Geophysical Data Analysis: Volume 45

Preface.Introduction.DESCRIBING INVERSE PROBLEMSFormulating Inverse Problems.The Linear Inverse Problem.Examples of Formulating Inverse Problems.Solutions to Inverse Problems.SOME COMMENTS ON PROBABILITY THEORYNoise and Random Variables.Correlated Data.Functions of Random Variables.Gaussian Distributions.Testing the Assumption of Gaussian StatisticsConfidence Intervals.SOLUTION OF THE LINEAR, GAUSSIAN INVERSE PROBLEM, VIEWPOINT 1:THE LENGTH METHODThe Lengths of Estimates.Measures of Length.Least Squares for a Straight Line.The Least Squares Solution of the Linear Inverse Problem.Some Examples.The Existence of the Least Squares Solution.The Purely Underdetermined Problem.Mixed*b1Determined Problems.Weighted Measures of Length as a Type of A Priori Information.Other Types of A Priori Information.The Variance of the Model Parameter Estimates.Variance and Prediction Error of the Least Squares Solution.SOLUTION OF THE LINEAR, GAUSSIAN INVERSE PROBLEM, VIEWPOINT 2: GENERALIZED INVERSESSolutions versus Operators.The Data Resolution Matrix.The Model Resolution Matrix.The Unit Covariance Matrix.Resolution and Covariance of Some Generalized Inverses.Measures of Goodness of Resolution and Covariance.Generalized Inverses with Good Resolution and Covariance.Sidelobes and the Backus-Gilbert Spread Function.The Backus-Gilbert Generalized Inverse for the Underdetermined Problem.Including the Covariance Size.The Trade-off of Resolution and Variance.SOLUTION OF THE LINEAR, GAUSSIAN INVERSE PROBLEM, VIEWPOINT 3: MAXIMUM LIKELIHOOD METHODSThe Mean of a Group of Measurements.Maximum Likelihood Solution of the Linear Inverse Problem.A Priori Distributions.Maximum Likelihood for an Exact Theory.Inexact Theories.The Simple Gaussian Case with a Linear Theory.The General Linear, Gaussian Case.Equivalence of the Three Viewpoints.The F Test of Error Improvement Significance.Derivation of the Formulas of Section 5.7.NONUNIQUENESS AND LOCALIZED AVERAGESNull Vectors and Nonuniqueness.Null Vectors of a Simple Inverse Problem.Localized Averages of Model Parameters.Relationship to the Resolution Matrix.Averages versus Estimates.Nonunique Averaging Vectors and A Priori Information.APPLICATIONS OF VECTOR SPACESModel 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.Inequality Constraints.LINEAR INVERSE PROBLEMS AND NON-GAUSSIAN DISTRIBUTIONSL1 Norms and Exponential Distributions.