Index Akaike information criterion (AIC) 105, 290 analysis of variance 35, 44, 127 132 angular transformation 22 anisotropy 59, 99 affine or geometric 59, 100 101 anisotropy ratio 101 exploring and displaying 70, 73 transects 70 zonal 59 ASReml 203 a priori variance 52, 58, 84, 92, 98, 102 asymmetry 17, 110 112 long-tail, outliers 110 asymmetric covariance 221 authorized models 80, 82 95 autocorrelation 3, 53 autocorrelation coefficients 55, 66, 76 autocorrelation function 55, 57 autocovariance 53, 56 autocovariance function 57 balanced differences 33 34 Bessel functions 92, 98, 99 bias 43, 144, 186, 199 binary variables 11, 246 block kriging 159, 167 171, 175 180, 188 189 Borders Region of Scotland 24, 105 107 factorial kriging analysis 216 217 histogram 24 variogram 216 bounded variation 84 bounded models, bounded variograms 84 94 bounded linear variogram model 85 box plots, box-and-whisker plots 14, 25 Broom s Barn Farm 257 263, 278 283 maps 262, 264, 279, 282 ph 160 posting 25, 27 potassium 13, 23, 25, 67, 278, 280, 283 variograms 102 103, 175 179, 278, 280, 283 Brownian motion 83 capacity variables 15 Caragabal transect 140 142, 150 151 variogram 142, spectral analysis 150 151 CEDAR Farm 219 220, 226 228 coregionalization 226 228 central limit theorem 32 chi-square distribution 31 circular variogram model 87 circular scales 12 classical sampling theory 28 33, 43, 124, 127 CNSD 80, 224 codispersion coefficient 222 coefficient of variation 17, 287 cokriging 228 234 benefits 231 234 fully sampled case 231 undersampled case 231 variance 229, 231, 234 Geostatistics for Environmental Scientists, 2nd Edition R. Webster and M. A. Oliver 2007 John Wiley & Sons, Ltd. ISBN: 978-0-470-02858-2
310 Index combining models 95 97 conditional negative semidefinite, see CNSD conditional probability 256 confidence intervals of variograms 119 125 confidence limits 29 31 continuity 57 continuous function 57 continuous lag 57 continuous variables 12 continuous scales 12 coregionalization 219 et seq. CEDAR Farm 219 220, 226 228 linear model 222 224 matrices 235 correlation 19 20 correlation coefficient 20, 111 correlation range, see range correlograms 74 cross-correlogram 222 covariance 19 20, 47 60 covariance function 53 55 covariance matrix 57 equivalence with variogram 54 estimation 74 cross-correlation 219 cross-correlation coefficient 221 estimating 222 modelling 222 224 cross-covariance 220 222 cross-covariance function 220 221 cross-indicator variograms 248 249 cross-validation 191 193 cross-variograms 220 cross-semivariance 220 cubic trend surface 40 cubic variogram function 93 cumulative distribution 15, 19, 23, 24, 26, 247, 250 260 cumulative distribution function 250, 256, 258, 260 degrees of freedom 128, 132 design-based estimation 28 Dirac function 58 Dirichlet tesselation, tiles, see Thiessen polygons discontinuity 81, 177 disjunctive kriging 243 et seq. Gaussian 251 Hermite polynomial 252 255 variance 256 dispersion 16 17 dispersion variance 60 64, 102, 120 distance parameters 89, 91 96, 224, 237, 298 double spherical variogram model 97 double spherical examples 96, 107, 216, 237 238 drift 59, 195 205 external drift 203 211 E-BLUP 202 efficiency 32 33 environmental data, notation 12 environmental variables 11 12 binary 11 continuous 12 ergodicity 53 estimation 26 30, 32 classical, design-based 26 30, 33 estimation variance 29, 33 local 153 181 regional 181 183 simple random samples 28 32 stratification 32 33 systematic sampling 33 35 exhaustive variogram 122 experimental covariance function 73 74 experimental spectrum, see spectral analysis experimental variogram 60, 68 73, 288, 295 experimental semivariances 60, 68 73, 288, 295 exploratory data analysis and display 22 25, 285 288 exponential variogram model 88, 91, 92, 94, 95 F ratio 130 factorial kriging analysis 212 217 first moment 52
Index 311 fitting models 101 107, 290, 296, 298 complexity 105 computer programs 103 difficulties 101 102 GenStat 290, 296 recommended procedure 102 Fourier transform, see spectral analysis frequency distribution 13 15, 286 frequency domain, see spectral analysis gamma function 31 Gaussian diffusion process 251 Gaussian disjunctive kriging 251 Gaussian distribution 18 Gaussian variogram model 93 Gaussian simulation 273 274, 278 281 GenStat 293 298 geometric or affine anisotropy 100 101 geometric mean 21 geostatistics, general 1 6 history 6 8 overview 1 6 roles 2 goodness-of-fit criterion 104 GSLIB 274, 275, 277 heavy metals 235 241 Hermite transformation 252 257 Hermite polynomials 252 257 hierarchical analysis of variance 127 132 nested sampling and analysis 127 128, 131 132 histogram 13, 286, 294 hole effect 56, 58 hole effect models 98 99 inclined plane 40, 206 indictor variables (indicators) 246 indicator coding 246 indicator covariance function 249 indicator kriging 249 251 indicator variograms 247 intensity variables 15 interpolation 37 42 intraclass correlation 44 intrinsic corgionalization 224 225 intrinsic hypothesis 54 intrinsic random function of order k 59 intrinsic variation 54 inverse functions of distance 40 isarithmic chart 73, 75 isotropic variation 70, 82, 124, 160, 187, 289, 297 isotropic variograms 70, 73 joint cdf 52 joint distribution 20, 52, 66 joint pdf 20 Krige s relation 60 61, 63 kriging 153 et seq., 291, 297 Bayesian 155 block kriging 156 159 cokriging 228 234 disjunctive kriging 243 et seq. factorial kriging 212 217 general characteristics 154 general theory 155 159 indicator kriging 249 kriging with external drift 203 205 kriging equations 172 173 kriging neighbourhood 172 173 kriging variance 158, 159, 163, 178 180, 182, 184, 185, 188 189, 198, 209, 211, 256 kriging weights 159 160 kriging with trend 195 211 E-BLUP 202 kriging with external drift 203 205 universal kriging 196 203 lognormal kriging 184 185 mapping 173 174, 181 191 ordinary kriging 155 160 ordinary kriging equations probability kriging 155 regression kriging 199 simple kriging 183 184 universal kriging 196 203 universal kriging equations 197 198 Kronecker delta 57, 95 kurtosis 18 lag 53, 57 increments, interval 69 73
312 Index Lagrange multiplier 157 least-squares methods 40, 102 103, 105, 199, 290 Levenberg Marquardt method 103 linear drift 197 linear mixed model 134, 200 linear sequences 139 140 local estimation 153 et seq. logarithmic transformation 21, 259, 287 logit transformation 22 log-likelihood 202 lognormal kriging 184 185 long-range trend 82, 198, 215 mapping 291 292 interpolation 37 42 kriging 173 174 optimal sampling 185 191 posting 27 Marcuse model II 127 Matérn variogram function 94 MATLAB 277 mean 15 mean error (ME) 191 mean squared deviation ratio (MSDR) 192 mean squared difference (MSE) 191 mean squared error, prediction (MSE) 45 46 mean squared residual (MSR) 107 measurement error in kriging 180 median 16 missing values 68, 70, 286 mode 16 model fitting, see fitting models Monte Carlo methods 121, 270 multiple regression 40 multi-stage sampling 127 natural neighbours 39 interpolation 39 negative exponential variogram model, see exponential variogram model nested sampling and analysis 127 138 balanced designs 127, 128 components of variance 127, 128 REML estimation 132 138 unbalanced designs 131 132 Wyre Forest 132 138 nested spherical variogram model, see double spherical variogram model non-ergodic variogram 60, 120 non-linear regression 103 normal distribution 18 20, 252, 287 random variables 49 normalized difference vegetation index (NDVI) 47 notation 12 nugget, nugget variance 56 58, 79 84, 131 nugget:sill ratio 110, 161 163 nugget variogram 95 Occam s razor 77 ordinary kriging 155 160 outliers 22, 65, 113 118 Pearson product-moment correlation coefficient 20 pentaspherical variogram model 84, 88 periodic variation 97 99, 139 152 amplitude 97 98 periodic variogram model 97 99 phase 97 98 point samples 3 Poisson process 87, 90 positive intercept 79, 81 positive semidefiniteness 57, 79 posting 5, 25, 27, 285, 295 power function variogram 83 power spectrum, see spectral analysis prediction 37, 153 194 general formula 37 kriging 153 et seq. prediction error 43 prediction variance 43 purposively chosen sample 45 random sample 44 probability density 18, 49 probability density function 49 process control 5
Index 313 projection matrix 201 pseudo-cross-variogram 241 242 punctual kriging 155 et seq. pure nugget, pure nugget variogram 56, 95 quadratic trend surface 40, 41 quadratic drift 197, 207 quasi-stationarity 55 random effects model 127, 200 random sample, prediction 28 random variables 49 et seq. random functions, random process 49 random variation 49, 59, 79 random walk model 83 range 84 effective range 89 realization 49 regional estimation 181 183 regional variogram 49, 60, 121 regionalized variables, theory 48 60 regression 40 44 regression kriging 199 regression surfaces, see trend surfaces regular sampling for variogram in one dimension 68 69 in two dimensions 70 71 regularization 63 65 regularized variogram 64 relative precision 33 residual maximum likelihood (REML) 132 134, 200 202 components of variance 133 134 variogram estimation 202 sample correlogram 74 sample mean 15, 29 sample variogram, see experimental variogram sampling 26 et seq. design, plan 28, 186 intensity, density, spacing 164, 186 et seq. sample size for variogram estimation 119 126 theory 28 et seq. SAS 103 scatter diagram 22, 66, 193, 210, 263 Schwarz s inequality 223 screening 285 second moments 17, 52 second-order polynomial, see quadratic second-order stationarity 52 semivariance 54 et seq. estimation 65 et seq. short-range drift 59, 81 sill 56, 79 sill variance 84 simple kriging 183 184 simple random sampling 28 30 estimation 28 estimation variance 29 standard error 29 simulation, stochastic 267 283 case study, illustration 278 283 Cholesky (LU) decomposition 272 273 conditional 270 271 purpose 271 sequential Gaussian 273 274 simulated annealing 274 276 turning bands 276 unconditional 270 sinusoidal function 97 98 skewness 17, 287 skewed histogram 24, 287 smoothing function, see spectral analysis soil classification 42 44 soil maps 42 44 spatial analysis, aide mémoire 285 292 spatial classification 42 44 spatial correlation 55 et seq. spatial correlation functions, characteristics 55 60 spatial covariance 50 spatial dependence 58 spatial distribution 288 spatial domain, see spectral analysis spatial estimation, see kriging spatial interpolation 37 40 spatial prediction 37 46 spatial processes 47 et seq.
314 Index spectral analysis 139 152 Bartlett windows 145 Caragabal transect 140 142, 147, 150 151 confidence limits and intervals 149 estimation 144 Fourier transformation 143 frequency domain 143 Parzen windows 146 power spectrum 143 144 smoothing 145 146 spatial domain 140, 142 spectral density 140 theory 140 143 spectrum see spectral analysis spherical variogram, spherical model 84, 87 88, 100, 164, 166 splines 42 SPOT 50 S-Plus 277 square root transformation 21 stable variogram models 91, 93 standard deviation 13 standard error 29 standard normal deviate 30 standard normal distribution 31 stationarity 52 et seq. statistical fitting 102 statistics, basic 11 et seq. stochastic process 49 stratified sampling 32 estimates 32 precision 32 stratification 32 full stationarity 53 structural variance covariance matrices, see coregionalization matrices Student s t 30 sum of squares 31, 130 summary statistics 13 et seq., 293 support 61 Swiss Jura 235 241 coregionalization 237 240 principal component analysis 236 trace metals 235 241 variograms 238 239 symmetric distributions 16 systematic sampling 33 35 target population 28 theoretical variogram 60, 288 Thiessen polygons 38 transformations 20 22, 99 back-transformation 185 Fourier transformation 143 et seq. Hermite transformation 252 254 trend 40, 59, 81, 195 et seq. trend surfaces 40 42 triangulation 38 two-dimensional variogram, see variogram unaligned sampling 34 unbalanced sampling design 131 132 unbounded random variation 83 unbounded variogram 58 units, see sampling universal kriging 196 203 variance 16, 29 variance ratio 130 variogram 54 76, 288, 295 behaviour near the origin 80 82 behaviour towards infinity 82 block-to-block integration 64 definition 54 equivalence with covariance 54 estimation 65 76, 295 linear approach to origin 81 modelling 77 107, 296 parabolic approach to origin 81 regularized variogram 63 65 reliability 109 et seq. variogram cloud 65 66 variogram functions, limitations on 79 80 Voronoi polygons, see Thiessen polygons weak stationarity 52 weighted average 37 weighted least squares 102, 104
Index 315 weighting function 252 weights interpolation weights 38 40 kriging weights 159 172 white noise 58, 83 Whittle elementary correlation 92 Whittle variogram model 92 within-class variance 44 Wyre Forest survey 134 138 nested sampling and analysis 134 138 Yattendon 205 211 kriging with drift 207 211 REML estimation 208 211 variograms 207 zonal anisotropy 59