Inverse Theory and Applications in Geophysics Second Edition by Michael Zhdanov ISBN 978-0444626745 0444626743
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ISBN 10: 0444626743
ISBN 13: 978-0444626745
Author: Michael S. Zhdanov
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.
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
Table of contents:
PARTI Introduction to Inversion Theory.
Chapter 1: Forward and Inverse Problems in Science and Engineering.
1.1 Formulation of Forward and Inverse Problems for Different Physical Fields…
1.1.1 Gravity Field.
1.1.2 Magnetic Field….
1.1.3 Electromagnetic Field
1.1.4 Seismic Wavefield..
1.2 Existence and Uniqueness of the Inverse Problem Solutions.
1.2.1 Existence of the Solution….
1.2.2 Uniqueness of the Solution…
1.2.3 Practical Uniqueness
1.3 Instability of the Inverse Problem Solution.
References.
Chapter 2: III-Posed Problems and the Methods of Their Solution…
2.1 Sensitivity and Resolution of Geophysical Methods.
2.1.1 Formulation of the Inverse Problem in General Mathematical Spaces.
2.1.2 Sensitivity.
2.1.3 Resolution.
2.2 Formulation of Well-Posed and Ill-Posed Problems.
2.2.1 Well-Posed Problems..
2.2.2 Conditionally Well-Posed Problems.
2.2.3 Quasi-Solution of the III-Posed Problem.
2.3 Foundations of Regularization Methods of Inverse Problem Solution.
2.3.1 Regularizing Operators.
2.3.2 Stabilizing Functionals.
2.3.3 Tikhonov Parametric Functional.
2.4 Family of Stabilizing Functionals….
2.4.1 Stabilizing Functionals Revisited.
2.4.2 Representation of a Stabilizing Functional in the Form of a Pseudo-Quadratic Functional
2.5 Definition of the Regularization Parameter..
2.5.1 Optimal Regularization Parameter Selection.
2.5.2 L-Curve Method of Regularization Parameter Selection.. References..
PART II Methods of the Solution of Inverse Problems
Chapter 3: Linear Discrete Inverse Problems
3.1 Linear Least-Squares Inversion.
3.1.1 The Linear Discrete Inverse Problem.
3.1.2 Systems of Linear Equations and Their General Solutions.
3.1.3 The Data Resolution Matrix..
3.2 Solution of the Purely Underdetermined Problem…
3.2.1 Underdetermined System of Linear Equations.
3.2.2 The Model Resolution Matrix
3.3 Weighted Least-Squares Method..
3.4 Applying the Principles of Probability Theory to a Linear Inverse Problem…..
3.4.1 Some Formulae and Notations from Probability Theory.
3.4.2 Maximum Likelihood Method.
3.4.3 Chi-Square Fitting.
3.5 Regularization Methods.
3.5.1 The Tikhonov Regularization Method.
3.5.2 Application of SLDM Method in Regularized Linear Inverse Problem Solution.
3.5.3 Integrated Sensitivity
3.5.4 Definition of the Weighting Matrices for the Model Parameters and Data…
3.5.5 Controlled Sensitivity
3.5.6 Approximate Regularized Solution of the Linear Inverse Problem..
3.5.7 The Levenberg-Marquardt Method.
3.5.8 The Maximum a Posteriori Estimation Method (the Bayes Estimation).
3.6 The Backus-Gilbert Method..
3.6.1 The Data Resolution Function.
3.6.2 The Spread Function..
3.6.3 Regularized Solution in the Backus-Gilbert Method. References..
Chapter 4: Iterative Solutions of the Linear Inverse Problem.
4.1 Linear Operator Equations and Their Solution by Iterative Methods..
4.1.1 Linear Inverse Problems and the Euler Equation…
4.1.2 The Minimal Residual Method.
4.1.3 Linear Inverse Problem Solution Using MRM.
4.2 A Generalized Minimal Residual Method.
4.2.1 The Krylov-Subspace Method..
4.2.2 The Lanczos MRM.
4.2.3 The Generalized Minimal Residual Method
4.2.4 A Linear Inverse Problem Solution Using GMRM.
4.3 The Regularization Method in a Linear Inverse Problem Solution
4.3.1 The Euler Equation for the Tikhonov Parametric Functional.
4.3.2 MRM Solution of the Euler Equation..
4.3.3 GMRM Solutions of the Euler Equation for the Parametric Functional. References..
Chapter 5: Nonlinear Inversion Technique.
5.1 Gradient-Type Methods.
5.1.1 Method of Steepest Descent.
5.1.2 The Newton Method
5.1.3 The Conjugate Gradient Method.
5.2 Regularized Gradient-Type Methods in the Solution of Nonlinear Inverse Problems…
5.2.1 Regularized Steepest Descent..
5.2.2 The Regularized Newton Method..
5.2.3 Approximate Regularized Solution of the Nonlinear Inverse Problem.
5.2.4 The Regularized Preconditioned Steepest Descent Method…
5.2.5 The Regularized Conjugate Gradient Method..
5.3 Regularized Solution of a Nonlinear Discrete Inverse Problem..
5.3.1 Nonlinear Least-Squares Inversion….
5.3.2 The Steepest Descent Method for Nonlinear Regularized Least-Squares Inversion.
5.3.3 The Newton Method for Nonlinear Regularized Least-Squares Inversion.
5.3.4 Numerical Schemes of the Newton Method for Nonlinear Regularized Least-Squares Inversion.
5.3.5 Nonlinear Least-Squares Inversion by the Conjugate Gradient Method..
5.3.6 The Numerical Scheme of the RCG Method for Nonlinear Least-Squares Inversion.
5.3.7 Nonlinear Least-Squares Inversion in the Complex Euclidean Space.
5.4 Conjugate Gradient Re-Weighted Optimization.
5.4.1 The Tikhonov Parametric Functional with a Pseudo-Quadratic Stabilizer….
5.4.2 Re-Weighted Conjugate Gradient Method.
5.4.3 Minimization in the Space of Weighted Parameters..
5.4.4 The RRCG Method in the Space of Weighted Parameters.
5.4.5 Inversion in Logarithmic Model Parameter Space.. References…
Chapter 6: Multinary Inversion.
6.1 Level Set Method.
6.1.1 Shape Reconstruction Inverse Problem.
6.1.2 Evolution Equation
6.1.3 Regularization of the Level Set Inversion.
6.2 Multinary Inversion…
6.2.1 Representation of the Model Parameters by the Multinary Functions..
6.2.2 Continuous Parameterization of the Multinary Inverse Problem.
6.2.3 Regularized Conjugate Gradient Inversion in the Space of the Transformed Model Parameters.
References..
Chapter 7: Resolution Analysis of Regularized Geophysical Inversion
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. References.
Chapter 8: Monte Carlo Methods.
8.1 Random Search Methods
8.1.1 Sampling Method..
8.1.2 Metropolis Algorithm
8.2 Simulated Annealing.
8.2.1 Process of Annealing.
8.2.2 SA Method
8.3 Genetic Algorithm.
8.3.1 Selection of the Search Subspace and Creating Initial Population and Individuals
8.3.2 Selection of Intermediate Population..
8.3.3 Crossover and Mutation
8.3.4 Convergence and Termination Conditions. References.
Chapter 9: Generalized Joint Inversion of Multimodal Data..
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.
9.3.1 Gramian Space of Model Parameters.
9.3.2 Gramian Space of Model Parameter Gradients.
9.3.3 Gramian Spaces of Different Transforms of the Model Parameters.
9.3.4 Joint Regularized Inversion of Multiple Datasets with the Gramian Stabilizers.
9.3.5 Model Study
References.
PART III Geopotential Field Inversion
Chapter 10: Integral Representations of 2-D Gravity and Magnetic Fields..
10.1 Basic Equations for Gravity and Magnetic Fields.
10.1.1 Gravity and Magnetic Fields in Three Dimensions.
10.1.2 Two-Dimensional Models of Gravity and Magnetic Fields.
10.2 Integral Representations of Potential Fields Based on the Theory of Functions of a Complex Variable.
10.2.1 Complex Intensity of a Plane Potential Field.
10.2.2 Complex Intensity of a Gravity Field..
10.2.3 Complex Intensity and Potential of a Magnetic Field.
10.3 Gradient Methods of 2-D Gravity Field Inversion..
10.3.1 Steepest Ascent Direction of the Misfit Functional for the Gravity Inverse Problem.
10.3.2 Application of the Re-Weighted Conjugate Gradient Method
10.4 Migration of 2-D Gravity Field
10.4.1 Physical Interpretation of the Adjoint Gravity Operator.
10.4.2 Gravity Field Migration in the Solution of the Inverse Problem.
10.4.3 Iterative Gravity Migration..
10.5 Gradient Methods of 2-D Magnetic Anomaly Inversion..
10.5.1 Magnetic Potential Inversion…
10.5.2 Magnetic Potential Migration.. References.
Chapter 11: Migration of 3-D Gravity, Gravity Tensor, and Total Magnetic Intensity Data..
11.1 Gravity Gradiometry Data..
11.2 Migration of 3-D Gravity and Gravity Gradiometry Data….
11.2.1 Adjoint Operators for Gravity and Gravity Gradiometry Inversion.
11.2.2 Adjoint Operator for 3-D Gravity Fields.
11.2.3 Adjoint Operator for 3-D Gravity Tensor Fields.
11.3 Fast Density Imaging Based on Migration.
11.3.1 Principles of Fast Inverse Imaging.
11.3.2 Migration of Gravity and Gravity Tensor Fields and 3-D Density Imaging.
11.3.3 Integrated Sensitivity of 3-D Gravity Fields.
11.3.4 Integrated Sensitivity of 3-D Gravity Tensor Fields.
11.4 Migration of Total Magnetic Intensity Data..
11.4.1 Adjoint Operator for the Total Magnetic Intensity
11.4.2 Migration of the Total Magnetic Intensity.
11.4.3 Integrated Sensitivity of the Total Magnetic Intensity.
11.4.4 Model Study. References.
Chapter 12: Numerical Methods in Forward and Inverse Modeling of Geopotential Fields….
12.1 Numerical Methods in Forward and Inverse Modeling.
12.1.1 Discrete Forms of 3-D Gravity and Gravity Gradiometry Forward Modeling Operators.
12.1.2 Discrete Forms of 3-D Magnetic Forward Modeling Operators.
12.1.3 Discrete Form of 2-D Forward Modeling Operator.
12.2 Regularized Inversion of Gravity and Gradiometry Data..
12.2.1 Numerical Examples of 2-D Gravity Inversion..
12.2.2 3-D Inversion of Synthetic Gravity Gradiometry Data. References.
PART IV Electromagnetic Inversion
Chapter 13: Foundations of Electromagnetic Theory
13.1 Electromagnetic Field Equations.
13.1.1 Maxwell’s Equations.
13.1.2 Field in Homogeneous Domains of a Medium.
13.1.3 Boundary Conditions.
13.1.4 Field Equations in the Frequency Domain…..
13.1.5 Quasi-Static (Quasi-Stationary) Electromagnetic Field..
13.1.6 Field Wave Equations…
13.1.7 Field Equations Allowing for Magnetic Currents and Charges..
13.1.8 Stationary Electromagnetic Field….
13.1.9 Fields in Two-Dimensional Inhomogeneous Media and the Concepts of E- and H-Polarization.
13.2 Electromagnetic Energy Flow.
13.2.1 Radiation Conditions..
13.2.2 Poynting’s Theorem in the Time Domain.
13.2.3 Energy Inequality in the Time Domain.
13.2.4 Poynting’s Theorem in the Frequency Domain
13.3 Uniqueness of the Solution of Electromagnetic Field Equations
13.3.1 Boundary-Value Problem.
13.3.2 Uniqueness Theorem for the Unbounded Domain..
13.4 Electromagnetic Green’s Tensors.
13.4.1 Green’s Tensors in the Frequency Domain..
13.4.2 Lorentz Lemma and Reciprocity Relations.
13.4.3 Green’s Tensors in the Time Domain.
References.
Chapter 14: Integral Representations in Electromagnetic Forward Modeling…
14.1 IE Method.
14.1.1 Background (Normal) and Anomalous Parts of the EM Field…
14.1.2 Poynting’s Theorem and Energy Inequality for an Anomalous Field.
14.1.3 IE Method in Two Dimensions…..
14.1.4 Calculation of the First Variation (Fréchet Derivative) of the EM Field for 2-D Models..
14.1.5 IE Method in Three Dimensions.
14.1.6 Calculation of the First Variation (Fréchet Derivative) of the EM Field for 3-D Models.
14.1.7 Fréchet Derivative Calculation Using the Differential Method.
14.2 Family of Linear and Nonlinear Integral Approximations of the EM Field….
14.2.1 Born and Extended Bom Approximations..
14.2.2 QL Approximation and TQL Equation.
14.2.3 QA Solutions for a 3-D EM Field..
14.2.4 QA Solutions for 2-D EM Field.
14.2.5 LN Approximation.
14.2.6 Localized QL Approximation.
14.3 Linear and Nonlinear Approximations of Higher Orders.
14.3.1 Born Series
14.3.2 Contraction Green’s Operator.
14.3.3 Contraction Born Series.
14.3.4 QL Approximation of the Contraction Green’s Operator.
14.3.5 QL Series..
14.3.6 Accuracy Estimation of the QL Approximation of the First and Higher Orders..
14.3.7 QA Series.
14.4 Integral Representations in Numerical Dressing..
14.4.1 Discretization of the Model Parameters…
14.4.2 Galerkin Method for EM Field Discretization.
14.4.3 Discrete Form of EM IEs Based on Boxcar Basis Functions.
14.4.4 Contraction Integral Equation (CIE) Method..
14.4.5 CIE as the Preconditioned Conventional IE.
14.4.6 Matrix Form of Born Approximation.
14.4.7 Matrix Form of QL Approximation…
14.4.8 Matrix Form of QA Approximation.
14.4.9 The Diagonalized Quasi-Analytical (DQA) Approximation.. References.
Chapter 15: Integral Representations in Electromagnetic Inversion
15.1 Linear Inversion Methods.
15.1.1 Excess (Anomalous) Current Inversion.
15.1.2 Born Inversion
15.1.3 Conductivity Imaging by the Born Approximation.
15.1.4 Iterative Born Inversions.
15.2 Nonlinear Inversion..
15.2.1 Formulation of the Nonlinear Inverse Problem.
15.2.2 Fréchet Derivative Calculation.
15.3 Quasi-Linear Inversion.
15.3.1 Principles of Quasi-Linear Inversion.
15.3.2 Quasi-Linear Inversion in Matrix Notations.
15.4 Quasi-Analytical Inversion.
15.4.1 Fréchet Derivative Calculation
15.4.2 Inversion Based on the Quasi-Analytical Method References.
Chapter 16: Electromagnetic Migration Imaging.
16.1 Electromagnetic Migration in the Frequency Domain.
16.1.1 Formulation of the Electromagnetic Inverse Problem as a Minimization of the Energy Flow Functional.
16.1.2 Integral Representations for Electromagnetic Migration Field.
16.1.3 Gradient Direction of the Energy Flow Functional.
16.1.4 Migration Imaging in the Frequency Domain..
16.1.5 Iterative Migration.
16.2 Electromagnetic Migration in the Time Domain..
16.2.1 Time Domain Electromagnetic Migration as the Solution of the Boundary Value Problem.
16.2.2 Minimization of the Residual Electromagnetic Field Energy Flo
16.2.3 Gradient Direction of the Energy Flow Functional in the Time Domain..
16.2.4 Migration Imaging in the Time Domain.
16.2.5 Iterative Migration in the Time Domain.. References…
Chapter 17: Differential Methods in Electromagnetic Modeling and Inversion
17.1 Electromagnetic Modeling as a Boundary-Value Problem…
17.1.1 Field Equations and Boundary Conditions…
17.1.2 Formulation of the EM Field Equations with Respect to Anomalous Field in Anisotropic Medium
17.1.3 Electromagnetic Potential Equations and Boundary Conditions.
17.2 Finite Difference Approximation of the Boundary-Value Problem….
17.2.1 Discretization of Maxwell’s Equations Using a Staggered Grid.
17.2.2 Discretization of the Second Order Differential Equations Using the Balance Method.
17.2.3 Discretization of the Electromagnetic Potential Differential Equations.
17.2.4 Application of the Spectral Lanczos Decomposition Method (SLDM) for Solving the Linear System of Equations for Discrete Electromagnetic Fields….
17.3 Finite Element Solution of Boundary-Value Problems..
17.3.1 Galerkin Method.
17.3.2 Exact Element Method.
17.3.3 Edge-Based Finite Element Method.
17.4 Inversion Based on Differential Methods..
17.4.1 Formulation of the Inverse Problem on the Discrete Grid.
17.4.2 Fréchet Derivative Calculation Using Finite Difference Methods.. References…
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