اسم المؤلف
William Menke
التاريخ
11 يوليو 2024
المشاهدات
59
التقييم
(لا توجد تقييمات)

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
11.10 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
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
14.6 The Fourier slice theorem 253
14.7 Linear operators 255
14.8 The Frechet derivative 258
14.9 The Frechet derivative of error 258
14.10 Back-projection 260
14.11 Frechet 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
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
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
Backward recursion, 144–145
Banana-doughnut kernels, 307–309, 308f
Bayesian inference, 62
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
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
in case of first-order dynamics, 143–144
using Thomas recursion, 144–145
Data covariance, 110
Data kernel, 36
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
continuous inverse problem, 263f
Frcehet derivative, 261–262
linear, 256
Digital filter design, 275–276
Dirac delta function, 256
Dirac impulse function, 301
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
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 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
Frechet 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)
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
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
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
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
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
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
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
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
Quality factor, 311
R
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
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
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
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
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
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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