GEOPHYSICAL INVERSE THEORY AND REGULARIZATION PROBLEMS

Methods in Geochemistry and Geophysics, 36 GEOPHYSICAL INVERSE THEORY AND REGULARIZATION PROBLEMS Michael S. ZHDANOV University of Utah Salt Lake City UTAH, U.S.A. 2OO2 ELSEVIER Amsterdam - Boston - London - New York - Oxford - Paris - Tokyo San Diego - San Francisco - Singapore - Sydney

Contents Preface XIX I Introduction to Inversion Theory 1 1 Forward and inverse problems in geophysics 3 1.1 Formulation of forward and inverse problems for different geophysical fields 3 1.1.1 Gravity field 6 1.1.2 Magnetic field 7 1.1.3 Electromagnetic field. 9 1.1.4 Seismic wavefield 14 1.2 Existence and uniqueness of the inverse problem solutions 16 1.2.1 Existence of the solution 16 1.2.2 Uniqueness of the solution 17 1.2.3 Practical uniqueness 23 1.3 Instability of the inverse problem solution 24 2 Ill-posed problems and the methods of their solution 29 2.1 Sensitivity and resolution of geophysical methods 29 2.1.1 Formulation of the inverse problem in general mathematical spaces 29 2.1.2 Sensitivity 30 2.1.3 Resolution 31 2.2 Formulation of well-posed and ill-posed problems 32 2.2.1 Well-posed problems 32 2.2.2 Conditionally well-posed problems 33 2.2.3 Quasi-solution of the ill-posed problem 34 2.3 Foundations of regularization methods of inverse problem solution.. 36 2.3.1 Regularizing operators 36 2.3.2 Stabilizing functionals 39 2.3.3 Tikhonov parametric functional 42

VIII CONTENTS 2.4 Family of stabilizing functionals 45 2.4.1 Stabilizing functionals revisited 45 2.4.2 Representation of a stabilizing functional in the form of a pseudoquadratic functional 50 2.5 Definition of the regularization parameter 52 2.5.1 Optimal regularization parameter selection 52 2.5.2 L-curve method of regularization parameter selection 55 II Methods of the Solution of Inverse Problems 59 3 Linear discrete inverse problems 61 3.1 Linear least-squares inversion 61 3.1.1 The linear discrete inverse problem 61 3.1.2 Systems of linear equations and their general solutions... 62 3.1.3 The data resolution matrix 64 3.2 Solution of the purely underdetermined problem 66 3.2.1 Underdetermined system of linear equations 66 3.2.2 The model resolution matrix 67 3.3 Weighted least-squares method 68 3.4 Applying the principles of probability theory to a linear inverse problem 69 3.4.1 Some formulae and notations from probability theory 69 3.4.2 Maximum likelihood method 71 3.4.3 Chi-square fitting 73 3.5 Regularization methods 74 3.5.1 The Tikhonov regularization method 74 3.5.2 Application of SLDM method in regularized linear inverse problem solution ;. 75 3.5.3 Definition of the weighting matrices for the model parameters and data 77 3.5.4 Approximate regularized solution of the linear inverse problem 79 3.5.5 The Levenberg - Marquardt method 81 3.5.6 The maximum a posteriori estimation method (the Bayes estimation) "82 3.6 The Backus-Gilbert Method 84 3.6.1 The data resolution function 84 3.6.2 The spread function 86 3.6.3 Regularized solution in the Backus-Gilbert method 88 4 Iterative solutions of the linear inverse problem 91 4.1 Linear operator equations and their solution by iterative methods.. 91 4.1.1 Linear inverse problems and the Euler equation 91 4.1.2 The minimal residual method 93

CONTENTS IX 4.1.3 Linear inverse problem solution using MRM 99 4.2 A generalized minimal residual method 101 4.2.1 The Krylov-subspace method 101 4.2.2 The Lanczos minimal residual method 103 4.2.3 The generalized minimal residual method 108 4.2.4 A linear inverse problem solution using generalized MRM... 112 4.3 The regularization method in a linear inverse problem solution... 113 4.3.1 The Euler equation for the Tikhonov parametric functional.. 113 4.3.2 MRM solution of the Euler equation 115 4.3.3 Generalized MRM solutions of the Euler equation for the parametric functional 117 5 Nonlinear inversion technique 121 5.1 Gradient-type methods 121 5.1.1 Method of steepest descent 121 5.1.2 The Newton method 131 5.1.3 The conjugate gradient method 137 5.2 Regularized gradient-type methods in the solution of nonlinear inverse problems 143 5.2.1 Regularized steepest descent 143 5.2.2 The regularized Newton method 145 5.2.3 Approximate regularized solution of the nonlinear inverse problem... 147 5.2.4 The regularized preconditioned steepest descent method... 147 5.2.5 The regularized conjugate gradient method 148 5.3 Regularized solution of a nonlinear discrete inverse problem 149 5.3.1 Nonlinear least-squares inversion 149 5.3.2 The steepest descent method for nonlinear regularized leastsquares inversion 150 5.3.3 The Newton method for nonlinear regularized least-squares inversion 151 5.3.4 Numerical schemes of the Newton method for nonlinear regularized least-squares inversion 152 5.3.5 Nonlinear least-squares inversion by the conj ugate gradient method 153 5.3.6 The numerical scheme of the regularized conjugate gradient method for nonlinear least-squares inversion 153 5.4 Conjugate gradient re-weighted optimization 155 5.4.1 The Tikhonov parametric functional with a pseudo-quadratic stabilizer 155 5.4.2 Re-weighted conjugate gradient method 157 5.4.3 Minimization in the space of weighted parameters 160

X CONTENTS 5.4.4 The re-weighted regularized conjugate gradient (RCG) method in the space of weighted parameters 161 III Geopotential Field Inversion 167 6 Integral representations in forward modeling of gravity and magnetic fields 169 6.1 Basic equations for gravity and magnetic fields 169 6.1.1 Gravity and magnetic fields in three dimensions 169 6.1.2 Two-dimensional models of gravity and magnetic fields... 170 6.2 Integral representations of potential fields based on the theory of functions of a complex variable 171 6.2.1 Complex intensity of a plane potential field 171 6.2.2 Complex intensity of a gravity field 174 6.2.3 Complex intensity and potential of a magnetic field 175 7 Integral representations in inversion of gravity and magnetic data 177 7.1 Gradient methods of gravity inversion 177 7.1.1 Steepest ascent direction of the misfit functional for the gravity inverse problem 177 7.1.2 Application of the re-weighted conjugate gradient method.. 179 7.2 Gravity field migration 181 7.2.1 Physical interpretation of the adjoint gravity operator 181 7.2.2 Gravity field migration in the solution of the inverse problem. 184 7.2.3 Iterative gravity migration 186 7.3 Gradient methods of magnetic anomaly inversion 188 7.3.1 Magnetic potential inversion.:. 188 7.3.2 Magnetic potential migration. 189 7.4 Numerical methods in forward and inverse modeling 190 7.4.1 Discrete forms of 3-D gravity and magnetic forward modeling operators 190 7.4.2 Discrete form of 2-D forward modeling operator 193 7.4.3 Regularized inversion of gravity data 193 IV Electromagnetic Inversion 199 8 Foundations of electromagnetic theory 201 8.1 Electromagnetic field equations 201 8.1.1 Maxwell's equations 201 8.1.2 Field in homogeneous domains of a medium 202 8.1.3 Boundary conditions 203

CONTENTS XI 8.1.4 Field equations in the frequency domain 204 8.1.5 Quasi-static (quasi-stationary) electromagnetic field 209 8.1.6 Field wave equations 210 8.1.7 Field equations allowing for magnetic currents and charges.. 211 8.1.8 Stationary electromagnetic field 212 8.1.9 Fields in two-dimensional inhomogeneous media and the concepts of E- and //-polarization 213 8.2 Electromagnetic energy flow 215 8.2.1 Radiation conditions 216 8.2.2 Poynting's theorem in the time domain 216 8.2.3 Energy inequality in the time domain 218 8.2.4 Poynting's theorem in the frequency domain 220 8.3 Uniqueness of the solution of electromagnetic field equations 222 8.3.1 Boundary-value problem 222 8.3.2 Uniqueness theorem for the unbounded domain 223 8.4 Electromagnetic Green's tensors 224 8.4.1 Green's tensors in the frequency domain 224 8.4.2 Lorentz lemma and reciprocity relations 225 8.4.3 Green's tensors in the time domain 227 Integral representations in electromagnetic forward modeling 231 9.1 Integral equation method 231 9.1.1 Background (normal) and anomalous parts of the electromagnetic field V " 231 9.1.2 Poynting's theorem and energy inequality for an anomalous field233 9.1.3 Integral equation method in two dimensions 234 9.1.4 Calculation of the first variation (Frechet derivative) of the electromagnetic field for 2-D models 237 9.1.5 Integral equation method in three dimensions 239 9.1.6 Calculation of the first variation (Frechet derivative) of the electromagnetic field for 3-D models 240 9.1.7 Frechet derivative calculation using the differential method.. 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 CONTENTS 9.3.2 Modified Green's operator 257 9.3.3 Modified Born series 259 9.3.4 Quasi-linear approximation of the modified Green's operator. 261 9.3.5 QL series 263 9.3.6 Accuracy estimation of the QL approximation of the first and higher orders 263 9.3.7 QA series 266 9.4 Integral representations in numerical dressing 267 9.4.1 Discretization of the model parameters 267 9.4.2 Galerkin method for electromagnetic field discretization... 269 9.4.3 Discrete form of electromagnetic integral equations based on boxcar basis functions 271 9.4.4 Contraction integral equation method 275 9.4.5 Contraction integral equation as the preconditioned conventional integral equation 276 9.4.6 Matrix form of Born approximation 278 9.4.7 Matrix form of quasi-linear approximation 278 9.4.8 Matrix form of quasi-analytical approximation 280 9.4.9 The diagonalized quasi-analytical (DQA) approximation... 281 10 Integral representations in electromagnetic inversion 287 10.1 Linear inversion methods 288 10.1.1 Excess (anomalous) current inversion 288 10.1.2 Born inversion 290 10.1.3 Conductivity imaging by the Born approximation 292 10.1.4 Iterative Born inversions 296 10.2 Nonlinear inversion 297 10.2.1 Formulation of the nonlinear inverse problem '.. 297 10.2.2 Frechet 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 matrix notations 301 10.3.3 Localized quasi-linear inversion 306 10.4 Quasi-analytical inversion 311 10.4.1 Frechet derivative calculation 311 10.4.2 Inversion based on the quasi-analytical method 312 10.5 Magnetotelluric (MT) data inversion 314 10.5.1 Iterative Born inversion of magnetotelluric data 315 10.5.2 DQA approximation in magnetotelluric inverse problem... 317 10.5.3 Frechet derivative matrix with respect to the logarithm of the total conductivity 319 10.5.4 Regularized smooth and focusing inversion of MT data... 320

CONTENTS XIII 10.5.5 Example of synthetic 3-D MT data inversion 321 10.5.6 Case study: inversion of the Minamikayabe area data 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 338 11.1.5 Iterative migration 343 11.2 Electromagnetic migration in the time domain 344 11.2.1 Time domain electromagnetic migration as the solution of the boundary value problem 345 11.2.2 Minimization of the residual electromagnetic field energy flow 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 boundary conditions 365 12.2 Finite difference approximation of the boundary-value problem... 366 12.2.1 Discretization of Maxwell's equations using a staggered grid. 367 12.2.2 Discretization of the second order differential equations using the balance method 371 12.2.3 Discretization of the electromagnetic potential differential equations 376 12.2.4 Application of the spectral Lanczos decomposition method (SLDM) for solving the linear system of equations for discrete electromagnetic fields 379 12.3 Finite element solution of boundary-value problems 380 12.3.1 Galerkin method 380 12.3.2 Exact element method 384 12.4 Inversion based on differential methods 385 12.4.1 Formulation of the inverse problem on the discrete grid... 385 12.4.2 Frechet derivative calculation using finite difference methods. 386

XIV CONTENTS V Seismic Inversion 393 13 Wavefield 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 (WKBJ) 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 Lame 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 Lame 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 propagation problem based on radiation conditions 429 13.4.6 Kirchhoff formula for an unbounded domain 434 13.4.7 Radiation conditions for elastic waves 437 14 Integral representations in wavefield theory 443 14.1 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 445 14.1.3 Reciprocity theorem 447 14.1.4 Calculation of the first variation (Frechet derivative) of the acoustic wavefield 448 14.2 Integral approximations of the acoustic wavefield 449 14.2.1 Born approximation 449

CONTENTS XV 14.2.2 Quasi-linear approximation 450 14.2.3 Quasi-analytical approximation... 451 14.2.4 Localized quasi-linear approximation 452 14.2.5 Kirchhoff approximation 453 14.3 Method of integral equations in vector wavefield analysis 456 14.3.1 Vector wavefield separation 456 14.3.2 Integral equation method for the vector wavefield 457 14.3.3 Calculation of the first variation (Frechet derivative) of the vector wavefield 458 14.4 Integral approximations of the vector wavefield 460 14.4.1 Born type approximations 460 14.4.2 Quasi-linear approximation 461 14.4.3 Quasi-analytical solutions for the vector wavefield 461 14.4.4 Localized quasi-linear approximation 462 15 Integral representations in wavefield inversion 467 15.1 Linear inversion methods 467 15.1.1 Born inversion of acoustic and vector wavefields 468 15.1.2 Wavefield imaging by the Born approximations 470 15.1.3 Iterative Born inversions of the wavefield 475 15.1.4 Bleistein inversion 475 15.1.5 Inversion based on the Kirchhoff approximation... 491 15.1.6 Traveltime inverse problem 494 15.2 Quasi-linear inversion...-.- 496 15.2.1 Quasi-linear inversion of the acoustic wavefield 496 15.2.2 Localized quasi-linear inversion based on the Bleistein method 497 15.3 Nonlinear inversion 499 15.3.1 Formulation of the nonlinear wavefield inverse problem... 499 15.3.2 Frechet derivative operators for wavefield problems 500 15.4 Principles of wavefield migration 503 15.4.1 Geometrical model of migration transformation 503 15.4.2 Kirchhoff integral formula for reverse-time wave equation migration 507 15.4.3 Rayleigh integral 510 15.4.4 Migration in the spectral domain (Stolt's method) 514 15.4.5 Equivalence of the spectral and integral migration algorithms. 516 15.4.6 Inversion versus migration 517 15.5 Elastic field inversion 518 15.5.1 Formulation of the elastic field inverse problem 518 15.5.2 Frechet derivative for the elastic forward modeling operator.. 520 15.5.3 Adjoint Frechet derivative operator and back-propagating elastic field 522

XVI CONTENTS A Functional spaces of geophysical models and data 531 A.I Euclidean Space 531 A. 1.1 Vector operations in Euclidean space 531 A.1.2 Linear transformations (operators) in Euclidean space 534 A.I.3 Norm of the operator 534 A. 1.4 Linear functionals 536 A.1.5 Norm of the functional 536 A.2 Metric space 537 A.2.1 Definition of metric space 537 A.2.2 Convergence, Cauchy sequences and completeness 538 A.3 Linear vector spaces 539 A.3.1 Vector operations 539 A.3.2 Normed linear spaces 540 A.4 Hilbert spaces 541 A.4.1 Inner product 541 A.4.2 Approximation problem in Hilbert space 544 A.5 Complex Euclidean and Hilbert spaces 546 A.5.1 Complex Euclidean space 546 A.5.2 Complex Hilbert space 547 A.6 Examples of linear vector spaces 547 B Operators in the spaces of models and data.. 553 B.I Operators in functional spaces 553 B.2 Linear operators 555 B.3 Inverse operators. 556 B.4 Some approximation problems in the Hilbert spaces of geophysical data 557 B.5 Gram - Schmidt orthogonalization process 559 C Functionals in the spaces of geophysical models 563 C.I Functionals and their norms 563 C.2 Riesz representation theorem 564 C.3 Functional representation of geophysical data and an inverse problem 565 D Linear operators and functionals revisited 569 D.I Adjoint operators 569 D.2 Differentiation of operators and functionals 571 D.3 Concepts from variational calculus 573 D.3.1 Variational operator 573 D.3.2 Extremum functional problems 574

CONTENTS XVII E Some formulae and rules from matrix algebra 577 E.I Some formulae and rules of operation on matrices 577 E.2 Eigenvalues and eigenvectors 578 E.3 Spectral decomposition of a symmetric matrix 579 E.4 Singular value decomposition (SVD) 580 E.5 The spectral Lanczos decomposition method 582 E.5.1 Functions of matrices 582 E.5.2 The Lanczos method 583 F Some formulae and rules from tensor calculus 589 F.I Some formulae and rules of operation on tensor functions 589 F.2 Tensor statements of the Gauss and Green's formulae 590 F.3 Green's tensor and vector formulae for Lame and Laplace operators. 591 Bibliography 593 Index 604