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 ffirs 28 January 2012; 13:1:56 WILEY SERIES IN PROBABILITY AND STATISTICS Established by WALTER A SHEWHART and SAMUEL S WILKS Editors: David J Balding, Noel A C Cressie, Garrett M Fitzmaurice, Harvey Goldstein, Iain M Johnstone, Geert Molenberghs, David W Scott, Adrian F M Smith, Ruey S Tsay, Sanford Weisberg Editors Emeriti: Vic Barnett, J Stuart Hunter, Joseph B Kadane, Jozef L Teugels A complete list of the titles in this series appears at the end of this volume ffirs 28 January 2012; 13:1:56 Geostatistics Modeling Spatial Uncertainty Second Edition JEAN-PAUL CHILE`S BRGM and MINES ParisTech PIERRE DELFINER PetroDecisions ffirs 28 January 2012; 13:1:56 Copyright r 2012 by John Wiley & Sons, Inc All rights reserved Published by John Wiley & Sons, Inc., Hoboken, New Jersey 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 Section 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, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permission Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose No warranty may be created or extended by sales representatives or written sales materials The advice and strategies contained herein may not be suitable for your situation You should consult with a professional where appropriate Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002 Wiley also publishes its books in a variety of electronic formats Some content that appears in print may not be available in electronic formats For more information about Wiley products, visit our web site at www.wiley.com ISBN: 978-0-470-18315-1 Library of Congress Cataloging-in-Publication Data is available Printed in the United States of America 10 ffirs 28 January 2012; 13:1:56 Contents Preface to the Second Edition ix Preface to the First Edition xiii Abbreviations xv Introduction Types of Problems Considered, Description or Interpretation?, Preliminaries 1.1 1.2 1.3 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 ftoc 28 January 2012; 13:4:27 vi CONTENTS Kriging 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 238 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 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 4.9 147 299 Introduction, 299 Notations and Assumptions, 300 Simple Cokriging, 302 Universal Cokriging, 305 Derivative Information, 320 Multivariate Random Functions, 330 Shortcuts, 360 Space Time 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 ftoc 28 January 2012; 13:4:27 vii CONTENTS 6.6 6.7 6.8 6.9 6.10 6.11 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 ftoc 28 January 2012; 13:4:27 In memory of Georges MATHERON (1930 2000) ftoc 28 January 2012; 13:4:27 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 (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 (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 (Kriging) as a variant of the universal kriging ix fpref 28 January 2012; 13:3:53 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 (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 Y previously in the deleted chapter survives in this chapter Chapter (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 (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 fpref 28 January 2012; 13:3:53 ...WILEY SERIES IN PROBABILITY AND STATISTICS Established by WALTER A SHEWHART and SAMUEL S WILKS Editors: David J Balding, Noel A C Cressie, Garrett M Fitzmaurice, Harvey Goldstein, Iain M Johnstone,... appears at the end of this volume ffirs 28 January 2012; 13:1:56 Geostatistics Modeling Spatial Uncertainty Second Edition JEAN- PAUL CHILE `S BRGM and MINES ParisTech PIERRE DELFINER PetroDecisions... 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