Inverse Problems: Basics, Theory and Applications in Geophysics

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