Geophysical Data Analysis and Inverse Theory with MATLAB® and Python

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 a ``resized´´ 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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