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أخواني في الله أحضرت لكم كتاب Geophysical Data Analysis and Inverse Theory with MatLAB and Python - Fifth Edition William Menke Department of Earth and Environmental Sciences, Columbia University, New York, NY, United States
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Contents Preface xi 1. Getting started with MATLAB or Python Part A. MATLAB as a tool for learning inverse theory 1 Part B. Python as a tool for learning inverse theory 16 References 31 2. Describing inverse problems 2.1 Forward and inverse theories 33 2.2 Formulating inverse problems 35 2.3 Special forms 36 2.4 The linear inverse problem 36 2.5 Example: Fitting a straight line 37 2.6 Example: Fitting a parabola 38 2.7 Example: Acoustic tomography 39 2.8 Example: X-ray imaging 40 2.9 Example: Spectral curve fitting 42 2.10 Example: Factor analysis 42 2.11 Example: Correcting for an instrument response 43 2.12 Solutions to inverse problems 44 2.13 Estimates as solutions 45 2.14 Bounding values as solutions 45 2.15 Probability density functions as solutions 46 2.16 Ensembles of realizations as solutions 46 2.17 Weighted averages of model parameters as solutions 46 2.18 Problems 47 References 47 3. Using probability to describe random variation 3.1 Noise and random variables 49 3.2 Correlated data 52 3.3 Functions of random variables 54 3.4 Normal (Gaussian) probability density functions 58 3.5 Testing the assumption of normal statistics 60 3.6 Conditional probability density functions 61 3.7 Confidence intervals 63 3.8 Computing realizations of random variables 63 3.9 Problems 65 References 66 4. Solution of the linear, Normal inverse problem, viewpoint 1: The length method 4.1 The lengths of estimates 67 4.2 Measures of length 67 4.3 Least squares for a straight line 70 4.4 The least-squares solution of the linear inverse problem 70 4.5 Example: Fitting a straight line 72 4.6 Example: Fitting a parabola 73 4.7 Example: Fitting of a planar surface 74 4.8 Example: Inverting reflection coefficient for interface properties 74 4.9 The existence of the least-squares solution 76 4.10 The purely underdetermined problem 78 4.11 Mixed-determined problems 79 4.12 Weighted measures of length as a type of prior information 80 4.13 Weighted least squares 81 4.14 Weighted minimum length 81 4.15 Weighted damped least squares 81 4.16 Generalized least squares 82 4.17 Use of sparse matrices in MATLAB and Python 83 4.18 Example: Using generalized least squares to fill in data gaps 87 4.19 Choosing between prior information of flatness and smoothness 88 4.20 Other types of prior information 88 4.21 Example: Constrained fitting of a straight line 89 4.22 Prior and posterior estimates of the variance of the data 90 4.23 Variance and prediction error of the least-squares solution 91 4.24 Concluding remarks 93 4.25 Problems 93 References 94 5. Solution of the linear, Normal inverse problem, viewpoint 2: Generalized inverses 5.1 Solutions versus operators 95 5.2 The data resolution matrix 95 5.3 The model resolution matrix 96 5.4 The unit covariance matrix 97 vii5.5 Resolution and covariance of some generalized inverses 98 5.6 Measures of goodness of resolution and covariance 99 5.7 Generalized inverses with good resolution and covariance 99 5.8 Sidelobes and the Backus-Gilbert spread function 101 5.9 The Backus-Gilbert generalized inverse for the underdetermined problem 102 5.10 Including the covariance size 103 5.11 The trade-off of resolution and variance 106 5.12 Reorganizing images and 3D models into vectors 107 5.13 Checkerboard tests 108 5.14 Resolution analysis without a data kernel 110 5.15 Problems 110 References 111 6. Solution of the linear, Normal inverse problem, viewpoint 3: Maximum likelihood methods 6.1 The mean of a group of measurements 113 6.2 Maximum likelihood applied to inverse problems 115 6.3 Prior pdfs 116 6.4 Maximum likelihood for an exact theory 118 6.5 Inexact theories 120 6.6 Exact theory as a limiting case of an inexact one 122 6.7 Inexact theory with a normal pdf 123 6.8 Limiting cases 125 6.9 Model and data resolution in the presence of prior information 125 6.10 Relative entropy as a guiding principle 127 6.11 Equivalence of the three viewpoints 128 6.12 Chi-square test for the compatibility of the prior and observed error 128 6.13 The F-test of the significance of the reduction of error 130 6.14 Problems 133 References 134 7. Data assimilation methods including Gaussian process regression and Kalman filtering 7.1 Smoothness via the prior covariance matrix 135 7.2 Realizations of a medium with a specified covariance matrix 135 7.3 Equivalence of two forms of prior information 137 7.4 Gaussian process regression 139 7.5 Prior information of dynamics 141 7.6 Data assimilation in the case of first-order dynamics 143 7.7 Data assimilation using Thomas recursion 144 7.8 Present-time solutions 145 7.9 Kalman filtering 146 7.10 Case of exact dynamics 147 7.11 Problems 149 References 149 8. Nonuniqueness and localized averages 8.1 Null vectors and nonuniqueness 151 8.2 Null vectors of a simple inverse problem 152 8.3 Localized averages of model parameters 152 8.4 Averages versus estimates 153 8.5 “Decoupling” localized averages from estimates 153 8.6 Nonunique averaging vectors and prior information 154 8.7 End-member solutions and squeezing 156 8.8 Problems 157 References 157 9. Applications of vector spaces 9.1 Model and data spaces 159 9.2 Householder transformations 159 9.3 Designing householder transformations 162 9.4 Transformations that do not preserve length 163 9.5 The solution of the mixed-determined problem 164 9.6 Singular-value decomposition and the natural generalized inverse 165 9.7 Derivation of the singular-value decomposition 169 9.8 Simplifying linear equality and inequality constraints 170 9.9 Inequality constraints 171 9.10 Problems 177 References 177 10. Linear inverse problems with non-Normal statistics 10.1 L1 norms and exponential probability density functions 179 10.2 Maximum likelihood estimate of the mean of an exponential pdf 180 10.3 The general linear problem 182 10.4 Solving L1 norm problems by transformation to a linear programming problem 182 10.5 Solving L1 norm problems by reweighted L2 minimization 186 10.6 Solving L∞ norm problems by transformation to a linear programming problem 189 10.7 The L0 norm and sparsity 192 10.8 Problems 193 References 195 11. Nonlinear inverse problems 11.1 Parameterizations 197 11.2 Linearizing transformations 198 11.3 Error and log-likelihood in nonlinear inverse problems 199 11.4 The grid search 199 11.5 Newton’s method 203 11.6 The implicit nonlinear inverse problem with Normally distributed data 206 viii Contents11.7 The explicit nonlinear inverse problem with Normally distributed data 208 11.8 Covariance and resolution in nonlinear problems 210 11.9 Gradient-descent method 212 11.10 Choosing the null distribution for inexact non-Normal nonlinear theories 213 11.11 The genetic algorithm 213 11.12 Bootstrap confidence intervals 220 11.13 Problems 222 Reference 222 12. Monte Carlo methods 12.1 The Monte Carlo search 223 12.2 Simulated annealing 224 12.3 Advantages and disadvantages of ensemble solutions 225 12.4 The Metropolis-Hastings algorithm 227 12.5 Examples of ensemble solutions 228 12.6 Trans-dimensional models 229 12.7 Examples of trans-dimensional solutions 230 12.8 Problems 234 References 234 13. Factor analysis 13.1 The factor analysis problem 235 13.2 Normalization and physicality constraints 240 13.3 Q-mode and R-mode factor analysis 244 13.4 Empirical orthogonal function analysis 245 13.5 Problems 248 References 248 14. Continuous inverse theory and tomography 14.1 The Backus-Gilbert inverse problem 249 14.2 Trade-off of resolution and variance 250 14.3 Approximating a continuous inverse problem as a discrete problem 251 14.4 Tomography and continuous inverse theory 252 14.5 The Radon transform 252 14.6 The Fourier slice theorem 253 14.7 Linear operators 255 14.8 The Frechet derivative 258 14.9 The Frechet derivative of error 258 14.10 Back-projection 260 14.11 Frechet derivatives involving a differential equation 261 14.12 Case study: Heat source in problem with Newtonian cooling 262 14.13 Derivative with respect to a parameter in a differential operator 264 14.14 Case study: Thermal parameter in Newtonian cooling 266 14.15 Application of the adjoint method to data assimilation 268 14.16 Gradient of error for model parameter in the differential operator 270 14.17 Problems 271 References 272 15. Sample inverse problems 15.1 An image enhancement problem 273 15.2 Digital filter design 275 15.3 Adjustment of crossover errors 277 15.4 An acoustic tomography problem 279 15.5 One-dimensional temperature distribution 280 15.6 L1, L2, and L∞ fitting of a straight line 282 15.7 Finding the mean of a set of unit vectors 284 15.8 Gaussian and Lorentzian curve fitting 287 15.9 Fourier analysis 289 15.10 Earthquake location 291 15.11 Vibrational problems 294 15.12 Problems 296 References 296 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 297 16.2 Moment tensors of earthquakes 299 16.3 Adjoint methods in seismic imaging 300 16.4 Wavefield tomography 303 16.5 Seismic migration 303 16.6 Finite-frequency travel time tomography 305 16.7 Banana-doughnut kernels 307 16.8 Velocity structure from free oscillations and seismic surface waves 309 16.9 Seismic attenuation 311 16.10 Signal correlation 312 16.11 Tectonic plate motions 312 16.12 Gravity and geomagnetism 312 16.13 Electromagnetic induction and the magnetotelluric method 313 16.14 Problems 314 References 314 17. Important algorithms and method summaries 17.1 Implementing constraints with Lagrange multipliers 317 17.2 L2 inverse theory with complex quantities 317 17.3 Inverse of a “resized” matrix 319 17.4 Method summaries 321 References 326 Index 327 Index Note: Page numbers followed by f indicate figures and t indicate tables. A Acoustic tomography, 39–40, 39f, 279–280, 280–281f Adjoint differential equation, 265, 271 equation, 263 fields, 265 linear operator, 257 method, 147, 299, 313 to data assimilation, 268–270 in seismic imaging, 300–302 operator, 259 self-adjoint, 257 source, 302 Algebraic eigenvalue problem, 10, 26 Amin and amax, 155 Amplitude spectral density, 291 Antiidentity matrix, 83 Armijo’s rule, 212 Arrival times, 291–292 Assumption, of Normal statistics, 60 Atlantic Rock data set, 238, 244f Attenuation tomography, 311 Autocorrelation, 276 function, 135–136 Auto-covariance function, 135, 137f Auxiliary information, 37 B Back-projection, 260–261, 260–261f Backus-Gilbert generalized inverse, underdetermined problem, 102–103 Backus-Gilbert inverse problem, 249–250 Backus-Gilbert spread function, 101, 106f Backward recursion, 144–145 Banana-doughnut kernels, 307–309, 308f Bayesian inference, 62 Biconjugate gradient algorithm, 83 Block diagonal/lower bidiag, 144 Block tri-diagonal matrix, 144 Bootstrap confidence intervals, 220–221, 221f, 324 Bordering method, 320 Born approximation, 265, 301 Bound, 154–155, 154f, 157 Boundary conditions, 142, 256 Bounding values, as solutions, 45–46 C Cellstr, 13 Central limit theorem, 58 Centroid moment tensor (CMT), 300, 311 Centroid, of source, 299 Characteristic values, 10–11, 26 Characteristic vectors, 10–11, 26 Character strings and lists MATLAB , 12–13 Python, 27–28 Checkerboard tests, 108–109 Chi-square test, 128–130, 131f Circular random variable, 318 Clipping vector, 29 Cluster analysis, 244 CMT. See Centroid moment tensor (CMT) Column vector, 4–5 Complex least squares, 318 Computed tomography (CT) medical scanner, 41f Computing realizations, of random variables, 63–65 Conditional commands, 29 Conditional probability density functions, 61–62 Condition of detailed balance, 227 Confidence intervals/limits, 63, 71, 91, 220–221, 221f Continuous inverse problem data kernel, 263f differential equation, 263f as discrete problem, 251–252 solution of, 259f, 263f Continuous inverse theory, 249 tomography and, 252 Convolution, 275–276, 277f operation, 44 Covariance, 53–54, 53f, 71, 73, 79–81, 90, 92 generalized inverses with good, 99–101 matrix, 115, 120, 123, 129 specified, 135–136 measures of goodness, 99 and resolution, in nonlinear inverse problems, 210–212, 211f size, 103–105 of some generalized inverses, 98–99 Crossover errors, adjustment of, 277–279, 278f Cumulative chi-squared distribution, 60 Cumulative sum, 50 D Dagger symbol, 257 Damped least-squares solution, 80 Damped minimum-length, 101 Data assimilation, 143 adjoint method to, 268–270, 270f in case of first-order dynamics, 143–144 using Thomas recursion, 144–145 Data covariance, 110 Data kernel, 36 Data loading from file MATLAB , 14 Python, 30 Data plotting MATLAB , 15–16 Python, 30–31 Data resolution matrix, 95–96, 96f Data writing to file MATLAB , 14–15 Python, 30 Deblurring problem, 275f, 276 Deconvolution, 44, 276 Degrees of freedom, 60 Differential equation, 267–270 adjoint, 265, 271 continuous inverse problem, 263f Frcehet derivative, 261–262 linear, 256 Digital filter design, 275–276 Dirac delta function, 256 Dirac impulse function, 301 Dirichlet spread functions, 99, 101 Dispersion curve, 309 Dispersion function, 309 Displacement, of ground, 299 Dot product, 7, 23 Double-difference method, 298 Dynamics matrix, 143 E Earthquakes, 297 locations, 291–294, 293f, 300 moment tensors of, 299–300, 312 Earth’s gravity field, 313 Eigenfrequencies, 294 Eigenvectors, 236f Electromagnetic induction, 313–314 Element, 21–22 El Nino-Southern Oscillation climate instability, 246–247 Empirical orthogonal function (EOF) analysis, 245–247, 245–248f End-member solutions and squeezing, 156–157, 156f 327Ensemble solutions advantages and disadvantages of, 225–227 examples of, 228–229 of Laplace transform problem for model function, 229f of same nonlinear curve-fitting problem, 226f Entropy, 117 EOF. See Empirical orthogonal function (EOF) analysis Equality constraints, 116 Error injecting, 302 and log-likelihood, in nonlinear inverse problems, 199, 200f propagation, 57, 80, 91 Euler’s formula, 289 Euler’s method, 267 Euler vector, 312 Even-determined problems, 77 Exact dynamics, 147–148 Exact theory, limiting case of inexact one, 122 Explicit linear form, 36 Explicit nonlinear form, 36 Explicit nonlinear inverse problems, 208–210 Exponential probability density functions, 179, 180f F Factor analysis, 42–43, 42f, 325 problem, 235–239, 236f variability of rock data set, 239 Factor loadings, 236 Factor matrix, 244 Fast Fourier Transform (FFT), 289 Fermat’s Principle, 299 Finite bounds, 181–182 Finite-frequency travel time tomography, 305–307, 307f Fisher distribution, 285f Fisher probability density function, 284, 285–286f Fisher-Snedecor pdf, 131 Folder (directory) structure MATLAB , 2–3, 2f Python, 18–19, 18f Force-couples, 299 Format string, 12, 28 Forward recursion, 144 Forward theory, 33–35 Fourier analysis, 289–291, 291f Fourier slice theorem, 253–254, 254f Frechet derivative, 258, 299–300 differential equation, 261–262 of error, 258–259 Free oscillations and seismic surface waves, velocity structure, 309–311 F-test, 134 reduction of error significance, 130–133, 132f Function analysis, empirical orthogonal, 245–247 Fundamental theorem of calculus, 256 G Gaussian curve fitting, 287–289 Gaussian process regression (GPR), 139–141, 140–141f gdabox() function, 239 Geiger’s method, 292 Generalized inverse, 95 Generalized Least Squares (GLS), 82, 87, 87f, 119–120, 135, 142f Genetic algorithm, nonlinear inverse problems, 213–219, 215t, 215–216f Geomagnetism, 312–313 GLS. See Generalized Least Squares (GLS) Gradient-descent method, nonlinear inverse problems, 212–213, 212f Gradient of error, model parameter in differential operator, 270–271 Gravitational field, 313 Gravity and geomagnetism, 312–313 Green function, 256 Grid search, 322 nonlinear inverse problems, 199–203, 201f Ground displacement, 299 H Heat diffusion equation, 142 Heat source, in problem with Newtonian cooling, 262–264 Hermitian matrix, 318 Hermitian transpose, 318 Householder rotation, 318–319 Householder transformations, 159–162 designing, 162–163 Hypocenter, 291, 297 Hypocentral parameters, 297 I Identity matrix, 8, 23 Imaging principle, 303–305 Impedance, 313 Implicit linear form, 36 Implicit nonlinear inverse problems, 206–208, 207f Inequality constraint, 154–155, 169–176 Inexact theories, 120–122, 121f with Normal pdf, 123–125 Information gain, 117–118, 118f, 127, 133 Initial condition, 142 Inner product, 7 of function, 256–257 Instrument response, 43–44 Inverse problem, 33, 312 formulating, 35–36 linear, 36–37 of “resized” matrix, 319–321 solution (answer) to, 44 Inverse theory, 1, 33, 312 formulating problems, 35–36 forward and, 33–35 Inverting for interface properties, 74–76 K Kalman filtering, 146–147 Kepler’s third law, 73, 74f Kriging, 141 Kronecker delta symbol, 8, 23 Kuhn-Tucker theorem, 171–172, 176 Kurile-Kamchatka subduction zone, 287f L Lagrange multiplier, 78–79, 89, 317, 318f LAMBDA, 11 Lamè parameter, 300 Least squares generalized, 321 solution, 76–78 of linear inverse problem, 70–71 for straight line, 70 variance and prediction error, 91–93 weighted, 81 Length method of estimates, 67 measures of, 67 Linear equality and inequality constraints, simplifying, 170–171 Linear inverse problem, 36–37 non-Normal statistics, 179–196 Linearizing transformations, nonlinear inverse problems, 198, 199f Linear mixture, 235 Linear operator, 255–258 Linear programming problem, 154–155, 182–186 L2 inverse theory, with complex quantities, 317–319 Lists MATLAB , 12–13 Python, 20–27 L1, L2, and L∞ fitting, of straight line, 282–284, 284f L ∞ norm problems, transformation to linear programming problem, 189–191 L1 norms and exponential probability density functions, 179, 180t, 180f Localized averages, 250, 256 “decoupling” from estimates, 153–154 of model parameters, 152 nonuniqueness and, 151–158 Logical addressing, 14, 29 Log-likelihood function, 113–115, 114f Loops MATLAB , 13–14 Python, 28–29 Lorentzian curve fitting, 287–289, 288f Love wave, 309 L2 problem, 186–188 M Magnetotelluric (MT) method, 313–314 Magnetotelluric problem, 313 Mapping function, 312 Markov chain, 227 Markov Chain Monte Carlo (MCMC) method, 227 inversion, 323 MATLAB , 1 character strings and lists, 12–13 folder (directory) structure, 2–3, 2f function gda_FTFrhs(), 86 getting started with, 1–2 328 Indexloading data from file, 14 loops, 13–14 matrix differentiation, 11 plotting data, 15–16 simple arithmetic, 3–4 transpose, 5–7 vectors and matrices, 4–11 writing data to file, 14–15 Matrices derivative, 11 differentiation, MATLAB , 11 MATLAB , 4–11 norms, 69 Python, 20–27 Maximum likelihood estimate, of mean of exponential pdf, 180–182, 181f Maximum likelihood method applied to inverse problems, 115 for exact theory, 118–120 Maximum likelihood point, 50–51, 50f Maximum relative entropy method, 127 Median, 181, 187–188 Method of least squares, 67 Method of maximum likelihood, 113, 114f Metropolis-Hastings algorithm, 64, 227–228 Migration, seismic, 303–305, 304–305f Minimum-length solution, 79 Minimum relative entropy method, 127 Mixed-determined problems, 77, 79–80 solution of, 164–165 Mixture of components, 245 linear, 235 simple, 235 Model and data spaces, 159 Model parameters, 33 Model resolution, 249–250 matrix, 96–97 Moment-rate tensor, 299 Moment tensor, 299–300 Monte Carlo methods, 223 ensemble solutions advantages and disadvantages of, 225–227 examples of, 228–229 of Laplace transform problem for model function, 229f of same nonlinear curve-fitting problem, 226f Metropolis-Hastings algorithm, 227–228 Monte Carlo search, 223–224, 224f trans-dimensional models, 229–230 curve-fitting example, 232f examples of solutions, 230–234, 231f of Laplace transform problem, 232f Mossbauer spectroscopy experiment, 42f N Natural solution, 164, 168, 174f Newtonian cooling heat source in problem with, 262–264 thermal parameter in, 266–268 Newton’s method, nonlinear inverse problems, 203–206, 204f, 206f Newton’s Second Law for the motion, 143 Noise and random variables, 49–52 Nonlinear inverse problems bootstrap confidence intervals, 220–221, 221f covariance and resolution in, 210–212, 211f error and log-likelihood in, 199, 200f explicit with Normally distributed data, 208–210 genetic algorithm, 213–219, 215t, 215–216f gradient-descent method, 212–213, 212f grid search, 199–203, 201f implicit with Normally distributed data, 206–208, 207f linearizing transformations, 198, 199f Newton’s method, 203–206, 204f, 206f null distribution for inexact non-Normal, 213 parameterizations, 197–198 Nonlinear least squares, 323 Nonnegative least squares, 172–174, 173–174f, 177 Nonuniqueness and localized averages end-member solutions and squeezing, 156–157, 156f null vectors and, 151 Norm, 67–68 Normalization and physicality constraints, 240–244 Normal pdf, 115–117, 118f, 119, 123 Normal (Gaussian) probability density functions, 58–59 Null distribution for inexact non-Normal nonlinear inverse problems, 213 Null hypothesis, 129, 131 Null pdfs, 117 Null solution, 151–152 Null space, 164–165, 177 Null vectors and nonuniqueness, 151 of simple inverse problem, 152 O One-dimensional temperature distribution, 280–282, 282–283f Operators, solutions vs., 95 Optical sensor, 273 Origin time, 291 Outer product, 7, 23 Outlier, 179, 189, 192f Overdetermined problems, 77 Overfit, 129 Overlap integral, 250 P Parabola, 38–39 fitting problem, 73 Parameterizations, nonlinear inverse problems, 197–198 Pearson’s chi-squared test, 61f Placeholders, 12 Planar surface, fitting of, 74 Point-spread function, 109 Posterior (a posteriori) variance, 90 Precision parameter, 285–286, 286f Prediction error, 67, 68f Present-time solutions, 145–146 Prior covariance matrix, 137–139 Prior information, 146 of dynamics, 141–143 flatness and smoothness, 88 and posterior estimates, variance of data, 90–91 types of, 88–89 Prior joint probability density function, 119f Prior pdfs, 116–118, 116–117f Prior solution, 138 Prior variance, 90 Probability density function, 49, 50f, 114f, 116f, 118f, 317–318 conditional, 61–62, 64f joint, 52–53 long-tailed, 69f multivariate, 52–53f Normal (Gaussian), 58–59 properties of, 46 as solutions, 45–46f, 46 uniform, 55–57f Pure path approximation, 310 Pythagoras’s law for right triangles, 69 Python character strings and lists, 27–28 folder (directory) structure, 18–19, 18f getting started with, 16–17 lists, 20–27 loading data from file, 30 loops, 28–29 matrix, 20–27 differentiation, 27 plotting data, 30–31 simple arithmetic, 19–20 transpose, 21 tuples, 20–27 vectors, 20–27 writing data to file, 30 Q Q-mode factor analysis, 244 Quadratic form, 7, 23 Quality factor, 311 R Radon’s problem, 252, 253f Radon transform, 253, 253–254f Random variables computing realizations of, 63–65 functions of, 54–58 Ray, 297 Rayleigh wave, 309 Ray tracing, 291–292 Reflection coefficient, 74–76 Relative entropy, 117 as guiding principle, 127–128 maximum, 127 minimum, 127 Rescaled generalized inverse, 154 Resolution analysis without data kernel, 110 generalized inverses with good, 99–101 measures of goodness, 99 of some generalized inverses, 98–99 and variance, trade-off, 250–251, 251f Resolving kernel, 249–250 Index 329Reweighting process, 187 Riemann approximation, 50 R-mode factor analysis, 244 Robust, 69 Row vector, 4–5 Rule for error propagation, 57 S Sample matrix, 237, 240, 244, 246 Sample mean, 52 Sample median, 115 Sample standard deviation, 52 Scale parameter, 179 Schultz method, 321 Secular variation, 312–313 Seismic attenuation, 311 Seismic imaging, adjoint methods in, 300–302 Seismic migration, 303–305, 304–305f Seismometers, 299 Self-adjoint, 257 Sidelobes and Backus-Gilbert spread function, 101 Signal correlation, 312 Signal processing techniques, 298 Simple arithmetic MATLAB , 3–4 Python, 19–20 Simulated annealing method, 224–225, 226f Singular-value decomposition, 79, 165–169, 237–238, 237f, 240, 244, 298 derivation of, 169–170 and natural generalized inverse, 165–169 Smoothness via prior covariance matrix, 135 Solution (answer), 44 estimation, 45 to inverse problems, 44 vs. operators, 95 probability density functions as, 46 weighted averages of model parameters as, 46–47 Sparse matrices, MATLAB and Python, 83–86 Sparseness, 68 Spectral curve fitting, 42 Spread function Backus-Gilbert, 101, 106f Dirichlet, 99, 101 Square root of variance, 51–52 Squeezing, 156–157, 156f Static variables, 4, 28 Stationary, 135 Steepness/roughness of solution, 80–81 Straight-line problem, 72–73 constrained fitting of, 89–90 Surface wave, 309 tomography, 310 Sylvester equation, 100–101 T Tectonic plate motions, 312 Tee-star, 311 Thermal diffusivity, 142 Thermal parameter in Newtonian cooling, 266–268, 268f Thomas method, 144 Three-dimensional exponential covariance, 135 Three-dimensional Gaussian covariance, 135 Toeplitz matrix, 44, 45f Tomography, 40 acoustic, 39–40, 39f, 279–280, 280–281f attenuation, 311 continuous inverse theory and, 252 finite-frequency travel time, 305–307, 307f inversions, 299 surface wave, 310 wavefield, 303, 304f Total joint probability density function, 130f Total variation (TV) regularization, 188 Trade-off curve, 106–107, 106–107f, 251 Trade-off, resolution and variance, 106–107, 107f, 250–251, 251f Trans-dimensional models, 229–230 curve-fitting example, 232f examples of solutions, 230–234, 231f of Laplace transform problem, 232f Transformation matrix, 159 of variables, 230 Transpose MATLAB , 5–7 Python, 21 Triangle inequalities, 69 Tuples, Python, 20–27 U Underdetermined problem, 77–79 Unit covariance matrix, 97–98, 98f Unit vector, 7 V Variance, 51 Varimax procedure, 241 Varimax rotation, 242–244 Vectors MATLAB , 4–11 Python, 20–27 Vector spaces, applications of, 159–178 householder transformations, 159–162 designing, 162–163 inequality constraints, 171–176 model and data spaces, 159 simplifying linear equality and inequality constraints, 170–171 singular-value decomposition derivation of, 169–170 and natural generalized inverse, 165–169 solution of mixed-determined problem, 164–165 transformations, length preservation, 163–164 Velocity structure, 297 free oscillations and seismic surface waves, 309–311 Vibrational problems, 294–295 Voxels, 251 W Wave equation, 301 Wavefield tomography, 303, 304f Wave number, 253 Weighted averages, 151 of model parameters, as solution, 46–47 Weighted damped least squares, 81–82 Weighted minimum length, 81 Welch-Satterthwaite approximation, 129 Woodbury formula/identity, 123–124, 319 X X-ray imaging, 40–41 #ماتلاب,#متلاب,#Matlab,#مات_لاب,#مت_لاب,
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