Troyan V., Kiselev Y. Statistical Methods of Geophysical Data Processing
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Index
Absolute deviation (mean), 27
Algebra of events, 3
Amplitude spectrum, 374
Basis
Hilbert space, 251
Korhunen–Loeve, 282
Bayes
formula, 11, 18
strategy, 88, 280
Bias, 285
Bit, 62
Brightness, 132
Coefficient
asymmetry (nondimensional), 28
correlation, 274
variation, 27
Complement, 3
Condition number, 259
Conditional
mathematical expectation, 49
variance, 49
Confidence
bounds, 59
interval, 191
probability, 191
Correlation
coefficient, 25
function, 274
Markovian, 108
operator, 273
Covariance, 25
Cross-correlator, 274
Debye expansion, 136
Decile, 26
Decision rule, 84
admissible, 88
Density function
marginal, 17
of random vector, 16
standard normal, 46
Design matrix, 103
Design of experiment
D-optimal, 290
E-optimal, 291
G-optimal, 291
L-optimal, 291
continuous, 290
standardized, 290
Deviation
mean-square, 24
standard, 24
Dilatation, 119
Distance
10 — 20 percent, 27
interquartile, 27
Distribution
a priori, 277
of probabilities, 12
Distribution function
cumulative, 12
of random vector, 16
standard normal, 46
Dynamic (Kalman) filtration, 382
Effective solution, 285
Eigenfunction, 253
Eigenvalue, 253
Ellipse
of equiprobability, 50
433
434 STATISTICAL METHODS OF GEOPHYSICAL DATA PROCESSING
of dispersion, 50
Entropy
differential, 66
mean conditional, 66
total, 66
Equation
characteristic system, 140
diffusion, 130
dispersion, 137
eikonal (general), 139
Euler, 254
Lame, 120
telegraph, 130
transport, 132, 306
wave, 126
scalar, 130
vector, 129
Wiener–Hopf, 390
Ergodic hypothesis, 76
Estimate
asymptotic efficient, 96
asymptotically bias, 93
efficient, 96
maximum a posteriori probability, 280
maximum a posteriori probability , 90
Event
certain, 3
elementary, 1, 2
impossible, 3
random, 2
Events
independent, 9
nonindependent, 9
Fisher matrix, 69
Flow of events, 41
ordinary, 42
Poisson, 42
stationary, 42
without a consequence, 41
Frechet derivative, 253
Function
autocorrelation, 74
correlation, 74
decision, 84
likelihood, 89, 164, 279
loss, 84
sufficient reception, 212
Functional, 251
convex, 268
Rayleigh, 254
stabilizing, 271
tomographic, 297
Functional gradient, 253
Fundamental sequence, 249
Generating function
central moments, 32
initial moments, 31
Geometrical optics, 135
Green’s theorem
first, 257
second, 257
Independent random variables, 18
Inequality
Cauchy–Bunyakovskii, 250
Schwarz, 249
Information
a priori, 265
complete mutual, 65
Kullback, 209
partial, 65
Interpretation
combined, 88
interval-quantitative, 88
point-quantitative, 87
qualitative, 87
quantitative, 87
quantitative-qualitative, 88
Intersection, 2
Lame parameters, 119
Law
Coulomb, 127
Faraday, 127
Oersted, 128
of averages, 34
Maxwell hypothesis, 128
Median, 26
Method
likelihood ratio, 90
maximum likelihood, 89, 279
steepest descent, 267
Minimax procedure, 89
Mode, 27
Moment
central, 28
first, 73
Index 435
initial, 28
Newton generalized scheme, 267
Norm
of function, 248
of linear operator, 252
Operator, 251
compact, 252
conjugate (Lagrange), 253
Fisher (information), 281
Hermite, 253
Hilbert–Schmidt, 252
Lame, 120
limited, 252
linear, 251
linear homogeneous, 77
linear regression, 286
non-negative, 255
positive, 255
positive defined, 255
self-conjugate, 253
unbounded, 263
unitary, 253
unlimited, 252
Pareto optimality, 291
Percentile, 26
Phase spectrum, 374
Poisson ratio, 119
Poynting’s vector, 130
Principle
Fermat, 143
Huygens–Kirchhoff, 143
Probability
conditional, 9
elementary, 45
model, 4
space, 4
statistical, 5
Probability density
a posteriori, 278
conditional, 277
marginal, 277
Problem
ill-posed, 264
well-posed, 264
well-posed by Tikhonov, 264
Quartile, 26
Quasisolution, 272
Radon transform, 315
Random field (homogeneous), 274
Random fields (independent), 277
Random function
Markovian, 76
stationary, 74
wide sense, 75
Random variable
centered, 28
continuous, 14
discrete, 12
Range, 27
Rate of occurrence, 41
Refraction vector, 146
Regression model, 290
Regularization
n-th order, 270
Ivanov–Lavrentiev, 270
parameter , 271
Tikhonov, 270
Resolvability condition, 137
Risk
Bayes, 85
function, 85
Robust estimation, 98
Sample, 83
repeated, 83
size, 83
Shortwave asymptotic, 146
Significance level, 59
Singular numbers, 180
Slowness, 160
Slowness vector, 122
Space
Banach, 249
decision, 84
elementary events, 1
Hilbert, 249
metric, 249
complete, 249
observation, 84
outcomes, 1
parameter, 84
Spatially-time ray, 142
Spline functions, 217
Statistics, 83
Stress, 116
436 STATISTICAL METHODS OF GEOPHYSICAL DATA PROCESSING
tenzor, 117
System of estimates
jointly asymptotic efficient, 96
jointly efficient, 96
System of normal equations, 168
Tomographic
experiment, 294
Uniform pressure modulus, 120
Union, 2
Vector potential, 129
Wavefront, 146
Young modulus, 119