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Geophysical Data Analysis Discrete Inverse Theory

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ISBN-10: 0123971608

ISBN-13: 9780123971609

Edition: 3rd 2012

Authors: William Menke

List price: $61.99
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Please use extracts from reviews of first editionKey Features* Updated and thoroughly revised edition* additional material on geophysical/acoustic tomography* Detailed discussion of application of inverse theory to tectonic, gravitational and geomagnetic studies
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Book details

List price: $61.99
Edition: 3rd
Copyright year: 2012
Publisher: Elsevier Science & Technology
Publication date: 6/26/2012
Binding: Hardcover
Pages: 330
Size: 6.00" wide x 9.00" long x 1.00" tall
Weight: 1.540
Language: English

Describing Inverse Problems
Formulating Inverse Problems
Implicit Linear Form
Explicit Form
Explicit Linear Form
The Linear Inverse Problem
Examples of Formulating Inverse Problems
Example 1: Fitting a Straight Line
Example 2: Fitting a Parabola
Example 3: Acoustic Tomography
Example 4: X-ray Imaging
Example 5: Spectral Curve Fitting
Example 6: Factor Analysis
Solutions to Inverse Problems
Estimates of Model Parameters
Bounding Values
Probability Density Functions
Sets of Realizations of Model Parameters
Weighted Averages of Model Parameters
Some Comments on Probability Theory
Noise and Random Variables
Correlated Data
Functions of Random Variables
Gaussian Probability Density Functions
Testing the Assumption of Gaussian Statistics
Conditional Probability Density Functions
Confidence Intervals
Computing Realizations of Random Variables
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 Straight Line Problem
Fitting a Parabola
Fitting a Plane Surface
The Existence of the Least Squares Solution
Underdetermined Problems
Even-Determined Problems
Overdetermined Problems
The Purely Underdetermined Problem
Mixed-Determined Problems
Weighted Measures of Length as a Type of A Priori Information
Weighted Least Squares
Weighted Minimum Length
Weighted Damped Least Squares
Other Types of A Priori Information
Example: Constrained Fitting of a Straight Line
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
Least Squares
Minimum Length
Measures of Goodness of Resolution and Covariance
Generalized Inverses with Good Resolution and Covariance
Overdetermined Case
Underdetermined Case
The General Case with Dirichlet Spread Functions
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
Techniques for Computing Resolution
Solution of the Linear, Gaussian Inverse Problem, Viewpoint 3: Maximum Likelihood Methods
The Mean of a Group of Measurements
Maximum Likelihood Applied to Inverse Problem
The Simplest Case
A Priori Distributions
Maximum Likelihood for an Exact Theory
Inexact Theories
The Simple Gaussian Case with a Linear Theory
The General Linear, Gaussian Case
Exact Data and Theory
Infinitely Inexact Data and Theory
No A Priori Knowledge of the Model Parameters
Relative Entropy as a Guiding Principle
Equivalence of the Three Viewpoints
The F-Test of Error Improvement Significance
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
Linear Equality Constraints
Linear Inequality Constraints
Inequality Constraints
Linear Inverse Problems and Non-Gaussian Statistics
L<sub>1</sub> Norms and Exponential Probability Density Functions
Maximum Likelihood Estimate of the Mean of an Exponential Probability Density Function
The General Linear Problem
Solving L<sub>1</sub> Norm Problems
The L<sub>&#8734;</sub> Norm
Nonlinear Inverse Problems
Linearizing Transformations
Error and Likelihood in Nonlinear Inverse Problems
The Grid Search
The Monte Carlo Search
Newton's Method
The Implicit Nonlinear Inverse Problem with Gaussian Data
Gradient Method
Simulated Annealing
Choosing the Null Distribution for Inexact Non-Gaussian Nonlinear Theories
Bootstrap Confidence Intervals
Factor Analysis
The Factor Analysis Problem
Normalization and Physicality Constraints
Q-Mode and R-Mode Factor Analysis
Empirical Orthogonal Function Analysis
Continuous Inverse Theory and Tomography
The Backus-Gilbert Inverse Problem
Resolution and Variance Trade-Off
Approximating Continuous Inverse Problems as Discrete Problems
Tomography and Continuous Inverse Theory
Tomography and the Radon Transform
The Fourier Slice Theorem
Correspondence Between Matrices and Linear Operators
The Frechet Derivative
The Frechet Derivative of Error
Frechet Derivatives Involving a Differential Equation
Sample Inverse Problems
An Image Enhancement Problem
Digital Filter Design
Adjustment of Crossover Errors
An Acoustic Tomography Problem
One-Dimensional Temperature Distribution
L<sub>1</sub> L<sub>2</sub>, and L<sub>&#8734;</sub> Fitting of a Straight Line
Finding the Mean of a Set of Unit Vectors
Gaussian and Lorentzian Curve Fitting
Earthquake Location
Vibrational Problems
Applications of Inverse Theory to Solid Earth Geophysics
Earthquake Location and Determination of the Velocity Structure of the Earth from Travel Time Data
Moment Tensors of Earthquakes
Waveform "Tomography"
Velocity Structure from Free Oscillations and Seismic Surface Waves
Seismic Attenuation
Signal Correlation
Tectonic Plate Motions
Gravity and Geomagnetism
Electromagnetic Induction and the Magnetotelluric Method
Implementing Constraints with Lagrange multipliers
L<sub>2</sub> Inverse Theory with Complex Quantities