| Preface | p. xi |
| The General Discrete Inverse Problem | p. 1 |
| Model Space and Data Space | p. 1 |
| States of Information | p. 6 |
| Forward Problem | p. 20 |
| Measurements and A Priori Information | p. 24 |
| Defining the Solution of the Inverse Problem | p. 32 |
| Using the Solution of the Inverse Problem | p. 37 |
| Monte Carlo Methods | p. 41 |
| Introduction | p. 41 |
| The Movie Strategy for Inverse Problems | p. 44 |
| Sampling Methods | p. 48 |
| Monte Carlo Solution to Inverse Problems | p. 51 |
| Simulated Annealing | p. 54 |
| The Least-Squares Criterion | p. 57 |
| Preamble: The Mathematics of Linear Spaces | p. 57 |
| The Least-Squares Problem | p. 62 |
| Estimating Posterior Uncertainties | p. 70 |
| Least-Squares Gradient and Hessian | p. 75 |
| Least-Absolute-Values Criterion and Minimax Criterion | p. 81 |
| Introduction | p. 81 |
| Preamble: e[subscript p]-Norms | p. 82 |
| The e[subscript p]-Norm Problem | p. 86 |
| The e[subscript 1]-Norm Criterion for Inverse Problems | p. 89 |
| The e[subscript infinity]-Norm Criterion for Inverse Problems | p. 96 |
| Functional Inverse Problems | p. 101 |
| Random Functions | p. 101 |
| Solution of General Inverse Problems | p. 108 |
| Introduction to Functional Least Squares | p. 108 |
| Derivative and Transpose Operators in Functional Spaces | p. 119 |
| General Least-Squares Inversion | p. 133 |
| Example: X-Ray Tomography as an Inverse Problem | p. 140 |
| Example: Travel-Time Tomography | p. 143 |
| Example: Nonlinear Inversion of Elastic Waveforms | p. 144 |
| Appendices | p. 159 |
| Volumetric Probability and Probability Density | p. 159 |
| Homogeneous Probability Distributions | p. 160 |
| Homogeneous Distribution for Elastic Parameters | p. 164 |
| Homogeneous Distribution for Second-Rank Tensors | p. 170 |
| Central Estimators and Estimators of Dispersion | p. 170 |
| Generalized Gaussian | p. 174 |
| Log-Normal Probability Density | p. 175 |
| Chi-Squared Probability Density | p. 177 |
| Monte Carlo Method of Numerical Integration | p. 179 |
| Sequential Random Realization | p. 181 |
| Cascaded Metropolis Algorithm | p. 182 |
| Distance and Norm | p. 183 |
| The Different Meanings of the Word Kernel | p. 183 |
| Transpose and Adjoint of a Differential Operator | p. 184 |
| The Bayesian Viewpoint of Backus (1970) | p. 190 |
| The Method of Backus and Gilbert | p. 191 |
| Disjunction and Conjunction of Probabilities | p. 195 |
| Partition of Data into Subsets | p. 197 |
| Marginalizing in Linear Least Squares | p. 200 |
| Relative Information of Two Gaussians | p. 201 |
| Convolution of Two Gaussians | p. 202 |
| Gradient-Based Optimization Algorithms | p. 203 |
| Elements of Linear Programming | p. 223 |
| Spaces and Operators | p. 230 |
| Usual Functional Spaces | p. 242 |
| Maximum Entropy Probability Density | p. 245 |
| Two Properties of e[subscript p]-Norms | p. 246 |
| Discrete Derivative Operator | p. 247 |
| Lagrange Parameters | p. 249 |
| Matrix Identities | p. 249 |
| Inverse of a Partitioned Matrix | p. 250 |
| Norm of the Generalized Gaussian | p. 250 |
| Problems | p. 253 |
| Estimation of the Epicentral Coordinates of a Seismic Event | p. 253 |
| Measuring the Acceleration of Gravity | p. 256 |
| Elementary Approach to Tomography | p. 259 |
| Linear Regression with Rounding Errors | p. 266 |
| Usual Least-Squares Regression | p. 269 |
| Least-Squares Regression with Uncertainties in Both Axes | p. 273 |
| Linear Regression with an Outlier | p. 275 |
| Condition Number and A Posteriori Uncertainties | p. 279 |
| Conjunction of Two Probability Distributions | p. 285 |
| Adjoint of a Covariance Operator | p. 288 |
| Problem 7.1 Revisited | p. 289 |
| Problem 7.3 Revisited | p. 289 |
| An Example of Partial Derivatives | p. 290 |
| Shapes of the e[subscript p]-Norm Misfit Functions | p. 290 |
| Using the Simplex Method | p. 293 |
| Problem 7.7 Revisited | p. 295 |
| Geodetic Adjustment with Outliers | p. 296 |
| Inversion of Acoustic Waveforms | p. 297 |
| Using the Backus and Gilbert Method | p. 304 |
| The Coefficients in the Backus and Gilbert Method | p. 308 |
| The Norm Associated with the 1D Exponential Covariance | p. 308 |
| The Norm Associated with the 1D Random Walk | p. 311 |
| The Norm Associated with the 3D Exponential Covariance | p. 313 |
| References and References for General Reading | p. 317 |
| Index | p. 333 |
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