Spatial and Spatio-Temporal Geostatistical Modeling and Kriging
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Content
List of figures xi
List of tables xvii
Foreword xix
Preface xxi
The companion website xxiii
1 From classical statistics to geostatistics 1
1.1 Not all spatial data are geostatistical data 1
1.2 The limits of classical statistics 5
1.3 A real geostatistical dataset: data on carbon monoxide in Madrid, Spain 7
2 Geostatistics: preliminaries 10
2.1 Regionalized variables 10
2.2 Random functions 11
2.3 Stationary and intrinsic hypotheses 13
2.3.1 Stationarity 13
2.3.2 Stationary random functions in the strict sense 14
2.3.3 Second-order stationary random functions 15
2.3.4 Intrinsically stationary random functions 16
2.3.5 Non-stationary random functions 18
2.4 Support 19
3 Structural analysis 20
3.1 Introduction 20
3.2 Covariance function 21
3.2.1 Definition and properties 21
3.2.2 Some theoretical isotropic covariance functions 23
3.3 Empirical covariogram 26
3.4 Semivariogram 27
3.4.1 Definition and properties 27
3.4.2 Behavior at intermediate and large distances 30
3.4.3 Behavior near the origin 31
3.4.4 A discontinuity at the origin 33
3.5 Theoretical semivariogram models 35
3.5.1 Semivariograms with a sill 36
3.5.2 Semivariograms with a hole effect 46
3.5.3 Semivariograms without a sill 47
3.5.4 Combining semivariogram models 50
3.6 Empirical semivariogram 52
3.7 Anisotropy 64
3.8 Fitting a semivariogram model 69
3.8.1 Manual fitting 70
3.8.2 Automatic fitting 71
4 Spatial prediction and kriging 80
4.1 Introduction 80
4.2 Neighborhood 83
4.3 Ordinary kriging 84
4.3.1 Point observation support and point predictor 84
4.3.2 Effects of a change in the model parameters 90
4.3.3 Point observation support and block predictor 99
4.3.4 Block observation support and block predictor 110
4.4 Simple kriging: the special case of known mean 113
4.5 Simple kriging with an estimated mean 115
4.6 Universal kriging 116
4.6.1 Point observation support and point predictor 116
4.6.2 Point observation support and block predictor 121
4.6.3 Block observation support and block predictor 121
4.6.4 Kriging and exact interpolation 122
4.7 Residual kriging 122
4.7.1 Direct residual kriging 123
4.7.2 Iterative residual kriging 124
4.7.3 Modified iterative residual kriging 125
4.8 Median-Polish kriging 125
4.9 Cross-validation 134
4.10 Non-linear kriging 138
4.10.1 Disjunctive kriging 138
4.10.2 Indicator kriging 142
5 Geostatistics and spatio-temporal random functions 145
5.1 Spatio-temporal geostatistics 145
5.2 Spatio-temporal continuity 146
5.3 Relevant spatio-temporal concepts 147
5.4 Properties of the spatio-temporal covariance and semivariogram 157
6 Spatio-temporal structural analysis (I): empirical semivariogram
and covariogram estimation and model fitting 162
6.1 Introduction 162
6.2 The empirical spatio-temporal semivariogram and covariogram 163
6.3 Fitting spatio-temporal semivariogram and covariogram models 170
6.4 Validation and comparison of spatio-temporal semivariogram and covariogram models 174
7 Spatio-temporal structural analysis (II): theoretical covariance models 178
7.1 Introduction 178
7.2 Combined distance or metric model 180
7.3 Sum model 183
7.4 Combined metric-sum model 184
7.5 Product model 187
7.6 Product-sum model 191
7.7 Porcu and Mateu mixture-based models 192
7.8 General product-sum model 194
7.9 Integrated product and product-sum models 198
7.10 Models proposed by Cressie and Huang 201
7.11 Models proposed by Gneiting 207
7.12 Mixture models proposed by Ma 211
7.12.1 Covariance functions generated by scale mixtures 211
7.12.2 Covariance functions generated by positive power mixtures 212
7.13 Models generated by linear combinations proposed by Ma 215
7.14 Models proposed by Stein 222
7.15 Construction of covariance functions using copulas and completely monotonic functions 223
7.16 Generalized product-sum model 223
7.17 Models that are not fully symmetric 236
7.18 Mixture-based Bernstein zonally anisotropic covariance functions 237
7.19 Non-stationary models 241
7.19.1 Mixture of locally orthogonal stationary processes 241
7.19.2 Non-stationary models proposed by Ma 242
7.19.3 Non-stationary models proposed by Porcu and Mateu 246
7.20 Anisotropic covariance functions by Porcu and Mateu 247
7.20.1 Constructing temporally symmetric and spatially anisotropic covariance functions 247
7.20.2 Generalizing the class of spatio-temporal covariance functions proposed by Gneiting 248
7.20.3 Differentiation and integration operators acting on classes of anisotropic covariance functions on the basis of isotropic components: 'La descente étendue' 251
7.21 Spatio-temporal constructions based on quasi-arithmetic means of covariance functions 253
7.21.1 Multivariate quasi-arithmetic compositions 255
7.21.2 Permissibility criteria for quasi-arithmetic means of covariance functions in Rd 256
7.21.3 The use of quasi-arithmetic functionals to build non-separable, stationary, spatio-temporal covariance functions 259
7.21.4 Quasi-arithmeticity and non-stationarity in space 264
8 Spatio-temporal prediction and kriging 266
8.1 Spatio-temporal kriging 266
8.2 Spatio-temporal kriging equations 267
9 An introduction to functional geostatistics 274
9.1 Functional data analysis 274
9.2 Functional geostatistics: The parametric vs. the non-parametric approach 279
9.3 Functional ordinary kriging 283
9.3.1 Preliminaries 283
9.3.2 Functional ordinary kriging equations 284
9.3.3 Estimating the trace-semivariogram 288
9.3.4 Functional cross-validation 289
A Spectral representations 295
B Probabilistic aspects of Uij = Z(si)-Z(sj) 300
C Basic theory on restricted maximum likelihood 302
D Most relevant proofs 304
Bibliography and further reading 327
Index 351
List of figures
1.1 Location of the pollution monitoring stations in Madrid and map of predicted NOx levels (10 pm; average of the week days; 50th week of 2008) using geostatistical techniques. 1.2 Percentage of households with problems of pollution and odors in Madrid, Spain, 2001 (census tracts). 1.3 Fires in Castilla-La Mancha, Spain, 1998. 1.4 Location of the monitoring stations in the city of Madrid. 2.1 Simulation of a regionalized variable. 2.2 Four pairs of points separated by a distance h in a 2D domain. 2.3 Stationary and intrinsic hypotheses. 2.4 Top panel: Realization of a Wiener-Levy process. Bottom panel: First-order increments of the realization of the above Wiener-Levy process. 3.1 Spherical, exponential, and Gaussian covariance models with and different values of the scale parameter. 3.2 Spherical, exponential, and Gaussian models with and . 3.3 Bounded semivariogram and its covariogram counterpart. 3.4 Simulations of a rf having semivariograms that only differ in the range: (a) ; (b) ; (c) . 3.5 2D representation of simulations of two rf's with a semivariogram that only differs in the behavior near the origin: (a) linear, (b) parabolic. 3.6 Simulated fields of values using a semivariogram with scale parameter : (a) nugget effect = 0; (b) nugget effect = 0.25. 3.7 Nested semivariogram. 3.8 Upper panel: Spherical model. Left: . Right: . Middle panel: Simulation of a rf having a spherical semivariogram (2D representation). Left: . Right: . Bottom panel: Simulation of a rf having a spherical semivariogram (3D representation). Left: . Right: . 3.9 Left: Pure nugget semivariogram. Right: Simulation of a non-spatially correlated rf (2D representation). 3.10 Upper panel: Exponential model. Left: . Right: . Middle panel: Simulation of a rf having an exponential semivariogram (2D representation). Left: . Right: . Bottom panel: Simulation of a rf having an exponential semivariogram (3D representation). Left: . Right: . 3.11 Upper panel: Gaussian model. Left: . Right: . Middle panel: Simulation of a rf having a Gaussian semivariogram (2D representation). Left: . Right: . Bottom panel: Simulation of a rf having a Gaussian semivariogram (3D representation). Left: . Right: . 3.12 Cubic model with different ranges and the same sill (). 3.13 Simulation of a rf with cubic semivariogram model (3D representation). Left: . Right: . 3.14 Left: 3D representation of a simulation of a rf having a Gaussian semivariogram with . Right: 3D representation of a simulation of a rf having a cubic semivariogram with . 3.15 Upper panel, left: Stable model with the same sill () and scale parameter () but different shape parameter . Upper panel, right: 3D representation of a simulation of a rf having a stable semivariogram (). Bottom panel, left: 3D representation of a simulation of a rf having a stable semivariogram (). Bottom panel, right: 3D representation of a simulation of a rf having a stable semivariogram (). 3.16 Cauchy models with the same sill () and scale parameter () but different shape parameter . 3.17 K-Bessel model with the same scale parameter () and the same sill () but different shape parameter . 3.18 Cardinal sine models with the same sill () and different values of the scale parameter: (a) plot of the models; (b), (c) and (d) simulation of a rf having a cardinal sine model with , respectively (2D representation). 3.19 Power models. 3.20 Linear model. 3.21 Logarithmic model (). 3.22 Nested model composed of: (a) Pure nugget semivariogram (); (b) Spherical semivariogram (); (c) Spherical semivariogram (); (d) Nested semivariogram (3.40). 3.23 Tolerance region on . 3.24 Effect of the tolerance angle on a North-South empirical semivariogram. Tolerance angle: (a) , (b) , (c) , (d) . The observed regionalization was simulated with an spherical model (). 3.25 Twenty-five observed values in a grid. 3.26 Left: Empirical semivariogram (classic estimator). Right: Semivariogram cloud. 3.27 Observed values of logCO* at the 23 monitoring stations operating in Madrid, week 50, 10 pm. Left panel: 2D representation (black: higher values; white: lower values). Right panel: 3D representation. 3.28 Data on carbon monoxide in Madrid, week 50, 10 pm: (a) Classical empirical semivariogram; (b) Semivariogram cloud. 3.29 Observed points and data values. 3.30 Observed points and data values (without the outlier). 3.31 Left panel: Geometric anisotropy. Right panel: Zonal anisotropy. 3.32 Simulation of two rf's. Left panel: The geometric anisotropy case; Right panel: The zonal anisotropy case. 3.33 Simulation of a isotropic rf. 3.34 Semivariogram maps: (a) The isotropic case (circular contours); (b) The anisotropic case (elliptic contours, ). The axes depict lag distances in the corresponding coordinate system. 3.35 3D representation of zonal anisotropy. Left panel: Pure zonal anisotropy in vertical direction. Right panel: Directional semivariograms in horizontal directions (), vertical () and in an intermediate direction (). Source: Emery (2000, p. 111). Reproduced with permission of Xavier Emery. 3.36 Spherical models resulting from the automatic fitting (data on carbon monoxide in Madrid, week 50, 10 pm). 4.1 Location of eight observation points used for prediction at the non-observed point . 4.2 New location of the prediction point . 4.3 Prediction and prediction standard deviation (SD) maps of logCO*: January 2008, 2nd week, 10 am. 4.4 Prediction and prediction standard deviation (SD) maps of logCO*: January 2008, 2nd week, 3 pm. 4.5 Prediction and prediction standard deviation (SD) maps of logCO*: January 2008, 2nd week, 9 pm. 4.6 Six points, , discretizing the block V, and an observed point, . 4.7 Location of seven observation points used for prediction over the block . 4.8 Six points, discretizing , and six points, discretizing . 4.9 Coal-ash data. Location (reoriented) and observed values. 4.10 Coal-ash data. 3D scatterplot. 4.11 Contour plot of coal-ash percentages. 4.12 Coal-ash percentages surface interpolation (via triangulation). 4.13 Coal-ash percentages: Column and row summaries. 4.14 Prediction point. 4.15 Coal-ash percentages: Median-polish residuals. 4.16 Coal-ash data: Original data, median-polish drift and median-polish residuals (the lighter the color, the higher the value). 4.17 Coal-ash residuals: Classical semivariogram. 4.18 Exponential, spherical, cubic, and Gaussian models resulting from the WLS fitting. (Data on carbon monoxide in Madrid, week 50, 10 pm). 5.1 A spatio-temporal dataset on : 7 spatial locations observed at 3 moments in time (adapted from Luo 1998). 5.2 (a) Three...
