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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 | |