Geophysical Data Analysis: Discrete Inverse Theory

PrefaceIntroduction1 Describing Inverse Problems 1.1 Formulating Inverse Problems 1.2 The Linear Inverse Problem 1.3 Examples of Formulating Inverse Problems 1.4 Solutions to Inverse Problems2 Some Comments on Probability Theory 2.1 Noise and Random Variables 2.2 Correlated Data 2.3 Functions of Random Variables 2.4 Gaussian Distributions 2.5 Testing the Assumption of Gaussian Statistics 2.6 Confidence Intervals3 Solution of the Linear, Gaussian Inverse Problem, Viewpoint 1: The Length Method 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 A Priori Information 3.10 Other Types of A Priori Information 3.11 The Variance of the Model Parameter Estimates 3.12 Variance and Prediction Error of the Least Squares Solution4 Solution of the Linear, Gaussian Inverse Problem, Viewpoint 2: Generalized Inverses 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 Variance5 Solution of the Linear, Gaussian Inverse Problem, Viewpoint 3: Maximum Likelihood Methods 5.1 The Mean of a Group of Measurements 5.2 Maximum Likelihood Solution of the Linear Inverse Problem 5.3 A Priori Distributions 5.4 Maximum Likelihood for an Exact Theory 5.5 Inexact Theories 5.6 The Simple Gaussian Case with a Linear Theory 5.7 The General Linear, Gaussian Case 5.8 Equivalence of the Three Viewpoints 5.9 The F Test of Error Improvement Significance 5.10 Derivation of the Formulas of Section 5.76 Nonuniqueness and Localized Averages 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 A Priori Information7 Applications of Vector Spaces 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 Constraints8 Linear Inverse Problems and Non-Gaussian Distributions 8.1 L1 Norms and Exponential Distributions 8.2 Maximum Likelihood Estimate of the Mean of an Exponential Distribution 8.3 The General Linear Problem 8.4 Solving L1 Norm Problems 8.5 The L∞ Norm9 Nonlinear Inverse Problems 9.1 Parameterizations 9.2 Linearizing Parameterizations 9.3 The Nonlinear Inverse Problem with Gaussian Data 9.4 Special Cases 9.5 Convergence and Nonuniqueness of Nonlinear L2 Problems 9.6 Non-Gaussian Distributions 9.7 Maximum Entropy Methods10 Factor Analysis 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 Analysis11 Sample Inverse Problems 11.1 An Image Enhancement Problem 11.2 Digital Filter Design 11.3 Adjustment of Crossover Errors 11.4 An Acoustic Tomography Problem 11.5 Temperature Distribution in an Igneous Intrusion 11.6 L1, L2, and L∞ Fitting of a Straight Line 11.7 Finding the Mean of a Set of Unit Vectors 11.8 Gaussian Curve Fitting 11.9 Earthquake Location 11.10 Vibrational Problems12 Numerical Algorithms 12.1 Solving Even-Determined Problems 12.2 Inverting a Square Matrix 12.3 Solving Underdetermined and Overdetermined Problems 12.4 L2 Problems with Inequality Constraints 12.5 Finding the Eigenvalues and Eigenvectors of a Real Symmetric Matrix 12.6 The Singular-Value Decomposition of a Matrix 12.7 The Simplex Method and the Linear Programming ProblemAppendix A: Implementing Constraints with Lagrange MultipliersAppendix B: Inverse Theory with Complex QuantitiesReferencesIndex