H. Wackernagel Multivariate Geostatistics

Springer-Verlag Berlin Heidelberg GmbH

Hans Wackemagel

Multivariate Geostatistics An Introduction with Applications

Third, completely revised edition

with 117 Figures and 7 Tables

Springer

DR.HANSVVACKERNAGEL Centre de Geostatistique Beole des Mines de Paris 35 rue Saint Honore 77305 Fontainebleau France

ISBN 978-3-642-0791 -5 ISBN 978-3-662-05294-5 (eBook) DOI 10.1007/978-3-662-05294-5

Library of Congress Cataloging-in-Publication Data

Wackernagel. Hans. Multivariate geostatistics : an introduction wilh app lications I Hans WackemageL -3rd, complet~ly rev.ed.

p.cm. lncludes bibliographical re(~rences

1. Geology - Staistical .medtods. 2. Multivariate analysis. I. Title.

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1

L'analyse des donnees est "un outil pour degager de la gangue des don­nees le pur diamant de la veridique nature".

IP BENZECRI (according to [333])

Multivariate analysis is "a tool to extract from the gangue of the data the pure diamond of truthful nature".

Preface to the 3rd edition

Geostatistics has become an important methodology in environmental, c1imatological and ecological studies. So a new chapter demonstrating in a simple manner the ap­plication of cokriging to conductivity, salinity and chlorophyll measurements as well as numerical model output has been incorporated. Otherwise some recent material on collocated cokriging has been added, leading to aseparate chapter on this topic. Time was too short to inc1ude further results at the interface between geostatistics and data assimilation, which is a promising area of future developments (see reference [27]).

The main addition, however, is a detailed treatment of geostatistics for selection problems, which features five new chapters on non linear methods.

Fontainebleau, September 2002 HW

Preface to the 2nd edition

"Are you a statistician?" I have been asked. "Sort of ... " was my answer. A geostatis­tician is probably as much a statistician as a geophysicist is a physicist. The statisti­cian grows up, intellectually speaking, in the culture of the iid (independent identically distributed random variables) model, while the geostatistician is confronted with spa­tially/temporally correlated data right from the start. This changes radically the basic attitude when approaching a data set.

The present new edition has benefited from a dozen reviews in statistical, applied mathematics, earth and life science journals. The following principal changes have been made. The statistical introductory chapter has been split into three separate chap­ters for improved darity. The ordinary kriging and cokriging chapters have been re­shaped. The part on non-stationary geostatistics was entirely rewritten and rearranged after fruitful discussions with Dietrich Stoyan. I have also received interesting com­ments from Vera Pawlowsky, Tim Haas, Gerard Biau and Laurent Bertino. Last but not least I wish to thank Wolfgang Engel from Springer-Verlag for his editorial advice.

Fontainebleau, May 1998 HW

Preface to the first edition

Introducing geostatistics from a multivariate perspective is the main aim of this book. The idea took root while teaching geostatistics at the Centre de Geostatistique (Ecole des Mines de Paris) over the past ten years in the two postgraduate programs DEA and CFSG. A first script of lecture notes in French originated from this activity.

A specialized course on Multivariate and Exploratory Geostatistics held in September 1993 in Paris (organized in collaboration with the Department of Statistics of Trinity College Dublin) was the occasion to test some of the material on a pluridis­ciplinary audience. Another important opportunity arose last year when giving a lec­ture on Spatial Statistics during the summer term at the Department of Statistics of

the University of Washington at Seattle, where part of this manuscript was distributed in an early version. Short accounts were also given during COMETI and TEMPUS courses on geostatistics for environmental studies in Fontainebleau, Freiberg, Rome and Prague, which were sponsored by the European Community.

I wish to thank the participants of these various courses for their stimulating ques­tions and comments. Among the organizers of these courses, I particularly want to acknowledge the support received from Georges Matheron, Pierre Chauvet, Margaret Armstrong, John Haslett and Paul Sampson. Michel Grzebyk has made valuable com­ments on Chapters 29 and 30, which partly summarize some of his contributions to the field.

Fontainebleau, May 1995 HW

Contents

1 Introduction 1

A From Statistics to Geostatistics 7

2 Mean, Variance, Covariance 9 The mean: center of mass . 9 Distribution function 11 Expectation 12 Variance .. 13 Covariance 14

3 Linear Regression and Simple Kriging 15 Experimental covariance 15 Linear regression ..... 16 Variance-covariance matrix 20 Multiple linear regression . 21 Simple kriging. . . 24

4 Kriging the Mean 27 Arithmetic mean and its estimation variance 27 Estimating the mean with spatial correlation 28 No systematic bias ....... 29 Variance of the estimation error . 30 Minimal estimation variance 30 Kriging equations . . . 31 Case of no correlation . . . . 32

B Geostatistics 35

5 Regionalized Variable and Random Function 39 Multivariate time/space data ...... 39 Regionalized variable . . . . . . . . . . 40 Random variable and regionalized value 41

x

Random funetion . . . . Probability distributions. Striet stationarity .. . . Stationarity of first two moments

6 Variogram Cloud Dissimilarity versus separation . . . . . . . . . . . . . Experimental variogram. . . . . . . . . . . . . . . . . Replacing the experimental by a theoretical variogram .

7 Variogram and Covariance Function Regional variogram . . Theoretieal variogram. . Covariance funetion . . . Positive definite funetion Conditionally negative definite funetion Fitting the variogram with a covariance funetion .

8 Examples of Covariance Functions Nugget-effeet model ..... . Exponential covarianee funetion . . . Spherieal model . . . . . . . . . . . . Derivation of the spherical eovariance

9 Anisotropy Geometrie Anisotropy. . . . . . . Rotating and dilating an ellipsoid . Exploring 3D spaee for anisotropy Zonal anisotropy ....... . Nonlinear deformations of space .

10 Extension and Dispersion Variance Support ..... . Extension variance Dispersion variance Krige's relation .. Change of support effeet Change of support: affine model Application: acoustic data ... Comparison of sampling designs

11 Ordinary Kriging Ordinary kriging problem . Simple kriging of inerements Bloek kriging . . . . . . . .

41 42 43 44

45 45 47 48

50 50 50 52 53 53 55

57 57 57 58 58

62 62 62 64 65 65

66 66 67 68 69 70 71 73 76

79 79 81 82

Simple kriging with an estimated mean . Kriging the residual . . . . . . . . . . . Cross validation . . . . . . . . . . . . . Kriging with known measurement error variance .

12 Kriging Weights Geometry ......... . Geometrie anisotropy . . . . Relative position of sampies Screen effect. . . . . . . . . Factorizable covariance functions . Negative kriging weights . . . . .

13 Mapping with Kriging Kriging for spatial interpolation. Neighborhood . . . . . . . . . .

14 Linear Model of Regionalization Spatial anomalies . . . . . . . . . . Nested variogram model . . . . . . Decomposition of the random function . Second-order stationary regionalization Intrinsic regionalization . . . . . . . . . Intrinsic regionalization with mostly stationary components . Locally stationary regionalization .............. .

15 Kriging Spatial Components Kriging of the intrinsic component . . . . . . . Kriging of a second-order stationary component Filtering .................... . Application: kriging spatial components of arsenic data

16 The Smoothness of Kriging Kriging with irregularly spaced data Sensitivity to choiee of variogram model . Application: kriging topographie data ..

C Multivariate Analysis

17 Principal Component Analysis Uses ofPCA ........... . Transformation into factors . . . . Maximization of the variance of a factor Interpretation of the factor variances . . Correlation of the variables with the factors

xi

84 85 87 88

89 89 91 91 92 93 94

96 96 97

101 101 102 103 104 105 105 106

107 107 108 110 111

113 113 115 117

121

123 123 123 125 126 127

Xll

18 Canonical Analysis Factors in two groups of variables Intermezzo: singular value decomposition Maximization of the correlation. . . . . .

19 Correspondence Analysis Disjunctive table ........... . Contingency table . . . . . . . . . . . . Canonical analysis of disjunctive tables Coding of a quantitative variable . . . . Contingencies between two quantitative variables Continuous correspondence analysis . . . . . . .

D Multivariate Geostatistics

20 Direct and Cross Covariances Cross covariance function . Delay effect . . . . . . Cross variogram . . . . . . Pseudo cross variogram . . Difficult characterization of the cross covariance function

21 Covariance Function Matrices Covariance function matrix Cramer's theorem Spectral densities Phase shift . . . .

22 Intrinsic Multivariate Correlation Intrinsic correlation model Linear model .. . . . . Codispersion coefficients

23 Heterotopic Cokriging Isotopy and heterotopy Ordinary cokriging Simple cokriging . . .

24 Collocated Cokriging Cokriging neighborhood Collocated simple cokriging . Collocated ordinary cokriging Simplification with a particular covariance model

137 137 138 138

140 140 140 141 141 141 142

143

145 145 145 146 149 150

151 151 151 152 153

154 154 155 156

158 158 159 161

165 165 166 167 168

25 Isotopic Cokriging Cokriging with isotopic data Autokrigeability . . . . . . . Bivariate ordinary cokriging

26 Multivariate Nested Variogram Linear model of coregionalization ..... Bivariate fit of the experimental variograms Multivariate fit. . . . . . . . . . . . . . . . The need for an analysis of the coregionalization

27 Case Study: Ebro Estuary Kriging conductivity ........ . Cokriging of chlorophyll ...... . Conditional simulation of chlorophyll

28 Coregionalization Analysis Regionalized principal component analysis. Generalizing the analysis ......... . Regionalized canonical and redundancy analysis . Cokriging regionalized factors . . Regionalized multivariate analysis

29 Kriging a Complex Variable Coding directional data as a complex variable Complex covariance function . . . . . . . Complex kriging .............. . Cokriging of the real and imaginary parts .. Complex kriging and cokriging versus aseparate kriging Complex covariance function modeling ........ .

30 Bilinear Coregionalization Model Complex linear model of coregionalization . Bilinear model of coregionalization. . . . .

E Selective Geostatistics

31 Thresholds and Selectivity Curves Threshold and proportion . . . . . . . . . . . . . . Tonnage, recovered quantity, investment and profit. Selectivity . . . . . . . . . . . . . . . . . . Recovered quantity as a function of tonnage Time series in environmental monitoring . .

xiii

170 170 171 173

175 175 177 178 181

183 183 185 187

194 194 195 196 196 197

200 200 200 201 202 203 205

207 207 208

211

213 213 213 216 217 219

xiv

32 Lognormal Estimation Information effect and quality of estimators

Logarithmic Gaussian model . . . .

Moments of the lognormal variable .

Lognormal simple kriging .

Proportional effect .....

Permanence of lognormality

Stable variogram model . . .

Lognormal point and block ordinary kriging

33 Gaussian Anamorphosis with Hermite Polynomials Gaussian anamorphosis . . . . . . . . . . . . .

Hermite polynomials ............. .

Expanding a function into Hermite polynomials Probabilistic interpretation . . . . . . . . . . .

Moments of a function of a Gaussian variable .

Conditional expectation of a function of a Gaussian variable

Empirical Gaussian anamorphosis . . .

Smoothing the empirical anamorphosis .

Bijectivity of Gaussian anamorphosis

34 Isofactorial Models Isofactorial bivariate distribution

Isofactorial decomposition . . .

Isofactorial models . . . . . . .

Choice of marginal and of isofactorial bivariate distribution .

Hermitian and Laguerre isofactorial distributions ....

Intermediate types between diffusion and mosaic models

35 Isofactorial Change of Support The point-block-panel problem . .

Cartier's relation and point-block correlation .

Discrete Gaussian point-block model . . . . .

General structure of isofactorial change-of-support

36 Kriging with Discrete Point-Bloc Models Non-linear function of a block variable . . . . . . . . . .

Conditional expectation and disjunctive kriging of a bloc

Disjunctive kriging of a panel.

Uniform conditioning . . . . . . . . . . . . . . . . . . .

221 221 223 224 226 228 228 233 233

238 238 239 240 240 241 243 244 245 247

250

250 251 252 253 254 258

262 262 262 266 267

273 273 274 275 277

F Non-Stationary Geostatistics

37 External Drift Depth measured with drillholes and seismie Estimating with a shape funetion . . Estimating external drift eoefficients . . Cross validation with external drift . . . Regularity of the external drift funetion Cokriging with multiple external drift . Ebro estuary: numerieal model output as external drift. Comparing results of eonditional simulations and kriging

38 Universal Kriging Universal kriging system Estimation of the drift . . Underlying variogram and estimated residuals From universal to intrinsie kriging . . . . . .

39 Translation Invariant Drift Exponential-polynomial basis funetions Intrinsie random funetions of order k . Generalized eovariance funetion Intrinsie kriging . . . . . . . . . . . . Trigonometrie temporal drift . . . . . Filtering trigonometrie temporal drift . Dual kriging . Splines ................ .

APPENDIX

Matrix Algebra

Linear Regression Theory

Covariance and Variogram Models

Additional Exercices

Solutions to Exercises

References

Bibliography

Index

xv

281

283 283 284 285 290 294 296 297 297

300 300 302 303 306

308 308 309 310 311 312 312 313 314

317

319

329

334

337

339

353

358

381