Index. Geostatistics Modeling Spatial Uncertainty. Edited by JEAN-PAUL CHILkS and PIERRE DELFINER Copyright by John Wiley & Sons, Inc

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1 Geostatistics Modeling Spatial Uncertainty Edited by JEAN-PAUL CHILkS and PIERRE DELFINER Copyright by John Wiley & Sons, Inc Index Accumulation, 149, 200 Additive renormalization, 252 Additivity dispersion variances, 132 kriging and drift, , , 310 Affine correction. see Change-of-support models Agriculture, 382, 419 Aliasing, 499, 501 Allowable linear combination, 60, 169 of order k (ALC-k), variance and covariance, 6 I, 62 Alternating process, 99, 149 random set, 541 Ambartzumian process, 489 Anamorphosis, 381, See also Transformation function Anisotropy, 35, 52, 93-97, 107, 123, 571 geometric, 93-95, other models, 96-97, 476 zonal (stratified), 95-96, 156, 364, 57 I Annealing, see Simulated annealing Area of influence, 48 ARIMA, see Autoregressive ARMA, see Autoregressive Asynchronous data, Atom, 637 Autoregressive integrated moving-average (ARIMA), 249 moving-average (ARMA), 205, 249, process, I 15, I Barker's algorithm, 528 Bathymetry, 80, 210, 217, 449 Bayesian conditional simulation, I covariance estimation, 110, 190 kriging. see Kriging Bessel covariances, see Variogram models functions, 638 Beta distribution, 403 Bi-gamma distribution, 404, 443, 448 Bi-gaussian distribution, 399 random function, 17 Bias in sample covariance, 33 in variograrn of residuals, I2 I, 124 Bilinear model of coregionalization, 350 Bilognormal, 103, 645 Binomial isofactorial model, 403 Birth-and-death process, 402, 560 spatial, 560 Bivariate distribution, see Isofactorial models Black-Scholes model, 229 Block average, 195 distribution, see Change of support, estimation, , 197 kriging, , 197 simulation, BLUEPACK, 283 Bochner's theorem, 64 Boolean objects, 545 random function, 401, , 561 random set, Bootstrap, I10 Bore1 set,

2 688 INDEX Boundary conditions, 600, Brownian motion. 89, 172, 23 I, 29 I, , 635. See also Brownian random function fractional, 89, 259, 592 integral of, 249, 5 18 representation of, 257, 268 Brownian random function, 236 fractional, 89, 149, 510 Cardinal-sine covariance, see Variogram models Cartier s relation. 426 Categorical variable, 458, Cauchy algorithm, 63 Change of support , 376 by disjunctive kriging, 445 by indicator kriging, 436 by maximum, 549 afline correction, 43 I, 436 isofactorial models, See also Discrete change-of-support models mosaic. 433 multi-gaussian, 435, 446 permanence of lognormality, 433, 436 Channel tunnel, Charlier polynomials, 403 Cholesky decomposition, 169, 464 Choquet capacity, 546 Circular Civil engineering, 128, , 578 Cluster process. 553, 554 Clustered data. 45, 378 Codispersion, 330, 334 Coherency spectrum, 328 Cokriging and inverse problems, 610, 630 and regression, I, 339 collocated, , 360 disjunctive, 419 indicator, indicator-rank, of errors, see Filtering ordinary , 307 simple, universal, with nugget effect, 307 Cokriging equations simple, 296 universal, 299 with cross-variograms, 330 with derivatives, 315 Cokriging variance, 296, 299, 305 Cokriging weights, 300, 301, 613, 620 Collocated cokriging, see Cokriging Completely monotone covariance, 70, ,493 Compositional data, 350 Concave variograms, 68, S14, 537 Conditional distribution, 378 approximation to, 379, 394 Conditional expectation as ideal estimator, 154, 191, 229, defined. 14 Conditional simulation, Conditioning by indicator values or inequalities , by kriging, histogram, I Confidence interval, 164, 177, 191, 214, 220, 379 Conformal transformation, 626 Connectivity, 520, , 602 Consistency change-of-support models, 430 variogram models, , 107 Continuity in the mean square, 13, 58 of realizations, 58, 148 Contour mapping, 8, 207, 212 Conventional income. see Selectivity curves Convex covariances, 66, 537 Carrelation length, 74, 148 Correlogram, 30 Cosirnulation, , 629, 630 Covariance function. 13, 30 isotropic, 3 I, 69 spectral representation, see Spectral representation Covariance function matrix, 325 Covariance matrix decomposition, , 51 I Covariance models, see Variogram models Covariogram, 25,70-73 geometric, 7 1 Cox process, Cross-covariance, of indicators, 102, 387, 399 Cross-spectral measure, 325 Cross-validation, , 209, , Cross-variogram, , 332 Cubic

3 INDEX 689 generalized covariance. see Generalized covariance models: polynomial Curvilinear distance, 45 Darcy s law, , 616, 634 De Wijs formula, 9 I variograni, see Variogram models Dead-leaves random function, 549, 56 I tessellation, 549 Derivative, 58 covariance of. 59, 316 directional, 314 estimation of, estimation with, physical validity, I9 Design, see Network design, Sampling design Destructuring, 40 I, 402, 404, 4 I2 Differentiability in the mean square, 58, 68, 72, 73, 85, 160, 260 of realizations, 59 Diffusive random function. 402, 459, 542 Dilation, 592 Dilution simulation method, , Dip. 313, 318 Dirac measure (delta-function), 637 Discrete change-of-support models gamma, 443 Gaussian, , 435, , , 573 Hermite, 442 Laguerre, 448 Disjunctive cokriging, 419 Disjunctive kriging, lognormal, 4 I6 practice, 4134 I6 with change of support, see Change of support with unknown mean, 417, 446 Disjunctive kriging equations general case, 390 with isofactorial model, 392 Disjunctive kriging variance, 39 I, 394 Dispersion absolute. 9 I indicator, 407, 409, 424 variance, Distribution empirical, , 38 I, 408 estimation, See dso Change of support, regional, 376 Doubly stochastic Poisson process, see Cox process Drift, defined, 13, 165, 260, 270 estimation nonpolynomial, I27 random, 187 Dual kriging, see Kriging Dummy points, 318 Econometrics, 423 Entropy. I 15, 149, 412 Environmental studies. 48, 137, 365, 419, 448, 632 Epistemology. 3, 22-24, 137 Equidistributed sequence, 575 Ergodicity, 1 Y-22, 74, 543 Error contrast, 239 Estimation terminology, I5 1 Estimation variance, Exploratory data analysis, 34, 39, 4 I, 112 Exponential covariance, see Variogram models Extension variance, 130 External drift model, External knowledge. 35 I, 472 Extreme value prediction, 214, , 416, 454 Facies,.see Categorical variable Factor, 3Y I, 39.5 nonpolynomial, polynomial, 397 Factorial kriging analysis, Factorized covariance, , 364, 589 Fast Fourier transform (FFT), Fault, 54, 208, 219, 357 Filtering, see olso Factorial kriging analysis polynomial, see ALC-k random error, , 266, systematic error, Filtration velocity, 596 Finite-dimensional distributions, 29 Flow steady-state, 595 transient. 629, 632 Flow equation. 616 perturbation approximation, 617, Forestry, 138, , 308 Fractals. 89, Y I, 99, , 149, 507, 509, Fractures, 279, Fuzzy, see Kriging, Variogram

4 690 INDEX Gamma distribution, 403, 406, 641 function (Euler). 638 Gaussian distribution, 639 random function, 17 transformation. see Transformation Generalized covariance, estimation, spectral representation, see Spectral representation Generalized covariance models polynomial, , power law, , 5 17 spline, 264, , 609 Generalized cross-covariance, Generalized increment, see ALC-k Generalized least squares, see Least squares Generalized random process, 78, , 606,609,619 Generalked variogram, 121, Genetic algorithm, 567 Geochemistry, 423 Geology, 8,45, , 318, 562. See also Fault, Fractures Geophysics, see Gravimetry, Magnetism, Seismic Geostatistics, 2 Geothermal activity, Germs, 486, 541, 546 Gibbs sampler, 229, 529, 531, 566 Global estimation, 25-26, 131, , 196, I99 of block distribution, of point distribution, Global warming, 372 Gold deposits, 52, 91, 212, 214, 412, 436 Gradient, see Derivative Grains, see Boolean objects Gravimetry, 85, 128, Green function, 605, 610, 623, 635 Guess field model, h-scattergram, 41 1 h". see Generalized covariance models, Power law model, Variogram models Hankel transform, 68 Hard data, 468, 525 Hard-core process, 553 Hastings algorithm, 528 Hausdorff dimension, see Fractals Heat equation, 402, 635 Hermite isofactorial model, 404, 409 change of support, polynomials, 399, 402, 448, 640 Hilbert space, 161, 185, 201,250, 297, 390 defined, 14 Histogram. see Distribution Hitting functional, 546 Hole effect, 55, 69, 92 Homoscedasticity, 164 Hydraulic conductivity, 595, 634. See also Permeabili ty head, 595, 633 resistivity, 599 Hydrogeology, 123, 137, , 382, , Hyperbolic covariance, see Variogram models Icosahedron, 35,477 Ill-posed problem, 612, 614 lmage restoration, 53 I, 566 Impulse function, see Green function Indicator, see Cokriging, Kriging, Sequential indicator simulation, Variogram Inequality constraints, see Conditioning, Kriging Infill asymptotics, see Micro-ergodicity Information effect, 376, 427, 429 Infrared catastrophe, 67, 259 Inhibition process, see Hard-core process Integral range, 73-74, 544, 619 Interpolation, I 50, See also Kriging Interpretive models, 8 Intrinsic correlation model, see Proportional covariance model Intrinsic properties, 25 I, 266 Intrinsic random function (IRF), 17, 31 Intrinsic random function of order k (IRF-k), abstract, lnvariance under linear transformation of the f', 127, under shifts, see Shift invariance Inverse Gaussian distribution, 553 Inverse problem, Isofactorial models, 391, , 543. See also Binomial, Hermite, Jacobi, Laguene, Meixner, Negative binomial, Poisson, Uniform barycentric, 410 beta, 403, 41 1

5 INDEX change of support, See also defined, 395 diffusive, mosaic, 400, 4 I0 orthogonal indicator residuals, 401, 405, 549 properties, pure, , 409 Iterated exponential covariance, see Variogram models It6 stochastic integral, 635 Jackknife, 41, 284 Jacobi isofactorial model, 403 change of support, 444 polynomials, 403 Jump model, 459,537 K-function, 554 Krawtchouk polynomials, 403 Krige's regression effect, , 200 relationship, 132 Kriging and regression, and splines, Bayesian, disjunctive, see Disjunctive kriging dual, , 226 fuzzy, 176 indicator, , 387, 419, 436 interpolation properties, , 171 intrinsic (IRF-k), lognormal, , 382, 436 minimax, 176 of complex variable, 350 of spatial average, ordinary, 165, 170, 185 positive, 225 probability, 385 random, 197 robust, sensitivity to variogram simple, trigonometric, under boundary conditions, I under inequality constraints, universal, with nugget effect, 197, Kriging equations Bayesian, I89 dual, 186 intrinsic (IRF-k). 265 dual, 272, 275 ordinary, see universal simple. 155 universal, with variogram, I70 Kriging neighborhood, 153 global, 187, 209 moving, 202, 234 selection, , 367 Kriging variance, 156, 170, 265 use, Kriging weights, , L' norm, 42 L2 norm, Lagged scatterplots, 41 I Lagrange multiplier, 167, 299 Laguerre isofactorial model, 404, 409 change of support, 443 polynomials, 403, 642 Laplace equation, 607 Layer cake model, Least squares, 27 I generalized, 180 Lebesgue measure, 637 Legendre polynomials, 403 Levinson algorithm, 479 Linear congruential method, 574 Linear model of coregionalization, Linear predictive coding, 204 Linear variogram, see Variogram models Lipschitz condition, 634 Local average subdivision, 506, 5 I7 Local estimation, see Block average, Cokriging, Disjunctive kriging. Kriging Locally equivalent stationary covariance, I stationary random function, 107, 4 17, 446, 57 1 Logarithmic variogram. see Variogram models Lognormal blocks,,see disjunctive kriging, see Disjunctive kriging distribution, 644 grades, 56, 91, 423 kriging, see Kriging random function, 104, 108, 142, 145, 382, 448,599, variogram, I03 LU decomposition, see Covariance matrix decomposition

6 692 INDEX Magnetism, 85, 363, Marked point process, 545 Markov chain, 402, 527, 537, 542, 560 process, property, , , 364 random field, 529 Markov chain Monte Carlo, Markov-type model, 305, 326 Master points, see Pilot points Maximum likelihood, 109, 630 restricted, 110, 281 Measure theory, Measurement error, 78-79, 2 10, 2 18, 468. See also Filtering Median unbiased, 19 I Meixner isofactorial model, 404, 409 polynomials, 403, 644 Meteorology, 35, 90, 110, , , 352 Metropolis algorithm, 528, 566, 570 Micro-ergodicity, , 278 Microstructure, Migration process, , 477, 613 Minimax, see Kriging Minimum stress surface, 291. See also Splines Mining. 56, 108, 129, 194, 214, 307, 412, , 553, Misclassification, 388 Monitoring data, 362, 368, 371 Monte Carlo, 453 Montee, see Radon transform Mosaic isofactorial model, change of support, 444 random function, 384, 459, 535 Moving-average process, , Multi-gaussian model, 381. See also Multidimensional scaling, 37 I Multivariate covariance model, 325 estimation,.see Cokriging recovery, 447 spectral representation, 325 Navier-Stokes equations, 596, 634 Nearest neighbor, 554 Negative binomial distribution, 407, 643 isofactorial model, 403, 405 change of support, 444 Nested structures, 48, 105. See also Factorial kriging analysis Network design, 136, 366, 448 Neural networks, 567 Neyman-Scott process, see Cluster process Nonstationary covariance, I88 estimation, I Nonstationary mean, see Drift Normality index, 424 Nugget effect , 104 multivariate, 310, 327 Objective property, 23, 144, 234 Octant search, see Kriging neighborhood: moving Order relations, 384 Ordinary kriging, see Kriging Ordinary least squares, see Least squares Ore tonnage, see Selectivity curves Orthogonal complex random measure, 637 Orthogonal indicator residuals, , 549 change of support, 444 Orthogonal polynomials, 397 Orthogonality property, 161, 184, 297, 390 Outliers, 212, 215 Palm distribution, 554 Parent-daughter process, see Cluster process Pareto distribution, 423 Percolation, 592 Periodicities, 56 Periodogram, 114 Permanence of lognormality, see Permeability, 525, 586, 595 block, 600, 628 effective, equivalent. 594, inter-block, 601, 628 intrinsic, 634 upscaled, Perturbation approximation. see Flow equation Petroleum, 319, 354, 563, 567, 568, Phase spectrum, 328 Piemmetric head, 595, 633 Pilot points, 629 Point process, Poisson alignment point process, 554 equation, isofactorial model, 403 lines, 268, 513, , 543 point process, , 536, 554, 638

7 INDEX 693 doubly stochastic, see Cox process with regionalized intensity. 55 I, 559 polygons. 54 I, 546 process, 89, 507 Pdya s theorem, 66, 68 Polygons of influence, 429, 539 Polynomial generalized covariance, see Generalized covariance models Positioning uncertainty, 79-80, 222 Positive definite, 55, 61, 148 conditionally, 6 1 k-conditionally, 258 strictly, 60, 64, , I68 Power law model, see Generalized covariance models, Variogram models Power-averaging. 600 Prediction terminology, 15 I Principal component analysis, 345 Prior distribution, 110, 456, 568 Probability-field simulation, Prqjection theorem, 14 Proportional covariance model, , 384 Proportional effect, 45, 56, 107 Pseudorandom numbers, Quadratic Quantity of metal. see Selectivity curves Quasi-random numbers, 477, Radon transform, 72-73, 82, 85, 475, 612 Random coins method, 49 I Random error, see Filtering Random error model, Random function (field), 12 Random midpoint displacement, 5 10 Random noise, see White noise Random sct, 99. See cdsu Boolean Random spectral measure, 18, 65.68, 494, 638 Random walk, 509 Rangc, 47, 106, I I I. 157, 160, 179, 205 integral, see Integral range practical, 84 Rational quadratic covariance, see Cauchy covariance Reality (comparison with), 198, 220, 418, 436, 447 Recharge. 616 Recovery functions, 420 duality formulas, 423 isofactorial expansion, 415 Regional distribution, see Distribution variograni, see Variograrn Regionalized variable, 2 Regression, 14, 151, 155, 179, 185, 282. See nlso Kriging Regularization, 75-77, 628 Relay effect, 205 Renewal process, Representation of an IRF-k, 2.50 internal, 252, 267, 610 locally stationary, 267, 291 stationary, 255, 210 Reserves, 214,428, 429, 431, 447, 578, 585 Residual, 165, 229, 250 Resistant. see Robust Resistivity, 599 Retrodeformable, 230 Risk assessment, 218 Robust, 39. See ulso Kriging, Variograni Sample variance, 32, 38 Sampling design, error, 79 random, 134 square grid, I35 stratified random, 135 Scale effect, 593, 628. See iilso Upscaling Scale parameter, 82, , 86 Screening effect, , 306, 338, 364 Seismic, 190, 207, 217, 351, , 362 Seismic inversion, 563 Selectivity, curves, 420 index. 424 of a distribution, 425 Self-affinity, I49 Self-similarity, 89, I49 Semivariograin, 32. See UIXJ Variograin Separable covariance, see Factorized covariance Separable random function, 13, 148 Sequential Gaussian simulation, I I Sequential indicator simulation, , 567 Sequential random function, 549 Sequential self-calibration, 629 Shift invariance, 239 of kriging, 156, of polynomials, Shooting process, see Cluster process Sichel distribution, 553, 592 Sill, , 102 Simple kriging, see Kriging Simulated annealing, Singleton s algorithm, 503

8 694 INDEX Skewed distribution, 41, 140, 192, 212, 214, 406 Slutsky s condition, 21, 74 Smoothing. See also Splines filter, 207 relationship, , 185 Soft data, I, 525, 530 Soil science, see Agriculture Space-time models, Spatial distribution, 12 Spatial uncertainty, 2 Spectral density, 65, 70, I 13 filtering, 344 measure, 18, 2 I, 64, 67 modeling, I I5 Spectral representation covariance, 18, isotropic, cross-covariance, 325 generalized covariance, intrinsic random function, 68 stationary random function, 18, 64 variograrn, isotropic, 69 Spectral simulation method continuous, discrete, , , 519, 520 Spectrum, see Spectral density Spherical Spin exchange, Spline generalized covariance, see Generalized covariance models Splines, Square root method, see Covariance matrix decomposition Stability properties (covariances and variograms), 60, 61 Stable distribution, 98 random function, 98, 484, 493 Standard error, 156 Stationarity second-order (weak), strict (strong), 16 Statistical tests, I I I Steady-state flow, see Flow Steiner compact set, 95 Stochastic differential equations, Stochastic integral, 63 Stochastic process, see Random function Structure function, 34. See ulso Variogram Subordinated process, 542 Substitution random function, 402, Successive random additions, 5 10 Support, 6, 153, 637. See ulso Change of support random, 200,385 Support effect, 425, 429 Surface area estimation, Systematic effect, 165, 222, 236 Systematic error, see Filtering Tessellation, Poisson polyhedra, Voronoi, Three perpendiculars theorem, 200 Time series, , 155, 164, 21 1, 479, 481,483, Toeplitz matrix, 465, 479 Topography. 125, 285 Total uncertainty, 110, 176, 455 Training image, 562, 567 Transformation conformal, see Conformal transformation gamma, 406,4 13 Gaussian, 193, 380, 469, 580 negative binomial, 407 uniform, Transformation function, 381 modeling, 381, 405, 407, 413, 580 reverse, 470 Transient flow, see Flow Transition models, 72 phenomenon, 24,48 Transitive theory, Transmissivity, 616. See also Permeability Transport, 632 Trend, 57, 165,234236, 259, Triangle Triangular inequality, 98, 99 Truncated Gaussian simulation, , 567, Truncated pluri-gaussian simulation. 533 Turbulence, 90 Turning bands, , 513,517, 520 Unbiasedness conditional, , 185, 193, , 430 conditions, 166, 299, 312, 315, 355, 367 price for, 183 Uniform conditioning, 446, 447 isofactorial model, 405 Unit sphere, 639

9 INDEX 695 Universal kriging, see Kriging Universal kriging model, 165, , 239, 2.59, 270, 355 Universality conditions. see Unbiasedness Upscaling, 594 Van Jer Corput sequence, 477, 576 Variogram, 31 anisotropic, see Anisotropy as generalized covariance, 2.55 behavior at origin, 50, 104, 1 I1 box plot, 35 cloud, concave, see Concave variograms estimation, , fitting, I fluctuation, fuzzy, 44, 1 I 1, 176 generalized, see Generalized variograni Huberized, 43 indicator, , 383, 400 map, 37 median, 42 misspecification, 105 nugget effect, see Nugget effect of order I, , 149, 41 1 of order 1, 44 of order a of residuals, 116, 120, proportional, 56 quantile, 42 raw, 116, 120, regional, 30, 38, 122, 137 regularized, 75, 90 robust, 39-44, 100, 215 sample, 30, 35-38, 137 slope at origin. 33, 95, 106 spectral representation, see Spectral representation theoretical, 30, 31, 38, 57 underlying, I 18, 120 Variogram models, cardinal-sine, 93, 505 Cauchy, 70, 85, 343,493 circular, 82, 490, 49 I cubic, 84, 491 exponential, 70, 84-85, 101, 202, 203, 225, 402, 453, , 489, , 537, 541, 542, 606, 607 gamma, 70 Gaussian, 69, 85, 175, 203, 3 16, 490, 500, 502, 505 hyperbolic, 70, 505 iterated exponential, 70 J-Bessel, 68, 93 K-Bessel, 86, 202, 489, 493, 607 linear, 87, 172, 182, 225, , 609 logarithmic (de Wijsian), 90-91, 202, 609 pentaniodel, 84 periodic, 92 power law, 87-90, 125, , 538 quadratic, 268 spherical, , 225, 48 1, 490, 49 I, 505 stable, YO, 505 triangle, 6 I, 66, 8 I Voronoi polygons, , 546 Wavelets, , 5 17 Weierstrass-Mandelbrot random function, 5 16 Weight on the mean, 186 White noise, 52, 78, 609, 635, 638 Wiener bounds, 598, 600 Wiener-Lkvy process, see Brownian motion Wild horse, 229 Winsorizing, 214 Yule-Walker equations, 155, 479 Zero effect, 406, 433, 434, 435