Inverse Problems: Basics, Theory and Applications in Geophysics

✍ Scribed by Mathias Richter

• Publisher: Springer
• Year: 2021
• Tongue: English
• Leaves: 281
• Series: Lecture Notes in Geosystems Mathematics and Computing
• Edition: 2
• Category: Library

✦ Table of Contents

Preface of the First Edition
Scope
Content
Acknowledgments
Preface of the Second Edition
Contents
1 Characterization of Inverse Problems
1.1 Examples of Inverse Problems
1.2 Ill-Posed Problems
Definition and Practical Significance of Ill-Posedness
Amendments to Definition 1.5
1.3 Model Problems for Inverse Gravimetry
1.4 Model Problems for Full-Waveform Seismic Inversion
2 Discretization of Inverse Problems
2.1 Approximation of Functions
Approximation in One Space Dimension
Approximation in Two Space Dimensions
2.2 Discretization of Linear Problems by Least Squares Methods
Description of the Method
Application to Model Problem 1.12: Linear Waveform Inversion
Analysis of the Method
2.3 Discretization of Fredholm Equations by Collocation Methods
Description of the Method
Application to Model Problem 1.12: Linear Waveform Inversion
Analysis of the Method
2.4 The Backus–Gilbert Method and the Approximative Inverse
Description of the Backus–Gilbert Method
Application to Model Problem 1.12: Linear Waveform Inversion
Analysis of the Method
The Approximative Inverse
2.5 Discrete Fourier Inversion of Convolutional Equations
Description of the Method
Application to Model Problem 1.10 of Inverse Gravimetry
Analysis of the Method
2.6 Discretization of Nonlinear Inverse Gravimetry
Multiscale Discretizations
Further Reading
2.7 Discretization of Nonlinear Waveform Inversion
Discretization of the Initial/Boundary-Value Problem
Discretization of the Operator T
Discrete Inverse Model Problem
Further Reading
3 Regularization of Linear Inverse Problems
3.1 Linear Least Squares Problems
3.2 Sensitivity Analysis of Linear Least Squares Problems
Matrices with Deficient Rank
Generalized Least Squares Problems
3.3 The Concept of Regularization
3.4 Tikhonov Regularization
Analysis of Problem 3.20 and Its Practical Solution
Generalization of Tikhonov Regularization
3.5 Discrepancy Principle
Random Data Perturbations
Examples
Other Heuristics to Determine Regularization Parameters
3.6 Reduction of Least Squares Regularization to Standard Form
Summary
3.7 Regularization of the Backus–Gilbert Method
3.8 Regularization of Fourier Inversion
Application to Linear Inverse Gravimetry
3.9 Landweber Iteration and the Curve of Steepest Descent
3.10 The Conjugate Gradient Method
Conjugate Gradient Method for Linear Systems of Equations
Conjugate Gradient Method for Linear Least Squares Problems
4 Regularization of Nonlinear Inverse Problems
4.1 Tikhonov Regularization of Nonlinear Problems
4.2 Nonlinear Least Squares Problems
4.3 Computation of Derivatives
4.4 Iterative Regularization
4.5 Solution of Model Problem for Nonlinear Inverse Gravimetry
Reconstruction by Smoothing Regularization
Reconstruction by Variation Diminishing Regularization
Reconstruction by Iterative Regularization
Further Reading
4.6 Solution of Model Problem for Nonlinear Waveform Inversion
Specification of an Example Case for Waveform Inversion
Further Reading
A Results from Linear Algebra
A.1 The Singular Value Decomposition (SVD)
B Function Spaces
B.1 Linear Spaces
B.2 Operators
B.3 Normed Spaces
B.4 Inner Product Spaces
B.5 Convexity, Best Approximation
C The Fourier Transform
C.1 One-Dimensional Discrete Fourier Transform
C.2 Discrete Fourier Transform for Non-equidistant Samples
C.3 Error Estimates for Fourier Inversion in Sect.2.5
D Regularization Property of CGNE
E Existence and Uniqueness Theorems for Waveform Inversion
E.1 Wave Equation with Constant Coefficient
E.2 Identifiability of Acoustic Impedance
References
Index