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Geophysical data analysis : Discrete inverse theory / William Menke

By: Menke, William [autor].
Publisher: ámsterdam, Boston : Elsevier, ©2012Edition: 3th ed.Description: xxxvi, 293 páginas : ilustraciones, tablas ; 24 cm.Content type: texto Media type: no mediado Carrier type: volumenISBN: 9780123971609.Subject(s): Geofisica -- Medición | Oceonografía -- Medición | Problemas inversos (ecuaciones diferenciales) -- Soluciones numéricasDDC classification: 551
Contents:
Describing Inverse Problems Chapter. -- Some Comments on Probability Theory Chapter. -- Solution of the Linear, Gaussian Inverse Problem, Viewpoint 1: The Length Method Chapter. -- Solution of the Linear, Gaussian Inverse Problem, Viewpoint 2: Generalized Inverses Chapter. -- Solution of the Linear, Gaussian Inverse Problem, Viewpoint 3: Maximum Likelihood Methods. -- Nonuniqueness and Localized Averages. -- Applications of Vector Spaces. -- Linear Inverse Problems and Non-Gaussian Statistics. -- Nonlinear Inverse Problems Chapter. -- Factor Analysis. -- Continuous Inverse Theory and Tomography. -- Sample Inverse Problems. -- Applications of Inverse Theory to Solid Earth.
Summary: The treatment of inverse theory in this book is divided into four parts. Chapters 1 and 2 provide a general background, explaining what inverse problems are and what constitutes their solution as well as reviewing some of the basic concepts from linear algebra and probability theory that will be applied throughout the text. Chapters 3-7 discuss the solution of the canonical inverse problem: the linear problem with Gaussian statistics. This is the best understood of all inverse problems; and it is here that the fundamental notions of uncertainty, uniqueness, and resolution can be most clearly developed. Chapters 8-11 extend the discussion to problems that are non-Gaussian, nonlinear and continuous. Chapters 12-13 provide examples of the use of inverse theory and a discussion of the steps that must be taken to solve inverse problems on a computer.
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Colección general 551 M545 (Browse shelf) 3rd ed. 2012 1 Available 0000043045
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Enhanced descriptions from Syndetics:

<p>Since 1984, Geophysical Data Analysis has filled the need for a short, concise reference on inverse theory for individuals who have an intermediate background in science and mathematics. The new edition maintains the accessible and succinct manner for which it is known, with the addition of:</p> MATLAB examples and problem sets Advanced color graphics Coverage of new topics, including Adjoint Methods; Inversion by Steepest Descent, Monte Carlo and Simulated Annealing methods; and Bootstrap algorithm for determining empirical confidence intervals

Describing Inverse Problems Chapter. -- Some Comments on Probability Theory Chapter. -- Solution of the Linear, Gaussian Inverse Problem, Viewpoint 1: The Length Method Chapter. -- Solution of the Linear, Gaussian Inverse Problem, Viewpoint 2: Generalized Inverses Chapter. -- Solution of the Linear, Gaussian Inverse Problem, Viewpoint 3: Maximum Likelihood Methods. -- Nonuniqueness and Localized Averages. -- Applications of Vector Spaces. -- Linear Inverse Problems and Non-Gaussian Statistics. -- Nonlinear Inverse Problems Chapter. -- Factor Analysis. -- Continuous Inverse Theory and Tomography. -- Sample Inverse Problems. -- Applications of Inverse Theory to Solid Earth.

The treatment of inverse theory in this book is divided into four parts. Chapters 1 and 2 provide a general background, explaining what inverse problems are and what constitutes their solution as well as reviewing some of the basic concepts from linear algebra and probability theory that will be applied throughout the text. Chapters 3-7 discuss the solution of the canonical inverse problem: the linear problem with Gaussian statistics. This is the best understood of all inverse problems; and it is here that the fundamental notions of uncertainty, uniqueness, and resolution can be most clearly developed. Chapters 8-11 extend the discussion to problems that are non-Gaussian, nonlinear and continuous. Chapters 12-13 provide examples of the use of inverse theory and a discussion of the steps that must be taken to solve inverse problems on a computer.

Table of contents provided by Syndetics

  • Introduction (p. xv)
  • 1 Describing Inverse Problems
  • 1.1 Formulating Inverse Problems (p. 1)
  • 1.1.1 Implicit Linear Form (p. 2)
  • 1.1.2 Explicit Form (p. 2)
  • 1.1.3 Explicit Linear Form (p. 3)
  • 1.2 The Linear Inverse Problem (p. 3)
  • 1.3 Examples of Formulating Inverse Problems (p. 4)
  • 1.3.1 Example 1: Fitting a Straight Line (p. 4)
  • 1.3.2 Example 2: Fitting a Parabola (p. 5)
  • 1.3.3 Example 3: Acoustic Tomography (p. 6)
  • 1.3.4 Example 4: X-ray Imaging (p. 7)
  • 1.3.5 Example 5: Spectral Curve Fitting (p. 9)
  • 1.3.6 Example 6: Factor Analysis (p. 10)
  • 1.4 Solutions to Inverse Problems (p. 11)
  • 1.4.1 Estimates of Model Parameters (p. 11)
  • 1.4.2 Bounding Values (p. 12)
  • 1.4.3 Probability Density Functions (p. 12)
  • 1.4.4 Sets of Realizations of Model Parameters (p. 13)
  • 1.4.5 Weighted Averages of Model Parameters (p. 13)
  • 1.5 Problems (p. 13)
  • 2 Some Comments on Probability Theory
  • 2.1 Noise and Random Variables (p. 15)
  • 2.2 Correlated Data (p. 19)
  • 2.3 Functions of Random Variables (p. 21)
  • 2.4 Gaussian Probability Density Functions (p. 26)
  • 2.5 Testing the Assumption of Gaussian Statistics (p. 29)
  • 2.6 Conditional Probability Density Functions (p. 30)
  • 2.7 Confidence Intervals (p. 33)
  • 2.8 Computing Realizations of Random Variables (p. 34)
  • 2.9 Problems (p. 37)
  • 3 Solution of the Linear, Gaussian Inverse Problem, Viewpoint 1: The Length Method
  • 3.1 The Lengths of Estimates (p. 39)
  • 3.2 Measures of Length (p. 39)
  • 3.3 Least Squares for a Straight Line (p. 43)
  • 3.4 The Least Squares Solution of the Linear Inverse Problem (p. 44)
  • 3.5 Some Examples (p. 46)
  • 3.5.1 The Straight Line Problem (p. 46)
  • 3.5.2 Fitting a Parabola (p. 47)
  • 3.5.3 Fitting a Plane Surface (p. 48)
  • 3.6 The Existence of the Least Squares Solution (p. 49)
  • 3.6.1 Underdetermined Problems (p. 51)
  • 3.6.2 Even-Determined Problems (p. 52)
  • 3.6.3 Overdetermined Problems (p. 52)
  • 3.7 The Purely Underdetermined Problem (p. 52)
  • 3.8 Mixed-Determined Problems (p. 54)
  • 3.9 Weighted Measures of Length as a Type of A Priori Information (p. 56)
  • 3.9.1 Weighted Least Squares (p. 58)
  • 3.9.2 Weighted Minimum Length (p. 58)
  • 3.9.3 Weighted Damped Least Squares (p. 58)
  • 3.10 Other Types of A Priori Information (p. 60)
  • 3.10.1 Example: Constrained Fitting of a Straight Line (p. 62)
  • 3.11 The Variance of the Model Parameter Estimates (p. 63)
  • 3.12 Variance and Prediction Error of the Least Squares Solution (p. 64)
  • 3.13 Problems (p. 67)
  • 4 Solution of the Linear, Gaussian Inverse Problem, Viewpoint 2: Generalized Inverses
  • 4.1 Solutions Versus Operators (p. 69)
  • 4.2 The Data Resolution Matrix (p. 69)
  • 4.3 The Model Resolution Matrix (p. 72)
  • 4.4 The Unit Covariance Matrix (p. 72)
  • 4.5 Resolution and Covariance of Some Generalized Inverses (p. 74)
  • 4.5.1 Least Squares (p. 74)
  • 4.5.2 Minimum Length (p. 75)
  • 4.6 Measures of Goodness of Resolution and Covariance (p. 75)
  • 4.7 Generalized Inverses with Good Resolution and Covariance (p. 76)
  • 4.7.1 Overdetermined Case (p. 76)
  • 4.7.2 Underdetermined Case (p. 77)
  • 4.7.3 The General Case with Dirichlet Spread Functions (p. 77)
  • 4.8 Sidelobes and the Backus-Gilbert Spread Function (p. 78)
  • 4.9 The Backus-Gilbert Generalized Inverse for the Underdetermined Problem (p. 79)
  • 4.10 Including the Covariance Size (p. 83)
  • 4.11 The Trade-off of Resolution and Variance (p. 84)
  • 4.12 Techniques for Computing Resolution (p. 86)
  • 4.13 Problems (p. 88)
  • 5 Solution of the Linear, Gaussian Inverse Problem, Viewpoint 3: Maximum Likelihood Methods
  • 5.1 The Mean of a Group of Measurements (p. 89)
  • 5.2 Maximum Likelihood Applied to Inverse Problem (p. 92)
  • 5.2.1 The Simplest Case (p. 92)
  • 5.2.2 A Priori Distributions (p. 92)
  • 5.2.3 Maximum Likelihood for an Exact Theory (p. 97)
  • 5.2.4 Inexact Theories (p. 100)
  • 5.2.5 The Simple Gaussian Case with a Linear Theory (p. 102)
  • 5.2.6 The General Linear, Gaussian Case (p. 104)
  • 5.2.7 Exact Data and Theory (p. 107)
  • 5.2.8 Infinitely Inexact Data and Theory (p. 108)
  • 5.2.9 No A Priori Knowledge of the Model Parameters (p. 108)
  • 5.3 Relative Entropy as a Guiding Principle (p. 108)
  • 5.4 Equivalence of the Three Viewpoints (p. 110)
  • 5.5 The F-Test of Error Improvement Significance (p. 111)
  • 5.6 Problems (p. 113)
  • 6 Nonuniqueness and Localized Averages
  • 6.1 Null Vectors and Nonuniqueness (p. 115)
  • 6.2 Null Vectors of a Simple Inverse Problem (p. 116)
  • 6.3 Localized Averages of Model Parameters (p. 117)
  • 6.4 Relationship to the Resolution Matrix (p. 117)
  • 6.5 Averages Versus Estimates (p. 118)
  • 6.6 Nonunique Averaging Vectors and A Priori Information (p. 119)
  • 6.7 Problems (p. 121)
  • 7 Applications of Vector Spaces
  • 7.1 Model and Data Spaces (p. 123)
  • 7.2 Householder Transformations (p. 124)
  • 7.3 Designing Householder Transformations (p. 127)
  • 7.4 Transformations That Do Not Preserve Length (p. 129)
  • 7.5 The Solution of the Mixed-Determined Problem (p. 130)
  • 7.6 Singular-Value Decomposition and the Natural Generalized Inverse (p. 132)
  • 7.7 Derivation of the Singular-Value Decomposition (p. 138)
  • 7.8 Simplifying Linear Equality and Inequality Constraints (p. 138)
  • 7.8.1 Linear Equality Constraints (p. 139)
  • 7.8.2 Linear Inequality Constraints (p. 139)
  • 7.9 Inequality Constraints (p. 140)
  • 7.10 Problems (p. 147)
  • 8 Linear Inverse Problems and Non-Gaussian Statistics
  • 8.1 L 1 Norms and Exponential Probability Density Functions (p. 149)
  • 8.2 Maximum Likelihood Estimate of the Mean of an Exponential Probability Density Function (p. 151)
  • 8.3 The General Linear Problem (p. 153)
  • 8.4 Solving L 1 Norm Problems (p. 153)
  • 8.5 The L ∞ Norm
  • 8.6 Problems (p. 160)
  • 9 Nonlinear Inverse Problems
  • 9.1 Parameterizations (p. 163)
  • 9.2 Linearizing Transformations (p. 165)
  • 9.3 Error and Likelihood in Nonlinear Inverse Problems (p. 166)
  • 9.4 The Grid Search (p. 167)
  • 9.5 The Monte Carlo Search (p. 170)
  • 9.6 Newton's Method (p. 171)
  • 9.7 The Implicit Nonlinear Inverse Problem with Gaussian Data (p. 175)
  • 9.8 Gradient Method (p. 180)
  • 9.9 Simulated Annealing (p. 181)
  • 9.10 Choosing the Null Distribution for Inexact Non-Gaussian Nonlinear Theories (p. 184)
  • 9.11 Bootstrap Confidence Intervals (p. 185)
  • 9.12 Problems (p. 186)
  • 10 Factor Analysis
  • 10.1 The Factor Analysis Problem (p. 189)
  • 10.2 Normalization and Physicality Constraints (p. 194)
  • 10.3 Q-Mode and R-Mode Factor Analysis (p. 199)
  • 10.4 Empirical Orthogonal Function Analysis (p. 199)
  • 10.5 Problems (p. 204)
  • 11 Continuous Inverse Theory and Tomography
  • 11.1 The Backus-Gilbert Inverse Problem (p. 207)
  • 11.2 Resolution and Variance Trade-Off (p. 209)
  • 11.3 Approximating Continuous Inverse Problems as Discrete Problems (p. 209)
  • 11.4 Tomography and Continuous Inverse Theory (p. 211)
  • 11.5 Tomography and the Radon Transform (p. 212)
  • 11.6 The Fourier Slice Theorem (p. 213)
  • 11.7 Correspondence Between Matrices and Linear Operators (p. 214)
  • 11.8 The Frechet Derivative (p. 218)
  • 11.9 The Frechet Derivative of Error (p. 218)
  • 11.10 Backprojection (p. 219)
  • 11.11 Frechet Derivatives Involving a Differential Equation (p. 222)
  • 11.12 Problems (p. 227)
  • 12 Sample Inverse Problems
  • 12.1 An Image Enhancement Problem (p. 231)
  • 12.2 Digital Filter Design (p. 234)
  • 12.3 Adjustment of Crossover Errors (p. 236)
  • 12.4 An Acoustic Tomography Problem (p. 240)
  • 12.5 One-Dimensional Temperature Distribution (p. 241)
  • 12.6 L 1 L 2 , and L ∞ Fitting of a Straight Line (p. 245)
  • 12.7 Finding the Mean of a Set of Unit Vectors (p. 246)
  • 12.8 Gaussian and Lorentzian Curve Fitting (p. 250)
  • 12.9 Earthquake Location (p. 252)
  • 12.10 Vibrational Problems (p. 256)
  • 12.11 Problems (p. 259)
  • 13 Applications of Inverse Theory to Solid Earth Geophysics
  • 13.1 Earthquake Location and Determination of the Velocity Structure of the Earth from Travel Time Data (p. 261)
  • 13.2 Moment Tensors of Earthquakes (p. 264)
  • 13.3 Waveform "Tomography" (p. 265)
  • 13.4 Velocity Structure from Free Oscillations and Seismic Surface Waves (p. 267)
  • 13.5 Seismic Attenuation (p. 269)
  • 13.6 Signal Correlation (p. 270)
  • 13.7 Tectonic Plate Motions (p. 271)
  • 13.8 Gravity and Geomagnetism (p. 271)
  • 13.9 Electromagnetic Induction and the Magnetotelluric Method (p. 273)
  • 14 Appendices
  • 14.1 Implementing Constraints with Lagrange multipliers (p. 277)
  • 14.2 L 2 Inverse Theory with Complex Quantities (p. 278)
  • Index (p. 281)

Author notes provided by Syndetics

William Menke is a Professor of Earth and Environmental Sciences at Columbia University, USA. His research focuses on the development of data analysis algorithms for time series analysis and imaging in the earth and environmental sciences and the application of these methods to volcanoes, earthquakes and other natural hazards.

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