Preface
1 The General Discrete Inverse Problem
1.1 Model Space and Data Space
1.2 States of Information
1.3 Forward Problem
1.4 Measurements and A Priori Information
1.5 Defining the Solution of the Inverse Problem
1.6 Using the Solution of the Inverse Problem
2 Monte Carlo Methods
2.1 Introduction
2.2 The Movie Strategy for Inverse Problems
2.3 Sampling Methods
2.4 Monte Carlo Solution to Inverse Problems
2.5 Simulated Annealing
3 The Least-Squares Criterion
3.1 Preamble: The Mathematics of Linear Spaces
3.2 The Least-Squares Problem
3.3 Estimating Posterior Uncertainties
3.4 Least-Squares Gradient and Hessian
4 Least-Absolute-Values Criterion and Minimax Criterion
4.1 Introduction
4.2 Preamble:ln-Norms
4.3 The ln-Norm Problem
4.4 The l1-Norm Criterion for Inverse Problems
4.5 The ln-Norm Criterion for Inverse Problems
5 Functional Inverse Problems
5.1 Random Functions
5.2 Solution of General Inverse Problems
5.3 Introduction to Functional Least Squares
5.4 Derivative and Transpose Operators in Functional Spaces
5.5 General Least-Squares Inversion
5.6 Example: X-Ray Tomography as an Inverse Problem
5.7 Example: Travel-Time Tomography
5.8 Example: Nonlinear Inversion of Elastic Waveforms
6 Appendices
6.1 Volumetric Probability and Probability Density
6.2 Homogeneous Probability Distributions
6.3 Homogeneous Distribution for Elastic Parameters
6.4 Homogeneous Distribution for Second-Rank Tensors
6.5 Central Estimators and Estimators of Dispersion
6.6 Generalized Gaussian
6.7 Log-Normal Probability Density
6.8 Chi-Squared Probability Density
6.9 Monte Carlo Method of Numerical Integration
6.10 Sequential Random Realization
6.11 Cascaded Metropolis Algorithm
6.12 Distance and Norm
6.13 The Different Meanings of the Word Kernel
6.14 Transpose and Adjoint of a Differential Operator
6.15 The Bayesian Viewpoint of Backus (1970)
6.16 The Method of Backus and Gilbert
6.17 Disjunction and Conjunction of Probabilities
6.18 Partition of Data into Subsets
6.19 Marginalizing in Linear Least Squares
6.20 Relative Information of Two Gaussians
6.21 Convolution of Two Gaussians
6.22 Gradient-Based Optimization Algorithms
6.23 Elements of Linear Programming
6.24 Spaces and Operators
6.25 Usual Functional Spaces
6.26 Maximum Entropy Probability Density
6.27 Two Properties of ln-Norms
6.28 Discrete Derivative Operator
6.29 Lagrange Parameters
6.30 Matrix Identities
6.31 Inverse of a Partitioned Matrix
6.32 Norm of the Generalized Gaussian
7 Problems
7.1 Estimation of the Epicentral Coordinates of a Seismic Event
7.2 Measuring the Acceleration of Gravity
7.3 Elementary Approach to Tomography
7.4 Linear Regression with Rounding Errors
7.5 Usual Least-Squares Regression
7.6 Least-Squares Regression with Uncertainties in Both Axes
7.7 Linear Regression with an Outlier
7.8 Condition Number and A Posteriori Uncertainties
7.9 Conjunction of Two Probability Distributions
7.10 Adjoint of a Covariance Operator
7.11 Problem 7.1 Revisited
7.12 Problem 7.3 Revisited
7.13 An Example of Partial Derivatives
7.14 Shapes of the ln-Norm Misfit Functions
7.15 Using the Simplex Method
7.16 Problem 7.7 Revisited
7.17 Geodetic Adjustment with Outliers
7.18 Inversion of Acoustic Waveforms
7.19 Using the Backus and Gilbert Method
7.20 The Coefficients in the Backus and Gilbert Method
7.21 The Norm Associated with the 1D Exponential Covariance
7.22 The Norm Associated with the 1D Random Walk
7.23 The Norm Associated with the 3D Exponential Covariance
References and References for General Reading
Index
