GEOPHYSICAL INVERSE THEORY AND REGULARIZATION PROBLEMS

Transcription

1 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

2 Contents Preface XIX I Introduction to Inversion Theory 1 1 Forward and inverse problems in geophysics Formulation of forward and inverse problems for different geophysical fields Gravity field Magnetic field Electromagnetic field Seismic wavefield Existence and uniqueness of the inverse problem solutions Existence of the solution Uniqueness of the solution Practical uniqueness Instability of the inverse problem solution 24 2 Ill-posed problems and the methods of their solution Sensitivity and resolution of geophysical methods Formulation of the inverse problem in general mathematical spaces Sensitivity Resolution Formulation of well-posed and ill-posed problems Well-posed problems Conditionally well-posed problems Quasi-solution of the ill-posed problem Foundations of regularization methods of inverse problem solution Regularizing operators Stabilizing functionals Tikhonov parametric functional 42

3 VIII CONTENTS 2.4 Family of stabilizing functionals Stabilizing functionals revisited Representation of a stabilizing functional in the form of a pseudoquadratic functional Definition of the regularization parameter Optimal regularization parameter selection L-curve method of regularization parameter selection 55 II Methods of the Solution of Inverse Problems 59 3 Linear discrete inverse problems Linear least-squares inversion The linear discrete inverse problem Systems of linear equations and their general solutions The data resolution matrix Solution of the purely underdetermined problem Underdetermined system of linear equations The model resolution matrix Weighted least-squares method Applying the principles of probability theory to a linear inverse problem Some formulae and notations from probability theory Maximum likelihood method Chi-square fitting Regularization methods The Tikhonov regularization method Application of SLDM method in regularized linear inverse problem solution ; Definition of the weighting matrices for the model parameters and data Approximate regularized solution of the linear inverse problem The Levenberg - Marquardt method The maximum a posteriori estimation method (the Bayes estimation) " The Backus-Gilbert Method The data resolution function The spread function Regularized solution in the Backus-Gilbert method 88 4 Iterative solutions of the linear inverse problem Linear operator equations and their solution by iterative methods Linear inverse problems and the Euler equation The minimal residual method 93

4 CONTENTS IX Linear inverse problem solution using MRM A generalized minimal residual method The Krylov-subspace method The Lanczos minimal residual method The generalized minimal residual method A linear inverse problem solution using generalized MRM The regularization method in a linear inverse problem solution The Euler equation for the Tikhonov parametric functional MRM solution of the Euler equation Generalized MRM solutions of the Euler equation for the parametric functional Nonlinear inversion technique Gradient-type methods Method of steepest descent The Newton method The conjugate gradient method Regularized gradient-type methods in the solution of nonlinear inverse problems Regularized steepest descent The regularized Newton method Approximate regularized solution of the nonlinear inverse problem The regularized preconditioned steepest descent method The regularized conjugate gradient method Regularized solution of a nonlinear discrete inverse problem Nonlinear least-squares inversion The steepest descent method for nonlinear regularized leastsquares inversion The Newton method for nonlinear regularized least-squares inversion Numerical schemes of the Newton method for nonlinear regularized least-squares inversion Nonlinear least-squares inversion by the conj ugate gradient method The numerical scheme of the regularized conjugate gradient method for nonlinear least-squares inversion Conjugate gradient re-weighted optimization The Tikhonov parametric functional with a pseudo-quadratic stabilizer Re-weighted conjugate gradient method Minimization in the space of weighted parameters 160

5 X CONTENTS The re-weighted regularized conjugate gradient (RCG) method in the space of weighted parameters 161 III Geopotential Field Inversion Integral representations in forward modeling of gravity and magnetic fields Basic equations for gravity and magnetic fields Gravity and magnetic fields in three dimensions Two-dimensional models of gravity and magnetic fields Integral representations of potential fields based on the theory of functions of a complex variable Complex intensity of a plane potential field Complex intensity of a gravity field Complex intensity and potential of a magnetic field Integral representations in inversion of gravity and magnetic data Gradient methods of gravity inversion Steepest ascent direction of the misfit functional for the gravity inverse problem Application of the re-weighted conjugate gradient method Gravity field migration Physical interpretation of the adjoint gravity operator Gravity field migration in the solution of the inverse problem Iterative gravity migration Gradient methods of magnetic anomaly inversion Magnetic potential inversion.: Magnetic potential migration Numerical methods in forward and inverse modeling Discrete forms of 3-D gravity and magnetic forward modeling operators Discrete form of 2-D forward modeling operator Regularized inversion of gravity data 193 IV Electromagnetic Inversion Foundations of electromagnetic theory Electromagnetic field equations Maxwell's equations Field in homogeneous domains of a medium Boundary conditions 203

6 CONTENTS 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 //-polarization Electromagnetic energy flow Radiation conditions Poynting's theorem in the time domain Energy inequality in the time domain Poynting's theorem in the frequency domain Uniqueness of the solution of electromagnetic field equations Boundary-value problem Uniqueness theorem for the unbounded domain Electromagnetic Green's tensors Green's tensors in the frequency domain Lorentz lemma and reciprocity relations Green's tensors in the time domain 227 Integral representations in electromagnetic forward modeling Integral equation method Background (normal) and anomalous parts of the electromagnetic field V " Poynting's theorem and energy inequality for an anomalous field Integral equation method in two dimensions Calculation of the first variation (Frechet derivative) of the electromagnetic field for 2-D models Integral equation method in three dimensions Calculation of the first variation (Frechet derivative) of the electromagnetic field for 3-D models Frechet derivative calculation using the differential method Family of linear and nonlinear integral approximations of the electromagnetic field Born and extended Born approximations Quasi-linear approximation and tensor quasi-linear equation Quasi-analytical solutions for a 3-D electromagnetic field Quasi-analytical solutions for 2-D electromagnetic field Localized nonlinear approximation Localized quasi-linear approximation Linear and non-linear approximations of higher orders Born series 256

7 XII CONTENTS Modified Green's operator Modified Born series Quasi-linear approximation of the modified Green's operator QL series Accuracy estimation of the QL approximation of the first and higher orders QA series Integral representations in numerical dressing Discretization of the model parameters Galerkin method for electromagnetic field discretization Discrete form of electromagnetic integral equations based on boxcar basis functions Contraction integral equation method Contraction integral equation as the preconditioned conventional integral equation Matrix form of Born approximation Matrix form of quasi-linear approximation Matrix form of quasi-analytical approximation The diagonalized quasi-analytical (DQA) approximation Integral representations in electromagnetic inversion Linear inversion methods Excess (anomalous) current inversion Born inversion Conductivity imaging by the Born approximation Iterative Born inversions Nonlinear inversion Formulation of the nonlinear inverse problem ' Frechet derivative calculation Quasi-linear inversion Principles of quasi-linear inversion Quasi-linear inversion in matrix notations Localized quasi-linear inversion Quasi-analytical inversion Frechet derivative calculation Inversion based on the quasi-analytical method Magnetotelluric (MT) data inversion Iterative Born inversion of magnetotelluric data DQA approximation in magnetotelluric inverse problem Frechet derivative matrix with respect to the logarithm of the total conductivity Regularized smooth and focusing inversion of MT data

8 CONTENTS XIII Example of synthetic 3-D MT data inversion Case study: inversion of the Minamikayabe area data Electromagnetic migration imaging Electromagnetic migration in the frequency domain Formulation of the electromagnetic inverse problem as a minimization of the energy flow functional Integral representations for electromagnetic migration field Gradient direction of the energy flow functional Migration imaging in the frequency domain Iterative migration Electromagnetic migration in the time domain Time domain electromagnetic migration as the solution of the boundary value problem Minimization of the residual electromagnetic field energy flow Gradient direction of the energy flow functional in the time domain Migration imaging in the time domain Iterative migration in the time domain Differential methods in electromagnetic modeling and inversion Electromagnetic modeling as a boundary-value problem Field equations and boundary conditions Electromagnetic potential equations and boundary conditions Finite difference approximation of the boundary-value problem Discretization of Maxwell's equations using a staggered grid Discretization of the second order differential equations using the balance method Discretization of the electromagnetic potential differential equations Application of the spectral Lanczos decomposition method (SLDM) for solving the linear system of equations for discrete electromagnetic fields Finite element solution of boundary-value problems Galerkin method Exact element method Inversion based on differential methods Formulation of the inverse problem on the discrete grid Frechet derivative calculation using finite difference methods. 386

9 XIV CONTENTS V Seismic Inversion Wavefield equations Basic equations of elastic waves Deformation of an elastic body; deformation and stress tensors Hooke's law Dynamic equations of elasticity theory for a homogeneous isotropic medium Compressional and shear waves Acoustic waves and scalar wave equation High frequency approximations in the solution of an acoustic wave equation Green's functions for wavefield equations Green's functions for the scalar wave equation and for the corresponding Helmholtz equation High frequency (WKBJ) approximation for the Green's function Green's tensor for vector wave equation Green's tensor for the Lame equation Kirchhoff integral formula and its analogs Kirchhoff integral formula Generalized Kirchhoff integral formulae for the Lame equation and the vector wave equation Uniqueness of the solution of the wavefield equations Initial-value problems Energy'conservation law Uniqueness of the solution of initial-value problems Sommerfeld radiation conditions Uniqueness of the solution of the wave propagation problem based on radiation conditions Kirchhoff formula for an unbounded domain Radiation conditions for elastic waves Integral representations in wavefield theory Integral equation method in acoustic wavefield analysis Separation of the acoustic wavefield into incident and scattered (background and anomalous) parts Integral equation for the acoustic wavefield Reciprocity theorem Calculation of the first variation (Frechet derivative) of the acoustic wavefield Integral approximations of the acoustic wavefield Born approximation 449

10 CONTENTS XV Quasi-linear approximation Quasi-analytical approximation Localized quasi-linear approximation Kirchhoff approximation Method of integral equations in vector wavefield analysis Vector wavefield separation Integral equation method for the vector wavefield Calculation of the first variation (Frechet derivative) of the vector wavefield Integral approximations of the vector wavefield Born type approximations Quasi-linear approximation Quasi-analytical solutions for the vector wavefield Localized quasi-linear approximation Integral representations in wavefield inversion Linear inversion methods Born inversion of acoustic and vector wavefields Wavefield imaging by the Born approximations Iterative Born inversions of the wavefield Bleistein inversion Inversion based on the Kirchhoff approximation Traveltime inverse problem Quasi-linear inversion Quasi-linear inversion of the acoustic wavefield Localized quasi-linear inversion based on the Bleistein method Nonlinear inversion Formulation of the nonlinear wavefield inverse problem Frechet derivative operators for wavefield problems Principles of wavefield migration Geometrical model of migration transformation Kirchhoff integral formula for reverse-time wave equation migration Rayleigh integral Migration in the spectral domain (Stolt's method) Equivalence of the spectral and integral migration algorithms Inversion versus migration Elastic field inversion Formulation of the elastic field inverse problem Frechet derivative for the elastic forward modeling operator Adjoint Frechet derivative operator and back-propagating elastic field 522

11 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 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

12 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