Geophysical Inverse Theory and Regularization Problems 1st Edition by Michael S Zhdanov ISBN 0444510893 9780444510891

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ISBN 10: 0444510893 
ISBN 13: 9780444510891
Author: Michael S Zhdanov

This book presents state-of-the-art geophysical inverse theory developed in modern mathematical terminology. The book brings together fundamental results developed by the Russian mathematical school in regularization theory and combines them with the related research in geophysical inversion carried out in the West. It presents a detailed exposition of the methods of regularized solution of inverse problems based on the ideas of Tikhonov regularization, and shows the different forms of their applications in both linear and nonlinear methods of geophysical inversion. This text is the first to treat many kinds of inversion and imaging techniques in a unified mathematical manner.

The book is divided in five parts covering the foundations of the inversion theory and its applications to the solution of different geophysical inverse problems, including potential field, electromagnetic, and seismic methods. The first part is an introduction to inversion theory. The second part contains a description of the basic methods of solution of the linear and nonlinear inverse problems using regularization. The following parts treat the application of regularization methods in gravity and magnetic, electromagnetic, and seismic 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 methods. The book also includes chapters related to the modern technology of geophysical imaging, based on seismic and electromagnetic migration.This volume is unique in its focus on providing a link between the methods used in gravity, electromagnetic, and seismic imaging and inversion, and represents an exhaustive treatise on inversion theory.

Geophysical Inverse Theory and Regularization Problems 1st Table of contents:

I: Introduction to Inversion Theory
Chapter 1: Forward and Inverse Problems in Geophysics
1.1 Formulation of forward and inverse problems for different geophysical fields
1.2 Existence and uniqueness of the inverse problem solutions
1.3 Instability of the inverse problem solution
Chapter 2: Ill-posed Problems and the Methods of their Solution
2.1 Sensitivity and resolution of geophysical methods
2.2 Formulation of well-posed and ill-posed problems
2.3 Foundations of regularization methods of inverse problem solution
2.4 Family of stabilizing functionals
2.5 Definition of the regularization parameter
II: Methods of the Solution of Inverse Problems
Chapter 3: Linear Discrete Inverse Problems
3.1 Linear least–squares inversion
3.2 Solution of the purely underdetermined problem
3.3 Weighted least-squares method
3.4 Applying the principles of probability theory to a linear inverse problem
3.5 Regularization methods
3.6 The Backus-Gilbert Method
Chapter 4: Iterative Solutions of the Linear Inverse Problem
4.1 Linear operator equations and their solution by iterative methods
4.2 A generalized minimal residual method
4.3 The regularization method in a linear inverse problem solution
Chapter 5: Nonlinear Inversion Technique
5.1 Gradient-type methods
5.2 Regularized gradient-type methods in the solution of nonlinear inverse problems
5.3 Regularized solution of a nonlinear discrete inverse problem
5.4 Conjugate gradient re-weighted optimization
III: Geopotential Field Inversion
Chapter 6: Integral Representations in Forward Modeling of Gravity and Magnetic Fields
6.1 Basic equations for gravity and magnetic fields
6.2 Integral representations of potential fields based on the theory of functions of a complex variable
Chapter 7: Integral Representations in Inversion of Gravity and Magnetic Data
7.1 Gradient methods of gravity inversion
7.2 Gravity field migration
7.3 Gradient methods of magnetic anomaly inversion
7.4 Numerical methods in forward and inverse modeling
IV: Electromagnetic Inversion
Chapter 8: Foundations of Electromagnetic Theory
8.1 Electromagnetic field equations
8.2 Electromagnetic energy flow
8.3 Uniqueness of the solution of electromagnetic field equations
8.4 Electromagnetic Green’s tensors
Chapter 9: Integral Representations in Electromagnetic Forward Modeling
9.1 Integral equation method
9.2 Family of linear and nonlinear integral approximations of the electromagnetic field
9.3 Linear and non-linear approximations of higher orders
9.4 Integral representations in numerical dressing
Chapter 10: Integral Representations in Electromagnetic Inversion
10.1 Linear inversion methods
10.2 Nonlinear inversion
10.3 Quasi-linear inversion
10.4 Quasi-analytical inversion
10.5 Magnetotelluric (MT) data inversion
Chapter 11: Electromagnetic Migration Imaging
11.1 Electromagnetic migration in the frequency domain
11.2 Electromagnetic migration in the time domain
Chapter 12: Differential Methods in Electromagnetic Modeling and Inversion
12.1 Electromagnetic modeling as a boundary-value problem
12.2 Finite difference approximation of the boundary-value problem
12.3 Finite element solution of boundary-value problems
12.4 Inversion based on differential methods
V: Seismic Inversion
Chapter 13: Wavefield Equations
13.1 Basic equations of elastic waves
13.2 Green’s functions for wavefield equations
13.3 Kirchhoff integral formula and its analogs
13.4 Uniqueness of the solution of the wavefield equations
Chapter 14: Integral Representations in Wavefield Theory
14.1 Integral equation method in acoustic wavefield analysis
14.2 Integral approximations of the acoustic wavefield
14.3 Method of integral equations in vector wavefield analysis
14.4 Integral approximations of the vector wavefield
Chapter 15: Integral Representations in Wavefield Inversion
15.1 Linear inversion methods
15.2 Quasi-linear inversion
15.3 Nonlinear inversion
15.4 Principles of wavefield migration
15.5 Elastic field inversion

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Tags: Michael S Zhdanov, Geophysical Inverse, Theory, Regularization