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Introduction | |

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Describing Inverse Problems | |

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Formulating Inverse Problems | |

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Implicit Linear Form | |

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Explicit Form | |

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Explicit Linear Form | |

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The Linear Inverse Problem | |

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Examples of Formulating Inverse Problems | |

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Example 1: Fitting a Straight Line | |

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Example 2: Fitting a Parabola | |

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Example 3: Acoustic Tomography | |

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Example 4: X-ray Imaging | |

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Example 5: Spectral Curve Fitting | |

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Example 6: Factor Analysis | |

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Solutions to Inverse Problems | |

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Estimates of Model Parameters | |

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Bounding Values | |

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Probability Density Functions | |

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Sets of Realizations of Model Parameters | |

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Weighted Averages of Model Parameters | |

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Problems | |

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Some Comments on Probability Theory | |

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Noise and Random Variables | |

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Correlated Data | |

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Functions of Random Variables | |

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Gaussian Probability Density Functions | |

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Testing the Assumption of Gaussian Statistics | |

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Conditional Probability Density Functions | |

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Confidence Intervals | |

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Computing Realizations of Random Variables | |

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Problems | |

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Solution of the Linear, Gaussian Inverse Problem, Viewpoint 1: The Length Method | |

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The Lengths of Estimates | |

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Measures of Length | |

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Least Squares for a Straight Line | |

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The Least Squares Solution of the Linear Inverse Problem | |

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Some Examples | |

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The Straight Line Problem | |

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Fitting a Parabola | |

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Fitting a Plane Surface | |

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The Existence of the Least Squares Solution | |

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Underdetermined Problems | |

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Even-Determined Problems | |

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Overdetermined Problems | |

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The Purely Underdetermined Problem | |

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Mixed-Determined Problems | |

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Weighted Measures of Length as a Type of A Priori Information | |

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Weighted Least Squares | |

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Weighted Minimum Length | |

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Weighted Damped Least Squares | |

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Other Types of A Priori Information | |

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Example: Constrained Fitting of a Straight Line | |

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The Variance of the Model Parameter Estimates | |

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Variance and Prediction Error of the Least Squares Solution | |

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Problems | |

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Solution of the Linear, Gaussian Inverse Problem, Viewpoint 2: Generalized Inverses | |

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Solutions Versus Operators | |

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The Data Resolution Matrix | |

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The Model Resolution Matrix | |

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The Unit Covariance Matrix | |

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Resolution and Covariance of Some Generalized Inverses | |

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Least Squares | |

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Minimum Length | |

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Measures of Goodness of Resolution and Covariance | |

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Generalized Inverses with Good Resolution and Covariance | |

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Overdetermined Case | |

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Underdetermined Case | |

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The General Case with Dirichlet Spread Functions | |

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Sidelobes and the Backus-Gilbert Spread Function | |

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The Backus-Gilbert Generalized Inverse for the Underdetermined Problem | |

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Including the Covariance Size | |

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The Trade-off of Resolution and Variance | |

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Techniques for Computing Resolution | |

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Problems | |

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Solution of the Linear, Gaussian Inverse Problem, Viewpoint 3: Maximum Likelihood Methods | |

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The Mean of a Group of Measurements | |

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Maximum Likelihood Applied to Inverse Problem | |

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The Simplest Case | |

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A Priori Distributions | |

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Maximum Likelihood for an Exact Theory | |

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Inexact Theories | |

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The Simple Gaussian Case with a Linear Theory | |

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The General Linear, Gaussian Case | |

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Exact Data and Theory | |

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Infinitely Inexact Data and Theory | |

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No A Priori Knowledge of the Model Parameters | |

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Relative Entropy as a Guiding Principle | |

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Equivalence of the Three Viewpoints | |

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The F-Test of Error Improvement Significance | |

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Problems | |

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Nonuniqueness and Localized Averages | |

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Null Vectors and Nonuniqueness | |

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Null Vectors of a Simple Inverse Problem | |

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Localized Averages of Model Parameters | |

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Relationship to the Resolution Matrix | |

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Averages Versus Estimates | |

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Nonunique Averaging Vectors and A Priori Information | |

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Problems | |

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Applications of Vector Spaces | |

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Model and Data Spaces | |

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Householder Transformations | |

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Designing Householder Transformations | |

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Transformations That Do Not Preserve Length | |

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The Solution of the Mixed-Determined Problem | |

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Singular-Value Decomposition and the Natural Generalized Inverse | |

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Derivation of the Singular-Value Decomposition | |

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Simplifying Linear Equality and Inequality Constraints | |

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Linear Equality Constraints | |

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Linear Inequality Constraints | |

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Inequality Constraints | |

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Problems | |

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Linear Inverse Problems and Non-Gaussian Statistics | |

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L<sub>1</sub> Norms and Exponential Probability Density Functions | |

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Maximum Likelihood Estimate of the Mean of an Exponential Probability Density Function | |

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The General Linear Problem | |

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Solving L<sub>1</sub> Norm Problems | |

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The L<sub>∞</sub> Norm | |

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Problems | |

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Nonlinear Inverse Problems | |

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Parameterizations | |

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Linearizing Transformations | |

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Error and Likelihood in Nonlinear Inverse Problems | |

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The Grid Search | |

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The Monte Carlo Search | |

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Newton's Method | |

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The Implicit Nonlinear Inverse Problem with Gaussian Data | |

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Gradient Method | |

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Simulated Annealing | |

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Choosing the Null Distribution for Inexact Non-Gaussian Nonlinear Theories | |

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Bootstrap Confidence Intervals | |

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Problems | |

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Factor Analysis | |

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The Factor Analysis Problem | |

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Normalization and Physicality Constraints | |

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Q-Mode and R-Mode Factor Analysis | |

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Empirical Orthogonal Function Analysis | |

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Problems | |

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Continuous Inverse Theory and Tomography | |

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The Backus-Gilbert Inverse Problem | |

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Resolution and Variance Trade-Off | |

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Approximating Continuous Inverse Problems as Discrete Problems | |

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Tomography and Continuous Inverse Theory | |

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Tomography and the Radon Transform | |

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The Fourier Slice Theorem | |

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Correspondence Between Matrices and Linear Operators | |

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The Frechet Derivative | |

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The Frechet Derivative of Error | |

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Backprojection | |

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Frechet Derivatives Involving a Differential Equation | |

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Problems | |

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Sample Inverse Problems | |

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An Image Enhancement Problem | |

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Digital Filter Design | |

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Adjustment of Crossover Errors | |

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An Acoustic Tomography Problem | |

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One-Dimensional Temperature Distribution | |

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L<sub>1</sub> L<sub>2</sub>, and L<sub>∞</sub> Fitting of a Straight Line | |

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Finding the Mean of a Set of Unit Vectors | |

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Gaussian and Lorentzian Curve Fitting | |

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Earthquake Location | |

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Vibrational Problems | |

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Problems | |

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Applications of Inverse Theory to Solid Earth Geophysics | |

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Earthquake Location and Determination of the Velocity Structure of the Earth from Travel Time Data | |

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Moment Tensors of Earthquakes | |

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Waveform "Tomography" | |

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Velocity Structure from Free Oscillations and Seismic Surface Waves | |

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Seismic Attenuation | |

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Signal Correlation | |

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Tectonic Plate Motions | |

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Gravity and Geomagnetism | |

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Electromagnetic Induction and the Magnetotelluric Method | |

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Appendices | |

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Implementing Constraints with Lagrange multipliers | |

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L<sub>2</sub> Inverse Theory with Complex Quantities | |

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Index | |