Preface. -- Introduction. -- DESCRIBING INVERSE PROBLEMS -- Formulating Inverse Problems. -- The Linear Inverse Problem. -- Examples of Formulating Inverse Problems. -- Solutions to Inverse Problems. -- SOME COMMENTS ON PROBABILITY THEORY -- Noise and Random Variables. -- Correlated Data. -- Functions of Random Variables. -- Gaussian Distributions. -- Testing the Assumption of Gaussian Statistics -- Confidence Intervals. -- SOLUTION OF THE LINEAR, GAUSSIAN INVERSE PROBLEM, VIEWPOINT 1:THE LENGTH METHOD -- 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 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 INVERSES -- Solutions 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 METHODS -- The 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 AVERAGES -- Null 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 SPACES -- 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. -- Inequality Constraints. -- LINEAR INVERSE PROBLEMS AND NON-GAUSSIAN DISTRIBUTIONS -- L1 Norms and Exponential Distributions.