} is the orthonormal basis of the Cartesian system of coordinates with their origin on the earth's surface. This field is time harmonic as e~*'^^ We also consider a quasi-stationary model of the electromagnetic field, so that displacement currents are neglected. For this model, the electromagnetic field can be described by a single function Ey satisfying the equations
333
Electromagnetic migration in the frequency domain
(111) V^Ey
-h iujiiQCjEy = 0,
z > 0,
and the magnetic field components can be expressed by the equations ZCJ/XQ OZ
icjfiQ
ox
The expression (8.104) for the complex Poynting vector P in two dimensions takes the form P =^E X H* = ^EyH:d, - \EyH:d„ (11.3) where in the case of E-polarization E =Eydy , H.*=H*dx + H*dz and the asterisk * indicates a complex conjugate value. The real part of the vector P describes the intensity of the electromagnetic field energy flow. The divergence of the real part of P determines the energy dissipated in heat per unit volume per second: V - R e P = -^crE-E*.
(11.4)
It can be shown, using Green's theorem in two dimensions, that the total energy F^ dissipated throughout any region S bounded by a contour L is equal to F^ = -Re
[ P'ndl
= ^ ff
aE' E*ds > 0,
(11.5)
where n is the unit outward normal vector, and the contour is traversed counterclockwise. The proof of formula (11.5) is similar to the proof of the Poynting theorem in the frequency domain. When the region S coincides with the lower half-plane (z > 0), the contour L can be composed of an infinitely large semicircle, CR, in the lower half-plane, and the part bfi of the horizontal axis z = 0 bounded by CR (Figure 11-1). Since the electric and magnetic fields, E and H*, vanish exponentially at infinity, the contour integral over an infinitely large semicircle tends to zero (if R -^ oo). Thus, the total energy Fuj dissipated in the lower half-plane can be calculated using the formula +00
/ OO
-j
P • d,dl = -j
^
/'+00
J—oo
{EyHl + E;H,) dx',
(11.6)
where x^ is the integration variable and we use the formula: Re{EyH:) = ^{EyH: + E;H^). Let us denote the electromagnetic field components observed at the point x' on the surface of the earth {z = 0) by Ey^obs {x', 0, u), Hx^ohs {x\ 0, u) and the theoretical
334
Electromagnetic migration imaging
electromagnetic field components calculated for a given geoelectrical model a{x,z), by Ey^pr {cf) = Ey^pr {x\ 0, oj), Hx^pr (cr) = Hx^pr (x', 0, uj) (predicted field). Obviously, observed and predicted fields can be represented as the sums of the background field, E^, Hi, and the corresponding anomalous parts of the observed and predicted fields: ^2/' 77'
Tpb
I
Tr
TT
•
Tjb
Tja
Ey^pr = Ey^pr (cr) = Ey-\- Eyp^ ( a ) , Hx,pr = Hx,pr (cr) = H^-\- H^j^ {a).
(11.7)
We can introduce the residual fields as the diff'erence between the predicted theoretical fields, Ey^pr, Hx^pr, for the given conductivity model cr, and the observed field: E^A ip ^y — ^y,pr rjA ^x
rpobs rpa J^y — J^y^pr
L7 — ^x,pr -
TTobs ^x -
rpa,obs ^y i
(11.8)
Tja zja.obs ^ x , p r "" ^ x •
The total energy fiow F^ of the residual field through the earth's surface (2: = 0), is calculated by the formula V^.d.dl = - / •00
^
(11.9)
{E^H^* + E^*H^) dx',
J—00
where we use the H- sign, opposite to the sign of the expression (11.6), because the sources of the residual field, excess currents in the inhomogeneity F, are located in the lower half-plane. In Chapter 9 we proved an important theorem, according to which the energy fiow of the anomalous field is always non-negative. The same result holds, obviously, for the energy flow F^ of the residual field, which is a diflFerence between two anomalous fields, predicted and observed (see formula (11.8)): F^ > 0.
(11.10)
Moreover, if the conductivity of the upper half-plane is assumed not to be identically equal to zero (a^ > 0), the energy flow of the residual field is always positive (for a residual field not identically equal to zero, E^ ^0). This result can be obtained from equation (11.5) applied to the upper half-plane:
F^ = \ J J atE^-E^*ds = \jj
'^b |E^f ds > 0.
(11.1
Based on this theorem we can introduce the measure $ of the diflFerence between the observed and the background theoretical fields as energy flow of the residual field, integrated over the frequency range Vt :
$ (a) = / F^doj = \ f
r^
{E^H^* + E^^H^) dx'dw.
(11.12)
Electromagnetic
335
migration in the frequency domain
The residual field components E!^ {X',0,UJ) and H^ {x',0,uj) depend on the conductivity distribution a{x,z) in the given geoelectrical model and, therefore, ^ is a functional of the conductivity model: $ = $ (a). The functional $ (a) can be treated as an analog of the misfit between the observed and theoretical fields. The advantage of this new functional in comparison with the traditional misfit functional is that $ (cr) has a clear physical meaning of the residual field energy fiow through the profile of observations, and it automatically involves both electric and magnetic fields. Thus, the 2-D electromagnetic inverse problem can be reduced to the minimization of the energy flow functional: $(a-)=min.
(11.13)
In the following sections we will discuss an approach to the solution of this problem. However, first we introduce the migration transformation, which plays a critical role in minimization of the energy flow functional. 11.1.2 Integral representations for electromagnetic migration field In the case of E-polarization and in the frequency domain, the electromagnetic migration field is introduced, similar to gravity migration (Chapter 7), as the downward analytical continuation of the complex conjugate electric field, £"*, in the lower half-plane. This transformation is equivalent to the solution of the boundary value problem for the Helmholtz equation. The complex conjugate component of the electromagnetic migration field, E^* ^ satisfies the equation V^E"^' - iujfio(^E^* = 0, z > 0
(11.14)
everywhere in the lower half-plane, vanishes according to the radiation conditions at infinity, and is equal to the observed field on the surface of observation z' — 0 : E^* (x^ z = 0,uj) = Ey {x\ z = 0,uj), dE'^^^x'.z'.uj) dz'
dEy
{x',z',ijj)
dz'
(11.15) \z'=0
We have the same conditions for the migrated residualfieldE'^ y 2 TT'Am*
•, , , ,
^
TTIATTI*
V'E^"^* - iwMo'^^y
= 0, z > 0
(11.16)
and E^"^*{x',0,oj)=E^{x',0,oj),
z'=0
dE^{x',zf,uj) dz'
(11.17) z'=0
336
Electromagnetic migration imaging
The complex conjugated of the Green's function G* satisfies the following equation in the lower half-plane: V^G* (x,z \x',z') - iuj/j.QaGl (x,z |x',z') = iujfiS{x -x',z-
z').
(11.18)
We can now apply Green's formula to the contour L formed by the part hfi of the horizontal axis z = 0 and an infinitely large semicircle, CR, in the lower half-plane (Figure 11-1). Taking into account equations (11.16) and (11.18) and repeating the derivations similar to the one described in Chapter 9 for the 2-D integral equation method, we find the expression for the migrated residual electric field as an integral over the profile of observations on the horizontal axis x of the residual electric field and its derivative:
-ia-M^f™*(x,2) y + 00
/
G:(x,zK.')^^^-^,^(:r',.'
Taking the complex conjugate of the left-hand and right-hand parts of equation (11.19), we determine
iu^E^^{x,z)
The last two equations give integral representations for the migrated residual field. Note that in practice the migrated field can be calculated not only by using the integral representation (11.20), but also by means of the diff"erential equation method. For example, we can solve the boundary value problem (11.16) and (11.17) by the finite diflFerence or finite element method (Lee et al., 1987; Zhdanov et al., 1996). Computationally this solution will be similar to numerical forward modeling in an inhomogeneous model with the conductivity a (x, z). Thus, migration transformation is equivalent to a single forward modeling calculation. 11.1.3 Gradient direction of the energy flow functional Now we will show how the electromagnetic field migration introduced above is related to minimization of the energy flow functional. The important step in the solution of the functional minimization problem (11.13) is calculating the steepest ascent direction (or the gradient) of the functional. To solve this problem, let us perturb the conductivity distribution: crj, (x, z) = cr^ (x, z)-^6o- (x, z). Actually, we have to perturb the conductivity only within the inhomogeneous domain F of the lower half-plane: ^cr(x,z) = 0, ( x , z ) ^ F .
(11.21)
Electromagnetic
337
migration in the frequency domain
The first variation of the misfit functional with respect to the perturbation of the background conductivity can be calculated as <5$ (a, 6a) = - f
/ ^ " {6E^H^* + E^SH^
+ SE^'H^
+ E^*8H^)
dx'cko.
(11.22) Here SE^, SH^* are the first variations of the residual electric and magnetic fields: 6E^ = 6E^- SHt
= SH:\
(11.23)
because 6Ef' = SH^^'* = 0. According to the integral representations (9.25) and (9.26), the first variations of the anomalous electric and magnetic fields can be calculated as
6E^ = II GaSaEy^^ds, 6H^,{x',{),u)
= — ^
f[ ^SaEy^prds,
(11.24) (11.25)
where Ga is the Green's function of the geoelectrical model with conductivity G — G (x, z), and £^j/,pr = Ey.w [p) is the predicted theoretical field calculated for the same conductivity G = G{X/Z). Substituting (11.24) and (11.25) into (11.23) and (11.22) and changing the order of integrations, we obtain
At the same time, the residual magnetic field H^ can be expressed as the vertical derivative of the residual electric field E^ using the equation i/A ^ - ^ ^ . ^
lujfiQ dz'
Taking this last equation into account, we can modify equation (11.26):
(11.27) ^
^
338
Electromagnetic migration imaging
According to equations (11.20) and (11.19), the integrals over the x axis are equal to the migrated residual field and its complex conjugate values:
and
Substituting expressions (11.29) and (11.30) back into (11.28), we obtain 6^ (a, Sa) = ^Jj6aJ
{Ey,^E^^
+ ^ . V < " * ) dcjds (11.31)
= ^ ff 6aRe f Ey^prE^'^dwds.
Therefore, to make the first variation of the misfit functional negative, we have to select 6a as da (x, z) = -kol (x, z), (x, z) G T, (11.32) where the gradient direction / (x, z) (or direction of the steepest ascent) is calculated using the expression / (x, z)=Re
(11.33)
[ Ey^prE^'^dcj, Jn
and ko is a positive number (length of a step). This choice of 6a makes 6^ {a,6a) = -^o
/ / |Re / Ey^prE^'^diul
ds,
which is indeed negative. Thus, we can see that the gradient direction of the residual field energy flow functional for the given conductivity is equal to the integral over frequency of the product of the predicted field for this conductivity and the migrated residual electric field. In other words, calculation of the gradient direction can be based on the migration of the residual field. 11.1.4 Migration imaging in the frequency domain We have learned in the previous chapters of this book, that an electromagnetic inverse problem is ill-posed. Therefore, direct minimization of the energy flow functional (11.13) could lead to an unstable solution. In order to generate a regularized solution, we have to minimize the parametric functional: P - (a) = ^{a)+a{W
(a-aapr),
W {(T-aajrr))M = mi^'
(11-34)
339
Electromagnetic migration in the frequency domain
where M is a Hilbert space of the conductivity functions, distributed within domain r , with the inner product (a, P)M = / / ^ W P (r) ds; a.peM,
(11.35)
and H/^ is a hnear weighting operator of multiphcation by the function w ( r ) . Let us find an approximate regularized solution of the minimization problem (11.13), similar to one introduced in Chapter 5 (formula (5.90)). According to the basic principles of the regularized Newton method, we consider one iteration: ai=cro-hAcr,
(11.36)
where the initial approximation, CTQ, may be selected to be equal to the background conductivity, CTQ = cr^,. We assume also, that the a priori model is equal to the background conductivity, aapr = cr^, and calculate the parametric functional for the first iteration: F " (ai) = ^ (ai) + a {WAa, WAa)j^ . (11.37) The Newton method is based on the idea that one can find the minimum of the parametric functional in one iteration. We now perturb the iteration step. ACT, and find the corresponding variation of the parametric functional (11.37). According to (11.31) and (11.35), it is equal to 5 P " {ai, 6Aa) = 6^ {a^, 6Aa) -f 2a{W6Aa,
WAa) M
= 6^ {ab -h A(j, 6Aa) + 2a ff 6Aaw^Aads.
(11.38)
The first variation of the energy flow functional at the point (cr^, -h ACT) can be evaluated using linearization of the forward modeling operator for the predicted electric field £'y,pr (^^6 + ACT) : 6^ {ab + ACT, 6Aa) = ^ / / ^ ^ ^ ^ ^ / ^^'^^ ^^^ "^ ^ ^ ^ ^^"^ ^^^ ^ ^ ^ ) ^^^^ ^ l
ff
^Aa Re / [E'y + FE{Aa)] [E^^^' + F]^{Ac7)] duds,
where the Frechet diff'erential FsiAa)
(11.39)
is determined according to formula (9.28):
FE{Aa) = ff CAaE'yds,
(11.40)
electric field E^^ is the diflference between the background and observed field, E^° = E'y - Ey,,,„
(11.41)
340
Electromagnetic migration imaging
and the upper subscript "m" denotes the migration transformation apphed to the corresponding fields. Substituting expression (11.39) back into (11.38), and taking into account that, according to Theorem 81 of Appendix D, the first variation of the parametric functional at the minimum must be equal to zero, we have (5P"(ai,(5Aa) / / 6/\a I i Re / ElE^'^^du
-h DE{/^(T) + 2aw^/^(T ds = 0,
(11.42)
where DE{/^CJ)
- ^ Re / [EIF^{/\G)
+ FE{1^CT)E^'^
+ F^(A(7)F^(Aa)] dw.
Note that equation (11.42) must hold for any variation (5ACT. Therefore, from equation (11.42) we find at once the regularized normal equation for the optimum step, ACT : DE{^(T)
+ 2aw^^a
= - i Re / E'yE^'^duj = -ho,
(11.43)
where IQ is the gradient direction for the initial (background) conductivity distribution. Note that the first term on the left-hand side of the last equation can be treated as a Hessian operator H(j^ applied to ACT : DsiAa)
= H,, ( A a ) .
We can find an approximate solution of the regularized normal equation (11.43) for the optimum step, using the same idea we applied for approximate solution of the general nonlinear inverse problem in Chapter 5. Let us assume that the regularization parameter a is big enough to neglect the term DE (ACT) with respect to the term 2oLW^[S.a in equation (11.43). Then we obtain 2aw^[^G « - i Re / E^E^'^'^du = - J/Q. Applying the inverse weighting operators to both sides of the last equation, we find A a (x, z) ^ -]a-^w-^
Re / E^E^'^^duj = -^a'^w-^lo
(x, z).
(11.44)
Note that the coefficient a~^/4 in the previous formula can now be treated as just a scaling factor, which can be found by minimizing the misfit between the observed and predicted data. Thus, on the basis of expression (11.44), we can introduce the so-called migration apparent conductivity, Aama (^j z),
341
Electromagnetic migration in the frequency domain
Aama {x, z) = -kow-^ Re / E^yEf'^^cLu = -kow-% Jn
(x, z),
(11.45)
where the coefRcient ko is determined by a Une search for the minimum of the functional: $ [ab (x, z) - koW~'^lo (x, z)] = $ {ko) == min . (11.46) Derivations presented below show that in the case of a linear line search, the coefRcient ko can be determined by the formula
where E!^ and H!^ are the electric and magnetic fields, calculated for the model perturbed in the gradient direction normalized by the square weights, w~'^. Formula (11.45) plays an important role in fast inverse imaging. We can see from (11.45) that the migration apparent conductivity Acr^a(x, z) is proportional (with some constant coefficient ko and weights w'"^) to the frequency stacked values of the product of the background (incident) field Ey (the field that corresponds to the background distribution of conductivity at (x, z)) and the migrated residual field ^Aom ^ j-jg;6 ^x'^o.u) - Ef' {x'.o.u)^. We will show in the next sections of this chapter that, in the time domain, the stacking formula corresponds to the convolution of the background and migrated electric field. Definition of the scalar coefRcient ko. We solve now the minimization problem (11.46) of the energy flow functional. We denote by Ey \ Hx the electromagnetic field components corresponding to the geoelectrical model with the conductivity distribution ai = a^ (x, z) — koW~'^ (x, z) I (x, z). Let us substitute E\J\ HX into equation (11.12): $ = i /• r ° ° (4i)A^(i)A* + £;(i)A*;^a)A) dx'du,
(11.48)
^ JQ J —oo
where E^^''^ = E^y'> - E°^', H^^^^ = Hi^^ - Hf\ field Ey',Hx'iS
and the calculation of theoretical
linearized, using a Born approximation:
+ / / G^CTfc {x - x'> z) ^CT {x, z) Ey {x, z) ds = Elix',0,uj)
- koE'y" {x',0,oj),
(11.49)
342
Electromagnetic migration imaging
1
ff
dGa, ix-x',z-
J
z')
= i/,^ {x', 0,a;) - fco/f^ (x', 0,uj),
(11.50)
where ^CT {X, Z) = —kow''^ {x, z) lo{x, z) and Ey°,H'^ are the fields, calculated using a Born approximation for the model, perturbed in the direction w~^ {x, z) lo{x, z):
E[' {x', o, u) = j j Ga, {x - x', z) w-Ho {x, z) El (x, z) ds,
^x = —.
1 BF'^ ^ -
(11.51)
(11.52)
Substituting (11.49) and (11.50) into (11.48), we obtain
4 Jfi 7-00
= i / /^" {(^^ - ^°^^) (^-""^ - ^0^^)* -^ (^^ - ^0^^)* (^-""^ - ^0^^)} ^^'^^ (11.53) Now we can find the first variation of ^ (ko) with respect to /CQ : 4 o * (fco) = --5fco / / { [Ei'H^'* + E^^'H'r + El^'H^" ^ JnJ-oo -\- E^'^Hl']
- 2ko [E^'H^r + E'fHi')
] dx'duj.
(11.54)
After some algebraic transformations, we obtain
= --6h
/
/
{Re {E[^Ht°* + E^-H'r)
- 2fco Re {E[-Hi^*) ] dx'dw.
The necessary condition for minimizing $ (fco) is Sko^{ko)=0. Therefore, we have /
/
Jo, J —oo
Re { ( £ ; ^ / / f * + E^Hl"*) + 2fco Re ( 4 ° / / ^ * ) } dx'cL; = 0.
(11.55)
343
Electromagnetic migration in the frequency domain
From the last equation we find at once formula (11.47). The weighting operator W = Wa is a linear operator of multiplication of the conductivity a by the square root of the integrated sensitivity, a function we denote by WcT (see formula (3.78) in Chapter 3): w^ = \ / 5 .
(11.56)
In accordance with definition (3.75), the integrated sensitivities for an electromagnetic field, are calculated by formulae (9.34) and (9.36). 11.1.5 Iterative migration We have demonstrated above that migration imaging can be treated as the first iteration in the solution of some specific electromagnetic inverse problem based on minimizing the energy flow functional. Obviously, we can obtain better imaging results if we repeat the iterations. We can now apply a general scheme (5.151) of the re-weighted, regularized, conjugate gradient method in the space of the weighted parameters, to form an iterative process for electromagnetic inversion. According to this scheme, we introduce a space of weighted conductivities: a^ = WG,
(11.57)
where the weighting operator W is the linear operator corresponding to multiplication of the conductivity a by the function w. The general iterative process can be described by the formulae: < + l = < + S< =<where
«(^n),
(11-58)
_ ''w\^n)
Co — Co'
— ^wn — ^wn'^Pn^w{n-l)^ not _ ||7a ||2 h'n ~~ W^WUWM /
/ ||7a
(11.59)
||2
\rw{ri-l)\\f^
Uj^KSd^ \JJjlZ^„_,)fds
-1
The weighted regularized gradient direction on the n-th iteration l^^ can be calculated by a formula, derived from (11.33) by analogy with the formula (5.142):
= W-' Re / E;E^-"^
+ a{aZ-
where In is a gradient direction on the n t h iteration: In = Re I
Jo.
E^E^-'^dio,
(11.60)
344
Electromagnetic migration imaging
Ey = Ey (crj^) is the field calculated by forward modeling for the geoelectrical model with the conductivity distribution cFn {x^z)^ and £^^""^ is the migrated residual field E^", computed as the difference between the theoretical predicted field Ey, found on the nth iteration, and the observed field: £;f" (x', 0, uj) =
E;
ix', 0, Lj) - El"^ {x', 0, uj), (11.61)
H^- ix', 0, Lo) = H^ [x', 0, a;) - //f»{x', 0, a;). The optimal length of the step kn can be determined by a formula similar to (11.47):
where E}j' is the electric field, calculated for the model crn(x,z), perturbed in the direction [w~'^ln-\-OL{(Jn —(^apr)]- Note that on each step we recompute the real conductivities from the weighted conductivities at the nth iteration: dn = W-'a^,
(11.63)
Thus, we can describe the developed method of electromagnetic inversion as the process of iterative migration. On every iteration we calculate the theoretical electromagnetic response for the given geoelectrical model cr^ (x, z), obtained on the previous step, calculate the residual field between this response and the observed field, and then migrate the residual field. The gradient direction is computed as an integral over the frequencies of the product of the migrated residual field and the theoretical response Ey. Using this gradient direction and the corresponding value of the optimal length of the step fc^, we calculate the new geoelectrical model cr^+i (x, z) on the basis of expressions (11.58) and (11.63). The iterations are terminated when the functional $ (a) reaches the level of the noise energy. The optimal value of the regularization parameter a is selected using conventional principles of regularization theory, described in Chapter 2. The migration apparent resistivity, introduced above, appears as the first iteration in this iterative process. 11.2
Electromagnetic migration in the time domain
Time domain electromagnetic (EM) migration is based on downward extrapolation of the residual field in reverse time. In this section I will show that electromagnetic migration, as the solution of the boundary value problem for the adjoint Maxwell's equation, can be clearly associated with solution of the inverse problem in the time domain. In particular, I will demonstrate that the'gradient of the residual field energy flow functional with respect to the perturbation of the model conductivity is equal to the vector cross-correlation function between the predicted field for the given
345
Electromagnetic migration in the time domain
geoelectrical model and the migrated residual field, calculated as the solution of the boundary value problem for the adjoint Maxwell's equation. This result clearly leads to a construction of a rigorous method of solving the inverse electromagnetic problem, based on iterative electromagnetic migration in the time domain, and a gradient (or conjugate gradient) search for the optimal geoelectrical model. 11.2.1
Time domain electromagnetic migration as the solution of the boundary value problem
Let us formulate a general time domain electromagnetic inverse problem. Consider a 3-D geoelectrical model consisting of a homogeneous atmosphere and an inhomogeneous earth with conductivity a (r) = cr^, (r)-f ACT (r), where a^ (r) is some background (normal) distribution of the conductivity, and A a (r) is the anomalous conductivity, which is non vanishing only within some domain D. We will denote the surface of the earth by E (Figure 11-2). We will confine ourselves to consideration of non magnetic media and, hence, assume that fi = /2Q = An x 10~'^H/m, where fiQ is the free-space magnetic permeability. The electromagnetic field in this model is generated by a source, located with the current density j ^ in the upper half-space (Figure 11-2). Receivers are located on the surface of the earth. We assume also, that the electromagnetic field is varying in time relatively slowly, so that in the equations for this field the second derivative with respect to time, d'^/dt'^, can be discarded. In other words, we consider the so-called quasi-stationary model of the electromagnetic field (without displacement currents). We can represent the total electromagnetic field, observed in this model, as the sum of the background (normal) field {E^,H^} , generated by the given source in the model with the background conductivity distribution, and the anomalous field {E^, H^}, due to an inhomogeneity a a (r) : E = E^ + E ^ H = H^ -f- H ^
(11.64)
The total electromagnetic field satisfies Maxwell's equations VxH=((j6 4-Aa)E+f, (11.65) V XE =
-fxdU/dt,
while the anomalous field satisfies the equations V X H"=(j5E"-fA(j (E^ -f E^) , (11.66) V X E^ -
-fidW/dt.
The general electromagnetic inverse problem can be formulated as follows. We are given the observed total electromagnetic field on the surface of the earth and
Electromagnetic migration imaging
346
Figure 11-2 3-D geoelectrical model consisting of a honnogeneous atmosphere and an inhomogeneous earth with conductivity a (r) = cr^ (r) + ACT ( r ) , where a^ (r) is some background (normal) distribution of the conductivity, and ACT (r) is the anomalous conductivity, which is non vanishing only within some domain D.
the background (normal) distribution of the conductivity cr^ ( r ) . The problem is to determine the conductivity a (r) = cr^ (r) -f ACT (r) within the anomalous domain D. In this section we introduce first the migrated anomalous electromagnetic field and show how it can be calculated from the anomalous field. In the following sections we will demonstrate the connections between the migrated electromagnetic fields and the solution of the electromagnetic inverse problem. We can extend the integral representations in the frequency domain, formulae (9.37) of Chapter 9, to the time domain. As a result, the anomalous electromagnetic field in the model can be expressed as an integral over the anomalous domain D of the product of the corresponding Green's tensors and excessive currents ACT ( E ^ 4- E^): E"(r,i) = r
[ff
J-oo J J JD
G's (r, t I r',i') • Aa (r') [ E " (r',«') + E" (r',i')] dv'dt',
(11.67)
and H°(r,i) = r
[ff
7-00 J J JD
G'H (r, t I r',i') • Aa (r') [E^ {r',t') + E" {r',t')] dv'dt',
(11.68)
where &^ and G ^ are electric and magnetic Green's tensors for the background conductivity a^ ( r ) , whose vector components relate the electric and magnetic fields
347
Electromagnetic migration in the time domain
excited at the point r by an electric dipole source of unit intensity located at the point r' of the domain D. The basic equations for Green's tensors and their properties were summarized in Chapter 8. The definition of the electromagnetic migration field in time domain was introduced in the monograph by Zhdanov (1988). According to this definition, the migration field is the solution of the boundary value problem for the adjoint Maxwell's equations. For example, we can introduce the migration anomalous field E""^, H^"^ as the field, determined in reverse time r = —t, whose tangential components are equal to the anomalous field in reverse time at the surface of the earth E n X E"^ (r,r) = n x E^ (r, - t ) , (11.69) n X H"'^ (r,T) = n X H^ (r, - t ) , r G E, where n is the unit vector of the normal to E directed into the upper half-space. Figure 11-2, and satisfying Maxwell's equations in reverse time within the earth with the background conductivity ab : V X H"^=(76E^^, (11.70) V X E"^ =
-fiOW'^/dr.
From (11.70) we can obtain the separate equation for the migrated anomalous electric field: V X V X E^^ = -iiab-^—. (11.71) or Therefore, in reverse time r the electric migrated field satisfies the ordinary vector diff"usion equation. However, in direct time, t = —r, the migrated anomalous electric field satisfies the equation adjoint to (11.71): Q^am
V X V X E"^ = fiat—^.
(11.72)
While the ordinary diffusion equation describes the development of the process of electromagnetic field propagation in increasing time from the source to the receiver, equation (11.72) reflects the same process in a reverse order, i.e. from the final distribution of the field at the receivers to its initial distribution at the sources. That is why, following Wang et al. (1994), equation (11.72) can be called the vector concentration equation. As a result, the electromagnetic migration field can be treated as the field converging into the sources of the anomalous field, which actually coincide with the geoelectrical inhomogeneities. The solution of the boundary value problem (11.69) and (11.70) for the concentration equation can be obtained with the aid of Green's tensor formula (F.IO) (see Appendix F). We assume that the volume V is bounded by the surface 5, which
348
Electromagnetic migration imaging
is composed of an infinitely large hemisphere in the lower half-space, CR, and of a part E H of the surface of the earth E, bounded by CR, Figure 11-2. Since the electromagnetic field satisfies the radiation conditions, i.e. functions E"'^ and H^"^ vanish exponentially at infinity, the surface integral over the infinitely large hemisphere tends to zero, if i^ ^ oo. If we substitute in the formula (F.IO) from Appendix F the migrated field E^"^ (r,r) for the field F and the adjoint Green's tensor G ^ (r, r | v',T') determined in Chapter 8 for the tensor P , we obtain | [ V X V X E^^ (r,r)] • G^+ (r,T | r'.r') E"^ (r,r) • [ v X V x G ^ + (r, r | r',r')] } dv j j n. { E « - (r,T) X [ v x G ^ + ( r , r | r',rO] V X E"^ ( r , r ) ] x G ^ + ( r , r | r V ) } r f 5 .
(11.73)
Integrating the left-hand and the right-hand sides of the expression (11.73) over time r and taking into account equations (11.71) and (8.148), after some algebraic calculations we obtain* E - ( r V ' ) ^ j " ^ ^ j j n- { E - - (r,r) x [&^ (r, r | r ' , / ) ] - H ^ ^ (r,r) X G^+ ( r , r | r ' , r ' ) } dsdr.
(11.74)
Returning from the reverse time r to the ordinary time t = — r, and taking into account the reciprocal relations (8.145), (8.146) from Chapter 8 and the boundary conditions (11.69) for the migration field, we finally can write Earn (^,^ _ ^,) ^ j ^ ^ j j
^ . l ^ a (^^t) X [ G J , (r, t \ v',t')
- H ^ ( r , t ) X G ^ ( r , t | Y\t')\dsdt.
(11.75)
Integral formula (11.75) describes the solution of the concentration equation for the migrated anomalous electric field. The corresponding integral representation for the migrated anomalous magnetic field can be obtained from (11.75), using the second Maxwell's equation. In a similar way we can construct the integral representations for the migration through an arbitrary distribution of conductivity cr. The only difference will *Note that, in the quasi-stationary model, equation (8.148) does not contain the term with the second derivative d"^ /dt^.
Electromagnetic migration in the time domain
349
be that we should use the corresponding Green's tensors for the conductivity distribution cr, instead of the Green's tensor for the background model ab in (11.75). These integral transformations describe the conversion of the anomalous electromagnetic field, generated by the excess currents in the geoelectrical inhomogeneities and diverging in real media, into the migration field, converging to the corresponding inhomogeneities. This process is actually equivalent to the field transformation in ordinary optical holography. That is why this technique also can be called an electromagnetic holography (Zhdanov, 1988, 2001). In the next sections I will show how this converging field can be used for the solution of the electromagnetic inverse problem. Meanwhile, let us consider a few simple examples, illustrating the spatial behavior of the migrated electromagnetic fields. The first example represents the results of migration of the magnetic component Hx of the field generated by a local horizontal electric dipole, located at some depth, ZQ, in the homogeneous lower half-space of conductivity a^. The current in the dipole is described by the delta-pulse:
i = iHt), where 6 {t) is the Dirac delta-function, and j is some constant horizontal vector. We apply now the migration transformation, outlined above, to the electromagnetic field of an electric dipole, observed on the surface of the earth, z = 0. Note, however, that following Zhdanov et al. (1988), we will use a conductivity crj,in the migration transformation equal to half of the background conductivity crj, = |cr6, in order to obtain the correct image of the dipole source. The original electromagnetic field has a singular point in the lower half-plane, associated with the electric dipole, so the downward analytic continuation can only be done down to this singularity, and it is an ill-posed, unstable procedure (Zhdanov, 1988). On the contrary, migration results in the field being an analytical function and satisfying the adjoint Maxwell's equations everywhere in the lower half-plane, and it is a well-posed, stable transformation. From the physical point of view, the migration field is obtained, similar to the case of gravity migration (Chapter 7), by moving the electric dipole source above the observational surface. Nevertheless, as we will see below, the migrated electromagnetic field contains some "remnant" information about the original electric dipole position. For example, Figures 11-3 and 11-4 show the isolines of the migrated horizontal component of the field, H^, for time t = 0 in two mutually orthogonal planes. Figure 11-3 presents the map of H^ {t — 0) in the vertical plane orthogonal to the dipole axis and passing through its center (0,0, zo) • Figure 11-4 shows the map of H"^ (t — 0) in the horizontal plane z = Z\^ containing the dipole. One can see that the migrated field has a local extremum which coincides with the position of the dipole. Figure 11-5 shows similar migration result for an electromagnetic field generated by two electric dipoles located in the lower half-space. Once again, we can see that the positions of these dipoles are manifested as the local extremes of the migrated field at time t — ^. Note that these results were reached for the modified background conductiv-
350
Electromagnetic migration imaging
Figure 11-3 The map of the migrated horizontal component of the magnetic field H^ at time ^ = 0 in the vertical plane, orthogonal to the dipole axis and passing through its center (0,0, ZQ) .
Figure 11-4 The map of the migrated horizontal component of the magnetic field H^ at time ^ = 0 in the horizontal plane z — ZQ containing the dipole.
Electromagnetic migration in the time domain
351
X/Zl
-1.5 -1.0 -0.5 Dipole
—1 0.5 0.5
1 1.0
\ 1.5 Dipole
Z/Zi Figure 11-5 The map of the migrated horizontal component of the magnetic field H^ generated by two electric dipoles located in the lower half-space, at time t = 0 in the vertical plane, orthogonal to the dipole axis.
ity (jj, = ^ab- The extensive analytical study, presented by Zhdanov et al. (1988), shows that this choice of background conductivity is the optimum for the homogeneous medium in a 3-D case in order to image properly the sources of the anomalous field by electromagnetic migration. However, in general cases, it is much more convenient to apply special weighting functions to the migrated field to reach the same result - correct imaging of geoelectrical inhomogeneities. We will outline the general principles of this technique later. Note, in conclusion of this section, that the migrated field, especially in the case of an arbitrarily inhomogeneous conductivity distribution, can be calculated using numerical differential methods (finite difference or finite element) to solve the boundary value problem (11.69) - (11.70). For example, one can use a finite difference time domain solution based on the DuFort-Frenkel scheme which is unconditionally stable (Oristaglio and Hohmann, 1984; Zhdanov and Li, 1997). Thus, computationally, migration transformation in the time domain through an inhomogeneous background medium is equivalent to one forward modeling of the time domain electromagnetic field in this background medium. If the background conductivity is constant or has a layered structure, it is convenient to use the integral representation (11.75). If the background conductivity is inhomogeneous, it is more practical to apply finite difference or finite element methods. 11.2.2 Minimization of the residual electromagnetic field energy flow The energy flow of the electromagnetic field can be calculated using the Poynting vector P (see Chapter 8), introduced by the following formula: P = E X H. The Poynting vector P may be interpreted as the intensity of electromagnetic
352
Electromagnetic migration imaging
energy flow at a given point, that is the energy per second crossing a unit area whose normal is oriented in the direction of the vector E x H. For example, the total energy flow of the anomalous electromagnetic field through the surface of the Earth E is equal to / / P-nds=
ff
(E X H) 'lids,
where n is the unit vector of normal to the surface E, directed to the upper half-space (assuming that the sources of the anomalous field are located in the lower half-space). We denote the observed electromagnetic field as {Eo6s,Ho6s}. The theoretical electromagnetic field, calculated for the given geoelectrical model a (r) = a^ (r) -fAcr(r), we denote as {Ep^,Hpr} (predicted field). According to equation (11.64) T7«6 _|_ 17«a
rr
T^b _i TT^a
Tj
JJfb _i_ TTO
(11.76) Tj-6 _i_ TJa
Now we determine the residual field {E"^, H ^ } as the diflPerence between the predicted and observed fields: E ^ (r, t) = Epr (r, t) - Eobs (r, t) = E^, (r, t) - E^,, (r, t), (11.77) H ^ (r, t) = Hpr (r, t) - H^bs (r, t) = H^, (r, t) - H^,, (r, t). We can introduce the energy fiow of the residual field through the surface of the earth:
F,^ = JJ
[E^{r,t)xn^{r,t)]-nds.
Similar to frequency domain cases, it can be proved, based on the energy inequality (8.100), that the energy flow Ff^ of the residual field is positive: Ff" > 0,
(11.78)
and if and only if the residual field is identically equal to zero: E ^ = 0 , and H ^ = 0 . With this theorem we can introduce the measure $ of the diflFerence between the observed and predicted fields as the energy fiow of the residual field through the surface of observations, integrated over time t : ^{a)=
f
ff
7-00 J 7 s
[E^ (r, t) x H ^ (r, t)] -ndsdt > 0.
353
Electromagnetic migration in the time domain
Obviously, the theoretically predicted fields Eipr (r, t ) , Upr (r, t) depend on the conductivity distribution a{x^ z) in the given geoelectrical model and, therefore, $ can be treated as a functional of the conductivity model: $ = $ (a). The advantage of the residual energy flow functional in comparison with the traditional misfit functional is that in a natural way it estimates simultaneously the closeness of both electric and magnetic predicted fields to the observed fields. Thus, the electromagnetic inverse problem can be reduced to the minimization of the residual field energy flow functional: $(a)=min.
(11.79)
In the following section we will discuss the solution of this problem. 11.2.3 Gradient direction of the energy flow functional in the time domain We apply a gradient type method to the solution of the minimum energy flow problem, which is based on computing the gradient direction for the misfit functional and decreasing this functional by moving iteratively "down the hill" (Chapter 5) in the space of the inverse problem solutions. Following the conventional notions of gradient methods, we calculate the first variation of the energy flow functional in order to find the gradient direction: 8^^
f
II
n [E^ (r, t) x6li^^ (r, t) - H ^ (r, t) x SE^^ (r, t)] dsdt,
(11.80)
where we have noted that, according to (11.77), SE^ =
6E;,,
6H^
=
sn;^.
The perturbations of anomalous electric and magnetic fields in the time domain can be expressed through the perturbation of the conductivity 6a using the integral formulae similar to equations (9.51) and (9.52) for electromagnetic field perturbation in the frequency domain: 6 E ^ (r, t) = r
Iff
G ^ (r, t \ r',t') • 6a (r') E^, (r',i') dv'dt',
6 H ^ (r, t) = r
[ff
G ^ (r, t | r',t') • 6a (r') Ep. {v',t') dv'dt',
J-oo J J JD
J-oo J J J D
where G|; and G ^ are electromagnetic Green's functions for the given conductivity distribution a (r), and E^r is the predicted electric field calculated for the same conductivity a ( r ) . Substituting the last equations into (11.80), we find 6^ = Iff
JJJD
6a (r') H
H
ff
J~oo J-oo J JT,
n-
{E'^ ^
(r, t) x G ^ (r, t \ r',i'
354
Electromagnetic migration imaging
- H ^ (r, t) X G^ (r, t \ r\t') ds\ dt • E^^ (r',t') dt'dv',
(11.81)
According to equation (11.75) the integral over the earth's surface can be treated as the migration of the residual field through the medium with conductivity a (r): j ^ ^ j j
n- { E ^ (r,i) x G ^ (r, t \ r ' / )
- H ^ (r,i) X G ^ (r, t \ T',t')\ dsdt = E ^ " (r', - t').
(11.82)
Substituting equation (11.82) into (11.81), and taking into account (11.76), we obtain: (5$ {a, 6a) = / / / 6a (r) f
E ^ ' " (r, -t)-Epr
(r,i) dtdv,
(11.83)
where we have omitted the prime superscript over r and t to simplify the formula. We have to find a perturbation of the anomalous conductivity Sa (r) that will reduce the energy fiow functional. In this case we go "down the hill" in the space of the inverse problem solutions. The obvious choice is oo
/
E^"'{T,-t)-Eprir,t)dt,
(11.84)
•CO
where oo
(11.85) / •OO E ^ ' " ( r , - t ) - E p , ( r , * ) d ^ is the gradient direction, and /CQ > 0 is the length of a step. In this case the first variation of the energy flow functional is indeed negative: 5^ (a, 6a) = -ko fff
[Sa {r)f dv < 0.
Thus, we have proved that the gradient of the residual field energy fiow functional with respect to the perturbation of the model conductivity is equal to the vector cross-correlation function between the predicted field for the given geoelectrical model cr (r) and the migrated residual field. 11.2.4 Migration imaging in the time domain We develop now a migration imaging method based on an approximate regularized solution of the time domain inverse problem by introducing the parametric functional P " {a) = ^{a)-^a{W
( a - a ^ p . ) , W ( ( 7 - a « ^ ) ) ^ = min,
(11.86)
where M is a Hilbert space of the conductivity functions, distributed within domain D, with the inner product (^. P)M = jjf
^ W P (r) dv, a.peM,
(11.87)
355
Electromagnetic migration in the time domain
and W is the linear weighting operator of multiphcation by the function w ( r ) . The approximate regularized solution of the minimization problem (11.79) can be introduced based on the regularized Newton method. We consider just one iteration: cri=ao + Aa,
(11.88)
where the initial approximation, CTQ, may be selected to be equal to the background conductivity, CTQ = cTfe. We assume also, that the a priori model is equal to the background conductivity, aapr — ^6, and calculate the parametric functional for the first iteration: (11.89) p « (cji) = $ {(Ti) -f a {WAa, WAa)j^ . Following the basic principles of the Newton method, we try to find the minimum of the parametric functional in one iteration. To do so, we perturb the iteration step. ACT, and find the corresponding variation of the parametric functional (11.89). According to (11.83) and (11.87), it is equal to (5P" (ai.SAa)
= 6^ {(JuSAa) -f- 2a{W6Aa,
WAa)M
^ 6^ {ab + Aa, SAa) + 2a / / / SAa w^Aadv.
(11.90)
As in the frequency domain case, considered above, we use linearization of the forward modeling operator for the predicted electric field Ep^, {a^ -}- ACT) : Ep, {at + Aa; r,t) « E^ (r,^) -f F ^ (Aa; r , t ) ,
(11.91)
where F^; (A(j;r,^) is the corresponding Frechet differential of the forward modeling operator, determined according to the formula [ff
F^; (ACT; r , ^ = /
G^ (r, t \ T',t') • 6a (y') E^ {r',t') dv'dt'.
(11.92)
J-oo J J JD
Taking into account expansion (11.91), we can evaluate the first variation of the energy flow functional at the point {at + ACT) as 6^ {at + Aa, SAa) « Iff
/ / / SAa /
JJJD
SAa r
7-00
E^"" {at + Aa; r, - ^) • Epr {at + Aa; r,t) dtdv
[E^"™ (r, - i) + F ^ {Aa; r, - t)] • [E' (r,i) + F g (Aa; r,i)] dtdv,
(11.93) where the residual field E^° is the difference between the background and observed field, E^° (r, - i) = E" (r, - t) - E^ts (r, - t), (11.94)
356
Electromagnetic migration imaging
and the upper subscript "m" denotes the migration transformation of the corresponding fields in the time domain. Substituting expression (11.93) back into (11.90), we have (5P"(ai,5Acr) -
[ff
6Aa \ f
E^°"^ (r, -t)-E^
where
oo
/
(r,t) dt + DsiAa)
+
2aw^Aa\ dv
(11.95)
{E^<'™(r,-i)-FB(Aa;r,i)
-CO
+ F ^ (Aa; r, - ^) • E^ (r,t) + F ^ (Aa; r, -t) • F ^ (Aa; r,t)} dt. Observing that according to Theorem 81 of Appendix D, the first variation of the parametric functional at the minimum must be equal to zero for any variation SAa, we find at once the regularized normal equation for the optimum step, ACT : oo
/
gAom (j.^ _tyE'>
(r,t) dt + DsiAa)
+ 2aw^Aa = 0
-OO
or
oo
/ oo
n
E ^ ^ (r, - ^) • E^ (r,^) dt = --IQ.
(11.96)
^
Note that the first term on the left-hand side of the last equation can be treated as a Hessian operator Hf^^^ applied to ACT : DE{Aa) = H,, {ACT) . We can find an approximate solution of the regularized normal equation (11.96) for the optimum step, using the same idea that we applied for the approximate solution of the general nonlinear inverse problem in Chapter 5. Let us assume that the regularization parameter a is big enough to neglect the term i/^^^ (ACT) with respect to the term 2aw'^A(T in equation (11.96). Then we obtain oo
/
-.
gAom ^^^ _ ^) . E^ (i.,t) dt = - - / o . •oo
^
Applying the inverse weighting operators to both sides of the last equation, we find /"OO
1
ACT (r) w --a-^w-^
/
-j
E^°™ (r, - ^ • E*- (r,i) dt = --a-'w-\
(r),
(11.97)
where we can treat the coefficient a~^/2 as a scalar factor, and find it by minimizing the misfit between the observed and predicted data. Thus, on the basis of expression (11.97) we introduce the migration apparent conductivity in the time domain, Aama ( r ) ,
357
Electromagnetic migration in the time domain oo
/
E^"* (r, - t) • E* (r,i)
rfi,
(11.98)
•OO
where the coefRcient /CQ is determined by a hne search for the minimum of the functional: $ [at (r) - kow-H (r)] = $ (fco) = min (11.99) with respect to ko. Derivations, similar to the one conducted in the frequency domain, show that in the case of the linear line search, the coefRcient ko can be determined by the formula ko = l r
[H'° (r', to X E^° ( r , 0 + H^° (r, t) x E^° {r\t') -n] ds'dt'x
ff
-1
r
E'° (r', t') X H^° (r', t') -nds'dt'
ff
(11.100)
where the field {E^°,H^°} is the electromagnetic field calculated using the Born approximation for the geoelectrical model, perturbed in the gradient direction, and normalized by the square weights, w"'^: E' (r, t) = / "
/ / / G^^ (r, t I r', t') • ^ " ^ (r') I, {y') E^ (r', t') dv'dt\
H'o (r, t) = r
[ff
J ~oo J J J D
G5, (r, t \ r', t') • w'' (r') /Q (r') E* (r', t') dv'dt'.
J -OO J J JD
11.2.5
Iterative migration in the time domain
Obviously Aa^a (r), the migration apparent conductivity in the time domain, provides only the first approximation to the real anomalous conductivity. To improve the resolution of the method, we can repeat the same procedure, which results in an iterative time domain migration. It can be implemented using the general scheme (5.151) of the re-weighted, regularized, conjugate gradient method in the space of the weighted conductivities, introduced according to formula (11.57). The general iterative process can be described by the standard formulae, which we reproduce here for convenience: <+i
= < + ^< = <-
KtiO,
(11-101)
where ^wi^n)
— ^wn — C n + / ^ n C ( n - l ) 5 C o — C o ' Pn — W^wnUM / | r w ( n - l ) | | j ^
(11.102)
Electromagnetic migration imaging
358
///J':»|'H [///.
-1
K{n-l)\^d^
The weighted regularized gradient direction on the nth iteration l^^ can be calculated by a formula derived from (11.85), by analogy with the formula (11.60):
OO
E A . - (r, - i) . E ; . (r,t) dt + a {a^ - cr^ ,) / •OO where In is the gradient direction at the nth iteration,
w
(11.103)
OO
/
•OO
Ei^^ (r,t) is the field calculated by forward modeling for the geoelectrical model with the conductivity distribution an (r), and E^""^ (r, — t) is the migration of the residual field E^"", H^", which is the difference between the observed field and the theoretical predicted field E^^ = E (a^): E ^ - (x', o, u) = E-, (r,t) - Eots (r,0 , (11.104) The optimal length of the step kn can be determined from a formula similar to (11.100) kn = ^ r
11 [H^- (r', t') X E ^ - (r, t) + H ^ - (r, t) x E^- (r', t') -n] ds'dt'x
r
[[ E'- (r',tO X H^'^ (r^tO -nds'dt]
J-OOJJE
(11.105)
J
where {E^^,H^"} is the electromagnetic field calculated for the model an and perturbed in the direction w~^ln-\-o^ {an — cTapr)- Note that, on each step, we recompute the real conductivities from the weighted conductivities at the nth iteration:
w-
(11.106)
Thus, we can describe the developed method of electromagnetic inversion as a process of iterative migration. On every step of iteration we calculate the theoretical electromagnetic response for the given geoelectrical model an (r) obtained on the previous step, calculate the residual field between this response and the observed field, and then migrate the residual field. The gradient direction is computed as a vector cross-correlation between the migrated residual field and the theoretical predicted field E^^. Using this gradient direction and the corresponding value of the
Electromagnetic migration in the time domain
359
optimal length of the step fc^, and applying the corresponding weights, we calculate the new geoelectrical model cr^+i (r) on the basis of expressions (11.101), (11.102), and (11.106). The iterations are terminated when the functional $ (a) reaches the level of the noise energy. The optimal value of the regularization parameter a is selected using the conventional principles of regularization theory discussed in Chapter 2. The results of the theoretical analysis presented in this chapter demonstrate that there is a very close connection between the method of electromagnetic migration and the solution of the conventional electromagnetic inverse problem. Actually, we can say that migration is an approximate inversion. It realizes the first iteration in the inversion algorithm with a minimization of the residual field energy flow through the profile or surface of observations. Iterative migration off"ers a method of interpreting electromagnetic data which combines the ideas of downward continuation and traditional inversion. Numerical modeling demonstrates that migration generates images of the subsurface structures in order of magnitude faster than traditional inversion, because of the availability of fast algorithms for downward continuation (Zhdanov et al., 1996; Zhdanov and Traynin, 1997). However, there are some limitations in using migration for the interpretation of electromagnetic data. The main problem is that the background conductivity distribution used for migration should be known a priori, while in the case of conventional inversion it is usually generated in the process of solving the inverse problem. Diff'erent ways of solving this problem have been discussed in Zhdanov et al. (1995, 1996).
References to Chapter 11 Claerbout, J. F, 1985, Imaging the Earth's interior: Blackwell Scientific Publications, Oxford, London, Edinburgh, 399 pp. Lee, S., McMechan, G.A., and S.L.V. Aiken, 1987, Phase-field imaging: The electromagnetic equivalent of seismic migration: Geophysics, 52, 679-693. Oristaglio, M., and G. Hohmann, 1984, Diffusion of electromagnetic fields into a two-dimensional Earth: A finite-difference approach: Geophysics, 49, 870-894. Schneider, W. A., 1978, Integral formulation for migration in two and three dimensions: Geophysics, 43, 49-76. Wang, T., and G. W. Hohmann, 1993, A finite-difference time domain solution for three-diraensional electromagnetic modeling: Geophysics, 58, 797-809. Wang, T., M. Oristaglio, Tripp, A., and G. W. Hohmann, 1994, Inversion of diffusive electromagnetic data by a conjugate-gradient method: Radio Science, 9, No. 4, 1143-1156. Zhdanov, M. S.., and M. A. Frenkel, 1983a, The solution of the inverse problems- on the basis of the analytical continuation of the transient electromagnetic field in reverse time: J. Geomag. Geoelectr., 35, 747-765. • Zhdanov, M. S.., and M. A. Frenkel, 1983b, Electromagnetic migration: in Hjelt, S. E.., Ed., The development of the Deep Geoelectric model of the Baltic Shield,
360
Electromagnetic migration imaging
Part 2, Univ. of Oulu, Oulu, 37-58. Zhdanov, M. S.., Matusevich, V. Yu.., and M. A. Prenkel, 1988, Seismic and electromagnetic migration: Nauka (in Russian), 376 pp. Zhdanov, M. S., 1988, Integral transforms in geophysics: Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo, 367 pp. Zhdanov, M. S., and G. Keller, 1994, The geoelectrical methods in geophysical exploration: Elsevier, Amsterdam, London, New York, Tokyo, 873 pp. Zhdanov, M. S., Traynin, P., and O. Portniaguine, 1995, Resistivity imaging by time domain electromagnetic migration: Exploration Geophysics, 26, 186-194. Zhdanov, M. S., Traynin, P., and J. Booker, 1996, Underground imaging by frequency domain electromagnetic migration: Geophysics, 6 1 , 666-682. Zhdanov, M. S., and P, Traynin, 1997, Migration versus inversion in electromagnetic imaging technique: J. Geomag. Geoelctr., 49, 1415-1437. Zhdanov, M. S., and O. Portniaguine, 1997, Time domain electromagnetic migration in the solution of the inverse problems: Geophysical Journal International, 131, 293-309. Zhdanov, M. S., and Li, W., 1997, 2-D finite difference time domain electromagnetic migration: 67th SEG Annual International Meeting, Dallas, Texas, 370-373. Zhdanov, M. S., 1999, Electromagnetic migration: in Deep Electromagnetic Exploration, Springer-Verlag, Narosa Pubhshing House, New Delhi, 283-298. Zhdanov, M. S., 2001, Method of broad band electromagnetic holographic imaging: US Patent # 6,253,100 Bl.
Chapter 12 DIFFERENTIAL METHODS IN ELECTROMAGNETIC MODELING AND INVERSION In the previous chapters we developed numerical methods of electromagnetic modeling and inversion based on discretization of the corresponding integral representations for the electromagnetic field. However, we can arrive at numerical solutions directly based on the differential form of Maxwell's equations. This approach has its own advantages and disadvantages over the integral equation method. The advantage is that one can apply a very flexible grid for discretization of the model parameters using differential methods. The disadvantage is that we now have to discretize the entire modeling domain, while in the integral equation method one should discretize only the domain with the anomalous conductivity distribution. Differential methods have found wide applications in geophysics because of the ease with which they handle a very complicated model of geological structures. I will discuss in this chapter the basic ideas of using differential methods in electromagnetic forward modeling and inversion.
12.1
Electromagnetic modeling as a boundary-value problem
In most geophysical applications of electromagnetic methods, it is necessary to model geoelectrical structures of quite arbitrary shape and size, with anomalous conductivity varying in an arbitrary manner and not necessarily restricted to a local region. The most widely used approach to forward modeling of such problems is through the use of finite difference and finite element methods to find numerical solutions to Maxwell's equations written in differential form (Weaver, 1994; Zhdanov et al., 1997). 12.1.1
Field equations and boundary conditions
Let us assume that the region of modeling, V, is bounded by a surface dV. For numerical modeling we can use the original coupled first-order Maxwell's equations (8.23), or the corresponding second order partial differential equations (8.29) for electric or magnetic fields. In the first case we should formulate and solve the corresponding boundary-value problem for electric and magnetic fields simultaneously. In the second case, one can solve separately the corresponding equations for the electric or
362
Differential methods in electromagnetic modeling and inversion
magnetic fields, which we reproduce here for convenience: ^ V x ( - V X E j - fc^E =iujfir, 5Vx flv
X H j - k^H =aVx fljA
,
(12.1)
where j ^ is the density of extraneous electric current;fc^is the corresponding wave number: _ k^=iu;/2a = icufia 4- ou'^fxe. (12.2) In the case of the quasi stationary field, which is the common case in geophysical exploration, we usually ignore the displacement currents (see Chapter 8), which results in a simplification of system (12.1): / i V x ( - V X E j -fc^E=zu;/ij^, aVx
( - V XH j - P H =aVx ( - A ,
(12.3)
where the wave number k'^ is given by k^=iuj/jLa.
Solving the corresponding equation for the electric field, we can determine the magnetic field from the second Maxwell's equation: H - — V X E.
(12.4)
In a similar way, solving the corresponding equation for the magnetic field, we can determine the electric field from the first Maxwell's equation, E=ivxH-ir, a a which is simplified for the quasi-stationary field outside the source: E = i v X H.
(12.5)
(12.6)
(J
Maxwell's equations (8.23), or the second order diflFerential equations (12.1) and (12.3), are supplemented with a boundary-value condition, i.e. with the additional equations for the electric or magnetic fields on the boundary dV of the volume V, UE = RE, or UH = R//, (12.7)
Electromagnetic modeling as a boundary-value problem.
363
where L5 is the corresponding boundary differential operator, and R^; and R// are the boundary values for the electric or magnetic fields. Note that the traditional statements of the boundary-value problems are based on application of Dirichlet boundary-value conditions of the first, second, or the third order, formed by means of linear combinations of the field itself and its derivative normal to the boundary. Dirichlet boundary conditions of the first order fix the values of the field at the boundary. Dirichlet boundary conditions of the second order, or Neumann boundary conditions, fix the value of the gradient of the field normal to the boundary; and Dirichlet boundary conditions of the third order, or Cauchy boundary conditions, fix both the value and the normal gradient of the field at the boundary (Morse and Feshbach, 1953). Usually, the boundary of the modeling volume is set so far from the conductivity anomaly that it is possible to neglect the anomalous field there. In this case, the simplest Dirichlet boundary conditions of the first order can be implemented by choosing the background (normal) field values as the boundary values. The tackground field is usually selected as one-dimensional, but it can be two-dimensional, if the background distribution of the electromagnetic parameters is two-dimensional. Another approach is based on the simplest Neumann boundary conditions which requires that the normal gradient of the field is zero on the boundary. Note however, that application of the aforementioned simple conditions requires the size of the modeling region to exceed the size of the inhomogeneous region Va many times over, in order to be able to neglect the effect of the anomalous field at the boundary. To overcome this limitation one can use asymptotic boundary conditions, developed for 2-D models by Weaver and Brewitt-Taylor (1978), and extended to 3-D models by Zhdanov et al. (1982) and Berdichevsky and Zhdanov (1984). These conditions are based on the analysis of the asymptotic behavior of the electromagnetic field far away from the geoelectrical anomalies. For example, in the three-dimensional case the asymptotic boundary condition for a magnetic field in free space is written in the form L6 = (1 + r . V) [H (r) - H ^ (r)] = 0. (12.8) The accuracy of condition (12.8) is estimated as O ( l / |r| ) . One can find expressions for asymptotic boundary conditions of higher order of accuracy with respect to distance, |r|, in Berdichevsky and Zhdanov (1984). In high frequency modeling the absorbing boundary conditions (ABC), based on the Perfect Matched Layer (PML) can be used as well (Berenger, 1993; Chew and Weedon, 1994). In the framework of this approach we introduce an artificial layer at the boundary, filled with PML material, which absorbs the outgoing electromagnetic waves, so that almost no reflections occur at the boundary. The absorbing boundary conditions dramatically reduce the required modeling region in microwave frequency applications. The system of equations defined by (8.23) and (12.1) or (12.3), and by the corresponding Dirichlet (12.7), asymptotic (12.8), or absorbing boundary conditions.
364
Differential methods in electromagnetic modeling and inversion
forms a boundary-value problem for the electromagnetic field. In geophysical apphcations, it is important to incorporate different types of sources in electromagnetic modeling. The most convenient way to do this is based on the separation of the field into the background (normal) and anomalous parts, considered in Chapter 9: E = E^ + E ^ H = H^ H- H ^
(12.9)
where the background field is a field generated by the given sources j ^ in the model with the background (normal) distribution of conductivity a^, a,nd the anomalous field is produced by the anomalous conductivity distribution A ? . The advantage of this approach is that we can use known analytical solutions for the background field and solve numerically the boundary-value problem for the anomalous field only. According to equations (9.3), the corresponding Maxwell's equations for the anomalous field have the form (12.10) V X E^ = zcj/iH",
where r (r) = A 5 (r) E (r) = A 5 (r) [E^ (r) + E« (r)]
(12.11)
is the density of excess electric currents within the inhomogeneity D. Substituting the excess electric current j ^ for the extraneous electric current j ^ , and the background conductivity at, for the total conductivity a in equation (12.1), we obtain the separate equations for the electric and magnetic anomalous fields: /iVx ( - V X E"j -klE''=iufii^,
?6Vx f^V
X nA - klH^^atVx f ^ A ,
(12.12)
where kl=iujfj,ab = iooficFiy -\- uj^iie. Taking into account (12.11), we finally find after some algebra, / / V x ( - V X E ^ j - fc2E«==^a;/iA5E^
Vx (^V
X w\
- iujfiW - Vx (^w\
= Vx (^E^)
•
(12.13)
Calculating the anomalous electric field from the second Maxwell's equation, and substituting it back into (12.13), we can cast the magnetic field equation in the form
Electromagnetic modeling as a boundary-value problem
V x flv
X H " j - 2Ct;/iH^= V x ( ^ E M
365
(12.14)
In the case of the quasi-stationary field, equations (12.13) and (12.14) can be written as follows: /xVx ( - V X E^ j - A:2E^-^a;/xAc^E^ o-Vx ( - V X H.A - A^^H^-aVx ( — E M .
(12.15)
We can use Dirichlet boundary-value conditions (12.7), asymptotic boundary conditions (12.8), or absorbing boundary conditions in the formulation of the boundary-value problem for the anomalous electromagnetic field. 12.1.2 Electromagnetic potential equations and boundary conditions Another approach to the formulation of electromagnetic boundary-value problem is to use the electromagnetic potentials introduced in Chapter 8. This approach has been used in a number of pubhcations on numerical electromagnetic methods (Biro and Preis, 1990; Everett and Schultz, 1996; Everett, 1999; Haber et al., 1999). In the framework of this approach we solve the corresponding second order partial diff"erential equations (8.42) and (8.39) for electromagnetic potentials, A and U: V^A -h IIQCJ {iujA - VU) = -MoJ%
(12.16) V-[5(ia;A-V[/)] = - V - f . The electromagnetic field is expressed in terms of these potentials by: E = ZCJA - Vf/,
(12.17)
H = - ^ V X A,
(12.18)
where the vector potential A satisfies the Coulomb gauge condition V-A=:0.
(12.19)
The advantage of this approach is in the simplicity of both the potential equations and the boundary-value conditions. Biro and Preis (1990) demonstrated that the Coulomb gauge can be enforced by the following boundary-value condition on the surface dV of the modeling region: A . n|^^ = 0.
(12.20)
This condition is motivated by the integral identity / / / V • Ac/^= / / A . nds. J J Jv J JdV
(12.21)
366
Differential methods in electromagnetic modeling and inversion
Due to the Coulomb gauge, the volume integral on the left-hand side of (12.21) is equal to zero. The boundary condition (12.20) provides the same result as well. Similar to the field separation into the background and anomalous parts, one can represent the electromagnetic potentials as the sums of the corresponding potentials for the background and anomalous fields: A = A^ + A ^ U = U^ + f/^,
(12.22)
where E^ =
IOJA!'
- VU\
W =
ZCJA"
- Vt/^
(12.23)
The anomalous field potentials satisfy the equations obtained from equations (12.16) by substituting the excess electric current j ^ , equation (12.11), for the extraneous electric current 'f, and the background conductivity a^ for the total conductivity 5: V ' A " + /io?, (ia;A« - VC/") = -/XoJ% V- [ab (icjA^ - V[/")] = - V • j ^
(12.24)
Taking into account expressions (12.11) and (12.23), we obtain V^A" -h /io5 (za;A« - VC/^) - -/XoA5E^ V- [G (ZCJA^ - VC/")] = - V • A 5 E ^
(12.25)
The formulation of the boundary-value problem for potentials is completed by adding the corresponding boundary conditions: A . n|^^ = 0, and
r)TJ\
—
dV
= 0.
(12.26)
Thus, in this case we have to solve the boundary-value problem for two electromagnetic potentials, vector and scalar, simultaneously. 12.2
Finite difference approximation of the boundary-value problem
The finite difference method provides a simple but effective tool for numerically solving the electromagnetic boundary-value problem (Weaver and Brewitt-Taylor, 1978; Zhdanov et al., 1982, Zhdanov and Spichak, 1992; Weaver, 1994; Mackie et al., 1993, 1994; Smith, 1996; Zhdanov et al.,1997; Spichak, 1999). It is based on a replacement of the differential operators in equations (12.1), (12.3), (12.13), (12.15), (12.16), and (12.25) by the finite difference operators, acting on the discrete values of the electromagnetic field. There are several ways to obtain the discrete analogs of Maxwell's equations. In the vast majority of finite difference algorithms, the model region is discretized into a number of prisms as shown in Figure 12-1. A Cartesian coordinate system is defined with the 2:-axis directed downward, and the x-axis directed to the
Finite difference approximation of tfie boundary-value problem
y ^ ^
367
X
'\^M
^
1
0
a
i
i
1
i
2 3
1: /
1 Figure 12-1 The model region is discretized into a number of prisms. The indices i,k, and / are used to number the grid point in the x,y, and z directions, respectively. Electromagnetic parameters, cr, / i , and e, are assumed to be constant within each elementary prism.
right. The indices z, A:, and / are used to number the grid point in x,y, and z directions, respectively. The electromagnetic parameters, cr, //, and e, are assumed to be constant within each elementary prism. We denote this grid by ^ : X\ — X ,
Xj\ij — X ,
Xi^i
— Xi T" /AX^,
I
z = l,2,...,7V,
E
yi = y\ {xi,yk,zi)
VNK
= y"^
y^+i = ^//c + Ay^, k = l,2,...,NK
>.
/ = l,2,...,iV^ 12.2.1 Discretization of Maxwell ^s equations using a staggered grid One common technique of field discretization is based on a staggered-grid scheme (Yee, 1966; Wang and Hohmann, 1993), which is effective in solving the coupled first order Maxwell's equations. On a staggered grid, the electric field components, parallel to the corresponding edges, are sampled at the centers of the prism edges, while the magnetic field components, orthogonal to corresponding faces, are sampled at the centers of the prism faces, as shown in Figure 12-2. The advantage of the staggered grid is that the corresponding electric field E components and magnetic field B =/iH components, selected above, are continuous on the edges and faces of the homogeneous prisms. Thus, the staggered grid is composed of two elementary loops (Wang and Hohmann, 1993): the electric loop is formed by four E components
Differential methods in electromagnetic modeling and inversion
368
Axi
(U,l)
£^V
(i+U,l)
Ex
Ey
(U+U) Ex ^
Ez y
y
By y
(Ttiti)
ET
Bx f
K Ez
Ez a(i,k,l)
|Li(i,k,l)
Figure 12-2 Discretization of the electromagnetic field on a staggered grid, according to Yee (1966) and Wang and Hohman (1993). The electric field components, parallel to the corresponding edges, are sampled at the centers of the prism edges, while the magnetic field components, orthogonal to corresponding faces, are sampled at the centers of the prism faces.
parallel to the edges of one face of a prism, and the magnetic loop is composed of four adjacent B components, as shown in Figure 12-3. The concept of electric and magnetic loops in a staggered grid is useful in discretizing the Maxwell's equations. For example, from the second Maxwell's equation (12.4) we have B = — V X E.
(12.27)
Integrating this equation over a face S of the prism, bounded by a rectangular contour L, and using Stokes's theorem, we find / / B . nd5 - — / / (V X E) -nds = — / E • rdl,
(12.28)
where n is a unit vector, normal to the prism face, 5, and r is the corresponding unit vector tangential to the prism edges, L; the directions of n and r are consistent with the corkscrew rule, i.e., with a corkscrew rotating in the direction r , the point moves in the direction n. We can evaluate the integrals in equation (12.28) approximately in terms of the discretized functions of the electric and magnetic fields. For node (z,A:,Z), the x,?/, and z components of electric field E are sampled at (i -h |,A:,/), (z, /c -f ^, O5 ^^d (^7 ^5^+2)' ^respectively, and the x, y, and z components of magnetic field B are sampled at (i,/c-f^,/-(-|), (z-|-|. A:, Z-h|), and (i-h|,/c + | , / ) , respectively. For example, at a prism face parallel to the vertical plane xz (see Figure 12-3) the
Finite difference approximation of the boundary-value problem
369
Axi
Bx (i,k, 1) N
1
1
1
1
1 i Bi N < r- — — *^^^———'
Ez^
r
Bx
Figure 12-3 The concept of electric and nnagnetic loops in a staggered grid in discretizing Maxwell's equations (according to Wang and Hohman, 1993). The electric loop is formed by four E components parallel to the edges of one face of a prism, and the magnetic loop is composed of four adjacent B components.
370
Differential methods in electromagnetic modeling and inversion
discrete version of the integral equation (12.28) is as follows: iujB:, ( ^, ^ + 2 ' ^ "^ 2 ' ^y^^^^
+
^
1 M -Ey(i,k-\--,l-\-l
Ey[i,k-^-Jj
EAi,k,l-\-
EAi,k-{-l,l-^-
Azi.
(12.29)
Similar algebraic expressions can be obtained for other faces of an elementary prism. We derive analogous formulae by numerically integrating the first Maxwell's equation. Combining all these algebraic equations together, we arrive at a full system of linear equations for the discretized values of the electromagnetic field on a staggered grid. Note that in practical implementation it is useful to transform an unequally gridded model into one that is equally gridded by making appropriate transformations of the size of the cells, the conductivity and permeabihty, and the field values (Madden and Mackie, 1989). Following Mackie et al. (1993), we denote the transformed parameters by a prime superscript, and introduce the scaling factors for the transformation as Ax Ay Az (12.30) l.Pz Px Ax'' Py Az' Au We can introduce now the transformed sors as _ crPyPz/Px 0 0 ^^PyPz/Px 0 0
A^
conductivity and magnetic permeability ten0 0 crPxPz/Py I, 0 (ypxPy/Pz 0 ^PxPz/Py 0
(12.31)
0
(12.32) fJ^PxPy/Pz J
In the majority of geophysical applications we assume that the medium is isotropic. However, in some important practical cases, for example in well-logging, anisotropic properties of the rock formations can play a significant role. In this case we can easily generalize the staggered grid technique for anisotropic models by taking into account the tensor character of the conductivity and permeability and applying formulae similar to (12.31)-(12.32): CTxxPyPz/Px
0
0
/^
0
0
l^xxPyPz/Px
0
0
0
0
(12.33)
(^yyPxPz/Py CTzzPxPy/Pz 0
(12.34)
^yyPxPz/Py
0
fi^^p^Py/p^
Finite difference approximation of tfie boundary-value problem
371
In the last formulae we assume that (Txx^cryy^o-zz and l^xx^ f^yy^ l^zz ^^^ ^^e principal values of the conductivity and permeability tensors, and the axes of the grid are oriented along the principal axes of these tensors. As a result of scaling we obtain a system of linear equations for discretized values of the electromagnetic field on a staggered grid, (D5-[-za;/LAcr) e = zcj/xc,
(12.35)
where e is the 3N vector (A'' is the number of the mesh nodes) of unknown values of the electric field; a and /x are the diagonal matrices of the conductivities and permeabilities in the cells of the grid; Dg is the complex-symmetric SN x 3N matrix of coefficients for the system, which is independent of frequency, u and sparse with 13 nonzero entries per row; and c is the vector of free terms, determined by the source and the boundary-value conditions. 12.2.2 Discretization of the second order differential equations using the balance method Another approach to discretization of electromagnetic field equations is based on the balance method (Zhdanov et al., 1982; Samarsky, 1984; Zhdanov and Spichak, 1989, 1992; Spichak, 1999; Mehanee and Zhdanov, 2001). This method involves integrating the original diflferential equations over each cell of the rectangular grid ^ and discretizing the corresponding system of integral equations. The advantage of this approach is that it automatically preserves the current balance in the volume and the corresponding charge conservation law. We will illustrate this method by the solution of the second order partial differential equation (12.15) for the quasi-stationary anomalous electric field in a medium with the constant magnetic permeability, /XQ: V x (V X E") - ^cc;AXoC^E^=^^MAc^E^
(12.36)
Using the vector identity given earher in equation (8.14), we can write V (V • E") -V^E^=icj/XocrE" + ^cJ/iAc^E^
(12.37)
The magnetic field H^ is expressed in terms of E" : H^ - — V X E".
(12.38)
100(1
Taking the divergence of the first Maxwell's equation for total and background field (outside a domain with the extraneous currents j^), V • a E =0, and V • a^E^ = 0, we obtain V • (E"+E^) = - (E" + E^) . — = - (E" + E^) - V i n a ,
372
Differential methods in electromagnetic modeling and inversion
and V-E^=-E^-Vlna6. Therefore V • E ^ - E ^ • V In at = - E ^ - V In a - E^-V In a, and V - E " = - E ^ - V l n c r - E ', -6V- l^ n —^.
(12.39)
As a result, equation (12.37) can be rewritten as - V (E^-Vlna) -V^E"-iu;/io^E^ = iujfiAaE^ + V ( E ^ - V l n — j .
(12.40)
In the balance method, the fields are discretized on a rectangular, uneven three-dimensional mesh ^ consisting of the nodal points at the centers of each cell of the original mesh ^ , Xi^^ = Xi + Axi/2,
i = 1,..., Nj - 1 1
zi^i =zi-\- Azi/2,
I = 1,..., NL - 1
We introduce the discretized vector function E-^i^,^!^.^! = E{x^_^i, y^_^i .Zj^i) and the discretized scalar functions a^_^i j^_^i i_^i = cr{x^_^i, y^^i, Zi_^i) and Acr^_^i f._^i i_^ = Aa{x^^,yk^,zi^) on the mesh ^ . In constructing a proper finite diff'erence scheme for solution of this problem by the balance method, we do not use equation (12.40), but rather, an integral identity obtained by integrating equation (12.40) over an elementary cell Viki of the mesh Yl based on the vector statements of the Gauss theorem (Zhdanov, 1988): ///
V ( E " - V l n ( j ) d t ; = [[
[ff
V^E''dv=
[[
(E^-Vlna) nds, (n-V)E"d5.
(12.41) (12.42)
As the result, we obtain -
//
(E^-V In a) nds -
If
ff
(E^-V\n—)nds
(n • V ) E«d5 - iuufx^ [ff
+ iuj/io I
I
AaE^dv,
aE^'dv =
(12.43)
Finite difference approximation
of tlie boundary-value
373
problem
where Siki is the rectangular boundary of the cell Viki, formed by six sides, right and left, S^^i and S~^i, front and back, S^^i and 5"^^, and bottom and top, S^^i and Sm = 5 i ^ , U Sr,^, U S^l U 5^^, U 5^^, U 5 - , , and n is the unit vector normal to it and directed out of the volume. We can evaluate the volume and surface integrals in equation (12.43) approximately in terms of the discretized vector and scalar functions E^^i^_^i^_^i and CTj+i ^_f.i /_^i. This corresponds to assigning the electric field to the center of the prismatic cells with homogeneous conductivity. In particular, we can use a simple relationship, ^^Mo / / /
^E«^t;^2a;/Xoa,+ i,fc+i,/+iE^^^^^^i^^^il/,fcz,
(12.44)
where Viki is the volume of the corresponding cell. In a similar way we can write for the background field ^^Mo JJJ
^(J^'dv
= za;MoAa,+ i^,+ i,,^iEj^i^^^i^^^iy,fc/.
(12.45)
The surface integrals are computed using a simple difference form. For example: /£,„.V,E"., = / £ f . . ,
(1.46)
and the derivative dFi^/dn is estimated as
aE"
Ea
rpa
^^2'^^2'^^2
dx
t-t-2 >«-t-2'^^ 2
/-J2
A7)
AXi + AXi+l
In a similar way we can evaluate the second surface integral / / .Sik
(E"-Vlna)nd5.
(12.48)
The derivatives of Incr are calculated using a three-point finite-difference scheme. The values of the electric field on the sides S^^f^, '^tjk^ ^"tjk^ ^f ^^^ ^^11 ^ijk ^^e approximated by the corresponding average values of the field in the nodes of the grid
E" S - , = 5 (E--,i..+i,+| + E«^i^,^i,^i) , etc.
(12.49)
374
Differential methods in electromagnetic modeling and inversion
71
"
'
/
7
/
1 1 .Vx
/ •m/
/ m//
1
/
i
/ Figure 12-4 Within the framework of the balance method, the resulting stencil for electric field E has seven points, located at the center of the prismatic conductivity cell, and at the centers of six adjacent cells.
The surface integrals are calculated using the rectangular rule. The resulting stencil for the electric field E has seven points, located at the center of the prismatic conductivity cell, and at the centers of six adjacent cells, as shov^n in Figure 12-4. The resulting system of linear algebraic equations and the accompanying boundary conditions can be expressed in matrix notation as (D-fia;/iocr) e" —
ZUJIIQ
[Acre^ -f b)
(12.50)
where e" and e^ are 3A^ vectors (where A^ is a number of the mesh nodes) of unknown values of the anomalous electric field and known values of the background electric field at the centers of the cells; a and ACT are the^iagonal matrices of the total and anomalous conductivities in the cells of the grid E, D is the 3N x 3N matrix of coefficients for the system, which is independent of frequency, LU; and b is the vector of free terms, determined by the boundary-value conditions. The structure of the matrix D essentially depends on the method used to order the vector e"^ and on the choice of boundary conditions. In the simplest case, the nodes of the mesh are numbered consecutively along the horizontal and vertical directions. Note that for the given numbering of the nodes, n = 1,2,3...., AT, (A^ = IKL) one can establish a simple one-to-one relationship between the index n and the triple number
n
i + (A: - 1) / -h (/ - 1) IK.
(12.51)
Finite difference approximation
of the boundary-value
problem
375
In this case, matrix D has a septa-block-diagonal structure:
m di"^
d^-(+^) 0....
0
^3
i(-^)
D =
2(-y)
J(0)
•^3
(+z)
0.. li+y)
0...
0...
0
..dil(+2/) ^N-IJ
"3
..0 (+2/) ..d N-I ..0
0..
0.. 0... 0.. ..0
..0 ..0
..0
\{-y)
0
l(0)
.0
..d N
and the vector e" has a structure e" = [ Ex,i
Ey^i
Ez^i
Ex,2
Ey^2
E^fi
E:, 'x,N
E.,N J^y,N
Ez ,N
In this form, the system (12.50) is readily solved using the methods of linear algebra, presented in Chapter 4 of this book. An important consideration in numerical modeling is control of the accuracy. Errors in modeling, in general, are caused by errors of three types: • 1) those due to the approximation involved in replacing a continuous function by a discretized function, • 2) those that arise in the solution of the finite diff'erence equations, and • 3) those involved in numerical differentiation of the numerical results to obtain the field components that are not discretized (the magnetic field in the case of electric field equation solution, and the electric field in the case of electric field equation solution). The most eff"ective check on computational errors of all three types is a criterion based on the integral relationships between the fields within the region of anomalous conductivity Va and at the surface of the earth. For example, the electric field at the observation points, r^, must satisfy the condition E(r,)
J J JVa
GE
(r, I r) • A a (r) E (r) dv + E^ (r,),
(12.52)
where G^; is the electric Green's tensor for the background (normal) conductivity model. Substituting the values for the electric field obtained from a finite difference computation scheme into the right hand part of equation (12.52) and then integrating numerically, we can compare the result obtained E'"^* with the finite difference solution W~^ at the observation points, and establish a criterion for the accuracy of modehng:
376
Differential methods in electromagnetic modeling and inversion
II E^nt _ ^f-d II / II ^f-d 11^ ^
^12.53)
In essence, the use of this criterion establishes agreement between a solution obtained by finite difference modeling and a solution obtained by the integral equation method. 12.2.3 Discretization of the electromagnetic potential differential equations We can use a staggered grid approach in combination with the balance method for discretizing the equations for electromagnetic potentials. We begin with the analysis of equations (12.25). Following to the basic principles of the balance method, we integrate these equations over an elementary cell Viki of the mesh ^ , and apply the Gauss theorem to the volume integral containing Laplacian and divergence operators: / / .SM
(n • V) A^ds + fJ-o
^ (i^A." - Vf/") dv =-^o
JJJVM
ff
// /
AaE^dv,
JJJVM
a {iojA" - VU") •nds = - ff
A5E'' • nds,
(12.54)
where Siki is the rectangular boundary of the cell Viku and n is the unit vector normal to it, directed outward. For the field and potential discretization, we will use the same staggered technique on the grid ^ introduced above. However, now we assume that the normal components of electric field E"^ are defined at the centers of the cell faces, while the tangential components of magnetic field H" are defined on the cell edges, as it shown in Figure 12-5. In other words, we introduce the discrete electric field functions
and discrete vector potential and gradient of the scalar potential functions,
and
uv
-vu^ix
.V.Z A
u'i^^^^y^^^^^l^)-u^i^^^^y^^^l^)
Finite difference approximation of the boundary-value problem
Axi
(i+l,k,l)
Hx (U+1,1) f-
1
Hz y
y
Ex f
K Hz
-'(U.l+l)
ir
kL
Hy
••^ Hx
p
^
377
Hz a(i,k,l) |^(U,1)
Figure 12-5 The staggered grid for electromagnetic field discretization in the electromagnetic potential method. The normal components of electric field E are defined at the centers of the cell faces, while the tangential components of magnetic field H are defined on the cell edges.
t/f+i fc+i / = ^zU\x,^.
y^^x.zi)
=
at the centers of the cell faces. The discrete conductivity functions cr^_|_i ;.^i^_^i = 5(x.^ 1,^^+1,2:^+1) and A5^+i^fc+i^^_^i = A5(x^i,yfc_^ 1,2^+1) are defined on the dual rectangular, uneven three-dimensional mesh ^ , introduced above. We can evaluate now the volume and surface integrals in equation (12.54) approximately in terms of the discretized functions. This can be achieved by using the relationships, similar to (12.44) and (12.46). In particular: Mo / / / A^ X
lU
*i+^,fc,/+i
5(^CJA" A- A^
- VU'')dv 1
\
^ -Mo5,+ i,fc+i,i+i X
/ ^ik^y^i
-^ f^r+i,fc+i,/+i
' ^ i+i,A:+l,Z+i
\ ^ r + i fc+i / + f^r+i fc+i/+i /
In the same way as in (12.46), we obtain (12.56) where the derivative dA°'/dn can be computed using a simple finite difference form.
Differential methods in electromagnetic modeling and inversion
378
Finally, J JSi,
a {iujA." - VU") • nds :
5J1kl ^ikh
where 5f^^, 5f^^, S'f^/ are the areas of the x,y, and z faces of the elementary prismatic cell of the grid J2 • The resulting system of linear algebraic equations and the accompanying boundary conditions for vector and scalar potentials can be expressed using matrix notations as CA
cu
(12.57)
where a is the vector of unknown values of the vector potential
and u is the vector of unknown values of the scalar potential
i,ULi ^t+^,fc,Z+i' ^ z + i fc+•4..) {^tA;+i,Z4-^',[/", ^lA 5 ^2A ,^3A jDit/, and T>2U ^-re the corresponding matrices of coefficients for the system, which are independent of frequency, a;; and c^, Cjj are the vectors of free terms, determined by the background field and the anomalous conductivity. The system (12.57) is solved using one of the methods described in Chapter 4. The electric field components are determined from the simple formula: E"" 'i,k+li+i
^^AVi/+^2 ~^iMh.i+h: *,"'-r2V
E\ E
iuA
— TJ^
The magnetic field on the staggered grid is computed using formulae similar to (12.29). The calculation for the magnetic field completes the numerical algorithm of electromagnetic modeling based on electromagnetic potentials.
Finite difference approximation of tfie boundary-value problem
12.2.4
379
Application of the spectral Lanczos decomposition method (SLDM) for solving the linear system of equations for discrete electromagnetic fields
The spectral Lanczos decomposition method is designed for effective calculation of the matrix functions. This method has found a useful application in the solution of linear equations in electromagnetic modeling (Druskin and Knizhnerman, 1994; Druskin et aL, 1999)). Let us consider, for example, the matrix equation (12.35). The formal solution of this equation has the form e = 20; (D5-f-z(x;/icr)~ /xc.
(12.58)
Equation (12.58) can be rewritten in the form e =iujf^ (D,) ^ c ,
(12.59)
where the function /^ is defined by: /^ = (D,+zcj/xo-)-'.
(12.60)
According to the basic formula (E.39) of SLDM method (see Appendix E) we can write (12.61) e^iuj \\^c\\ Qf^ (T) e[^^ = iuj ||/Ltc|| Q (T+iu;/xcr)~^ e[^\ In the last formula Q is the matrix of the orthogonal basis, Q = [qi,q2,
,qL],
of Krylov subspace, spanned over the vectors obtained as the result of multiplication of the vector /xc by the power of the square L x L matrix D5: XL = span{/xc, Ds/Ltc, . . . . , D ^ " V c } . Thus, matrix Q depends only on the coefficients of the matrix Dg. Vector e^ Ms an L vector, consisting of all zeros with exception of the first component, equal to one: e^ ^ = (1,0, ...,0, ...0). T is the tri-diagonal symmetric matrix, ai
/?!
...
Pi
OL2 P2
0
PL-1
0
O^L
obtained by matrix multiplication T - Q*D,Q.
(12.62)
380
Differential methods in electromagnetic modeling and inversion
The elements QJ, PJ and qj^i of matrices Q and T are obtained by the Lanczos algorithm, developed in Appendix E : pQ = 1, qo = 0, qi = /xc/ ||/xc|| ,
(a)
while Pj ^ 0, q^+i = rj/Pj,
(6)
ajl - q*D,q^-,
(12.63) ^3 = {^s-ocjl^)
^j - Pj-iqj-u
Pj = W^jW,
(c)
j = l, 2 , . . . , L - 1 , where the asterisk * indicates transposed complex conjugate matrices (note that in Appendix E we present the same algorithm for real matrices only). Note that the most expensive part of the numerical calculations is the determination of the matrix Q using the Lanczos method. This matrix depends only on the coefficients of the matrix Dg and the vector /xc. Therefore, due to the fact that matrix D5 does not depend on frequency, we should apply this decomposition only once for all frequency ranges (if also vector c does not depend on frequency, which is typical for many practical problems). The calculation of the inverse of the matrix (T-f-ia;/LAcr) is computationally a much simpler problem, because T is a tri-diagonal matrix, and /u and a are diagonal matrices. As a result, one application of SLDM allows us to solve forward problems for the entire frequency range. That is why SLDM increases the speed of solution of the forward problem by an order for multifrequency data. This is the main advantage of this method over any other approach. 12.3
Finite element solution of boundary-value problems
The method of finite elements is one of the most powerful techniques for numerical solution of boundary-value problems. The mathematical literature on the finite element method contains hundreds of publications. Foundations of this technique can be found in Tong and Rosettos (1977), Zienkiewicz and Taylor (1989), and Beltzer (1990). The method is based on a division of the modeling region into a number of comparatively small sub-regions (finite elements) and an approximation of the unknown function within each element as a combination of some basis functions. There are many different versions of the finite element method, depending on the type of the basis functions and the techniques for determining the coefficients of the basis function expansions. I will present here two useful versions. One is based on the Galerkin method, which we already discussed in Chapter 9, and the other is based on the exact element (EE) method (Godunov, 1959; Harten, et al, 1997). 12.3.1 Galerkin method We have discussed already the application of the Galerkin method to discretization of integral operator equations (see Chapter 9). The same technique can be used for discretization of electromagnetic field diff'erential equations as well.
381
Finite element solution of boundary-value problems
We introduce again a complex Hilbert space L2 {V) of the vector functions determined in the modehng region V and integrable in V with inner product:
(X' y)L2{v) = 111 X i^)' y* i^) dvV
In accord with standard practice, the region V in which calculations are to be made is divided into elements. For the two-dimensional case, the common practice is to divide the region into triangular elements, involving three points in the plane at a time. For three-dimensional problems, the equivalent element would be a tetrahedral element, De, involving four points in space. We introduce also a system of basis functions {vi ( r ) , V2 ( r ) , ....Vjv (r)} in the Hilbert space L2 (V^). In the simplest case, we can use as basis functions the linear functions of coordinates, x, y, z, which are determined at any point within one of the elements, De, but are zero outside. We introduce a finite dimensional Hilbert subspace, L2 , of the space L2 (D) \ L2 C L2 (D) spanned by the basis functions {vi ( r ) , V2 ( r ) , ....VTV (r)} . We will be looking for an approximate solution of the electromagnetic field equations in the form N
E(r)^^a„v,(r),
(12.64)
n=l N
H(r)«^6,v,(r),
(12.65)
n=l
where a^ bn {n = 1,2,..., N) are the scalar coefficients of the expansions. The second order electric or magnetic field equations (12.1) can be written in operator form as (L^ — iuj/ia) E =zct;/i-j®, (L^ - iiufia) H = 5 V x ( l A
,
(12.66)
where L^ and L^ are the second order differential operators: L^ = / i V x ( - V x L"" = aVx
(4vxj .
(12.67)
We substitute expansion (12.64) into the first equation (12.66) AT
^an
{L^ - iu)iia)Yn = 10011]^,
(12.68)
382
Differential methods in electromagnetic modeling and inversion
and, following the conventional technique of the Galerkin method, take the inner product of both sides of equation (12.68) with the basis function Vp : N
N
Y^an (L^v^, Vp)^^(^) -iu^^an n=l
(//5v^,'^P)L2{D) = ^^ (/^J^^P)L2{D) '
(12.69)
n=l
Introducing the notations (L^Vn, vp)^^(^) = L ^ , ( / i 5 v „ Vp)^^(^)
— (^pn
and if^i"^^p)L2iD)=Jp^
we can write equation (12.69) in the form N
N
y ^ ^ ^ ^ n - ^^ X ! ^pndn = lUJJp^
(12.70)
Let us evaluate the component L^ : L ^ = (L^v„, vp)j,^(^) = JJJ^ L''v„ (r) • v ; (r) dv
^IIIv^^ iw)^ "" ^^ ^""0 • ^ ^""^ ^^ ^""^ '^' Note that the common practice is to compute the inner product of the differential operator L^ and the basis function, using integration by parts, which in the 3-D case is based on the vector identity R V x P - P V x R
= V(PxR)
and the Gauss theorem: [ff R-V J J Jv
xPdv=
[[[ P'V J J Jv
xRdv^
[[ (P X R ) ds. J JdV
(12.72)
Taking into account (12.72), we can modify (12.71) as ^ ^ = III
( ^ ^ + 11 J JdV
^ ^ " W ) • ( V x / i ( r ) v ; ( r ) ) dv
( V X v„ (r)) x v ; (r) ds,
(12.73)
where the integral over dV is a surface integral over the boundary of the domain. The integrals in the last expression are usually calculated element by element due to
383
Finite element solution of boundary-value problems
the special selection of the basis functions in the finite-element method. Also, the surface integrals are mutually cancelled out for the neighboring elements with the joint faces. Only the surface integrals over the element faces forming the boundary of the modeling region are left, and are determined by the boundary condition of the boundary-value problem under consideration. As a result, we arrive at the following expression for the internal elements of the matrix Lpn : i^'"*' = 11 J^ ( ^ V
X v„ ( r ) ) • ( V x ; . (r) v ; (r)) dv.
(12.74)
Finally, we can cast equation (12.70) for the basis function coefficients in the matrix form {U-iuj^)ai=iujf, (12.75) where the matrices L^ and a, and vectors, a and j ^ , are given by: L^= [L^] , 5 = [ a ^ ] , a = [a„], j ^ = [j^] . Note that the structure of equation (12.75) is similar to the structure of the finite difference matrix equation (12.35). Thus one can use, for example, spectral Lanzcos decomposition to solve this equation: a =iu; {U-iuay'f=iuj
\\f\\ Q^ {T^-iuja)-'
e[^\
(12.76)
where Q^ and T^ are the matrices of the QT decomposition of the matrix L^. Substituting the coefficients a = [ai,a2, ...aiv] into expansion (12.64) we determine the electric field, and then, from the second Maxwell's equation, find the magnetic field: B = — V X E.
(12.77)
In a similar way we can find the magnetic field using expansion (12.65) and the corresponding second order differential equation. The matrix equation for the coefficients of magnetic field expansion has the form {L''-iLua)h=ioof,
(12.78)
where the matrix L^, and vectors, b and j ^ , are given by, ^"=[1;^],
h =
[b„],f=[j^],
and LU = III
( ~ | i : y V X v „ ( r ) ) . ( V x 5 ( r ) v ; ( r ) ) dv,
The solution of equation (12.78) by the spectral Lanzcos decomposition method is as follows: b =iu; {L''-iuja)-'f=iuj \\f\\ Q " {T'^-iua)-' ef\ (12.79) where Q"^ and T"^ are the matrices of the QT decomposition of the matrix U^.
384
Differential methods in electromagnetic modeling and inversion
12.3.2 Exact element method In the Galerkin method we could, in particular, select the basis functions as the exact analytical solutions of Maxwell's equations within each element (Harten, et al, 1997). We will call this technique the exact element (EE) method. In this case the volume integrals within each element equal zero. All discretization errors, therefore, exist only on the boundaries between elements. As a result, the EE method permits accurate representation of field discontinuities. In addition, since the scheme consists of many analytical solutions joined together, the approximation of the electromagnetic field in the form of expansions (12.64) and (12.65) over the basis functions, satisfies the full set of Maxwell's equations rather than separate equations of electric or magnetic field only. This property increases the accuracy of the numerical solution obtained by the EE method. Another advantage of the EE finite element scheme is that it accurately represents the field in cells with a high ratio of cell dimensions and with a high conductivity contrast between the different cells. The disadvantage of the EE scheme is its complexity. Typically, the EE scheme requires more coefficients per element than the conventional finite element method. Similar to the conventional finite element method, we divide the modeling region V into tetrahedral or prismatic elements. The electromagnetic parameters a and /i are constant within each element. We select within each element the basis functions of the Galerkin method as plane waves of different polarizations traveling in orthogonal directions. These basis functions are equal to zero outside a given element. Thus, within each element we have twelve different basis functions, which approximate the electric field according to the following formulae: Jbx = aiC ^ -f a2e tjy = a^e
^ -{- a^e
+ a^e
+ a-jc
4- a^^e -f a^e
, ,
E, = age^^" + aioe-^^^ + ane'^^ + ai2e-^^^
(12.80)
Applying a curl operator to both sides of (12.80) and dividing by iu/j. we derive the expressions for the magnetic field components: H, = —(ane^'^ - aue-'~'y - aye^^^ + age"^^^), LUjil
Z Jtiy = — [ a ^ e LJfl
^
^ — a^e
^ — age
-\- aioe
H, - —(ase^^^ - age-'^^ - aie^^^ + ase"'^^).
),
(12.81)
Direct calculations show that within any homogeneous domain the electromagnetic field described by formulae (12.80) and (12.81) satisfies the system of Maxwell's
385
Inversion based on differential methods
equations (8.23). Thus, in the EE method we approximate an electromagnetic field within each homogeneous element by a linear combination of twelve plane waves of diff'erent polarizations, which represent an analytical solution of Maxwell's equations,
E(r)=
E
EKK/3).
(12-82)
where the corresponding plane wave basis functions are equal to: p ^ ^ ( r ) - d , e x p ( ( - l ) ^ z ^ / 3 ) ; g = 0,l; a , / ? - x , 7 / , z ; a ^ / ? .
(12.83)
This expansion is described by twelve constant coefficients for each element Ve \ i^2% ^ — O5I5 ^^P — ^^y^^'i <^ 7^ P)^ ^^ equations (12.80) and (12.82), which have to be determined by the Galerkin method. The elements are joined together by continuity of the four tangential components of the electromagnetic field on each boundary. Omitting the long derivation, we can write the resulting system of algebraic equations for the coefficients of the EE scheme as follows: G a - c,
(12.84)
where G is the matrix of the EE scheme depending on the wave number distribution in the elements, and on the geometrical parameters of the elements; a is the global vector of the unknown coefficients of the analytical approximations (12.80); and c is the vector containing nonzero values only on the external boundary dV of the modeling region V. The linear system (12.84) can be solved by any numerical method outlined in Chapter 4. 12.4
Inversion based on differential methods
12.4-1 Formulation of the inverse problem on the discrete grid In differential methods we consider the solution of both forward and inverse electromagnetic problems on the discrete grid. This grid can be formed by rectangular cells, as in the case of the finite difference method (Figure 12-1), or by an arbitrary sub-region (finite elements), as in the case of the finite element method. We will number the grid cells, V^, within the modeling and inverting region V consecutively along the horizontal and vertical directions, q = 1,2,3, ...A/^m, where A^^ is the total number of the grid cells equal to the number of the model (anomalous conductivity) parameters. The conventional assumption is that the background and anomalous conductivities are constant within each cell. We denote by Aa^^^ the anomalous conductivity of cell Vq. We assume, for simplicity, that the receivers are located at the centers of some cells of the grid, so that the observed anomalous electric field E"^ ( r ) , for example, is given at the points r^, j = 1,2,3, ...A^d, where A^^ is the total number of the observation points.
386
Differential methods in electromagnetic modeling and inversion
The discrete electromagnetic inverse problem is described, as usual, by the relationship d = A(m), (12.85) where A is a nonlinear matrix operator, determined by the finite diflPerence or finite element solution of the Maxwell's equations; m is an Nm vector formed by an unknown set of anomalous conductivities (model parameters) within cells V^, d is the ?>Nd vector of data (for example, given values of the anomalous electric field at the receiver positions). The parametric functional for the discrete inverse problem (12.85) with a minimum norm stabilizer, can be written in the form P " ( m , d ) = (WrfA(m) - Wrfd)*(WrfA(m) - W^d) + a ( W ^ m - W^mapr)*(W^m - W^niapr),
(12.86)
where W^ and W ^ are the corresponding weighting matrices of data and model parameters (see Chapter 5), and vciapr is some a priori model. We can solve this minimization problem using one of the techniques developed in Chapter 5. For example, we can apply the RCG method, which for discrete model parameters and data can be cast in the form Rn = A(mn) - d,
(a)
1^ = P ^ ( m J ^ F ; W 2 R , + a . W ^ ( m , - m « , , ) ,
PT = l|in1lV||C--/|p,
(6)
I-=c+/3-fc\ c = c , (c)
nin+i = m ^ - f c ^ - I ^ - .
(12.87)
{d)
k^ is determined from the minimization problem F - (m,+i) = P - ( m , - ^-I^) = min .
(e)
The most critical part of the RCG algorithm (12.87) is computing the Frechet derivative matrix, F^, or applying the adjoint Frechet derivative matrix to the weighted residual field, F* W^R^. We will discuss below the solution of this problem in a discrete case. Note that one can use different stabilizers in the parametric functional (12.86), for example, the minimum support or minimum gradient support functionals introduced in Chapter 2. As a result, we will arrive at focusing inversion of EM data, similar to the method of focusing inversion of MT data, developed in Chapter 10. 12.4.2 Frechet derivative calculation using finite difference methods The problem of the Frechet derivative, or sensitivity matrix calculation for electromagnetic field has been examined in many pubhcations and was discussed in Chapter
387
Inversion based on differential methods
9. We can find the equations for the Frechet derivative by differentiating the corresponding electromagnetic field and potential equations. According to (9.48), the equations have the form
Vx6H = a6E-^f, (12.88) V x ^ E = iujfiSH, where 5a is the conductivity variation; 5H, 6E are the corresponding magnetic and electric field variations; j^=6aEi is the excess electric current within a domain of the perturbed conductivity. Note that equation (12.88) has exactly the same structure as equation (12.10). Therefore, we can apply any of the differential methods, introduced in this chapter, to find the magnetic and electric field sensitivities (variations 6H and ^E). For example, the corresponding system of finite difference equations is similar to system (12.35): (Ds+ioofio-) 6e = iujiidae^, (12.89) where^ Sa is the diagonal matrix of the perturbed conductivities in the cells of the grid E, and we assume that perturbations of magnetic permeability /x are equal to zero. If we would like to find the sensitivity of the electric field to the perturbation of the conductivity da in one cell, Vq, we have to substitute for e^ a vector e^ which has only three nonzero components, E^^ i , ^ i . i, E'^^ i . , , ., E^^ i . ^ i ,^ i, in cell (Ds-^iuj^a) ^e^^) = iujfi8ae\.
(12.90)
We denote by ^e^^^ the element of the vector 5e^^^ corresponding to the r^th receiver position. It can be treated as an electric field, generated by an electric source ^cre^, located in the cell Vq. The Frechet derivative matrix, F, is formed by the components 6ej /6a. Therefore, the direct, "brute force" method of computing the Frechet matrix would require A^"^ forward modeling solutions for each inversion iteration. We can reduce significantly the number of the required iterations using the reciprocity principle (Madden, 1972; Rodi, 1976; Madden and Mackie, 1989; McGillivray and Oldenburg, 1990; de Lugao and Wannamaker, 1996; and de Lugao et al., 1997) for Frechet derivative calculations. Due to reciprocity, the field 5e^^^ is equal to the J
Aj) electric field 6eq at the center r^ of cell V^, generated by the electric source Sae,9 ' located at the r^-th receiver position. Thus, solving one forward modeling for each receiver position gives us the required sensitivity matrix. Therefore, instead of solving the system of equations (12.90) Nm times (the number of unknown parameters grid cells) we can run forward modeling just Nd times (for every receiver position). As a result, the use of the reciprocity principle warrants a huge reduction in the required computations because, as a rule, the number of the receiver positions is much
388
Differential methods in electromagnetic modeling and inversion
smaller than the total number of the grid cells, Nd « Nm- The corresponding formulae for the sensitivities were developed in Chapter 9. We reproduce them here for convenience: E ^ ( r , ) - E ( r q ) , a = x,y,z,
(12.91)
where the auxiliary field E^ is generated by an elementary electric current, located at the receiver position, and polarized along x,y, or z axis; ^ ^ ^
= -E-K)-EK),
(12.92)
where the auxiliary field E ^ is generated by an elementary magnetic current, located at the receiver position, and polarized along x,y, or z axis. We can reduce the number of calculations even more dramatically by invoking the fact that we do not need to know the Frechet derivative matrix itself on the n-th iteration, but rather the result of its application to the weighted residual field F*W^Rn. In Chapter 10 (see formula (10.60)) we demonstrated that this term is equal to the scalar product between the complex conjugate electric field E^, computed at the n-th iteration, and the auxiliary complex conjugate electric field E^* due to the reciprocal sources on the observation surface: F : W X = E ; • E^:.
(12.93)
According to formula (10.59), the reciprocal field E^^ can be computed as the solution of the following system of equations: {"Dsn^ujfjian) e l = ia;/xRL,
(12.94)
where an is a diagonal matrix of conductivity in the cells of the grid E on the n-th iteration; D^^ is the matrix of coefficients for the system, determined for the conductivity model o^n] ^wn is a vector formed by the values of the field E ^ (r) in the cells of the grid E; and 'R^n is the vector of free terms, determined by the weighted residual field W^R^ in the receiver positions. Thus, we can conclude that the calculation of the term F* W^R^ requires just one additional forward modeling solution on each iteration for the auxiliary electric field E ^ due to the reciprocal sources.
References to Chapter 12 Berdichevsky, M. N., and M. S. Zhdanov, 1988, Advanced theory of deep geomagnetic sounding: Elsevier, Amsterdam, London, New York, Tokyo, 408 pp. Beltzer, A. I., 1990, Variational and Finite Element Methods, A symbolic computational approach: Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo, 254 pp.
Inversion based on differential methods
389
Biro, O., and K. Preis, 1990, Finite element analysis of 3-D eddy currents: IEEE Trans. Magn., 26, 418-423. Berenger, J., 1993, A perfectly matched layer for absorption of electromagnetic waves: J. Comput. Phys., 114, 185-200. Black, K. 1998, Spectral elements on infinite domain, SIAM J. on Sc. Comp., 1667-1681. Chew, W. C , and W. H. Weedon, 1994, A 3-D perfectly matched medium from modified Maxwell's equations with stretched coordinates: Microwave and Optical Tech. Lett., 7, 599-604. Coggon, J. Z., 1971, Electromagnetic and electrical modeling by the finiteelement method: Geophysics, 36, 132-155. de Lugao, P., and P. E. Wannamaker, Calculating the two-dimensional magnetotelluric Jacobian in finite elements using reciprocity, Geophys. J. Int., 127, 806-810, 1996. de Lugao, P., Portniaguine, O., and M. S. Zhdanov, 1997, Fast and stable two-dimensional inversion of magnetotelluric data: J. Geomag. Geoelectr., 49, 14691497. Druskin, V., and L. Knizhnerman, 1994, Spectral approach to solving threedimensional Maxwell's diffusion equations in the time and frequency domains: Radio Science, 29, No. 937-953. Druskin, V., Knizhnerman, L., and P. Lee, 1999, New spectral Lanczos decomposition method for induction modeling in arbitrary 3D geometry: Geophysics, 64, 701-706. Everett, M. E., and A. Schultz, 1996, Geomagnetic induction in a heterogeneous sphere: Azimuthally symmetric test computations and the response of an undulating 660-km discontinuity: J. Geophys. Res., 101, 2765-2783. Everett, M. E., 1999, Finite element formulation of electromagnetic induction with coupled potentials: Three-dimensional electromagnetics. Published by the Society of Exploration Geophysics, Tulsa, OK, 444-450. Godunov, S. K., 1959, Difference method of numerical calculating the break solutions of hydrodynamic equations (in Russian): Mat. Sb., 47, No. 3, 271-306. Haber, E., Ascher, U. M., Aruhah, D. A., and D. Oldenburg, 2000, Fast simulation of 3-D electromagnetic problems using potentials; Technical report. Institute of Applied Mathematics, University of British Columbia. Haber, E., 1999, Modeling of 3-D EM using potentials and mixed finite elements: Proc. 2nd Int. Symp. of Three-Dimensional Electromagnetics, Salt Lake City, UT, 12-17. Harten, A., Engquist, B., Osher, S., and S. R. Chakravarthy, 1997, Uniformly high-order accurate essentially non-oscillatory schemes, III: Journal of Computational Physics, 131, 3-47. Madden, T. R., 1972, Transmission system and network analogies to geophysical forward and inverse problems: Report 72-3, Department of Earth and Planetary
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Sciences, MIT, Cambridge, MA. Madden T. R., and R. L. Mackie, 1989, Three-dimensional magnetotelluric modeling and inversion: Proc. IEEE, 77, 318-333. Mackie, R. L., Madden, T. R., and R E. Wannamaker, 1993, Three-dimensional magnetotelluric modeling using difference equations - theory and comparisons to integral equation solutions: Geophysics, 58, 215-226. Mackie, R. L., Smith, J. T., and T. R. Madden, 1994, Three-dimensional electromagnetic modeling using finite difference equations: The magnetotelluric example: Radio Sci., 29, 923-935. McGillvray, P. R., and D. W. Oldenburg, 1990, Methods for calculating Frechet derivatives and sensitivities for the nonlinear inverse problem: a comparative study: Geophysical Prospecting, 38, 499-524. Mehanee, S., and M. S. Zhdanov, 2001, 3-D finite-difference forward modeling based on the balance method: 71st SEG Annual International Meeting, San Antonio, Texas, 1443-1446. Newman, G. A., and D. L. Alumbaugh, 1997, Three-dimensional massively parallel inversion - I. Theory: Geophys. J. Int., 128, 355-363. Newman, G. A., and D. L. Alumbaugh, 2000, Three-dimensional magnetotelluric inversion using non-linear conjugate gradients: Geophys. J. Int., 140, 410-424. Rodi, W. L., 1976, A technique for improving the accuracy of finite element solutions for magnetotelluric data: Geophys. J. R. Astr. Soc, 44, 483-506. Samarsky, A.A., 1984, Theory of the difference schemes (in Russian): Nauka Publishing House, Moscow, 656 pp. Smith, J. T., 1996, Conservative modehng of 3-D electromagnetic fields; Part II: Bi-conjugate gradient solution and an accelerator: Geophysics, 6 1 , 1319-1324. Spichak, V. V., 1999, Magnetotelluric fields in three-dimensional geoelectrical models (in Russian): Scientific World, Moscow, 204 pp. Tong, P., and J. N. Rossettos, 1977, Finite-element Method: MIT Press, Cambridge, 332 pp. Wang, T., and G. W. Hohmann, 1993, A finite difference time-domain solution for three dimensional electromagnetic modeling: Geophysics, 58, No. 6, 797-809. Wannamaker, P. E., Stodt, J. A.., and L. Rijo, 1987, A stable finite element solution for two-dimensional magnetotelluric modeling: Geoph. J. Roy. Astr. Soc, 88, 277-296. Weaver, J. T., and C. R. Brewitt-Taylor, 1978, Improved boundary conditions for the numerical solution of E-polarization problems in geomagnetic induction: Geophys. J. R. Astron. Soc, 87, 917-948. Weaver, J. T., 1994, Mathematical methods for geo-electromagnetic induction: Research Studies Press, Taunton, UK, 316 pp. Yee, K. S., 1966, Numerical solution of initial boundary problems involving Maxwell's equations in isotropic media: IEEE Trans. Ant. Prop., A P - 1 4 , 302-309. Zhdanov M. S., Varentsov, I. M., Weaver, J. T., Golubev, N. G., and V. A.
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Krylov, 1997, Methods for modeling electromagnetic fields. Results from COMMEMI - the international project on the comparison of modeling methods for electromagnetic induction: Journal of Applied Geophysics, 37, 1-273. Zhdanov, M. S., Golubev, N. G., Spichak, V. V., and I. M. Varentsov, 1982, The construction of effective methods for electromagnetic modeling: Geophys. J. R. Astr. Soc, 68, 589-607. Zemanian, A. H., 1987, Generalized Integral Transforms: Dover, New York, 300 pp. Zhdanov, M. S., 1988, Integral transforms in geophysics: Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo, 367 pp. Zhdanov, M. S., Varentsov, I. M., Golubev, N. G., and V. A. Krylov, 1990, Methods of modeling electromagnetic fields (in Russian): Nauka, Moscow, 199 pp. Zhdanov, M. S., and V. V. Spichak, 1989, Mathematical modeling of threedimensional quasi-stationary electromagnetic fields in geoelectrics (in Russian): DAN SSSR, 309 (1), 57-60. Zhdanov, M. S., and V. V. Spichak, 1992, Mathematical modeling of electromagnetic fields in three-dimensional inhomogeneous media (in Russian): Nauka, Moscow, 188 pp. Zienkiewicz, O. C., and R. L. Taylor, 1989, The finite element method. Fourth Edition: McGraw-Hill, New York.
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Part V Seismic I n v e r s i o n
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Chapter 13 WAVEFIELD EQUATIONS In this part of the text I will consider several applications of inversion theory to the solution of seismic inverse problems. It is impossible, of course, to cover in a book on general inversion theory all the variety of modern methods of seismic inversion. The selection of the material included in this part is governed by the primary goal outlined in the introduction - to demonstrate the close linkage between forward modeling and inversion methods for gravity, electromagnetic, and seismic fields. In particular, we will consider diff"erent wavefield migration and inversion techniques similar to electromagnetic migration and inversion methods discussed in the previous chapters. Elastic wavefields represent one of the most important types of geophysical fields. These fields are widely used in seismology and in seismic exploration to gain important information about the internal structure of the earth and the location of mineral deposits. In general cases, the elastic wave equations describe the propagation of these fields in the media. In this chapter I will discuss the general principles of elastic wave propagation. I have found it necessary to include this chapter in the monograph to provide a more complete account of the seismic inverse problem and to work out the standard forms for integral representations of the seismic field equations which we will frequently use in this part of the book. 13.1 13.1.1
Basic equations of elastic waves Deformation of an elastic body; deformation and stress tensors
Applying external forces to an elastic body we change the relative position of its different parts which results in a change in body size and shape, i.e. under stressed conditions an elastic body undergoes deformation. As the particles of a body are shifted with respect to each other, the body develops elastic forces, namely stresses, opposing the deformation. In the course of deformation these forces increase and at a certain instant of time they can even counter-balance the effect of the external stress. At this moment the deformation process comes to an end, and the body is in a state of elastic equilibrium. As the stress is removed gradually, the elastic body returns to its initial state; however, the abrupt disappearance of the outside force causes the particles inside the body to oscillate. To describe these oscillations, it is necessary to quantify the relationships between the forces arising at each point of the deformed
Wavefield equations
396
Before deformation
Figure 13-1
Small deformations of an elastic body.
elastic body and the magnitude of the deformation. Following Landau and Lifshitz (1970), we first describe small deformations of an elastic body. Let us assume that r is the radius-vector of some point M inside the elastic body prior to a deformation. Upon deformation, the point M is displaced by some vector U (the displacement vector or the deformation vector)^ so that the radius vector r' of the new position of the point M' is related to r by a simple formula (see Figure 13-1): r ' = r + U. (13.1) Evidently, the displacement vector U depends generally on the position of the initial point M (that is on the radius-vector r). In other words, we can consider a field of displacement vectors U = U(r) within a deformed elastic body. To describe the deformation in an elastic body, we will introduce the distances, |(ir| and |dr'|, between two adjacent points M and L prior to the deformation and after it. We can find the relationship between these two distances by differentiating expression (13.1) dr' = dr -f dU, and calculating the square of the distance |c/rf = |dr-f dU|^ = Y^
{drc^-^dUc^f
(13.2)
a=x,y,z
where dra and dUa {o^ = x,y,z) respectively.
are the scalar components of vectors dr and d\J,
397
Basic equations of elastic waves
In the framework of elasticity theory we consider, as a rule, continuous deformations, in which the displacement vectors are smooth (differentiable) functions of coordinates. Therefore, we can calculate the differential of the displacement vector as dUa = ~^drf3
= Ua^fsdrp, a, /? = x, y, z,
(13.3)
where Ua,/3 = dUa/dVjs, and the summation sign over the repeated index /? is omitted (this rule is known as the summation convention). In accordance with formulae (13.2) and (13.3) we have |c/rf
=
^
\dr + dVf=
(dr« + dC/^)^ =
a=x,y,z
\drf + 2Ua,(3dradrp + Ua^f^Ua.-ydr^dr^,
(13.4)
where we use again the summation convention for repeated indices. Since in the second term of the right-hand side of (13.4) summation is over both indices a and /3, we can exchange those indices as follows: '^Ua,f3dradrf3 = Ua^^dradrjs -h Uf^^otdradrp.
(13.5)
The indices in the third term of the right-hand side of relation (13.4) can be changed by substituting 7 for a, a for /?, and /? for 7: Ua,^Ua,^dri3drj — U^^aUj^isdradrp.
(13.6)
Substituting relations (13.5) and (13.6) into (13.4), we can finally write \dr'\ = \drf -f 2Uaf3dradr(3, where Uap = -{Ua,(3 + f//3,a + U^,aU^,^).
(13.7)
The tensor U formed by the elements [7^/3, U = [U^(}] --
^ XX
^ xy
^yx
^yy
^ xz ^yz
^ zx
^ zy
^ zz
is called an elastic medium deformation tensor. This tensor depends on the location of the point M in whose vicinity we consider the displacements of points in the deformed body; i.e., the tensor U forms a tensor field U = U(r).
398
Wavefield equations
Since elastic oscillation theory confines itself, normally, to small deformations only, the last term in relation (13.7) can be neglected as a small quantity of the second order, compared to the first two terms. As a result, the matrix of the deformation tensor takes the form C/a/3 = ^(f/a,/3 + C/Ac.).
(13.8)
Now consider the stresses arising inside a deformed elastic body. The forces acting on an elastic body fall into two types: volume and surface forces. The volume forces act on the various elements of the body volume. For example, we assume that the force on an infinitesimal element of the volume dv is equal to Fdv^ where F is the density of the volume force. The surface forces act on the surface elements conceived inside an elastic body or on its boundary. Let us imagine a flat infinitesimally small site ds at some point in an elastic body, and let us draw a unit normal n towards it, taking some direction of that normal as positive. If there is an elastic stress in the body, the parts of the body positioned on the different sides of the site nds will act on one another with a certain force. Let us denote by d^ the force acting on the site nds. Within the framework of elastic oscillation theory, this force is linearly proportional to the site nds: d * =T'nds, (13.9) where the linear operator r is called the stress tensor. Thus, the surface force on an infinitely small element of the surface ds is equal to d * = Pnds, where (13.10)
Pn=Tn
is the normal surface force per unit area, i.e. the stresses on a given area. We can describe the equilibrium condition for an arbitrary volume D in a body, which is bounded by a surface 5, by the following equation [[[ JJJD
F (r) dv = / / d * (r) = / / P „ (r) ds = [[T (r) -nds, JJS
JJS
(13.11)
JJS
where n is the unit vector of an outward pointing normal to S. Based on the tensor statement of the Gauss theorem (F.8), we can introduce the divergence of the tensor field r (r) as follows -.,x fffn V - r (r)di; {LT{v)-nds V • r r ' - lim ^^^—^ ^ ^ - lim U^-^ ,
13.12
where the volume D contracts in the point r'. Substituting equation (13.11) into (13.12), and using the mean value theorem for integrals we find V-T{r')=YimJ-^Is^^^^=Y{v').
(13.13)
Basic equations of elastic waves
399
The last formula shows that the density of the volume forces is equal to the divergence of the stress tensor. Thus, the stress tensor enables us to define both the density of volume forces and the stresses applied to a given area ds with a unit vector of the normal n: F = V-T, (13.14) Pn-r-n. 13.1.2
(13.15)
Hooke's law
One of the important problems in elasticity theory is finding the relationship between the stress and the deformation of the elastic body. The first (and the simplest) formulation of this relationship belongs to the English scientist R. Hooke who established experimentally that the deformation of the elastic body is proportional to the applied force. Hooke himself interpreted that dependence as follows: if a tensile stress is applied to the ends of a thin rod, then the increment in the rod length A/ will be proportional to the force apphed. The present-day formulation of Hooke's law was given as early as the 19th century by Cauchy and Poisson and is read as follows: if a small deformation occurs in^ an isotropic body, the stress tensor r is a linear function of the deformation tensor U (and vice versa). This law can be written for scalar components of the stress and deformation tensors as follows ra/3 = 2flUa(3 + Af/^^5a/3,
(13.16)
where The coefficients A and /i were introduced by Lame and named after him. They are related to elastic parameters of a medium, the Young's modulus E and the Poisson's ratio cr, by the formulae (Love, 1944; Udias, 1999)
and
These parameters characterize the elasticity properties of the medium at a given point in the body. As seen from experience, in cases of small deformations the linearity of the stresses with respect to deformations is ensured with an acceptable degree of accuracy.
400
Wavefield equations
13.1.3 Dynamic equations of elasticity theory for a homogeneous isotropic medium Let us recall the general form of the equations of motion of a homogeneous isotropic elastic medium under the influence of its own elastic forces. These equations are of utmost importance for seismology and seismic exploration and lie at the foundation of the majority of their theoretical constructions. To obtain those equations according to Newton's law, let us represent the volume forces as a product of acceleration d'^XJ/dt'^ and elastic medium density p : F + F^=p-^,
(13.19)
where F^ is the strength of the external force per unit volume applied to the elastic body. Using representation (13.14) for the volume force density F as the divergence of the stress tensor, we can write V•r = p - ^ - F ^
(13.20)
This equation can be written for scalar components of the stress tensor and deformation vector as follows (Udias, 1999)
where [/«, F^ (a = x, y, z) are components of the vectors U and F^ respectively. Assuming all deformations to be small, we can employ Hooke's law (13.16) and the deformation tensor expressed through the displacement vector (13.8). Substitution of (13.16) into (13.21) yields the following form for the equation of motion of a homogeneous isotropic elastic medium:
Taking into account expression (13.8) for the deformation tensor, we find
Formulae (13.23) form a system of linear partial differential equations of motion of a homogeneous isotropic elastic medium. These equations can be presented in more compact form using vector notation. First of all, it is clear that the bracketed expression on the left-hand side of relation (13.23) is equal to the divergence of the displacement vector: dUf3 ^ dU, ^ dUy ^ dU, ^ ^ drp drj, dry 9r^
^
401
Basic equations of elastic waves
Next, the values
Avu,
Avu,
Or^
OTy
Avu OTz
are just the components of the vector V (V • U). Finally, the second term on the left-hand side of (13.23) results from an application of the Laplacian operator to the components of the vector U, i.e. it is a component of the vector V^U. Hence, the desired vector form of the equations of motion will be as follows: (A + / i ) V V • U + / i V ' U = p
^ - F^
(13.24)
Taking into account the known identity V^U = V V - U - V x V x U ,
(13.25)
one can write the equation of motion in the form c^ V V - U - c ^ V x V x U - d'^V/dt'
- -F^
(13.26)
where the constants Cp and Cg are expressed in terms of the density p and elastic parameters of the medium (Lame's coefficients) by means of the following formulae: cl = {\ + 2/x) /p,
cl = li/p.
(13.27)
The vector form of the equations of motion (13.26) is called the Lame equation. The constants Cp and Cs have clear physical meaning. We will see below that equation (13.26) characterizes the propagation of two types of so called body waves in an elastic medium, compressional and shear waves, while the constants Cp and Cg are the velocities of those waves respectively. We will call them Lame velocities. Let us also introduce a differential operator L = c^VV- - c^V X V X .
(13.28)
Then equation (13.26) takes the form: LU = d'^JJ/dt^ - - F ^
(13.29)
The operator L is called Lame's operator. When Cp = Cs = c, the Lame's operator L becomes the Laplacian operator, L = c^V\
(13.30)
while equation (13.29) turns into the classical vector wave equation: c^V^U = d^V/dt^
- -F^
(13.31)
402
Wavefield equations
Note that in the time domain, the displacement vector U(r, t) and the external volume force F^(r, t) can be represented as superpositions of the waves, tx(r,Lc;), and forces, f^(r,a;), in the frequency domain using inverse Fourier transform: U(r,t)
-
— / u{r, uj)e-'^'(L;, 27ry_oo
(13.32)
F^(r,t)
=
— /
(13.33)
r{r,uj)e-'^'du;,
Therefore, one can write the Lame equation (13.29) and the vector wave equation (13.31) in the frequency domain as follows: Lu(r,u;) +a;2u(r,u;) = - - P (r,a;), P
(13.34)
V2u(r,cj) + ^ u ( r , u ; ) = — ^ P (r,a;).
(13.35)
and
13.1.4 Compressional and shear waves Let us analyze the space and time structure of the elastic displacement field in detail*. We will demonstrate that equation (13.26) describes the propagation of two types of body waves in an elastic medium, i.e., compressional and shear waves travelling at different velocities and featuring different physical properties. To this end, let us recall the well-known Helmholtz theorem according to which an arbitrary vector field, in particular an elastic displacement field U(r), may be represented as a sum of a potential, Up(r), and a solenoidal, Us(r), field (Zhdanov, 1988): U ( r ) = U ^ ( r ) + U,(r),
(13.36)
where V X Up - 0,
V • Up = V • U,
V X U , = V X U,
V • U , = 0.
(13.37) (13.38)
We can apply the same decomposition to the strength of an external volume force F^: F'=(r) = F;(r) + F:(r), (13.39) where V X F ; = 0,
V • F ; = V . F^
V X F^ = V X F ^
V • F^ - 0.
(13.40) (13.41)
*Note that we will consider the body wave theory only. One can find the detailed analysis of the surface wave theory, important in seismological applications, in Dahlen and Tromp, 1998.
Basic equations of elastic waves
403
Now let us substitute relations (13.36) and (13.39) into the equation of motion of a homogeneous isotropic medium (13.26): cl V V . (U^+U,) -dVxVx = d^{Up^lJs)/dt^
(U^+U,)
- - (F; + F:) .
(13.42)
We will apply the divergence operation to both sides of this equation. Since V • Uc = 0 and V • F ! = 0, we obtain V-
clV'V,
- _ /
-f - F ; U
0.
(13.43)
The curl of the expression enclosed in parentheses is also zero. Thus the field (cpV^Up — d^JJp/dt'^ + Fp/p) has neither sources nor vortices in the whole space. Imposing a natural requirement that this field should vanish at infinity, one can prove that it is equal to zero in the whole space, i.e., 4 V ^ U , - ^
(13.44)
= --F;.
Similarly, applying the curl operator to both sides of (13.42), and recalling that V X Up = 0 and V X F^ = 0, we find V X ^clV'Us
- ^
+ ^ F : ) = 0.
(13.45)
Since the expression enclosed in parentheses has a zero divergence, we arrive at the following equation for the solenoidal field dV'U.
- ^
= --/I.
(13.46)
As we can see here, both the potential and solenoidal components of the elastic displacement field satisfy wave equations and therefore represent waves traveling in space at velocities Cp and Cg respectively. Let us examine them in detail. First of all, let us establish the relationship between the direction of displacement and the direction of propagation of waves in the first and the second cases. To solve this problem, we consider partial solutions of both equations, for which it is considerably easier to find the direction of displacement and the direction of wave propagation: Up,, = b / ( a • r - Cp,, t),
|a| = 1,
(13.47)
where / ( . . . ) is an arbitrary differentiable function, the unit vector a shows the direction of wave propagation, and vector b determines the direction of the particle
404
Wavefield equations
displacement in the wave. These solutions can be treated as plane waves traveling in the vector direction a at a velocity of Cp or Cg respectively. In this case the displacement of the particles occurs in the direction of vector b . The potential components of the elastic displacement Up should satisfy the following equation: V X Up = / ' ( a • r - Cpt) [a X b] = 0. (13.48) Since the function / is arbitrary, the cross product a x b must be equal to zero in order to satisfy the last condition: a X b = 0.
(13.49)
This means that the vectors a and b are parallel! Thus, the potential waves Up are characterized by the fact that in their case particle displacement occurs in the direction of wave propagation. Therefore, the particle motion associated with this type of waves is always in the direction of wave propagation, and it consists of alternating condensations and rarefactions of the particles within the elastic body. Such waves are described as compressional waves. The waves of the second type, corresponding to the solenoidal component of the elastic displacement Us, should have zero divergence, i.e., V • Us = / ' ( a • r - Cst)(a • b) = 0.
(13.50)
To satisfy equation (13.50), the vectors a and b must be orthogonal: a - b = 0.
(13.51)
In other words, the particle displacement in waves Us occurs in a direction orthogonal to that of wave propagation. As a result of field Us being solenoidal, the volume expansion coefficient of the medium during deformations involving the propagation of these waves is equal to zero. This means that Us waves are not related to volume changes. That is why such waves are described as shear waves. I would like to point out in conclusion that the velocity of compressional waves is always greater than that of shear waves. This is quite obvious from the following expressions: Cp = V(A4-2/x)/p, Cs = ^/jljp. (13.52) In fact, a strict inequality holds true: Cp > V2cs.
(13.53)
This inequality may be obtained by analyzing the limits of changes of the magnitudes A and /x, characterizing the elastic properties of various substances.
Basic equations of elastic waves
405
13.1.5 Acoustic waves and scalar wave equation We now consider a simple model of seismic waves, which is widely used in applications. Let us assume that the earth can be treated as an acoustic medium and the influence of variations in density can be ignored. In this case the propagation of seismic waves in the earth can be described by the acoustic wave equation V ' ^ ( r , t) - ^ ^ ^ ( r , t) = -F^ ir,t),
(13.54)
where P(r, t) is the pressure field, F^{r,t) is the strength of an external source of energy, and c(r) is the velocity of wave propagation. The use of the scalar wave equation (13.54) instead of vector wave equations (13.44) and (13.46) simplifies significantly the theory of seismic wave propagation. An acoustic wave in the time domain, P ( r , t), can be represented as a superposition of waves in the frequency domain, p(r,u;), using the inverse Fourier transform: 1
/'+00
P(r, i) = ^ J
Pi^^ uj)e-'^'dw.
(13.55)
In the frequency domain, equation (13.54) can be written as the Helmholtz equation V2p(r, cu) + - ^ P ( r , uj) = - r (r, uj),
(13.56)
where /^(r,a;) is the Fourier transform of F^{r,t) : +00
/
•oo
In concluding this section, I introduce a simple 1-D equation for acoustic waves. This equation arises in a case where the wavespeed, c, the pressure field, p, and the source field, /^, are functions of only one coordinate, say, the vertical coordinate z. In practice this model can be used, for example, in acoustic well logging when one is interested in wave propagation along the borehole only. In this situation the general 3-D scalar Helmholtz equation (13.56) is reduced to the 1-D Helmholtz equation —p{z,
uj) + -^P{z,
uj) = - r (z, u^) •
(13.57)
13.1.6 High frequency approximations in the solution of an acoustic wave equation The reader, familiar with the background of seismic exploration methods, should recall that many successful seismic interpretation algorithms are based on the simple principles of "geometrical seismics," which resembles the ideas of "geometrical optics." The question is how this simple but powerful approach is connected with the
Wavefield equations
406
general wavefield equation solution, which is the subject of this text. The answer is that a "geometrical optics or seismics" approach to wavefield theory appears when we consider the high frequency asymptotics of the corresponding wavefield. In this case one can use the wavepacket description of wavefield phenomena, which describes a narrow bundle of plane waves with hmited ou values (Felsen and Marcouvitz, 1994). The wavepacket propagates along certain paths or trajectories, called rays. The rays are straight in a homogeneous medium but curved within inhomogeneous domains. The rays' behavior at the acoustic boundaries are governed by a simple Snell's law. The wavepackets, in turn, define specific surfaces called wavefronts, which are orthogonal to the rays. To understand better the connection between the geometrical optics approach and wave equation solutions, we will discuss in this section the basic equations describing high frequency scalar wavefield propagation. Following Bleistein (1984) and Bleistein et al. (2001), we represent the solution of the scalar wave equation (13.56) outside of the source in the form of the Debye series (13.58) n=0 ^
'
where function r (r) can be treated as the traveltime of the wavepacket, and An (r) are the coefficients of the series. We now substitute the Debye series into the left-hand side of the scalar wavefield equation (13.56):
VVr,a;) +
c2(r)
n=0 ^
'
^
^ - ( V r ( r ) ) 2 A„(r) c2(r)
+ iuj [2Vr (r) • VA„ (r) + A„ (r) V V (r)] + V M „ (r)} .
(13.59)
Outside the source, the series on the right hand side of expression (13.59) must be equal to zero. Obviously, the terms with different powers of a; cannot cancel each other, which requires that all coefficients of the series at different powers of u) must vanish. In particular, from the first two terms (n = 0,1), we find all expressions containing the second and the first power of frequency: LO
1 c2(r)
(Vr(r))^ A^ (r) + iuj [ 2 V T (r) • V ^ o (r) + A^ (r) V V (r)]
+-
c2(r)
(Vr(r))^
Al (r) = 0.
(13.60)
As a result, we arrive at the following two equations for travel time r (r) and coefficient Ao (r): c2(r)
- ( V r ( r ) ) 2 = 0,
(13.61)
Green's functions for wavefield equations
407
and 2 V r (r) • V ^ (r) + AQ (r) V V (r) - 0.
(13.62)
The first equation, (13.61), is the eikonal equation. It characterizes the traveltime behavior of the wavefield. The second equation, (13.62), is the (first) transport equation. It describes the amplitude attenuation of the wavefield. After solving these two equations we can find all other coefficients. An ( r ) , recursively, using transport equations of higher orders. These higher-order transport equations can be obtained by making the coefficients equal to zero at the higher-order inverse frequency, 1/ {iuj)^, in series (13.59). Solving the eikonal equation, we can find the ray trajectories of the wavefield propagation. Using the transport equations, we can determine the amplitudes of the wavefield along the rays. One can find a detailed discussion of ray theory in Udias (1999) and Cerveny (2001). 13.2 13.2.1
Green's functions for wavefield equations Greenes functions for the scalar wave equation and for the corresponding Helmholtz equation
Green's functions appear as the solutions of seismic field equations (acoustic wave equation or equations of dynamic elasticity theory) in cases where the right-hand side of those equations represents the point pulse source. These solutions are often referred to as fundamental solutions. For example, in the case of the scalar wave equation (13.54), the density of the distribution of point pulse forces is given as a product, Fl{v,t) = 6{v)6{t), where 6{t) is one-dimensional delta-function, while (5(r) is three-dimensional Dirac delta-function ^(r) = 6{x)6{y)6{z), such that
and V = V -\- S is domain V with its boundary S. The fundamental solution or the scalar Green's function G^ in this case satisfies the following equation: V'G^{r,t\T\t')-^^G''{T,t\T\t') = -6{r-r')6{t-t').
(13.64)
The importance of studying Green's functions is dictated by the following considerations. First of all, according to the basic property of the (5-function (13.63), any source F^(r, t) can be represented as a sum of point pulse sources appropriately distributed
408
Wavefield equations
in space and time, i.e., as an expression ^(r — r')6{t — t') with suitable amplitudes prescribed by the given function F^(r,t): F^(r', t') = [ff
f °^ F^(r, t)6{v' - T)6{t' - t)dtdv,
(13.65)
where the volume integral is calculated over the entire infinite space T4o. Second, if we assume that the process is hnear (which can be seen immediately from the original wave equations), the sum of effects on an acoustic medium must be equal to the sum of its responses. In other words, the wavefield generated by the source presented as a linear combination of some other sources is equal to the linear combination (with the same coefficients) of the fields generated by the corresponding individual sources. Therefore, we can conclude that with the knowledge of the Green's function G^(r, t\Y',t') for the wave equation one can also find the solution to this equation for an arbitrary right-hand side F^(r,t), (i.e. for an arbitrary allocation of sources), as the convolution of the Green's function with the function F^(r, t), i.e., v{Y\t')=
[[[
f
F\Y,t)G'^{Y\t'\Y,t)dtdv.
(13.66)
A similar result holds true for any general dynamic equation of elasticity theory and for equations of compressional and shear waves as well. Under the assumption that the point pulse source is located at the origin of coordinates and generates a pulse at time f = 0, one can find a general solution of equation (13.64) in the form G^{Y,t\Y' = 0,^' = 0) = G-(r,t) = ^g, (^ - ^ ) + i ^ ^ (^ + ^ ) ,
(13.67)
where r = |r|, and gi and ^2 are arbitrary differentiable functions, which will be specified below. We can see that the first term in equation (13.67) characterizes a spherical wave traveling from the origin of the coordinates to infinity, while the second one is a spherical wave traveling from infinity to the origin of the coordinates. Since the physical formulation of the problem contains only one source at the origin of coordinates, the function ^2 in equation (13.67) must be taken as equal to zero. Thus, the Green's function G^(r,t), as one could anticipate, represents a divergent spherical wave: G-{v,t) = -gAt--). (13.68) r \ cJ Note that at this point we selected the function gi arbitrarily, because we did not specify the shape of the source-generated pulse. To determine the function ^1, we should substitute solution (13.68) into equation (13.64): [1 / r\\ 1 9^ [1 / r\^ = -6{r)S{t). (13.69) V \-r'A'--c)\-7^wA~r''\'~~c)\
Green's functions
409
for wavefield equations
Using the well-known identity V^(l/r) = —47r5(r), we can reduce equation (13.69) to the form (assuming t » r/c and c » r/t) -47rgi(t)^(r) =
-6{T)6{t),
from which we find at once an expression for gi : g,{t) = i - 5 ( f ) .
(13.70)
Substituting (13.70) into (13.68), we obtain the final form of the Green's function^ : Thus, the wavefield of a point pulse source, or Green's function of the wave equation in three-dimensional space, is a sharp impulsive wavefront, traveling with velocity c, and passing across the point M located at a distance of r from the origin of coordinates at the moment t = r/c. The magnitude of the wavefield is equal to zero at the point M prior to arrival of the wavefront and thereafter. The wave amplitude decays proportionally to multiplier 1/r as a result of geometrical divergence. We have derived equation (13.71) under the assumption that the point pulse source is located at the origin of coordinates and generates a pulse at time f = 0. In the general case of an arbitrary position of the source r' and arbitrary time of its operation f, formula (13.71) takes the form:
Let us now find the Green's function of the corresponding Helmholtz equation (13.56). We can solve this problem by applying the Fourier transform to expression (13.71): oo
/
G'"{r,t)e'^'dt
-OO
1 r 6{t- r/c) e'^^'dt = -^^^^l\ (13.73) 47rr y_o 47rr where, to simplify the notation, we use the same symbols for Green's functions in time and frequency domains. However, it is easy to distinguish between these two functions by the presence of the corresponding arguments: t in the time domain or uj in the frequency domain. In the general case of an arbitrary position of the source r', the Green's function in the frequency domain can be presented in the form G"(r| r'; a;) = — ^ .'''V-'Mc^ 47r r - r'
(13 74)
^One can find the detailed derivation of this formula in Morse and Feshbach (1953, page 838).
410
Wavefield equations
The Green's function G^(r| r'; c*;) is the fundamental solution of the Helmholtz equation: V2G"(r| r'; uj) + ^ G " ( r | r'; cu) = -6{r - r') .
(13.75)
The integral representation formula (13.66) in the frequency domain takes the form p(r', Lj) = Iff
rir, a;)G'"(r'| r; u)dv = G„ (D ,
(13.76)
where G^ (/^) is called the scalar Green's integral wave operator in the frequency domain. Note that the definition of the Green's function can be extended in the case of a variable velocity distribution: c = c ( r ) . The corresponding Green's function ^ ^ ( r l r';a;) is the fundamental solution of the Helmholtz equation for the given velocity c(r): V'G"(r|r»-f-^^G"(r|r';a;)=:-(5(r-r') . (13.77) c^(r) We can also introduce the Green's function g'^{z\ z';ij) for the 1-D Helmholtz equation (13.57). This function is a solution of the following equation ^g-'{z\z';uj)
= -6{z-z')
+ -^^g^{z\z';u)
.
(13.78)
For example, for the constant wavespeed c {z) = CQ this function is (Morse and Feshbach, 1953): ^iuj\z-z'\/co
g-iz\z';u;)
= -co
^.^
.
(13.79)
13.2.2 High frequency (WKBJ) approximation for the Green's function There is an important class of methods for wavefield inversion, based on high-frequency asymptotics (Bleistein, 1984; Bleistein et al., 2001). Most of these methods employ a high frequency approximation for the Green's function. This approximation arises from the Debye series (13.58) if we preserve only the leading-order term in the series: G^(r| r'; uo) ^ A (r, r') e'-^(^'"'\
(13.80)
where A is the amplitude term given by the transport equation (13.62), 2 V r (r, r') • VA (r, r') -h A (r, r') V V (r, r') = 0, and r is the ray traveltime obtained from the eikonal equation (13.61)
(13.81)
Green's functions for waveEeld equations
411
Numerically, the above ray equation can be solved by a set of ray-tracing algorithms (Cerveny, 2001). Expression (13.80) is called "WKBJ (ray theoretic) Green's function," because it is associated with the names of several physicists, G. Wentzel, H. A. Kramers, L. Brillouin, and H. Jeffreys, who independently introduced this approximation in connection with the solution of different physical problems (Morse and Feshbach, 1953). I would also recommend an excellent book by Bleistein et al. (2001), where the interested reader can find a more thorough mathematical analysis of the WKBJ approximation. In the special case of a 1-D model, the WKBJ approximation of Green's function takes the form (Bleistein et al., 2001, p. 69) Vc{z)c{z')
g{z\z']uj)
2iuj
exp
lU
i:
dz
(13.83)
J z'
for z > z'. Note that the 1-D formula (13.83) contains one additional negative power of cj in comparison with a 3-D WKBJ approximation (13.80). This term was introduced by analogy with the expression (13.79). Certainly, in the model with a constant wavespeed, c{z) = CQ, formula (13.83) naturally reduces to the expression for a 1-D Green's function in a homogeneous medium: g{z\z';ij)
2iu) exp
lUJ
r— Jz' Co
-Co
exp[iuj{z- z') /co] , — -, ioY z> z . 2i(jj
13.2.3 Greenes tensor for vector wave equation Now let us analyze the fundamental solutions of the vector wave equation:
1 a^u
V^U
c2 df^
_$-
(13.84)
where $^ = ¥^/c^p. We found in the previous section that this equation describes separately the propagation of the compressional and shear waves in a homogeneous medium. Similarly to the scalar case (equation (13.65)), an arbitrary source field *^ in the vector equation can be represented as the sum of point pulse sources: ^\T',t')
= III
[
^^(r, t)6{r'~r)6{t'
- t)dtdv.
(13.85)
Note that the last equation can be rewritten in the equivalent form r'+oo
(r', t') -
///
/
4^^(r, t) • I6{T'-T)6{t' - t)dtdv,
(13.86)
412
Wavefield equations
where I is the identity tensor. Equation (13.86) indicates that the density of the distribution of point pulse forces in a vector case should be described by the tensor expression 16{r'—)6{t' — t). According to the linearity of the wave equation, the vector field of an arbitrary source can be represented as the sum of elementary fields generated by the point pulse sources. However, the polarization (i.e., direction) of the vector field does not coincide with the polarization of the source, F^. For instance, the elastic displacement field generated by an external force directed along axis x may have nonzero components along all three coordinate axes. That is why in the vector case not just one scalar but three vector functions are required. The combination of those vector functions forms a tensor object G^(r, t), which we call the Green's tensor of the vector wave equation. The Green's tensor can be treated as the solution of the vector wave equation with the right-hand side given, according to formula (13.86), by the product of the identity tensor and delta functions: V 2 G - ( r , t\ r', t') - ^ ^ ^ ^^2
'
= -I'5(^ - r')^(^ - *')•
(13.87)
As in the scalar case (equation (13.66)), we can conclude that, using Green's tensor G'^ of the vector wave equation, one can find the solution to this equation with an arbitrary right-hand side F^(r,^), as the convolution of the Green's tensor G^ with the function F^(r,t), i.e., U(r', t') = Iff
*"(r, t) . G^(r', t'\ r, t)dtdv,
[
(13.88)
where ^ ^ • G^ should be interpreted as the inner product of vector ^ ^ by tensor G^. Using considerations similar to the one discussed above for a scalar wave equation, one can demonstrate that the Green's tensor for the vector wave equation is G"^(r,^|r',0 = I G ^ ( r , t | r ' , t ' ) ,
(13.89)
where G^ is the Green's fundamental function for the wave equation. The matrix of the Green's tensor in any Cartesian coordinate system is diagonal and can be represented as follows: [G"'(r,t|r',i')] =
47r|r
6{t-t' 0
0
-\Y-T'\/C)
0 8{t~t'
0
-\Y-T'\/C)
0 0
8{t-t'
-\v- r'l/c) J (13.90)
413
Green's functions for wavefield equations
One can also introduce the Green's tensor of the vector wave equation in the frequency domain oo
/
G'"{r,t\r',t')e''^^'-^'^dt
•OO
^ ^ia;|r-r'|/c
= IT-I n(13-91) 47r |r - r'l This tensor can be used for an integral representation of the frequency domain wavefield ct>%T,u) • G - ( r ' | r',uj)dv = 0 ^ ( 0 ^ ) , (13.92) u(r^a;) = fff where Gyj {cff) is called the vector integral Greenes wave operator in the frequency domain^ and (jf{Y^ijo) is the Fourier transform of $^(r,t). Note again that we use the same symbols for Green's tensors in the time and frequency domains to simplify the notation. One can easily recognize the corresponding tensor by checking its arguments, t ox u. 13.2.4 Greenes tensor for the Lame equation Now let us consider the Green's tensor for the system of equations of dynamic elasticity theory, the vector form of which is called a Lame equation. We will call this tensor an elastic oscillation tensor G^ or Green's tensor for the Lame equation. As in the case of the vector wave equation, discussed above, the components of the elastic oscillation tensor describe the propagation of elastic waves generated by a point pulse force. In other words, it satisfies the following Lame equation (see equation (13.29)): LG^(r, i|r', t') - ^'^'^i^^^^l''''^')
= _is(^r - r')6{t - t').
(13.93)
We have learned already that any elastic oscillation can be represented as a superposition of the compressional and shear waves, which correspond to the potential and solenoidal parts of the elastic displacement field. Therefore, it is clear that the elastic tensor G^ can also be represented as the sum of the potential and solenoidal components, described by tensor functions G^^^^ and G^(*) respectively: G^ = G ^ ( r , t | r ' , 0 - G^(^)(r,t|r',tO + G^(^)(r, t | r ^ t ' ) .
(13.94)
Omitting the long derivation, which can be found in the relevant text books on the theory of elastic waves (see, for example, Morse and Feshbach, 1953; Aki and Richards, 2002), I present here the final expressions for functions G^^^^ and G^^^^ in a homogeneous medium: G^(P)(r,^|r',^0
Wavefield equations
414
G^(^)(r,^|r',0 V X Vx
1 47r|r - r'l
r —r
x{t-t')-x(t-t'-
where
Cp
(13.95)
0, ^ < 0 t, t>0
H (r) is the Heaviside step function,
and 6 (r) is the delta-function. Using Green's tensor G^{r^t) for the Lame equation, we can express the solution of this equation for an arbitrary right-hand side F^(r, t) as the convolution of the Green's tensor G^(r,t) with the function F^(r,t), i.e., U ( r , t') = If I
- F " ( r , t) • G^(r', t'\ r, t)dtdv.
f
(13.96)
Similar to the vector wavefield, we can obtain the integral representation of the frequency domain elastic wavefield u(r',a;) = jjj
- ^ ( r , u ) • G^(r'|r; w)dv = G i ( f 7 p ) ,
(13.97)
where G L is called the integral Greenes Lame operator in the frequency domain and G^(r'|r;cj) is the corresponding Green's tensor in the frequency domain: G^(r'
+00 _
r
G^(r',0|r,t)e^^*dt.
(13.98)
J —o
Note that both representations (13.96) and (13.97) hold true for a medium with variable Lame parameters (Aki and Richards, 2002). However, in a general case one should use the corresponding Green's tensors for the Lame equation with variable coefficients. 13.3
KirchhofF integral formula a n d its analogs
In this section we consider a number of identities relating the wavefield values within a domain to its values on the domain boundary. These identities provide a tool for solving the boundary-value problem of the wavefield, which is extremely important in the solution of the forward and inverse problems. First, let us consider the case of a scalar wavefield where the classical Kirchhoff formula holds true.
415
Kirchhoff integral formula and its analogues
13.3.1 Kirchhoff integral formula Let us consider a limited volume domain D, bounded by a surface S. A scalar wavefield P (r, t) is assigned within the bounds of that domain and satisfies the following equation: V 2 p ( r , i ) - 4 ^ P ( r , i ) = 0,
v&D.
(13.99)
According to Gauss's theorem the relation
/ / / V •¥dv= II ¥• nds
(13.100)
holds true for any vector field F continuous with all its partial derivatives in domain D up to its boundary S, and n is the unit vector of the outward normal to surface S For any two, twice continuously differentiable functions P and Q, the following identity is easy to establish: V • {PVQ - QVP)
= PV^Q - QV^P.
(13.101)
Let us assume that in formula (13.101) the function P satisfies equation (13.99), while Q is taken to be the Green's function of the wave equation: Q(r,t) = G " ' ( r ' , i ' | r , t ) ,
r 6 D.
(13.102)
Then W'P{r,t)
=
i^P(r,i),
(13.103)
V^Q{r,t)
=
V2G™(r',t'ir,i)
=
_5(r'-r)6(t'-i) + l ^ G " ' ( r ' , i ' | r , i )
(13.104)
and relation (13.101) takes the form V • [P(r, i) VG"' (r', i'|r, t) - G'" (r', t'\r, t) V P ( r , t)] = -P{v,t)S{r'-T)6{t'
- t) +
,2
P(r,i)|^G-(r',t'|r,i)
-G-(r',t'|r,i)^P(r,t)
(13.105)
After integrating both sides of relation (13.105) over t with infinite limits, we have oo
/
•OO
[P(r, i) VG"" (r', i'|r, t) - G^ (r', t'\r, t)) V P ( r , t)] dt
Wavefield equations
416 -P(r,t')5(r'-r) + l
r P(r,t)^G-(r^t'|r,t)
-G-(r',t'|r,O^P(r,0
d^.
(13.106)
Integrating by parts and taking into account the finite character of function G^, we find that the integral on the right-hand side of equation (13.106) is equal to zero. Therefore, introducing a function F ( r , t ' ) , oo
/
\dL [P(r, t)VG^ (r', t'\v, t)) - G"" (r', t'\r, t) VP(r, t)],
•CX)
we have V • F(r, t') = - P ( r , i ' ) ^ ( r ' - r ) ,
(13.107)
and
III V-Fir, t')dv
P(r,i')<5(r'-r)dz; = - P ( r ' , i ' ) , if r' e D.
(13.108)
Thus, according to Gauss's theorem (13.100), we obtain the following integral identity: P(r
•'•' = - / / /F'uds^ "* = / i /
G^{r',t'\v,t)-^P{r,t) dtds^
-Pir,t)—G^ir\t'\r,t)
T'
eD.
(13.109)
Expression (13.109) is called the Kirchhoff integral formula. This formula shows that the wavefield inside domain D can be determined by the values of this field (and its normal derivative as well) on the domain boundary S. In the case where point r' lies outside domain D, we have fff
V - F{r, t')dv - - fff
P{r,t')6{Y'-r)dv
- 0, if r' ^ D,
(13.110)
where D = D -{- S is domain D with its boundary 5. Consequently, the boundary values of the wavefield must satisfy the following condition: J Js J-o
P{r,t)-^G^{r',t'\r,t) (13.111)
d
-G^{T\t'\r,t)—Pir,t)
dtds — 0,
r'
^D.
KirchhofF integral formula and its analogues
417
Relations (13.109) and (13.111) can be combined into one:
J Js J-c
G- (r', t'\r, t) | ^ P ( r , t) - P(r, t)-^G^ r'
(r', t'\r, t) dtds
eD
(13.112)
0,
As one can see from relation (13.112), the wavefield at the point r' e D may be viewed at the moment of time f as the sum of elementary fields of point and dipole sources distributed over the surface S with densities dP{r^t)/dn and P{r,t) respectively. The interference of these fields beyond the domain D results in complete suppression of the total wavefield. Thus, the Kirchhoff integral formula can be treated as the mathematical formulation of the classical physical Huygens-Fresnel principle. In the case of a homogeneous medium we can simplify the expression on the left-hand side of relation (13.112) by substituting the analytical formula (13.72) for the Green's function G^ into it, and integrating with respect to t, to obtain the final form: 1
47r yy^
c
-r|
d | r ' - r \dP L 1 ) p ( r t') + - ^ ( ^ dn ' dt' ' dn V|r'-- r | |r'—r dn
_l
P{r',t'),
"i . o>
r' r'
eD iD
t
^'
t')
ds
(13.113)
where P{Y,t') = P {r,t' — |r'—r| /c) is called a retarded wave. The obtained relation is precisely the classical integral formula of Kirchhoff for the scalar wavefield P. 13.3.2
Generalized Kirchhoff integral formulae for the Lame equation and the vector wave equation
Now, following Zhdanov (1988) and Zhdanov et al. (1988), we generalize the Kirchhoff formula for vector fields characterizing arbitrary elastic oscillations of the medium, that is for fields satisfying the Lame equation (13.29) or the vector wave equation (13.31). Consider the following boundary value problem: express the displacement vector field U in some domain V in terms of the values of U and of its normal derivative dU/dn on the inner side of the surface S bounding this domain. The constant elastic parameters of a homogeneous medium, Cp and Cg, are assumed to be known. The external volume forces F^ are distributed within some domain D, which is located inside V {D C V), so the field U in D satisfies the Lame equation LJJ-d^U/dt^
=
--¥',
(13.114)
Wavefield equations
418 where L is Lame's operator,
L^c^VV'-c^Vx Vx,
(13.115)
and Cp, Cs are the Lame velocities characterizing the speed of compressional and shear waves in the elastic body. Now let Q (r,t) be an arbitrary tensor field twice continuously differentiable in D and absolutely integrable over the whole time axis. Then we have U (t) -LQ (i) - L U (t) • Q (t) dt J —c
f
U(t).LQ(^)-^-^.Q(t)
dt-h
Q{t)dt,
J-ooP
(13.116)
J —C
where we temporarily omit the argument r to simplify the notations. Integrating the term in (13.116) containing d'^TJ (t) /dt^ twice by parts, we arrive at the following identity U(i)-LQ(i)-LU(i)-Q(0
J —c
U(i).
LQ(i)
dt^
J —c
dt +
dt
Q (t) dt.
J-ooP
(13.117)
Now we will integrate identity (13.117) over the domain V and apply the Green's tensor formula for the Lame's operator (F.16) to the left-hand side of the resulting formula: /
/ / / ( U • LQ - L U • Q ) dvdt
+ c:n- U x ( V X Q ) -f (V X U) X Q I } dsdt
J-oo J J Jv
+
I
LQ{t)
d'Qit) dt^
III -F' • Q^^'^*.
dvdt
(13.118)
where in the last volume integral we take into account equation (13.114) and integrate only over the domain D containing the external forces F^.
Kirchhoff integral formula and its analogues
419
We can select the tensor field Q to be equal to the Green's tensor G^(r', t'|r, t) for the Lame equation satisfying the equation (13.93): 2nL
Q = G^(r',t'|r,t), L G '
-i6{v'-r)6{t'
-t).
(13.119)
Substituting expression (13.119) into (13.118), and taking into account the basic property of the delta-function (13.63), we find:
+ cln- [Ux ( v X G^) + (V X U) x G ^ l } dsdt
In particular, assuming that the domain D with the external forces is located outside the volume V^ we arrive at the generalized Kirchhoff integral formula for the Lame equation (Zhdanov, 1988):
-f c^n U x ( v X G^) + (V X U) xG^l } dsdt U(r',f),
0,
T'_^V
(13.121)
r' iV
Let us consider in conclusion of this section the case where Cp = Cs = c and U satisfies a homogeneous vector wave equation: (13.122)
P In this case formula (13.118) takes the form
+ n- [Ux ( v X Q ) + (V x U) X Q I I dsdt U(i)J —C
c'V'Q
(t)
d'Qjt)
dvdt+
r fff 7-00
IF^ Qdvdt.
JJJDP
(13.123)
Wavefield equations
420
Now if the tensor function Q is provided by the Green's tensor for the wave equation G^(r,t|r',t'), 2^11)
Q = G"'(r,t|r',i'), V " G
1 a^G" c2 df^
-l8{T-v')8{t-t'),
=
formula (13.123) transforms into the following expression:
[Jf/H^{^o')-&'(v.v)]
f
+ n- U x ( v X G ^ ) + (V X U)
XGH
}dsdt
(13.124) Dp 10' ^ f^ J —o Assuming that all external forces are located outside the homogeneous domain V {c = const), we arrive at the Kirchhoff integral formula for the vector wave equation
+
H-n- U x ( v X G ^ ) + ( V X U) x G ^ j } dsdt U ( r ' , f ) , v'_eV T' iV 0,
(13.125)
The generalized Kirchhoff integral formulae enable the values of the elastic displacement field (equation (13.121)) or vector wavefield (equation (13.125)) to be reconstructed everywhere inside the domain V from the known values of these fields and their normal derivatives at the domain boundary S. 13.4
Uniqueness of the solution of the wavefield equations
13.4-1 Initial-value problems Many practical problems of wavefield theory require examining propagation of waves generated by a given source in a medium with given material property distributions. This is a typical example of the forward geophysical problem. There are two major types of these problems: 1) so-called "initial-value problems", and 2) problems without initial values. In the framework of the initial-value problems we examine wavefield propagation within a domain of an acoustic or elastic medium starting from a certain moment of time t = to, assuming that we know the initial wavefield distribution at this starting moment. This condition is called a domain initial-value condition.
Uniqueness of the solution of the wavefield
421
equations
Let us consider the case of a homogeneous isotropic elastic medium within a local domain V, bounded by the surface S. Various physical processes may take place on the boundary of the domain V and in the space around it. For example, the oscillation energy can flow freely through the boundary, or energy can stay within the domain, being reflected from the boundary 5, etc. Therefore to make the problem of wavefleld propagation of the initial disturbance mathematically specific, we also should determine the boundary conditions of the oscillations of the elastic medium. This condition is called a boundary-value condition. In summary, we can characterize the initial-value problems by assigning conditions of two kinds: 1) domain initial-value conditions, and 2) boundary-value conditions. Let us consider these conditions in more detail. The domain initial-value conditions for an elastic displacement field can be formulated as follows: U(r,to) = UO(r),
au(r,t) dt
U^(r),
rev,
(13.126)
t=to
where U^ and U^ are the assigned vector functions of the radius-vector r. Relations (13.126) indicate that at the initial moment of time the spatial distribution of the wavefield and its time derivative within a domain V are known. In addition to the domain initial value conditions (13.126), we can introduce three different types of boundary-value conditions, so that the type of problem is practically dependent on the type of boundary conditions. In the first problem, we fix the stresses acting at the boundary S on the infinitesimally small fiat sites nds, tangential to the surface 5, i.e.. Pn(r,t) = F(i)(r,t),
r€5,
(13.127)
where F^^^(r,t) is the vector function assigned at the points of surface 5; n is the unit vector of the outwards normal at the point of the surface 5, and Pn is a vector of stresses acting in the direction of vector n. From the physical point of view condition (13.127) corresponds to the case of a known external load applied to the boundary S. In the second problem, we specify the displacement vector itself on the boundary 5, i.e., U ( r , t ) = F(2Hr,t), r e 5, (13.128) where F^^)(r, t) is a known function on the surface S. Physically this corresponds to a situation where the required displacements are imparted to the points of the surface 5 by a certain force. The third problem is a combination of the first two where the displacement is specified on one part of the surface 5, and the external stresses are given on the other.
422
Wavefield equations
It can be proved that the equations of motion have a unique solution for each particular problem formulated above. The proof is based on utilization of energy relations which will be treated in the next section. 13.4-2 Energy conservation law Let us calculate the work R performed by external forces during the motion of an elastic medium. For an infinitesimal interval of time dt the elastic body particle is shifted by a vector dJJ = U dt,
(13.129)
where the dot over vector U denotes its differentiation with respect to time. Hence, the work done by an external stress P ^ to displace an infinitesimal element of surface dS, and the work accomplished by volume force F^dv to shift an infinitesimal volume element dv, are equal respectively to Pnds- \] dt = Pn- U dsdt,
(13.130)
Y^dv \]dt = F"- U dvdL
(13.131)
The integration of relations (13.130) and (13.131) makes it clear that the work accomplished by the external surface and volume forces for the duration of an infinitesimal period of time dt may be calculated as follows: dR= I ff
Pn U d5 + / / / F"- IJdvl
(13.132)
dt,
from which it follows that dR = / / Pn IJ ds-i- [ff dt
F' U dv,
(13.133)
Now let us transform the surface integral in (13.133) into a volume integral, using the Gauss formula. According to equation (13.15), the normal stresses, P ^ , can be expressed by the stress tensor as follows: Pn == r n .
(13.134)
Substituting expression (13.134) into the surface integral and applying the Gauss formula, we find: / / Pn
=
\jds=
/ / ( P , n Ux +Pyn
Uy -^Pzn
\{rxf3n^) Ux +{ry^nf3) Uy -^-ir^pup) [/J ds = = / / {raf3 Ua)nf3ds =
Uz)
ds
{ra(3n^) Ua ds
'dr'^^'^^ Ua)dv
Uniqueness of the solution of the wavefield
=///
dTa3
- ^
423
equations
•
d
*
. ,
^
(13.135)
I/a +Ta/3^— Uoc dv, a, 0 = X, y, z;
where we use again the agreement on summation over twice repeated Greek indices. Taking into account the symmetry of the stress tensor and the expression (13.8) for the deformation tensor, we can write d
1,
'
. d
d
'
d
(13.136)
where [/«/? are the components of the deformation tensor U. Next, using the equation of motion as given in (13.21), we recall that (13.137)
dr^(3/drp = pUa-F^. Substituting equations (13.136) and (13.137) into (13.135), we find
-m
/ / P n - IJ ds=
[ ( p U^ - F ^ ) Ua, + r a ^ Ua(3
I I I
dv
,2 ,i d_ . 2 dv - III F"- ijdv+ III Ta/} Ua0 dv. 2^dt Finally, substitution of this expression into relation (13.133) yields
dR dt
1 d_ Y 2 dt
• 2
%1
%2
dv + III
Tap Uccn dv.
(13.138)
(13.139)
The first term on the right-hand side of (13.139) may be rewritten as follows:
IIL~2'i
.2
.2
,2-l
U, + Uy + U,
'"^^Jt^^
(13.140)
where K is the kinetic energy of the elastic medium, K
nil
• 2
^2
«2
•
dv = 2
2
p |U| dv.
(13.141)
Thus, the first term, dK/dt, is the rate of change of the kinetic energy of the elastic medium. We can understand the physical meaning of the second term in expression (13.139) by introducing the following function:
424
Wavefield equations
+M {U',, + U^y + Ul + Ul + Ul + Ul + Ul + Ul + C/,2J .
(13.142)
One can see that T is a positively determined quadratic form of the stress tensor components, while its partial derivatives with respect to those components coincide with the stress tensor components themselves. Indeed, dT/dUxx dT/dUyy dT/dU,,
= A(C4a, + Uyy + Uzz) + 2fiUxx = XO -\- 2/iC4x == r^x, = \{Uxx + Uyy-^U,,)-h2flUyy = Xe-\-2flUyy = Tyy, = \{Uxx-^Uyy + U,,)-\-2fiU,, = X9-{-2fxU,,=r,,,
dT/dUaf3 = 2fxUc.f3 = ra/3, if a 7^ /?; a,/? - x,y,z,
(13.143)
where 9 = Uxx + Uyy + UzzBased on relationship (13.143), we can show that the rate of change of function T coincides with the second term under the integral sign in formula (13.139):
Thus, equation (13.139) may finally be transformed as follows: dR^dI£dW_ dt dt dt where function W is given by the integral of T , W = fffrdv,
(13.146)
and function K is determined by formula (13.141). The work accomplished by the external surface stresses and volume forces from the initial time to to the current time t is equal to the integral of equation (13.145): ^ = / ^dt = K{t) -f W{t) - [K{to) + W{to)]. Jto dt
(13.147)
If the body underwent no deformation at the initial time to, i.e., Uap{to) = 0 and W{to) = 0 respectively, and if it was also at rest at to, i.e. K{to) — 0, then the relationship (13.147) takes the form R^K(t)^-W{t).
(13.148)
It is quite clear from definition (13.142) of function T that it depends only on the deformation tensor components at any given time and at any given point in the elastic body. This means that function W obtained by the integration of T over the entire volume V^ occupied by the elastic medium, depends only on its deformations at a given moment of time.
425
Uniqueness of the solution of the wavefield equations
Using formula (13.148), one can demonstrate that function W describes the potential energy of deformations, i.e. the work performed by the external stresses and volume forces in order to produce the given deformations in the elastic body. Indeed, if the body, affected by those forces, changes from its initial state of rest (in the absence of deformations) into a new state of rest under deformations, then, as seen from (13.148), it appears that (13.149)
R=W{t),
since the kinetic energy K equals zero when the body is in a state of rest. Thus, relations (13.148) and (13.149) express the law of energy conservation: the work performed by external forces is equal to the sum of the kinetic energy K and the potential deformation energy W of the elastic body. 13.4-3 Uniqueness of the solution of initial-value problems We will now prove the uniqueness of the solution of the three initial-value problems formulated at the beginning of this section. Let us assume that any of these problems admits two solutions. Then we can obtain the difference between those solutions U'(r,i). It is evident that U'(r,t) satisfies the equations of motion in the absence of volume forces; besides, this solution characterizes the elastic medium which was initially in a state of rest: U'(r, *)!,=*„ = 0,
dlJ'{r,t)/dtl^,^
= 0.
(13.150)
The boundary conditions for U'(r, t) in all three problems can be described as follows: P , - U ' = 0, r G 5 . (13.151) Indeed, in the case of the first problem (the case of the known external load applied to the boundary S) the stresses acting at the boundary S for the difference solution U'(r,^) must be equal to zero: Pn = 0. In the second problem (the case of the known displacement vector U at the boundary S) the difference solution itself is equal to zero on the boundary S for all time, from which it follows that U ' = 0 on the surface S. The third problem is a combination of the first two. Therefore condition (13.151) holds at the corresponding parts of the boundary for the same reasons as in the first or second problems. Using equation (13.132), we can calculate the work done by external stresses in the absence of external volume forces :
R=
/ / P^-V' dsdt. JJ to to JJ Js Js
(13.152)
Substituting condition (13.151) into this equation, we find that the work of the external stresses is equal to zero: R = 0.
426
Wavefield equations
Hence, in accordance with the law of energy conservation (13.148), it follows that: K + W = 0.
(13.153)
Each summand on the left-hand side of formula (13.153) is non-negative. Therefore this relation holds true only in the case where K = W = 0 ior all time t>to. Taking into account that
III>-^
K=
III
^PIU'I dv,
we can conclude that the condition K = 0 leads to |U'|= 0, i.e. U'x=U'y=U'z= 0, from which we deduce that U' is constant. Eventually, since U' = 0 initially, we conclude that U' = 0 for all subsequent time. This means that the two possible solutions introduced at the beginning of this section identically coincide within the domain V. This completes the proof of the uniqueness of the solution of three principal initial-value problems for a bounded domain. A similar approach can be used to prove the uniqueness theorem for the solution of the initial value problem in an unbounded domain. Firstly, as seen from the previous consideration, relation (13.133) is valid for any bounded domain, in particular, for a domain K- which is bounded by a sphere Or with radius r and centre at the origin of coordinates. In this case expression (13.152) takes the form R=
f
ff
P^.IJ'dsdt.
(13.154)
J to J J Or
Secondly, we can use the known fact that the disturbance (the displacement field) propagates in the elastic medium with a finite velocity. This means that if the disturbance is local at initial moment of time to (U (r, to) 7^ 0 only within some local domain), then it will be contained within a sphere of some finite radius r at any subsequent moment. The same observation holds true for the diflFerence field U ' ( r , t ) as well. Therefore, considering the diff'erence displacement field U ' ( r , t ) at some specified moment of time t > to, one can select an auxiliary sphere of the radius r large enough for the integral over the sphere Or in formula (13.154) to be equal to zero. From here, as it was previously discussed in the case of the bounded domain, it follows that relation (13.153) holds true, and the solution of the corresponding initial-value problem is unique. 13.4-4 Sommerfeld radiation conditions In quite a number of practically important problems of wavefield propagation the initial state of the medium happens to be unknown. This situation occurs, for example, when the direct measurements can be carried out, for technical reasons, only
427
Uniqueness of the solution of the wavefield equations
on the boundary of the examining region, which is typical for geophysical observations. In order to characterize the wavefield distribution at the current moment of time in such a case, it is necessary to take into account the wavefield behavior for all previous time. In other words, we should solve the wave propagation problem for an infinite time interval from minus to plus infinity, ~oo < t < -hoc. If we also select an unbounded domain of field propagation, the corresponding wavefield equation may have non-unique solutions. These non-unique solutions are represented by both convergent (arriving from infinity) and divergent (going out to infinity) waves, which are equivalent from the mathematical point of view. However, it is clear that only divergent waves are physically meaningful. That is why it is important to formulate an analytical principle permitting selection of a physically meaningful solution. This principle, described as the "radiation condition," was first introduced by Sommerfeld (1912, 1954). We derive first the Sommerfeld radiation conditions for the scalar wave equation. Let us analyze a scalar wavefield, satisfying the following equation, V ' P ( r , t) -
1 9^
^^:^P{T,
t) = -F'
(r,t),
r G i?3,
- o o < t < oo,
(13.155)
(i?3 is entire space) and decaying at infinity: P(r, t) -> 0,
r -> oo,
- o o
(13.156)
As a typical example of this type of a wavefield, we can consider the following elementary time-harmonic spherical waves: Pi(r,t) = ~cosa ft - -V Airr \ c/ P2(r,t) = — cosa (t + - V
(13.157) (13.158)
where r = |r| , and a is some constant. Functions Pi{r,t) and P2{r,t) satisfy equation (13.155) (for F^ (r,t) = 0) throughout the entire space except at the origin of the coordinates, where they have a singularity of type 1/r. The first function, Pi(r, t), corresponds to a time-harmonic wave with its wavefront (defined as the constant phase surface) described by a sphere expanding with time, i.e. Pi(r, t) is nothing else but a divergent spherical wave. The function P2{r, t) characterizes a convergent, i.e. arriving from infinity, spherical wave. Adding and subtracting the functions Pi(r, t) and P2{r,t)^ we construct two standing waves: Ps{r, t) = ^^
cos (a-)
ZTTT
\
cos {at),
(13.159)
c/
and P4(r, i) = Y~ ^^^ ( ^ " ) ^^^ ^^^^ •
(13.160)
428
Wavefield equations
At the origin of coordinates the first wave, P3(r, t), features the same singularity as waves Pi(r,t) and P2{r,t), i.e. it is a constrained standing wave. It is easy to see that the second wave, P4(r,t), has no singularity throughout the space where it satisfies equation (13.155) (with zero right-hand side) and condition (13.156), i.e. it decays at infinity. This solution of equation (13.155) represents a free standing wave. It can easily be seen that adding function P4(r, t) to any solution of the problem defined by (13.155) and (13.156) provides another solution of the same problem. Thus, it becomes clear that there exists a set of diflFerent mathematical solutions of (13.155) and (13.156). In other words, we cannot uniquely define the real physical process by assigning the sources of oscillations and the condition of field decay at infinity. At the same time, in the real world there is only one physical solution of this problem. It becomes clear now that we should impose an additional constraint to obtain a unique, physically meaningful, solution. This constraint should be able to reject convergent waves, acquiring and carrying energy from infinity, which is physically impossible. Note that the divergent and convergent spherical waves, (13.157) and (13.158), are characterized by diflFerent asymptotic behavior at infinity. The following relation is valid for the divergent wave
but is invalid for the convergent waves; vice versa, in the case of convergent waves, the following relation holds:
which is invalid for divergent waves. Note that the expression o(l) in (13.161) and (13.162) denotes a function which asymptotically goes to zero, when r -^ oo : lim o(l) = 0.
r—•oo
Intuitively, one can imagine that the field generated by an arbitrary source, located within a bounded domain of space, can be approximated accurately enough by a spherical wave at a large distance from the source. We can expect, therefore, that the asymptotic behavior of this field at infinity in the case of divergent and convergent waves can be characterized by formulae (13.161) and (13.162) as well. These heuristic considerations provide a basis for choosing only those solutions of the problem defined by (13.155) and (13.156) which satisfy additionally condition (13.161). Examining the wavefield in the frequency domain, we can apply the Fourier transform to relation (13.161). As a result, we obtain: r (^—p(r,cj) - i^p{T,ujyj
= o(l),
r -> GO,
(13.163)
Uniqueness of the solution of the waveReld equations
429
where p(r,a;) is the Fourier spectrum of the wavefield P(r, t). Condition (13.163) was introduced by Sommerfeld (1912) to characterize the divergent waves. This condition in combination with the requirement that the product of the field p(r,u;) itself and the distance r = |r| is bounded, p(r,a;)-0(l/r),
(13.164)
are called Sommerfeld radiation conditions. In the last formula the function 0 ( l / r ) stands for function which asymptotically goes to zero as fast as 1/r : V 0{l/r) ^ hm ——;— < const.
r-^oo
(13.165)
l/r
In summary, Sommerfeld radiation conditions for a scalar wavefield can be written as follows:
rp{r,(jj) is bounded, lim r —p(r,cj) or
-i-p{r,u;) = 0, c
r = r
(13.166)
Note in conclusion of this section that these conditions can be extended to the case of an elastic wavefield U as well (Kupradze, 1933, 1934, 1963; Zhdanov et al., 1988; Aki and Richards, 2002). We will discuss this generalization later in this chapter. 13.4-5
Uniqueness of the solution of the wave propagation problem based on radiation conditions
In the case of the unbounded domain coinciding with the entire space, the radiation conditions ensure the uniqueness of the solution of the wave propagation problem. In order to prove this statement, we will write the internal integral in the Kirchhoff formula (13.112) in the frequency domain, using the convolution theorem, as follows:
f
P(r, t)-^G^ On
(r', t'\r, t) -G^ (r', t'\v, t) | - F ( r , t) dt dn
J —c
p ( r , u ; ) | - G - (r>;a;) ~G^ (r'|r;u;) | ^ p ( r , a ; ) exp(-iu;t')(ia;, (13.167) where p(r,cj) is the time spectrum of wavefield P{r,t) and G'^(r'|r;a;) is the time spectrum of the Green's function of the wave equation. After substituting relation (13.167) into (13.112) and changing the order of integrations, the Kirchhoff formula takes the form
'•''^-rj:
P{r'
G-{v'\v;oj)-^p{v,u)
Wavefield equations
430
-p(r,a;)—G^(r'|r;a;) ds\ exp{-iujt')duj,
r' e D.
(13.168)
Based on the last expression, we can write the KirchhofF integral formula in frequency domain as " " > ' = / / » [C"(r'|r;u.)|;p(r,i^) -p(r,a;)—G-(r'|r;a;)
ds,
Y'
€ D.
(13.169)
Note that formula (13.169) holds true for any distribution of the wavespeed in the media, assuming that we use the corresponding Green's function in this formula. In the simplest case of a constant wavespeed, we can use expression (13.74) and write formula (13.169) as follows: p(r',cj) ,
//.
, d
exp(2a;|r'—r|/c)^ d 47r|r'—r|
/exp(ia;|r'—rl/c)
ds,
T' G D ,
(13.170)
where we assume that the field p(r, a;) satisfies the homogeneous Helmholtz equation throughout the space UJ"
V>(r,a;)-f — p ( r , a ; ) = 0
(13.171)
and approaches zero at infinity in such a manner that p(r,cj) = 0 ( l / r ) ,
r -^ GO.
(13.172)
We assume that this field satisfies the radiation conditions at infinity as well: d -p(r,cj) - i-p{v,u) or c
= o(l/r),
(13.173)
where the function o ( l / r ) stands for a function which asymptotically goes to zero faster than 1/r : lim
r—>oo
\
l/r
^ = 0.
(13.174)
Our goal is to prove that this field is identically equal to zero in the entire space:
p(r,u;) = 0.
Obviously we can use the Kirchhoff formula in the frequency domain (13.170) to represent the field p(r,u;) within any bounded domain of the space. For example, we can apply the Kirchhoff formula to a domain K bounded by a sphere Or of radius r and center at the origin of coordinates (Figure 13-2). In this case the direction
Uniqueness of the solution of the wavefield
431
equations
Figure 13-2 Application of the Kirchhoff integral formula to a domain K bounded by a sphere Or of a radius r with the center at the origin of coordinates. The direction of the normal to the boundary of the domain at a point M e Or coincides with the direction of the radius-vector at that point.
of the normal to the boundary of the domain at a point M e Or coincides with the direction of the radius-vector of that point, i.e. v^ith the direction of vector r, hence d/dn = d/dr.
(13.175)
R=: | r ' - r | ,
(13.176)
Introducing the notation and taking into account (13.175), we can transform relation (13.170) as follows:
^<''"^'^/I ,
, d
^exp{iujR/c)\
d p(r,cj) dr
ATTR
fexp(iujR/c)
ds,
r' e Vr.
(13.177)
It is clear enough that the value of the integral on the right-hand side of (13.177) should not depend on the radius r, since the left-hand side of this equation is independent of r also. Therefore, taking the limit as r —> oo, we can write
H^'-)=j™//„,[(
exp {iujR/c) \ d
p(r,uj)
Wavefield equations
432
,
, d
/expUuR/c)
ds.
(13.178)
Let us denote the inner integral in (13.178) as Io^{p;r',uj) : exp(iujR/c)\
IoAp;r',uj)^ [f
d
,
,
.
.d
/expliLuR/c)
ds.
J J Or
(13.179) We will prove that the limit of this integral goes to zero if the radius of a sphere Or tends to infinity: lim 7o,(7?;r',a;) == 0. (13.180) Indeed, taking into account that d fexp{iLjR/c)\ dr ATTR
d fexp{iujR/c)\
dR
OR dr
ATTR
.LJ exp{iLuR/c)
exp{iujR/c)
dr ' we may rearrange expression loXv'^^'^^) ,
IL
ff
^ follows: (
.dRexp{iujR/c)
d .uj dR exp {iuuR/cj ds. -p(r,u;)-z-Kr,a;) — ATTR
(13.181)
Now, let us evaluate each of the integrals on the right-hand side of equation (13.181) separately. Proceeding from the cosine law, we have R = | r ' - r | = r^ — 2rr' cos (r, r') H- r''
1/2
(13.182)
From here, first, we find, that
JL_ _ 1 1 - 2—cos (r,r') + -:'R~r
-1/2
-1/2
i + o( r
i+o(-
oo.
r We can evaluate also an asymptotic behavior of dR/dr : dR _ r-r' dr
cos (lyr') _ r[l + 0 ( l / r ) ] R R
(13.183)
Uniqueness of the solution of the wavefield
433
equations
i2
l +O
l +O
(13.184)
oo.
From relations (13.172) and (13.184) we find that
OR
(13.185)
oo.
Therefore, taking into account (13.183), we can evaluate the expression under the integral sign in the first of the integrals in (13.181) as follows: P(r,
dR exp {iujR/c) dr
^ + ^1-
ATTR^
O
-
,
r - . oo,
(13.186)
where we use the equation O
rJ
\r'^
In a similar way we can evaluate the asymptotic behavior of the integrand in the second term of equation (13.181). Using (13.185) and the radiation condition (13.173): d
.u
OR exp {iuR/c)
.|:p(r,c.) - z^^pir^u) + o ( 1 ) ] O ( 1 ) ^ o ( 1 ) ,
r ^ oo.
(13.187)
Substituting asymptotic formulae (13.186) and (13.187) into (13.178), and taking into account that the area of the sphere Or is equal to 47rr^, we finally obtain p{r',u) = lim / /
o (l/r^) ds = ATT lim
o(l/r') l/r2
(13.188)
Since we selected an arbitrary point r', we conclude that the wavefield satisfying conditions (13.172) and (13.173) is identically zero throughout the entire space. We can take as a function p(r', a;) a difference between any two solutions of the problem (13.171), (13.172), (13.173). Therefore, we have found that this difference is identically equal to zero. This completes the proof of the uniqueness of the solution of the wave propagation problem using radiation conditions. The radiation conditions make it also possible to prove uniqueness of the solution of the wave propagation problem in more general situations with an arbitrary distribution of wavespeed and when the wavefield is given not in the entire space, but only within some external domain CD. Note that an external domain CD is a complement of some bounded domain D in the entire space (see Figure 13-3).
Wavefield equations
434
Figure 13-3
An example of the external domain CD.
In this situation one is required to complement the problem (13.171), (13.172), (13.173), with certain boundary conditions on the boundary S of the domain CD. Such a formulation of the wave propagation problem appears in the analysis of wave scattering by local inhomogeneities of the medium. The proof of the uniqueness of the solution of this problem is similar to the one outlined above for the wave propagation in the full space (Kupradze, 1935, 1950, 1963). 13.4-6 Kirchhoff formula for an unbounded domain Note that we have derived the Kirchhoff formula (13.112) only for bounded domains. Equipped now with the radiation conditions, we can apply this formula to unbounded domains as well. Let us consider a domain V^ which may include a point at infinity. The domain V is bounded by the surface S (Figure 13-4). Let wavefield F(r, t) satisfy the homogeneous wave equation in that domain: V 2 p ( r , t ) - i ^ P ( r , t ) = 0,
v&V,
-oo < ^ < oo.
(13.189)
The corresponding field p(r, oj), obtained as a result of a Fourier transform from the time to the frequency domain, satisfies the homogeneous Helmholtz equation: VV(r, u) -h ^ p ( r , cj) = 0,
r GK
(13.190)
We also assume that the frequency domain field p(r, u) decays at infinity, p(r,a;) = o(l),
r - ^ oo,
Y eV,
(13.191)
and satisfies the radiation condition: —p{v,u)
- i%{v,uj)
^
oi-
oo,
rGK
(13.192)
We will show that under these assumptions the field P{r,t) may be represented inside the domain V by the Kirchhoff formulae (13.112) or (13.168).
Uniqueness of the solution of the wavefield
a.
equations
435
b.
Figure 13-4 Some examples of unbounded domains, a. The boundary 5 of a domain y is a quasi-horizontal surface extending to infinity; F is a lower half-space, b. The boundary 5 of a domain F is a closed surface; V is the complement of some bounded domain CV in the entire space.
Let us draw a sphere Or with a large enough radius r and centre at the point r'. We denote by symbols K- and 5^ the parts of the domain V and surface S respectively which happen to be inside the sphere Or- Note that 5^ may be a part of surface S (Figure 13-4a) or it may coincide with it (Figure 13-4b) (The latter case occurs when the entire surface S is inside the sphere O^). For example, the geometrical situation presented in Figure 13-4b may occur if the sources of the wavefield are located inside domain CV, the complement of domain V for the full space. In this case, obviously, the wavefield satisfies the homogeneous wave equation ever3rwhere outside domain CV containing the sources. It is easy to see that the boundary of domain K in the case shown in Figure 13-4a consists of the surface S!^ and of that part of the surface of sphere Or which happens to be inside domain V, and may be denoted as S^!. In the case presented in Figure 13-4b, the sphere Or itself forms the outer boundary of the domain K. Since domain Vr is bounded, the field P{Y,t) can be represented by the Kirchhoff formula (13.170): p(r',u;) = /5;(7?;r',u;)-h/5;'(p;r',cj), r' G K-.
(13.193)
where we have used the notation of (13.179). We can make the radius r in expression (13.193) go to infinity: r —^ oo. Then, obviously, limy^-^oo h'^V] r',a;) = Is{p] r ' , ^ ) and, from the asymptotic formulae
Wavefield equations
436
(13.183)-(13.187) given above, it follows that: lim IS"{P\Y',uj)
= 0.
^
r—^oo
Thus, exp(ia;|r'-r|/c)\ d
,
.
,
/ e x p (zcjlr'-rl/c)
.d
//.
ds.
(13.194)
A similar result holds true for the case shown in Figure 13-4b. Applying the inverse Fourier transform to both sides of the last expression, we arrive at the Kirchhoff integral formula for an unbounded domain: 1
P ( r ' , t ' ) = —• /
/'+00
27r 7_oo
-I
p{r\uj)exp{-iujt')duj
rj:{jL .
J J s J-o
=-—
27r
/»+00
/
Is{p]r',uj)exp{-iut')duj
J_^
exp(za;|r'—r|/c)\ d 47r|r'-r|
, d /exp(ia;|r'
exp(—za;t')(ia;
G-{v',t'\v,t)—P{v,t)
d
-P{r,t)^G^{r',t'\T,t)
dtds,
r'
eV.
(13.195)
If r' ^ V, the integral on the right-hand side of equation (13.195) vanishes. Note that formula (13.195) holds for a solution of the wave equation with variable wavespeed, if G ^ ( r ' , f | r , t ) is taken as the corresponding Green's function for this equation, satisfying the corresponding radiation condition at infinity:
J Js J-c
G- (r', t'\v, t) ~P{v,
t) - P{v, t)^G^
^ r P{r',t')^T' ' \0,T' ^V
(r', t'\v, t) dtds
£V
(13.196)
We can also write a similar representation in the frequency domain, using the convolution theorem (Arfken and Weber, 1995):
II
G - (r'|r;a;) l-p{r,u;)
- p{r,uj)^G^
(r'|r;c.) ds
Uniqueness of the solution of the wavefield
437
equations
(13.197)
0, r' ^ 1/
Let us compare relation (13.195) with the Kirchhoff formula for domain K shown in Figure 13-4b: 1 /"^°° P(r', ^0 = 5 ^ / Is{p] r', uj) exp{-iujt')cLj +00
+
Ior{p;v',uj) exp{—iujt^)d(jj, r' e K-.
(13.198)
Equating the right-hand sides of formulae (13.195) and (13.198), we find that integral IO^{P]T',LU) and the Kirchhoff integral over surface Or are both equal to zero for a sufficiently large but finite value of r (for which S'^ = S): /o,(7?;r',a;) -
IL
exp (za;|r'—r|/c) 47r|r'—r|
—p{v,u)-p{v,uj)
—
exp(zcc;|r'—r|/c) 47r|r'—r|
ds = 0, (13.199)
and
ILL
G - (r', t'\v, t) | - P ( r , t) - P(r, t)^G-
(r^ t > , t) dtds = 0.
(13.200)
One can assume that the Kirchhoff integral over the surface Or characterizes the superposition of waves traveling across that surface from the surrounding outer space inside domain K-. Therefore, equations (13.199) and (13.200) show that in the given model, only outward waves travel across the surface O^., providing the radius r is large enough. In fact, based on the radiation conditions, one can prove (Kupradze, 1950; Courant and Hilbert, 1962) that in this case the wavefield energy flux across the surface Or is directed out of the domain Vr. Taking into account that in our case the sources of the wavefield are located inside domain CV (Figure 13-4b), one can conclude that the total energy radiated out of the domain containing the sources of the wavefield is always non-negative. Thus, the wavefield sources continuously radiate energy away from these sources! The last fact corresponds well to the basic energy theorems we discussed in Chapter 8 for the electromagnetic field. In particular, one can prove energy inequahties for the wavefield, which are similar to the electromagnetic energy inequalities (8.100) and (8.111). 13.4-7
Radiation conditions for elastic waves
In the conclusion of this section we demonstrate that the Sommerfeld radiation conditions can be extended to the case of an elastic wavefield U (Kupradze, 1933, 1934,
438
Wavefield equations
1963; Zhdanov et al., 1988; Aki and Richards, 2002). Let us consider an elastic wavefield U ( r , t ) characterized by the Lame equation in an unlimited domain: LU - d'^V/dt^ - - F ^ P
(13.201)
Since this field may be represented as a superposition of two types of waves, compressional and shear waves, we are faced with the problem of formulating a certain analytical criterion (similar to the Sommerfeld radiation conditions) that provides for the exclusion from the solution of the elastic field equations of compressional and shear waves that are convergent at infinity. It should also be pointed out that the radiation conditions are not included as some kind of heuristic principle in the initial mathematical formulation of the problem. In the scalar case we obtained these conditions by analyzing (for the sake of simplicity) the asymptotic behavior of spherical waves. At the same time, as it has been demonstrated above, the same result could be obtained by analyzing the conditions required to ensure that the corresponding Kirchhoff integral goes to zero over a large sphere expanding to infinity. Following Kupradze (1963), we will demonstrate that the radiation conditions for the potential (compressional wave) and solenoidal (shear wave) components of the elastic field can be formulated as follows: LnUp,s(r,a;) - za;Cp,sUp,5(r,a;) = o ( - j ,
Up,,(r, u;) • G^(^'^) (r|r'; uo) = o (^ Up,,(r,u;) = o(l),
r -^ oo,
,
r -> oo,
r -> oo,
(13.202)
(13.203) (13.204)
where Up(r,a;), Us(r,u;), G^(^) (r|r';a;), G^(^) (r|r';a;) are respectively the potential and solenoidal components of the elastic wavefield u(r,a;) and the Green's tensor G^ (r|r';a;) of the Lame equation in the frequency domain; while Ln is the stress operator introduced in Appendix F (see formula F.17), Ln -= c^nV- - c^n X V X .
(13.205)
Indeed, let us consider again the model shown in Figure 13-4b. We assume that the elastic field U ( r , t ) satisfies the Lame equation (13.201) everywhere in full space, while the sources of the wavefield are located inside the domain CV^ the complement of domain V for the full space. We can write the integral representation (13.120) of the elastic wavefield within a domain D^^ bounded by a sphere Or of radius r as: - / ^ jj^
{ c > - [U(r, t) ( V - G ^ (r', t'\v, t)) - G^ (r', i'lr, t) ( V • U(r, t))
439
Uniqueness of the solution of the wavefield equations
-hc^n- hj{r,t)x
[ v x G ^ ( r ' , t ' | r , d + [V x U(r, t)] x G ^ (r', t'|r,^)l } dsdt
+- /
/ / / - F " ( r , t) • G^ (r', t'|r, t) dvdt - U (r', t'), r' G Dr. (13.206) 7-00 J J JcV P Applying the Fourier transform to both sides of this equation, we can obtain the corresponding Kirchhoff formula in the frequency domain. According to the convolution theorem (Arfken and Weber, 1995), the Fourier transform of the convolution of two functions is equal to the product of spectra of these functions. Therefore we obtain: - j j
{ c > . [u(r, u) ( V - G ^ (r'|r; u)) - G^ (r|r'; uj) (V • u(r, a;))]
-\-cln- [u(r,a;)x {v x G^(r'|r;a;)) + (V x u(r,u;)) x G ^ (r'|r;a;)] } ^5 +
/ / /
-r{Y,Uj)'G^{Y'\Y',Uj)dv
= M{Y,Uj),
Y' e Dr,
(13.207)
where G^ (r|r';a;) is the Green's tensor for the Lame equation in the frequency domain. We can formulate the radiation conditions for elastic waves as certain requirements for the field behavior at large distances from the source region CV. These requirements should ensure that the Kirchhoff integral over the sphere Or in formula (13.207) goes to zero as r —> oo. In order to determine the corresponding conditions, let us analyze the surface integral in formula (13.207) : /5^(u;r',cc;)//
{c>- [u(r,a;)(v-G^(r'|r;a;)) -G^(r1r;a;)(V.u(r,a;))]
+ cln- [u(r,w)x ( ' v x G ^ ( r ' | r ; w ) ) + ( V x u(r,a;)) x G ^ (r'|r;a;)l } ds.
(13.208)
Using the stress operator (13.205), we can write expression (13.208) as follows: /^^(u;r',w)= / /
|u(r,w)-L„G^(r'|r;w)-G^(r'|r;w)-L„u(r,w)|ds.
(13.209)
Now we recall that u = Up+u„ and G^ = G^^^) + G^(^). Substituting (13.210) into (13.209) we find: I^^ (u;r',a;) = j j
[(u^+u.) \.„ (G^^^) + G^W)
(13.210)
Wavefield equations
440
(^G^ip) + G^(^)) . L„ (up+u,)]ds.
(13.211)
Opening the parentheses in the integrand, we can represent the last integral as a sum of four surface integrals:
-IL -IL
ds L„Up-G^(^) + L„u, • G^(^) ds u,-L„G^(^)+Up.L„G^(^)' ds
-IL
LnUp.G^(^) + LnU,.G^(p)| ds.
In this equality, we add and subtract terms iojCpUp-G^^^^ and iuCsUs m the first and second integrals, and terms icuCsUs-G^^^^ iujCpUp-G^^^\ and iujCsUp'G^^^\ iuuCp Q^p) in the third and fourth integrals. As a result, we have /o.(u;r',u;) / / [up- ( L „ G ^ ( ^ ) - zc^CpG^(^)) - (LnUp - iLJCpUp) •G^(P)"ds
+
/ / [u, . (LnG^(^) - za;c,G^(^)) - (LnU, - ia;c,u,) • G^^^^ ds
+
/ / [u,. (LnG^^P) - 2cjCpG^(^)) - (L„Up - iujCpUp) .G^(^) ds I
+ ff iuj ff
[up . (LnG^(^) - ia;c,G^(^)) - (L^u, - za;c,u,) • G^^^) ds
Lup.G^(^) + CsUs • G^(P)] ds 4- iuj ff
Lu^.G^^^^ + c,Up • G^^^^l ds.
(13.212) Extending the radius of sphere Or in formula (13.212) to infinity, r —> oo, and taking into account conditions (13.202) - (13.204), which hold true for the tensors G^^P^ and G^^") as well, we find that lim/^^(u;r',a;)=0. Thus, we have the unique solution of the Lame equation (13.201) for the infinite elastic space, provided by integral formulae (13.207) and (13.206): u(r, cu) = / / /
-f^(r, uj) • G^ (r'|r; uj) dv in the frequency domain,
(13.213)
Uniqueness of the solution of the wavefield equations
441
and U (r^ t') = I
III
-F^(r, t) ' G^ (r', t'|r, t) dvdt in the time domain. (13.214)
J-OQ J J JCV P
It may be pointed out that radiation conditions (13.202)-(13.204) provide the uniqueness of the elastic problem solution in the case of an arbitrary external domain.
References to Chapter 13 Arfken, G., and H. J. Weber, 1995, Mathematical methods for physicists. Fourth Edition: Academic Press, San Diego, New York, Boston, London, Tokyo, Toronto, 1028 pp. Aki, K., and P. G. Richards, 2002, Quantitative seismology, 2nd Edition: Columbia University Press, 576 pp. Bleistein, N., 1984, Mathematical methods for wave phenomena: Academic Press Inc., (Harcourt Brace Jovanovich Publishers), New York, 341 pp. Bleistein, N., Cohen, J. K., and J. W. Stockwell, Jr., 2001, Mathematics of multidimensional seismic imaging, migration, and inversion: Springer, New York, Berlin, London, Tokyo, 510 pp. Cerveny, V., 2001, Seismic ray theory: Cambridge University Press, Cambridge, New York, 696 pp. Courant, R. and D. Hilbert, 1962, Methods of mathematical physics. Vol. II: Partial differential equations (Vol. II by R. Courant): Interscience Publishers (a division of John Wiley & Sons), New York, London, 830 pp. Dahlen, F.A. and J. Tromp, 1998, Theoretical global seismology, Princeton University Press, Princeton, 1025 pp. Felsen, L., and N. Marcuvitz, 1994, Radiation and scattering of waves: IEEE Press, New York, 888 pp. Kupradze, V. D., 1933, Uber das Ausstrahlungsprinzip von A. Sommerfeld: DAN SSSR, 1, 2. Kupradze, V. D., 1934, Method of integral equations in the diffraction theory (in Russian): Mathematical Series, 4 1 , 4. Kupradze, V. D., 1935, Main problems of the diffraction mathematical theory (in Russian): ONTI, Leningrad, Moscow, 111 pp. Kupradze, V. D., 1950, Boundary problems of the oscillation theory and integral equations (in Russian): Gostekhteoizdat, Moscow, Leningrad, 280 pp. Kupradze, V. D., 1963, Methods of potential in the elastic theory (in Russian): Fizmatgiz, Moscow, 472 pp. Landau, L. D., and E. M. Lifshitz, 1970, Theory of elasticity. Second Edition: Pergamon Press, Oxford, New York, 165 pp. Love, A. E. H., 1944, A treatise on the mathematical theory of elasticity, Fourth Edition: Dover, New York, 643 pp.
442
Wavefield equations
Morse, P. M., and H. Feshbach, 1953, Methods of theoretical physics, McGraw Hill Book Co., New York, 1978 pp. Sommerfeld, A., 1912, Die Greensche Function der Schwingungsgleichung. Jahresher: D. D. Math. Vereinigung, 21, 309 pp. Sommerfeld, A., 1954, Optics. Lectures on theoretical physics, vol. 4: Academic Press, New York, 383 pp. Udias, A., 1999, Principles of seismology: Cambridge University Press, Cambridge, 475 pp. Zhdanov, M. S., 1988, Integral transforms in geophysics: Springer-Verlag, New York, Berlin, London, Tokyo, 367 pp. Zhdanov, M. S., Matusevich, V. U., and M. A. Frenkel, 1988, Seismic and electromagnetic migration (in Russian): Nauka, Moscow, 376 pp.
Chapter 14 INTEGRAL REPRESENTATIONS IN WAVEFIELD THEORY In this chapter we will derive the basic integral representations for scalar and vector wavefields similar to those introduced in Chapter 9 for an electromagnetic field. These representations play an important role in forward modeling and in calculating the Frechet derivative of the forward modeling operator, thus providing the foundation for solution of the inverse problem. Our discussion will be restricted to fields satisfying wave equations only. However, based on generalized Kirchhoff integral formulae and Green's functions for the Lame equation, it is possible to extend practically all the methods outlined in this Chapter to the case of elastic vector fields. While this extension is straightforward, there are very comprehensive mathematical problems that should be solved to address the problem of modeling elastic field data in full detail. Thus, these advanced issues are left beyond the scope of this book. Nevertheless, some general ideas of elastic field integral representations will be discussed in Chapter 15 in connection with elastic field inversion. 14.1
Integral equation method in acoustic wavefield analysis
H.1.1
Separation of the acoustic wavefield into incident and scattered (background and anomalous) parts
Let us consider an acoustic medium. described by the scalar wave equation
The propagation of acoustic waves can be
V ' ^ ( r , t) - ^ ^ i ' C r , t) = -F^ (r,i),
(14.1)
where P{r,t) is the pressure field, F^(r,t) is the strength of an external source of energy located within some bounded domain Q, and c(r) is the velocity of the wave propagation. The time domain pressure variations, P ( r , t ) , can be represented as a superposition of frequency domain signals, p(r,a;), using the inverse Fourier transform: P(r,^) = — y
p{v,uj)e-'^'dw,
(14.2)
444
Integral representations in wavefield theory
where
+00 +00
Pir,t) iujt dt. / -00 In the frequency domain, equation (14.1) can be written as the Helmholtz equation u^
V'p{T, u) + ^ r ^ p ( r , u;) = - r (r, u;), where
+00 r+00
(14.3)
F%r,t)e'^'dt.
/ •00 We also assume that the pressure field satisfies the radiation conditions at infinity: rpCr.ou) is bounded, lim r I -;r-p{r,(jj) — i—p{r,uj) = 0, - — I or c
r = |r|.
(14.4)
Let us represent the distribution of the velocity in the model as follows: 1
1
cf(r)~4(r)^^^^^^'
(14-5)
where Cb{r) is the background (normal) velocity distribution, and function a(r) ^ 0 appears within the local anomalous zone D with a velocity diff'erent from the given background values. The inhomogeneity is considered to be local, so there is some radius R such that for y^xM-^^-fz^ > R we have a medium with the background velocity ct (see Figure 14-1). Note that inverse velocity is often referred to as the slowness s of the wave propagation, c (r)
Cft(r)
Therefore, function a(r) represents the normahzed anomalous square slowness As^ (r) ^W =
s2 (r) - si (r) _ As^ (r) ^FTT^ = ZT' slir) ~ -JT slir)
(14-^)
where As'{v)
= s'{r)-sl{r).
(14.8)
Similar to an electromagnetic field (see Chapter 9), the total wavefield in the model described above can be represented as a sum of two parts, the incident p^ (background) field p'(r,a;) and the scattered (anomalous) field p*(r,u;), p{T,uj)=p'{T,u;)-^p'{r,uj),
(14.9)
where the incident field is a field generated by the given sources in a model with the background (normal) distribution of velocity Cb{r), and the scattered field is produced by the anomalous square slowness distribution A5^(r).
Integral equation method in acoustic wavefield
analysis
445
Figure 14-1 Separation of the acoustic wavefield into incident and scattered (background and anomalous) parts. The external source is located within some bounded domain Q. The wavespeed is equal to the background velocity everywhere with exception of the local domain D where the velocity is different from the given background values.
The incident field p'(r,a;) is the solution of the Helmholtz equation with the background velocity distribution CJ'
(14.10)
The scattered field p*(r, w) satisfies the same Helmholtz equation, but with a diflferent right-hand part, V V ( r , a ; ) + ^—-p^(r,w) = -oj'^As'^ (r)p(r,a;).
(14.11)
where As^ (r) is the anomalous square slowness determined by equation (14.8). Both the incident and the scattered fields satisfy the radiation conditions (14.4) at infinity. Let us denote by f^{r,uj) the strength of the anomalous source of energy (14.12) Then the equation for the scattered field takes the form UJ
(14.13)
14-1-2 Integral equation for the acoustic wavefield As in the electromagnetic case, the differential equation (14.13) can be transformed to an integral equation for the scattered field. Certainly, let us recall the integral
446
Integral representations in wavefield theory
representation formula (13.76) for the solution of the Helmholtz equation (14.10) for the incident wavefield: p\vi,oj)
= ///
nT,oj)G'"{v^\v-uj)dv
= G^ i n ,
(14.14)
J J JVoo
where G^ (/^) is the scalar integral Green's wave operator, and the Green's function G^(r| Tj] uj) is the fundamental solution of the Helmholtz equation for the background velocity distribution + - i ^ G - ( r , | r ; a ; ) = -6{r,-r)
V'G^ir,\r;u)
.
(14.15)
Note that formula (14.14) can be derived directly applying the same technique which has been used for the Kirchhoff integral formula derivation in Chapter 13. Indeed, let us select an arbitrary point r' and around it draw a sphere, On, with a radius great enough that the region Q falls within the region VR bounded by the sphere 0^] that is, Q e VR (Figure 14-1). We now apply Green's theorem to functions p^ and G^ within the region VR :
/ / / VR
J J OR
[j9'(r,a;)V^G" (r,| r;cj) - G" (r,| r;a;) V V ( r , ^ ) ] dv
(r,| T;UJ) - G^ (r,| r;^;) |^P^(r,a;) ds,
p\r,u;)^G-
(14.16)
where n is the unit vector along the outwards normal to the surface OR. Substituting V V and V^G"^ from equations (14.10) and (14.15) into the last equation, we find
III
G'^{vj\v;uj)nT,u;)dv
VR
IL
[ p i | . G - - G^§-PVS
= p'{Tj,cj) if Tj e Vn.
(14.17)
We have demonstrated in a previous chapter that, according to the Sommerfeld radiation conditions (14.4), if the radius R is expanded without limit, the surface integral over OR tends to zero. As a result, we arrive at expression (14.14). Using the same technique, we can derive a similar integral representation for the scattered field as well: f{rj,u;)
= III
G''{rj\v;co)r{r,uj)dv
= G„ ( / « ) .
(14.18)
447
Integral equation method in acoustic wavefield analysis
Substituting the expression for the anomalous source (14.12) into (14.18), we finally obtain the well-known representation of the scattered field as an integral over the anomalous source in inhomogeneous domain D (Bleistein, 1984): p'iVj.Lj) = u^G^ (As^ {v)v{v,uj)) .
(14.19)
Adding the incident field to both sides of this equation, we obtain an integral expression for the total field: p(r,-,u;) = u^G^ {As^ (r)p(r,u;)) +P^(r,-,u;).
(14.20)
Using integral formulae (14.19) and (14.20), one can calculate the wavefield at any point Fj, if this field is known within the inhomogeneity. Expression (14.20) becomes an integral equation with respect to the wavefield p ( r ) , if r,, G D. Equations (14.19) and (14.20) form the basis of forward modeling of the scalar wavefield. 14'1-3 Reciprocity theorem The reciprocity principle plays an important role in the theory of wavefield propagation and in the inversion of wavefield data. It is based on an application of the integral formula (14.17) to two Green's functions, G^ (r"|r;a;) and G^ (r'|r;cc;), satisfying the equations V2G^(r'| r;a;) + ^G"^(r'|r;a;) - -6{r'-r)
,
(14.21)
V'G^(r"| r]uj) 4- ^ G " ( r " | r;u;) - -(5(r"-r) ,
(14.22)
and
2 UJ'
C2
and the corresponding radiation conditions at infinity. Indeed, assuming that /^(r,a;) — 5(r''—r) and, thereforep*(r, a;) = G^(r"| r; CJ), we can write G-(r"l r'-Lo) Iff
G^{r'\ r;u;)6{r"-T)dv-
J J JVR
ff
G-(r"| r ; a ; ) | - G - ( r ' | r;a;)
J J OR
-G-{v'\
v-u;)^G-{v"\
r-uj) ds, if r' G Vft.
(14.23)
Note that, due to radiation conditions, the surface integral over the sphere OR goes to zero if the radius R tends to infinity, and the volume integral is equal to G^(r'| r'';a;). As a result we arrive at the following important theorem. T h e o r e m 27 (The reciprocity theorem) The wavefield at a point r" generated by a point source located at a point r' is equal to the wavefield at a point r' generated by a point source located at a point T" : G"(r''| Y']U) = G"(r'| r'-.tu).
(14.24)
448
Integral representations in wavefield theory
The reciprocity theorem shows that for any wavefield we can switch between the receiver and source positions without changing the values of the observed field. This result plays an extremely important role in wavefield imaging and inversion, especially in wavefield migration, which will be discussed in Chapter 15. 14'1-4 Calculation of the first variation (Frechet derivative) of the acoustic wavefield We can find the equations for the Frechet derivative by differentiating the corresponding scalar wave equation (14.3), which we state here using the slowness function s (r) instead of inverse velocity l/c(r) : V2p(r, Lu) + ujh^ (r) p(r, oo) = -f
(r, ou).
(14.25)
We will denote by Aa the forward modeling acoustic operator providing the solution of this equation for any given source field /^ . This operator transforms the slowness distribution s (r) into the corresponding acoustic wavefield: P = Aa{s').
(14.26)
Let us perturb the square of slowness distribution 5^ ( r ) . Applying the perturbation operator to both sides of equation (14.25), we obtain the equation for corresponding variations of the acoustic wavefield: V^6p{v,uj) -{-u^s^ {r)6p{r,uj) = -ou'^Ss^ {r)p{r,uj),
(14.27)
where ^5^ is the square slowness variation, 6p is the corresponding acoustic wavefield variation, and we take into account that the perturbation of an external source is equal to zero: 6f^ = 0. Applying the integral equation method (formula (14.19)) to the solution of equation (14.27), we find that the variation of the acoustic wavefield at a point r is related to the variation of the slowness distribution by the following integral operator: Sp (?, uu) = uj^G^ {6s^ (r) p (r, a;)) .
(14.28)
Using the last equation, we calculate the Frechet diff'erential (for definition see Appendix D) of the forward modeling acoustic operator: Fais^Ss^) = 6Aa{s^6s^) = Sp{r,uj) = uj^ III
G'"[Y\Y]uj)6s\v)p{Y,uj)dv.
(14.29)
Note that the argument in the expression for the Frechet differential, Fa(s^,55^), consists of two parts. The first part, 5^, is a square slowness distribution, for which we calculate the variation of the forward modeling operator; the second part, ^5^, is the corresponding variation of the square slowness. Function p (r, cj) in expression (14.29) represents the acoustic wavefield for the given slowness 5.
Integral approximations
449
of the acoustic waveReld
We can estimate the sensitivity of acoustic data to model parameters by means of formula (14.29). In particular, substituting Ss"^ (r) = 6 {r'—r)6s'^ (r') in equation (14.29), we find the perturbation of the acoustic wavefield, 6p(r,uj), corresponding to the local perturbation of the square slowness, ^5^ (r'), within a differential element of the volume dv located at a point r' : 6p (?, Lu) = a;2G"^(r| r'; uj)6s^ {r') p (r', u).
(14.30)
Thus, we obtain the differential sensitivity Sa (?|r';ct;) of the scalar wavefield at the point r to the square slowness perturbation* at the point r': s„(r|r';u;) =
^j^
= u;'G-{r\T';cj)p{r',u).
(14.31)
The integrated sensitivity Sa (r') of the acoustic data, collected over some surface E of observations over a frequency interval Jl, in accordance with the definition (3.75), is equal to
where the L2 norm
H-.-HQ^^
is determined by the formula
\m\a,^
= J m^\Sp{r,u:)\'
dScLj.
Therefore, the integrated sensitivity of the acoustic wavefield to the local perturbation of the square slowness at a point r is equal to Sa{r') = J J^u^ ll^\G-'iv\r'-w)piv',u)fdSdco,
(14.33)
where the surface integration is conducted over the variable r. Formula (14.33) plays an important role in the inversion of scalar wavefield data. 14.2
Integral approximations of the acoustic wavefield
14-2.1 Born approximation We consider again the basic integral equation of the acoustic forward modeling problem (14.19), written for the scattered wavefield
*Note that using the simple equaUty ^5^ (r') = 2s (r') ^s (r'), we can obtain the differential sensitivity of the scalar wavefield at the point r to the slowness perturbation itself at the point r ' : 6p (?,Lu) /6s (rO - 2uj'^s {r') G'"{v\ r';Lu)p
{r\uj).
450
Integral representations in wavefield theory
a;2
G'"(r,|r;a;)As2 (r) [p\v,w) +f{v,uj)\
jjj
dv.
(14.34)
The conventional Born approximation P^{YJ^UJ) for the scattered field can be obtained from (14.34), if inside D the scattered field is neghgibly small in comparison with the incident field:
u? fff
G''{rj\r;uj)As^r)p\r,cj)dv.
(14.35)
One can see that the Born approximation reduces the forward modeling solution to simple quadrature calculation. However, this approximation holds only for a weak scatterer (when As^ (r) is relatively small). We can improve the accuracy of integral approximation significantly if, following the ideas described in Chapter 9 for an electromagnetic field, we introduce a quasi-hnear approximation. 14-2.2 Quasi-linear approximation Using expression (14.9), we can represent the total field in the form P{T,UJ)=P'{T,UJ)
+P'{T,UJ)
= [1 + A(r,a;)]p^(r,a;),
(14.36)
where A is a reflectivity coefficient, equal to the ratio of the scattered and incident fields:
Xir,u) = ?p4.
(14.37)
Substituting expression (14.36) for the total wavefield in the integral formula (14.34), we obtain P [^j',,u;)=u'
III
G-{r,\r',uj)As^v)[l-^X{T,uj)]p^{r,uj)dv
= cj^G^ {As^ (r) [1 + A(r,cu)] p\v, u)] .
(14.38)
Note that formula (14.38) is exact if we use expression (14.37) for the reflectivity coefficient A. This formula transforms into a Born approximation if we select A = 0. However, as in the electromagnetic case, we can chose an approximate value of A, which is easy to calculate without a priori knowledge of the scattered field. First of all, we can write the equation in terms of the reflectivity coefficient by substituting (14.37) into the left-hand part of (14.38): \{YJ,UJ)P\YJ,UJ)
= uj^G^ {As^ (r) [1 + X{r,uj)]p'{r,Lu)} .
(14.39)
By analogy with the electromagnetic case, we call formula (14.39) the quasi-linear (QL) integral equation.
451
Integral approximations of the acoustic waveReld
Following the original QL approximation, introduced by Zhdanov and Fang (1996), we can find A(r, uj) as the numerical solution of a minimization problem arising from the QL equation (14.39): ||A(r,-,u;)p^(r„a;) - UJ'G^ {AS^ (r) [1 + A(r,a;)]p^(r,a;)}||^^^ = min,
(14.40)
where the L2 norm ||-..||^£, is determined by the formula
\\p\\u,D = J
mj^\p{^->^)fdvdw.
The advantage of this approach is that we can determine the reflectivity coefficient A by solving a minimization problem (14.40) on a coarse grid. The accuracy of the QL approximation depends only of the accuracy of this discretization of A, and, in principle, can be made arbitrarily good. For example, one can select A to be independent of frequency a; : A == A ( r ) . This situation corresponds well to the case of high frequency asymptotics of the acoustic field, in which wave propagation can be described by the simple principles of the geometric theory of diffraction and reflection. In this case expression (14.38) becomes a quasi-linear (QL) approximation, pQ^{rj,uj), f{vj,u)^fQ:^{Vi,^)=u?
jjj
G-(r,|r;cc-)As2(r)[l + A(r)]p'(r,a;)dt;,
(14.41)
and the minimization problem (14.40) takes the form | | A ( r , y (r„a;) - UJ'G^ { A S ^ (r) [1 + A(r)]p'(r,a.)}||„_^ = min.
(14.42)
14-2.3 Quasi-analytical approximation By full analogy with the electromagnetic case, one can consider different ways of introducing the reflectivity coefficient A. In particular, two of these solutions play an important role in inversion theory. One is the so-called quasi-analytical (QA) solution, and the other is the localized quasi-linear (LQL) approximation. In this section I will introduce the QA approximation for the acoustic wavefield. Note that the QL integral equation (14.39) can be cast in the form X{rj,u;)p\Tj,u)
= Lj^G^ [As^Xp'] 4-p^(r,-,a;),
(14.43)
where p^ stands for the Born approximation (14.35). Following ideas of the extended Born approximation (Habashy et al., 1993), we recall that the Green's function G^(r^| r; u) exhibits either singularity or a peak at the point where r^ = r. Therefore, one can expect that the dominant contribution to the integral G^ [As^Ap'] in equation (14.43) is from some neighborhood of the point Tj = r. In fact, we can expand A (r,a;) into a Taylor series about r = r^: A(r,a;) = A(r„c^) -f (r - r,) • VA(r„a;) + O (|r - r , f ) ,
(14.44)
Integral representations in wavefield theory
452
where O (|r — r j | ) represents an error, which is of the order of |r — r^l in the neighborhood of Vj. Assuming also that the variation of the reflectivity coefficient A (r^, c^;) is fairly smooth within domain D, we approximate its gradient to zero order. On these assumptions, we can write
(14.45) Solving the last equation with respect to A, we find A(r,-,a;) =
.,
r, r,- e D.
^y^'^^J,
(14.46)
Substituting expression (14.46) back into (14.38), we obtain the quasi-analytical approximation, PQA, for the scattered acoustic wavefield: PQA{^J>'^)
=^'^GU
p'{r,oj)
p'{r,Lo) ^^ WZ7 p'{r,w) -pS(r,ct))
(14.47)
This formula can be re-written in a form similar to the electromagnetic case (expressions (9.90)-(9.91)), assuming that p'{rj,uj) ^^ 0 in D: PQA(ri,w)=w^G„ A . ^ ( r ) - ^ ^ l-g{Y,uj)
(14.48)
where 9(r,w) =
P'{Y,UJ)
P\Y,UJ)^0,
YeD.
(14.49)
Formulae (14.47) and (14.48) give the quasi-analytical (QA) solutions for a 3-D acoustic wavefield. Note that the basic difference between the QA approximation and the Born approximation (14.35) is in the presence of the scalar function [1 — ^ (r, uj)]~ . That is why the computational expenses to generate the QA approximation and the Born approximation are practically the same. 14-2.4 Localized quasi-linear approximation The general expression for QL approximation, according to formula (14.38), can be cast in the form PQii^v^) = ^^Gw [As^ (1 + \)p'] • (14.50) Following the same principle that was used in deriving the localized quasi-linear approximation for an electromagnetic field, we can assume that the dominant contribution to the integral Gw [As^Ap'] in equation (14.50) is from the neighborhood of the point YJ = r. We can expand p' (r,a;) into a Taylor series about r = r^, similar to (14.44):
Integral approximations of the acoustic wavefield
453
p ' (r,a;) = p' (r,-,a;) + (r - r , ) • V p ' (r,-,a;) + O (|r - r , f ) . Assuming also that p* (r,a;) is a slowly varying function within domain D and approximating its gradient to zero order, we rewrite equation (14.50) as p'Qi^{rj,co) « cj'G^ [As' (1 + A)] p' (r,-,a;),
(14.51)
where G^ [As^ (1 4- A)] = fff
G^irjl r;uj)As^ (r) [1 + A(r,a;)] dv.
(14.52)
On the other hand, from (14.37) we have firj,uj)^X{rj,cj)p'{rj,u^).
(14.53)
Comparing (14.53) and (14.51), we find that X{rj,uj)p'{rj,uj) ^ uj^G^ [As^ (1 -f A)] p' {TJ.UJ) .
Cancelling p^ (r^, cj) on both sides of the last equation, we conclude that the reflectivity coefficient can be determined, as in the electromagnetic case, from the solution of the minimization problem: ||A(r,-,a;) - OJ'G^ [AS' (r) (1 + A(r,a;))] ||^_^ = min.
(14.54)
The solution of minimization problem (14.54) gives us a localized reflectivity coefficient AL(r,a;), which is obviously source independent. Expression (14.50) with AL(r,a;), is called a localized quasi-linear (LQL) approximation for the acoustic wavefield: PlQiirj,^)
~ ^^G^ [As' (1 + \L) f] •
(14.55)
14-2.5 Kirchhoff approximation The ideas of quasi-linear approximation can be expressed in the most clear and simple way using the integral representation of the scattered field based on the Kirchhoff integral formula. Following Bleistein et al., (2001), we consider now a model of an acoustic medium which contains an infinite interface, denoted by B, across which the wavespeed may change (Figure 14-2). The interface B can be treated as a refiecting boundary. The pressure field p(r,cj) is generated by an external source of energy, distributed with the strength of /^(r,u;) in some bounded domain Q in the upper half-space above the surface B. This field satisfies the Helmholtz equation with different right-hand parts above and below the reflecting surface B : r72 / \ , '^^ / \ [ -/^(^^j^)^ above .B, V P(r, - ) + ^ j ^ p ( r , a.) = | ^^^ ^^^^^ ^
... ^_. (14.56)
454
Integral representations in wavefield theory
We can represent the total field as a superposition of the incident and scattered fields above the refiecting boundary B, while we identify it with the "transmitted wave," p^(r,a;), below the reflector:
^•-•'{^^'(rJr*"''"---'''
(""'
below B.
The incident and scattered fields satisfy the following equations everywhere in the upper half-space above the reflector: VV(r,a;) +
-^P'{T,UJ)
c^(r)
= -f
(r,a;), above B,
(14.58)
V V ( r , a ; ) + -^--p^(r,a;) = 0, above B, (14.59) c^(r) while the transmitted wave satisfies the homogeneous Helmholtz equation everywhere below the reflector: ^2
V V ( r , cj) + -^rr-p^{r, LJ) - 0, below B. c^ (r)
(14.60)
We assume also that all wavefields - the total, the incident, the scattered, and the transmitted fields - satisfy the Sommerfeld radiation conditions, which are rp{r,uj) is bounded, lim r —p(r,a;) or
-i-p{r,uj) = 0, c
r-|r|.
(14.61)
The Kirchhoff approximation is based on expressing the scattered field p^(r', ct;) at some point r' in the upper half-space, using its values at the refiecting boundary (see Figure 14-2). To solve this problem we apply the KirchhoflF integral formula (13.197) to the scattered field in the upper half-space:
P'ir',u;)
JL
G-(r'|rB;u;)^/(rB,c^)
-p'{vB,uj)-^Cr{r'\vB;u;) OUB
dsB, TB e B,
(14.62)
where dsB denotes the differential element of the surface 5 , and n ^ is a unit vector defining the downward-pointing normal to B. This result can be obtained in a way similar to that presented in Chapter 13 for the construction of the Kirchhoff integral formula for an unbounded domain. We just introduce a sphere Or of a radius r with center at the observation point r', and apply the original Kirchhoff integral formula to the domain K, bounded by the closed surface formed by the part of the sphere Or located within the upper half-space, and the part Br of the surface J5, which happens
Integral approximations of the acoustic wavefield
455
Figure 14-2 A model of an acoustic medium which contains an infinite interface, B, across which the wavespeed may change abruptly. The Kirchhoff approximation is based on expressing the scattered field p^{r',uj) at some point r' in the upper half-space, using its values at the reflecting boundary.
to be inside the sphere Or- Thus, proceeding to the hmit as r —> oo, and taking into account the radiation conditions (14.61), we arrive at the integral formula (14.62). Note that in most cases we prefer to use the normal direction pointing upwards from the surface of integration, n = —n^. By changing the direction of the normal in expression (14.62), we arrive at the following integral representation for the scattered field: d p'W.uj)
IL
-G' '{r'\rB;uj)—p'{rB,u;)\
dsi
(14.63)
We may recall now that the surface B is a reflecting surface. This means that the incident field experiences a reflection at this boundary, which can be characterized approximately by the reflection coefficient A providing the relationship between the leading order (high frequency asymptotics) incident and scattered fields (Bleistein et a l , 2001): p'{rB,uj)=A{TB)p\TB,uj).
(14.64)
Note that expression (14.64) resembles formula (14.53) for quasi-linear approximation. However, the reflection coefficient A now has a much clearer physical meaning. It has been demonstrated by Bleistein et al. (2001), that to leading order (high frequency asymptotics), this reflection coefficient is the same as that derived for plane waves reflected by planar interfaces in the media with a piecewise-constant distribution of the parameters. Note also that to leading order in the frequency a;, the
456
Integral representations in wavefield theory
normal derivative of the scattered wavefield (which actually represents the reflected waves in this model) is just the negative of the normal derivative of the incident field, multiplied by the same reflection coefficient:
We can now use formulae (14.64), (14.65), and the Kirchhoff integral formula (14.63), for constructing an integral representation of the scattered field based on the known reflection coefficient A (r^) at the boundary B: f{v',uj)
=
jj^K{vs) p'ivB,cj)^G^{v'\vB;oj)
+G^{v'\vB;oj)-^f{rB,u;)
IL
dsf
^^^^)aip'(r„^)G-(r-|r.;.)|^^^
(1466,
The integral approximation (14.66), introduced by Bleistein et al., (2001), is called the Kirchhoff approximation. Note that if the reffector has edges and does not extend at infinity, than according to Bleistein et al., (2001), we can still use this approximation by setting the refiection coefficient to zero where there is no actual reflector. The Kirchhoff approximation has proved to be especially useful in forward and inverse wave propagation problems where the distribution of the reffecting boundaries is the main target. This is the case, for example, in seismic exploration. 14.3
M e t h o d of integral e q u a t i o n s in vector wavefield analysis
14^3.1 Vector wavefield separation Let us assume now that the propagation of seismic waves in the earth can be described by the vector wave equation which, according to (13.35), can be written in the frequency domain as V2u(r,u;) -H ^ u ( r , a ; ) = - 0 ^ (r,a;),
(14.67)
where 0^ (r, u) = f^/c^p characterizes the distribution of the external volume forces within some bounded domain Q. We assume also that the wavefleld satisfies the radiation condition at infinity (Felsen and Marcuvitz, 1994). This model describes well, for example, the propagation of compressional elastic waves. We represent the distribution of velocity in the model in the same way as we did for the acoustic wave equation: ^ cHr)
^ +A52(r), cliv)
(14.68)
Method of integral equations in vector wavefield analysis
457
where Cb{r) is the background (normal) velocity distribution, and the function As'^ (r) is the anomalous square slowness As^ (r), which is non-vanishing only within some local inhomogeneous domain D. Similar to the acoustic field, the total vector wavefield in the model described above can be represented as a sum of two parts, the incident (background) field u*(r,a;) and the scattered (anomalous) field u*(r,6t;), u(r,a;) = u\r, UJ)-h u'{r, u),
(14.69)
where the incident field is a field generated by the given sources in a model with the background distribution of velocity Cb{r), and the scattered (anomalous) field is produced by the domain D with the anomalous velocity distribution. The incident field u'(r,ct;) is the solution of the Helmholtz equation with the background velocity distribution: V V ( r , a ; ) + ^u'{v,u)
= -' (r,^;),
(14.70)
while the scattered field u*(r,a;) satisfies the same Helmholtz equation, but with a different right-hand part: V V ( r , a ; ) + ^ u ^ ( r , a ; ) = - C J ' A S ^ (r) [u'(r,a;) + u^(r,a;)] .
(14.71)
Both the incident and the scattered fields satisfy the corresponding radiation conditions at infinity. Let us denote by
{T)\I{T,UJ).
(14.72)
Then the equation for the scattered field takes the form ,2 uj-
V V ( r , a ; ) + —u^(r,a;) - -
(14.73)
14-3.2 Integral equation method for the vector wavefield In Chapter 13 we derived a general integral representation of the frequency domain wavefield (13.92), which can be written for an incident field as u'(r.,a;) = jjj
0^(r,a;) • G^(r,|r;a;)dt; = G^(<^^),
(14.74)
where G^y {(ff) is the vector Green's integral wave operator in the frequency domain, and where the Green's tensor G^^ (r| rj;ct;) for the background velocity distribution satisfies the Helmholtz equation V^G^{v\ r,;a;) + ^ G ^ ( r | r,;a;) = -16 (r - r,).
(14.75)
458
Integral representations in wavefield theory
Note that we can show, by methods analogous to those used j^n deriving the reciprocity theorem 27 (formula (14.24)) for a scalar wavefield, that G^ satisfies the reciprocity relationship: G;r(r"| r';a;) = G^(r'| r ' » .
(14.76)
It can be also shown that G^ is a symmetric tensor (Morse and Feshbach, 1953); in other words: G^(r,| r;a;) • 0(r,a;) = 0(r,u;) • G^(r,| r;a;) for any 0 .
(14.77)
Based on these properties of the Green's tensor, we can write expression (14.74) in the equivalent form: u'(r.,a;) = JJJ
Gr(r,-|T;UJ) • 4,\v,u)dv.
(14.78)
Applying the integral representation (14.78) to the scattered field satisfying equation (14.73), we obtain u\v^,oj)
= JJj
G r ( r , | r ; a ; ) • ct>%T,Lu)dv = G^((/>^).
(14.79)
Substituting expression (14.72) for the strength of the anomalous source into (14.79), we arrive at an integral equation for the scattered vector wavefield: n%r.,uj) = u? jjj
Gjr(r,| r,a;) • As^ (r) [u'(r,a;) + u^(r,a;)] dv = w^G^ [As^ (u' + u^)] .
(14.80)
This integral equation can be rewritten for the total vector wavefield by adding the incident field to both sides of (14.80)
= uj'G^ [As'u] +u'(r,-,a;).
(14.81)
14'S.3 Calculation of the first variation (Frechet derivative) of the vector wavefield We begin this section with an analysis of the equation for the vector wavefield variation. This equation can be derived by applying the perturbation operator to both sides of the vector Helmholtz equation (14.67), expressed in terms of the slowness function 5(r), V'6u{r,uj)^ujh\r)6u{r,uj)
= -uj'6s\r)u{r,u;),
(14.82)
459
Method of integral equations in vector waveReld analysis
where Ss"^ is the square slowness variation, and 6u is the corresponding wavefield variation. Using integral representation formula (14.79) for the solution of equation (14.82), we obtain the following expression for the vector wavefield variation: 6u{r,uj)=iu'
///
G]^{r\r;uj) - 6s\r)u{T,Lj)dv
= uj'G^{6s\r)u{r,uj)).
(14.83)
Therefore, the Frechet differential of the forward modeling wavefield operator is given by the following expression: Fy{s^,6s^) =6u = uj^ IIj
G n ? | r;a;) • 6s\T)n{Y,uj)dv.
(14.84)
Note that vector u in expression (14.84) represents the wavefield for the given slowness s. As in the electromagnetic and acoustic cases, we can calculate the sensitivity of the vector wavefield to the model parameters using formula (14.84). In particular, substituting 8s^ (r) = ^ (r — r') 6s^ (r') in equation (14.84), we determine the perturbation of the vector wavefield corresponding to the local perturbation of the square slowness, ^5^ ( r ' ) , within a differential element of the volume dv : (5u(r,cj) = a;2Gn?| v']u) • 8s'^{Y')n{Y',u).
(14.85)
The differential sensitivity s^ (r|r';a;), of the wavefield at the point r to ta local perturbation of the square slowness at the point r', is calculated by the formula s, (?lr';u;) =
^-^^
OS (r'j
= u:''G'",{v\v'-w) • u(r',c^).
(14.86)
Note that the differential sensitivities are vector functions, because they characterize the sensitivity of the vector wavefield to the square slowness variation. From the last formula we see that the Green's tensor G^ provides the sensitivity estimation of the vector wavefield. We also introduce an integrated sensitivity S^ (r') of the vector wavefield, recorded over some surface S of observations over a frequency interval f), in accordance with the definition (3.75), as
where the L2 norm ||...||^5] is determined by the formula 5u|ln,E = W /
/ / \6xi.{r,uj)\^dsduj.
Integral representations in wavefield theory
460
Therefore, the integrated sensitivity of the vector wavefield to a local perturbation of the square slowness at the point r is given by Sv{r') =
Jj^u;^JIjG-{r\r-^u;)-u{T',uj) ;) dsdu,
(14.88)
where integration over the surface E is conducted with respect to the variable ? G E. 14.4
I n t e g r a l a p p r o x i m a t i o n s of t h e vector wavefield
14-4'^ Born type approximations The basic ideas of constructing Born type approximations for the vector wavefield are very similar to those considered in Chapter 9 for an electromagnetic field. I will provide below a brief description of the corresponding approximations. The conventional Born approximation u ^ (r^, u) for the scattered field is obtained from (14.80) by setting the scattered field under the integral to zero: u^(r.,a;)=a;2Q^[AsV]
(14.89)
We can derive the extended Born approximation of Habashy et al. (1993) and Torres-Verdin and Habashy (1994) by replacing the total field in the integral (14.80), not by the incident field, as in the Born approximation, but by its projection onto a scattering tensor F (r,a;) : u (r, uj) = u' (r, u) + u* (r, LU) =f (r, uj) • u' (r, cj)
(14.90)
We obtain the expression for the scattering tensor by approximating u(r,a;) in the integral (14.80) by its value at the peak point r = r^ of the Green's tensor: u ' {TJ,U) = u (YJ.UJ) - u^ {YJ,U)
IIL
or
u{rj,u>),
G^{rj\r;Lo)As'^{r)dv
T - 1
u{rj,oj)
I-oj'
G^{Tj\r;Lo)As'^{r)dv
u'(r,-,w),
(14.91)
where I is the identity tensor. Therefore, we arrive at the analytical expression for the scattering tensor, r(r,-,a;) =
7///"''|^^
Lu)As'^ (r) dv
which is independent of the external volume forces.
-1
(14.92)
Integral approximations of the vector waveReld
461
14-4'2 Quasi-linear approximation We can introduce the quasi-linear (QL) approximation of the vector wavefield assuming that the scattered field u* inside the inhomogeneous domain is linearly proportional to the incident field u* (see Chapter 9 for comparison): u'{r,u;)^X{r,uj)'U'{r,u;),
(14.93)
where A(r,a;) is the reflectivity tensor. Substituting formula (14.93) into (14.80), we obtain the QL approximation Ug^ (r) for the scattered field: UQL(rj.^) = ^ ^ G ^ As- (I + A) -U^ = LU^G, As^X • u' + u^(r,-,a;).
(14.94)
Following the basic ideas of the original QL approximation for an electromagnetic field, introduced by Zhdanov and Fang (1996), we determine the reflectivity tensor by solving the minimization problem: A {vj, cj) u' (r^, uj) — uj'^Gn As^X • u^ - u ^ ( r , - , a ; ) where the L2 norm
||...||Q
n,D
= mm,
(14.95)
^ is determined by the formula
Mu,D = J I J 11
\u{r;u)\^dv(kj.
We can determine the reflectivity tensor A by solving the minimization problem (14.95) on a coarse grid. The accuracy of the QL approximation depends only on the accuracy of this discretization of A , and, in principle, can be made arbitrarily good. 14-4-3 Quasi-analytical solutions for the vector wavefield As for the electromagnetic case, the reflectivity tensor, can be selected to be a scalar, A=AI, in the quasi-linear approach. In this case, formula (14.93) can be cast in the form: u^ (r, u)^X (r, cj) u^ (r, u). (14.96) Substituting this expression for the scattered field on the left-hand side of the integral representation (14.94), we obtain the following integral equation for the reflectivity coefficient A : A {vj^ou) u' (r,-,(x;) = u^G^ [As^Xu'] -f u^ (r,-,a;).
(14.97)
Following the ideas of the extended Born approximation outlined above, we can take into account that the Green's tensor G'^{rj \ r, u) exhibits either a singularity or a peak at the point where Vj = r. Therefore, one can expect that the dominant contribution
Integral representations in wavefield theory
462
to the integral G^ [As'^Xu^] in equation (14.97) is from some neighborhood of the point Vj = r. Assuming also that A (r^, cj) is slowly varying within domain D, one can write (14.98)
X{rj,uj)u^{Tj,uj)^u^{rj,uj).
Calculating the scalar product of both sides of equation (14.98) with the incident wavefield and dividing the resulting equation by u* • u \ we finally obtain an analytical expression for the reflectivity coefficient: (14.99)
l-g{rj,uj)
where 5(r>,w)
u^{rj,u))-u'{rj,u))
(14.100)
and we assume that u' (rj,w) • u' (r^, w) 7^ 0. Substituting (14.99) into (14.69), we find u {rj,uj) = u' {rj,u)) + u' (r^, w) ~[A(r,-,a') + l]u^(r,-,w) =
1 l-g{rj,uj)
u^(r,-,a;).
(14.101)
Therefore from (14.80) we finally determine u^^ (r,, w) = w' jjj^
G r ( r , | r;uj) • As' (r) u(r,u;)dv
A52 (r)
/ / / .D
l-^(r,^)
u' (r, uj) dv = cj^Gi,
A52
-u
(14.102)
Formula (14.102) gives the quasi-analytical (QA) approximation for a 3-D vector wavefield, which can be treated as an analog of the corresponding QA approximation for the electromagnetic case (Zhdanov, et al., 2000) 14-4-4 Localized quasi-linear approximation Let us analyze again the equations of the QL approximation (14.93) and (14.94). Following the same principle that was used in deriving the localized QL approximation for electromagnetic and acoustic fields, we can rewrite equation (14.94) as
^QL i^j.^) « G^ [AS^ ( i + A)] • U' (r,-,a;)
(14.103)
Integral approximations of the vector waveReld
463
where Gi,
As^(i+\j
G^(r,| r-Lu) • As"^ {r) (l + \{r,io))
dv.
(14.104)
Comparing (14.93) and (14.103), we find that UQL (r>, W) « A {rj,u) • u' (r^-, w) •u^G^
A S 2 ( T + A ) ] •u^(rj,a;),
where the reflectivity tensor can be determined, as in the original QL approximation, from the solution of the minimization problem:
n,D
min,
(14.105)
which can be substituted by the problem X{rj,CLi)
-uj'^Gy
As-
(UX)
Q,D
mm.
(14.106)
The solution of equation (14.106) gives us a localized reflectivity tensor X^ (r, u;), which is obviously source independent. Expression (14.94) with XL (r,<^) determined according to (14.106), is called a localized quasi-linear (LQL) approximation for a vector wavefield (Zhdanov and Tartaras, 2002): ^IQL
(i-j, cj) « cj^G^ As^I + AL (r, cj) • u^ (r, u).
(14.107)
As in the electromagnetic case, one can choose different types of reflectivity tensors. For example, one can introduce a scalar or diagonal reflectivity tensor. In the case of a scalar reflectivity tensor, XL = A^I, and u ' ( r , a ; ) ^ AL (r,cj) u^ (r,u;).
(14.108)
Substituting formula (14.108) into (14.107), we obtain the scalar LQL approximation u^LQL (r, (JU) for the anomalous field:
(14.109)
where the scalar reflectivity coefficient A^ (r, a;) is determined by the solution of the optimization problem:
XL{r,uj)I-u;^G^\As^l-^XL)I
Q,D
mm.
(14.110)
464
Integral representations in wavefield theory
In the special case of a diagonal reflectivity tensor A^ = diagA^, XL = diagA^==
r Aa, 0 0 0 A^^ 0 [ 0 0 A,
where AL is a reflectivity vector Aa; A7I
we have
(14.111)
Substituting formula (14.111) into (14.106), we obtain a diagonal LQL approximation u^LQL (r,(^) for the scattered field: (14.112)
where the reflectivity vector A^ (r, a;) is determined as the solution of the optimization problem diagA^ (r,a;) - cj^G^ \AS'^ ^ T + diagA^ (r,^))
n,D
mm.
(14.113)
References to Chapter 14 Bleistein, N., 1984, Mathematical methods for wave phenomena, Academic Press Inc., (Harcourt Brace Jovanovich Publishers), New York, 341 pp. Bleistein, N., Cohen, J. K., and J. W. Stockwell, Jr., 2001, Mathematics of multidimensional seismic imaging, migration, and inversion: Springer, New York, BerUn, London, Tokyo, 510 pp. Felsen, L., and N. Marcuvitz, 1994, Radiation and scattering of waves: IEEE Press, New York, 888 pp. Habashy, T. M., Groom, R. W., and B. R. Spies, 1993, Beyond the Born and Rytov approximations: a nonlinear approach to electromagnetic scattering: J. Geophys. Res., 98, No. B2, 1759-1775. Morse, P. M., and H. Feshbach, 1953, Methods of theoretical physics: McGraw Hill Book Co., New York, 1978 pp. Torres-Verdin C , and T. M. Habashy, 1994, Rapid 2.5-dimensional forward modeling and inversion via a new scattering approximation: Radio Science, 29, No. 4, 1051-1079. Zhdanov, M. S. and S. Fang, 1996, Quasi-linear approximation in 3-D EM modeling: Geophysics, 6 1 , 646-665.
Integral approximations of the vector waveReld
465
Zhdanov, M. S., Dmitriev, V. I., Fang, S., and G. Hursan, 2000b, Quasianalytical approximations and series in electromagnetic modeling: Geophysics, 65, 1746-1757. Zhdanov, M. S. and E. Tartaras, 2002, Inversion of multi-transmitter 3-D electromagnetic data based on the localized quasi-linear approximation: Geophys. J. Int., 148, No 3.
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Chapter 15 INTEGRAL REPRESENTATIONS IN WAVEFIELD INVERSION In this chapter I will outline the general principles of seismic inversion and imaging based on integral representations of scalar and vector wavefields developed in Chapter 14. The integral equation representations and the corresponding linear and nonlinear approximations introduced in the previous chapter constitute a powerful tool for seismic inversion and for imaging seismic cross-sections. We will see that this method makes it easier to determine the sensitivity matrix, which lies at the cornerstone of any inversion algorithm. Also, the integral equation approach provides an analytical insight into the migration transformation, which is now one of the most widely used methods of seismic imaging. As in the previous chapter our discussion will be restricted mostly to the fields satisfying wave equations. However, in the last section I will outline the basic principles of nonlinear elastic inversion. 15.1
Linear inversion methods
Consider a 3-D seismic model with a background (normal) slowness distribution ^^(r) and a local inhomogeneity D with an arbitrarily varying square of slowness 5^(r) = 5^(r)-|-A5^(r). We will examine, in parallel, two cases: the propagation of the acoustic field and of the vector wavefield in this model. The acoustic field in the frequency domain, p(r,a;), satisfies the equation VMv,oj)+uh'iv)p{v,uj)
= -r{T,oj),
(15.1)
where /^(r,(x;) is the strength of an external source of energy. The vector wavefield in the model is excited by an arbitrary source with the distribution of volume force characterized by cf)^ (r, uj). In frequency domain this field satisfies the vector Helmholtz equation: V^u{r,Lu)+uj^s'^{r)u{r,uj)
= - 0 ^ (r,a;).
(15.2)
Both the acoustic and the vector wavefields satisfy the corresponding radiation conditions at infinity (see Chapter 13).
468
Integral representations in wavefield inversion
The scattered acoustic field and the vector wavefield in this model can be represented, according to formulae (14.19) and (14.80), by the following integrals p'{rj,u;)=iu^
III
G'^{vj\v',uj)/^s\v)p{v,uj)dv
= u^G^ {As^
,
(15.3)
and u^(r^.,c.;) = Lu^ fff
G^{vj\r;ij)
• A^^ (r) u{T,uj)dv = uj^G^ [As^u] ,
(15.4)
where G'^{rj\ T;UJ) and G^(rj| r;a;) stand for the Green's function and tensor defined for an unbounded medium with the background slowness s^. We assume that the scattered fields, p^{rj,uj) or u*(r.,a;), are given on some surface of observation S. The goal is to find the anomalous square slowness As^ ( r ) . 15.1.1 Born inversion of acoustic and vector wavefields The most common approach to the solution of the wavefield inverse problem is based on linearization of the integral representations (15.3) and (15.4) using Born approximations: P'{TJ,UJ)^UJ^G^
[A^V]
=^' ///
G^{rj\T;u;)As^T)p'{T,uj)dv
(15.5)
and u'{r.,uj)^uj^G^[As^u']
= uj^ fff
G^{rj\r',uj)
• As^ {r)u'{r,uj)dv.
(15.6)
Comparing formulae (15.5), (15.6) and (14.29), (14.84), we see that Born approximations can be expressed as the Frechet derivative operators (or Frechet differentials) calculated for the background square slowness 5^ and the anomalous square slowness As'^: p'(Yj,uj) ^ Lu^G^ [AsY] = Fa{slAs^), (15.7) u%r^,uj) ^ uj^G^ [As^u'] = F,{slAs^).
(15.8)
The argument in the expressions for the Frechet diff'erentials Fa(5^,As^) and F^(5^,A5^) consists of two parts. The first part, 5^, is the background square slowness distribution, from which we calculate the forward modeling operator variation; the second part, As^, is the anomalous square slowness. We will use below the following simplified notations for the Frechet differentials:
and
469
Linear inversion methods
where the upper subscript "6" denotes that the differential is calculated for the background slowness. Substituting the Born approximations (15.7) and (15.8) into formulae (15.3) and (15.4), we arrive at the linearized form of the forward modeling operators p^ = F^{As''),
(15.9)
u^ = Ft{As').
In the inverse problem solution we assume that the scattered fields, p^ or u*, and the background slowness st are given. The goal is to find the anomalous square slowness distribution, As'^. As a result, we have hnear equations (15.9) with respect to A 5 ^ Following the general principles of regularization theory (Chapter 2), we solve the linear inverse problems (15.9) by regularization methods, imposing additional conditions on the class of inverse models with corresponding stabilizing functionals. We introduce a Hilbert space Da of the acoustic field data, given on the surface of observations S with the metric {Pi Q)Da^^^
/ / P(r,cc;)g'*(r,a;)d5da;; p, q e Da, oo e ft,
(15.10)
and a Hilbert space Dy of the vector wavefield data, given on the same surface with the metric (u, v)^^ = Re / / / u(r,c<;) • V* {r,uj)dsduj] u , v G Dy, cj G f2,
(15.11)
where ds denotes a differential element of the surface S, and the integration is conducted over the variable r e S. We also consider a Hilbert space M of models m (square slowness m = 5^) with the metric {m^'\ m(2))^ = / / / m(^) (r) m^^) (r) dv; m^'\m^^^ G M.
(15.12)
According to the conventional Tikhonov regularization method, we replace the solution of the linear inverse problems (15.9) by a minimization of the corresponding parametric functionals, for example, with a minimum norm stabilizer: P: = (F^iAs^) - da, F^iAs') + a {Wa {As'-Asl^), or
P: =
- da)^^
Wa (As'-Asl^))^
= min,
(15.13)
{Fl{As')-d,,Fl{As')-d^)^
+a {W. {As'-Asl^), W, {As'-Asl^))^ = min,
(15.14)
470
Integral representations in wavefield inversion
where da = p^ and d^ = u* are observed acoustic and vector wavefield data, As^p^ is some a priori model of the anomalous square slowness distribution, and Wa, Wy are the weighting operators. The minimization problems (15.13) and (15.14) can be solved, for example, by the conjugate gradient or re-weighted conjugate gradient methods introduced in Chapter 5. 15.1.2 Wavefield imaging by the Born approximations We develop now a fast wavefield imaging technique based on the Born approximations. This technique employs formula (5.91) for an approximate regularized solution of the linear inverse problem. For acoustic or vector wavefield inverse problems (15.9), this formula takes the forms As^^k{W:WaVF',*{f), (15.15) or As^^k
{W:Wy)-^ F ^ (u^),
(15.16)
where F^* and F^* are the adjoint Frechet derivative operators, and the scalar coefficient A: is a solution of the minimization problems: = min,
(15.17)
||/cFt {{W^Wy)-' F ^ (u^)) - u^ll' = min.
(15.18)
\\kFl {{W:Wa)-'
Ft if))
- pf
or For a practical realization of the imaging algorithms (15.15) or (15.16) we have to find explicit expressions for the adjoint Frechet derivative operators F^* and F^*, according to these formulae: {FUAs'),q)^^
=
{As',Ft{q))^,aud
(Ft {As') , v)^^ = (A5^ Ft ( v ) ) ^ .
(15.19)
Using the definitions (15.10), (15.11) and (15.12) of inner products and the expressions (15.7), (15.8) for the forward operators, we can rewrite the first formula in (15.19) as {F^ {As') ,q)^^=ReJ u;^Re
u? j j G^ [A^V] q*dsdw
HI III ^"^^"'"''^^^'' ^""^^^'''^^'^'"^* ^ dw
= / / / As' (r) Re [ / oj'p"{r, w) ff JJJD
[JU
JJS
G r ( r | r; w)? (r, uj) dSdu; dv
= (A.^F^(^))^.
(15.20)
Linear inversion methods
471
From the last formula we have (AS^
\
IF^
L
(q) -Re f ^Y'{T,U;) Jn
0.
ff Gr{r\ v-u:)q {T,UJ) dSdw J Js
(15.21)
Equation (15.21) should hold for any As^, for example for As^ = F^ (q) -Re f ct;V*(r, cu) [f G^*(?| r; uj)q (?, ou) dsdou. Jn J Js Therefore, 0.
Ft {q) -Re f u;V*(r, a;) / / Gr*(?l r; ^)9 (?, a;) dScLv M
From the last formula we conclude that the adjoint Frechet derivative operator F^* is given by the formula F^' {q) = Re [ UJ^P'\T, UJ) [[ G^ (?| r; cu) q (?, UJ) dSduj. Jn J Js
(15.22)
In a similar way we can obtain the formula for the adjoint Frechet derivative operator F^'*' :
F,^* (v) = Re / U'^U'\Y,UJ) • / / G^ (?! r;a;) • v (?,a;) d ? ^ . Jn J Js
(15.23)
Note that we can drop the symbol Re in the above formulae if we integrate over a symmetrical frequency interval ft (for example, from — oo to -foo) because the imaginary part of the integrand is anti-symmetric in UJ. On this understanding, and substituting expressions (15.22) and (15.23) into formulae (15.15) and (15.16), we finally arrive at the following imaging formulae: As^^k{W:Way' f u^v'\v,u) ffGr{v\Y',uj)f{v,uj)dSdw, JVL J Js
(15.24)
or As^^kiW^Wy)-^
/ u ; V * ( r , a ; ) - [[ G^ {r\r;uj) - u'{7,LU) dSduj. J JS
(15.25)
JQ
The weighting operators Wa and Wy for acoustic and vector wavefields are Unear operators of the multiplication by the functions Wa (r) and Wy (r) equal to the square root of the integrated sensitivity (see formula (3.78) in Chapter 3): Wa
sSa, and Wy = y/Sy.
(15.26)
Integral representations in wavefield inversion
472
The integrated sensitivity, Sa^ in accordance with the definition (3.75) and formula (14.33), is calculated by the formula:
\yG^{r\T;u;)p\r,u;)\\^^. Formula (15.27) acoustic field to vector r. Similarly, field to the local
(15.27)
should be treated as the integrated sensitivity of the scattered the local slowness anomaly located at a point with the position we can find the integrated sensitivity of the scattered vector waveslowness anomaly, using formula (14.88): Oy
u;^G^(?|r;a;)-u^(r,a;)
(15.28)
Dr,
The weighting functions, according to (15.26), are equal to Wa (r) =
Wy{r)
^\\UJ^G-^{T\T;U)P'{T,UJ)\\J,^
c.;2G^(r|r;a;)-u^(r,a;) Dy
(15.29)
Substituting formulae (15.29) into (15.24) and (15.25), we finally find 2 ,.^ ^^ J ^ a ; V * ( r , a ; ) / / ^ G r {r\r;uj)f {v.u)dSdu As^ (r) ^k ||a;2G-(r|r;a;)pHr,^)||p.
(15.30)
and 2 ,.^ ^ ^ J ^ c ^ V * ( r , a ; ) - / / 5 G r ( ? | r ; c c ; ) - u M ? , a ; ) r f ? c L ; As' (r) ^k cj2G^(r|r;a;) •u^(r,a;)
(15.31)
Dv
We can give a clear physical interpretation of formulae (15.30) and (15.31). According to the reciprocity theorem 27 (formula (14.24)), the surface integral term in equation (15.30) can be treated as a complex conjugate acoustic field p ^ . p ^ (r,a;) = j j G^ (?| r; a;) p - (?, a;) ds,
(15.32)
generated at the point r by the fictitious acoustic source, distributed over the surface of observations S with the surface density / | , equal to the complex conjugate scattered acoustic field: rs{v,uj)=p'*{t,uj). (15.33)
473
Linear inversion methods
Note that this interpretation is similar to one given in Chapter 10 for electromagnetic Born imaging (formulae (10.33) and (10.34) ). The approximate anomalous square slowness in formula (15.30) is calculated as an integral over the frequency range Q of the product of this auxiliary field p^* and the complex conjugate incident field at the point r, normalized by the norm of the product of the incident field and the corresponding Green's function at the same point r: As' (r) ~ kAs' (v) As {r)^kAs,(v),
and As' (v) In'^'p'*i^''^)p''* i^'^)^ a n d A 5 i ( r ) - ||^2G»(y|r.^)pi(r,a;)||^/
.15 34) ^^^•^^>
where the scalar coefficient k is the solution of the minimization problem
Formula (15.31) has a similar interpretation, except in this case we should introduce, by reciprocity, a vector wavefield u^, u ^ (r, uj) = 11 G^ (?| r; LJ) • u ^ (?, u) ds, generated at the point r by the fictitious vector wavefield source, distributed over the surface of observations S, with a surface density H?,u;)-u^*(r,u;).
(15.35)
As a result, we arrive at the following formula for anomalous square slowness As^ (r) ^ kAsl ( r ) , and Asl (r) =f'—::—-^—^ -^^—, |a;2G-(r|r;c^)-uHr,a;)
(15.36)
ID.
where the scalar coefficient k is the solutionQ ofof the the minimization mini: problem | / c F ^ A 5 2 ) - u ^ | | ' = min. Note that in the definition of the auxiliary fields p^* (r,a;) and u^* (r, uj) we use the complex conjugate Green's functions. It can be demonstrated that the transition to the complex conjugate function in frequency domain corresponds to changing a sign in front of time variable t in time domain. Thus, using the convolution theorem (Arfken and Weber, 1995, p. 863-865) and taking into account reciprocity theorem 27, we can write the inverse Fourier transform P^ (r,t') of the auxiliary scalar wavefield in the time domain as follows: 1
+00 r+oo
iujt'
du
474
Integral representations in wavefield inversion r+oo
27rJ_^ JJss
-ILL 5."
(?| r; iu) p' (?, Lu) dse-'^^^'dcj
or
+00 _
G^ (r', t'|r, t) p' (r, t) dtds,
(15.37)
where we use the notation
G^iv',t'\ r,t) = G^ir',-t'\
r,-t),
and the wave above the acoustic field P^ reflects the fact that we change a sign in front of time variable t'. The minus sign in front of time variable t in the expression for the Green's function indicates that the wave propagation occurs backward in time. Therefore, the field P^ (r,t') is a so-called "back-propagated wavefield." We will discuss the physical properties of this field in more details below in the section on seismic migration. Taking complex conjugate of both sides of equations (15.34) and (15.36), and noting that anomalous square slowness As^ (r) is a real function, we present these formulae in equivalent form: o/ X
L(jo'^pHr,(jj)p^ (r,(ju)duj
^^
||a;2G-(r|r;a;)pHr,a;)|lz)/
'
, ^
2/ ^ _ /^a;V(r,u;)-u^(r,u;)dc^ As^ (r) ^ k-p '-^-^ ' ' \. . ||a;2G^(?|r;c^).u^(r,u;)||^
^
(15.39)
According to the correlation theorem (Press et al., 1987, p. 383), the inverse Fourier transform of the product of spectrum of one function and the complex conjugate spectrum of another function is equal to correlation of these functions. Therefore, we can write the numerator in formula (15.38) as a cross correlation of the time derivatives of the back-propagated scattered field and the incident wavefield: 4-00
/
r+oo
uP'p'{Y,uj)p^{Y,uj)duj -00
=
= I
[-za;p^* (r,a;)] * do;
[-iujp\Y,ijo)\
J —00
/.+00 , i
/
J-00
P {r,t)P
1^
I
{r,t-t')dt\
r+oo
= l,^^
^i
•^R
P (r,t)P
{r,t)dt,
J-00
where the dot over the functions P* and P^ denotes their diff"erentiation with respect to time. Formula (15.39) for the vector wavefield permits a similar physical interpretation. We will show below in this chapter that the back-propagated field corresponds to a migration transformation of seismic data. Thus we see that Born imaging can be treated as an algorithm from a family of migration transformations.
475
Linear inversion methods
15.1.3 Iterative Born inversions of the wavefield The ideas of iterative Born inversion, introduced in Chapter 10 for an electromagnetic field, can be extended to wavefield inversion as well. Following the basic principles of this method, we first write the original integral equations for the acoustic or vector wavefields (14.20) and (14.81) as the domain equations for the wavefield inside the anomalous domain D: P{T',UJ)
= uj^G^ [As^ (r)p(r,a;)] +p'(r',a;), r' G D,
u(r ,0;)
= U ; 2 Q ^ [AS^
(r) u(r,
CJ)]
+ u ' ( r ' , a ; ) , r' G D,
(15.40) (15.41)
where Gyj and G^^ are the scalar and vector Green's operators determined by formulae (14.18) and (14.74). We can now use similar equations to connect the scattered acoustic [da = p^ = p — p'^ ) or vector (dv = u® = u — u*) wavefield observed on the surface S with the corresponding wavefield within the anomalous domain da{Tj,Lu) = LO^G^ [As^ (r)p(r,u;)] , r^- G 5, dv(rj,a;) =a;2G^ [As^ (r) u(r,cj)] , r^- G S.
(15.42) (15.43)
Note that each equation, (15.40), (15.41) and (15.42), (15.43), contains the product of unknown functions, As^ and p, or As^ and u. Therefore, these equations are bi-linear with respect to the corresponding unknowns. However, if we specify one of the unknowns, the equations become linear. For example, we can subsequently find As^ from equations (15.42), (15.43) for specified p or u, and then update p or u from equations (15.40) or (15.41) for predetermined As^, etc. Within the framework of this method the Green's functions G'^ and G^^ and the incident wavefields p^ or u* stay unchanged. As in the electromagnetic case, there is also another technique, the distortedBorn iterative method, which is based on updating the incident field and Green's functions after each iteration, according to the updated parameter As^ (Chew, 1990). Note in conclusion of this section that iterative Born inversion requires the application of the regularizing methods to make the solution stable. 15.1.4 Bleistein inversion In the case of a high frequency scalar wavefield it is possible to find an explicit inverse formula for Born approximation. The method of solving this problem has been systematically developed and studied in the works of Bleistein, Cohen and Stockwell (Bleistein, 1984; Bleistein et al., 2001). It is based on a high frequency WKBJ approximation for the Green's functions (13.80) and (13.83). I will outline here only the basic ideas of this technique, referring the reader to the books cited above. The Bleistein method can be more easily formulated for a 1-D wavefield inverse problem, which will be the starting point of our discussion.
Integral representations in wavefield inversion
476
One dimensional inversion Let us assume again that the wavespeed, c, the pressure field, p, and the source field, /^, are functions of the vertical coordinate z only. We also assume that the wavespeed in a 1-D model can be represented as follows:
where Cfe(z) is the background (normal) 1-D velocity distribution, and A5^(2:) is some anomalous square slowness which is non-vanishing only within some interval of positive z : A52(z) = 0 , if z < 0 . * The 1-D field p may describe, for example, plane acoustic wave propagation in the vertical direction in the medium. This field, according to (13.57), satisfies the equation — p ( z , z', a;) -h ^273^(^' ^'' ^ ) = -^{z-
z'),
(15.45)
where we use the impulse source function /^ [z) = 6 {z — z') located at the point z = z'. As one can see, the function p{z^ z' ^uo) depends on three variables: frequency a;, source position z'^ and receiver position z. We assume also that p(z,z'^uj) is bounded for all z and satisfies the 1-D radiation condition -—^(z, z', uj) =f —j-^p{z, z', cj) —^ 0, as z —> ±oo. (15.46) dz c[z) The total field in this model, as in the general 3-D case, can be represented as the sum of the incident and the scattered fields: p{z, z\ UJ) = p'{z, z', uj) + p'{z, z', u). The incident field p*(z, z', uS) satisfies the 1-D Helmholtz equation with the background velocity distribution -J{z,z',uj) + -^j-^p\z,z\uj) = -6{z- z'),
(15.47)
and the radiation condition -—p'(z, z', uo) =f —7-TP'('2^5 ^^ (x;) —> 0, as z —> ±00.
dz
Cb (z)
The scattered field p^{z, z' ^uj) also satisfies the radiation condition and the same 1-D Helmholtz equation, but with a different right-hand part, -^f{z,z',uj)^^--f{z,z\u)
= -a;2As2(z)p(z,z',a;),
(15.48)
*In the section on Bleistein inversion and later on we will direct the vertical axis z downward.
477
Linear inversion methods
where As^ (z) is the anomalous square of slowness determined by an equation similar to (14.8):
Differential equation (15.48) can be transformed into an integral equation for the scattered field, using the 1-D Green's function g'^{z\ z'\uj) for the 1-D Helmholtz equation, which is dependent on the position of the source z', the observation point z, and the frequency u. This function satisfies the 1-D Helmholtz equation
'^l^S^^
+ - i ^ , n - l -';-) = -S{z - z%
(15.49)
and the corresponding radiation condition at infinity. Evidently in our case the Green's function is equal to the incident wave: V\z,u)=p\z,z',u)=gl^{z\z'-u)
(15.50)
.
Note that Green's theorem in one dimension can be written as / '°{u{z)Hv{z) J-oo
- v{z)Hu{z)}dz
=
u{z)^^ dz
*°°
du{z)
-oo
"^
l+oo
(15.51)
;re H is the 1-D Helmholtz operator: „
(f
w'
We choose functions u{z) and v{z) to be equal to the scattered field p^{z, z\ uj) and the Green's function g'^{^z\ z'\u). Using Green's theorem, equations (15.48) and (15.49), radiation condition (15.46), and the fact that lS.s^{z) = 0 for 2 < 0 we obtain an integral equation for the scattered field p^{z^ z\uj): p+00
f{z,,z',uj)=u^
/
^s\z)[f{z,z\uo)^v\z,z\u)]g]^{z\zy,uj)dz,
(15.53)
We can also use the 1-D Green's theorem, equation (15.51), to prove the reciprocity theorem in one dimension, similar to the general reciprocity theorem 27: gl^{z'\z";u)
= gt{z"\z';iv).
(15.54)
Therefore, equation (15.53) can be written in equivalent form as r+oo
p\zj,z',Lj)=u?
/ Jo
i\s''{z)\p\z,z',uj)+p\z,z',uj)]gl^{zj\z-u)dz.
(15.55)
478
Integral representations in wavefield inversion
In the case of the Born approximation we neglect terms containing the scattered field in (15.53) and leave only the incident field: /•4-00
/ /^s^z)v\z,z\u)g^{z\zj',uj)dz. (15.56) Jo Taking into account that, according to equation (15.50), the incident field coincides in our case with the Green's function, we can write: f{zj,z',uj)=u^
/ As^z)g^{z\z';uj)g^iz\zj;Lu)dz. (15.57) Jo The original Bleistein method is based on direct inversion of integral equation (15.57) with respect to anomalous square slowness As'^ (z). In a general case, this equation is ill-posed, as is any geophysical inverse problem, and its solution requires application of the corresponding regularization methods outlined in the first part of this book. However, in the Bleistein method the Green's functions are represented by their high frequency (WKBJ) approximations. This reduces equation (15.57) to a specific class of integral equations which do have stable inverse solutions. Actually, this class is formed by Fourier integral operators, or by so-called pseudo-differential operators (Bleistein et al., 2001). We will see in the next section how this reduction can be done, and what analytic methods can be used for inversion of the corresponding integral equations. Note that numerical implementation of these theoretically stable inverse solutions will still require an application of the corresponding regularization methods. p%zj,z\uj)=uj^
1-D inversion in a constant-background medium We will discuss in this and the following sections the solution of the inverse problem for two cases: a) a constant background wavespeed, and b) a variable background wavespeed. We consider a simple "zero-offset problem" with source and receiver at the same location at the origin: z' = Zj = 0. In this case equation (15.57) can be simplified as r+oo
/
p'{0,u;)=p'{0,0,uj)=u;^
As^ {z)[g^{z\0]u)]^
Jo
dz.
(15.58)
According to equation (13.79), the Green's function in a model with a homogeneous background, c^ (z) = CQ, takes the form iu;\z-z'\/co
iuj\z\/co
^"(z| / ; a;) = -c^—-.
= -CQ-ZICJ
, if z' = 0.
(15.59)
ZILU
Substituting the last expression in equation (15.58), we obtain p'{0,uj)=p'{0,0,u;)
= - ^
/
4 Jo
As^z)e'^'^'^''dz,
(15.60)
479
Linear inversion methods
where \z\ was replaced by z, because the variable of integration is positive. Introducing an auxiliary "wavenumber" 2a; /c = — , Co
and taking into account that As^ (z) = 0 for z < 0, we can view the last formula as a Fourier transform from the vertical coordinate z domain to the wavenumber k domain: p'{0,Lo)=p'{0,cok/2)
= —j
I
As''{z)e'^'dz.
(15.61)
Applying the inverse Fourier transform to the last formula, we arrive at the following expression 2
/'+00
1
^As^ {z) = -— / p^(0, cok/2)e-"''dk. (15.62) 4 27r J_^ From the last formula we find at once the explicit integral expression for anomalous square slowness: As^ (z) =
. / TTCQ
f{0,ij)e-^'^'^''duj.
(15.63)
J-OO
Reflectivity function and bandlimiting of data The Bleistein method helps to determine not only the anomalous slowness distribution, but also another important characteristic of a seismic model, a reflectivity function. In a 1-D case this function is a spike train with spikes located at the positions of reflectors in the seismic model, whose heights are proportional to the reflection coefficients of the corresponding reflectors. It was demonstrated by Bleistein et al. (2001) that the reflectivity function can be represented as a combination of bandlimited delta-functions. The observed seismic data are usually bandlimited, because the frequency range is limited by the natural time constant of the source process, by receiver and survey design, by seismic preprocessing required for noise removal, etc. (see Aki and Richards, 2002, and Bleistein et al., 2001 for further discussions). The bandlimiting is an important characteristic of observed seismic data which should be taken into account, especially if we use the high frequency asymptotics for data inversion. Mathematically the bandlimited data can be introduced by filtering the theoretical signals. We consider, for example, the effect of this filtering on a delta-function. There is a well-known integral representation for a delta-function: 6{z-z')
1 /'+°° = — / exp [-iio {z - z')] duj. 27r 7_oo
(15.64)
480
SB {Z
Integral representations in wavefield inversion
-
Following Bleistein et al. (2001), we define the bandlimited Z') , as SQ{Z-Z')
1
= —
/*+°°
F (CJ) exp
[-iuo {z - z')] dcu,
delta-function,
(15.65)
where F (LJ) is some symmetric and non-negative function, characterizing the bandlimited filter in frequency domain. Some examples of bandlimited delta-functions can be found in mathematical textbooks on the Fourier transform. The main difference between the original delta-function, 6 {z — z'), and its bandhmited analogue, bs {z — z'), is that instead of the infinitely narrow and infinitely high spike oi6{z — z') at the point z', the bandhmited delta-function has a spike of limited width and height, located at the same point. Using a simple 1-D model, we will illustrate below the basic principles of the reflectivity function definition based on bandlimited data. Let us assume first that the anomalous square slowness distribution As^ {z) is described by a piecewise constant function as shown in Figure 15-1. This means that the wavespeed is constant within each layer, equal, say, to CQ within the first layer, ci within the second layer, etc. We can describe, for example, the first step on this curve by the Heaviside step function As^{z) = bH{z-h),
(15.66)
where h is the depth of the first reflecting boundary and b is the size of the step in square slowness, which is determined according to equation (15.44) by the wavespeed change on the first boundary: 6 = 1 - 1
(15.67)
Note that in this case we can express the wave speed c^ as co/^l
(15.68)
+ bcl
The Heaviside step function in (15.66) is equal to zero for z < /i, and to 1 for z > h.lt \s well known that it can be expressed by the following Fourier integral: 1
+00 r /•+°°
27ry_oo
L +00
1 ^^
exp [—iuj {z — h)] duo
1 1 r"^ 1 = T; — TT '^exp[—iuj{z — h)]duj. (15.69) 2 27r y_oo luj It is easy to find the exact solution of the wavefield equation within the first layer. As usual, it can be written as a superposition of the incident and scattered fields,
Linear inversion
methods
481
slowness
reflectivity
z=h
Figure 15-1 A model with piecewise constant distribution of slowness (left panel). The corresponding bandlimited reflectivity function (right panel) is represented by a combination of the bandlimited delta-functions with the positions of extremum at the reflecting boundaries, and the sizes of the peaks proportional to the corresponding reflection coefficients
p (z, uj) = p' (z, uj) H- p' (z, cj), z
and the scattered field can be calculated as p^ (z, (jj) = —A—- exp {—ioj {z — 2h) /CQ) .
(15.70)
The reflection coefficient A in the last formula is calculated based on the usual definition: Ci - C o A = (15.71) Cl + C o
We assume now that the scattered field is observed at the surface of the model at 2; = 0 : Co (15.72) p^ (0,0;) = -A^r— exp {2iu;h/co). Substituting formula (15.72) into inverse formula (15.63) of the Bleistein method, we obtain
482
Integral r e p r e s e n t a t i o n s in wavefield i n v e r s i o n
4
f^^^ Cn
A3^ (z) = —^A / TTCQ
- ^ exp [-2iuj {z - h) /CQ] du, 220;
J-OO
(15.73)
where we denote by AS^ {z) the anomalous square slowness recovered by inversion. Using the Fourier representation of the Heaviside function (15.69), we write: A?" {z) = — ^ A /
exp [~2iuj {z - h) /CQ] d (2a;/co)
^
4A
/f(z-/i)-i
(15.74)
Comparing formula (15.74) for the predicted anomalous square slowness AS^ {z) with the expression (15.66) for the original As^ {z), we find that the inverse result has the step at the right depth, but the magnitude of the step is not exact and the entire curve AS^ {z) is shifted with respect of the true curve As^ {z) by a constant (2A/co). The shift of the inversion curve actually depends on the value of the observed scattered field at zero frequency, which can never be practically determined in a real-world experiment (see Bleistein et al., 2001, for more details). The discrepancy in the step sizes can be easily resolved if we evaluate the reflection coefficient value for a small step 6, taking into account formula (15.68) ^^ ^ 1 - v T T 6 ^ ^ 1 - (1 + \bcl)+0{bcl) l + yrT6^ l + {l + | 6 c 2 ) + 0 ( 6 c 2 ) IhfP.
= -^ S °
1
+ O {bcl) = —bcl + O (6c2) , for small bcl
(15.75)
Comparing the leading order term in (15.75) with the step value in (15.74), we see that the inverse formula provides the correct estimate of a step, 6, with the accuracy determined by the leading term in the representation (15.75). However, Bleistein inversion is based on a Born approximation formula which is valid only to leadingorder in hc^. Thus the theoretical inversion result for this simple model fits well the basic assumption of the Bleistein method. The main significance of this example for the solution of our goal, outlined in the beginning of this section, is that it illustrates how one can find the reflection coefl^icient itself from the observed scattered field data. Indeed, let us consider the bandlimited data, v%{^^^)^ by including a bandlimited filter F {u) in expression (15.70): p% [z, uj) = -F [uj) A ^ exp {-iu {z - 2h) /CQ) . Applying the Bleistein inverse formula to this data, we obtain A
A 4 (Z) = : r ^ A /
r+oo
F{uj)-^
exp [-2iuj {z - h) /CQ] d (2a;/co)
(15.76)
Linear inversion methods
483 HB {Z
(15.77)
- h)
^0
where HB {Z — /i) is a bandlimited version of the Heaviside step function: 1
f^"^
HB{z-h) = - J ^ F{u;)
7T6 (UJ)
— ^ - exp [—iuj {z — h)] dw.
(15.78)
Differentiating function HB {Z — h) with respect to variable z, we arrive at the bandhmited delta-function (15.65) dHB {z-h)/dz
1 /"^^ = — F (uj) exp [-iou {z - h)] duj = 6B{z-h). 27r7-oo
(15.79)
Therefore, differentiating the function ASQ (Z) , we obtain dAs^B i^) /dz = — 2 ^ ^ ^ {z-
h).
(15.80)
Thus, the true reflection coefficient can be found from the formula ^2
"—i^m"^^'^'^""^.'^-
(15.81)
This result gives us an idea how one can modify the original inversion method to find the distribution of the reflecting boundaries and the corresponding reflection coefficients in the medium. Note also that differentiating the anomalous square slowness with respect to z in the Bleistein inversion formula (15.63) is equivalent to multiplying the scattered field in this formula by —2iuj/co. Based on these observations, Bleistein and his co-authors (Bleistein et al., 2001) introduced a bandlimited reflectivity function^ determined by the following formula: 2
4
PB (^) = - y — 4
r+oo
/
{2iuj/co)p'{0,u;)e-''-^/^du;
TTCQ J _ O O
+00
2
iujp'{0,u;)e-^'^'^''duj, (15.82) / -oo where they excluded a term 6B (0) in the denominator, assuming that the corresponding values of the reflection coefficient were scaled by the area under the filter F (UJ) . Indeed, applying formula (15.82) to the bandlimited scattered field (15.76), we obtain TTCI
PB {Z) = A6B
(z-h).
(15.83)
Formula (15.83) justifies the introduction of the reflectivity function. In the case of a multilayered model, this function represents a combination of the bandlimited delta-functions with the positions of extremum at the reflecting
Integral representations in wavefield inversion
484
boundaries, and the sizes of the peaks proportional to the corresponding reflection coefficients (see Figure 15-1). Thus, to transform the inverse formula for the anomalous square slowness into the inverse formula for corresponding reflectivity function distribution we have to multiply the observed scattered field by the factor {—2iuj/co) • (—Co/4) = iu;co/2. We will use this simple rule in subsequent sections on Bleistein inversion as well. 1-D inversion in a variable-background medium We would like to develop a similar approach to 1-D inversion in a medium with variable background wavespeed. Unfortunately, in a general case there is no simple analytical expression for the Green's function. However, when investigating a high frequency acoustic field, one can use the WKBJ approximation (13.83) for the calculation of the Green's function: 9b{^\^'^^)
exp
= --—^—
ZUJ
i:
dz
Cb{z)
(15.84)
Considering again "zero-offset data", / = Zj = 0,
we can write equation (15.57) as follows: r+oo
/
f{0,uj)=p'{0,0,uj)=u;'
As'{z)[g^{z,0,uj)]'dz.
Jo
(15.85)
In the last formula ^^^''''^^
=
^(^)
^
'JUJT{Z,0)
(15.86)
where A{z) = and T{Z.
-^Cb{z)cbiO) dz Jo Cb(z)
(15.87) (15.88)
Using notations (15.86) and (15.88), we can write (15.89) Taking into account that As^ (z) = 0 for z < 0, we can express the last formula in the form +00
/
-oo
4>{z')As'^ {z') e^^""'^'°Uz'
(15.90)
485
Linear inversion methods
where 4>iz') =
(15.91)
-\A\Z'),
and we use a new notation z' for the depth (this will be more convenient for the further algebraic transformations). Note that expression (15.90) has an appearance similar to a Fourier transform from the spatial z domain to the frequency LJ domain. Of course, this is only an apparent similarity, because we have a travel time r in the argument of the exponential function, instead of the depth, z. However, we can exploit this apparent similarity and construct an expression similar to the inverse Fourier transform (15.62): H-oo
/
p'{0,u;)e-^^^^'-°'^doj,
(15.92)
•OO
where ip (z) denotes the result of this "inverse Fourier transform." Let us substitute (15.90) into (15.92) and change the order of integration over the depth, z\ and frequency, LU: +00
/ where
/•+00
(l){z')As\z') OO
/
e'^'^^^^'^'^dwdz' ,
(15.93)
J —OO
iz,z')= f
dz
J z'
We know that
f
g2ia,r(z,.')^ = 2TT6{2T{Z, Z')) .
(15.94)
J —c
Therefore equation (15.93) can be written as follows: +00
/
(t>{z')As\z')6{2T{z, z'))dz'. •OO
On the other hand, according to the property of the delta function, we have 6{2T{Z, z'))dz' = 5(2
r
- ^ )
dz'
By substituting equation (15.96) into (15.95), we obtain
V'
/• ++ 00 00
/
-OO
(l){z')As'^{z')cb{z')6{z - z') dz' = TT4>{z)As'^{z)cb{z).
(15.95)
486
Integral representations in wavefield inversion
From the last equation we have at once
^s'{z)= J['\
..
(15.97)
Substituting equations (15.91) and (15.92) into (15.97), we finally arrive at an explicit formula for the anomalous square slowness:
It can be easily shown that the same formula holds true for bandlimited wavefield data (Bleistein et al., 2001). Note also that for a model with constant background wavespeed, Cb{z) = CQ, the last formula naturally reduces to formula (15.63) derived in the previous subsection. As in the case of constant background velocity, we can obtain the inverse operator for the reflectivity function from formula (15.98) by multiplying the observed scattered field by the factor icjci, (z) /2 : (3{z) =
—--
/
iujp\{),uj)e-^'''^^'^^^du.
(15.99)
Thus while the inverse operator (15.98) determines the distribution of the anomalous square slowness within the medium, formula (15.99) solves for the reflectors and the corresponding reflection coefficients. In geophysical applications, for example in seismic methods, the reflecting boundaries are the main target of exploration. That is why inversion formula (15.99) plays an important role in the interpretation of seismic data. In the next section we will show that this method can be extended to a 3-D case. This technique provides the basis for modern methods of seismic data interpretation. T h r e e - d i m e n s i o n a l inversion Consider now a 3-D acoustic model with the arbitrary distribution of the wavespeed c (r) given by the formula c^ (r)
c^(r)
where Cb{r) is the background (normal) velocity distribution, and As'^{r) is some anomalous square slowness, which is unequal to zero only within the local anomalous zone D. The acoustic field in this model is excited by a point source located at r'. According to (14.3), this field satisfies the equation Vyr,a;) + — - p ( r , a ; ) - - ( 5 ( r - r O ,
(15.100)
487
Linear inversion methods
and the corresponding radiation conditions. The total wavefield in the model can be expressed, as usual, as the sum of the incident field p*(r,c<;) and the scattered field p*(r,cj) (see equation (14.9)), where the incident field p*(r,a;) is the solution of the following Helmholtz equation with background velocity distribution: V V ( r , Lo) + ^ p X r , a;) = - 6 (r - r').
(15.101)
Note that the incident field generated by a point source is nothing more than the corresponding Green's function for the model with background wavespeed: p\r,ij)
= p\r, r',uj) = G'"(r| r';oj).
(15.102)
The Bleistein method is based on a Born approximation of the scattered field, which has the form p'{rj,uj)=uj^
fff
G''{rj\r;uj)As\r)p\T,uj)dv.
(15.103)
Substituting (15.102) into (15.103), we have fff
p'{rj,uj)^p'{rj,r\uj)=iu^
As^ (r) G^(r,| r;cc;)G^(r| r';cj)dt;.
(15.104)
Taking into account the reciprocity theorem (14.24), we can write the last formula in the form III
p\rj,uj)=p\rj,r',uj)=uj'
As' {T)G^{v\Tf,u;)G^{r\v';uj)dv.
(15.105)
The solution of the inverse problem can be simplified for the case of "zero-offset data," when r^ = r': P'{V\LU)=P'{T\T\LU)=UJ^
fff
A52(r)[G"(r|r';u;)f dt;.
(15.106)
Thus, we arrive at integral equation (15.106) for the anomalous square slowness. The key to the solution of this equation for a general variable-background wavespeed is in using the WKBJ approximation (13.80) for the Green's function. By substituting (13.80) into (15.106), we finally find P'{T\UJ)
= u^ fff
As^ (r) A" (r, r') e'^'^^^'^'^di;.
(15.107)
This equation becomes relatively simple in a model with constant background velocity.
I n t e g r a l r e p r e s e n t a t i o n s in w a v e f i e l d i n v e r s i o n
Let us consider the case where Cb{r) = CQ. Then, according to (13.74), the Green's function is equal to G-(r,|
1 iuj\r-r'\/co 47r|r — r'l
r »
(15.108)
Therefore equation (15.106) takes the form UJ"
fir^cu)
1
(r) •
As'
^2iuj\r-r'\/co^^^
(15.109)
Using the new notation TC = | r — r' |, we can rewrite the last equation: 1 (jj^
^2iujrc/co
fir'^u)
As^ (r)
(4^)2
—dv.
(15.110)
D
Differentiating both sides of equation (15.110) with respect the frequency a;, we have
> ( ^ »
^//X-^«
o2iu;rc/co
-dv.
(4
(15.111)
From the last equation we find at once -27rzco^OUJ
1 -y{r\cu)
III
UJ''
(15.112)
As\v)g^{T,,u)dv,
where
^2iojrc/co
g'^{x - x \ y - y^z-
(15.113)
z'.uj) = go{rc,uj) ATTTC
We assume now that the scattered field is observed on the horizontal surface z = 0 and the domain D with nonzero As^ belongs to the lower half-space. In this situation we can extend the integration volume on the right-hand side of equation (15.112) to cover the entire lower half-space 2: > 0 : -27rzco
_d_
1
duo
UJ^
V\x\y\{),u)
A5^ (x, 2/, z) g^ix -x',yy', z, u)dxdydz. (15.114) J J Jz>0 It is now clear that we have a convolution integral on the right-hand side of equation (15.114) over the horizontal variables x and y. The natural way of solving a convolution integral equation of this type is based on a Fourier transform over the spatial variables x and y.
489
Linear inversion methods
Following this idea, let us introduce a pair of forward and inverse Fourier transforms in the spatial domain: n r+oo
SfiK. ky) = II f{x, y) = ^
/(x, y)e-^'^^-^^^yy^dxdy,
ff " Sfih,
ky)e^'^^^^^^yy^dk,dky.
(15.115) (15.116)
Applying this Fourier transform to (15.112), we obtain 1 1 /*+°° SAs^{kx,ky,z)SgQ{kx,ky,z,uj)dz, (15.117) du —:^Sps{kx,ky,^,uj)\ = I /^ J Jo where S^s'^i^x^ ky, z) and SgQ{kx, ky, z,u)) are the spatial Fourier spectra of the functions As^ (r) and g^{rc,uj) respectively: —27rico
r
r+oo
SAs^kx^ ky^ z) = / /
As2(x, y,
z)e-^'^'^^^'yy^dxdy,
and r /•+00
Our next task is to find Sg^. It can be shown that this function satisfies a 1-D Helmholtz equation: Sgoikz^z) = -6{z)
^+(«.)'•
(15.118)
where kl = ujycl-kl-kl.
(15.119)
Thus SgQ is the 1-D Green's function for the medium with the constant velocity c=l/2: 1 ^2ikz\z\ Sg^ykz^z) (15.120) Substituting (15.120) into (15.117), we find -SnkzCo
d r 1 duo \-oSj,s{kx,ky,^,u)\
I r"^ =/
SAs^{kx.ky,z)e^'^^'dz,
(15.121)
where \z\ was replaced by z because the variable of integration is positive. Taking into account that As^ (x, y, z) = 0 for z < 0, we can present the right-hand side of the last formula as a Fourier transform from the z domain to the kz domain: -STTkzCo—OUJ
1
—Sj,s{kx,ky,0,uj)\ ^
1 J
/*"^°°
- /
J —oo
SAs'^(kx,ky,z)e^'^^'dz.
(15.122)
Integral representations in wavefield inversion
490
Note, however, that we should be careful in selecting the proper branch of the square root function when calculating kz based on expression (15.119). The standard choice for k^ is always to select the real-valued and positive square root: kz = sign u^u'^/cl
- k^ - k^ for |a;/co| > ^A:^ -f k^,
(15.123)
for \uj/co\ < ^Jkl + kl
(15.124)
and kz = i^kl
+ kl-u'^/cl
The last formulae show that kz is real-valued only for |ct;/co| > yjk^ -f- k"^. The multipHer (sign a;) ensures that Sg^ is an "outgoing " Green's function, which satisfies the radiation condition at infinity. The imaginary values of kz in formula (15.124) correspond to so-called evanescent waves, which are usually neglected in analyses of geophysical data. At the same time, we see that the slowness distribution, As^ ( r ) , can be found from the purely real kz data. That is why we will assume that our data are free from evanescent waves. Under this assumption, we can invert formula (15.123) with respect to UJ : uj = u {kx, ky, kz) = cosign kzJkl
-f A;^ -h /c^, for \UJ/(^\ > Jkl
-h k^.
(15.125)
Formulae (15.123) and (15.125) show that the interval —oo < CJ/CQ < \/^i~-i-~^ of a; variations corresponds to the interval — oo < /c^ < 0 of/c^ variations, and, respectively, interval ^k^ + /c^ < (JO/CQ < -f-oo is translated into the interval 0 < fc^ < H-oo of kz variations. Therefore, we can use fc^ as a parameter conjugate to the variable z in a Fourier transform, and introduce another pair of forward and inverse Fourier transforms: +CXD
/
m
7r7-oo
me
•CXD
(15.126)
Sf{kz)e-'''''dkz.
Applying the inverse Fourier transform (15.126) to (15.122), we write + 00
S^s'^{kx,ky,z)
= -8co
/
-OO
k.
d_
1
bps U' 2-p-
-2ikz
'dkz
(15.127)
Substituting (15.127) into (15.116), we finally arrive at the solution of our inverse problem: As^{r)=As'^{x,y,z) =
'-^ICL +00
/»+00
O
kz~^
2^p^ \kx) ky^v, UJ) e^'if^^^^f^yy^e-^^'^'dkxdkydkz.
LU
(15.128)
491
Linear inversion methods
According to the basic principles of the Bleistein method outhned above for a 1-D case, we can introduce a 3-D reflectivity function by multiplying the observed scattered field by the factor iujco/2 :
-'^II:L?^ +CXD
/•4-00
O
duj
1
e^^(f^^-+^yy)e-^'^^'dk,dkydk,.
UJ
(15.129)
Note in conclusion that a similar result can be derived for a variable-background velocity model. I refer readers to the text by Bleistein et al. (2001) for more details about this method. 15.1.5 Inversion based on the Kirchhoff approximation As we have discussed already in the previous section on Bleistein inversion, the goal of inversion in many applications is the reconstruction of the reflectivity function, which provides information about the distribution of the reflecting boundaries in the medium under investigation. This problem can also be addressed by using the Kirchhofl" approximation (14.66), which we reproduce here for convenience:
p-(r',.) = K(A) = / / ^ A ( r . ) ^ M £ - ^ > | l M £ ^ . . . ,
(15.130)
where n is upward-directed normal to 5 , and i^(A) denotes the corresponding integral operator applied to the reflection coeflHicients A ( r ^ ) . In particular, we can select as surface B an arbitrary horizontal plane, B^ {z = h). in the lower half-space. Then we have p'{r ,uj) = -
A{rB)-^-^
g^
-dsB,
(15.131)
where we take into account that d/dn= —d/dzB, assuming that the axis z is directed downward. The Kirchhoff approximation represents a linear Kirchhoff integral operator K{A) with respect to the unknown reflection coefficients A (r^): da=p'{r\Lj)=^K{A),
(15.132)
where da denotes the observed scattered field. In an inverse problem we assume that the scattered field, da = p^, and the background wavespeed distribution are given. The goal is to find the reflection coefficients A ( r ^ ) . As a result, we have a linear equation with respect to the unknown function A (r^) • As in the Born inversion method, we solve the linear inverse problem (15.132) by regularization methods, imposing additional conditions on the class of inverse models with corresponding stabilizing functionals.
492
Integral representations in wavefield inversion
We introduce again a Hilbert space Da of the acoustic field data, given on the surface of observations S with the metric determined by formula (15.10) and a Hilbert space M of reflection coefficients A, given on the surface B with the metric (A(i), A(2))^ = 1 ^ AW (ra) A(2) {rs)dss.
(15.133)
The corresponding parametric functional for Kirchhoff inversion with a minimum norm stabilizer has the form P^ = {K{A) - da, K{A) - da)o^ +a {Wa (A'-A^p,) , Wa {K^-hl^))^
- min,
(15.134)
where da = p^ are the observed acoustic wavefield data, Aapr represents some a priori values of the reflection coefficients, and Wa is the weighting operator. Note that we use the minimum norm stabilizer in this example just for convenience. In principle, any stabilizer introduced in Part I can be apphed. The minimization problem (15.134) is solved, for example, by the conjugate gradient or re-weighted conjugate gradient methods introduced in Chapter 5. One possible iterative scheme of the conjugate gradient method is given below: Rn = K{An) - da,
(a)
/ - = P ( A , ) =K^ {Rn) +aW'W{An-Aapr),
{b)
k^ is determined from the minimization problem P - (An+i) - P - ( A , - k^l^^ = min.
(e)
For practical realization of this iterative inversion scheme we have to flnd the explicit expressions for the adjoint Kirchhoff operator K* according to formula {KiA),q)j,^
= {A,K*{q))^.
(15.136)
Using the definitions (15.10) and (15.133) of inner products and expression (15.130) for the forward operator, we can rewrite formula (15.136) as {K (A) , q)j^^ =Re
f [[ K (A) q^'dSdu;
493
Linear inversion methods
= [ //Re / / A M g[p'(rB,a>)G"'(r|rB;a>)] dsBq* (r, uj) dsdu) On = ff A{rs)Re\f ff d\p'*{vB,co)G^*{T\rB;iv)] q (r, uj) dsdu lUJJs dn JJB IJnJJs =
dsf
(15.137)
{A,K*iq))^.
From the last formula we have A, K'
5[p-(r5,a;)G-*(?|rB;a;)] q (r, (jj) dsduj 9n
{q) -Re f ff Jn JIsJs
-0.
(15.138)
M
Equation (15.138) should hold for any A, for example for A = K^ (q) - Re
q{r,uj) dsdu.
dn
JnJJs/s
Therefore,
K*{q)-Re f ff —-—^ JnJJss
^-^
—q (r, u) dsdu
dn
M
0.
From the last formula we conclude that the adjoint Kirchhoff operator K^ is given by the formula K"^ {q) = Re Jn J Js
9[p-*(rB,a;)G"*(?kB;^)] g(r,a;) dsdu. dn
(15.139)
Assuming that we integrate over a symmetrical frequency interval f2 (for example, from — oo to -hoc), we can drop the symbol Re in the above formulae because the imaginary part of the integrand is anti-symmetric in UJ. Note also that we calculate the normal derivative in expression (15.139) with respect to the variable r^ G B, while the integration is conducted along the variable r on the surface of observation S. Therefore we can take the normal derivative operator outside of the integration symbols: K' (g) = ^
y p'^iTB.u;) 11 G-*
{VIVB; UJ)
q (?,u) d^du.
(15.140)
According to formula (15.32) we can introduce an auxiliary acoustic field p^* (r,u;) : p^* (r,a;) - / / G^* (?| r]uj)f
{r,uj) d:S.
(15.141)
Substituting equation (15.141) into (15.140), we obtain K* ( / ) = - ^ 1 p'*{vB, co)p^* {TB,CO) du.
(15.142)
494
Integral representations in wavefield inversion
Finally, if surface B coincides with the horizontal plane, Bh {z = h), we have K*{f)
= -^^Jp'*{vB,uj)p''*{rB,oj)(Lj,
(15.143)
where we take into account that d/dn= —d/dz because the axis z is directed downward. Note that, as in the case of the Born imaging method, jwe can use the convolution theorem to write the expressions for the auxiliary field P^ in the time domain as follows: i'^(r/) = ^ y
/ / . /
P^*(r,a;)e--''dc;
G"" (r', t'\T, t)p' (r, t) dtds,
(15.144)
where (5"(r,t| r ' , 0 = G " ( r , - t | r ' , - 0 . Therefore, the auxihary field P^ (r,f) is the back-propagated field, and the adjoint Kirchhoff" operator K* for a residual field can be calculated by back propagation of the residual field. We will show in the next sections that this procedure is similar to Stolt's Fourier-based migration transformation. Thus, this result demonstrates that migration is similar to applying an adjoint Kirchhoff operator to observed scattered wavefield data. In the general case of an iterative inversion scheme (15.135), we should apply the adjoint Kirchhoff operator K^ to the residual field Rn = K{An) — da computed on each iteration. Each of these applications is equivalent to the standard migration of the residual field. Thus, iterative Kirchhoff inversion can be treated as an iterative migration algorithm. 15.1.6 Traveltime inverse problem The traveltime inverse problem provides another typical example of application of the linearization method to the solution of inverse problems for wave phenomena. In the framework of the "geometric optics" approach to seismic problems, the traveltime T(r', r^) of the seismic ray can be related to the local seismic velocity c(r) by the relationship r(r',r,)=/ -^, (15.145) A(r',r,) C(r) where L{r',rj) denotes the raypath between source r' and receiver r^. The last equation can be written in another form, using slowness s(r) = l/c(r): r(r',r,) = /
s{r)dl.
(15.146)
495
Linear inversion methods
Note that this equation looks Hnear, but is actually nonlinear, because the raypath L{r\rj) also depends on the slowness 5(r). However, we can calculate the variation of the traveltime using Fermat's principle, which states that the traveltime is stationary with respect to a variation of the raypath L{r',rj) : 5r(r',r^)- /
6s{r)dL
(15.147)
Let us suppose that we know some background model ^^(r) of the slowness distribution and that the current model S{T) is obtained by a small perturbation of Sb{r): s{r) =Sb{r)-\-As{r). Denoting by r^ the traveltime along the unperturbed raypath Lt in the model 55 (r), we can use (15.147) as a linear integral equation for the difference A5(r) with respect to the difference of traveltime AT = T — r^: Ar(r',r^) = /
As{r)dL
(15.148)
Note that this equation has been obtained using Fermat's principle in which the result of a raypath perturbation is set equal to zero. Equation (15.148) can be considered the linearization of the nonlinear equation (15.146). For a whole vector of A^ travel time observations, one obtains a system of linear integral equations which after quadrature discretization gives a linear system of equations for slowness perturbations As. For practical applications a rectangular grid is generally chosen and the wave slowness is discretized by treating it as constant in the cells determined by the grid. Thus one forms a vector s of L slowness perturbations: S = (A5i,A52,
ASL)-
Usually the first arrival traveltimes r for waves propagated between sources and receivers are also considered. Differences of N travel times for theoretical models and actual observations produce the vector d of the data: d = (Ari,Ar2,....ATAr). Thus we obtain a linear system of equations for slowness perturbations: d = As.
(15.149)
Now we can use any of the algorithms described in Chapters 3 and 4 for regularized inversion of this matrix equation (15.149).
496
15.2
Integral representations in wavefield inversion
Quasi-linear inversion
In full analogy with the electromagnetic case, we can apply the quasi-linear (QL) approximations introduced in Chapter 14 for acoustic and vector wavefield inversion. We begin our discussion of the basic principles of QL inversion with the simpler scalar case of acoustic waves. 15.2.1 Quasi-linear inversion of the acoustic wavefield In the QL approximation, we assume that the scattered acoustic field p^ inside the inhomogeneous domain D is linearly proportional to the incident field p^ through some reflectivity coefficient A: P\Y,UJ)
(15.150)
= \{r,uj)p\r,uj).
At the same time, according to (14.38), p^ can be calculated as an integral involving A and the incident field: p'{Tj,uj) = u^G^ { A52 (r) [1 + A(r, UJ)]P\Y, U)] .
(15.151)
Following the basic principles of electromagnetic QL inversion, we can introduce a material property parameter m, equal to m(r, iu) - As^ (r) [1 -h A(r, u)].
(15.152)
Substituting expression (15.152) into (15.151), we arrive at a linear equation for the material property parameter m: P'{TJ,UJ)
= u^G^ [m{Y,u)p\Y,u)\
.
(15.153)
The reflectivity coefficient A can be determined from the following linear equation inside the inhomogeneous domain L>, as long as we know m : \{Y,U)P\Y,U)
= u'^Gy, [m{Y,uj)p\Y,u)\
.
(15.154)
After determining m and A it is possible to evaluate the anomalous square slowness distribution As^ (r) from equation (15.152). Note that equation (15.152) should hold for any frequency, because in a general case the reflectivity. A, and the material property parameter, m, are functions of frequency as well. In reality, of course, it holds only approximately. Therefore the square slowness As^ (r) can be found by using the method of least squares to solve equation (15.152)) with respect to A52(r): ||m(r,a;) - As^ (r) [1 + A(r,cj)]||^ = min, (15.155) where the L2 norm
||...||Q
is determined as
.-)L = y ^
l ^ ( r , ^ ) L = \\ I |m(r,a;)|^cL;. In
497
Quasi-linear inversion
In the case of high frequency asymptotics, we can select the reflectivity coefficient to be frequency independent, A = A ( r ) . In this case the material property parameter is also frequency independent, m(r) = As'^ (r) [1 + A(r)],
(15.156)
and equation (15.153) takes the form p'ivj.ij)
= cu'^G^ [m(r)p'(r,a;)] .
(15.157)
Now we can find A(r) by solving the minimization problem: \\X{r)p\rj,u;)
- uj^G^ [m{r)p'{r, u;)] \\^ = min.
The anomalous square slowness is then determined by the simple expression A,= (r) = ^
.
(15.158)
Note that the quasi-linear inversion method, outlined above, can be easily extended to the case of the vector wavefield. We leave the detailed derivation of the corresponding formulae (which look very similar to the analogous formulae for electromagnetic field QL inversion) as an exercise for the interested reader. 15.2.2 Localized quasi-linear inversion based on the Bleistein method We have noticed already in electromagnetic sections of the book that the quasilinear inversion, introduced above, cannot be used for interpretation of multi-source data, because both the reflectivity coefficient A and the material property parameter m depend on the illuminating incident wavefield. However, in many geophysical applications, for example in seismic exploration or in cross-well tomography, the data are collected using moving transmitters. In this case one can build an effective inversion scheme based on the localized quasi-linear approximation introduced in Chapter 9, which is source independent (Zhou and Liu, 2000; Zhdanov and Tartaras, 2002). Indeed, we can introduce a new function, ruL (r) = A^' (r) [1 + AL(r)],
(15.159)
which we call a localized material property parameter. Note that the function TUL (r) is independent of the source position because we select a localized reflectivity coefficient XL which does not depend on the incident field (and frequency). This is the main difference between the localized QL inversion and the original QL inversion considered in the previous section. We assume now that the scattered acoustic field p^{rj,cj) (generated by a source with one or different positions) is measured at a number of observation points.
498
Integral representations in wavefield inversion
Tj. Using LQL approximation for the observed field, p*(rj,a;), we arrive at the following equation, similar to (15.157): p'{rj,uj) = uj^G^ {rriL
{T)P\V,UJ))
,
(15.160)
which is linear with respect to the localized material property function TTIL ( r ) . The main difference between equations (15.160) and (15.157) is that in the LQL approximation, the material property function rriL (r) does not depend on an illuminating source. We can solve the linear equation (15.160) with respect to rriL ( r ) , which is source independent. Now, a scalar reflectivity coefficient \L (r) can be determined, based on condition (14.54): \\XL{V) - J'G^
K ( r , u ; ) ] | | ^ - min .
(15.161)
Knowing A^ (r) and m^ (r), we can find As^ (r) from equation (15.159):
This inversion scheme can be used for a multi-source technique because A^ and vfiL are source independent. It reduces the original nonlinear inverse problem to the same Born-type inversion we discussed in the previous sections. That is why some methods of Born-type inversion can be applied in this situation as well. In particular, the Bleistein method can be very useful in combination with localized QL inversion. For example, in the case of zero-offset data, we can use a formula similar to (15.106) to describe integral equation (15.160) v\r'M
= c^' / / / m^ (r) [G^(r| x'',uj)fdv.
(15.163)
Thus, we arrive at integral equation (15.163) for the material property function TTiL (r). Applying the Bleistein method to the solution of this equation, we use the WKBJ approximation (13.80) for the Green's function. By substituting (13.80) into (15.163), we obtain V\v\u)^ij
fff
mL{r)A^r,r')e^'^^^'^'''^dv.
(15.164)
We can now use the standard technique of Bleistein inversion to find mi ( r ) . The corresponding formulae become especially simple for a model with constant background wavespeed. In this case, according to equation (15.128), we have rriLir)
_8co fr°°+ 00 f/•+00
n' JJ-^ J-.
^ duo
=mL{x,y,z)
^«bps(A:x, r!,yi 0 , to) enk'^+''yy)e-^'''^'dk^dkydk„
(15.165)
499
Nonlinear inversion
where Sps{kx, ky, 0,uj) is the spatial Fourier transform of the observed scattered field on the ground: p r+oo
Sps{k:,,ky,0,uj) = / /
p'{x,y,0,Lj)e-'^'^''^''-^^yy^dxdy,
(15.166)
Thus, localized QL inversion is reduced to Bleistein inversion with respect to the material property function TTIL (X, y, z) and then to a simple correction of this inversion result by solving the minimization problem (15.161) and applying the algebraic transformation (15.162). Localized quasi-linear inversion increases the accuracy and efficiency of wavefield data interpretation because it is based on a much more accurate forward modeling solution than the Born approximation, used in the original Bleistein method. An example of successful apphcation of the localized QL approximation in radardiffraction tomography can be found in (Zhou and Liu, 2000). 15.3
N o n l i n e a r inversion
15.3.1 Formulation of the nonlinear wavefield inverse problem In this section, we consider general nonlinear acoustic and vector wavefield inverse problems in parallel : da = Aa{As^), (15.167) d^ =- A^ {As^) ,
(15.168)
where Aa and A^ are nonlinear forward modeling operators, given by the formulae (15.3) and (15.4): ^ . ( A ^ ^ ) = uj^G^ {As' {r)p{T,Lu)) ^cj^ fjf A, {As') =u'G^[As'u]
=u'
fff
G^{vj\r;uj)
G]S!{rj\T;uj)As'{r)p{T,u;)dv, ' As' {r)u{r,u)dv,
(15.169)
and da, d-^ represent the scattered acoustic or vector wavefield observed on the surface S. In the last formulae G'^{rj\r]Lu) and G^(rj|r;a;) stand for the Green's function and tensor respectively, defined for an unbounded medium with background slowness 56.
The nonlinearity of the operators Aa and A^ with respect to the anomalous slowness As' is related to the fact that the wavefields p(r, uj) and u(r, uj) also depend on As'. As usual, following the conventional Tikhonov regularization method, we substitute for the solution of the inverse problems (15.167) or (15.168) a minimization of the corresponding parametric functional with, for example, a minimum norm stabilizer: P - {As')
= {AaA^s')
- da,.,
AaA^s')
- <„)^^_^ +
Integral r e p r e s e n t a t i o n s in wavefield i n v e r s i o n
500
a {W (As2-A5^^) , W ( A s ^ - A s ^ ^ ) ) ^ ,
(15.170)
where W is the corresponding weighting operator, and the inner products (.., .)D and (.., .)^ are determined by formulae (15.10), (15.11) and (15.12). The minimization problem (15.170) can be solved by the conjugate gradient or re-weighted conjugate gradient methods introduced in Chapter 5. Let us describe, for example, the algorithm based on the regularized conjugate gradient method (5.92), which we reproduce here with small modifications^:
/ - = r ( A 4 ) =F*,^„ (i?„,„;„) +aW*W{/\sl-^sl^),
[b)
PZ = \\lnf I l I C i II'.
(c)
AsU,
t-l^^^Tu-x.
II = 1%.
= Asl-~k^t
(15.171)
(d)
k^ is determined from the minimization problem P - ( A 4 + : ) = P " {^sl
- k±)
= min,
(e)
where, as usual, we denote by A5^ the corresponding anomalous slowness distribution at the n-th iteration. We can see that practical implementation of this algorithm requires computing the Prechet derivative operator Fa,v;n of the corresponding forward modeling operators Aa^y on each n-th iteration. 15.3.2 Frechet derivative operators for wavefield problems In Chapter 14 we developed expressions for the Frechet differentials of the forward modeling wavefield operators (14.29) and (14.84), which we reproduce here for convenience: Fa{s\6s^) = Fair \ 6s^) = Lu^ fff
G'^{r\T;uj)6s^r)p{r,uj)dv.
(15.172)
G'^i^lv.u;) - 6s\r)u{r,uj)dv.
(15.173)
and Fy{s\8s^)=Fy{r\Ss^)=uj^
fff
^For conciseness, the scalar acoustic and vector wavefield equations have been combined into single equations in this summary, with alternative subscripts a and v to distinguish them; it is understood, of course, that d^, Ay, and Ky-n (but not the adjoint operator F*.^), would be printed in boldface in separate vector wavefield equations since they refer to 3-D vector elements in D^.
501
Nonlinear inversion
In these formulae, 5^ = 5^ + As'^ is the square slowness model for which we calculate the variation of the forward modeling operator; ^5^ is the corresponding variation of the square slowness 5^, which is obviously equal to the variation of the anomalous square slowness, 6s'^ = SAs"^; expressions G^(r|r;a;) and G ^ ( r | r ; a ; ) stand for the Green's function and tensor defined for the given square slowness 5^; and the function p{r,u) and vector u(r,a;) represent the total acoustic and vector wavefields for the given square slowness 5^. Note that in the RCG algorithm (15.171), the expression F*^.^ (Ra^y-n) denotes the result of an application of the adjoint Frechet derivative operator to the corresponding acoustic or vector residual field Ra,v;n = ^a,v(A5^) — da^y on the n-th iteration. The expressions for the adjoint Frechet derivative operators are given by formulae (15.22) and (15.23). Based on these formulae, we can write FL (Ran) = f ojY^ir, uj) ff GT (r| r; uj) Ran (r, a;) dScLj, Jn J Js
(15.174)
and
FL {Rvn) = / a;2u;(r, a;) • / / GZ* (?| r; a;) • R.„ (?, UJ) dScLj. Jo. J Js
(15.175)
where Pn(i*,^) = p'(r,a;) -f-p^(r,(x;) and Un(r,a;) = u'(r,(x;) -f u^(r,u;) are the wavefields, and G5^(r|r;a;) and G'^(T\Y,UJ) are the Green's functions and tensors, computed on the n-th iteration for the square slowness distribution, s^ (r) = s^^ (r) -|As^ ( r ) . Here the asterisk * denotes the operation of complex transposition. According to reciprocity theorem 27 (formula (14.24)), the surface integral terms in the last formulae can be treated as the complex conjugate acoustic, p ^ (r,(x;), or vector u^ (r,u;), wavefields, p^ {v,w) = J J G™ (r| r; uj) K „ (?, LJ) dS,
(15.176)
11 G - (r| r; cj) • R : „ (?, a;) ds,
(15.177)
u^ {r,oj)
generated at the point r inside the volume V, by the fictitious scalar or vector wavefield sources, distributed along the surface of observation S with the surface density determined by the complex conjugate residual fields: / ! „ ( ? , a;) =.i?:„(?,u.),
(15.178)
and (f>'snir,^) = Knir,^)-
(15.179)
Integral representations in wavefield inversion
502
Therefore, from (15.174), (15.175) and (15.176), (15.177) we find that the result of applying the adjoint Frechet operator to the residual field is equivalent to integrating over the frequency range Q of the product of the complex conjugate reciprocal fields and the corresponding wavefields p*(r,u;) and u*(r,c<;), computed at the n-th iteration: Kn (Ran) = J ujY„ir, u;)p^* {v,uj) dw, Jn
(15.180)
F:„ (R„„) = / a;2u:(r, LJ) • u f (r,u;) cko. (15.181) Jet Thus, we can conclude, that the calculation of the terms F^^ {Ran) a-nd F^^ (R^vn) requires just one additional forward modeling solution on each iteration for the auxiliary wavefields p^* (r,6t;) and u^* (T^LJ) due to the reciprocal sources. Note that, as in the case of the Born imaging method, we can use the convolution theorem to write the expressions for the auxiliary field P^ (r,t) in the time domain as follows:
-ILL
J JS J -oo
where
"
G^ (r, t|r, t') Ran (r, t) dtds, 1
Ran (?, t) = —
(15.182)
/'"^°° Ran {v,Uj)
e-'^'diJ
^7T J~oo and
_
G"(?,^|r,0 = G"(?,-t|r,-0Therefore, the auxiliary residual field P^ (r,^') is the back-propagated residual field! Taking complex conjugate of both sides of equations (15.180) and (15.181), and noting that the adjoint operators transform residuals into the real functions of square slowness distribution, we write these formulae in equivalent form: Kn (Ran) = / uj^Pn{r,u;)p^ (v^u) dw, Jn
(15.183)
F:^ {R,n) = / uj^Un{r,uj) . u ^ (r,a;) dou. (15.184) Jn We can also write the result of applying the adjoint Frechet operator to the residual field as a correlation of the time derivatives of the back-propagated residual field and the corresponding wavefield Pni'r, t) computed at the n-th iteration: Kn
{Ran)
=
/ UJ^Pn{r, Uj)p^ (r,Cj) diJ =
Jn
[-iujpn{T,
J—oo
Uj)] [-2CJp^* (r,Cj)] * doj
503
Principles of wavefield migration
Pn{r,t)P„{r,t)dt.
(15.185)
•CX)
Thus we can see that the adjoint Frechet derivative operator for the residual field can be found by back propagation (or "back projection," see Devaney (1984)) of the residual field. At the same time, this field can be obtained by using forward modehng calculations, because the wave equation is symmetric with respect to time (Tarantola, 1987; Mora, 1987). In the next section we will see that this backpropagated field can be reconstructed using migration of the residual data. However, in the inversion algorithm we should apply this migration iteratively on each step of the inversion. 15.4
Principles of wavefield migration
Migration algorithms are widely used in seismic exploration for imaging geological cross-sections. They became first known in the middle of the twentieth century as a facility for compensation of the structural distortions occurring in seismic time cross-sections as a result of seismic drifts.^ However, their utilization in seismic data processing became really widespread only in the late 1960s and 1970s, after the petroleum industry developed the corresponding technological basis required for migration imaging. Now seismic migration is one of the most powerful tools in the interpretation of seismic data. Originally, migration and inversion were treated as two diff^erent approaches to interpretation of the wavefield data. Traditional migration solves only for the traveltime parameter, thus providing the geometrical information about the position of the reflectors. In traditional inversion one searches for the physical properties of the medium, including its elastic parameters and wavespeed distributions, which requires a careful analysis of the true amplitudes of the wavefield. It has been demonstrated, however, that traditional migration can be treated as a first iteration in the full solution of the waveform inversion problem (Tarantola, 1987; Mora, 1987). This result is similar to one formulated in this text for potential (Chapter 7) and electromagnetic (Chapter 11) fields. Thus, modern developments in theoretical geophysics have led to dissolving the diff'erence between these two approaches to interpretation of seismic data (Bleistein et al., 2001). In this section of the book I will discuss the basic ideas underlying the principles of wavefield migration, and will show how these principles are related to the general inversion technique developed in the previous sections. 15.4-1 Geometrical model of migration transformation I will illustrate the basic idea of "geometrical migration" by considering a simple example of zero-off'set scattered wavefield data generated by an impulsive source and ^See the pioneer paper by Hagedoorn (1954), which explicitly formulates the migration problem, specifies some approaches to its solution, and introduces the terminology which is now widely used in geophysics.
504
Integral representations in wavefield inversion
observed on a horizontal surface of the earth. We assume that the earth can be treated as an acoustic medium with layered structure and constant wavespeed within each layer. We assume also that we can apply the "geometrical optics" approach to our data (which means that we restrict our analysis to high frequencies in this example). The observed seismic data within this model can be presented as the so-called "time cross-section," which represents the ensemble of responses from all source-receiver pairs. In the plot of time cross-section, the horizontal axis shows the location of each source-receiver pair, while the vertical axis corresponds to traveltime. In other words, we draw the seismic oscillations ("seismic trace") along the vertical lines going just beneath the corresponding source-receiver pair. Each pulse-like signal on a seismic trace (called a seismic "event") corresponds to a specific reflector in the medium. The continuous correlated distributions of the events in this plot are usually identified with the respective reflecting boundaries. However, the corresponding time shown on the vertical axis represents two-way traveltime. This is the time required for the pulse to travel from the source to the reflector and back to the receiver position, which coincides with the source position. Therefore, the depth to the reflector can be calculated by multiplying the traveltime by half the corresponding wavespeed. Applying this procedure to each seismic trace (which is equivalent to scaling the time on the time section by half the wavespeed), we arrive at the depth section, which could provide us with geometrical information about the location of the reflecting boundaries. The question is how accurate and reliable is this information? Unfortunately, this simple imaging technique works well only for horizontal or quasi-horizontal structures, because it is based on a simple model of vertically propagating waves and completely neglects the direction of their approach to the surface of observation. In seismic cross-sections, this presumption inevitably leads to considerable distortions of the curvilinear and steep-inclined reflecting boundaries which are typical for geological structures (Figure 15-2). We call this phenomenon a "seismic drift." The flrst migration algorithms were developed speciflcally to compensate for these distortions and to reconstruct the correct conflgurations and positions of the real reflecting boundaries in the seismic sections. Let us consider these distortions in detail. We denote by to (x) the traveltime of a wave travelling along the ray normal to the reflecting boundary, where x is the horizontal coordinate of the associated source-receiver pair. However, in the time-section this traveltime is plotted along the vertical axis, whereas in reality it may correspond to a propagation of the wave at some angle to the horizontal axis. It means that the reflection point may lie on a semicircle centered on the sourcereceiver position with radius equal to to{x). In a seismic cross-section, obtained by scaling the time by half the wavespeed c, the corresponding reflection point is located on a semicircle with radius equal to to (x) c/2. Note that the actual reflector must be tangent to this semicircle, so that the ray trajectory is orthogonal to the reflector (Figure 15-2). Considering the full ensemble of the source-receiver pairs and
Principles of waveReld migration
505
Figure 15-2 An example of the distortion in seismic section caused by a "seismic drift." a) The true reflecting boundary is shown by the bold solid line, b) The corresponding synthetic zero-offset seismic section. The image of the reflecting boundary is shown by the bold solid line. In order to generate a true reflecting point one should move the observed event (shown by squares, circles, or triangles) along the corresponding semicircles centered in the source-receiver positions with the radius equal to to (x) c/2. Note that the actual reflector must be tangent to this semicircle, so that the ray trajectory is orthogonal to the reflector. The actual reflecting boundary is drawn as an envelope of the ensemble of these semicircles.
506
Integral representations in wavefield inversion
Figure 15-3 An example of graphical migration. Reconstruction of the reflecting boundary by drawing an envelope of the ensemble of semicircles centered on the sourcereceiver positions with radius equal to to (x) c/2. The dotted line shows the position of the actual reflecting boundary.
the corresponding semicircles, we arrive at the conclusion that the actual reflecting boundary may be drawn as an envelope of the ensemble of these semicircles. Thus, to restore the correct configuration of the reflecting boundary in the seismic section, it is necessary to transform the position of each reflection point by moving it along the corresponding semicircle. Figure 15-2 illustrates the simple graphical method of this transformation. Following the previous discussion, we construct an envelope of the ensemble of semicircles centered on the source-receiver positions with radius equal to to (x) c/2 (Figure 15-3). The procedure just described represents the most elementary version of seismic migration, developed in the pioneer paper of Hagedoorn (1954). However, it demonstrates the basic idea behind the migration method: the geometrical images of seismic events are moved graphically or migrated from their original false positions in seismic cross-sections to the correct positions corresponding to true reflecting boundaries. In modern migration methods this fundamental idea is implemented using the sophisticated methods of ray tracing and reverse time wavefield extrapolation. Nevertheless,
Principles of wavefield migration
507
migration can still be treated as an imaging technique that provides a geometrical image of the earth's interior. 15.4-2 Kirchhoff integral formula for reverse-time wave equation migration We see from the previous discussion that the early migration schemes completely ignored the true amplitudes of the wavefield and their variation with time (the so-called dynamic characteristics of wavefield propagation). Introducing the dynamic characteristics of the seismic wavefield in the migration procedure has become a strong catalyst for the subsequent development of seismic interpretation methods. Remarkable achievements were made in this respect back in 1970-1972 by John Claerbout, who took advantage of the transition from pure kinematic models of wavefield propagation to a description of their dynamic characteristics with the help of the wave equation, and formulated the well-defined principles of seismic imaging (Claerbout, 1970, 1976, 1992; Claerbout and Doherty, 1972). Those principles were subsequently developed by Loewenthal et al. (1976) and many other researchers. In formulating the basic ideas of wave equation migration, I will follow mostly the paper by Loewenthal et al. (1976) and the text by Zhdanov et al. (1988). Let us consider some hypothetical wave traveling upwards to the surface of observation with its front coinciding in space at a certain moment of time (say at t = 0) with one of the reflecting boundaries. Then for time t > 0 this wave will obviously travel along the same rays as the actual zero-off"set scattered wavefield, refiected from the same boundary. This happens because, in a zero-oflPset case, the incident and reflected wave rays must coincide, which is possible only if the corresponding ray trajectory is normal to the reflecting surface. Thus, the actual zero-offset scattered wavefield and the hypothetical wave introduced above (which we call the Claerbout upgoing wave) would travel along the same raypaths. However, the Claerbout wave will take half the time to reach a certain point on the observation surface as will the actual scattered wave which has to travel the same distance twice. At the same time, if we assume that the Claerbout wave velocity in the medium is twice as slow as that of the real wave, then both waves will reach any point on the surface of observation simultaneously. These considerations form the basis of Claerbout's principle of seismic imaging, according to which the wavefronts of these fictitious upgoing waves at time t = 0 may be viewed as an image of the respective reflecting boundaries. Thus, according to Claerbout, the process of migration includes two elements: 1) backward extrapolation of the scattered wavefields (i.e. continuation of the waves in the direction opposite to that of their actual propagation) and 2) synthesis of the medium image as a snapshot (at time /; = 0) of the spatial structure of the wavefield produced by backward extrapolation. These principles form the foundation of the majority of algorithms of time section migration (Berkhout, 1980, 1984; Claerbout, 1985; Kozlov, 1986; Gardner, 1985). As we see from the previous discussion, the key element of this approach is the process of backward continuation of the scattered wavefield. The diversity of available migration algorithms is related mostly to the differences in the numerical
508
Integral representations in wavefield inversion
schemes of this continuation It is known from Chapter 13 that the accurate mathematical expression of the Huygens-Presnel principle in the wavefield case is the Kirchhoff formula (13.112) which makes it possible to continue (extrapolate) the wavefield from a certain surface S into the domain of space under examination. Subsequently, we generalized this formula for unbounded domains (13.196) using the Green's function for the wave equation in infinite space, characterizing the divergent spherical wave at infinity according to radiation conditions (13.163). With this choice of Green's function, the wavefields determined by equations (13.196) or (13.197) describe waves outgoing from the surface S in both upward and downward directions. According to the basic principles of migration transformation, outlined above, the backward extrapolation of the observed scattered field requires the reconstruction of upgoing Claerbout waves, i.e. waves traveling towards the surface of observation in the lower half-space. These waves should not satisfy the traditional Sommerfeld radiation condition at infinity, but rather the radiation condition (13.162) which corresponds to convergent waves. We can repeat the derivation of the Kirchhoff formula for an unlimited domain with the convergent wave radiation condition. In this case, however, we should use as the fundamental solution of the corresponding wave equation the function adjoint to the Green's function of the wave equation, G^(r,t| r ' , 0 = G ^ ( r , - t | r ' , - ^ ,
(15.186)
to characterize the propagation of convergent spherical waves. In the case of constant wavespeed c, the adjoint Green's function takes the form
which, according to (13.158), describes a convergent spherical wave, i.e. one arriving from infinity. In this case the wavefield represented by the corresponding Kirchhoff-type formula will be a combination of waves traveling towards the surface of observation as well: f/-(r', t) = 11 j ^ \G- (r', t'|r, t) ^U{r,
-U{T,t)^G^
{T\t'\T,t) dtds,
t)
T'eV"",
(15.188)
where [/^ is the migration field obtained as a result of backward extrapolation (continuation) of the Claerbout upgoing wave, 1/+ denotes the lower half-space, and the direction n in the differentiations is the upward pointing normal to the surface S. Thus, the selection of the fundamental solution in integral formula (13.196) determines the direction of propagation of the waves specified by that relation.
Principles of wavefield migration
509
A similar result takes place in the frequency domain. Substituting the Green's function G'^ (r'|r;a;) in equation (13.197) with its complex conjugate, G^* (r'|r;a;), which satisfies the radiation condition for convergent waves, we arrive at the following migration formula in the frequency domain: ^^^(r^a;)
//.
G'"*iv'\r-oj)-^uiT,u)
-u(r,a;)|^G-*(r'|r;a;) ds,
r'ev-^
(15.189)
I would like to point out that expression (15.188) may also be obtained from the original Kirchhoff formula (13.196) by replacing conventional time t by reverse time T = T — t, where T is time of observation of the^avefield on the surface of the earth. Indeed, let us introduce an auxiliary wavefield C/(r, r ) = C/(r, T — r) and write the Kirchhoff integral (13.196) for this field as follows: [/^(r^r) =
J Js J~c
G-(r',r'|r,r)—f/(r,r)
-[/(r,r)—G-(r',r'|r,r)
dr (is.
v' e F+,
(15.190)
where t/^'(r^ r) - C/^(r, T -r). Transforming from reverse time r to conventional time t = T — r in equation (15.190), we obtain expression (15.188). Note that with the convolution theorem, one can show that, in the frequency domain, the formula for backward extrapolation in reverse time takes the form
U^*{V',UJ) = jj^ G^{V'\V;LO)~U*{V,LO)
-u*{v,uj)—G^{v'\v-uj)
ds,
^' ev^
(15.191)
Thus, we draw the important conclusion that we can select upgoing waves with the aid of the integral Kirchhoff formula by making the transition from direct time t to reverse time r (or by replacing the observed scattered field by its complex conjugate in the frequency domain). That is why this transformation is referred to as reverse-time wave-equation migration. Note that the same idea can be applied to electromagnetic migration as well, ^ shown in Chapter 11.
510
Integral representations in wavefield inversion
15.4-3 Rayleigh integral It can be demonstrated that the boundary values of the wavefield and its normal derivative on surface S are not independent (Zhdanov, 1988). Indeed, if we substitute any two continuous functions, / (r, ^) and ^ (r, ^), instead of the wavefield U ir^t) and its normal derivative dU (r, t) jdn in expression (15.188), we will still generate some wavefield in the lower half-space. This wavefield can be continuously extrapolated back to the surface of observation S. However, this field and its normal derivative may not be equal to the original functions / (r, ^) and g (r^t) on 5. In order for formula (15.188) to reproduce the correct wavefield values in the lower half-space and on the surface S^ it is necessary and sufficient that integral (15.188) be identically equal to zero at the points of the upper half-space, from which we have at once
J Js J-c
G^{r",t'\r,t)—U{r,t)dtds an
^ILL
U{y,t)—G'"{Y",t'\T,t)dtds, Y" i 1/+, (15.192) sJ-oo on where r" is the position vector of a point lying in the upper half-space. This result follows immediately from the Kirchhoff integral formula (13.196). According to formulae (15.189), (15.191), and (13.197), similar relationships hold for the wavefield and its normal derivative in the frequency domain: 11 G'"'
{T"\Y]UJ)
^u{Y,uj)ds
= II
U(Y,UJ)^G''*
(Y"\Y]uj)ds,
Y"
iV^,
(15.193)
U\Y,UJ)^G'"
{Y"\Y]uj)ds,
Y"
^ V^.
(15.194)
and j l G'"{Y"\Y',uj)^u\Y,uj)ds=
jj
The solution of the integral equations represented by the relationships (15.192), (15.193), and (15.194) with respect to functions du{Y^t)/dn or du{Y,uj)/dn proves to be a difficult problem in the case of an arbitrary surface S. However, if the surface 5 is a horizontal plane, these integral formulae can be simplified using a simple geometrical method. Let us position the points r" and r' in the upper and lower half-spaces respectively, as shown in Figure 15-4, i.e. symmetrically with respect to the plane S. Then the following equalities hold true for any point r located on S: G^(r^t'|r,t)
-
G-(r',t'|r,t),
(15.195)
^G-(r^t>,t)
-
-Ag-(r',t'|r,t).
(15.196)
Principles of waveHeld migration
511
V ho) Figure 15-4 The points r' and r" are located in the lower, V'^, and upper half-spaces respectively, symmetrically with respect to the horizontal plane S.
Writing equality (15.192) for horizontal surface S{z = 0), and replacing the Green's function and its normal derivative at the point r" by the Green's function at the symmetrical point r', according to equations (15.195) and (15.196), we obtain the following identity:
ILL ILL
G'"{r',t'\r,t)—U{r,t)dtds
U{r,t)—G'^{r',t'\r,t)dtds, r' e V.
(15.197)
Substituting relation (15.197) into (15.188), we arrive at an important modification of the Kirchhoff integral known as the Rayleigh integral formula (Schneider, 1978; Berkhout, 1980): U'^{r\t') = 2 II
I
U{v,t)-^G'" {Y',t'\v,t)dtds,
(15.198)
where we have noted that d/dn = ~d/dz for the horizontal surface S (where the axis z is directed downward as shown in Figure 15-4, while the normal vector n is directed upward). By replacing conventional time t by reverse time r = T — t (where T is time of observation of the wavefield on the surface of the earth) and introducing an auxiliary
512
Integral representations in wavefield inversion
wavefield f/(r,r) = U{Y^T — r ) , we can write the Rayleigh integral (15.198) for this field as follows: f7(r', T') = 2 ff
r
[7(r, ^ ) ^ G ^ (r', / | r , r ) drds.
(15.199)
Thus, at points in the lower half-space V~^{z > 0), the Rayleigh integral describes an upgoing wavefield which assumes the given values U{r,t) on the observation plane 5, i.e. it may serve as an analytical tool for utilization of the main element of migration - backward extrapolation (continuation) of the observed scattered wavefield. In a similar way we can obtain the Rayleigh integral formula in the frequency domain: ^^(r',a;) = 2 f f uir^uj)—^^* {r'\r;iu)ds. (15.200) For reverse time backward extrapolation in the frequency domain, equation (15.199) can be written as u'^*{r\uj)=2
(15.201)
ffu*{T,u)-^G''{r'\v;uj)ds.
In accordance with the basic principles of migration imaging described above, we produce the image of the refiecting boundaries by plotting the distribution of the migrated wavefield in the lower half-space at time t' = 0. For example, in a homogeneous medium with the constant wavespeed c, the corresponding imaging condition takes the form t/"^(r', 0) = 2 / /
•ILL
+00
/ " C/(r, 0 ^ 5 " (r', 0|r, t) dtds
Q
[47r|r'-r| +°°
^
6[t'-t+''' \
'"'
c/2
H{2\Y'~v\/c-ty
dtds t'=0
dtds,
(15.202)
where we have used the half-wavespeed c/2 according to the principle of migration imaging discussed in the previous section. Relation (15.202) provides a theoretical basis for migration transformation in a homogeneous medium, but it is of little use for the construction of practical migration algorithms because of the presence of a singularity in the integrand. However, we can transform formula (15.202) to make it more suitable for practical implementation. Indeed, taking into account the following identity for the Green's function,
513
Principles of waveReld migration
and bringing the derivative with respect to the parameter z' outside the integral, we can rewrite formulae (15.198) - (15.199) as follows: t/-^(r', t') = - 2 ^ / / / U'^{Y\ T') = -2-^
ff
r
C/(r, t)G^ (r', t'|r, t) dtds,
(15.203)
f7(r, T ) G " (r', r ' | r , r ) drds,
(15.204)
The migration formulae in the frequency domain (15.200) - (15.201) can be transformed in a similar way: u'^ir^u)
= - 2 ^ Ij
tx^*(r',a;) = - 2 ^
u{Y,uj)G'"*{Y'\Y',uj)ds.
(15.205)
ds.
(15.206)
- t)dtds.
(15.207)
f f U%Y,U)G''{Y'\Y;U)
In particular, imaging conditions (15.202) are cast in the form [/-(r', 0) = - l - - ^ J 1^ ^
£ ^ U{r, t)6{2\v'-v\/c
The inner integral in relation (15.207) describes the convolution between the observed wavefield t/(r, t) and a delta-function. Therefore, this integral is equal to the values of the field at time t = 2|r'—r|/c. Thus, we can cast the last formula in the form
Expression (15.208) represents a simple algorithm of diffraction transformation, based on summation of the time section along the hodographs of the scattered wavefield (Timoshin, 1978). Another method of practical realization of the Kirchhoff type reverse-time migration is based on the Rayleigh integral formula in the frequency domain (15.200). Applying an inverse Fourier transform to both sides of the Rayleigh formula, we obtain for f = 0 [/^(r',0) =2
f
f I f
= 2 f f f
U(Y,UJ)^G'"''
{Y'\Y]uj)eyi^{-iujt')duj\
u{Y,Uj)^G'^^Y'\Y]Uj)dLUds.
ds
(15.209)
In a homogeneous medium this formula takes the form
^"•<''°'=-2^//jr»<''"'
exp(—z2u;|r'—r|/c)
— I ^^^^' 7 7 ' \ ^''^' I duds,
(15.210)
where again we use the half-wavespeed c/2 according to the principle of migration imaging discussed in the previous section.
514
Integral representations in wavefield inversion
15.4-4 Migration in the spectral domain (StolVs method) The backward extrapolation (continuation) of the wavefield is especially easy to reaUze in the spectral domain (Stolt, 1978; Gazdag, 1978; Chun and Jacewitz, 1981). This approach is based on representation of the migration field as a sum of plane harmonic waves characterized by the most elementary law of propagation: U'^{Y\
^) ^ -^
11
/ ^"^(^a;, ky, z', uj) exp \i(kx^ + kyy' - o;^)] dujdk^dky, (15.211)
where u^{kx,ky,z,uj)
= //
u^{x,y,z,uj)exp[—i{kxX-\-kyy)]dxdy.
(15.212)
Substituting expression (15.211) into the wave equation, we obtain a simple second-order differential equation:
dz^
u^{kx,ky,z,uj)
^f^^-kl-
k^] u^{kx^ky,z,uj)
= 0,
(15.213)
which determines the variation of the amplitudes and phases of the plane harmonic waves propagating along the z axis (which is directed vertically downward). The general solution of equation (15.213) can be written as follows: u^ikx, ky, z, oj) = A{kx, ky, cj) exp{ikzz) + B{kx, ky, ou) exp{—ikzz),
(15.214)
where k, = ^a;Vc2 - /c2 -
kl,
and the unknown coefficients A{kxjky,uj) and B{kx^ky,uj) are determined from the boundary condition and the condition at infinity z —> -hoc (with the positive z directed downward). The boundary condition requires that the migration wavefield found in the lower half-space from relations (15.214) and (15.211) should be equal to the observed scattered field on the plane z = 0 , i.e. A{kx,ky,uj) + B{kx,ky,uj)
= u{kx,ky,0,uj),
(15.215)
where u{kx, ky,0,uj) is the space-temporal spectrum of the scattered wavefield measured on the surface of observation. However, condition (15.215) is insufficient for unique determination of the coefficients A{kx,ky,(jij) and B{kx,ky,cj). Following Claerbout (1985), the additional condition may be introduced by specifying that the migration field should be a combination of only upgoing waves. For a real-valued and
515
Principles of waveReld migration
positive square root Ju'^/c^ — k^ — k"^ we choose parameter kz according to formula (15.123), i.e. fc, = sign uj^uyc^
for \uj/c\ > yJk[Vk^.
-kl-kl
(15.216)
With this choice of A:^, the following expressions, u^{kx, ky,0,uj) = A{kx,ky,Lj)exp[i
{k^x + kyy -\- k^z — cut)],
(15.217)
u~{kx^ky^O,(jj) — B{kx^ky^uj)exp [i {kxX + kyy — kzZ — out)]
(15.218)
represent, for a; > 0, the downgoing, u^^ and upgoing, u~^ waves respectively.^ From this it follows that, in accordance with the principles of migration transformation, only the second term should be retained on the right-hand side of equation (15.214), when cj > 0, i.e. coefficient A{kx^ ky, UJ) should be made equal to zero. Then constant B{kx, ky.uj) is found from relation (15.215), B{kx,ky,uj)
==
u{kx,ky,0,uj),
from which we deduce for \uj/c\ > yjk\ -h k^j and a; > 0 = u{kx, ky, 0,cj) exp{-iJuj'^/c^
ii!'\h,,,ky.^.^)
- kl - klz).
(15.219)
Conversely, for a; < 0 the terms on the right-hand side of (15.214) will correspond to upgoing and downgoing waves respectively, so that for |CJ/C| > ^Jk!^ 4- k'l, u; < 0 we have u^'^kx.ky.z.uo)
= u{kx,ky,0,uj)exp{iJuj^/c^
- ^I - k^z).
(15.220)
Combining formulae (15.219) and (15.220) into one expression, we find u^{kx^ ky^ z^uj) — u[kx, ky^ 0,cc;) exp {—% kzz)
(15.221)
where kz is given by expression (15.216). Thus, we conclude that formula (15.221) determines the spectrum of the regular upgoing wavefield in the lower half-space equal to the observed scattered field u on the surface of observation. In other words, it realizes the required principles of ^As usual, we exclude evanescent waves from consideration (arising when the squre root is imaginary, i.e. kz = i\ kl + ^^ — uj'^/c^ for \UJIC\ < Jk'^ + /c^) to simplify the discussion. However, the final migration formula will be the same even if we take evanescent waves into account (Zhdanov et al., 1988).
516
Integral representations in wavefield inversion
wavefield migration. Substituting expression (15.221) into (15.211), we arrive at the time domain representation of the migration field: f/^(r',t)
-I
p r-\-oo
= —- / /
STT*^ J J-OO
/
r+oo
u{kx, ky, 0, uj) exp [i{kxx' + kyy' — kzz' — uot)] dwdkxdky.
7-00
(15.222) In the spectral domain this formula can be written as U'^{Y', Lu) = —^ / /
u{kx, ky, 0, uj) exp [i{kxx' -f kyy' - k^z')] dk^dky.
(15.223)
We can produce an image of the medium under examination by reconstructing the migration field in the lower half-space at the initial time t = 0: ix-(x',2/',/,0) = ^
/ ^(^^' ^2/' 0'^) ^xp \-i sign u)yjAu'^/c^ - k^ - k^ z'j •
///
• exp [i{kxx' + kyy')] dkxdkydw,
(15.224
where we use the half-wavespeed c/2 according to the basic principles of migration imaging. Expression (15.224) represents Stolt's (1978) Fourier-based migration formula. 15.4-5 Equivalence of the spectral and integral migration algorithms In the previous sections we analyzed independently two different approaches to wavefield migration: 1) based on the Kirchhoff integral formula, and 2) Stolt's Fourierbased migration. It is important to understand that these two approaches are equivalent. Indeed, it can be demonstrated that, in z > 0, exp[-ik,z] where
= -2—g^\kx,ky,z',uj), 'dz
(15.225
n r+oo n-\-oo
g'^{kx,ky,z]uj)
= //
G'^{x,y,z]Lu)exp[-i{kxX-\-kyy)]dxdy,
and G^(x, y, z; u) is the Green's function of the Helmholtz equation G^{x,y,z;u;)
= G^{r;u;) = - ^ exp (iJ-^] , 47r |r| \ cJ
(15.226
517
Principles ofwavefield migration
where r is the position vector of the observation point (x, y^ z). Thus, in the spectral approach, wavefield extrapolation from one horizontal level to another is implemented by multiplying its spectrum by the complex conjugate of the vertical derivative of the spatial spectrum of the Green's function for the Helmholtz equation. Let us prove the equality (15.225). It can be shown that the function g'^{kx, ky, z; uj) satisfies the 1-D Helmholtz equation d^
.2'
g'"{k^,ky,z;u)) = -6{z).
(15.227)
Thus, g^ is the 1-D Green's function for the medium with the constant velocity CQ = 1 (see formula (13.79)): g'"{k^,ky,z;u)
= - ^ e x p ( i f c , \z\).
(15.228)
Therefore, for z > 0 —g'"\k:,,ky,z\Lu)
=
--exp{-ik^z),
which was to be proved. Substituting equation (15.225) into (15.223), we arrive at the integral formula
= - 2 — / /
u{k:r,ky,0,iu)—g'^*{kx,ky,z]Lj)exp[i{k:r:X
-\-kyy]dk:,dky.
It is well-known that the Fourier transform of the product of spectra of two functions is equal to the convolution of these functions; therefore we obtain ^i"^(r^cJ) = - 2 ^ IJu{Y,uo)G'"*{Y'\Y]u)ds,
(15.229)
which is exactly the Rayleigh integral formula (15.205) in the frequency domain. Thus we have demonstrated that the spectral extrapolation based on Stolt's method (formulae (15.222) and (15.223)) is equivalent to the reverse-time wave equation migration using the Rayleigh integral formulae (15.203) and (15.205). 15.4.6 Inversion versus migration In the section of this chapter describing the Kirchhoff inversion method and general non-linear inversion techniques, we have demonstrated that the calculation of the Kirchhoff adjoint operator (15.142) and the adjoint Frechet derivative operator
518
Integral representations in wavefield inversion
(15.180) requires back propagation of the residual field. This back propagation transformation can actually be performed by migration algorithms. Thus, conventional migration is equivalent to the calculation of the gradient direction (adjoint Frechet derivative) of inversion, except that it does not include convolution with the incident field (similar to one shown in formulae (15.142) and (15.180)). Actually, we can say that conventional migration and inversion algorithms employ diff'erent imaging conditions. This diff"erence can be easily explained by the fact that most of the inversion methods outlined above reconstruct the material property (slowness) distribution, while the migration method was designed to image the reflecting boundaries. In this sense, the closest similarities are between migration and the Bleistein and Kirchhoff inversions, because both of these methods recover the reflectivity function which helps to locate the reflection boundaries. For example, it was demonstrated by Bleistein et al., (2001) that inversion formula (15.129) can be reduced to Stolt's migration formula (15.224). Another important diff'erence between inversion and migration is that the general inversion algorithms are iterative. Without iterations, an inversion algorithm will not be able to reconstruct both geometrical and material properties of the medium. At the same time, iterative inversion, being a solution of an ill-posed problem, requires regularization. By contrast, migration is a stable procedure because it is equivalent to one forward modeling for computing the back-propagated field. However, we have demonstrated in this chapter that any inversion algorithm can be treated as an iterative migration of the residual fields obtained on each iteration. In this way, the difference between these two approaches to the interpretation of seismic data disappears. In conclusion, I should note that modern seismic migration algorithms are based mostly on finite-difference methods of the solution of back propagation problems. However, a description of these methods lies beyond the scope of this book. 15.5
Elastic field inversion
The problem of elastic field inversion is much more complicated than acoustic or vector wavefield inversion, considered in the previous sections of the book. However, the fundamental principles of elastic inversion resemble those discussed above for more simple models of seismic waves. I will present in this section a brief overview of the.basic ideas underlining the elastic field inversion. 15.5.1 Formulation of the elastic field inverse problem Consider a 3-D elastic medium. The propagation of elastic waves in the frequency domain can be described by the Lame equation (13.34) c^VV • u(r,a;) - c^V x V x M{Y,UJ)-{-iJ\i{v,uj)
= --f^
(r,cj),
(15.230)
where u(r,a;) is the elastic field, f^ ( r , ^ ) is the strength of an external source of energy located within some bounded domain Q, p is the density, and Cp (r) and Cs (r)
519
Elastic field inversion
are the Lame velocities. We also assume that the elastic field satisfies the radiation conditions (13.202) - (13.204) at infinity. We will consider a general nonlinear elastic inverse problem: d^ = AL ( m ) ,
(15.231)
m(r)=(|j^j)
(15.232)
where
is a column vector function describing the distribution of the model parameters (square velocities, Cp{r) and Cg(r)), and d^, represents the elastic field observed on the surface S. The goal is to find the unknown parameters Cp{r) and Cg(r). Operator Ai denotes the nonlinear forward modeling operator, given by the Lame equation (15.230) and radiation conditions (13.202) - (13.204). This operator can be calculated, for example, from the general integral representation of the elastic field in the frequency domain (13.97), which we write here in the form U{T\UJ) = III
- r ( r , a ; ) • G^{Y'\Y,u)dv
= A^. ( m ) ,
(15.233)
where G^'(r|r',a;) is the Green's tensor for the Lame equation in the frequency domain, which satisfies c^VV•G^^(r|r^a;)-c^V x V X&\Y\V',UO)^UJ'^&\Y\Y',uj)
= - I ( 5 ( r - r ' ) . (15.234)
Following our standard approach to regularized solution of the inverse problem, we substitute for the solution of the inverse problem (15.231) a minimization of the corresponding parametric functional with, for example, a minimum norm stabilizer: P^ (m) - (A^(m) - d^, Az.(m) - d^)^^ -\-a {W (m - mapr), W {m - mapr))^^ = min,
(15.235)
where W is the corresponding weighting operator, and the inner product (.., . ) ^ and (.., .)^^ in the Hilbert spaces D^ and M^ of data and model parameters are determined by formulae similar to (15.11) and (15.12): (u, v)^^ = R e / / / u(r,a;) - ^r* {Y,UJ) dsdw] u , v G Z^L, ^ e 1^, (15.236) Jn J Js where ds denotes a differential element of the surface S^ and integration is conducted over a variable ?, and (mi, m2)^^ = / / / m f (r)m2(r)c/^; m i , m 2 G ML,
(15.237)
520
Integral representations in wavefield inversion
where the superscript T denotes transposition, and rrii, m^ are the column vectors of the type, shown in formula (15.232). In particular, taking into account (15.232), we can write the last expression in explicit form: (mi, m s ) ^ ^ = j j j
[cl^{r)c%{T) + CI^{Y)CI^{T)\
dv.
The minimization problem (15.235) can be solved by the conjugate gradient or reweighted conjugate gradient methods introduced in Chapter 5. Let us describe, for example, the algorithm based on the regularized conjugate gradient method (5.92), which in the case of the elastic inverse problem takes the form: ^Ln = AL(m„) - dL,
(a)
m.+i=m.-fc^I^,
{d)
(15-238)
k^ is determined from the minimization problem P« (m,+i) = P " ( m , - fc-I-) = min,
n = 0,1, 2,... (e)
where, as usual, we denote by nin the distribution of the model parameters at the n-th iteration. As is usual in the inversion method, the practical implementation of this algorithm requires computing the Frechet derivative F^.^ of the corresponding forward modeling operator A^ on each n-th iteration. 15.5.2 Frechet derivative for the elastic forward modeling operator Let us assume that the Lame velocities are known and fixed everywhere in space with the exception of some local domain D. We can find the equations for the Frechet derivative by applying the variational operator to both sides of the corresponding Lame equation (15.230) c^VV-5u(r,a;) - c^V X V X 5u(r,u) H- J^6\I{Y,UJ)
- -f^(r,cj),
(15.239)
where f^(r,a;) - 6cl (r) V V • u(r,a;) - dc] (r) V x V x u(r,a;),
(15.240)
521
Elastic field inversion
6Cp (r) and Sc^ (r) are the square velocity variations, which are unequal to zero only within domain D, and Su is the corresponding elastic field variation. For the sake of simplicity, we assume that there are no density variations in equation (15.239), i.e. 5p = 0. We also take into account that the perturbation of an external source is equal to zero: 6f^ = 0. Applying integral representation (15.233) to the elastic field variation satisfying equation (15.239), we obtain <5u(r',a;) = JJJ
f^(r,a;) • G''{v'\r,uj)dv
= A^ {4,0',) .
Therefore, the Prechet differential of the forward modeling elastic field operator is given by the following expression: FL(m,5m) = ^u [6cl (r) V V • u(r,a;) - Sc^ (r) V x V x u(r,c^))] •
IIL
Srn^ (r) Du(r, uj) • G^(r'|r, Lj)dv
G^{r'\r,uj)dv. (15.241)
'D
where 6m is the 2 x 1 matrix given by the formula ,5m=,5m(r)=(^|j^j),
(15.242)
and the symbol D is the 2 x 1 matrix differential operator, formed of the gradient divergence V V - and curlcurl V x V x differential operators, such that ° = ( ^ V XVx ) •
(1^-243)
Thus the symbol D u can be regarded as the 2 x 1 matrix of "ordinary" vectors V V • u and —V X V X u (i.e. 3-D vectors in real physical space): \ - V XV X u Therefore, the first product 6m^ (r) Du(r,ct;) in (15.241) above is a (matrix) multiplication of a row vector and a column vector, while the second product -G^ is the tensorial inner product of this ordinary vector with the tensor G^. In particular, it is easy to show that [6ra^ (r) Du(r,a;)] • G^ = ^m^ (r) [Du(r,a;) • G^
Integral representations in >vavefield inversion
522
if we assume that
G^ = Note that vector u in expression (15.241) represents the elastic field for the given velocities Cp and Cg, and the Green's tensor is calculated for the same Lame velocities. We will use below the following simplified notation for the Frechet differential FL{m,6m) = F2^((5m). Thus, if we know the elastic field for the given velocity model, we can find the corresponding Frechet diff'erential using the integral representation (15.241). This formula can be treated as the Born approximation for the elastic field. Using this formula, one can construct a family of linear and nonlinear approximations of the elastic field similar to the corresponding approximations for electromagnetic, acoustic, and vector wavefields. However, this topic goes beyond the scope of our discussion. 15.5.3
Adjoint Frechet derivative operator and hack-propagating elastic field
Now we can address the problem of calculating the adjoint Frechet derivative operator. Note that in the RCG algorithm (15.238) expression F^^ (RLU) denotes the result of an application of the adjoint Frechet derivative operator to the corresponding residual field R^n =" A£,(inn) — AL on the n-th iteration. The explicit expression for the adjoint Frechet derivative operator F ^ can be found according to the definition: (Fr((5m),v)^^ = ( ^ m , F r ( v ) ) M , .
(15.244)
Using the definitions (15.236) and (15.237) of inner products and the expression (15.241) for the Frechet derivative operator, we can rewrite formula (15.244) as (Fr(^m),v)^^ = Re / / / [ iff Ju J Js U J
IIL
Sm'^{r)Rei
JD
6m^ (r) Du(r, w) • G^(^\v,uj)dv • V* (r, w) dsdu)
/ Du*(r,w)
JjG'^*{r\r,u;)-v{T,oj)ds]
= (^m,Fr(v)U.
duoy dv (15.245)
From the last formula we have (5m, F r (v) - Re f Bu%r,uj) • \ [[ G^%r\T, cu) - v {r, u) ds du\ ==0. J ML (15.246)
Elastic field
523
inversion
Equation (15.246) should hold for any 6m, for example for 6m = F r (v) - Re / Du*(r,cj) • | / / G^*(?|r,cj) • v (?,u;) dS du. Jn [J Js Therefore, = 0.
F r (v) - Re / Du*(r,u;) • [ / / G^*(?|r,a;) • v (?,a;) dS dw JQ
U Js
M
From the last formula we conclude that the adjoint Frechet derivative operator F^* is given by the formula F ^ * (v) = Re / Du*(r,a;) • | / / G^*(?|r,a;) • v (Y,U) d? du). Jn \_J Js
(15.247)
which can be displayed in the form of 2 x 1 matrix: F r (v) = Re
^[VV.u*(r,a;)].
//^G^*(?|r,a;)-v(?,c^)dS'
duj
/ J V X V X u*(r,c^)] . [//<,G^*(?|r,a;) • v(?,a.)dS-_ dw
Based on these formulae, we can write Fr,:(RLn) Rt
in [ V V • u;(r, w)] • [/J^G^f (r|r,w) • R L „ (r,c^) £&] duj - ^ [V X V X < ( r , Lo)] • [//^ G^-(?|r, uj) • K^n (?, uj) dsd(jj
, (15.248)
where u^(r,u;) and G^'(r|r,cj) are the elastic fields and Green's tensors, computed on the n-th iteration for the Lame velocities, c^^ (r) and c^^ (r) . The asterisk * denotes the operation of complex conjugate. The surface integral term in the last formula can be treated as the complex conjugate elastic field, u ^ (r,a;), u^ (r,o.) = II
G^{r\v,Lo) • R*„ {r,Lo) ds,
(15.249)
generated at the point r by fictitious sources distributed along the surface of observation S, with the surface density determined by the complex conjugate residual field: FL(?,cc')=RL(?,^), (15.250) Therefore, formula (15.248) can be cast as vm* . p
^^-n^f
In ["^^ • < ( r , w)] • < * (r>w) d-^
(15.251)
524
Integral representations in wavefield inversion
Note again that we can drop the symbol Re in the above formulae if we integrate over a symmetrical frequency interval f^ (for example, from — oo to +00) because the imaginary part of the integrand is anti-symmetric in a;,
or in equivalent from pm* (n
\ - f In [ ^ ^ • ^n{r, w)] • < (r.w) dw
\
We can get a better understanding of the physical meaning of the elastic adjoint Prechet operator if we consider the first iteration in the inversion scheme (15.238). Let us assume that the initial distribution of Lame velocities in the model is given by some background parameters Cpb (r) and Cst (r): mo = mo (r) = ( J^^j^j ") .
(15.253)
Then the first approximation of the model parameters, m i ( r ) , is mi = mo + Amo = mo - fco lo, where Amo = - ^ 0 lo and lo = F^o (RLO) - F*o (AL(mo) - dz.).
(15.254)
Application of the elastic forward modeling operator to the initial model with the background Lame velocities results in nothing more than the incident elastic field in the background model, u*(?, cj). The diff'erence between this incident field and the observed total field d/, gives us the scattered elastic field (with a minus sign): R;.o = AL(mo) - d^ = u^(?,a;) - dL{7,uj) = -u^(?,a;), r e S.
(15.255)
Substituting (15.255) into (15.254) and taking into account expression (15.252) for the adjoint Frechet operator, we find Amo = koFl^ (u^)
= ko(^^Z-''^Ml-''7Hf' ,, ) , where, by reciprocity the auxiliary field u^* {r,u) is given by the formula u^* (r,a;) = J J Gf *(r|r, oj) • u ' (?, LJ) dS,
(15.256)
525
Elastic field inversion
and G^(r|r,a;) is the elastic Green's tensor computed for the background velocity model. We may now recall that according to (15.253). the column vector ^mo represents corrections to the velocities Cpb (r) and Cgb (r) respectively, on the first iteration:
Therefore, formula (15.256) can be separated into two expressions Ac%{r) = kJ [ V V • vi»{vM] Jn
• u«* (r,a;)
= ko [ [VV-u^(r,a;)] - u^ {r,uj) du, Jn
(15.257)
and A 4 ( r ) = -/Co / [V X V X u^*(r,a;)] • u^* (r,a;) dw Jn = - ^ 0 / [V X V X u^(r, uj)] • u^ {r,uj) du. (15.258) Jn As in the case of the waveform inversion method, we note that by using the convolution theorem and a reciprocity, we can write the expressions for the inverse Fourier transform U^^ (r,^^ ^^ ^^^ auxiliary field u^* {Y.UJ) as follows: U^ {Y,t') = — / u^* (r,cj) e-'^^^'dw 27r7_oo = // /
W(y,t)-G^(i,t\v,t')dtd'S,
(15.259)
J JS J —oo
where Gj?,i| r,i')=G^(?,-t|
r,-0.
Therefore, the scattered field U ^ (r,t') is the back-propagated field. We can also write the result of an application of the adjoint Frechet operator to the scattered field as the correlation of this back-propagated scattered field and the incident field U*(r,t) acted on by the relevant difiPerential operators: +00
/ -OO
and
+00
/ -OO
_
[ V V • U' (r,i)] • U« (r,i) dt,
(15.260)
_
[V X V X U ' (r,i)] • U ^ (r,i) dt.
(15.261)
526
Integral representations in wavefield inversion
Thus, the corrections to the velocities Cpt (r) and Cgb (r) on the first iteration can be obtained by correlating the back-propagated scattered elastic field with the derivatives of the incident field U*(r,t). This transformation is similar to wavefield migration described previously. Thus we see that elastic field inversion can be constructed by iterative migration of the residual elastic data. Formulae (15.260) and (15.261) can be further simplified, if we assume that we have constant background velocities Cpb (r) = const and Csb (r) = const. In this case the incident field can be separated into the compressional and shear waves U'(r, t) = U;(r, t) + U:(r, t),
(15.262)
where according to (13.37) and (13.38) V X u ; - 0,
V • u; = V • u\
V X U*, - V X U \
V • U^, = 0.
(15.263) (15.264)
Substituting representation (15.262) into (15.260) and (15.261), and taking into account the properties of the compressional and shear waves (15.263) and (15.264), we arrive at a result that calculation of the anomalous compressional wavespeed requires only the compressional incident wavefield, while the anomalous shear wavespeed is determined solely by the shear incident wavefield: + CXD
/ and
•CO
_
[VV-U;M]-U«Mdi,
+00
(15.265)
_
/
[V X V X U ; {r,t)] • U ^ (r,i) dt. (15.266) We have found in Chapter 13, that the compressional and shear waves in a homogeneous domain satisfy the vector wave equations (13.44) and (13.46). Using the vector identity V^U = V V - U - V x V x U •OO
and properties of the compressional and shear waves (15.263) and (15.264), we can express these equations, in a domain free of external forces, in the form
cl,v'v; --gf dV'u: - ^
= cl,vv. u;- u^= o, = -civ
X V X U^,- u ; = . 0,
(15.267) (15.268)
where dots over the vectors U^ and U'^ denotes their differentiation with respect to time.
527
Elastic field inversion
Substituting the last two formulae into equations (15.265) and (15.266), we obtain Ac^(r) = ^
/
Up (r,i) • U « (r,i) dt,
(15.269)
Ad(r) = ^
/
U , (r,i) • U « (r,^) di.
(15.270)
and
Integrating by parts, we can move one time derivative from the incident field to the back-propagated residual field. As a result, we arrive at the following expressions for anomalous velocities ACp^(r) and Ac^5(r) Acl,{r) = - 4
/
U^ (r^^) • U
(r,^) dt,
(15.271)
(r,t) dt
(15.272)
and ka
Aclir) = -4
f^"^
^s6 J —oo
•^
-^
U , (r,t) • U
Thus the anomalous square velocities can be found as the correlations of the time derivatives of the back-propagated residual field and the corresponding incident wavefields - the compressional wavefield for determining ACp^(r) and the shear wavefield for determining Ac^^(r). This result is similar to one obtained previously for acoustic and scalar wavefield inversions.
References to Chapter 15 Arfken, G., and H. J. Weber, 1995, Mathematical methods for physicists. Fourth Edition: Academic Press, San Diego, New York, Boston, London, Tokyo, Toronto, 1028 pp. Aki, K., and P. G. Richards, 2002, Quantitative seismology, 2nd Edition: Columbia University Press, 576 pp. Berkhout, A. J., 1980, Seismic migration - Imaging of acoustic energy by wave field extrapolation: Elsevier, Amsterdam, Oxford, New York, 339 pp. Berkhout, A. J., 1984, Seismic migration - Imaging of acoustic energy by wave field extrapolation, B. Practical aspects: Elsevier, Amsterdam, Oxford, New York, 274 pp. Bleistein, N., 1984, Mathematical methods for wave phenomena: Academic Press Inc., (Harcourt Brace Jovanovich Publishers), New York, 341 pp. Bleistein, N., Cohen, J. K., and J. W., Stockwell, Jr., 2001, Mathematics of multidimensional seismic imaging, migration, and inversion: Springer, New York, Berlin, London, Tokyo, 510 pp. Chew, W. C , 1990, Waves and fields in inhomogeneous media: Van Nostrand Reinhold, New York, 608 pp.
528
Integral representations in wavefield inversion
Chun, Joong, H., and Ch. A. Jacewtiz, 1981, Fundamentals of frequency domain migration: Geophysics, 46, No. 5, 717-733. Claerbout, J. F., 1970, Coarse grid calculations of waves in inhomogeneous media with application to delineation of complicated seismic structure: Geophysics, 35, 3, 407-418. Claerbout, J. F., 1976, Fundamentals ofgeophysical data processing: McGrawHill, New York, 274 pp. Claerbout, J. F, 1985, Imaging the Earth's interior: Blackwell Scientific PubUcations, Oxford, London, Edinburgh, 399 pp. Claerbout, J. F., 1992, Earth soundings analysis. Processing versus inversion: Blackwell Scientific Publications, Oxford, London, Edinburgh, 304 pp. Claerbout, J. F. and S. M. Doherty, 1972, Downward continuation of moveout corrected seismograms: Geophysics, 37, No. 5, 741-768. Devaney, A. J., 1984, Geophysical diffraction tomography: IEEE, Geoscience and Remote Sensing, 22, 3-13. Gardner, G. H. F., Editor, 1985, Migration of seismic data: Geophysical reprint series, No. 4, Society of Exploration Geophysicists, Tulsa, Oklahoma, 462 pp. Gazdag, J., 1978, Wave equation migration with the phase-shift method: Geophysics, 43, No. 7, 1342-1351. Hagedoorn, J. G., 1954, A process of seismic reflection interpretation: Geophysical prospecting, 2, 85-127. Kozlov, Ye. A., 1986, Migration transformations in seismic prospecting (in Russian): Nedra, Moscow, 247 pp. Loewenthal, D., Lu, L., Roberson, R., and J. Sherwood, 1976, The wave equation applied to migration: Geophysical Prospecting, 24, 380-399. Mora, P. R., 1987, Nonlinear two-dimensional elastic inversion of multi-offset seismic data: Geophysics, 52, No. 9, 1211-1228. Press, W. H., Flannery, B. P., Teukolsky, S. A., and W. T. Vettering, 1987, Numerical recipes. The art of scientific computing: Cambridge University Press, Cambridge, 818 pp. Schneider, W. A., 1978, Integral formulation for migration in two and three dimensions: Geophysics, 43, No. 1, 49-76. Stolt, R. M., 1978, Migration by Fourier transform: Geophysics, 4 3 , No. 1, 23-49. Tarantola, A., 1987, Inverse Problem Theory: Elsevier, Amsterdam, Oxford, New York, Tokyo, 613 pp. Timoshin, Yu. V., 1978, Pulse seismic holography (in Russian): Nedra, Moscow, 321 pp. Zhdanov, M. S., 1988, Integral transforms in geophysics: Springer, New York, Berlin, London, Tokyo, 367 pp.
Elastic field inversion
529
Zhdanov, M. S., Matusevich, V. U., and M. A. Prenkel, 1988, Seismic and electromagnetic migration (in Russian): Nauka, Moscow, 376 pp. Zhdanov, M. S., and E. Tartaras, 2002, Inversion of multi-transmitter 3-D electromagnetic data based on the localized quasi-linear approximation: Geophys. J. Int., 148, No 3. Zhou, C , and L. Liu, 2000, Radar-difTraction tomography using modified quasi-linear approximation: IEEE Transactions on Geoscience and Remote Sensing, 38, No. 1, 404-415.
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Appendix A FUNCTIONAL SPACES OF GEOPHYSICAL MODELS AND DATA In this appendix we will discuss the mathematical tools necessary for developing the regularization theory of inverse problem solutions. They are based on the methods of functional analysis, which employ the ideas of functional spaces. Thus we should start our discussion by introducing the basic definitions and notations from functional analysis. Before doing so, I remind the reader of the basic properties of the simplest and, at the same time, the most fundamental mathematical space - Euclidean space. A.l
Euclidean Space
A. 1.1 Vector operations in Euclidean space Conventional physical space has three dimensions. Any point in this space can be represented by three Cartesian coordinates (xi, X2, x^) . The natural generalization of three dimensional (3-D) physical space is the n dimensional Euclidean space En (or Rn ), which can be described as the set of all possible vectors of order n : a = (ai,a2,a3,
a^),
where the scalars ai, a2, 03, a^ are usually real numbers. By analogy with the length of the vector in 3-D physical space, we can introduce a norm of the vector ||a|| as ^al -h a:1 + ^1 +
^al
(A.l)
It is easy to check that the norm introduced above satisfies the conditions ||a|| > 0 if a 7^ 0, ||a|| = 0 if a = 0,
(A.2)
||Aa|| = |A|||a||,
(A.3)
||a + b|| < ||a|| + ||b||.
(A.4)
The last inequality is called a triangle inequality. In 3-D physical space it has a very simple geometrical sense (Figure A-1): if you have any three points A, B and C, the
532
Functional spaces of geophysical models and data
a+b
Figure A-1
Triangle inequality.
distance between the two points AB is shorter than the sum of the distances between AC and CB : \AB\ < \AC\ -f \CB\. (A.5) Let us introduce the vectors a = AC, b = C 5 and a + b - AB.
(A.6)
Substituting equation (A.6) into (A.5), we obtain (A.4). In a general case of n dimensional Euclidean space, the triangular inequality comes from the Cauchy inequality, which we will discuss below. We can introduce also an operation on two vectors, called the inner (dot) product, as follows: n
3i'h
(A.7)
= ^aibi. i=l
Obviously, a norm can be determined as a square root of the dot product of the vector with itself:
||a|| = Va~^.
(A.8)
By analogy with conventional 3-D physical space, we can say that two vectors a and b are orthogonal if: a b = 0. The following vectors play a similar role in the space En as the vectors of the Cartesian basis in 3-D space: ei = (1,0,0,
0), 62 = (0,1,0, en-(0,0,0,
0), 63 = (0,0,1, 1).
0),
533
Euclidean Space
We will call these vectors ei, 62, e^, a basis of Euclidean space. Any vector SL G En can be represented as a linear combination of the basis vectors n
a = aiBi + a2e2 + where numbers ai, a2, as,
-f anOn = ^
a^e^,
(A.9)
an are the scalar components of the vector a. Evidently ei'ek
= 0] if 2 7^ fc, ek-ek = 1,
(A.10)
a • Gj = fli.
We can write equation (A. 10) in a short form: Bi-Bk = Sik=
<
i =k ' 0, z 7^ A: '
where symbol 6ik is called a symmetric Kronecker symbol. Using this symbol one can find that a^ can be treated as the projection of the vector a on the basis vector e^ ' a • efc = ^ tti (e^efc) = ^
aiSik = a^.
Using the dot product operation, we can prove the Cauchy inequality: a - b < ||a||||b||,
(A.ll)
which in scalar form can be written as
Proof: Let us introduce a non negative function ^{x) : n
(f{x) = ^^ {aiX + hif > 0. i=i
Opening the brackets in the last equation, we obtain
^ a M x' + 2 i^aA i=l
)
U + J ] ^ ' > 0.
\i = l
ir=l
J
(A.13)
Inequality (A.13) means that the equation
(J2ai) ^' + 2 fX:a.b)j x + J2b^=0
(A.14)
has only one real root, or no real roots at all, which is possible only if its discrimimamt is non-positive: /
\ 2
/
\
/
\
E«^M - E«n E^M^o,z=l
534
Functional spaces of geophysical models and data
A. 1.2 Linear transformations (operators) in Euclidean space Suppose that for any a. ^ En we can assign, according to a certain rule, some element a.' e En- We call this rule an operator A : a' = A ( a ) . Operator A is called linear if for any vectors ai,a2 G En and any scalars aia2 G Ei we have A (aiai + a2a2) = aiA (ai) -f a2A {^2).
(A.15)
Let us find the relationships between the components of the vectors a and a': n
n
a.' = A{ai) = A{Y^ a^e^) = J^ (^iM^i)i=l
(A.16)
i=l
At the same time, by applying operator A to the basis vector e^ we obtain a new vector e[, which in turn can be decomposed in terms of the basis vectors: A(e,)=e: = ^^fc,e,.
(A.17)
Substituting equation (A.17) into (A. 16), we obtain n
n
a! = A(a) = ^aiY^ z=l
AkiBk-
(A.18)
k=l
On the other hand, we can also express vector a' in the same basis by n
a' = A(a) = ^a'fcefc.
(A.19)
k=l
Comparing (A.18) with (A.19), we find a very important relationship between the scalar components of the vectors a and a' : n
o!k =
y^Akjaj. i=i
The matrix [Aki] is called the matrix of operator A. It describes the transformation of the scalar components of the vector by the linear operator A. A. 1.3
Norm of the operator
We can calculate a norm of the vector a' as follows l|a'||^ = X : a f = X ; ( X ; A , a f c r . i=l
i=l
k=l
(A.20)
535
Euclidean Space
From the Cauchy inequality we have n k=l
k=l
k=l
where Mi = X]fc=i "^Ik ^^^ some constants. Substituting equation (A.21) into (A.20), we obtain n
lla'll'< M^llalp, where
M2 = ^ M , 1=1
n
n
= ^ ^ A 2 ^ i=\ k=\
(A.22)
or lla'll < M | | a | | .
(A.23)
Let us introduce the definition: Definition 28 The norm of an operator A is the minimum value of all possible M that satisfy the inequality (A.23): \\A\\ = min {M > 0, \\A (a)|| < M||a||} , \\A\\ < M.
(A.24)
Thus we have ||A(a)||<||A||||a||<M||a||.
(A.25)
Based on the last formula, we can write an equivalent expression for the norm of operator A as \\A\\= sup P ( a ) | | / | | a | | . (A.26) llall^O Taking into account this definition and inequality (A.22), we can write
<
^lEE^?^V i=l
k=l
(A-27)
Note that the expression on the right hand side of the inequality (A.27) is called the Frobenius norm of a matrix, \\A\\p.:
WMi
^ i=l
k^l
Definition 29 A linear operator A is called a bounded operator if it has a bounded norm: \\A\\ < ^
536
Functional spaces of geophysical models and data
It is easy to show that a linear bounded operator is a continuous operator, i.e. that the small variations of the argument of the operator will result in a small variation of its values. Clearly, from equation (A.25) we have
P(a)-^(b)|| = P(a-b)||
(ai) + a 2 / (a2).
(A.28)
Consider as an example the following linear functional: /(a) = a - l ,
(A.29)
where a is an arbitrary vector, and 1 is some fixed vector. The remarkable fact is that any linear functional can be represented in the form (A.29). To prove this, we introduce scalars k = /(e^), i = 1,2,3, ....,n. Then n
n
/ ( a ) = / ( ^ aiei) = ^ 2=1
n
aif{ei) = ^
2=1
aik == a • 1,
2=1
which is exactly the equation (A.29). A. 1.5 Norm of the functional Consider the Cauchy inequality |/(a)|=|a-l|<||a||||l||.
(A.30)
From the last equation we have |/(a)|<M||a||.
(A.31)
Definition 30 The norm of the functional f is the minimum value of all possible M that satisfy the inequality (A.31): \\f\\ = min{M > 0, \f (a)| < M||a||} , ||/|| < M. On the other hand we have 1/(1)1 = 1 . 1 = | | 1 | | | | 1 | | < M | | 1 | | . So the minimum value of M that satisfies (A.31), is the norm of the constant vector 1: M>||1||. Therefore we have established that the norm of the functional is equal to the norm of the vector 1 given by its representation (A.29): ll/ll = Hill.
537
Metric space
A.2
Metric space
A.2.1 Definition of metric space We discussed in the previous section the fundamental mathematical space, the Euclidean space, which is a natural generalization of 3-D physical space to the case of n dimensions. Another way to generalize the mathematical idea of a space is to consider not only geometrical spaces, but spaces of more general nature. It means that we can use as an element of space not only geometrical points, but any mathematical objects, for example, mathematical functions. We will still be interested, however, in the geometrical properties of these spaces, but the geometry will be assigned to much more complicated objects than just mathematical points. Why do we need such a generalization? What would be the benefit for geophysicists in using these abstract spaces? The answer is very simple. In the first chapter of this text we have formulated an inverse problem as the solution of the operator equation d = ^(m),
(A.32)
where m is some function (or a 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 Cctse of discrete observations). The solution of the inverse problem consists in determining such a model nipr (predicted model), which generates the predicted data dpr fitting the observed data d well. We have discussed already in Chapter 1 that we do not want to fit the observed data exactly, because they always contain some noise, which we should not fit. Therefore, we are looking for some predicted data which will be close enough to the observed data (usually, within the accuracy of 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, which will help us to evaluate the accuracy of the inverse problem solution. That is why we need to introduce a geometry to measure the distance between actual and predicted data. The mathematical theory of function spaces provides us with guidance in the solution of this problem. The simplest and, at the same time, the most important space which contains a geometry (in the sense that there is a distance between any two elements of this space) is the metric space. A metric space is a set M of elements {h} for each two of which the nonnegative number /i(h, g) can be determined, called the distance between the two elements h and g or metric. The metric has to satisfy the following conditions: /i(h, g) = 0
if and only if h = g,
/i(h,g) = ^ ( g , h ) ,
(z)
(a)
538
Functional spaces of geophysical models and data
//(h, g) < /i(h, q) + /i(q, g)), for any h, g, q G M. (m) The last inequality is called triangle inequality. One important property of the metric space is that we can introduce an idea of convergence of a sequence of elements in this space. A.2.2 Convergence, Cauchy sequences and completeness We begin with several definitions. Definition 31 In a metric space an infinite sequence of elements fi,f2,f3, ....25 said to converge to the element g if as k ^^ 0, the distance between ik and g tends to zero: Definition 32 Any sequence in which the distance between any two elements tends to zero, /j,{{k,(j) —> 0, as k^j —> oo, is called a Cauchy sequence. It can be proved that any convergent sequence is a Cauchy sequence. In fact, from the triangle inequality, we can write M(ffc,fj) < /i(fA:,g) + /i(g,f^) ^ 0 , askj
-^ oo.
On the other hand, there can exist Cauchy sequences of elements which do not converge to an element in the metric space. For example, let us consider as a metric space the internal part of the geometric 3-D ball B without a boundary. We can introduce series of points Si, S2, S3,... which converge to the element SQ located at the boundary. Obviously, the set 81,82,83,... forms a Cauchy sequence, but it converges to the element SQ outside our metric space B. From this point of view we can call B an incomplete metric space. Now we give a rigorous mathematical definition. Definition 33 A metric space is said to be incomplete if there are Cauchy sequences in it that do not converge to an element of this metric space. Conversely, a space M is complete if every Cauchy sequence converges to an element of the space. We give below several additional definitions which play an important role in inversion theory. Definition 34 A subset C of the elements of the metric space M is called compact if any sequence of elements fi, f2, fs,... from C contains a convergent sequence, which converges to an element g G C. Definition 35 A subset N of the elements of the metric space M is called bounded if /i(f,g) < R = const for any {,g e N.
539
Linear vector spaces
For example, the metric 3-D ball B introduced above is a bounded subset of the physical 3-D space. There are several important theorems about compact sets of elements. Theorem 36 Any compact set is bounded. Theorem 37 For any subset S of Euclidean space E to be compact, it is necessary and sufficient that S be bounded. Thus we can see that a metric space contains one very important property of Euclidean space: there is a distance between any two points. However, the metric space is very amorphous; it has no rigid geometrical structure. In many applications we would like to have more specific geometrical properties than just a distance between two points. This goal can be reached by introducing a new operation on the elements of an abstract mathematical space. A.3
Linear vector spaces
A.3.1 Vector operations A linear vector space is a set L containing elements (vectors) which can be related by two operations, addition and scalar multiplication, satisfying the conditions f + g = g + f, f + (g + h) = (f -f g) + h 0 G L; f -f 0 - / , {a + /3)f - af 4- /3f, a(/3f) = {aP){ a{f -h g) = af + ag, where a.peE,,
f,gGL,
and element 0 is called a zero element of the linear vector space. We give now several definitions, which largely define the properties of linear spaces. Definition 38 A linear subspace of L is a subset of L that forms a linear vector space under the rules of addition and scalar multiplication defined for L. Definition 39 A linear combination form Qfifi -f a2h + (^^h + •••• + o^Jn-
of elements fi,f2,f3, ...f^ is any vector of the
540
Functional spaces of geophysical models and data
Definition 40 Elements fi,f2,f3, ...f„ are linearly dependent if it is possible to find a linear combination of them whose value is the zero element and not all the scalars of the combination are zero. Definition 41 Elements fi,f2,f3, ...f^ are linearly independent if a linear combination of them is equal to zero if and only if all the scalars of the combination are zero. Definition 42 The linear space L is called finite dimensional if there is a finite number of linearly independent elements ei, 63,63, ...ey^, a linear combination of which can determine any element of L. The elements ei, e2,63, ...e^ form a basis for L. Thus we can see that the Hnear vector space contains another very important property of the Euclidean space: it has a basis. However, there is no distance in a linear vector space. It would be extremely useful if we could combine these two properties of the Euchdean space, a distance and a basis, within one space. This space is called a normed linear space. A.3.2 Normed linear spaces A normed linear space is a Hnear space N in which for any vector f there corresponds a real number, denoted by ||f || and called the norm of f, in such a manner that ||f|| > 0 , and ||f|| = 0 < ^ f = 0, (z)
||af||Hc.|||f||,
M
l|f + g||
{ii)
where a G £^1. A normed linear space can be made a metric space if we introduce a metric by the formula /^(f,g) = l | f - g | | We introduce a special type of hnear space by the following definition: Definition 43 A Banach space B is a complete linear normed space. This means that every Cauchy sequence in a Banach space converges to an element of this space. Thus we can see that a normed linear vector space contains both a basis and a distance. It has two important properties of the Euclidean space, but not all of its properties. One property, which is still missing, is the analogue of the dot product of two vectors in the Euchdean space. This property is very important, because it actually provides the possibility not only to determine the distance between two points, but also to characterize qualitatively the direction from one point to another in abstract mathematical space. Therefore, the geometrical properties of the space become more rigid. We introduce the space with these properties below.
541
Hubert spaces
A.4
Hilbert spaces
A.4'^ Inner product Let us introduce a linear vector space L^ in which there is defined for every pair of elements f,g a functional, the inner product (f,g), with the properties (f,g) = (g,f)
(symmetry),
(f + g, h) = (f, h) + (g, h) (af, g) = a(f, g)
(linearity), (linearity).
(A.33) (A.34) (A.35)
This functional must also be positively determined, i.e. (f,f)>0
(A.36)
and (f, f) = 0 if and only if f = 0.
(A.37)
The operation of inner product can be treated as an analog of the dot product in Euclidean space. Evidently the space L^ comes equipped with the norm:
l|f|| = V M The linear normed space L^ equipped with the inner product is called a preHilbert space. In order to obtain a Hilbert space we require that the space L^ be complete, in other words, every Cauchy sequence of elements from L^ must converge to an element of this space. So we arrive at the following definition. Definition 44 A Hilbert space H is a complete linear normed space whose norm arises from the inner product defined above. We can prove now a very important inequality, which is the generalization of the Cauchy inequality for Euclidean space. Theorem 45 Schwarz inequality. If X and y are any two vectors in a Hilbert space, then (x,y)<||x||||y||.
(A.38)
Proof. When y = 0 the result is clear, for both sides vanish. When y 7^ 0 the inequality (A.38) is equivalent to
Functional spaces of geophysical models and data
542
Functional Spaces
Linear Space
Metric Space
Normed Space Hilbert Space Figure A-2
The hierarchy of functional spaces.
We may therefore prove only that if ||y|| = 1, then (x,y) < ||x|| for all x.
(A.39)
To prove the last inequality we note that: 0 < ||x - (x, y)y||2 = (x - (x, y)y, x - (x, y)y) = ( x , x ) - 2 ( x , y ) V ( x , y ) ^ = ||x||^-(x,y)^ since ||y|| = 1, from which statement (A.39) follows at once. Thus, a Hilbert space crowns the construction of different functional spaces. Figure A-2 illustrates the hierarchy of functional spaces. The simplest is a metric space, which possesses only geometrical properties: a distance (a metric) between any two points. A linear vector space possesses algebraic properties: addition and multiplication of the vectors. A normed vector space combines these geometrical and algebraic properties together. However, the geometry is still very amorphous, because there is no way to introduce direction or "angle" between two vectors in this space. The Hilbert space is the richest with geometrical properties. One can consider not only a distance between any two vectors from a Hilbert space, but also an angle, (^, between any two vectors, f and g, determined by the formula: cos<^
(f,g)
lgll
(A.40)
It follows from the Schwarz inequality that
l(f>g)l < 1. l|g|l Therefore, for any f and g, expression (A.40) determines some angle (^, and 0 < (p < n. If (f, g) = 0, than we have from (A.40) that ip = 7r/2, so that we can make the following definition.
543
Hubert spaces Definition 46 Two elements f and g of H are orthogonal if (f, g) = 0,
The geometrical structure of the Hilbert space makes it possible to build a basis in the Hilbert space, similar to the orthogonal basis in the Euclidean space. We will introduce a basis by the following sequence of definitions and theorems. Definition 4 7 A finite or countable set of elements {e^} of a Hilbert space H is called an orthonormal set if Bil. Bj for i ^ j (A.41) ||ei|| = 1 for every i Definition 48 An orthonormal set of elements {ej} is said to be complete if (x, e j = 0 (for any i) if and only i/ x = 0.
(A.42)
Theorem 49 / / { e j is an orthonormal set in a Hilbert space H , and if x is an arbitrary vector in H, then X-J](x,e-)e._Le^i
for each j . Proof. ( ^ - X ] ( ^ ' ^ ^ ) ^ ^ ' ^ ^ ) = (x,e^.)-X^(x,eJ(e^,e^) = ( x , e p - (x,e^.) = 0, i
i
from which equation the theorem statement follows at once. T h e o r e m 50 //{e^} is an orthonormal and complete set in a Hilbert space H, and if x 25 an arbitrary vector in H, then
i
and
||xf=5;(x,e,)^
(A.44)
i
Proof. From Theorem 49, x— ^ ^ (x, eje^ is orthogonal to {e^}, so (A.42) implies that x-^(x,e.)e^=0, i
or equivalently, that
544
Functional spaces of geophysical models and data
x = ^(x,e,)e,.
(A.45
i
By the joint continuity of the inner product, the expression in (A.44) yields | | x f = (x,x) = ( 5 ] ] ( x , e j e . , 5 ] ^ ( x , e ^ > ^ 0 = X ] E ( ( ^ ' ^ ^ ) ^ ^ ' ( ^ ' S ) ^ i ) i
^
j i
J
i
from which statement (A.44) follows at once. Definition 51 An orthonormal and complete set of elements {ei} in a Hilbert space H is called an orthonormal basis of Hilbert space. The numbers (x, e^) are called the Fourier coefficients of x, the expression x = ^ ( x , e i ) e i is called the Fourier expansion of x, and the equation (A.44) ^^ called ParsevaVs equation . These terms come from the classical theory of Fourier series. Theorem 52 Every non-zero Hilbert space contains a basis. We have thus demonstrated that the Hilbert space is a natural generalization of Euclidean space. It has almost the same properties as Euclidean space but, the elements of the Hilbert space are formed by much more complicated mathematical objects than simple geometrical points. This result, however, opens a way to work with these complicated objects in the same manner as we work with the geometrical points. For example, we can treat geophysical data as the elements of this space. Also, we can treat the geological and geophysical models as the elements of some Hilbert space as well. Therefore, we can easily introduce the distance between two different geological models and the distance between two different geophysical data sets. For example, we can measure the accuracy of fitting predicted data to observed data by using the distance between corresponding data sets. In other words, we can use all the power and simplicity of the geometrical structure of the Hilbert space to solve the geophysical inverse problem. I will consider below an example of solving a simple approximation problem in Hilbert space using the geometrical properties of the space. A.4-2 Approximation problem in Hilbert space In this section we will illustrate how one can use the geometrical properties of the Hilbert space to solve an approximation problem. Suppose that L is an n dimensional subspace of a Hilbert space H { L C H) and L is spanned by a linearly independent set of n vectors {di,d2, , d ^ } . The problem is to determine for any do e H the vector d e L closest to do (Figure A-3).
Hilbert spaces
545
d,
Figure A-3
Approximation problem in Hilbert space.
To solve this problem let us consider the norm of difference ||do—d||. Any vector d G 1/ can be represented in the form of a linear combination of basis vectors: d = aidi+a2d2+a3d3-f-....+andn. Thus we have the minimization problem ||do-d|| = ||do - (aidi+a2d2-fa3d3-h....-fandn)|| = min,
(A.46)
which can be written, using scalar product notation, in the form
l|do-df = lldo-^a^d^ i=i
= (do - ^
aidi, do - ^
a^d^) = min .
Let us calculate the derivatives of the ||do—d|p with respect to aj which must vanish at an extremum point:
(do-J]a,d,,d^.)=0. i=i
From the last equation we have the system of linear equations for the unknown coefficients a^: n
^ai{di,dj)
= (do,d^).
i=i
We may write the system more compactly as n
^ r , , a , = (do,d^.), i=l
(A.47)
546
Functional spaces of geophysical models £ind data
where the symmetric matrix Tji = (di,dj) is called the Gram matrix. It can be demonstrated that the linear independence of the elements dj guarantees that matrix Fji is nonsingular, which means that the solution to (A.47) {a^, i = l , 2 , . . . . n } always exists for any do and is unique. Note that we can assume that do are observed data, and {di,d2, ,d^} is the set of known theoretical data, which would correspond to some inverse problem solution. In this case, the minimization problem (A.46) is equivalent to the problem of observed data approximation by the given theoretical data set. We will discuss the different formulations of this problem in the next sections. A.5
Complex Euclidean and Hilbert spaces
A.5.1 Complex Euclidean space We can introduce a complex EucHdean space, where the scalar components of the vectors, the scalars ai, 02,03, a^, are the complex numbers. However, in this case we have to modify definitions (A.l) and (A.7) for the norm of the vector and the inner product in order to satisfy conditions (A.2)-(A.4). The norm of a vector in the complex Euclidean space is introduced as a
= yj\a\l + \a\l + \a\l^
+ \a\l
(A.48)
The inner (dot) product of two vectors is introduced as a complex value, determined by the formula n
(A.49)
ai-h = Y^aibl i=i
where the asterisk * means complex conjugate. Obviously, a norm can still be determined as a square root of the dot square of the vector: ||a|| = \ / a a = ^aiOi -f 0203 -f asa^ H-
+ ana^.
(A.50)
Note that in the complex Euclidean space the inner product operation is not symmetrical: a.b-(b-a)*. (A.51) It also follows from equation (A.51), that a-ab - {ah • a)* = a* (b • a)* = a*a • b .
(A.52)
Similar to the real Euclidean space, we can introduce linear operators and functionals in the complex space, however, the functionals in the complex space may have complex values.
547
Examples of linear vector spaces
A.5.2 Complex Hilbert space Similar to the complex Euclidean space, we can introduce a complex Hilbert space. Its construction is based on similar axioms, (A.33) - (A.37), to those for a real Hilbert space, but with one significant modification. The point is that the axioms (A.33) (A.37) cannot be satisfied simultaneously in a complex space. In fact, from (A.33) and (A.35) it follows that (af,af)=a2(f,f). In particular, if a = z, we have (if,if) = - ( f , f ) . From the last formula we see that if (if, if) > 0, then (f, f) < 0, and vice versa, which contradicts axiom (A.36). Therefore, we have to introduce a different definition for the inner product of two vectors in the complex space. It is defined as a complexvalued functional, (f,g), with the properties (f, g) = (g, f )* (complex symmetry),
(A.53)
where the asterisk * means complex conjugate, ( f + g,h) = (f,h) -f (g,h) (linearity),
(A.54)
(Q;f,g) = a(f,g) (linearity).
(A.55)
This functional has to be also positively determined, i.e. (f,f)>0,
(A.56)
and (f, f) = 0 if and only if f - 0.
(A.57)
Thus, we have corrected the first axiom of the real Hilbert space without changing the other axioms. Note that from (A.53) and (A.55) it follows that (f,ag) = a*(f,g). A.6
(A.58)
Examples of linear vector spaces
In this section we analyze some examples of the linear vector spaces, presented in Table Al. The simplest example of a linear space is Euclidean space E^, which we analyzed above. The next example can be constructed using a set of real functions, continuously differentiable to order n on the real interval [a,b]. Obviously, the sum of two
Functional spaces of geophysical models and data
548
Figure A-4 The plot of one function, / ( x ) , is shifted vertically with respect to the plot of the other function, g{x). The size of this shift is equal to the "distance" between two functions in the space with the uniform nornn.
difFerentiable functions is another differentiable function, and the multiphcation of the function by a constant scalar is a difFerentiable function as well. Therefore, this set of functions forms some linear space, denoted by Cn [a,b]. However, it is not a normed space yet, because we did not introduce a norm of the function. There are many different ways to introduce a norm in a function space. The simplest one is as follows:
||/IU=ma^|/(x)|.
(A.59)
a<.x<.b
It is easy to verify that the norm introduced in (A.59) satisfies all required conditions for the norm of the normed space, and I leave this proof to the reader as an exercise. This norm is called the uniform norm. The linear space of continuous functions, Co [a, ^], equipped with the uniform norm, forms a normed space, denoted by C [a, 6] . The distance between two functions, f{x) and g{x)^ in the space C [a, 6] with the uniform norm is equal to fi{f{x),g{x))
= m a x \f{x)a<x
g{x)\.
(A.60)
Figure A-4 gives an illustration of this distance as applied to functions f{x) and g{x). One can see that in this case the distance corresponds to the shift between the plots of these two functions. However, one can notice that the distance between two functions in this norm is determined by the extremum of the difference [f{x) — g{x)]. In other words, even if these two functions go very close to each other along the interval [a, 6], but are different only in one or two points (as shown in Figure A-5), these differences will determine the distance between these two functions in the metric of the space C[a, 6] . In geophysical applications it is much more convenient to have a metric that reflects the average discrepancy between two functions. This metric can be introduced by a so-called Li norm:
Examples of linear vector spaces
549
Figure A-5 The plots of two functions follow each other very closely along an entire interval [a, 6], with the exception of one outlier. These two functions are considered to be very different in the function space with a uniform norm. However, in the function space with an L i norm these two functions are close to each other.
II/IL, = / \f{^)\dx. Ja
In this case the distance between two functions is given by the formula M/(.x),.g(x))= /
\f{x)~g{x)\dx.
Ja
Thus, now two functions will be close to each other if the integral of their difference is small enough. The presence of one or two outliers will not affect the result significantly (see Figure A-5). The linear space of continuous functions on the real interval [a, 6] equipped with the Li norm, is called the Li[a, 6] space. This is a linear normed space, but it is not a Hilbert space because it has no inner product operation. In order to introduce the Hilbert metric in the space of real differentiable functions, we will define the norm as /2 (x) dx. This norm is called an L2 norm. It follows that the distance between two functions will be measured as
H{f{x),g{x)) =
\\f{x)-g{x)
IL2
[f{x)-g{x)fdx.
(A.61)
550
Functional spaces of geophysical models and d a t a
The advantage of this new norm is that one can derive it from the inner product of two functions, defined as {f{x),g{x))=
f
Ja
(A.62)
f{x)g{x)dx.
The hnear space of functions, integrable on the real interval [a, b] and equipped with the L2 norm, is called I/2[a, b] space. A linear normed space L2[a, b] is a Hilbert space, and therefore, possesses all the properties of the Hilbert space discussed above. One can introduce an inner product between two functions, f{x) and g{x), continuously differentiable to order n on the real interval [a, 6], using a different formula:
{m,g{x)) = •^"
TE^IW^^^X,
(A.63)
k=0
where ^0(^)5 ^1(^)5 ••••) ^n(^) are given real functions, (g'n(^) is not identically equal to zero). The corresponding Hilbert space is called a Sobolev space, WJ*. The metric in the space Wf (the distance between two functions) is determined according to the formula f.^,if{x),g{x)) = I fJ2Ql{x) I^IM^]
dx ) .
(A.64)
k=0
Thus, two functions in Sobolev space will now be close to each other if the integral of their difference, and all their derivatives up to the order n, is small enough. In other words, in Sobolev space not only the functions f{x) and g{x) themselves, but also all their derivatives (to order n) should be close to each other. Therefore, the Sobolev metric imposes more control on the function behavior, than the conventional L2 metric. Another example of the Hilbert space is a space Z/2^[a, b] formed by the sets of complex functions, integrable on the real interval [a, b] and equipped with the inner product f f{x)g*{x)dx. (A.65) Ja It is easy to check that expression (A.65) satisfies all axioms, (A.53) - (A.57), for the complex Hilbert space. {f{x),g{x))=
Examples of linear vector spaces
551
Table A l : Examples of linear vector spaces 1 Symbol | EN
Description 1 The set of real vectors of order A^: 1 (ai,a2, aiq)
Name Euclidean space
Cn[a,b]
1 The set of functions, continuously 1 differentiable to order n on the real 1 interval [a, 6]
Not a normed space
C[a,b]
1 Co [a, 6] equipped with the uniform norm H/IU =n}ax | / |
Normed space
1
Li [a, b]
L2 [a, b]
a<x
1 Co [a, 6] equipped with the L\ norm
111/11.,=ri/M^ 1 Set of real functions integrable 1 on the real interval [a, 6], 1 equipped with the inner product U.9)L.
Normed space
Hilbert Space
= lafddx
1 Cn [a, b] equipped with the norm WJi [a,/;]
^?k^]
\\\ffw.^-{i:ELo
2
"12
dx >
1 Set of complex functions integrable 1 on the real interval [a, 6], 1 equipped with the inner product {f^9)LC
=Iaf9*dx
Sobolev space
Complex Hilbert space
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Appendix B OPERATORS IN THE SPACES OF MODELS AND DATA B.l
Operators in functional spaces
We can treat the elements of functional spaces as geometrical points, and we can consider different transformations of these points from one into another. These transformations can be described by corresponding rules, which are called operators. We give now a more strict definition of an operator. Definition 53 Let X and Y be metric spaces and D some subdomain ofX: D C X. If for any x G D, we can assign according to a certain rule some element y G Y we say that the operator A is given on D with the values in Y : y = ^ (x), X G Z> C X, y G r . Figure B-1 gives an illustration of operator A acting from the space X into the space Y. Thus, we can see that the operator is a natural generalization of the function for an abstract mathematical space. We know that one of the very important properties of a function is whether it is continuous or discontinuous. This property can be applied to operators as well. Definition 54 Let X and Y be metric spaces with metrics /i^ (x', x'') and /i2 (y^ y") , and let A be an operator, transforming elements of X into Y. A is said to be continuous at a point XQ in X if for each real number e > 0 there exists a real number 6 > 0 such that, for any two elements x, XQ G X, satisfying the condition //^(x, XQ) < 6, the distance between the results of their transformation by an A operator is less than e : /i2(^ W , A (XQ)) < e. It is important to consider the sequence of elements Xi, X2, X3, ...x^,... G X in the metric space . This sequence is said to converge to the element XQ if /i (x^, XQ) —> 0 as n —> 00. Theorem 55 Let X and Y be metric spaces and A an operator of X into Y. Then A is continuous at XQ if and only if the convergence of some sequence of the elements from the metric space {x^} to element XQ (X^-^ XQ) results in the convergence of the transformed sequences of the elements {74 (x^)} to the element A(xo) : A (x^) -^ ^(xo).
554
Operators in the spaces of models and data
Operators y = Ax
Figure B-1
Introducing forward and inverse operators in metric spaces.
Definition 56 An operator C of X into itself is called a contraction (or contraction operator) if there exists a positive real number r < 1 with the property that fji{C (x), C (xo)) < rfM^x, XQ) for all points x and XQ in X. It is obvious that such an operator is continuous. Figure B-2 illustrates the definition of a contraction operator. The original points, x and XQ, are far apart one from another, while after application of the contraction operator the distance between the resulting points, A{x.) and A(xo), becomes smaller. Definition 57 The points x e X for which x = C (x) are called fixed points for the operator C. In other words, the contraction operator does not move the fixed point (Figure B-2). Lemma 58 If C is a contraction defined on a complete metric space X then C has a unique fixed point which can be determined by the method of successive approximations as follows. ProofLet XQ be an arbitrary point in X and write Xi = C (XQ), X2 = C (xo) ,
and in general Xn = C ^ ( x o ) - C ( x , _ i ) . Then we can prove that there exists a point x in X such that x^—>x and Cx = C(limXn) = limC (xn) = limx^+i = x.
Linear operators
555
Contraction Operator (A=C)
[i(Ax,Ax^ Figure B-2
)<
rjLifjt, ^
),
r
Introducing a contraction operator.
Indeed, if m < n, then /i(x^„ x j = /i(C^^'' (xo), C- (xo)) = / i ( C - (xo), C^-C-— (xo)) < r^fi{^o, = r ' X x o , x ^ _ ^ ) < r'^^[^(xo,Xi) +/x(xi,X2) -h < r'^Xx^, Xi)[l ^r^r^
^
-h :
C^-^XQ)
+//(x^-m-i, x^_^J] <
-/i(xo,Xi).
1 Since r < 1 it is evident from the last formula that {x^} is a Cauchy sequence, and by the completeness of X, there exists a point x in X such that x^-^x. We can use the continuity of C to infer that x is a fixed point: C (x) = C(limx^) = limC (x^) = limx^_^i=x. We conclude the proof by showing that x is the only fixed point. If y is also a fixed point, that is , if C (y) = y , then /x(x, y) = /i(C (x), C (y)) < r/x(x, y); and since r < 1 , this implies that /i(x, y) = 0 or y = x. B.2
Linear operators
Let X and Y be normed spaces with the same system of scalars. There is a very important class of operators in the normed linear spaces which are called linear operators .
556
Operators in the spaces of models and data
Definition 59 scalars ai:
An operator y =L (x) is called linear if for any x^ G X and any
L(aiXi + 0^2X2 +
4- OLnX^n) =
CXIL{KI)
+ Q;2l/(x2) -f
-I- anL{-Kn).
Definition 60 An operator y =L (x) is called bounded if there exists a real number M with the property that ||L(x)|| < M | | x | | (B.l) for every x G X . It is easy to prove the following theorem: Theorem 61 yl linear operator L is continuous if and only if it is bounded. Definition 62 The smallest constant M for which condition (B.l) :x. E X is called the norm of the operator: \\L\\ =mm{M
:M>0,
||L(x)|| < M||x||,
holds for any
foranyxeX}.
From the last formula we see at once that
ll^(x)||
Inverse operators
Let us consider the equation A(x)=y.
(B.2)
If the solution of equation (B.2) is unique, then we can assign to any y', for which equation (B.2) is solvable, the corresponding value x' (see Figure B-1). Thus we can determine some operator A~^: X-
A-^y,
which we call the inverse operator. Theorem 63 The inverse operator A~^ for a given linear operator A exists and is linear if and only if the equation Ax = 0 holds only for x = 0. Theorem 64 The inverse operator A~^ for a given linear operator A exists, and is linear and bounded if and only if there exists a real number m > 0 with the property that P(x)||>m||x||, for every x G X . In this case \\A-'\\<~
m
Some approximation
B.4
problems in the Hilbert spaces of geophysical data
557
Some approximation problems in the Hilbert spaces of geophysical data
Consider some metric space M of the geophysical models { m } . In the case of a gravity field, it could be some distribution of density p (r); in the case of a magnetic field, it could be some distribution of magnetization / ( r ) . In the electromagnetic case it is the distribution of conductivity a (r), and in the seismic case it is some distribution of velocity V (r) or slowness m ( r ) . For any geophysical model we can assign the corresponding geophysical data set d. The sets of the data form the space £). It is very convenient to equip the space D with some operation of inner product. Thus we have the Hilbert space of data D. We can write the operator of forward modeling: d = A(m), (B.3) where m e M, d e D. Suppose A is a linear operator and do are the observed data. We have a set of initial models {mi, m2, •••mn} ^^^ the corresponding data: di = .4(mi), d2 = ^ ( m s ) , dg = ^(mg), etc. We would like to construct a new model, m = ^o^zm,, i=i
such that the corresponding theoretical data d = A{in) are closest to the observed data do : ||d-do|| = P ( m ) - d o | | = m i n .
(B.4)
Consider the set of elements d, =:: A(m,) G D, 2 = 1,2, ...n. Suppose that {d^} is the linearly independent set of elements in D (if not we can select from them the linear independent subset). Evidently we can consider the subspace L spanned over n vectors {d^} . The problem of approximation is to find the vector da G L, closest to the observed data do E D. Evidently da-^a^d,. i=i
Thus we have the minimization problem : ||do-d|| = ||do - (aidi + a2d2 + asda 4-.... -h^ndn)!! = min.
(B.5)
558
Operators in the spaces of models and data
or, using inner product notation, n
||do-df = ||do-5^a,d,||2 n
= (do - ^
n
aidi, do - ^
a^d^) = min.
Using the same approach which I outhned in Appendix A, we may write a system of hnear equations for the unknown coefficients a^: n
J]r,,a, = (do,d^.) ,
(B.6)
z=l
where the symmetric matrix Tji = (dj,d^) is a Gram matrix. It can be demonstrated that the hnear independence of the elements dj guarantees that matrix Tji is nonsingular, which means that the solution to (B.4) {a^, z = 1, 2, ....n} always exists for any do and is unique. The Gram matrix can be calculated as follows: r , , = (d,,d^.) = (A(m,),A(m,)). (B.7) As soon as we have solved equation (B.6), we can express the approximate model ma as a sum: ma =
n
y^^aiiai.
Let us consider as an example the 2-D gravity problem, generated by some mass distributed in the domain 5 of a lower half plane. In this model the observed data set do is equal to observed gravity field gi ^. We can divide the domain S into a system of smaller subdomains (blocks Si) with constant densities pi inside each block, as shown in Figure B-3. The gravity field gz of each elementary block Si with the unit density can be calculated using the formula:
d, = d,{x') = 9^\x',0) = 271£ ^^ _ J^, ^ ^^d^dx. Total gravity field gz can be represented as a sum of contributions of each block with the coefficients proportional to their densities:
The Gram matrix can be calculated as Tji =
(di^dj)
Gram - Schmidt orthogonalization process
559
2D Gravity Problem
Figure B-3
Ja
Twodimensional gravity model.
J Js^ {x - x'y +
-dzdx
-dzdx \dx' ^ (x — x'Y + z
and J Jis^ Sn
Ja
-dzdxdx . [x-x'Y-^z^
In this formulation the unknown coefficients ai can be treated as the densities of the different blocks p- : Thus we have a system of linear equations with respect to the densities of the blocks: n
Solving this system, we find the density distribution in the domain S of the lower half plane. Figure B-4 provides an example of a gravity inverse problem solution using this technique. B.5
Gram - Schmidt orthogonalization process
Sometimes it is difficult to solve the system of equations (B.6). Evidently, this system can be very much simplified if the set of elements {di, d2, dg, ...d^} are orthonormal. Fortunately we can exhibit a constructive procedure for converting this set into the
Operators in the spaces of models and data
560
Vertical component of gravity field
Figure B-4
An example of a 2-D gravity inverse problem solution.
corresponding orthonormal set {ei,e2, ....,e^} with the property that the Hnear subspace of i7 spanned by {ei,e2, ....,e^} is the same as that spanned by {di,d2, ^d^}Our first step is ei
The next step is to construct the vector d2—(d2,ei)e^ , which is orthogonal to ei, and then to normahze it: _ d2-(d2,ei)ei ^ ~ ||d2-(d2,ei)ej| The next step is to construct the vector ds—(dg, 63^)6^ —(dg, 62)02, which is orthogonal to both vectors ei and e2, and then to normalize it: ^ ^
d3-(d3,ei)ej^-(d3,62)62 ||d3-(d3,ei)ei-(d3,e2)e2||
etc. This procedure is called Gram - Schmidt orthogonalization process. The advantage of having the orthonormal set {ei,e2, ....,e„} is that for this set the Gram matrix is reduced to the Kronecker matrix: Tji = ( e i , e )
=6i.
0, z 7^ 0; 1, i = j]
(B.8)
561
Gram - Schmidt orthogonalization process
Thus, equation (B.6) takes the form n
J]<5,,a| = (do,e^)
(B.9)
i=l
or c.- = (cio,e^).
(B.IO)
Therefore, we can determine immediately all the coefRcients of the approximate model: ma =
^{do,ei)m.. i=i
The orthogonalization process plays an important role in the solution of the inverse problem.
This Page Intentionally Left Blank
Appendix C FUNCTIONALS IN T H E SPACES OF GEOPHYSICAL MODELS C.l
Functionals and their norms
In the previous section we have discussed the operators which transform vectors into vectors. There is a special class of operators which play an exceptionally important role in the theory and applications. This class contains an operator transforming vectors from an arbitrary metric space into real numbers, which can be treated as the elements of one-dimensional Euclidean space Ei. The operators from this class are called functionals. We will now give a more rigorous definition of functionals. Let X be a metric space, D C X, and Ei be a one-dimensional vector space (a set of real numbers). We introduce the following definition. Definition 65 If for any :si e X we can assign according to a certain rule some real number y e Ei, we say that the functional f is given on D. A functional is a special case of an operator when Y = Ei. A linear functional is a special case of a linear operator. Definition 66 The functional y = f{x.) is called bounded if there exists a real number M with the property that
l/(x)|<M||x||,
(C.l)
for every x G X . Definition 67 The smallest constant M for which condition (C.l) holds for any X G X Z5 called the norm of the functional: -- min{M : M > 0, | / ( x ) |< M||x||, for any x G X}.
564
C.2
Functionals in t h e spaces of geophysical models
Riesz representation theorem
The properties of the linear functionals in Hilbert space are similar to the properties of the linear functionals in Euclidean space. For example, consider a fixed element 1 of a Hilbert space H. Then we can introduce a linear functional: / W = (l,x),
(C.2)
for any x G i / . Moreover, / ( x ) is a bounded linear functional: |/(x)i<M||x||, where M < oo. This follows from Schwarz's inequality: |/(x)N(l,x)|<||l||i|x||. We have demonstrated in Appendix A that any linear functional in Euclidean space can be represented in a form of (C.2). We can prove now that the same fact is correct in the general case of the functionals in Hilbert space. Theorem 68 Riesz representation theorem Every bounded linear functional / ( x ) in a Hilbert space may be represented as (1, x) and 1 is uniquely determined by / . Proof: Consider a basis {ei, e2, eg, e^,....} of the Hilbert space. We know that for any x G X x = X](x,eJe.. i
Thus, due to the linearity of the functional, we can write fix) = /(^(x,e,)ej = ^(x,e,)/(e.). i
(C.3)
i
Suppose that f{ei) = 1^ .
(C.4)
We can introduce a vector 1 with the scalar components k : \ = Y,li^i-
(C.5)
i
Then equation (C.3) can be written as / ( x ) = J ] ( x , e , ) / , = (x,l),
(C.6)
Functional representation of geophysical data and an inverse problem
565
from which the first statement of the theorem follows. Suppose now that there is another 1^^^ such that /(x) = (x,l«)
(C.7)
Then /(e,) = (e,,lW)=/W. In the same time, according to formula (C.6)
From the last two equations it follows that \ = l\ Thus, we have proved that the vector 1 is uniquely determined by / . C.3
Functional representation of geophysical data and an inverse problem
Assume that we have geophysical measurements in a fixed number of observation points dj, j = 1, 2, ...n; d^ G Ei. These measurements depend on parameters of the corresponding geophysical models and therefore can be treated as the functionals d,==/,(m),
(C.8)
j = 1,2, ...n; dj e El, m G M, where M is a Hilbert space of model parameters, and fj{m) are linear functionals, defined on M. According to the Riesz representation theorem there exist vectors 1^-^^ (elements of the space M) such that d,==(m,l(^)), (C.9) j = l,2,...n;,
P^eM.
Vectors l^-^^ are called "the data kernels" (Parker, 1994). Suppose that we know the data kernels 1^-^^. The problem is to determine the model m which fits the observed data. In other words we have to find the solution of the system of equations (C.9). To solve this problem we assume that {l^^\ j = l,2,...n;} is a system of linear independent vectors, which forms the subspace L C M. If the dimension of M is greater than L, the element m is not unequally defined by (C.9). So we can find the solution of (C.9) which possesses the additional properties, for example the smallest norm. First of all let us formulate the Decomposition Theorem.
Functionals in the spaces of geophysical models
566
Decomposition Theorem
Figure C-1
Decomposition Theorem.
T h e o r e m 69 The Decomposition Theorem For a given complete subspace L C M, any element m E M can be written as the sum of a part in L and a part in L^ : m=^l-hh,
(CIO)
where L^ is the orthogonal complement of L, such that if I € L, h € L-^, then (1, h) = 0 (Figure C-1). By definition: (C.ll) Substituting (C.IO) into (C.9), we have (C.12) j = l,2,...n; l^^^GM, so that only element 1 is to be determined to fit equation (C.9). An element h can be chosen from the other conditions, for example from the condition that the norm of m is minimum. Let us calculate this norm: | | m | | ' = (l + h , l + h) - | | l f + 2 ( l , h ) 4- ||h||'= | | l f + | | h f .
(CIS)
From (C.13) it follows that ||m|| = min if h = 0.
(C.14)
567
Functional representation of geophysical data and an inverse problem Thus we have the following solution for m: n
m = l= ^Al«,
(C.15)
i=l
where /?^ (z = 1, 2, ...n) are unknown coefficients which have to be determined from the observed data. By substituting (C.15) into (C.9), we have
d, = EA(i'^U'^'^) = Er,iA, i=l
(C.16)
2=1
where is the corresponding Gram matrix, which is nonsingular because the vectors 1^-^^ are assumed to be linear independent. Thus coefficients P^ (i = 1, 2, ...n) are found as the solution of the system of equations (C.16). As an example of practical application of the Riesz representation theorem, consider the 2-D gravity problem. The forward gravity operator is given by the formula
^,(x^0) = 27 / / , ^^^\fl ,dzdx. JJs [x-x'y + z^ Assume that observations were made at the surface of the earth at the points with the horizontal coordinates Xj :
-IL
Let us introduce a Hilbert space L2 {S) of the 2-D functions defined and integrable in the domain S. The inner product between two functions from this Hilbert space is defined by the formula (/(^,^),^(^,^)) = /
f{x,z)g{x,z)dzdx.
According to this formula, the integral in the equation (C.17) can be treated as an inner product between p{x, z) and a function l^-^^ = l^^\x, z) : lO) = /0)(a:, z) = 277 ^^j—^ . [x — XjY -]- z^
(C.18)
568
Functionals in the spaces of geophysical models
Therefore, we can write dj = 9z{xj, 0) = (p (x, z), l^^^ (x, z)) . From the last expression we can see that /^-^^ (x, z) are the data kernels for this problem. The Gram matrix takes the form
and the coefficients P^ satisfy the equation n
(C.19)
dj = g,{xj,0) = J2rjif3,. 2=1
Substituting (C.18) into (C.15), we find the density distribution which generates the observed data:
J—I
j_l
\
J/
Appendix D LINEAR OPERATORS AND FUNCTIONALS REVISITED In order to develop the appropriate technique for solution of the linear inverse problem we need to study more carefully the basic properties of linear operators and functiona l . This Appendix provides the necessary information about linear operators and functionals. D.l
Adjoint operators
We assume that X and Y are Hilbert spaces and yl is a linear operator from X y = ^x. Theorem 70 For any linear operator A on X and any y ^Y, element x* G X such that for allyi ^ X
toY: (D.l)
there exists a unique
(>lx,y)y = (x,x*)x,
(D.2)
||x*|U< Plll|y||r.
(D.3)
where
Proof: If element y is fixed, we can consider (Ax, y)y as a linear functional with respect to x: (^x,y)y = /(x). (D.4) According to the Riesz representation theorem, any linear functional can be represented as / ( x ) = (x,x*), (D.5) where x"^ G X exists and is unique and
11/11 = \W\\x.
(D.6)
On the other hand, according to (D.4) and the Schwarz inequality, we have ||/(x)||<||Ax||||y||<||A||||x||;,||y||y.
(D.7)
Linear operators and functionals revisited
570
Adjoint Operators
(Ax,y% =(x,AY),
(Ax, y% = (X, X*) Figure D-1
Introducing an adjoint operator.
Dividing the left-hand and the right-hand sides of (D.7) by ||x||, we have
ll/(x)ll
(D.8)
where x = x/||x|| and / ( x ) = / ( x ) / | | x | | by linearity. From the last inequality we have 11/11 = s u p { / ( x ) , | | x | | = l } <
y||y-
(D.9)
By comparison of (D.6) and (D.9), we see that
||x*||x
(D.IO)
From (D.5) and (D.IO) the statement of the Theorem 70 follows at once. Figure D-1 illustrates an idea of the adjoint operator. Definition 71 On the basis of Theorem 70 we can determine the operator A* which maps an element y G F into the proper element x* G X, according to formula (D.2): X* = A*y.
(D.ll)
The operator A* is called the adjoint operator of A: (>lx,y) = (x,yl*y)-
(D.12)
Theorem 72 The adjoint operator is a linear operator and \\A*\\ = ||A||.
(D.13)
571
Differentiation of operators and functionals
Definition 73 A linear operator A in a Hilhert space H is called self-adjoint (or symmetric ) if A = A\ (D.14) Thus for a self-adjoint operator we have (Ax,z) = (x,Az).
(D.15)
Definition 74 A symmetric operator is said to he positive on some subset S C H if for all:>c e S (Ax,x) > 0 (D.16) and
(Ax,x)-0,
if and only i / x = 0. Definition 75 A linear operator A in a real Hilbert space H is called positively determined in some subset S C H, if we can find a constant 7 > 0 such that, for all X G 5, the following relationship holds: (Ax,x)>7(x,x)=7l|xf .
(D.17)
The last definition can be extended to the case of the complex Hilbert space H. Definition 76 A linear operator A in a complex Hilbert space H is called an absolutely positively determined (APD) operator in some subset S C H, if we can find a constant 7 > 0 such that, for all x G 5, the following relationship holds: |(Ax,x)|>7(x,x)-7l|xf . D.2
(D.18)
Differentiation of operators and functionals
Assume that X and Y are two Banach spaces (complete normed linear spaces) and A is some operator from XtoY. Definition 77 The operator A is called differentiable at some point ^ ^ X if there exists a linear bounded operator F^, acting from X to Y, such that A(x-f(5x) - A(x) - F,((^x) + o(||(5x||), where
when ||(5x|| —^ 0.
o(IIM)
.
(D.19)
572
Linear operators and functionals revisited
The operator F^ is called the Frechet derivative of A at x and is written as (D.20)
F, = A{^).
The expression Fx{Sx.) is called the Frechet differential of ^ ( x ) at x and is written as F^((5x) = M(x,(5x).
(D.21)
In the particular case when we have a linear operator B, its derivative is equal to the operator B itself: 5'(x) = B. Suppose now that X is a Banach space and / ( x ) is a functional in it.
Definition 78 / / there exists such linear functional F / that in some point x G X, /(x+<5x) - / ( x ) =
FUSK)
+ o(||,5x||),
(D.22)
where
the functional / ( x ) is called differentiable at the point x.
The F / is called the Frechet derivative of / ( x ) at x and is written as Fi = / ' ( x ) .
(D.23)
The expression F/(5x) is called the Frechet differential of / ( x ) at x and is written as F/(5x) = df{x,Sx)-
(D.24)
E x a m p l e 79 Let us consider the functional / ( x ) defined on the Hilbert space X: / ( x ) = ||x||2 Then ||x+5x||2 - ||x||2 = 2(x,5x) + \\6Kf, from which we have at once Fi{6x) = dfixM)
=
2{K,6X).
(D.25)
573
Concepts from variational calculus
D.3
Concepts from variational calculus
D.3.1 Variational operator In the calculus of variations it is a common practice to use 6A or (5x to denote a variation of ^ or x: M x = A(x -h ^x) - A(x) ^ F;,(5x) - M ( x , (5x).
(D.26)
The operator 6 is called the variational operator and (5A(x, 5x) is called the first variation of A. Note that, in the expression (D.19), we can take into account the second order term with respect to \\6x\\ : A{x + (5x) - A(x) = F,((5x) + ^Fi2)((5x) + o{\\6^\\^).
(D.27)
Operator Fx is the operator of the second variation (second derivative) of the operator A. It has the sense of the second order derivative of the original operator A. Similar to expression (D.27) for an operator, we can write for a functional / ( x + 5x) - / ( x ) = F/(*x) + ^HliS^)
+ o(||5xf),
(D.28)
where H[ is the so-called Hessian operator, or second variation (second derivative) of the functional / . Example 80 Let MS consider the functional g(x.) defined on the Hilbert space X: ,g(x) = ||ylx-yo||2,
(D.29)
where A is some operator from X to Hilbert space F ; y = Ax. Then 5(x + <5x) - 5(x) = (A(x + ^x) - Yo, A(x + 5x) - yo) - (^x - yo, Ax - yo) = ( L x + F , (<5x) + iFi2)(5x) + o(||6x|n - yoj , Ax + F , (,5x) +
\F^'\6^)
+ o(||,5x|n - yo
- (Ax - yo, Ax - yo) = 2 (Ax - y^, F , {6x)) + (F, (5x), F , (5x)) + (Ax - yo, Fi2)(^x)) + o(||,5xf). From the last formula we obtain the expression for the Frechet differential of the functional g: F^{6x) = 6g{x, SK) = 2(Ax - yo, F,{6K)), (D.30) and for its Hessian: //f (5x) = 2 (F, (6x), F , (5x)) + 2 (Ax - yo, Fi'\6x))
.
(D.31)
574
Linear operators and functionals revisited
Note that, similar to the basic formulae of calculus for conventional functions, we can obtain simple rules and operations of the variational calculus. Actually, the variational operator acts like a differential operator. For example, let us consider the operators ^(x),5(x), and the complete operator G(x,z). We have d{A + B) = 6A + SB,
(D.32)
6{AB) = B8A + A6B,
(D.33)
8G = G ; ( 5 X , Z) + G;(X,(5Z).
(D.34)
Using these simple rules and the properties of the inner product in the Hilbert space, one can derive, for example, the expression for the Pr^chet derivative of the functional g{-x.) determined by formula (D.29) using the following calculations: 8g = 8{Ax. - Yo, ^Ix - yo) = ( M x , ^ x - yo) + ( ^ x - yo, M x ) « (F,(.5x), ^ x - yo) + (>lx - yo, F,(<5x)) = 2 ( ^ x - yo, F,(<5x))). Thus the first variation of the functional ^(x) is equal to (5^(x, (5x) = 2(Ax - Yo, F,((5x)) ,
(D.35)
which is the same as (D.30). D.3.2 Extremum functional problems T h e o r e m 81 A differentiable functional f {x.) has an extremum at some point XQ only if the first variation of the functional at this point is equal to zero for any variation ^x O / X Q ;
(5/(xo,(5x)=0.
(D.36)
,5/(xo, \8x) = /(xo + A^x) - /(xo) - o(||<5x||).
(D.37)
Proof According to the definition,
However, the first variation (5/(xo, 6x.) is a linear functional with respect to ^x; therefore ^/(xo, A^x) = A^/(xo, ^x). (D.38) Substituting (D.38) into (D.37) we have /(xo + A5x) - /(xo) = A5/(xo, (5x) + o(||(5x||).
(D.39)
575
Concepts from variational calculus
If (5/(xo,(5x)^0,
(D.40)
the sign of the right-hand side of (D.39) is governed by the sign of <5/(xo, ^x) and A. In this case, according to (D.39), the difference /(xo +A(5x) — /(xo) can be or positive or negative according to the choice of A which means that there is no extremum at the point XQ.
Example 82 Let us find the minimum of the functional g{x.) determined in example 80: ^(x) = P x - y o f . (D.41) According to (D.35) and (D.36), we have 6g{K,6x) = 2{Ax — yo, Fx{Sx.)) = 0, for any 6K E X. Note that the Frechet derivative F^ is a linear bounded operator. Therefore we can determine the linear and hounded adjoint operator F^, which satisfies the condition (D.42) (Ax - yo, F.(<5x)) = {F:{AX - y,), 6K) = 0. Equation (D.42) holds for any (5x if and only if F:{AK
- yo) =
F:{AX)
- F:(yo) = 0.
Thus we have the following equation for the extremum point XQ: F:A{XO)
= F:(yo).
Example 83 Suppose now that operator A is a linear operator. derivative F^ is equal to operator A itself:
(D.43)
Then its Frechet
F, = A. Substituting point XQ:
(D.44)
(D.44) ^^^^ (D.43), we have the following equation for the extremum AM(xo) = A*(yo).
It is important to notice that operator A^A is a self-adjoint (symmetric) operator. By inverting the operator A*A, we finally have Xo = {A*A)-'A*{yo).
(D.45) positive (D.46)
Example 84 Let us consider the function $ (k) of the complex variable k defined by the norm of difference between two vectors in the complex Hilbert space H : #(fc) = ||x-A;y||, x , y G i f .
(D.47)
Linear operators and functionals revisited
576
We would like to find the minimum of this function. It is clear that the minimum of (k) coincides with the minimum of its square, $^ {k). The first variation of this function is equal 6^^ (k) = 6 (x—/cy, X—A:y) = — {6ky, x—/cy) — (x—fcy, Sky) = - {6ky, x-fcy) - {Sky, x-A;y)* = -Sk (y, x-A:y) - [Sk (y, x-A:y)]* = - 2 Re [Sk (y, x - / c y ) ] ,
(D.48)
where we take into account the property of the inner product in the complex Hilbert space, (x,y) = (y,x)*, and asterisk * denotes the complex conjugate. The necessary condition for the minimum of function $^ (k) is ^$2 (^) ^ _ 2 Re [Sk (y, x-fcy)] = 0 for any Sk.
(D.49)
For exam,ple, we can select Sk as follows 6k = (y, x-fcy)*.
(D.50)
Substituting (D.50) into (D.49), we obtain:
i(y, x-fcy)|' = 0, and (y, x) - (y, fcy) = (x, y)* - k* (y, y) = 0, where we use another property of the inner product in the complex Hilbert space, (x,A;y) - /c*(x,y). Therefore we have the following equation for the minimum point ko : (x,y) (D.51) (y,y)' Substituting (D.51) into (D.47), we find the corresponding minimum of the function kn =
m i n $ (fc) = $ (ko) = A/(x-A;oy, x-fcoy) = [(x,x) - (fcoy, x) - (x, k^y) + (A;oy, koy)]^'"^ = [(x, x) - (x, A;oy)* - (x, koy) + k^ko (y,
y)f^^
= [(x, x) - 2 Re [k*o (x, y)] + k^ko (y, y)]'^^ (x, x) - 2 Re
"(x,yr (y>y)
(x, y) 1/2
(y,y)
(y,y)
fx v)*
1'/'
l(x,y)r = l|x|U 1 - l|x|| ||y||
(D.52)
Appendix E SOME FORMULAE AND RULES FROM MATRIX ALGEBRA E.l
Some formulae and rules of operation on matrices
A matrix is a rectangular table of numbers or other mathematical objects, A = [Aij\, where z = 1,2, ...A^; j = 1,2, ...L. If L = A^, the matrix is square. The replacement of the rows of a matrix by the corresponding columns results in a different matrix called the transpose of the initial one: A^ = [Aj,l A square matrix is called symmetric, if the transpose matrix is equal to the original matrix: A^^^ = A. The inverse matrix A~^ of the matrix A satisfies the condition A-^A-AA-^ =1,
(E.l)
where I is the identity matrix or unit matrix. The inverse matrix is defined if and only if A is a square matrix and the determinant of A does not vanish. A matrix satisfying these conditions is called nonsingular. A square matrix whose determinant vanishes is said to be singular. Definition 85 A square matrix V is called an orthogonal matrix if V ^ V = V V ^ - I.
(E.2)
From (E.l) and (E.2) we see that for the orthogonal matrix V ^ = V-^.
(E.3)
The definition of orthogonal matrix can be expanded for the case of rectangular matrices. Let U be a rectangular N x L matrix.
578
Some formulae and rules from matrix algebra
Definition 86 We say that matrix U is column-orthogonal if U ^ U = I.
(E.4)
Definition 87 We say that matrix R is row-orthogonal if RR^ - 1 . E.2
(E.5)
Eigenvalues and eigenvectors
We assume now that A is an A^ x A^ square matrix. In this case, there is at least one non-zero N vector v such that A v =Av, (E.6) where A is a scalar. The vector v is called an eigenvector of matrix A, and the scalar A is called an eigenvalue of A. The relationship (E.6) can be written in the form of a matrix equation, ( A - A I ) v = 0,
(E.7)
from which we see that the matrix (A—AI) is singular if A is an eigenvalue of A. Note also that multiplication of an eigenvector v by any scalar a produces another eigenvector v' = a v . Any real square matrix has N (complex) eigenvalues (not necessarily different), and at most N linearly independent eigenvectors. It can be proved that all eigenvalues of a symmetric matrix A are real numbers, and A has exactly N different eigenvectors. The normalized eigenvectors of a symmetric matrix form an orthonormal set of vectors {vi,V2,...v^} : v ' ^ v = 6where 8ij is Kronecker's delta symbol 8- = I "' '^~-^' In other words, the normalized eigenvectors of a symmetric matrix form an orthonormal basis in Euclidean space Ejsj. The set of eigenvalues is called the spectrum and is denoted as A ( A ) . The maximum eigenvalue of matrix A is called the spectral radius of A: pi A) = ^
^
max
z=l,2,.iV '
\\i\. '
The symmetric matrix A is called positive definite^ if all the eigenvalues of A are positive. In this case the following inequality holds for any non-zero vector x : x^Ax > 0.
Spectral decomposition of a symmetric matrix
E.3
579
Spectral decomposition of a symmetric matrix
Let us denote as A an A/" x A'' diagonal matrix, formed by the eigenvalues of the symmetric N x N matrix A,
A-
Ai 0 ... 0 0 A2 ... 0 0 0 0 A;v
v ^ ) an N x N matrix, formed by the corresponding eigenvecand as V = (vi, V2, tors. We know that all eigenvalues of a symmetric matrix are real numbers, and its eigenvectors form an orthonormal basis in Ej^. Therefore, matrix V is an orthogonal matrix: V^=V-^
(E.8)
According to the definition of the eigenvalues and the eigenvectors, we have: Av, = A,v„ z - 1 , 2 ,
A^.
(E.9)
We can rewrite formula (E.9) using matrix notation: AV-VA.
(E.IO)
Multiplying both sides of equation (E.IO) by V ' " \ and taking into account (E.8), we obtain A = VAV^, (E.ll) or in explicit form
Yl ^*^^^?
(E.12)
Formulae (E.ll) and (E.12) present a spectral decomposition of a matrix A. Note that from equation (E.ll) it follows immediately that det (A) - det (VAV'^) = det (V) det (A) det (V^) - det (A) = A1A2
A.^v,
because V is an orthogonal matrix, det (V) = 1, and A is a diagonal matrix. The spectral decomposition can be effectively used for matrix inversion, taking into account property (E.8) of the eigenvector matrix: .-1
= (VAV^)"' = VA-^V^ - Vdiag
V^
580
E.4
Some formulae and rules from matrix algebra
Singular value decomposition (SVD)
The SVD method is based on the following theorem of linear algebra, whose proof is beyond our scope. Theorem 88 (SVD Theorem) Any {N x L) matrix A whose number of rows is greater or equal to its number of columns can be represented as the product of an N X L column-orthogonal matrix JJ, an {L x L) diagonal matrix Q with positive or zero elements:
Q - [diag(ft)] and the transpose of a square {L x L) orthogonal matrix V ; A-UQV^.
(E.13)
The decomposition (E.13) can always be done, no matter how rectangular the matrix is. The numbers {Qi} are termed the singular values of A, and are generally assumed to be ordered so that Qi > Q2 ^ > 0. If ^ is of rank r, then Qr > 0 and Qr+i = 0. In other words, the matrix ^ of a rank r has r non-zero singular values. If the matrix A is a square {N x A^) matrix, then U, Q and V are all square matrices of the same size. In this case we can easily calculate the inverse matrix: A-i = V[diag(i-)]U^.
(E.14)
If one of the Qi is equal to zero, then the matrix A is singular. If one of the Qi is very small, then the matrix A is called ill-conditioned. So, SVD gives us a clear diagnosis of the type of matrix we work with. Figure EM provides a schematic illustration of SVD decomposition of a rectangular matrix. Note that if U is an A^ x L colurnn-orthogonal matrix, then there exists an N X {N — L) matrix U' such that matrix U composed of the columns of the matrices U and U ' is orthogonal: U^U = U U ^ =: I, where U = [U U ' ] . Using an orthogonal matrix U, we can rewrite the Singular Value Decomposition of a rectangular matrix A in the form: A-UQV^,
(E.15)
Singular value decomposition (SVD)
581
Singular Value Decomposition A = UQV^
A
u
L
L
[°>J['']
A-.v[d,a.(J.)]u'
;^ 1 0
A"'
U
N
Figure E-1 Singular value decomposition of a rectangular matrix A , using a columnorthogonal matrix U and a square diagonal matrix Q.
Singular Value Decomposition A = UQV Qi 0 ... 0
u
0 02... 0 0
0 ... QL
0
0 ... 0
0
0 ... 0
V
Figure E-2 Singular value decomposition of afreet angular matrix A , using an orthogonal matrix U and a rectangular diagonal matrix Q.
Some formulae and rules from matrix algebra
582
where Q is the A/'xL diagonal matrix, formed by the singular values {Qi, (52, Qi
0 0 0 0
0
. .. Q2 . .. 0 . .. 0 . .. 0 . .. 0
0
QL}'-
0 0 0 QL
0 0
A schematic illustration of SVD formula (E.15) is given in Figure E-2. For a symmetric matrix, the singular-value decomposition can be obtained from the spectral decomposition, and Qi = \^^\•
E.5
T h e spectral Lanczos decomposition method
E. 5.1 Functions of matrices In many practical situations we have to compute a function / (A) of an N x N matrix A. A popular way of computing a matrix function is through the truncated Taylor series approximation. The conditions under which a matrix function / (A) has a Taylor series representation are given by the following theorem (Golub and Van Loan, 1996). Theorem 89 / / a function f (A) has a power series
representation,
oo
/(A) = J^afcA^ k=0
on an open disk of the complex plane with its center at the origin of the coordinates, and with a radius p^ greater or equal to the spectral radius, PQ > p ( A ) , then the function of the matrix f (A) can be represented as an infinite power series as well: /(A) = ^ a , A ^
(E.16)
fc=0
where a/c, k = 0,1, 2,3,..., are the scalar coefficients. Consider the problem of computing a vector, d = /(A)c,
(E.17)
where A is a symmetric square N x N matrix, / is a function defined on a spectral interval of A, and c, d are vectors defined in Euclidean space EN-
583
The spectral Lanczos decomposition method
Denote the eigenvalues and corresponding normalized eigenvectors of A by A^ andvi (i-l,....7V;
||vi|| = 1):
Eigenvectors v^ (i = 1, ....A^) form an orthonormal basis in Ejs/. Therefore, vector c can be represented as N
c=^Qv,.
(E.18)
Substituting (E.18) into (E.17) and taking into account expansion (E.16), we obtain: CXD
N
OO
«fc A^c = ^ ttfc A^ ^
d = f{A)c=J2 k=0
k=0
dVi
i=l
N
OO
N
OO
N
i=l
A:=0
z=l
k=0
i=l
Let us introduce a polynomial function PL{\) of degree L < N — I, which approximates the function / (A^). Then we can approximate d by the vector d/, = /;/, (A) c. The accuracy of such an approximation is: |d - dz^ll = \\d-p,
(A) C|| =
Ef=l C^ [f {X^) - PL (A,)] V, (E.20)
^jT,L^nf{^i)^M^'Thus, the approximation problem has been reduced to discrete polynomial approximation of the function / with nodes A^ and weights c^. To solve this problem it is critical to find the eigenvalues of A. E.5.2
The Lanczos method
The most appropriate technique for solving a large, symmetric eigenproblem A v = Av is delivered by the Lanczos method (Golub and Van Loan, 1996). This method involves partial tridiagonalization of the given matrix. One advantage of the Lanczos method is that the estimation of the extremal engenvalues appears even before the tridiagonalization is complete. This makes the Lanczos method extremely useful in practical applications. We will introduce first the following definition. Definition 90 The finite dimensional subspace KL of the Euclidean space EN, spanned by the vectors c, Ac, ...., A^~^c, is called a Krylov space:
Some formulae and rules from matrix algebra
584
KL = sp8in[c, Ac, ....,A^ ^c} . The Lanczos method is based on generating the orthonormal basis in Krylov space KL =span{c, Ac, ...., A ^ ' ^ c } by applying the Gram-Schmidt orthogonalization process, described in Appendix A. In matrix notations this approach is associated with the reduction of the symmetric matrix A to a tridiagonal matrix TL and also with the special properties of T^ - This reduction (called also QT decomposition) is described by the formula QiAQi = Ti, (E.21) where T/, is the tri-diagonal symmetric matrix
Tr
=
ai
/?!
...
01
01.2
02
0
0
0L-1 O^L
Q L is the matrix of the orthogonal basis of Krylov space Q L = [qi,q2,
,qL],
and Q j is the transpose of Q^. Vectors q^ of the basis are called Lanczos vectors. All Lanczos vectors are A^-vectors, consisting of A^ scalar components. It is assumed also that /?^. > 0 , j = l, 2,...., L - 1 . (E.22) We consider first the tridiagonalization process using Krylov space of dimension N : KM =span{c, Ac, ...., A^~^c} . In this case, according to the definition, the matrix QA^ is orthogonal, Q ^ = Q,^- Therefore, the reduction formula (E.21) can be rewritten as AQ;v = QivT^. (E.23) For example, equating the j - t h column of each side of (E.23), we obtain a recursive formula, /3,qj+i = Aq^. - /3^_iq,-i - a^q^- = r^-, (E.24) which holds for j = 1,...., A^, if we define 0QC\Q = Pjsf^N = 0. The orthogonality of Q^v can be written in the form qf q^ = Sij^ where 6ij is Kronecker's delta symbol. Therefore, multiplying (E.24) by qj", we obtain 0 = qjAq^- or
o^j^N = qjAq^-, j
ajiN,
1, 2,.
A^-
(E.25)
585
The spectral Lanczos decomposition method
where I^^ is the N x N identity matrix. Also, using (E.22) and the orthogonahty of QTV, we have Pj = Pj
(E.26)
= \\rj\\ •
From equation (E.24) we also obtain q^+i =f,J
= 1. 2,...., iV - 1.
(E.27)
Thus, we have formulated the Lanczos algorithm to determine aj^Tj.Pj from the given values of q^-i, q^ and /3^_i, assuming that /3oqo - 0, qi = ^ ,
l|c||
and pj > 0.
and q^+i
(E.28)
The Lanczos algorithm can be summarized as follows: Po = 1. qo = 0, qi = c / | | c | | ,
(a)
while (3^ ^ 0, q^+i = Vj/p^, a^l = qjAq^-,
(6) (E.29)
r, = (A-ajlM)
q, - /:^,_iqj-i, Pj - ||r^|| ,
j = l, 2,....,
(c)
N-1.
As a result of the Lanczos algorithm we obtain matrices QN and TAT. Note that, in general cases, we can run the Lanczos algorithm until j = L— 1, where L < N. In this case we obtain L x L matrices Qi and T/^, which can still be effectively used to evaluate the eigenvalues and eigenvectors of the original matrix A (Golub and Van Loan, 1996). Let us introduce the eigenvalues and corresponding normalized eigenvectors of Tyv by 0, and s, {i = 1, ....TV; ||s,|| - 1): T;vs^ = e^s,.
(E.30)
The scalars 0^ are called the Ritz values. Note that the eigenvectors s^ form the orthonormal system [si,
, s ^ f [si,
,s;v]= [ e ^ ,
,e^)],
(E.31)
where e,^^^ is the i - t h unit A^ vector: e,^^^ = (0,0, ...1,0, ...0). Define also the Ritz vectors v, : QATSi = Vi.
(E.32)
586
Some formulae and rules from m a t r i x algebra
Substituting (E.32) into (E.23) and taking into account (E.30), we have A Q ^ s ^ = A v i = QivTATSi = QNOiSi = OiQ^Si = OiVi,
(E.33)
from which it follows that Av, = ^,v,.
(E.34)
Thus, we see that the Ritz values and vectors (^i, v^) form the eigenpairs of the matrix A, which are called the Rayleigh-Ritz pair. Note that, using the Lanczos algorithm, we can find in a similar way the Rayleigh-Ritz pair {9i^ v^), based on matrices QL and T^, L < N, which still provide a very good approximation to the eigenvalues and vectors of A from the subspace, span(qi,q2, JQL), spanned over the Lanczos vectors. The accuracy of this approximation is controlled by the Kaniel-Paige convergence theory (Golub and Van Loan, 1996). Taking into account (E.32), (E.28) and (E.31) and the formula N
i=l
we have c = ||c|| qi = ||c|| QNe[^^ = ||c|| QN Y1Z=I ^l^^^ (E.36) = \W\\ E z ^ l ^uQNSi
= ||C|| X],^i SiiVi = ^ ^ ^ ^ QVi,
where Ci = ||c|| 5^, and sf = (s^, , sm) • Comparing this result with (E.18) and (E.19), we obtain d = / (A) c = Er=o ^fcA'^c = Er=o «fcA'^ E . = i c,v, = l|c|| T,tl
Sli E ^ O akA'^Vi = ||c|| Yltl
Sli E ^ O O-kOi^i = l|c|| E j ^ l Suf {Oi) Vi
= l|c|| E i l i siif {Oi) QwSi = ||c|| QN E f e i suf {Oi) Si. (E.37) Considering expansion (E.35) for vector e[ ', we can write by analogy with (E.18) and (E.19), / (T;v) e r = Er=o a,T%e[''^= EZo '^kT% Ef=i ^l^^i (E.38) = E«^i sii EfeoakT%Si = J2ti
5H E^O(^kOiSi = X ) " i suf {Oi) Si.
Substituting (E.38) into (E.37), we finally obtain N
d = ||c|| Q;v X ; ^ n / (Oi) Si = l|c|| QNI {TN) ef\
(E.39)
The spectral Lanczos decomposition method
587
Thus we reduce the original problem of calculating the function / of matrix A to a much smaller problem of calculating the same function of the tridiagonal matrix TN' In many practical cases this problem can easily be solved numerically. Note that we can use L steps of the Lanczos method to generate matrices QL and T L , L < N, and to introduce a natural approximation to vector d as di = ||c|| Qjr J2 siif (^^) Si = M QLI (TL) ei'-l t=i
The error estimation of this approximation can be based on the equahty
Id-df,il = J^C,/(A,)v,-||c||X^5H/(^f')v, i=l
i=l
(E.40)
This Page Intentionally Left Blank
Appendix F SOME FORMULAE AND RULES FROM TENSOR CALCULUS This appendix briefly describes the fundamental theorems of tensor calculus, which are widely used in our book. The notation closely follows the monograph of Zhdanov (1988), where one can find further details. F.l
Some formulae and rules of operation on tensor functions
We define a tensor G as the "external" or dyadic product of two vectors a and b from a Euclidean space E^ by the formula G = ab =
^x ^x
^x ^y
QjyOx
^y^y
^x ^z
dz^x
(^zby azhz
^y^z
(F.l)
where a = (a^, (^y^ CLZ) Q^nd b = (6^, by^ b^) in the given Cartesian system of coordinates. Note that for the tensors represented in form (F.l), we have the following rules of vector operations (Zhdanov, 1988) c-G = (c • a) b = b (c • a) = b (a • c ) , G c = a (b c) = (b c) a = (c b) a, c x G = [c X a ] b , G X c = a [b X c]. We can introduce a transposed tensor, G, by the formula G = ba.
(F.2)
G - c = ( c - b ) a = c-G.
(F.3)
The following identity holds true
Some formulae and rules from tensor calculus
590
Let da {a = x,y,z) stand for the unit vectors of the Cartesian system of coordinates. Then tensor Gr, can be written in the form (F.4)
G — 2_^ G^dc^,
where Ga are the vectors formed by the columns of the matrix of the tensor G in this coordinate system, G = [Gsa] =
^xx
^xy
^xz
^yx
^yy
^yz
^zx
^zy
^zz
and Ga = 2_^ Gf3ad^'
(F.5)
f3=x,y,z
We can now define the tensor differentiation operations as follows:
V.G= Y. (V-GJd,,
(F.6)
and VxG=:
^
[V X Ga]da.
(F.7)
a=x,y,z
F.2
T e n s o r s t a t e m e n t s of t h e G a u s s a n d G r e e n ' s formulae
Let G = G (r) be a tensor field differentiable continuously everywhere in the domain D up to its boundary S. The tensor statement of the Gauss theorem can be expressed by the following formula:
IIL^^'^'lh
'Gds,
(F.8)
where n is the unit vector of an outward pointing normal to S. The Green's tensor formulae are derived from expression (F.8). Indeed, let us specify an auxihary tensor field G (r) : G = Fx V x P
-f [V X F] X P ,
where F , P are arbitrary vector and tensor fields respectively, twice continuously differentiable in the domain D (up to its boundary S). Algebraic calculations show that V-G - [V X V X F] • P - F- V X V x P (F.9)
Green's tensor and vector formulae for Lame and Laplace operators
591
Substituting equation (F.9) into the Gauss tensor formula (F.8), we write the Green's tensor formula in the final form:
IB
V X V X F] • P - F- V X V x P
IL"{
Fx
V x P + fVx
]}• \dv
F] xP} ds.
(F.IO)
If the vector field F is replaced by the tensor field Q, we arrive at another Green's tensor formula: V X VxQ
P - Q
VxP
+
V x VxP VxQ
] } •\dv
x P Us.
(F.ll)
Finally, if the tensor field P in equation (F.IO) is replaced by the vector field B , we obtain the Green's vector formula: {[V X V X F] • B - [V x V x B] • F } rft; n - { F x [ V x B ] - B x [V x F ] } d s . F.3
(F.12)
G r e e n ' s t e n s o r a n d vector formulae for L a m e a n d Laplace o p e r a t o r s
In this section we derive additional tensor and vector formulations of the Green's theorem, which are useful in seismic wave theory and applications. Following Zhdanov (1988), let us specify an auxiliary tensor field F (r) : F (r) = cl [U ( V - Q ) - Q (VU)] + c^ [Ux ( v x Q ) -f (V x U) X Q ] ,
(F.13)
where U = U (r) and Q = Q (r) are arbitrary vector and tensor functions, twice continuously differentiable in the domain D (up to its boundary S). Based on the known rules of performing algebraic and differential operations on the tensors, one can find that the divergence of tensor field F (r) is determined by the following expression (Zhdanov, 1988, p. 313): V • F = U- ( e ^ V V Q - c^V X V X Q ) - ( c ^ V V U - c^V x V x U) • Q.
(F.14)
Using Lame's operator (13.28), we can express the last identity in the form: V F = U LQ-LU
Q.
(F.15)
592
Some formulae and rules from tensor calculus
Formula (F.15) is the complete tensor analog of identity (13.101) for scalar functions. Substituting identity (F.15) into Gauss's formula (F.8), we arrive at the Green's tensor formula for Lame's operator: / / / ( U • LQ - LU • n {c^ [ u ( V - Q ) -
= II
Q
(V •
U)]
Q)
dv
-f c1 [Ux ( v x
- Q x ( V x U)] } ds.
Q)
(F.16) The surface integral in formula (F.16) can be written in a more compact form if we introduce the following vector differential operator: L„ = c ^ n ( V - ) - c 2 n x [ V x ] .
(F.17)
Indeed, using the identity n-14 [u ( V - Q ) -
Q
(V •
U)]
+ cl [Ux ( v x
Q)
= U- {c^n ( V - Q ) - c^nx [ v x
Q]
- Q x (V x U)] } }
- Q • {cln ( V • U) - c^n X [V X U]} = U • P „ Q - Q P „ • U, we obtain III
( u • LQ - LU • Q ) dv = II
(u- L„Q - QLnU) ds.
(FAS)
With the aid of equation (13.10) and Hooke's law, it can be demonstrated that the operator Ln transforms the displacement vector U at the given point into the stress vector arising on an infinitely small surface element with the normal vector n. That is why this operator is called a stress operator. For Cp =^ Cs = c formula (F.16) is reduced, by (13.30), to the Green's tensor formula for the Laplace operator: / / / ^ ( U A Q - A U Q ) dv = ff
|n- [u ( V - Q ) -
Q
(V -
U)1
+ n- [Ux ( v x
Q)
+ ( V x U)
XQI
} ds.
(F.19) If the tensor field Q is replaced by the vector field Q, we obtain the ordinary Green's vector formula (Morse and Feshbach, 1953):
[ff = ff
{lJ-V^Q-V'U-Q)dv
{n- [U ( V • Q ) - Q (V - U)] + n- [Ux (V x Q) - Q x ( V x U)]} ds.
(F.20)
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Index Adjoint Frechet derivative operator for acoustic field, 471, 502 for elastic field, 522 for vector wavefield, 471, 502 Adjoint Kirchhoff operator, 492 Ampere's law, 201 Backus-Gilbert data resolution function, 86 method, 84 regularized method, 88 spread function, 86 Balance method, 371 Banach space, 540 Bayes estimation, 82 theorem, 82 Bleistein inversion, 475 one-dimensional, 478, 484 three-dimensional, 487 Born approximation for acoustic field, 450 for electromagnetic field, 245, 246 extended, 246, 247 modified, 259 for vector wavefield, 460 extended, 460 Born inversion for acoustic field, 469 iterative, 475 for electromagnetic field, 290 iterative, 296 for vector wavefield, 469 iterative, 475 Born series
for electromagnetic field, 257, 260 Boundary conditions, 363 absorbing, 363 asymptotic, 363 for electromagnetic field, 203 for wavefield, 421 Boundary-value problem for electromagnetic field, 222, 361 Boxcar functions, 268, 271 Cauchy inequality, 532 integral formula, 173 Cauchy-Rieman conditions, 172 Complex electric conductivity, 204 Conjugate directions, 138 vectors, 141 Conjugate gradient method, 137, 143 elastic field inversion, 520 electromagnetic inversion, 298 Kirchhoff inversion, 492 re-weighted, 157, 161 regularized, 148, 155 wavefield inversion, 500 with adaptive regularization, 155 with the line search, 138 Constraint equations, 202 Continuity equation, 201 Contraction integral equation, 275 operator, 275, 554 Coulomb gauge, 208 law, 201 605
606
Data kernels, 565 Debye series, 406 Decomposition theorem, 565 Deformation tensor, 397 Delta-function, 224 two-dimensional, 235 Dielectric constant, 202 Displacement vector, 396 Eigenvalue, 578 Eigenvector, 578 Eikonal equation, 407 Electrical conductivity, 202 Electromagnetic field anomalous, 232 background (normal), 232 energy, 217 quasi-static, 209 stationary, 212 two-dimensional, 213 Electromagnetic potentials, 206, 365, 376 Energy conservation law for elastic field, 425 Energy inequality for anomalous field, 233 in frequency domain, 221 in time domain, 219 Euclidean space, 531 Euler equation, 92, 114 Evanescent waves, 490 Exact element (EE) method, 384 Fermat's principle, 495 Finite element method, 380 Finite-difference method, 366 Frechet derivative, 572 of acoustic field, 448 of elastic field, 521 of electromagnetic field for 2-D models, 237 of electromagnetic field for 3-D models, 240, 299 of vector wavefield, 459
INDEX
Frechet differential, 572 of acoustic field, 448 of elastic field, 521 of electromagnetic field for 2-D models, 238 of electromagnetic field for 3-D models, 241 of vector wavefield, 459, 468 Functional parametric, 43, 44, 51, 74, 88 stabilizing, 39, 45, 47, 49, 50 Galerkin method, 267, 270, 380, 384 Gauss law, 201 tensor formula, 590 Gram matrix, 546, 558 Gram-Schmidt orthogonalization process, 560 Gravity field, 6, 169 complex intensity, 174 inversion, 177 iterative migration, 187 migration, 184 two-dimensional, 170 Green's function for electromagnetic field, 224, 235 for the wave equation, 408 Green's tensor formula for Lame's operator, 592 for Laplace operator, 592 Green's tensors electromagnetic, 224, 227 for Lame equation, 413 of elastic field, 413 of vector wavefield, 412 Helmholtz equation, 206, 208, 215, 405 theorem, 402 Hessian operator, 133, 573 Hilbert space, 541, 549 complex, 547
INDEX
Hooke's law, 399 Huygens-Fresnel principle, 417 Ill-posed problem, 5, 32, 34, 36 Initial-value problem, 420 Inner product, 532, 541, 550 Integral equation method for acoustic field, 447 for electromagnetic field, 240 two-dimensional, 236 with contraction operator, 275 for vector wavefield, 458 Kirchhoff approximation, 456 integral operator, 491 inversion, 492 Kirchhoff integral formula, 416 for an unbounded domain, 436 for Lame equation, 419 for vector wave equation, 420 Krylov space, 76, 101, 583 Lame coefficients, 399 equation, 401 operator, 401 velocities, 401 Lanczos method, 103, 583 Least-squares method linear, 61 regularized nonlinear, 149 weighted, 68 Levenberg-Marquardt method, 81 Linear functional, 536, 563 operator, 534, 555 inverse, 556 vector space, 539 Linear line search, 127, 135, 137 Localized nonlinear approximation, 252 Lorentz gauge, 208
607
lemma, 225 Magnetic charges, 211 currents, 211 field, 7, 169 complex intensity, 175 two-dimensional, 170 permeability, 202 potential complex, 175 susceptibility, 7 Magnetic potential inversion, 188 migration, 189 Magnetotelluric impedance, 314 inversion iterative Born, 315 quasi-analytical, 318 regularized smooth and focusing, 320 method, 314 Matrix data resolution, 65 generalized inverse, 64 model resolution, 67 orthogonal, 577 pseudo inverse, 64 singular, 577 Maximum likelihood method, 72 point, 70 Maxwell's equations, 201 in frequency domain, 204 separation, 205 two-dimensional, 213 Migration apparent conductivity, 340 elastic field, 526 electromagnetic in frequency domain, 335
608
in time domain, 347 iterative, 343, 357 graphical method for seismic field, 504 gravity field, 184 KirchhoflF-type, 508 magnetic potential, 189 reverse-time wave equation, 507 Stolt's, 514 Minimal residual method, 94 generahzed (GMRM), 108 Lanczos minimal residual method, 103 regularized, 116 Misfit condition, 55, 77, 154 functional, 43, 52, 63, 73, 91 weighted, 68
INDEX
Pompei formula, 173 Poynting's theorem, 216 differential form, 217 for anomalous field, 233 integral form, 217 Preconditioned matrix equation, 277 steepest descent method, 136
Newton method, 131 regularized, 146, 152 with the line search, 135 Nonlinear inversion of acoustic field, 499 of elastic field, 519 of electromagnetic field, 297 of vector wavefield, 499 Norm of the functional, 536 of the operator, 535 of the vector, 531
Quadratic line search, 127, 131 Quasi-analytical approximation for 2-D electromagnetic field, 251 for acoustic field, 452 for electromagnetic field, 249 diagonalized, 281 discrete form, 280 modified, 266 for vector wavefield, 462 Quasi-analytical series, 266 Quasi-linear approximation for acoustic field, 451 localized, 453 for electromagnetic field, 248, 261 discrete form, 278 localized, 254 for vector wavefield, 461 locahzed, 463 Quasi-linear inversion of acoustic field, 496 locahzed, 497 of electromagnetic field, 300, 301 localized, 306
Operator absolutely positively determined (APD), 97, 571 adjoint, 92, 570 contraction, 554 differentiable, 571 inverse, 556 linear, 555 positively determined (PD), 94, 571 variational, 573
Radiation conditions for elastic waves, 438 for electromagnetic field, 216 for scalar wavefield, 427, 429, 433 Rays, 406 Reciprocity relations, 225, 228, 458 theorem, 447 Refiectivity coefficient, 450-452, 461
609
INDEX
electrical, 248 localized, 453 function, 479, 486, 491 bandlimited, 483 tensor, 461 electrical, 247 localized, 463 Regularization adaptive, 117, 152, 154, 159 algorithm, 36 method, 74, 82, 84 parameter, 36, 52, 55, 77, 115, 116, 154 theory, 32 Regularizing operator, 36, 37 Retarded wave, 417 Riesz representation theorem, 564 Ritz value, 585 vector, 585 Schwarz inequality, 94, 97, 541 Sensitivity of acoustic data, 449 integrated, 449 of electromagnetic data, 238, 241 integrated, 238, 243 of geophysical method, 30 of the vector wavefield data, 459 integrated, 459 Singular value decomposition, 580 Sobolev space, 550 Sommerfeld radiation conditions, 216, 427 Staggered grid, 367 Steepest descent direction, 125 method, 121, 125, 130 preconditioned, 136 regularized, 145, 151 Stress operator, 592
tensor, 398 Summation convention, 397 Telegraph equations, 203 Transport equation, 407 Traveltime, 406 inversion, 494 Triangle inequality, 531, 538 Uniqueness theorem for electromagnetic inverse problem, 20 for gravity inverse problem, 20 for seismic inverse problem, 22 for the solution of electromagnetic field equations, 222 for the solution of wavefield equations, 425 for the wave propagation problem, 429 Wave equation acoustic, 405, 406 electromagnetic, 210 vector wavefield, 401 Wave number, 205, 210 Wavefronts, 406 Wavepacket, 406 Waves acoustic, 405 compressional, 404 electromagnetic, 210 shear, 404 Well-posed problem, 32 conditionally well-posed, 33 WKBJ method for Green's function approximation, 411
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