Multivariate Geostatistics: An Introduction with Applications

388 Pages·2003·9.37 MB·English

by  Dr. Hans Wackernagel (auth.)

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 1-5 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 applications 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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Originally published by Kluwer Aeademic Publisher.; in 2003
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L'analyse des donnees est "un outil pour degager de la gangue des don
nees 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 ap
plication 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 geostatis
tician is probably as much a statistician as a geophysicist is a physicist. The statisti
cian grows up, intellectually speaking, in the culture of the iid (independent identically
distributed random variables) model, while the geostatistician is confronted with spa
tially/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 chap
ters for improved darity. The ordinary kriging and cokriging chapters have been re
shaped. The part on non-stationary geostatistics was entirely rewritten and rearranged
after fruitful discussions with Dietrich Stoyan. I have also received interesting com
ments 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 pluridis
ciplinary audience. Another important opportunity arose last year when giving a lec
ture 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 ques
tions 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 com
ments 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 . . . . 41
Probability distributions. 42
Striet stationarity .. . . 43
Stationarity of first two moments 44
6 Variogram Cloud 45
Dissimilarity versus separation . . . . . . . . . . . . . 45
Experimental variogram. . . . . . . . . . . . . . . . . 47
Replacing the experimental by a theoretical variogram . 48
7 Variogram and Covariance Function 50
Regional variogram . . 50
Theoretieal variogram. . 50
Covariance funetion . . . 52
Positive definite funetion 53
Conditionally negative definite funetion 53
Fitting the variogram with a covariance funetion . 55
8 Examples of Covariance Functions 57
Nugget-effeet model ..... . 57
Exponential covarianee funetion . . . 57
Spherieal model . . . . . . . . . . . . 58
Derivation of the spherical eovariance 58
9 Anisotropy 62
Geometrie Anisotropy. . . . . . . 62
Rotating and dilating an ellipsoid . 62
Exploring 3D spaee for anisotropy 64
Zonal anisotropy ....... . 65
Nonlinear deformations of space . 65
10 Extension and Dispersion Variance 66
Support ..... . 66
Extension variance 67
Dispersion variance 68
Krige's relation .. 69
Change of support effeet 70
Change of support: affine model 71
Application: acoustic data ... 73
Comparison of sampling designs 76
11 Ordinary Kriging 79
Ordinary kriging problem . 79
Simple kriging of inerements 81
Bloek kriging . . . . . . . . 82