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
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