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Geostatistics WlLEY SERIES IN PROBABILITY AND STATISTICS APPLIED PROBABILITY AND STATISTICS SECTION Established by WALTER A. SHEWHART and SAMUEL S. WILKS Editors: Vic Barnett, Noel A. C. Cressie, Nicholas I. Fisher, Iain M. Johnstone, J. B. Kadane, David G. Kendall, David W. Scott, Bernard W. Silver...

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Geostatistics

WlLEY SERIES IN PROBABILITY AND STATISTICS APPLIED PROBABILITY AND STATISTICS SECTION Established by WALTER A. SHEWHART and SAMUEL S. WILKS

Editors: Vic Barnett, Noel A. C. Cressie, Nicholas I. Fisher, Iain M. Johnstone, J. B. Kadane, David G. Kendall, David W. Scott, Bernard W. Silverman. Adrian F. M. Smith, Jozef L. Teugels; Ralph A. Bradiey, Emeritus, J. Stuart Hunter, Emeritus A complete list of the titles in this series appears at the end of this volume.

Geostatistics Modeling Spatial Uncertainty

JEAN-PAUL CHILES Bureau de Recherches Gkologiques et Miniires PTERRE DELFINER TOTAL Exploration Production

A Wiley-Interscience Publication JOHN WILEY & SONS, INC. New York Chichester Weinheim

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Singapore

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Toronto

This book is printed on acid-free paper. @ Copyright

(91999 by John Wiley & Sons, Inc. All rights reserved.

Published simultaneously in Canada. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4744. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 605 Third Avenue, New York, N Y 10158-0012, (212) 850-601 I , fax (212) 850-6008, E-Mail: PERMREQ8 WILEY.COM. Library of Congress Cataloging-in-Publication Data: Chiles, Jean-Paul. Geostatistics: modeling spatial uncertaintylJean-Paul Chiks, Pierre Delfiner. p. cm. - (Wiley series in probability and statistics. Applied probability and statistics section) “A Wiley-Interscience publication.” Includes bibliographical references and index. ISBN 0-471-08315-1 (alk. paper) I , Earth sciences-Statistical methods. I. Delfiner, Pierre. 11. Title. 111. Series: Wiley series in probability and statistics. Applied probability and statistics. QE33.2.S82C45 1999 98-3599 550’.72-&2 1 Printed in the United States of America 109876

Contents

Preface

ix

Abbreviations

xi

Introduction

1

Types of Problems Considered, 2 Description or Interpretation?, 7

1. Preliminaries

11

1.1. Random Functions, I 1 1.2. On the Objectivity of Probabilistic Statements, 22 1.3. Transitive Theory, 24

2. Structural Analysis

29

2.1. General Principles, 29 2.2. Variogram Cloud and Sample Variogram, 34 2.3. Mathematical Properties of the Variogram, 57 2.4. Regularization and Nugget Effect, 74 2.5. Variogram Models, 80 2.6. Fitting a Variogram Model, 104 2.7. Variography in Presence of a Drift, 115 2.8. Simple Applications of the Variogram, 128 2.9. Complements: Theory of Variogram Estimation and Fluctuation, 137

3. Kriging

150

3.1. Introduction, 150 3.2. Notations and Assumptions, 152 V

CONTENTS

vi

Kriging with a Known Mean, 154 Kriging with an Unknown Mean, 164 Estimation of a Spatial Average, 193 Selection of a Kriging Neighborhood, 201 Measurement Errors and Outliers, 210 Case Study: The Channel Tunnel, 215 3.9. Kriging under Inequality Constraints, 224

3.3. 3.4. 3.5. 3.6. 3.7. 3.8.

4. Intrinsic Model of Order k

231

4.1. IRF-0 and IRF-k, 231 4.2. A Second Look at the Model of Universal Kriging, 233 4.3. Allowable Linear Combinations of Order k, 236 4.4. Intrinsic Random Functions of Order k, 243 4.5. Generalized Covariance Functions, 252 4.6. Estimation in the IRF Model, 265 4.7. Generalized Variogram, 276 4.3. Automatic Structure Identification in the General Case, 281

5. Multivariate Methods 5.1. 5.2. 5.3. 5.4. 5.5. 5.6. 5.7. 5.8.

292

Introduction, 292 Notations and Assumptions, 293 Simple Cokriging, 296 Universal Cokriging, 298 Case of Gradient Information, 3 13 Multivariate Random Functions, 321 Shortcuts, 35 1 Space-Time Models, 362

6. Nonlinear Methods

6.1. Introduction, 375 6.2. Simple Methods for Estimating a Point Distribution, 376 6.3. Local Estimation of a Point Distribution by Disjunctive Kriging, 388 6.4. Simple Methods for Estimating a Block Distribution, 419 6.5. Local Estimation of a Block Distribution by Disjunctive Kriging, 437

375

vii

CONTENTS

7. Conditional Simulations

449

7.1. Introduction and Definitions, 449 7.2. Direct Conditional Simulation of a Continuous Variable, 462 7.3. Conditioning by Kriging, 465 7.4. Turning Bands, 472 7.5. Nonconditional Simulation of a Continuous Variable, 478 7.6. Nonconditional Simulation of an IRF-4, 506 7.7. Sirnulation of a Categorical Variable, 520 7.8. Object-Based Simulations: Boolean Models, 545 7.9. Constrained Simulations, 56 1 7.10. Practical Considerations, 57 1 7.11. Case Studies, 577

8. Scale Effects and Inverse Problems 8.1. 8.2. 8.3. 8.4.

593

Introduction, 593 Upscaling Permeability, 594 Stochastic Differential Equations, 602 Inverse Problem in Hydrogeology, 61 1

Appendix

636

References

650

Index

687

Preface

This book covers a relatively specialired 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, for 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 rnethodulogy. To this end we detail simple proofs when their understanding is deemed essential for geostatisticians, and 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. Makrn, Yaglom, Krige, de Wijs, and many others, it is from Matheron that geostatistics emerged as a discipline in its own righta 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 ix

X

PREFACE

Europe and North America, under the impetus of Michel David and Andk Journel, followed by numerous researchers trained in Fontainebleau first, and then elsewhere. This book 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 Lantkjoul, 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 the Bureau de Recherches Gologiques et Minikres, 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-lnterscience 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 thanks for their understanding and forbearance. La Vi1leter.tr.c. July 12, 1998

Abbreviations

c.d.f. i.i.d. IRF IRF-k m.s. OK p.d.f.

RF SK SRF UK

cumulative density function independent identically distributed intrinsic random function intrinsic random function of order k mean square ordinary kriging probability density function random function simple kriging stationary random function universal kriging

Xi

GeostatisticsModeling Spatial Uncertainty Edited by JEAN-PAUL CHILkS and PIERRE DELFINER Copyright 0 1999 by John Wiley & Sons, Inc

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 Cuttings Logging

0.000 000 001 0.000 000 007 0.000 001

By comparison, if we used the same proportions for an opinion poll of the 50 million US households, we would interview only between 0.05 and 50 households, while 1500 is standard. Yet the economic implications of sampling for natural resources development projects can be significant. The cost of an offshore development platform like that of the Brent field is about 2 billion 1

2

INTRODUCTION

dollars. Similarly in the mining industry “the decision to invest up to 1-2 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 reserves estimates because it did not look good-it meant incompetence. For job protection it was better to systematically underestimate reserves. (Ordered by his boss to get rid of uncertainty, an engineer once gave an estimate of proven oil reserves 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. Ceostatistics 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 hook geostatistics refers to a specific set of models and techniques, largely developed by C. Matheron, in the lineage of the works of L. S. Gandin in meteorology, B. Matim in forestry, D. G. Krige and H. J. de Wijs in mining, and A. Y. Khinchin, A. N. Kolmogorov, P. G v y , 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 applica-

TYPES OF PROBLEMS CONSIDERED

3

tions, but newcomers with different backgrounds and interests are encouraged to find equivalent formulations of the problems in their own disciplines. Geostatistical terms will be introduced and highlighted with 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 stoclzastic model in which the regionalized variable is regarded as one among many possible realizations of a random function. Does this make any sense? The objects we deal with-a mineral deposit, a petroleum reservoir-are perfectly deterministic. Probabilities and their experimental foundation in the famous “law of large numbers” require the possibility of repetitions, which are impossible with objects that are essentially unique. 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. 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 anisotropy? Do the data exhibit any spatial trend? Are there characteristic scales and what do they represent? Is the histogram symmetric or skewed‘! Answering these questions, among others, is known in geostatistics as srructurd analysis. One key tool is a structure function, the variogrum, which de-

4

INTRODUCTION

scribes statistically how the values at two points become different as the separation between these points increases. The variogram is the simplest way to relate uncertainty with distance from an observation. Other two-point structure functions can be defined that, when considered together, provide further clues for modeling. If the phenomenon is spatially homogeneous and densely sampled, it is even possible to go beyond structure functions and determine the complete bivuriate distributions of measurements at pairs of points. In applications there is rarely enough data to allow empirical determination of multiple-point statistics beyond two points, a notable exception being when the data originate, or are borrowed, from images. Unexpected difficulties arise when the data exhibit a systematic spatial effect, or trend, which in geostatistical theory is modeled as a space-varying mean called drijc. The determination of the variogram in the presence of a drift is often problematic due to the unclear separation between global and local scales. A special theory with structural tools insensitive to drifts has been developed to deal with such cases (intrinsic rundonz ,functions).

Survey Optimization In resources estimation probl...