Geophysical Data Analysis

Burlington : : Elsevier Science, , 1984.

1 online resource (273 p.)

Description based upon print version of record.. - Front Cover; Geophysical Data Analysis: Discrete Inverse Theory; Copyright Page; Table of Contents; PREFACE; INTRODUCTION; Chapter 1. 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 Problems; Chapter 2. 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 Intervals. - 3.12 Variance and Prediction Error of the Least Squares SolutionChapter 4. 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 Size4.11 The Trade-off of Resolution and Variance; Chapter 5. 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 5.7Chapter 6. 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 Information; Chapter 7. 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. - Chapter 3. SOLUTION OF THE LINEAR,GAUSSIAN INVERSE PROBLEM, VIEWPOINT 1: THE LENGTH METHOD3.1 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. - Geophysical Data Analysis: Discrete Inverse Theory

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