Geophysical Data Analysis: Discrete Inverse Theory
MATLAB Edition
- 3rd Edition, Volume 45 - August 19, 2016
- Author: William Menke
- Language: English
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 ma… Read more
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:
- 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
- Additional material on probability, including Bayesian influence, probability density function, and metropolis algorithm
- Detailed discussion of application of inverse theory to tectonic, gravitational and geomagnetic studies
- Numerous examples and end-of-chapter homework problems help you explore and further understand the ideas presented
- Use as classroom text facilitated by a complete set of exemplary lectures in Microsoft PowerPoint format and homework problem solutions for instructors
Graduate students and researchers in solid earth geophysics, seismology, atmospheric sciences and other areas of applied physics (e.g. image processing) and mathematics.
Chapter 1. Describing Inverse Problems
1.1 Formulating Inverse Problems
1.2 The Linear Inverse Problem
1.3 Examples of Formulating Inverse Problems
1.4 Solutions to Inverse Problems
1.5 Problems
REFERENCES
Chapter 2. Some Comments on Probability Theory
2.1 Noise and Random Variables
2.2 Correlated Data
2.3 Functions of Random Variables
2.4 Gaussian Probability Density Functions
2.5 Testing the Assumption of Gaussian Statistics
2.6 Conditional Probability Density Functions
2.7 Confidence Intervals
2.8 Computing Realizations of Random Variables
2.9 Problems
REFERENCES
Chapter 3. Solution of the Linear, Gaussian Inverse Problem, Viewpoint 1: The Length Method
3.1 The Lengths of Estimates
3.2 Measures of Length
3.3 Least Squares for a Straight Line
3.4 The Least Squares Solution of the Linear Inverse Problem
3.5 Some Examples
3.6 The Existence of the Least Squares Solution
3.7 The Purely Underdetermined Problem
3.8 Mixed-Determined Problems
3.9 Weighted Measures of Length as a Type of A Priori Information
3.10 Other Types of A Priori Information
3.11 The Variance of the Model Parameter Estimates
3.12 Variance and Prediction Error of the Least Squares Solution
3.13 Problems
REFERENCES
Chapter 4. Solution of the Linear, Gaussian Inverse Problem, Viewpoint 2: Generalized Inverses
4.1 Solutions Versus Operators
4.2 The Data Resolution Matrix
4.3 The Model Resolution Matrix
4.4 The Unit Covariance Matrix
4.5 Resolution and Covariance of Some Generalized Inverses
4.6 Measures of Goodness of Resolution and Covariance
4.7 Generalized Inverses with Good Resolution and Covariance
4.8 Sidelobes and the Backus-Gilbert Spread Function
4.9 The Backus-Gilbert Generalized Inverse for the Underdetermined Problem
4.10 Including the Covariance Size
4.11 The Trade-off of Resolution and Variance
4.12 Techniques for Computing Resolution
4.13 Problems
REFERENCES
Chapter 5. Solution of the Linear, Gaussian Inverse Problem, Viewpoint 3: Maximum Likelihood Methods
5.1 The Mean of a Group of Measurements
5.2 Maximum Likelihood Applied to Inverse Problem
5.3 Relative Entropy as a Guiding Principle
5.4 Equivalence of the Three Viewpoints
5.5 The F-Test of Error Improvement Significance
5.6 Problems
REFERENCES
Chapter 6. Nonuniqueness and Localized Averages
6.1 Null Vectors and Nonuniqueness
6.2 Null Vectors of a Simple Inverse Problem
6.3 Localized Averages of Model Parameters
6.4 Relationship to the Resolution Matrix
6.5 Averages Versus Estimates
6.6 Nonunique Averaging Vectors and A Priori Information
6.7 Problems
REFERENCES
Chapter 7. Applications of Vector Spaces
7.1 Model and Data Spaces
7.2 Householder Transformations
7.3 Designing Householder Transformations
7.4 Transformations That Do Not Preserve Length
7.5 The Solution of the Mixed-Determined Problem
7.6 Singular-Value Decomposition and the Natural Generalized Inverse
7.7 Derivation of the Singular-Value Decomposition
7.8 Simplifying Linear Equality and Inequality Constraints
7.9 Inequality Constraints
7.10 Problems
REFERENCES
Chapter 8. Linear Inverse Problems and Non-Gaussian Statistics
8.1 L1 Norms and Exponential Probability Density Functions
8.2 Maximum Likelihood Estimate of the Mean of an Exponential Probability Density Function
8.3 The General Linear Problem
8.4 Solving L1 Norm Problems
8.5 The L∞ Norm
8.6 Problems
REFERENCES
Chapter 9. Nonlinear Inverse Problems
9.1 Parameterizations
9.2 Linearizing Transformations
9.3 Error and Likelihood in Nonlinear Inverse Problems
9.4 The Grid Search
9.5 The Monte Carlo Search
9.6 Newton’s Method
9.7 The Implicit Nonlinear Inverse Problem with Gaussian Data
9.8 Gradient Method
9.9 Simulated Annealing
9.10 Choosing the Null Distribution for Inexact Non-Gaussian Nonlinear Theories
9.11 Bootstrap Confidence Intervals
9.12 Problems
REFERENCES
Chapter 10. Factor Analysis
10.1 The Factor Analysis Problem
10.2 Normalization and Physicality Constraints
10.3 Q-Mode and R-Mode Factor Analysis
10.4 Empirical Orthogonal Function Analysis
10.5 Problems
REFERENCES
Chapter 11. Continuous Inverse Theory and Tomography
11.1 The Backus-Gilbert Inverse Problem
11.2 Resolution and Variance Trade-Off
11.3 Approximating Continuous Inverse Problems as Discrete Problems
11.4 Tomography and Continuous Inverse Theory
11.5 Tomography and the Radon Transform
11.6 The Fourier Slice Theorem
11.7 Correspondence Between Matrices and Linear Operators
11.8 The Fréchet Derivative
11.9 The Fréchet Derivative of Error
11.10 Backprojection
11.11 Fréchet Derivatives Involving a Differential Equation
11.12 Problems
REFERENCES
Chapter 12. Sample Inverse Problems
12.1 An Image Enhancement Problem
12.2 Digital Filter Design
12.3 Adjustment of Crossover Errors
12.4 An Acoustic Tomography Problem
12.5 One-Dimensional Temperature Distribution
12.6 L1, L2, and L∞ Fitting of a Straight Line
12.7 Finding the Mean of a Set of Unit Vectors
12.8 Gaussian and Lorentzian Curve Fitting
12.9 Earthquake Location
12.10 Vibrational Problems
12.11 Problems
REFERENCES
Chapter 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
13.2 Moment Tensors of Earthquakes
13.3 Waveform “Tomography”
13.4 Velocity Structure from Free Oscillations and Seismic Surface Waves
13.5 Seismic Attenuation
13.6 Signal Correlation
13.7 Tectonic Plate Motions
13.8 Gravity and Geomagnetism
13.9 Electromagnetic Induction and the Magnetotelluric Method
REFERENCES
Chapter 14. Appendices
14.1 Implementing Constraints with Lagrange multipliers
14.2 L2 Inverse Theory with Complex Quantities
- Edition: 3
- Volume: 45
- Published: August 19, 2016
- Language: English
WM
William Menke
William Menke is a Professor of Earth and Environmental Sciences at Columbia University. 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. He has thirty years of experience teaching data analysis methods to both undergraduates and graduate students. Relevant courses that he has taught include, at the undergraduate level, Environmental Data Analysis and The Earth System, and at the graduate level, Geophysical Inverse Theory, Quantitative Methods of Data Analysis, Geophysical Theory and Practical Seismology.
Affiliations and expertise
Professor of Earth and Environmental Sciences ,Columbia UniversityRead Geophysical Data Analysis: Discrete Inverse Theory on ScienceDirect