# Inverse Theory and Applications in Geophysics

## 2nd Edition

**Authors:**Michael Zhdanov

**eBook ISBN:**9780444627124

**Hardcover ISBN:**9780444626745

**Imprint:**Elsevier Science

**Published Date:**10th July 2015

**Page Count:**730

## Description

*Geophysical Inverse Theory and Applications, Second Edition, *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. It’s the first book of its kind 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. Unique in its focus on providing a link between the methods used in gravity, electromagnetic, and seismic imaging and inversion, it represents an exhaustive treatise on inversion theory.

Written by one of the world’s foremost experts, this work is widely recognized as the ultimate researcher’s reference on geophysical inverse theory and its practical scientific applications.

## Key Features

- Presents state-of-the-art geophysical inverse theory developed in modern mathematical terminology—the first to treat many kinds of inversion and imaging techniques in a unified mathematical way.
- Provides a critical link between the methods used in gravity, electromagnetic, and seismic imaging and inversion, and represents an exhaustive treatise on geophysical inversion theory.
- Features more than 300 illustrations, figures, charts and graphs to underscore key concepts.
- Reflects the latest developments in inversion theory and applications and captures the most significant changes in the field over the past decade.

## Readership

Researchers working in all areas of geophysics, including but not limited to physics of the solid earth, exploration geophysics, gravity methods, electromagnetic methods, seismology, and broader areas of applied physics and geophysics.

## Table of Contents

- Dedication
- Preface to the Second Edition
- Preface
- Part I: Introduction to Inversion Theory
- Chapter 1: Forward and Inverse Problems in Science and Engineering
- Abstract
- 1.1 Formulation of Forward and Inverse Problems for Different Physical 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
- Abstract
- 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

- Chapter 1: Forward and Inverse Problems in Science and Engineering
- Part II: Methods of the Solution of Inverse Problems
- Chapter 3: Linear Discrete Inverse Problems
- Abstract
- 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
- Abstract
- 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
- Abstract
- 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

- Chapter 6: Multinary Inversion
- Abstract
- 6.1 Level Set Method
- 6.2 Multinary Inversion

- Chapter 7: Resolution Analysis of Regularized Geophysical Inversion
- Abstract
- 7.1 Resolution of a Linear Inverse Problem
- 7.2 Resolution Density
- 7.3 Resolution of a Nonlinear Inverse Problem
- 7.4 Application of the SLDM for Resolution Density Calculation

- Chapter 8: Monte Carlo Methods
- Abstract
- 8.1 Random Search Methods
- 8.2 Simulated Annealing
- 8.3 Genetic Algorithm

- Chapter 9: Generalized Joint Inversion of Multimodal Data
- Abstract
- 9.1 Joint Inversion Based on Functional Relationships Between Different Model Parameters
- 9.2 The Method of Cross-Gradients
- 9.3 Joint Inversion Based on Gramian Constraints

- Chapter 3: Linear Discrete Inverse Problems
- Part III: Geopotential Field Inversion
- Chapter 10: Integral Representations of 2-D Gravity and Magnetic Fields
- Abstract
- 10.1 Basic Equations for Gravity and Magnetic Fields
- 10.2 Integral Representations of Potential Fields Based on the Theory of Functions of a Complex Variable
- 10.3 Gradient Methods of 2-D Gravity Field Inversion
- 10.4 Migration of 2-D Gravity Field
- 10.5 Gradient Methods of 2-D Magnetic Anomaly Inversion

- Chapter 11: Migration of 3-D Gravity, Gravity Tensor, and Total Magnetic Intensity Data
- Abstract
- 11.1 Gravity Gradiometry Data
- 11.2 Migration of 3-D Gravity and Gravity Gradiometry Data
- 11.3 Fast Density Imaging Based on Migration
- 11.4 Migration of Total Magnetic Intensity Data

- Chapter 12: Numerical Methods in Forward and Inverse Modeling of Geopotential Fields
- Abstract
- 12.1 Numerical Methods in Forward and Inverse Modeling
- 12.2 Regularized Inversion of Gravity and Gradiometry Data

- Chapter 10: Integral Representations of 2-D Gravity and Magnetic Fields
- Part IV: Electromagnetic Inversion
- Chapter 13: Foundations of Electromagnetic Theory
- Abstract
- 13.1 Electromagnetic Field Equations
- 13.2 Electromagnetic Energy Flow
- 13.3 Uniqueness of the Solution of Electromagnetic Field Equations
- 13.4 Electromagnetic Green’s Tensors

- Chapter 14: Integral Representations in Electromagnetic Forward Modeling
- Abstract
- 14.1 IE Method
- 14.2 Family of Linear and Nonlinear Integral Approximations of the EM Field
- 14.3 Linear and Nonlinear Approximations of Higher Orders
- 14.4 Integral Representations in Numerical Dressing

- Chapter 15: Integral Representations in Electromagnetic Inversion
- Abstract
- 15.1 Linear Inversion Methods
- 15.2 Nonlinear Inversion
- 15.3 Quasi-Linear Inversion
- 15.4 Quasi-Analytical Inversion

- Chapter 16: Electromagnetic Migration Imaging
- Abstract
- 16.1 Electromagnetic Migration in the Frequency Domain
- 16.2 Electromagnetic Migration in the Time Domain

- Chapter 17: Differential Methods in Electromagnetic Modeling and Inversion
- Abstract
- 17.1 Electromagnetic Modeling as a Boundary-Value Problem
- 17.2 Finite Difference Approximation of the Boundary-Value Problem
- 17.3 Finite Element Solution of Boundary-Value Problems
- 17.4 Inversion Based on Differential Methods

- Chapter 13: Foundations of Electromagnetic Theory
- Part V: Seismic Inversion
- Chapter 18: Wavefield Equations
- Abstract
- 18.1 Basic Equations of Elastic Waves
- 18.2 Green’s Functions for Wavefield Equations
- 18.3 Kirchhoff Integral Formula and Its Analogs
- 18.4 Uniqueness of the Solution of the Wavefield Equations

- Chapter 19: Integral Representations in Wavefield Theory
- Abstract
- 19.1 Integral Equation Method in Acoustic Wavefield Analysis
- 19.2 Integral Approximations of the Acoustic Wavefield
- 19.3 Method of Integral Equations in Vector Wavefield Analysis
- 19.4 Integral Approximations of the Vector Wavefield

- Chapter 20: Integral Representations in Full Waveform Inversion
- Abstract
- 20.1 Linear Inversion Methods
- 20.2 Quasi-Linear Inversion
- 20.3 Nonlinear Inversion
- 20.4 Principles of Wavefield Migration
- 20.5 Full-Waveform Inversion of Elastic Field

- Chapter 18: Wavefield Equations
- Appendix A: Functional Spaces of Geophysical Models and Data
- A.1 Euclidean Space
- A.2 Metric Space
- A.3 Linear Vector Spaces
- A.4 Hilbert Spaces
- A.5 Complex Euclidean and Hilbert Spaces
- A.6 Examples of Linear Vector Spaces
- A.7 Gramian Spaces and Their Properties

- Appendix B: Operators in the Spaces of Models and Data
- B.1 Operators in Functional Spaces
- B.2 Linear Operators
- B.3 Inverse Operators
- B.4 Some Approximation Problems in the Hilbert Spaces of Geophysical Data
- B.5 Gram-Schmidt Orthogonalization Process

- Appendix C: Functionals in the Spaces of Geophysical Models
- C.1 Functionals and Their Norms
- C.2 Riesz Representation Theorem
- C.3 Functional Representation of Geophysical Data and an Inverse Problem

- Appendix D: Linear Operators and Functionals Revisited
- D.1 Adjoint Operators
- D.2 Differentiation of Operators and Functionals
- D.3 Concepts from Variational Calculus

- Appendix E: Some Formulae and Rules from Matrix Algebra
- E.1 Some Formulae and Rules of Operation on Matrices
- E.2 Eigenvalues and Eigenvectors
- E.3 Spectral Decomposition of a Symmetric Matrix
- E.4 Singular Value Decomposition (SVD)
- E.5 The Spectral Lanczos Decomposition Method

- Appendix F: Some Formulae and Rules from Tensor Calculus
- F.1 Some Formulae and Rules of Operation on Tensor Functions
- F.2 Tensor Statements of the Gauss and Green’s Formulae
- F.3 Green’s Tensor and Vector Formulae for Lamé and Laplace Operators

- Bibliography
- Index

## Details

- No. of pages:
- 730

- Language:
- English

- Copyright:
- © Elsevier Science 2015

- Published:
- 10th July 2015

- Imprint:
- Elsevier Science

- eBook ISBN:
- 9780444627124

- Hardcover ISBN:
- 9780444626745

## About the Author

### Michael Zhdanov

Michael S. Zhdanov is Professor of Geophysics in the Department of Geology and Geophysics at the University of Utah in Salt Lake City. Dr. Zhdanov is also Director of the Center of Electromagnetic Research at the same university. He has more than 30 years of experience in research and instruction in geophysical electromagnetic theory and he has authored more than 100 papers on the subject. He is the founding director of the Geoelectromagnetic Research Institute of the Russian Academy of Sciences and Member of the Russian Academy of Natural Science.

### Affiliations and Expertise

Professor, Department of Geology and Geophysics, University of Utah, Salt Lake City, USA

## Reviews

"...will be very beneficial for graduate students in global geophysics and geophysical prospection, as well as utilized by professionals dealing with mathematical geophysics." **--Pure and Applied Geophysics**