Geophysical Data Analysis: Discrete Inverse Theory

Transcription

1 Geophysical Data Analysis: Discrete Inverse Theory MATLAB Edition William Menke Lamont-Doherty Earth Observatory and Department of Earth and Environmental Sciences Columbia University. ' - Palisades, New York ELSEVIER AMSTERDAM BOSTON HEIDELBERG LONDON NEW. YORK OXFORD PARIS SAN DIEGO.' SAN FRANCISCO SYDNEY TOKYO Academic Press is an imprint of Elsevier

2 Introduction xv 1. Describing Inverse Problems Formulating Inverse Problems ' Implicit Linear-Form Explicit Form > Explicit Linear Form " \ ' The Linear Inverse Problem Examples of Formulating Inverse Problems Example 1: Fitting.a Straight Line ' Example 2: Fitting a Parabola Example 3: Acoustic Tomography ;4 ' Example 4: X-ray Imaging. ' ' s ' Example 5: Spectral Curve Fitting Example 6: Factor Analysis Solutions to Inverse Problems ~ Estimates of Model Parameters Bounding Values Probability Density Functions Sets of Realizations of Model Parameters Weighted Averages of Model Parameters, Problems Some Comments on Probability Theory 2.1 Noise and Random Variables Correlated Data Functions of Random Variables Gaussian Probability Density Functions \ Testing the Assumption of Gaussian Statistics Conditional Probability Density Functions Confidence Intervals ; Computing Realizations of Random Variables Problems 37

3 Contents 3. Solution of the Linear, Gaussian Inverse Problem, Viewpoint 1: The Length Method 3.1 The Lengths of Estimates Measures of Length Least Squares for a Straight Line The Least Squares Solution of the Linear,Inverse Problem Some Examples The Straight Line Problem " Fitting a Parabola Fitting a Plane Surface' The Existence of the Least Squares Solution Underdeterrnined Problems " Even-Determined Problems Overdetermined Problems-, The Purely Underdeterrnined Problem Mixed-Determined Problems Weighted Measures of Length as a Type of A Priori Information Weighted Least Squares Weighted Minimum Length Weighted Damped Least Squares Other Types of A Priori Information, Example: Constrained Fitting of a Straight Line i The Variance of the Model Parameter Estimates Variance and Prediction Error of the Least Squares Solution " Problems Solution of the Linear, Gaussian Inverse Problem, Viewpoint 2: Generalized Inverses 4.1 Solutions Versus Operators. s ' 69 ". 4.2 The Data Resolution Matrix The Model Resolution Matrix ' The Unit Covariance Matrix Resolution and Covariance of Some Generalized Inverses Least Squares, ' ' _ / Minimum Length ' Measures of Goodness of Resolution and Covariance Generalized Inverses with Good Resolution and Covariance Overdetermined Case Underdeterrnined Case The General Case with Dirichlet Spread Functions Sidelobes and the Backus-Gilbert Spread Function The Backus-Gilbert Generalized Inverse for the. Underdeterrnined Problem Including the Covariance Size 83

4 Contents 4.11 The Trade-off of Resolution and Variance Techniques for Computing Resolution Problems, Solution of the Linear, Gaussian Inverse Problem, Viewpoint 3: Maximum Likelihood Methods 5.1 The Mean of a Group of Measurements ^ Maximum Likelihood Applied to Inverse Problem The Simplest Case... " A Priori Distributions ' ". ' Maximum Likelihood for an Exact Theory Inexact Theories ' * The Simple Gaussian Case with, a Linear Theory "6 The General Linear, Gaussian Case Exact Data and Theory Infinitely Inexact Data and Theory No A Priori Knowledge of the Model Parameters Relative Entropy as a Guiding Principle Equivalence of the Three Viewpoints The F-Test of Error Improvement Significance ~ Problems * Nonuniqueness and Localized Averages 6.1 NullVectors and Nonuniqueness Null Vectors of a Simple Inverse Problem 116 ' 6.3 Localized Averages of Model Parameters 11 7' 6.4. Relationship to the Resolution Matrix Averages Versus Estimates Nonunique Averaging Vectors and A Priori Information Problems ' Applications of Vector Spaces 7.1 Model and Data Spaces ~ Householder Transformations, -, ' : Designing Householder Transformations \ Transformations That Do Not Preserve Length _ The Solution of the Mixed-Determined Problem Singular-Value Decomposition and the Natural Generalized Inverse Derivation of the Singular-Value Decomposition Simplifying Linear Equality and Inequality Constraints Linear Equality Constraints Linear Inequality Constraints Inequality Constraints ; Problems * 147

5 xii Contents 8. Linear Inverse Problems and Non-Gaussian Statistics 8.1 L-[ Norms and Exponential Probability Density Functions 149 y8.2 Maximum Likelihood Estimate of the Mean of an - Exponential Probability Density Function The General Linear Problem Solving Lt Norm Problems The Loo Norm - J ' Problems Nonlinear Inverse Problems 9.1 Parameterizatiqris, Linearizing Transformations Error and Likelihood in Nonlinear Inverse Problems The Grid Search The Monte Carlo Search Newton's Method The Implicit Nonlinear Inverse Problem with Gaussian Data Gradient Method Simulated Annealing Choosing the Null Distribution for Inexact Non-Gaussian Nonlinear Theories Bootstrap Confidence Intervals Problems Factor Analysis 10.1 The Factor Analysis Problem ' Normalization and Physicality Constraints J Q-Mode and R-Mode Factor Analysis Empirical Orthogonal Function Analysis Problems Continuous Inverse Theory and Tomography The Backus-Gilbert Inverse Problem Resolution and Variance Trade-Off Approximating Continuous Inverse Problems as Discrete Problems Tomography and Continuous Inverse Theory Tomography and the Radon Transform The Fourier Slice Theorem' Correspondence Between Matrices and Linear Operators The Frechet Derivative The Frechet Derivative of Error Backprojection Frechet Derivatives Involving a Differential Equation Problems ;. " 227

6 Contents. 12. Sample Inverse Problems 12.1 An Image Enhancement Problem Digital Filter Design ~ Adjustment of Crossover Errors An Acoustic Tomography Problem One-Dimensional Temperature Distribution L VL 2, and Loo Fitting of a Straight Line Finding the Mean of a Set of Unit Vectors > Gaussian and Lorentziari Curve Fitting Earthquake Location Vibrational Problems Problems " Applications of Inverse Theory to Solid Earth Geophysics ' 13.1 Earthquake Location and Determination of the Velocity Structure of the Earth from Travel Time Data Moment Tensors of Earthquakes Waveform "Tomography" : ' 265 ' 13.4 Velocity Structure from. Free Oscillations and Seismic Surface Waves Seismic Attenuation Signal Correlation, Tectonic Plate Motions Gravity and Geomagnetism Electromagnetic Induction and the MagnetoteJIuric Method Appendices 14.1 Implementing Constraints with Lagrange multipliers L 2 Inverse Theory with Complex Quantities 278 Index 281