Geophysical Inverse Theory and Regularization Problems (eBook)

Cover 1
Contents 8
Preface 20
Part I: Introduction to Inversion Theory 26
Chapter 1. Forward and inverse problems in geophysics 28
1.1 Formulation of forward and inverse problems for different geophysical fields 28
1.2 Existence and uniqueness of the inverse problem solutions 41
1.3 Instability of the inverse problem solution 49
Chapter 2. ILL-Posed problems and the methods of their solution 54
2.1 Sensitivity and resolution of geophysical methods 54
2.2 Formulation of well-posed and ill-posed problems 57
2.3 Foundations of regularization methods of inverse problem solution 61
2.4 Family of stabilizing functionals 70
2.5 Definition of the regularization parameter 77
Part II: Methods of the Solution of Inverse Problems 84
Chapter 3. Linear discrete inverse problems 86
3.1 Linear least-squares inversion 86
3.2 Solution of the purely underdetermined problem 91
3.3 Weighted least-squares method 93
3.4 Applying the principles of probability theory to a linear inverse problem 94
3.5 Regularization methods 99
3.6 The Backus-Gilbert Method 109
Chapter 4. Iterative solutions of the linear inverse problem 116
4.1 Linear operator equations and their solution by iterative methods 116
4.2 A generalized minimal residual method 126
4.3 The regularization method in a linear inverse problem solution 138
Chapter 5. Nonlinear inversion technique 146
5.1 Gradient-type methods 146
5.2 Regularized gradient-type methods in the solution of nonlinear inverse problems 168
5.3 Regularized solution of a nonlinear discrete inverse problem 174
5.4 Conjugate gradient re-weighted optimization 180
Part III: Geopotential Field Inversion 192
Chapter 6. Integral representations in forward modeling of gravity and magnetic fields 194
6.1 Basic equations for gravity and magnetic fields 194
6.2 Integral representations of potential fields based on the theory of functions of a complex variable 196
Chapter 7. Integral representations in inversion of gravity and magnetic data 202
7.1 Gradient methods of gravity inversion 202
7.2 Gravity field migration 206
7.3 Gradient methods of magnetic anomaly inversion 213
7.4 Numerical methods in forward and inverse modeling 215
Part IV: Electromagnetic Inversion 224
Chapter 8. Foundations of electromagnetic theory 226
8.1 Electromagnetic field equations 226
8.2 Electromagnetic energy flow 240
8.3 Uniqueness of the solution of electromagnetic field equations 247
8.4 Electromagnetic Green's tensors 249
Chapter 9. Integral representations in electromagnetic forward modeling 256
9.1 Integral equation method 256
9.2 Family of linear and nonlinear integral approximations of the electromagnetic field 270
9.3 Linear and non-linear approximations of higher orders 281
9.4 Integral representations in numerical dressing 292
Chapter 10. Integral representations in electromagnetic inversion 312
10.1 Linear inversion methods 313
10.2 Nonlinear inversion 322
10.3 Quasi-linear inversion 325
10.4 Quasi-analytical inversion 336
10.5 Magnetotelluric (MT) data inversion 339
Chapter 11. Electromagnetic migration imaging 356
11.1 Electromagnetic migration in the frequency domain 357
11.2 Electromagnetic migration in the time domain 369
Chapter 12. Differential methods in electromagnetic modeling and inversion 386
12.1 Electromagnetic modeling as a boundary-value problem 386
12.2 Finite difference approximation of the boundary-value problem 391
12.3 Finite element solution of boundary-value problems 405
12.4 Inversion based on differential methods 410
Part V: Seismic Inversion 418
Chapter 13. Wavefield equations 420
13.1 Basic equations of elastic waves 420
13.2 Green's functions for wavefield equations 432
13.3 Kirchhoff integral formula and its analogs 439
13.4 Uniqueness of the solution of the wavefield equations 445
Chapter 14. Integral representations in wavefield theory 468
14.1 Integral equation method in acoustic wavefield analysis 468
14.2 Integral approximations of the acoustic wavefield 474
14.3 Method of integral equations in vector wavefield analysis 481
14.4 Integral approximations of the vector wavefield 485
Chapter 15. Integral representations in wavefield inversion 492
15.1 Linear inversion methods 492
15.2 Quasi-linear inversion 521
15.3 Nonlinear inversion 524
15.4 Principles of wavefield migration 528
15.5 Elastic field inversion 543
Appendix A. Functional spaces of geophysical models and data 556
A.1 Euclidean Space 556
A.2 Metric space 562
A.3 Linear vector spaces 564
A.4 Hilbert spaces 566
A.5 Complex Euclidean and Hilbert spaces 571
A.6 Examples of linear vector spaces 572
Appendix B. Operators in the spaces of models and data 578
B.1 Operators in functional spaces 578
B.2 Linear operators 580
B.3 Inverse operators 581
B.4 Some approximation problems in the Hilbert spaces of geophysical data 582
B.5 Gram - Schmidt orthogonalization process 584
Appendix C. Functionals in the spaces of geophysical models 588
C.1 Functionals and their norms 588
C.2 Riesz representation theorem 589
C.3 Functional representation of geophysical data and an inverse problem 590
Appendix D. Linear operators and functionals revisited 594
D.1 Adjoint operators 594
D.2 Differentiation of operators and functionals 596
D.3 Concepts from variational calculus 598
Appendix E. Some formulae and rules from matrix algebra 602
E.1 Some formulae and rules of operation on matrices 602
E.2 Eigenvalues and eigenvectors 603
E.3 Spectral decomposition of a symmetric matrix 604
E.4 Singular value decomposition (SVD) 605
E.5 The spectral Lanczos decomposition method 607
Appendix F. Some formulae and rules from tensor calculus 614
F.1 Some formulae and rules of operation on tensor functions 614
F.2 Tensor statements of the Gauss and Green's formulae 615
F.3 Green's tensor and vector formulae for Lamé and Laplace operators 616
Bibliography 618
Index 630