Methods in Geochemistry and Geophysics, 36

GEOPHYSICAL INVERSE THEORYAND REGULARIZATION PROBLEMS

Michael S. ZHDANOVUniversity of UtahSalt Lake CityUTAH, U.S.A.

2OO2

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Contents

Preface XIX

I Introduction to Inversion Theory 1

1 Forward and inverse problems in geophysics 31.1 Formulation of forward and inverse problems for different geophysical

fields 31.1.1 Gravity field 61.1.2 Magnetic field 71.1.3 Electromagnetic field . 91.1.4 Seismic wavefield 14

1.2 Existence and uniqueness of the inverse problem solutions 161.2.1 Existence of the solution 161.2.2 Uniqueness of the solution 171.2.3 Practical uniqueness 23

1.3 Instability of the inverse problem solution 24

2 Ill-posed problems and the methods of their solution 292.1 Sensitivity and resolution of geophysical methods 29

2.1.1 Formulation of the inverse problem in general mathematicalspaces 29

2.1.2 Sensitivity 302.1.3 Resolution 31

2.2 Formulation of well-posed and ill-posed problems 322.2.1 Well-posed problems 322.2.2 Conditionally well-posed problems 332.2.3 Quasi-solution of the ill-posed problem 34

2.3 Foundations of regularization methods of inverse problem solution . . 362.3.1 Regularizing operators 362.3.2 Stabilizing functionals 392.3.3 Tikhonov parametric functional 42

VIII CONTENTS

2.4 Family of stabilizing functionals 452.4.1 Stabilizing functionals revisited 452.4.2 Representation of a stabilizing functional in the form of a pseudo-

quadratic functional 502.5 Definition of the regularization parameter 52

2.5.1 Optimal regularization parameter selection 522.5.2 L-curve method of regularization parameter selection 55

II Methods of the Solution of Inverse Problems 59

3 Linear discrete inverse problems 613.1 Linear least-squares inversion 61

3.1.1 The linear discrete inverse problem 613.1.2 Systems of linear equations and their general solutions . . . . 623.1.3 The data resolution matrix 64

3.2 Solution of the purely underdetermined problem 663.2.1 Underdetermined system of linear equations 663.2.2 The model resolution matrix 67

3.3 Weighted least-squares method 683.4 Applying the principles of probability theory to a linear inverse problem 69

3.4.1 Some formulae and notations from probability theory 693.4.2 Maximum likelihood method 713.4.3 Chi-square fitting 73

3.5 Regularization methods 743.5.1 The Tikhonov regularization method 743.5.2 Application of SLDM method in regularized linear inverse prob-

lem solution ;. 753.5.3 Definition of the weighting matrices for the model parameters

and data 773.5.4 Approximate regularized solution of the linear inverse problem 793.5.5 The Levenberg - Marquardt method 813.5.6 The maximum a posteriori estimation method (the Bayes esti-

mation) "823.6 The Backus-Gilbert Method 84

3.6.1 The data resolution function 843.6.2 The spread function 863.6.3 Regularized solution in the Backus-Gilbert method 88

4 Iterative solutions of the linear inverse problem 914.1 Linear operator equations and their solution by iterative methods . . 91

4.1.1 Linear inverse problems and the Euler equation 914.1.2 The minimal residual method 93

CONTENTS IX

4.1.3 Linear inverse problem solution using MRM 994.2 A generalized minimal residual method 101

4.2.1 The Krylov-subspace method 1014.2.2 The Lanczos minimal residual method 1034.2.3 The generalized minimal residual method 1084.2.4 A linear inverse problem solution using generalized MRM . . . 112

4.3 The regularization method in a linear inverse problem solution . . . . 1134.3.1 The Euler equation for the Tikhonov parametric functional . . 1134.3.2 MRM solution of the Euler equation 1154.3.3 Generalized MRM solutions of the Euler equation for the para-

metric functional 117

5 Nonlinear inversion technique 1215.1 Gradient-type methods 121

5.1.1 Method of steepest descent 1215.1.2 The Newton method 1315.1.3 The conjugate gradient method 137

5.2 Regularized gradient-type methods in the solution of nonlinear inverseproblems 1435.2.1 Regularized steepest descent 1435.2.2 The regularized Newton method 1455.2.3 Approximate regularized solution of the nonlinear inverse prob-

lem ... 1475.2.4 The regularized preconditioned steepest descent method . . . 1475.2.5 The regularized conjugate gradient method 148

5.3 Regularized solution of a nonlinear discrete inverse problem 1495.3.1 Nonlinear least-squares inversion 1495.3.2 The steepest descent method for nonlinear regularized least-

squares inversion 1505.3.3 The Newton method for nonlinear regularized least-squares in-

version 1515.3.4 Numerical schemes of the Newton method for nonlinear regu-

larized least-squares inversion 1525.3.5 Nonlinear least-squares inversion by the conj ugate gradient method 1535.3.6 The numerical scheme of the regularized conjugate gradient

method for nonlinear least-squares inversion 1535.4 Conjugate gradient re-weighted optimization 155

5.4.1 The Tikhonov parametric functional with a pseudo-quadraticstabilizer 155

5.4.2 Re-weighted conjugate gradient method 1575.4.3 Minimization in the space of weighted parameters 160

X CONTENTS

5.4.4 The re-weighted regularized conjugate gradient (RCG) methodin the space of weighted parameters 161

III Geopotential Field Inversion 167

6 Integral representations in forward modeling of gravity and mag-netic fields 1696.1 Basic equations for gravity and magnetic fields 169

6.1.1 Gravity and magnetic fields in three dimensions 1696.1.2 Two-dimensional models of gravity and magnetic fields . . . . 170

6.2 Integral representations of potential fields based on the theory of func-tions of a complex variable 1716.2.1 Complex intensity of a plane potential field 1716.2.2 Complex intensity of a gravity field 1746.2.3 Complex intensity and potential of a magnetic field 175

7 Integral representations in inversion of gravity and magnetic data 1777.1 Gradient methods of gravity inversion 177

7.1.1 Steepest ascent direction of the misfit functional for the gravityinverse problem 177

7.1.2 Application of the re-weighted conjugate gradient method . . 1797.2 Gravity field migration 181

7.2.1 Physical interpretation of the adjoint gravity operator 1817.2.2 Gravity field migration in the solution of the inverse problem . 1847.2.3 Iterative gravity migration 186

7.3 Gradient methods of magnetic anomaly inversion 1887.3.1 Magnetic potential inversion .: . 1887.3.2 Magnetic potential migration •. 189

7.4 Numerical methods in forward and inverse modeling 1907.4.1 Discrete forms of 3-D gravity and magnetic forward modeling

operators 1907.4.2 Discrete form of 2-D forward modeling operator 1937.4.3 Regularized inversion of gravity data 193

IV Electromagnetic Inversion 199

8 Foundations of electromagnetic theory 2018.1 Electromagnetic field equations 201

8.1.1 Maxwell's equations 2018.1.2 Field in homogeneous domains of a medium 2028.1.3 Boundary conditions 203

CONTENTS XI

8.1.4 Field equations in the frequency domain 2048.1.5 Quasi-static (quasi-stationary) electromagnetic field 2098.1.6 Field wave equations 2108.1.7 Field equations allowing for magnetic currents and charges . . 2118.1.8 Stationary electromagnetic field 2128.1.9 Fields in two-dimensional inhomogeneous media and the con-

cepts of E- and //-polarization 2138.2 Electromagnetic energy flow 215

8.2.1 Radiation conditions 2168.2.2 Poynting's theorem in the time domain 2168.2.3 Energy inequality in the time domain 2188.2.4 Poynting's theorem in the frequency domain 220

8.3 Uniqueness of the solution of electromagnetic field equations 2228.3.1 Boundary-value problem 2228.3.2 Uniqueness theorem for the unbounded domain 223

8.4 Electromagnetic Green's tensors 2248.4.1 Green's tensors in the frequency domain 2248.4.2 Lorentz lemma and reciprocity relations 2258.4.3 Green's tensors in the time domain 227

Integral representations in electromagnetic forward modeling 2319.1 Integral equation method 231

9.1.1 Background (normal) and anomalous parts of the electromag-netic field V " 231

9.1.2 Poynting's theorem and energy inequality for an anomalous field2339.1.3 Integral equation method in two dimensions 2349.1.4 Calculation of the first variation (Frechet derivative) of the elec-

tromagnetic field for 2-D models 2379.1.5 Integral equation method in three dimensions 2399.1.6 Calculation of the first variation (Frechet derivative) of the elec-

tromagnetic field for 3-D models 2409.1.7 Frechet derivative calculation using the differential method . . 243

9.2 Family of linear and nonlinear integral approximations of the electro-magnetic field 2459.2.1 Born and extended Born approximations 2469.2.2 Quasi-linear approximation and tensor quasi-linear equation . 2479.2.3 Quasi-analytical solutions for a 3-D electromagnetic field . . . 2489.2.4 Quasi-analytical solutions for 2-D electromagnetic field . . . . 2519.2.5 Localized nonlinear approximation 2529.2.6 Localized quasi-linear approximation 253

9.3 Linear and non-linear approximations of higher orders 2569.3.1 Born series 256

XII CONTENTS

9.3.2 Modified Green's operator 2579.3.3 Modified Born series 2599.3.4 Quasi-linear approximation of the modified Green's operator . 2619.3.5 QL series 2639.3.6 Accuracy estimation of the QL approximation of the first and

higher orders 2639.3.7 QA series 266

9.4 Integral representations in numerical dressing 2679.4.1 Discretization of the model parameters 2679.4.2 Galerkin method for electromagnetic field discretization . . . . 2699.4.3 Discrete form of electromagnetic integral equations based on

boxcar basis functions 2719.4.4 Contraction integral equation method 2759.4.5 Contraction integral equation as the preconditioned conven-

tional integral equation 2769.4.6 Matrix form of Born approximation 2789.4.7 Matrix form of quasi-linear approximation 2789.4.8 Matrix form of quasi-analytical approximation 2809.4.9 The diagonalized quasi-analytical (DQA) approximation . . . 281

10 Integral representations in electromagnetic inversion 28710.1 Linear inversion methods 288

10.1.1 Excess (anomalous) current inversion 28810.1.2 Born inversion 29010.1.3 Conductivity imaging by the Born approximation 29210.1.4 Iterative Born inversions 296

10.2 Nonlinear inversion 29710.2.1 Formulation of the nonlinear inverse problem '. . 29710.2.2 Frechet derivative calculation 298

10.3 Quasi-linear inversion 30010.3.1 Principles of quasi-linear inversion 30010.3.2 Quasi-linear inversion in matrix notations 30110.3.3 Localized quasi-linear inversion 306

10.4 Quasi-analytical inversion 31110.4.1 Frechet derivative calculation 31110.4.2 Inversion based on the quasi-analytical method 312

10.5 Magnetotelluric (MT) data inversion 31410.5.1 Iterative Born inversion of magnetotelluric data 31510.5.2 DQA approximation in magnetotelluric inverse problem . . . . 31710.5.3 Frechet derivative matrix with respect to the logarithm of the

total conductivity 31910.5.4 Regularized smooth and focusing inversion of MT data . . . . 320

CONTENTS XIII

10.5.5 Example of synthetic 3-D MT data inversion 32110.5.6 Case study: inversion of the Minamikayabe area data 324

11 Electromagnetic migration imaging 33111.1 Electromagnetic migration in the frequency domain 332

11.1.1 Formulation of the electromagnetic inverse problem as a mini-mization of the energy flow functional 332

11.1.2 Integral representations for electromagnetic migration field . . 33511.1.3 Gradient direction of the energy flow functional 33611.1.4 Migration imaging in the frequency domain 33811.1.5 Iterative migration 343

11.2 Electromagnetic migration in the time domain 34411.2.1 Time domain electromagnetic migration as the solution of the

boundary value problem 345

11.2.2 Minimization of the residual electromagnetic field energy flow 35111.2.3 Gradient direction of the energy flow functional in the time

domain 35311.2.4 Migration imaging in the time domain 35411.2.5 Iterative migration in the time domain 357

12 Differential me thods in electromagnet ic model ing and inversion 36112.1 Electromagnetic modeling as a boundary-value problem 361

12.1.1 Field equations and boundary conditions 361

12.1.2 Electromagnetic potential equations and boundary conditions 36512.2 Finite difference approximation of the boundary-value problem . . . . 366

12.2.1 Discretization of Maxwell's equations using a staggered grid . 36712.2.2 Discretization of the second order differential equations using

the balance method 37112.2.3 Discretization of the electromagnetic potential differential equa-

tions 37612.2.4 Application of the spectral Lanczos decomposition method (SLDM)

for solving the linear system of equations for discrete electro-magnetic fields 379

12.3 Finite element solution of boundary-value problems 38012.3.1 Galerkin method 380

12.3.2 Exact element method 384

12.4 Inversion based on differential methods 38512.4.1 Formulation of the inverse problem on the discrete grid . . . . 38512.4.2 Frechet derivative calculation using finite difference methods . 386

XIV CONTENTS

V Seismic Inversion 393

13 Wavefield equations 39513.1 Basic equations of elastic waves 395

13.1.1 Deformation of an elastic body; deformation and stress tensors 39513.1.2 Hooke's law 39913.1.3 Dynamic equations of elasticity theory for a homogeneous isotropic

medium 40013.1.4 Compressional and shear waves 40213.1.5 Acoustic waves and scalar wave equation 40513.1.6 High frequency approximations in the solution of an acoustic

wave equation 40513.2 Green's functions for wavefield equations 407

13.2.1 Green's functions for the scalar wave equation and for the cor-responding Helmholtz equation 407

13.2.2 High frequency (WKBJ) approximation for the Green's function 41013.2.3 Green's tensor for vector wave equation 41113.2.4 Green's tensor for the Lame equation 413

13.3 Kirchhoff integral formula and its analogs 41413.3.1 Kirchhoff integral formula 41513.3.2 Generalized Kirchhoff integral formulae for the Lame equation

and the vector wave equation 41713.4 Uniqueness of the solution of the wavefield equations 420

13.4.1 Initial-value problems -.- 42013.4.2 Energy'conservation law 42213.4.3 Uniqueness of the solution of initial-value problems 42513.4.4 Sommerfeld radiation conditions . 42613.4.5 Uniqueness of the solution of the wave propagation problem

based on radiation conditions 42913.4.6 Kirchhoff formula for an unbounded domain 43413.4.7 Radiation conditions for elastic waves 437

14 Integral representations in wavefield theory 44314.1 Integral equation method in acoustic wavefield analysis 443

14.1.1 Separation of the acoustic wavefield into incident and scattered(background and anomalous) parts 443

14.1.2 Integral equation for the acoustic wavefield 44514.1.3 Reciprocity theorem 44714.1.4 Calculation of the first variation (Frechet derivative) of the

acoustic wavefield 44814.2 Integral approximations of the acoustic wavefield 449

14.2.1 Born approximation 449

CONTENTS XV

14.2.2 Quasi-linear approximation 45014.2.3 Quasi-analytical approximation . . . 45114.2.4 Localized quasi-linear approximation 45214.2.5 Kirchhoff approximation 453

14.3 Method of integral equations in vector wavefield analysis 45614.3.1 Vector wavefield separation 45614.3.2 Integral equation method for the vector wavefield 45714.3.3 Calculation of the first variation (Frechet derivative) of the vec-

tor wavefield 45814.4 Integral approximations of the vector wavefield 460

14.4.1 Born type approximations 46014.4.2 Quasi-linear approximation 46114.4.3 Quasi-analytical solutions for the vector wavefield 46114.4.4 Localized quasi-linear approximation 462

15 Integral representations in wavefield inversion 46715.1 Linear inversion methods 467

15.1.1 Born inversion of acoustic and vector wavefields 46815.1.2 Wavefield imaging by the Born approximations 47015.1.3 Iterative Born inversions of the wavefield 47515.1.4 Bleistein inversion 47515.1.5 Inversion based on the Kirchhoff approximation . . . . . . . . 49115.1.6 Traveltime inverse problem 494

15.2 Quasi-linear inversion . . . - . - • 49615.2.1 Quasi-linear inversion of the acoustic wavefield 49615.2.2 Localized quasi-linear inversion based on the Bleistein method 497

15.3 Nonlinear inversion 49915.3.1 Formulation of the nonlinear wavefield inverse problem . . . . 49915.3.2 Frechet derivative operators for wavefield problems 500

15.4 Principles of wavefield migration 50315.4.1 Geometrical model of migration transformation 50315.4.2 Kirchhoff integral formula for reverse-time wave equation mi-

gration 50715.4.3 Rayleigh integral 51015.4.4 Migration in the spectral domain (Stolt's method) 51415.4.5 Equivalence of the spectral and integral migration algorithms . 51615.4.6 Inversion versus migration 517

15.5 Elastic field inversion 51815.5.1 Formulation of the elastic field inverse problem 51815.5.2 Frechet derivative for the elastic forward modeling operator . . 52015.5.3 Adjoint Frechet derivative operator and back-propagating elas-

tic field 522

XVI CONTENTS

A Functional spaces of geophysical models and data 531A.I Euclidean Space 531

A. 1.1 Vector operations in Euclidean space 531A.1.2 Linear transformations (operators) in Euclidean space 534A.I.3 Norm of the operator 534A. 1.4 Linear functionals 536A.1.5 Norm of the functional 536

A.2 Metric space 537A.2.1 Definition of metric space 537A.2.2 Convergence, Cauchy sequences and completeness 538

A.3 Linear vector spaces 539A.3.1 Vector operations 539A.3.2 Normed linear spaces 540

A.4 Hilbert spaces 541A.4.1 Inner product 541A.4.2 Approximation problem in Hilbert space 544

A.5 Complex Euclidean and Hilbert spaces 546A.5.1 Complex Euclidean space 546A.5.2 Complex Hilbert space 547

A.6 Examples of linear vector spaces 547

B Operators in the spaces of models and data .. 553B.I Operators in functional spaces 553B.2 Linear operators 555B.3 Inverse operators . 556B.4 Some approximation problems in the Hilbert spaces of geophysical data 557B.5 Gram - Schmidt orthogonalization process 559

C Functionals in the spaces of geophysical models 563C.I Functionals and their norms 563C.2 Riesz representation theorem 564C.3 Functional representation of geophysical data and an inverse problem 565

D Linear operators and functionals revisited 569D.I Adjoint operators 569D.2 Differentiation of operators and functionals 571D.3 Concepts from variational calculus 573

D.3.1 Variational operator 573D.3.2 Extremum functional problems 574

CONTENTS XVII

E Some formulae and rules from matr ix algebra 577E.I Some formulae and rules of operation on matrices 577E.2 Eigenvalues and eigenvectors 578E.3 Spectral decomposition of a symmetric matrix 579E.4 Singular value decomposition (SVD) 580E.5 The spectral Lanczos decomposition method 582

E.5.1 Functions of matrices 582E.5.2 The Lanczos method 583

F Some formulae and rules from tensor calculus 589F.I Some formulae and rules of operation on tensor functions 589F.2 Tensor statements of the Gauss and Green's formulae 590F.3 Green's tensor and vector formulae for Lame and Laplace operators . 591

Bibliography 593

Index 604