2012 (wiley series in probability and statistics) jean paul chiles, pierre delfiner(auth ), walter a shewhart, samuel s wilks(eds ) geostatistics modeling spatial uncertainty, second edition (2012)

Geostatistics

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Established by WALTER A. SHEWHART and SAMUEL S. WILKS
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Geostatistics
Modeling Spatial Uncertainty
Second Edition

JEAN-PAUL CHILE`S
BRGM and MINES ParisTech

PIERRE DELFINER
PetroDecisions

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Copyright r 2012 by John Wiley & Sons, Inc. All rights reserved.
Published by John Wiley & Sons, Inc., Hoboken, New Jersey
Published simultaneously in Canada
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ISBN: 978-0-470-18315-1
Library of Congress Cataloging-in-Publication Data is available.
Printed in the United States of America
10 9 8 7 6

5 4 3

2 1

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Contents

Preface to the Second Edition

ix

Preface to the First Edition

xiii

Abbreviations

xv

Introduction

1

Types of Problems Considered, 2
Description or Interpretation?, 8
1.

Preliminaries
1.1
1.2
1.3

2.

11

Random Functions, 11
On the Objectivity of Probabilistic Statements, 22
Transitive Theory, 24

Structural Analysis
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9

28

General Principles, 28
Variogram Cloud and Sample Variogram, 33
Mathematical Properties of the Variogram, 59
Regularization and Nugget Effect, 78
Variogram Models, 84
Fitting a Variogram Model, 109
Variography in the Presence of a Drift, 122
Simple Applications of the Variogram, 130
Complements: Theory of Variogram Estimation and
Fluctuation, 138

v

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vi
3.

CONTENTS

Kriging
3.1
3.2
3.3
3.4

3.5
3.6
3.7
3.8
3.9

4.

238

Introduction, 238
A Second Look at the Model of Universal Kriging, 240
Allowable Linear Combinations of Order k, 245
Intrinsic Random Functions of Order k, 252
Generalized Covariance Functions, 257
Estimation in the IRF Model, 269
Generalized Variogram, 281
Automatic Structure Identification, 286
Stochastic Differential Equations, 294

Multivariate Methods
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8

6.

Introduction, 147
Notations and Assumptions, 149
Kriging with a Known Mean, 150
Kriging with an Unknown Mean, 161
Estimation of a Spatial Average, 196
Selection of a Kriging Neighborhood, 204
Measurement Errors and Outliers, 216
Case Study: The Channel Tunnel, 225
Kriging Under Inequality Constraints, 232

Intrinsic Model of Order k
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9

5.

147

299

Introduction, 299

Notations and Assumptions, 300
Simple Cokriging, 302
Universal Cokriging, 305
Derivative Information, 320
Multivariate Random Functions, 330
Shortcuts, 360
SpaceÀTime Models, 370

Nonlinear Methods
6.1
6.2
6.3
6.4
6.5

386

Introduction, 386
Global Point Distribution, 387
Local Point Distribution: Simple Methods, 392
Local Estimation by Disjunctive Kriging, 401
Selectivity and Support Effect, 433

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vii

CONTENTS

6.6
6.7
6.8
6.9
6.10
6.11
7.

Multi-Gaussian Change-of-Support Model, 445
Affine Correction, 448
Discrete Gaussian Model, 449
Non-Gaussian Isofactorial Change-of-Support Models, 466
Applications and Discussion, 469
´
Change of Support by the Maximum (C. Lantuejoul),
470

Conditional Simulations
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
7.9
7.10

478

Introduction and Definitions, 478
Direct Conditional Simulation of a Continuous Variable, 489
Conditioning by Kriging, 495
Turning Bands, 502
Nonconditional Simulation of a Continuous Variable, 508
Simulation of a Categorical Variable, 546
Object-Based Simulations: Boolean Models, 574
Beyond Standard Conditioning, 590
Additional Topics, 606
Case Studies, 615

Appendix

629

References

642

Index

689

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In memory of Georges MATHERON
(1930À2000)

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Preface to the Second Edition

Twelve years after publication of the first edition in 1999, ideas have matured
and new perspectives have emerged. It has become possible to sort out material
that has lost relevance from core methods which are here to stay. Many new
developments have been made to the field, a number of pending problems have
been solved, and bridges with other approaches have been established. At the
same time there has been an explosion in the applications of geostatistical
methods, including in new territories unrelated to geosciences—who would
have thought that one day engineers would krige aircraft wings? All these
factors called for a thoroughly revised and updated second edition.
Our intent was to integrate the new material without increasing the size of the
book. To this end we removed Chapter 8 (Scale effects and inverse problems)
which covered stochastic hydrogeology but was too detailed for the casual
reader and too incomplete for the specialist. We decided to keep only the specific
contributions of geostatistics to hydrogeology and to distribute the material
throughout the relevant chapters. The following is an overview of the main
changes from the first edition and their justification.
Chapter 2 (Structural analysis) gives complements on practical questions
such as spatial declustering and declustered statistics, variogram map calculation for data on a regular grid, variogram in a non-Euclidean coordinate system
(transformation to a geochronological coordinate system). The Cauchy model

is extended to the Cauchy class whose two shape parameters can account for a
´ model
variety of behaviors at short as well as at large distances. The Matern
and the logarithmic (de Wijsian) model are related to Gaussian Markov random fields (GMRF). New references are given on variogram fitting and sampling design. New sections propose covariance models on the sphere or on a
river network. The chapter also includes new points on random function theory, such as a reference to the recent proof of a conjecture of Matheron on the
characterization of an indicator function by its covariogram. The introductory
example of variography in presence of a drift was removed to gain space.
The external drift model which was presented with multivariate methods is
now introduced in Chapter 3 (Kriging) as a variant of the universal kriging
ix

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x

PREFACE TO THE SECOND EDITION

model with polynomial drift. The special case of a constant unknown mean
(ordinary kriging) is treated explicitly and in detail as it is the most common in
applications. Dual kriging receives more attention because of its kinship with
radial basis function interpolation (RBF), and its wide use in the design and
analysis of computer experiments (DACE) to solve engineering problems.
Three solutions are proposed to address the longstanding problem of the
spurious discontinuities created by the use of moving neighborhoods in the case
of a large dataset, namely covariance tapering, Gaussian Markov random field
approximation, and continuous moving neighborhoods. Another important
kriging issue, how to deal with outliers, is discussed and a new, relatively

simple, truncation model developed for gold and uranium mines is presented.
Finally a new form of kriging, Poisson kriging, in which observations derive
from a Poisson time process, is introduced.
Few changes were made to Chapter 4 (Intrinsic model of order k). The
main one is the addition of Micchelli’s theorem providing a simple characterization of isotropic generalized covariances of order k. Another addition is
an analysis of the structure of the inverse of the intrinsic kriging matrix.
The Poisson differential equation ΔZ 5 Y previously in the deleted chapter 8
survives in this chapter.
Chapter 5 (Multivariate methods) was largely rewritten and augmented.
The main changes concern collocated cokriging and space–time models. The
chapter now includes a thorough review of different forms of collocated cokriging, with a clear picture of which underlying models support the approach
without loss of information and which use it just as a convenient simplification.
Collocated cokriging is also systematically compared with its common alternative, kriging with an external drift. As for space–time models, they were a real
threat for the size of the book because of the surge of activity in the subject. To
deal with situations where a physical model is available to describe the time
evolution of the system, we chose to present sequential data assimilation
and ensemble Kalman filtering (EnKF) in some detail, highlighting their links
with geostatistics. For the alternative case where no dynamic model is available,
the focus is on new classes of nonseparable space–time covariances that
enable kriging in a space–time domain. The chapter contains numerous other
additions such as potential field interpolation of orientation data, extraction of
the common part of two surveys using automatic factorial cokriging, maximum
´ cross-covariance model, layer-cake
autocorrelation factors, multivariate Matern
estimation including seismic information, compositional data with geometry
on the simplex.
Nonlinear methods and conditional simulations generally require a preliminary transformation of the variable of interest into a variable with a specified
marginal distribution, usually a normal one. As this step is critical for the
quality of the results, it has been expanded and updated and now forms a
specific section of chapter 6 (Nonlinear methods). More elaborate methods than

the simple normal score transform are proposed. The presentation of the change
of support has been restructured. We now present each model at the global scale

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xi

PREFACE TO THE SECOND EDITION

and then immediately continue with the local scale. Conditional expectation is
more detailed and accounts for a locally variable mean. The most widely used
change-of-support model, the discrete Gaussian model, is discussed in depth,
including the variant that appeared in the 2000s. Practical implementation
questions are examined: locally variable mean, selection on the basis of
future information (indirect resources), uniform conditioning. Finally this
chapter features a section on the change of support by the maximum, a topic
whose development in a spatial context is still in infancy but is important for
extreme-value studies.
Chapter 7 incorporates the numerous advances made in conditional simulations in the last decade. The simulation of the fractional Brownian motion
and its extension to IRF–k’s, which was possible only in specific cases (regular
1D grid, or at the cost of an approximation) is now possible exactly. A new
insight into the Gibbs sampler enables the definition of a Gibbs propagation
algorithm that does not require inversion of the covariance matrix. PluriGaussian simulations are explained in detail and their use is illustrated in the
Brent cliff case study, which has been completely reworked to reflect current
practice (separable covariance models are no longer required). New simulation
methods are presented: stochastic process-based simulation, multi-point simulation, gradual deformation. The use of simulated annealing for building
conditional simulations has been completely revised. Stochastic seismic inversion and Bayesian approaches are up-to-date. Upscaling is also discussed in

the chapter.

ACKNOWLEDGMENTS
´
Special acknowledgement is due to Christian Lantuejoul
for his meticulous
reading of Chapters 6 and 7, numerous helpful comments and suggestions, and
for writing the section on change of support by the maximum. We are also
greatly indebted to Jacques Rivoirard for many contributions and insights.
´
Thierry Coleou
helped us with seismic applications and Henning Omre with
Bayesian methods. Xavier Freulon provided the top-cut gold grades example
´ ` ne Beucher the revised simulation of the Brent cliff. Didier Renard
and Hele
carried out calculations for new figures and Philippe Le Cae¨r redrew the cover
figure. This second edition also benefits from fine remarks of some readers of
the first edition, notably Tilmann Gneiting, and from many informal discussions with our colleagues of the Geostatistics group of MINES ParisTech.
We remain, of course, grateful to the individuals acknowledged in the
Preface to the first edition, and especially to Georges Matheron, who left us in
2000, but continues to be a source of inspiration.
JEAN-PAUL CHILE`S
PIERRE DELFINER

Fontainebleau
October 23, 2011

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Preface to the First Edition

This book covers a relatively specialized subject matter, geostatistics, as it was
defined by Georges Matheron in 1962, when he coined this term to designate
his own methodology of ore reserve evaluation. Yet it addresses a larger audience, because the applications of geostatistics now extend to many fields in the
earth sciences, including not only the subsurface but also the land, the atmosphere, and the oceans.
The reader may wonder why such a narrow subject should occupy so many
pages. Our intent was to write a short book. But this would have required us to
sacrifice either the theory or the applications. We felt that neither of these
options was satisfactory—there is no need for yet another introductory book,
and geostatistics is definitely an applied subject. We have attempted to reconcile
theory and practice by including application examples, which are discussed
with due care, and about 160 figures. This results in a somewhat weighty
volume, although hopefully more readable.
This book gathers in a single place a number of results that were either
scattered, not easily accessible, or unpublished. Our ambition is to provide the
reader with a unified view of geostatistics, with an emphasis on methodology.
To this end we detail simple proofs when their understanding is deemed
essential for geostatisticians, and we omit complex proofs that are too technical. Although some theoretical arguments may fall beyond the mathematical
and statistical background of practitioners, they have been included for the
sake of a complete and consistent development that the more theoretically
inclined reader will appreciate. These sections, as well as ancillary or advanced
topics, are set in smaller type.
Many references in this book point to the works of Matheron and the Center
for Geostatistics in Fontainebleau, which he founded at the Paris School of
Mines in 1967 and headed until his retirement in 1996. Without overlooking the
´
contribution of Gandin, Matern,

Yaglom, Krige, de Wijs, and many others, it
is from Matheron that geostatistics emerged as a discipline in its own right—a
body of concepts and methods, a theory, and a practice—for the study of
spatial phenomena. Of course this initial group spawned others, notably in
xiii

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xiv

PREFACE TO THE FIRST EDITION

Europe and North America, under the impetus of Michel David and Andre´
Journel, followed by numerous researchers trained in Fontainebleau first, and
then elsewhere. This books pays tribute to all those who participated in the
development of geostatistics, and our large list of references attempts to give
credit to the various contributions in a complete and fair manner.
This book is the outcome of a long maturing process nourished by experience. We hope that it will communicate to the reader our enthusiasm for this
discipline at the intersection between probability theory, physics, and earth
sciences.

ACKNOWLEDGMENTS
This book owes more than we can say to Georges Matheron. Much of the
theory presented here is his work, and we had the privilege of seeing it in the
making during the years that we spent at the Center for Geostatistics. In later
years he always generously opened his door to us when we asked for advice on
fine points. It was a great comfort to have access to him for insight and support.

We are also indebted to the late Geoffrey S. Watson, who showed an early
interest in geostatistics and introduced it to the statistical community. He was
kind enough to invite one of the authors to Princeton University and, as an
advisory editor of the Wiley Interscience Series, made this book possible. We
wish he had been with us to see the finished product.
The manuscript of this book greatly benefited from the meticulous reading
´
and quest for perfection of Christian Lantuejoul,
who suggested many valuable
improvements. We also owe much to discussions with Paul Switzer, whose
views are always enlightening and helped us relate our presentation to mainstream statistics. We have borrowed some original ideas from Jean-Pierre
Delhomme, who shared the beginnings of this adventure with us. Bernard
Bourgine contributed to the illustrations. This book could not have been
´
completed without the research funds of Bureau de Recherches Geologiques
et
Minie`res, whose support is gratefully acknowledged.
We would like to express our thanks to John Wiley & Sons for their
encouragement and exceptional patience during a project which has spanned
many years, and especially to Bea Shube, the Wiley-Interscience Editor when
we started, and her successors Kate Roach and Steve Quigley.
Finally, we owe our families, and especially our dear wives Chantal and
Edith, apologies for all the time we stole from them, and we thank them for
their understanding and forebearance.
JEAN-PAUL CHILE`S
PIERRE DELFINER

La Villetertre
July 12, 1998

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Abbreviations

ALCÀk
c.d.f.
CK
DFT
DGM1, DGM2
DK
GCÀk
GLS
GV
i.i.d.
IRF
IRFÀk
KED
MM1, MM2
m.s.
m.s.e.
OK
PCA
p.d.f.
RF
SK
SRF
UK

allowable linear combination of order k
cumulative density function
cokriging
discrete Fourier transform
discrete Gaussian model 1, 2
disjunctive kriging
generalized covariance of order k
generalized least squares
generalized variogram
independent and identically distributed
intrinsic random function
intrinsic random function of order k
kriging with external drift
Markov model 1, 2
mean square
mean square error
ordinary kriging
principal component analysis
probability density function
random function
simple kriging
stationary random function
universal kriging

xv

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3.0
2.5
2.0
1.5
1.0
0.5
0.0
Ϫ0.5
Ϫ1.0
Ϫ1.5
Ϫ2.0
Ϫ2.5
Ϫ3.0
Ϫ3.5
Ϫ4.0
20 18

16 14

12 10

8 6
4 2

16
12 14
8 10
6

4
0 2

0

18 20

(a)

3.0
2.5
2.0
1.5
1.0
0.5
0.0
Ϫ0.5
Ϫ1.0
Ϫ1.5
Ϫ2.0
Ϫ2.5
Ϫ3.0
Ϫ3.5
Ϫ4.0
20 18

16 14

12 10

8 6
4 2

0

16
12 14
8 10
6
4
0 2

18 20

(b)
FIGURE 3.18 Continuous kriging neighborhood: (a) ordinary kriging surface obtained using a
standard moving neighborhood with radius R ¼ 10; (b) ordinary kriging surface obtained using
a continuous moving neighborhood with radii r ¼ 7.5 and R ¼ 12.5. [From Rivoirard and Romary
(2011), with kind permission of the International Association for Mathematical Geosciences.]

Geostatistics: Modeling Spatial Uncertainty, Second Edition. J.P. Chile`s and P. Delfiner.
r 2012 John Wiley & Sons, Inc. Published 2012 by John Wiley & Sons, Inc.

binsert

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(a)
1.00

0.80
0.60
0.40
0.20
0.00
(b)
5.0
4.0
3.0
2.0
1.0
0.0
(c)
10.0
5.0
2.0
1.0
0.5
0.0
(d)
10.0
5.0
2.0
1.0
0.5
0.0
(e)

FIGURE 3.21 Vertical cross section through the block model along a gold vein. (a) Top view of
the deposit showing the trace of the cross-section and the position of the blast holes (drilling mesh

of 2.5 m perpendicular to the section and 5 m along the section); (b) Map of the indicator associated
with Z(x) $ 5 g/t, estimated by ordinary cokriging from truncated and indicator data; (c) Map of
truncated grade below a 5 g/t cutoff, estimated by ordinary cokriging from truncated and indicator
data; (d) Map of the final cokriging estimates obtained by recombining the indicator and the truncated grades by (3.71); (e) Map of direct ordinary kriging estimates of grades. Notice that the scale
of (c) ranges from 0 to 5 whereas that of (d) and (e) ranges from 0 to 10. [From Rivoirard et al.
(2012), with kind permission of the International Association for Mathematical Geosciences.]

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0.3
γY//

Variogramme

γX//

γY⊥

0.2
γX⊥
0.1
γZ⊥
0.0
0

5000

10000

15000
Distance

20000

25000

FIGURE 5.5 Longitudinal and transverse variograms of potential field derivatives for the
Limousin dataset, Massif Central, France. Notations: γX// is the longitudinal variogram of @T/@x
and γX> the transverse. [From Aug (2004).]

Horizontal view
B

A

B

A

Vertical cross section
FIGURE 5.6 Potential field interpolation. Top: points at interfaces and structural data, sampled
on the topographic surface; bottom: vertical cross section through the 3D model. [From Courrioux
et al. (1998).]

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FIGURE 5.14 The common part S (center) of the two maps Z1 and Z2 (left) is extracted
by cokriging. Residuals (right) display noise and stripes due to acquisition footprints. [From
´
Coleou
(2002).]

10 m
Total
thickness
base map

1 km

(a)

(b)

(c)

FIGURE 5.16 Kriging of layer thicknesses constrained by a seismic total thickness map.
(a) Unconstrained ordinary kriging: total thickness is not reproduced; (b) constrained kriging
with uncertainty on total thickness: total thickness is partly reproduced; (c) constrained kriging with
no uncertainty on total thickness: total thickness is reproduced exactly. [From Haas et al. (1998).]

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(a)

(c)

(b)

FIGURE 7.28 Simulation of a substitution RF by Markov coding of an RF with discontinuities
equal to 61: (a) stationary isotropic RF; (b) stationary anisotropic RF; (c) nonstationary RF.
´
[From C. Lantuejoul,
personal communication.]

FIGURE 7.32 Realizations of mosaic random functions derived from dead-leaves models:
(a) single-dead-leaves model (black poplar) with independent assignment of a value to each leaf;
(b) multi-dead-leaves model with value assignment depending on leaf species (alder, elm, oak,
poplar).

binsert

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binsert

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0

0

100

100

200

200

300

300

400

400

600

500

600

conditional simulation

500

kriging

G2

G1

L3

L2

L1

C

´
FIGURE 7.45 Example of a vertical section through the Tiebaghi
nickel orebody. Top: kriged section; bottom: simulated section. [From Chile`s (1984), with
kind permission from Kluwer Academic Publishers.]

450

500

550

450

500

550


(a)

20

(b)

10

Lithotypes

0

Sandstone
Shaly sandstone
Ϫ10

Sandy shale
Shale
0.0

0.3

0.5

1.0

(c)
γ

0.2

0.1

h
0

5

10

15

20

FIGURE 7.47 Identification of the parameters of the pluri-Gaussian simulation model:
(a) vertical proportion curves; (b) facies substitution diagram; (c) sample variograms of facies
indicators and fits derived from the variograms of the two Gaussian SRFs. [Output from Isatiss.
From H. Beucher, personal communication.]

binsert

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N

W Up
S

Do E
wn

N

W Up
S

Do E
wn

FIGURE 7.48 3D conditional facies simulation of the Brent formation: 3D view and fence
diagram. Size of simulated domain: 1 km 3 1 km 3 30 m. [Output from Isatiss. From H. Beucher,
personal communication.]

binsert

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Introduction

Geostatistics aims at providing quantitative descriptions of natural variables
distributed in space or in time and space. Examples of such variables are
 Ore grades in a mineral deposit
 Depth and thickness of a geological layer
 Porosity and permeability in a porous medium
 Density of trees of a certain species in a forest
 Soil properties in a region
 Rainfall over a catchment area
 Pressure, temperature, and wind velocity in the atmosphere
 Concentrations of pollutants in a contaminated site

These variables exhibit an immense complexity of detail that precludes a
description by simplistic models such as constant values within polygons, or
even by standard well-behaved mathematical functions. Furthermore, for economic reasons, these variables are often sampled very sparsely. In the petroleum
industry, for example, the volume of rock sampled typically represents a minute
fraction of the total volume of a hydrocarbon reservoir. The following figures,
from the Brent field in the North Sea, illustrate the orders of magnitude of the
volume fractions investigated by each type of data (“cuttings” are drilling debris,
and “logging” data are geophysical measurements in a wellbore):
Cores
0.000 000 001
Cuttings 0.000 000 007
Logging 0.000 001
By comparison, if we used the same proportions for an opinion poll of the
100 million US households (to take a round number), we would interview only
Geostatistics: Modeling Spatial Uncertainty, Second Edition. J.P. Chile`s and P. Delfiner.
r 2012 John Wiley & Sons, Inc. Published 2012 by John Wiley & Sons, Inc.

1

cintro

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2

INTRODUCTION

between 0.1 and 100 households, while 1500 is standard. Yet the economic
implications of sampling for natural resources development projects can be

significant. The cost of a deep offshore development is of the order of 10 billion
dollars. Similarly, in the mining industry “the decision to invest up to 1À2
billion dollars to bring a major new mineral deposit on line is ultimately based on
a very judicious assessment of a set of assays from a hopefully very carefully
chosen and prepared group of samples which can weigh in aggregate less than
5 to 10 kilograms” (Parker, 1984).
Naturally, these examples are extreme. Such investment decisions are based
on studies involving many disciplines besides geostatistics, but they illustrate the
notion of spatial uncertainty and how it affects development decisions. The fact
that our descriptions of spatial phenomena are subject to uncertainty is now
generally accepted, but for a time it met with much resistance, especially from
engineers who are trained to work deterministically. In the oil industry there
are anecdotes of managers who did not want to see uncertainty attached to
resources estimates because it did not look good—it meant incompetence. For
job protection, it was better to systematically underestimate resources. (Ordered
by his boss to get rid of uncertainty, an engineer once gave an estimate of
proven oil resources equal to the volume of oil contained in the borehole!)
Such conservative attitude led to the abandonment of valuable prospects. In
oil exploration, profit comes with risk.
Geostatistics provides the practitioner with a methodology to quantify spatial
uncertainty. Statistics come into play because probability distributions are the
meaningful way to represent the range of possible values of a parameter of
interest. In addition, a statistical model is well-suited to the apparent randomness
of spatial variations. The prefix “geo” emphasizes the spatial aspect of the
problem. Spatial variables are not completely random but usually exhibit some
form of structure, in an average sense, reflecting the fact that points close in space
tend to assume close values. G. Matheron (1965) coined the term regionalized
variable to designate a numerical function z(x) depending on a continuous space
index x and combining high irregularity of detail with spatial correlation.
Geostatistics can then be defined as “the application of probabilistic methods

to regionalized variables.” This is different from the vague usage of the word in
the sense “statistics in the geosciences.” In this book, geostatistics refers to a
specific set of models and techniques, largely developed by G. Matheron, in the
´ in forestry, D. G.
lineage of the works of L. S. Gandin in meteorology, B. Matern
Krige and H. J. de Wijs in mining, and A. Y. Khinchin, A. N. Kolmogorov,
´
P. Levy,
N. Wiener, A. M. Yaglom, among others, in the theory of stochastic
processes and random fields. We will now give an overview of the various
geostatistical methods and the types of problems they address and conclude by
elaborating on the important difference between description and interpretation.

TYPES OF PROBLEMS CONSIDERED
The presentation follows the order of the chapters. For specificity, the problems
presented refer to the authors’ own background in earth sciences applications,

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3

INTRODUCTION

but newcomers with different backgrounds and interests will surely find
equivalent formulations of the problems in their own disciplines. Geostatistical
terms will be introduced and highlighted by italics.

Epistemology
The quantification of spatial uncertainty requires a model specifying the
mechanism by which spatial randomness is generated. The simplest approach is
to treat the regionalized variable as deterministic and the positions of the
samples as random, assuming for example that they are selected uniformly and
independently over a reference area, in which case standard statistical rules for
independent random variables apply, such as that for the variance of the mean.
If the samples are collected on a systematic grid, they are not independent and
things become more complicated, but a theory is possible by randomizing the
grid origin.
Geostatistics takes the bold step of associating randomness with the regionalized variable itself, by using a stochastic model in which the regionalized variable is regarded as one among many possible realizations of a random function.
Some practitioners dispute the validity of such probabilistic approach on the
grounds that the objects we deal with—a mineral deposit or a petroleum
reservoir—are uniquely deterministic. Probabilities and their experimental
foundation in the famous “law of large numbers” require the possibility of
repetitions, which are impossible with objects that exist unambiguously in
space and time. The objective meaning and relevance of a stochastic model
under such circumstances is a fundamental question of epistemology that
needs to be resolved. The clue is to carefully distinguish the model from the
reality it attempts to capture. Probabilities do not exist in Nature but only in
our models. We do not choose to use a stochastic model because we believe
Nature to be random (whatever that may mean), but simply because it is analytically useful. The probabilistic content of our models reflects our imperfect
knowledge of a deterministic reality. We should also keep in mind that models
have their limits and represent reality only up to a certain point. And finally, no
matter what we do and how carefully we work, there is always a possibility that
our predictions and our assessments of uncertainty turn out to be completely
wrong, because for no foreseeable reason the phenomenon at unknown places
is radically different than anything observed (what Matheron calls the risk of
a “radical error”).

Structural Analysis
Having observed that spatial variability is a source of spatial uncertainty, we
have to quantify and model spatial variability. What does an observation at
a point tell us about the values at neighboring points? Can we expect continuity
in a mathematical sense, or in a statistical sense, or no continuity at all? What
is the signal-to-noise ratio? Are variations similar in all directions or is there

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