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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 Brisbane Singapore - Toronto

This book is printed on acid-free paper. @ Copyright (9 1999 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, NY 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 550’.72-&2 1 98-3599 Printed in the United States of America 1 0 9 8 7 6

Contents Preface Abbreviations Introduction Types of Problems Considered, 2 Description or Interpretation?, 7 1. Preliminaries 1.1. Random Functions, I 1 1.2. 1.3. Transitive Theory, 24 On the Objectivity of Probabilistic Statements, 22 2. Structural Analysis 2.1. 2.2. 2.3. 2.4. 2.5. 2.6. 2.7. 2.8. 2.9. General Principles, 29 Variogram Cloud and Sample Variogram, 34 Mathematical Properties of the Variogram, 57 Regularization and Nugget Effect, 74 Variogram Models, 80 Fitting a Variogram Model, 104 Variography in Presence of a Drift, 115 Simple Applications of the Variogram, 128 Complements: Theory of Variogram Estimation and Fluctuation, 137 3. Kriging 3.1. Introduction, 150 3.2. Notations and Assumptions, 152 ix xi 1 11 29 150 V

vi CONTENTS 3.3. Kriging with a Known Mean, 154 3.4. Kriging with an Unknown Mean, 164 3.5. Estimation of a Spatial Average, 193 3.6. Selection of a Kriging Neighborhood, 201 3.7. Measurement Errors and Outliers, 210 3.8. Case Study: The Channel Tunnel, 215 3.9. Kriging under Inequality Constraints, 224 4. Intrinsic Model of Order k IRF-0 and IRF-k, 231 A Second Look at the Model of Universal Kriging, 233 Allowable Linear Combinations of Order k, 236 Intrinsic Random Functions of Order k, 243 Generalized Covariance Functions, 252 Estimation in the IRF Model, 265 Generalized Variogram, 276 Automatic Structure Identification in the General Case, 281 4.1. 4.2. 4.3. 4.4. 4.5. 4.6. 4.7. 4.3. 5. Multivariate Methods 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 5.1. 5.2. 5.3. 5.4. 5.5. 5.6. 5.7. 5.8. 231 292 6. Nonlinear Methods 375 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

CONTENTS 7. Conditional Simulations 7.1. 7.2. 7.3. 7.4. 7.5. 7.6. 7.7. 7.8. 7.9. 7.10. 7.11. Introduction and Definitions, 449 Direct Conditional Simulation of a Continuous Variable, 462 Conditioning by Kriging, 465 Turning Bands, 472 Nonconditional Simulation of a Continuous Variable, 478 Nonconditional Simulation of an IRF-4, 506 Sirnulation of a Categorical Variable, 520 Object-Based Simulations: Boolean Models, 545 Constrained Simulations, 56 1 Practical Considerations, 57 1 Case Studies, 577 8. Scale Effects and Inverse Problems 8.1. Introduction, 593 8.2. Upscaling Permeability, 594 8.3. Stochastic Differential Equations, 602 8.4. Inverse Problem in Hydrogeology, 61 1 Appendix References Index vii 449 593 636 650 687

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 au- dience, for the applications of geostatistics now extend to many fields in the earth sciences, including not only the subsurface but also the land, the atmo- sphere, 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 rnethodul- ogy. 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 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 ix

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