# Geophysical Data Analysis and Inverse Theory with MATLAB® and Python

- 5th Edition - February 22, 2024
- Author: William Menke
- Language: English
- Paperback ISBN:9 7 8 - 0 - 4 4 3 - 1 3 7 9 4 - 5
- eBook ISBN:9 7 8 - 0 - 4 4 3 - 1 3 7 9 5 - 2

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… Read more

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Request a sales quote*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.

- Cover image
- Title page
- Table of Contents
- Copyright
- Dedication
- Preface
- References
- Chapter 1 Getting started with MATLAB® or Python
- Abstract
- Part A MATLAB® as a tool for learning inverse theory
- Part B Python as a tool for learning inverse theory
- References
- Chapter 2 Describing inverse problems
- Abstract
- 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
- Abstract
- 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
- Abstract
- 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
- Abstract
- 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
- Abstract
- 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
- Abstract
- 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
- Abstract
- 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
- Abstract
- 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
- Abstract
- 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
- Abstract
- 11.1 Parameterizations
- 11.2 Linearizing transformations
- 11.3 Error and log-likelihood in nonlinear inverse problems
- 11.4 The grid search
- 11.5 Newton’s 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
- Abstract
- 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
- Abstract
- 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
- Abstract
- 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
- Abstract
- 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
- Abstract
- 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
- Abstract
- 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
- References
- Index

- No. of pages: 400
- Language: English
- Edition: 5
- Published: February 22, 2024
- Imprint: Academic Press
- Paperback ISBN: 9780443137945
- eBook ISBN: 9780443137952

WM

### William Menke

*Geophysical Data Analysis and Inverse Theory with MATLAB® and Python*on ScienceDirect