iyC& L&bJb J-cd/) JL Short Course in Ge ology: Volume 8 Fundamentals of Geostatistics in Five Lessons Andre G. Journel American Geophysical Union 91040:3018,8 910327 PFDR WASTE WM- 11 PDIR I Short Course in Geology: Volume 8 Fundamentals of Geostatistics in Five Lessons Andre G. Journel \Q Short Course Presentedat the 28th InternationalGeological Congress Washington, D.C. American Geophysical Union, Washington, D.C. Maria Luisa Crawford and Elaine Padovani Short Course Series Editors tibrary of Congress Cataloging-in.Pubhcation Data journel, A. G. Fundamentals of geostatistics in five lessons. Bibliography: p. 1. Geology-Statistical methods. I. Title. QE33.2.S82J68 1989 551'.072 89-14911 ISBN 0-87590-708-3 Copyright 1989 by the American Geophysical Union, 2000 Florida Avenue, NW, Washington, DC 20009, U.S.A. Figures. tables, and short excerpts may be reprinted in scientific books and journals if the source is properly cited. Authorization to photocopy items for internal or personal use, or the internal or personal use of specific clients, is granted by the American Geophysical Union for libraries and other users registered with the Copyright Clearance Center (CCC) Transactional Reporting Service. provided that the base fee of 51.00 per copy plus $0.10 per page is paid directly to CCC, 21 Congress Street, Salem, MA 10970. 0065-448/89/$01. + .10. This consent does not extend to other kinds of copying, such as copying for creating new collective works or for resale. The reproduction of multiple copies and the use of full articles or the use of extracts, including figures and tables, for commercial purposes requires permission __ from AGU. Printed in the United States of America. CONTENTS Preface v Introduction I Lesson I: Statistics Review and Notations 2 Lesson II: Linear Regression Theory, or Simple 10 Kriging Lesson III: Linear Regression under Constraints, 15 and Ordinary Kriging Lesson IV: Non-parametric assessment of local 21 uncertainty Lesson V: Stochastic imaging for imaging 30 Spatial Uncertainty Subject Index 39 Iii PREFACE From its inception as a separate discipline, geostatis accelerate the diffusion tics sought recognition from practitioners, process if they could go about ex not from math tracting ematicians or physicists. the essence of their favorite tools - a and rightfully so. Indeed, the very hum theory was essentially bling task - and deliver it in simple terms. established by the 1950's by Kol mogorov and Wiener Stressing the essence of one's finding may not and exposed by Matern (1960), Whit make it tle (1963), and into a publication list but would help the understanding Matheron (1965), among others. But there is a long, hard and correct application of the corresponding way between a concept expressed by matrix algorithm. notations in a Hilbert Behind most sophisticated concepts, there space and its implementation and is a simple routine application. idea sometimes so simple that we feel like dressing It is my opinion that the main con it up. tribution Practitioners face real data with their of geostatistics has been and still is implementa. extraordinary com tion, plexity that defies any pre-conceived an essential follow-up step much too often model, and they are forsaken the best positioned by theoreticians. to customize the algorithm to make Implementation it work. Thus, it is of great importance requires a prior effort of simplifica that they (the tion. A concept practitioners) understand what we (the or algorithm will take root only if un academics) are derstood by the proposing. user, who can customize it to the ever changing needs of his various projects. Practice over time is a merciless judge that will strip all concepts of their fancy dressing, whether wording or computer coding, and let the sole essence Andre G. Journel stand for itself. Geostatisticians would Stanford University v I Fundamentals of Geostatistics in Five Lessons Andre G. Journel Stanford Center for Reservoir Forecasting Department of Applied Earth Sciences Stanford University, Stanford, California 94035 Introduction weighting criteria. Associating kriging to distance weighting algorithms, These lessons, except for the or in a dual fashion to surface fourth, were "speed"-written fitting algorithms, as support for a "Geostatistics makes it more "ordinary" now that it for Reservoir Characteri is zation" course severed from that hazy random function source. given in Dallas, December of 1987. There Geo is definitely statistics may end up looking less prestigious (or mysteri a need for new books in Geostatistics that would ous?), but being better understood will be acknowledge the contribution of new application better applied. fields After much hard selling, the time for a fresh and sort the wheat from the tares, the theory and more that temperate look at geostatistics has come. did yield from that which remained but elegant. Geostatistics I know is foremost Data Analysis and Spatial of at least two such books in the mill. In Continuity Model the meantime, ing. Such analysis and course supports were needed modeling cannot be done without and I attempted this quick a clear understanding draw. I ask for the reader indulgence of the origin of the data, includ and patience until ing geological interpretation. the availability of The main reason for model official books. ing spatial continuity Textbooks are polished is to assess spatial uncertainty. As logical constructions which do for using not lend themselves probabilistic models, it is naive to think that to the spot-painting and diversions any that could statistical tool provides objectivity, it should sometimes enlighten a class. Lessons allow provide though consistency once a prior digressions, returns to fundamentals, parallels, model has been chosen. that could Geostatistics is a set reveal a hidden side of of numerical tools to be added to the theory being developed and, the large tool in the best chest of the geologist; it allows transporting case, the essence of an algorithm that which quantitatively a geological model all the makes it work. Building from rigorous random way to process function design and engineering. That geological theory, how could one tell that the essence model should not of ordinary stem from a blackbox package, kriging is: particularly if that pack age is cryptic. Good geology based on well understood data is 1. the usage of a structural still the only recipe for good reservoir/site/deposit distance, specific to the characterization. va iable being considered, which need not be a var These five iogram lessons address the fundamentals of geo statistical theory relevant to spatial interpolation, image 2. the reconstitution possibility of accounting for data redundancy, and uncertainty modeling. The practice of as figured by the data covariance matrix? geostatistics, although of paramount importance, is not covered here for lack of space. The book from Srivas If kriging sometimes tava works it is not because of its prob and Isaaks, which should be in the shelves by mid abilistic pedigree - in fact the algorithm could be estab 3989, will fulfill that need beautifully. The remarkably lished without a single reference to random variables user-friendly and yet complete software "Geostat Tool but because it extends well-proven and intuitive distance box", made public-domain by its author Roland Froide vaux, provides the tools for anyone to get started. Copyright 1989 American Geophysical Union Lesson I proposes a brief review of statistics and no- I I 2 FUNDAMENTALS OF GEOSTATISTICS tations; needed for developing the further lessons. The Too often, a Gaussian error distribution is casually taken reader is supposed to have a prior familiarity with statis as model for uncertainty although evidence for the in tics, integral and differential calculus at an introductory adequacy of such symmetric distribution model do college level. exist. Alternative models based on the actual distribution Lesson II presents the classical of linear regression theory neighboring data are proposed building on with the particular geostatistical twist, an indicator in the sense that data kriging paradigm. Construction the data used (the so-called of models of un "independent" variables of certainty precedes classical regression) the derivation of an estimate for the are themselves dependent one upon unknown, each other and thus which allows retaining non-least squares, i.e. needed to be made independent in non-kriging-type a first step. The simple estimates possibly better suited to the kriging system (SK) is shown to project at hand. be but a variant of the normal system of equations. A All moving average-type estimates, including dual interpretation of the simple kriging algorithm all krig shows ing estimates, provide a smooth it amounts to fit covariance-type interpolation image of the underlying functions reality: the variogram of these estimates to the data values at their locations. would not repro duce the data variogram. The concept of conditional Simple kriging requires that the mean of the variable sim ulation allows generating alternative, equiprobable, im over the field being estimated be constant and known. Or ages which honor data values at their locations and dinary kriging does not require knowledge of that mean, re flect a series of spatial continuity functions. The novel as long as it remains constant. Kriging with a trend model technique of Indicator conditional simulations, presented allows considering a variable mean, function of the coor in Lesson V allows generation of images dinates values. That function is everywhere unknown that do not suf but fer from the maximum entropy (maximum is of known functional form and could represent a local disorganiza tion for a given variograrn model) limitacin-i of trend component being added to residual values. In Les Gaussian related random function models. Also, son III, it is shown that ordinary kriging and kriging indicator simula with tions allows honoring, in a trend model are achieved by adding specific addition to hard data, soft infor constraints mation whether local (e.g.