Geophysical Data Analysis and Inverse Theory with MatLAB and Python – Fifth Edition

Geophysical Data Analysis and Inverse Theory with MatLAB and Python – Fifth Edition
اسم المؤلف
William Menke
التاريخ
11 يوليو 2024
المشاهدات
59
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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
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

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