Efnisyfirlit

• 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

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