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• Cover image
• Title page
• Table of Contents
• Copyright
• Preface
• Introduction
• I.1 Forward and Inverse Theories
• I.2 MATLAB as a Tool for Learning Inverse Theory
• I.3 A Very Quick MATLAB Tutorial
• I.4 Review of Vectors and Matrices and Their Representation in MATLAB
• I.5 Useful MatLab Operations
• Chapter 1: Describing Inverse Problems
• Abstract
• 1.1 Formulating Inverse Problems
• 1.2 The Linear Inverse Problem
• 1.3 Examples of Formulating Inverse Problems
• 1.4 Solutions to Inverse Problems
• 1.5 Problems
• Chapter 2: Some Comments on Probability Theory
• Abstract
• 2.1 Noise and Random Variables
• 2.2 Correlated Data
• 2.3 Functions of Random Variables
• 2.4 Gaussian Probability Density Functions
• 2.5 Testing the Assumption of Gaussian Statistics
• 2.6 Conditional Probability Density Functions
• 2.7 Confidence Intervals
• 2.8 Computing Realizations of Random Variables
• 2.9 Problems
• Chapter 3: Solution of the Linear, Gaussian Inverse Problem, Viewpoint 1: The Length Method
• Abstract
• 3.1 The Lengths of Estimates
• 3.2 Measures of Length
• 3.3 Least Squares for a Straight Line
• 3.4 The Least Squares Solution of the Linear Inverse Problem
• 3.5 Some Examples
• 3.6 The Existence of the Least Squares Solution
• 3.7 The Purely Underdetermined Problem
• 3.8 Mixed-Determined Problems
• 3.9 Weighted Measures of Length as a Type of Prior Information
• 3.10 Other Types of Prior Information
• 3.11 The Variance of the Model Parameter Estimates
• 3.12 Variance and Prediction Error of the Least Squares Solution
• 3.13 Problems
• Chapter 4: Solution of the Linear, Gaussian Inverse Problem, Viewpoint 2: Generalized Inverses
• Abstract
• 4.1 Solutions Versus Operators
• 4.2 The Data Resolution Matrix
• 4.3 The Model Resolution Matrix
• 4.4 The Unit Covariance Matrix
• 4.5 Resolution and Covariance of Some Generalized Inverses
• 4.6 Measures of Goodness of Resolution and Covariance
• 4.7 Generalized Inverses With Good Resolution and Covariance
• 4.8 Sidelobes and the Backus-Gilbert Spread Function
• 4.9 The Backus-Gilbert Generalized Inverse for the Underdetermined Problem
• 4.10 Including the Covariance Size
• 4.11 The Trade-Off of Resolution and Variance
• 4.12 Checkerboard Tests
• 4.13 Problems
• Chapter 5: Solution of the Linear, Gaussian Inverse Problem, Viewpoint 3: Maximum Likelihood Methods
• Abstract
• 5.1 The Mean of a Group of Measurements
• 5.2 Maximum Likelihood Applied to Inverse Problem
• 5.3 Model Resolution in the Presence of Prior Information
• 5.4 Relative Entropy as a Guiding Principle
• 5.5 Equivalence of the Three Viewpoints
• 5.6 Chi-Square Test for the Compatibility of the Prior and Posterior Error
• 5.7 The F-test of the Error Improvement Significance
• 5.8 Problems
• Chapter 6: Nonuniqueness and Localized Averages
• Abstract
• 6.1 Null Vectors and Nonuniqueness
• 6.2 Null Vectors of a Simple Inverse Problem
• 6.3 Localized Averages of Model Parameters
• 6.4 Relationship to the Resolution Matrix
• 6.5 Averages Versus Estimates
• 6.6 Nonunique Averaging Vectors and Prior Information
• 6.7 End-Member Solutions and Squeezing
• 6.8 Problems
• Chapter 7: Applications of Vector Spaces
• Abstract
• 7.1 Model and Data Spaces
• 7.2 Householder Transformations
• 7.3 Designing Householder Transformations
• 7.4 Transformations That Do Not Preserve Length
• 7.5 The Solution of the Mixed-Determined Problem
• 7.6 Singular-Value Decomposition and the Natural Generalized Inverse
• 7.7 Derivation of the Singular-Value Decomposition
• 7.8 Simplifying Linear Equality and Inequality Constraints
• 7.9 Inequality Constraints
• 7.10 Problems
• Chapter 8: Linear Inverse Problems and Non-Gaussian Statistics
• Abstract
• 8.1 L1 Norms and Exponential Probability Density Functions
• 8.2 Maximum Likelihood Estimate of the Mean of an Exponential Probability Density Function
• 8.3 The General Linear Problem
• 8.4 Solving L1 Norm Problems by Transformation to a Linear Programming Problem
• 8.5 Solving L1 Norm Problems by Reweighted L2 Minimization
• 8.6 The L∞ Norm
• 8.7 The L0 Norm and Sparsity
• 8.8 Problems
• Chapter 9: Nonlinear Inverse Problems
• Abstract
• 9.1 Parameterizations
• 9.2 Linearizing Transformations
• 9.3 Error and Likelihood in Nonlinear Inverse Problems
• 9.4 The Grid Search
• 9.5 The Monte Carlo Search
• 9.6 Newton’s Method
• 9.7 The Implicit Nonlinear Inverse Problem With Gaussian Data
• 9.8 Gradient Method
• 9.9 Simulated Annealing
• 9.10 The Genetic Algorithm
• 9.11 Choosing the Null Distribution for Inexact Non-Gaussian Nonlinear Theories
• 9.12 Bootstrap Confidence Intervals
• 9.13 Problems
• Chapter 10: Factor Analysis
• Abstract
• 10.1 The Factor Analysis Problem
• 10.2 Normalization and Physicality Constraints
• 10.3 Q-Mode and R-Mode Factor Analysis
• 10.4 Empirical Orthogonal Function Analysis
• 10.5 Problems
• Chapter 11: Continuous Inverse Theory and Tomography
• Abstract
• 11.1 The Backus-Gilbert Inverse Problem
• 11.2 Resolution and Variance Trade-Off
• 11.3 Approximating Continuous Inverse Problems as Discrete Problems
• 11.4 Tomography and Continuous Inverse Theory
• 11.5 Tomography and the Radon Transform
• 11.6 The Fourier Slice Theorem
• 11.7 Correspondence Between Matrices and Linear Operators
• 11.8 The Fréchet Derivative
• 11.9 The Fréchet Derivative of Error
• 11.10 Backprojection
• 11.11 Fréchet Derivatives Involving a Differential Equation
• 11.12 Derivative With Respect to a Parameter in a Differential Equation
• 11.13 Problems
• Chapter 12: Sample Inverse Problems
• Abstract
• 12.1 An Image Enhancement Problem
• 12.2 Digital Filter Design
• 12.3 Adjustment of Crossover Errors
• 12.4 An Acoustic Tomography Problem
• 12.5 One-Dimensional Temperature Distribution
• 12.6 L1, L2, and L∞ Fitting of a Straight Line
• 12.7 Finding the Mean of a Set of Unit Vectors
• 12.8 Gaussian and Lorentzian Curve Fitting
• 12.9 Earthquake Location
• 12.10 Vibrational Problems
• 12.11 Problems
• Chapter 13: Applications of Inverse Theory to Solid Earth Geophysics
• Abstract
• 13.1 Earthquake Location and Determination of the Velocity Structure of the Earth From Travel Time Data
• 13.2 Moment Tensors of Earthquakes
• 13.3 Adjoint Methods in Seismic Imaging
• 13.4 Wavefield Tomography
• 13.5 Finite-Frequency Travel Time Tomography
• 13.6 Banana-Doughnut Kernels
• 13.7 Seismic Migration
• 13.8 Velocity Structure From Free Oscillations and Seismic Surface Waves
• 13.9 Seismic Attenuation
• 13.10 Signal Correlation
• 13.11 Tectonic Plate Motions
• 13.12 Gravity and Geomagnetism
• 13.13 Electromagnetic Induction and the Magnetotelluric Method
• 13.14 Problems
• Chapter 14: Appendices
• 14.1 Implementing Constraints With Lagrange multipliers
• 14.2 L2 Inverse Theory With Complex Quantities
• 14.3 Method Summaries
• Index