Geophysical Data Analysis and Inverse Theory with MATLAB® and Python

✍ Scribed by William Menke

• Publisher: Academic Press
• Year: 2024
• Tongue: English
• Leaves: 344
• Edition: 5
• Category: Library

✦ Synopsis

Geophysical Data Analysis and Inverse Theory with MATLAB or Python, Fifth Edition is a revised and expanded introduction to inverse theory and tomography as it is practiced by geophysicists. The book demonstrates the methods needed to analyze a broad spectrum of geophysical datasets, with special attention given to those methods that generate images of the earth. Data analysis can be a mathematically complex activity, but the treatment in this volume is carefully designed to emphasize those mathematical techniques that readers will find the most familiar and to systematically introduce less-familiar ones. A series of "crib sheets" offer step-by-step summaries of methods presented. Utilizing problems and case studies, along with MATLAB and Python computer code and summaries of methods, the book provides professional geophysicists, students, data scientists and engineers in geophysics with the tools necessary to understand and apply mathematical techniques and inverse theory.

• Includes material on probability, including Bayesian influence, probability density function, and metropolis algorithm
• Offers detailed discussions of the application of inverse theory to seismological, gravitational, and tectonic studies
• Provides numerous examples, color figures, and end-of-chapter problems to help readers explore and further understand the presented ideas
• Includes both MATLAB and Python examples and problem sets

✦ Table of Contents

Geophysical Data Analysis and Inverse Theory with MATLAB® and Python [AN 3674882]
Front Cover
Geophysical Data Analysis and Inverse Theory with MATLAB® and Python
Copyright
Dedication
Contents
Preface
References
Chapter 1: Getting started with MATLAB® or Python
Part A.. MATLAB® as a tool for learning inverse theory
1A.1. Getting started with MATLAB®
1A.2. Effective use of folders
1A.3. Simple arithmetic
1A.4. Vectors and matrices and their representation in MATLAB®
1A.5. Matrix differentiation
1A.6. Character strings and lists
1A.7. Loops
1A.8. Loading data from a file
1A.9. Writing data to a file
1A.10. Plotting data
Part B.. Python as a tool for learning inverse theory
1B.1. Getting started with Python
1B.2. Effective use of folders
1B.3. Simple arithmetic
1B.4. Lists, tuples, vectors, and matrices
1B.5. Matrix differentiation
1B.6. Character strings and lists
1B.7. Loops
1B.8. Loading data from a file
1B.9. Writing data to a file
1B.10. Plotting data
References
Chapter 2: Describing inverse problems
2.1. Forward and inverse theories
2.2. Formulating inverse problems
2.3. Special forms
2.4. The linear inverse problem
2.5. Example: Fitting a straight line
2.6. Example: Fitting a parabola
2.7. Example: Acoustic tomography
2.8. Example: X-ray imaging
2.9. Example: Spectral curve fitting
2.10. Example: Factor analysis
2.11. Example: Correcting for an instrument response
2.12. Solutions to inverse problems
2.13. Estimates as solutions
2.14. Bounding values as solutions
2.15. Probability density functions as solutions
2.16. Ensembles of realizations as solutions
2.17. Weighted averages of model parameters as solutions
2.18. Problems
References
Chapter 3: Using probability to describe random variation
3.1. Noise and random variables
3.2. Correlated data
3.3. Functions of random variables
3.4. Normal (Gaussian) probability density functions
3.5. Testing the assumption of normal statistics
3.6. Conditional probability density functions
3.7. Confidence intervals
3.8. Computing realizations of random variables
3.9. Problems
References
Chapter 4: Solution of the linear, Normal inverse problem, viewpoint 1: The length method
4.1. The lengths of estimates
4.2. Measures of length
4.3. Least squares for a straight line
4.4. The least-squares solution of the linear inverse problem
4.5. Example: Fitting a straight line
4.6. Example: Fitting a parabola
4.7. Example: Fitting of a planar surface
4.8. Example: Inverting reflection coefficient for interface properties
4.9. The existence of the least-squares solution
4.10. The purely underdetermined problem
4.11. Mixed-determined problems
4.12. Weighted measures of length as a type of prior information
4.13. Weighted least squares
4.14. Weighted minimum length
4.15. Weighted damped least squares
4.16. Generalized least squares
4.17. Use of sparse matrices in MATLAB® and Python
4.18. Example: Using generalized least squares to fill in data gaps
4.19. Choosing between prior information of flatness and smoothness
4.20. Other types of prior information
4.21. Example: Constrained fitting of a straight line
4.22. Prior and posterior estimates of the variance of the data
4.23. Variance and prediction error of the least-squares solution
4.24. Concluding remarks
4.25. Problems
References
Chapter 5: Solution of the linear, Normal inverse problem, viewpoint 2: Generalized inverses
5.1. Solutions versus operators
5.2. The data resolution matrix
5.3. The model resolution matrix
5.4. The unit covariance matrix
5.5. Resolution and covariance of some generalized inverses
5.6. Measures of goodness of resolution and covariance
5.7. Generalized inverses with good resolution and covariance
5.8. Sidelobes and the Backus-Gilbert spread function
5.9. The Backus-Gilbert generalized inverse for the underdetermined problem
5.10. Including the covariance size
5.11. The trade-off of resolution and variance
5.12. Reorganizing images and 3D models into vectors
5.13. Checkerboard tests
5.14. Resolution analysis without a data kernel
5.15. Problems
References
Chapter 6: Solution of the linear, Normal inverse problem, viewpoint 3: Maximum likelihood methods
6.1. The mean of a group of measurements
6.2. Maximum likelihood applied to inverse problems
6.3. Prior pdfs
6.4. Maximum likelihood for an exact theory
6.5. Inexact theories
6.6. Exact theory as a limiting case of an inexact one
6.7. Inexact theory with a normal pdf
6.8. Limiting cases
6.9. Model and data resolution in the presence of prior information
6.10. Relative entropy as a guiding principle
6.11. Equivalence of the three viewpoints
6.12. Chi-square test for the compatibility of the prior and observed error
6.13. The F-test of the significance of the reduction of error
6.14. Problems
References
Chapter 7: Data assimilation methods including Gaussian process regression and Kalman filtering
7.1. Smoothness via the prior covariance matrix
7.2. Realizations of a medium with a specified covariance matrix
7.3. Equivalence of two forms of prior information
7.4. Gaussian process regression
7.5. Prior information of dynamics
7.6. Data assimilation in the case of first-order dynamics
7.7. Data assimilation using Thomas recursion
7.8. Present-time solutions
7.9. Kalman filtering
7.10. Case of exact dynamics
7.11. Problems
References
Chapter 8: Nonuniqueness and localized averages
8.1. Null vectors and nonuniqueness
8.2. Null vectors of a simple inverse problem
8.3. Localized averages of model parameters
8.4. Averages versus estimates
8.5. Decoupling´´ localized averages from estimates 8.6. Nonunique averaging vectors and prior information 8.7. End-member solutions and squeezing 8.8. Problems References Chapter 9: Applications of vector spaces 9.1. Model and data spaces 9.2. Householder transformations 9.3. Designing householder transformations 9.4. Transformations that do not preserve length 9.5. The solution of the mixed-determined problem 9.6. Singular-value decomposition and the natural generalized inverse 9.7. Derivation of the singular-value decomposition 9.8. Simplifying linear equality and inequality constraints 9.9. Inequality constraints 9.10. Problems References Chapter 10: Linear inverse problems with non-Normal statistics 10.1. L1 norms and exponential probability density functions 10.2. Maximum likelihood estimate of the mean of an exponential pdf 10.3. The general linear problem 10.4. Solving L1 norm problems by transformation to a linear programming problem 10.5. Solving L1 norm problems by reweighted L2 minimization 10.6. Solving L norm problems by transformation to a linear programming problem 10.7. The L0 norm and sparsity 10.8. Problems References Chapter 11: Nonlinear inverse problems 11.1. Parameterizations 11.2. Linearizing transformations 11.3. Error and log-likelihood in nonlinear inverse problems 11.4. The grid search 11.5. Newtons method 11.6. The implicit nonlinear inverse problem with Normally distributed data 11.7. The explicit nonlinear inverse problem with Normally distributed data 11.8. Covariance and resolution in nonlinear problems 11.9. Gradient-descent method 11.10. Choosing the null distribution for inexact non-Normal nonlinear theories 11.11. The genetic algorithm 11.12. Bootstrap confidence intervals 11.13. Problems Reference Chapter 12: Monte Carlo methods 12.1. The Monte Carlo search 12.2. Simulated annealing 12.3. Advantages and disadvantages of ensemble solutions 12.4. The Metropolis-Hastings algorithm 12.5. Examples of ensemble solutions 12.6. Trans-dimensional models 12.7. Examples of trans-dimensional solutions 12.8. Problems References Chapter 13: Factor analysis 13.1. The factor analysis problem 13.2. Normalization and physicality constraints 13.3. Q-mode and R-mode factor analysis 13.4. Empirical orthogonal function analysis 13.5. Problems References Chapter 14: Continuous inverse theory and tomography 14.1. The Backus-Gilbert inverse problem 14.2. Trade-off of resolution and variance 14.3. Approximating a continuous inverse problem as a discrete problem 14.4. Tomography and continuous inverse theory 14.5. The Radon transform 14.6. The Fourier slice theorem 14.7. Linear operators 14.8. The Fréchet derivative 14.9. The Fréchet derivative of error 14.10. Back-projection 14.11. Fréchet derivatives involving a differential equation 14.12. Case study: Heat source in problem with Newtonian cooling 14.13. Derivative with respect to a parameter in a differential operator 14.14. Case study: Thermal parameter in Newtonian cooling 14.15. Application of the adjoint method to data assimilation 14.16. Gradient of error for model parameter in the differential operator 14.17. Problems References Chapter 15: Sample inverse problems 15.1. An image enhancement problem 15.2. Digital filter design 15.3. Adjustment of crossover errors 15.4. An acoustic tomography problem 15.5. One-dimensional temperature distribution 15.6. L1, L2, and L fitting of a straight line 15.7. Finding the mean of a set of unit vectors 15.8. Gaussian and Lorentzian curve fitting 15.9. Fourier analysis 15.10. Earthquake location 15.11. Vibrational problems 15.12. Problems References Chapter 16: Applications of inverse theory to solid earth geophysics 16.1. Earthquake location and determination of the velocity structure of the earth from travel time data 16.2. Moment tensors of earthquakes 16.3. Adjoint methods in seismic imaging 16.4. Wavefield tomography 16.5. Seismic migration 16.6. Finite-frequency travel time tomography 16.7. Banana-doughnut kernels 16.8. Velocity structure from free oscillations and seismic surface waves 16.9. Seismic attenuation 16.10. Signal correlation 16.11. Tectonic plate motions 16.12. Gravity and geomagnetism 16.13. Electromagnetic induction and the magnetotelluric method 16.14. Problems References Chapter 17: Important algorithms and method summaries 17.1. Implementing constraints with Lagrange multipliers 17.2. L2 inverse theory with complex quantities 17.3. Inverse of aresized´´ matrix
17.4. Method summaries
Method summary 1, generalized least squares
Method summary 2, the grid search
Method summary 3, nonlinear least squares
Method summary 4, MCMC inversion
Method summary 5, bootstrap confidence intervals
Method summary 6, factor analysis
References
Index
Back Cover