An Introduction to Applied Geostatistics

587 Pages·1989·28.12 MB·English

by  Isaaks

An Introduction to
Applied Geostatistics
EDWARD H. ISAAKS
Department of Applied Earth Sciences,
Stanford University
R. MOHAN SRIVASTAVA
FSS International,
Vancouver, British Columbia
New York Oxford OXFORD UNIVERSITY PRESS 1989
Oxford New York Toronto
Delhi Bombay Calcutta Madras Karachi
PetalingJaya Singapore HongKong Tokyo
Nairobi DaresSalaam CapeTown
Melbourne Auckland
and associated companies in
Berlin Ibadan
Copyright Eg 1989 by Oxford University hess, Inc.
Published by Oxford University Press, Inc.,
198 Madison Avenue, New York. New York 10016-4314
Oxford is a registered trademark of Oxford University Press
All rights reserved. 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, or otherwise,
without the prior permission of Oxford University Press.
Iibrary of Congress Cataloging-in-PublicationD ata
Isaaks, Edward H.
Applied geostatistics / Edward H. lsaaks and R. Mohan Srivastava.
p. cm. Bibliography: p. Includes index.
ISBN 978-0-1 9-605013-4
1. Geology-Statistical methods. 1. Srivastava, R. Mohan. 11. ntle.
QE33.2.M3183 1989 551’.72-d~20 89-34891 CIP
29 28 21 26 25 24 23 22
Printed in the United States of America
on acid-free paper
FOREWORD
This is a book that few would have dared to write, a book that presents
the fundamentals of an applied discipline without resorting to abstract
concepts. There is a traditional approach to the definition of prob-
ability and random variables-one that dwells on the clean ergodic
properties of Gaussian, isofactorial, and factorable random processes.
Had the authors wanted simply to add to their list of publications, they
could have followed this safe and well-worn path, and produced yet one
more book on random fields, novel for its terminology, traditional and
unassailable, but not really useful. Instead, by questioning supposedly
unquestionable dogma and by placing practicality above mathematical
elegance, they have chosen a more difficult path-one that risks the
scorn of the self-ordained Keepers of the Tablets.
Geostatistics owes much to practice. It has evolved through what
appeared initially as inconsistent applications or ad hoc adaptations of
well-established models. As these adaptations established their practi-
cal utility through several successful applications, theoreticians belat-
edly granted them respectiblity and established their theoretical pedi-
gree. Despite having sprung from practice that was once dismissed
as theoretically inconsistent, many of these ideas are now presented
as clean and logical derivations from the basic principles of random
function theory. Two most enlightening examples are:
The practice introduced by Michel David of the general rela-
0
tive variogram (Chapter 7 of this book) whereby the traditional
experimental variogram Y(h) is divided by the squared mean
[m(h)I2 of the data used for each lag h. Though inconsistent
with the stationarity hypothesis, this practice proved very suc-
cessful in cleaning up experimental variograms and in revealing
X A n Introduction to Applied Geosta t ist ics
features of spatial continuity that were later confirmed by addi-
tional data. It was much later understood that the theoretical
objections to David’s proposal do not hold since all variogram
estimators are conditional to the available data locations and are
therefore nonstationary. Moreover, his general relative variogram
can be shown theoretically to filter biases due to preferential data
clusters, a feature commonly encountered in earth science data.
The practice of using a moving data neighborhood for ordinary
0
kriging (OK), with a rescaling of the kriging variance by some
function of the local mean data value. Though strictly inconsis-
tent with the stationarity hypothesis underlying OK, this prac-
tice is the single most important reason for the practical success
of the OK algorithm that drives geostatistics as a whole. It was
later understood that OK with moving data neighborhoods is in
fact a nonstationary estimation algorithm that allows for local
fluctuations of the mean while assuming a stationary variogram.
Rather than being motivated by theoretical considerations of ro-
bustness, the now common practice of OK with moving data
neighborhoods was motivated in the 1960s by trite considera-
tions of computer memory and CPU time.
Geostatistics with Mo and Ed, as this book is known at Stanford,
is remarkable in the statistical literature and unique in geostatistics in
that concepts and models are introduced from the needs of data anal-
ysis rather than from axioms or through formal derivations. This pre-
sentation of geostatistics is centered around the analysis of a real data
set with “distressing” complexity. The availability of both sparse sam-
pling and the exhaustive reference allows assumptions and their con-
sequences to be checked through actual hindsight comparisons rather
than through checking some theoretical property of the elusive random
function generator. One may argue that the results presented could
be too specific to the particular data set used. My immediate answer
would be that a real data set with true complexity represents as much
generality as a simplistic random function model, most often neatly
stationary and Gaussian-related, on which supposedly general results
can be established.
Applied geostatistics, or for that matter any applied statistics, is
an art in the best sense of the term and, as such, is neither completely
automatable nor purely objective. In a recent experiment conducted
Foreword xi
by the U.S. Environmental Protection Agency, 12 independent rep-
utable geostatisticians were given the same sample data set and asked
to perform the same straightforward block estimation. The 12 results
were widely different due to widely different data analysis conclusions,
variogram models, choices of kriging type, and search strategy. In the
face of such an experiment, the illusion of objectivity can be main-
tained only by imposing one's decisions upon others by what I liken to
scientific bullying in which laymen are dismissed as incapable of un-
derstanding the theory and are therefore disqualified from questioning
the universal expertise written into some cryptic software package that
delivers the correct and objective answer.
It bears repeating that there is no accepted universal algorithm
for determining a variogram/covariance model, whether generalized or
not, that cross-validation is no guarantee that an estimation procedure
will actually produce good estimates of unsampled values, that kriging
need not be the most appropriate estimation method, and that the
most consequential decisions of any geostatistical study are taken early
in the exploratory data analysis phase. An Introduction to Applied
Geostatistics delivers such messages in plain terms yet with a rigor
that would please both practitioners and mature theoreticians (i.e.,
from well-interpreted observations and comparative studies rather than
from theoretical concepts whose practical relevance is obscure).
This book is sown with eye-opening remarks leading to the most
recent developments in geostatistical methodology. Though academics
will be rewarded with multiple challenges and seed ideas for new re-
search work, the main public for this book will be undergraduates and
practitioners who want to add geostatistics to their own toolbox. This
book demonstrates that geostatistics can be learned and used properly
without graduate-level courses in stochastic processes. Mo and Ed
came to geostatistics not directly from academia but from the harsh
reality of the practice of resource estimation within producing compa-
nies. They returned to university to better understand the tools that
they found useful and are now back solving problems, sometimes us-
ing geostatistics. Their book puts geostatistics back where it belongs,
in the hands of practitioners mastering both the tools and the mate-
rial. Listen to their unassuming experience and remember: you are in
command!
May, 1989 Andre' G. Journel
CONTENTS
1 Introduction 3
. . . . . . . . . . . . . . . . .
The Walker Lake Data Set 4
. . . . . . . . . . . . . . . . .
Goals of the Case Studies 6
2 Univariate Description 10
. . . . . . . . . . . . .
Frequency Tables and Histograms 10
. . . . . .
Cumulative Frequency Tables and Histograms 12
. . . . . . . . .
Normal and lognormal Probability Plots 13
. . . . . . . . . . . . . . . . . . . . .
Summary Statistics 16
. . . . . . . . . . . . . . . . . . . . .
Measures of Spread 20
. . . . . . . . . . . . . . . . . . . . .
Measures of Shape 20
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
Notes 21
. . . . . . . . . . . . . . . . . . . . . .
Further Reading 23
3 Bivariate Description 24
. . . . . . . . . . . . . . .
Comparing Two Distributions 24
. . . . . . . . . . . . . . . . . . . . . . . . .
Scatterplots 28
. . . . . . . . . . . . . . . . . . . . . . . . .
Correlation 30
. . . . . . . . . . . . . . . . . . . . . .
Linear Regression 33
. . . . . . . . . . . . . . . . . .
Conditional Expectation 35
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
Notes 38
. . . . . . . . . . . . . . . . . . . . . .
Further Reading 39
4 Spatial Description 40
. . . . . . . . . . . . . . . . . . . . . . . .
Data Postings 40
. . . . . . . . . . . . . . . . . . . . . . . .
Contour Maps 41
. . . . . . . . . . . . . . . . . . . . . . . .
Symbol Maps 43
. . . . . . . . . . . . . . . . . . . . . . .
Indicator Maps 44
. . . . . . . . . . . . . . . . .
Moving Window Statistics 46
xiv CONTENTS
Proportional Effect . . . . . . . . . . . . . . . . . . . . . 49
Spatial Continuity . . . . . . . . . . . . . . . . . . . . . 50
. . . . . . . . . . . . . . . . . . . . . . . .
h-Scatterplots 52
Correlation Functions. Covariance Functions. and Vari-
. . . . . . . . . . . . . . . . . . . . . . . . . . .
ograms 55
Cross h-Scatterplots . . . . . . . . . . . . . . . . . . . . 60
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
Notes 64
Further Reading . . . . . . . . . . . . . . . . . . . . . . 65
5 The Exhaustive Data Set 67
The Distribution of V . . . . . . . . . . . . . . . . . . . 67
The Distribution of U . . . . . . . . . . . . . . . . . . . 70
The Distribution of T . . . . . . . . . . . . . . . . . . . 73
Recognition of Two Populations . . . . . . . . . . . . . . 75
The V-U Relationship . . . . . . . . . . . . . . . . . . . 76
Spatial Description of V . . . . . . . . . . . . . . . . . . 78
Spatial Description of U . . . . . . . . . . . . . . . . . . 80
Moving Window Statistics . . . . . . . . . . . . . . . . . 90
Spatial Continuity . . . . . . . . . . . . . . . . . . . . . 93
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6 The Sample Data Set 107
Data Errors . . . . . . . . . . . . . . . . . . . . . . . . . 109
The Sampling History . . . . . . . . . . . . . . . . . . .1 10
Univariate Description of V . . . . . . . . . . . . . . . . 1 20
Univariate Description of U . . . . . . . . . . . . . . . . 1 23
The Effect of the TType . . . . . . . . . . . . . . . . .1 27
The V-U Relationship . . . . . . . . . . . . . . . . . . .1 27
. . . . . . . . . . . . . . . . . . . . .
Spati a1 D esc rip t i on 129
Propo r t ional Effect . . . . . . . . . . . . . . . . . . . . . 136
Further Reading . . . . . . . . . . . . . . . . . . . . . . 138
7 The Sample Data Set: Spatial Continuity 140
Sample h-Scatterplots and Their Summaries . . . . . . . 141
. . . . . . . .
An Outline of Spatial Continuity Analysis 143
Choosing the Distance Parameters . . . . . . . . . . . . 146
Finding the Anisotropy Axes . . . . . . . . . . . . . . .1 49
Choosing the Directional Tolerance . . . . . . . . . . . . 154
. . . . . . . . . . . . . . . . .
Sample Variograms for U 154
Relative Variograms . . . . . . . . . . . . . . . . . . . . . 163
CONTENTS xv
. . . . . . . . . . . .
Comparison of Relative Variograms 166
. . . . .
The Covariance Function and the Correlogram 170
. . . . . . . . .
Directional Covariance Functions for U 173
. . . . . . . . . . . . . . . . . . . . . .
Cross-Variograms 175
. . . . . . . . . . . . . .
Summary of Spatial Continuity 177
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
Notes 181
. . . . . . . . . . . . . . . . . . . . . .
Further Reading 182
8 Estimation 184
. . . . . . . . . . . . . .
Weighted Linear Combinations 185
. . . . . . . . . . . . . . .
Global and Local Estimation 187
. . . . . . . . . . . .
Means and Complete Distributions 188
. . . . . . . . . . . . . . . . .
Point and Block Estimates 190
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
Notes 194
. . . . . . . . . . . . . . . . . . . . . .
Further Reading 194
9 Random Function Models 196
. . . . . . . . . . . . . . . . .
The Necessity of Modeling 196
. . . . . . . . . . . . . . . . . . . .
Deterministic Models 198
. . . . . . . . . . . . . . . . . . . .
Probabalistic Models 200
. . . . . . . . . . . . . . . . . . . . .
Random Variables 202
Functions of Random Variables . . . . . . . . . . . . . .2 04
. . . . . . . . . . . .
Parameters of a Random Variable 206
. . . . . . . . . . . . . . . . . .
Joint Random Variables 210
. . . . . . . . . . . . . . . . . . .
Marginal Distributions 211
. . . . . . . . . . . . . . . . .
Conditional Distributions 212
. . . . . . . . . .
Parameters of Joint Random Variables 213
.
Weighted Linear Combinations of Random Variables . 215
. . . . . . . . . . . . . . . . . . . . .
Random Functions 218
. . . . . . . . . . . .
Parameters of a Random Function 221
. . . .
The Use of Random Function Models in Practice 226
. . . .
An Example of the Use of a Probabalistic Model 231
. . . . . . . . . . . . . . . . . . . . . .
Further Reading 236
10 Global Estimation 237
. . . . . . . . . . . . . . . . . . .
Polygonal Declustering 238
. . . . . . . . . . . . . . . . . . . . . .
Cell Declustering 241
. . . . . . . . . . .
Comparison of Declustering Methods 243
. . . . . . . . . .
Declustering Three Dimensional Data 247
. . . . . . . . . . . . . . . . . . . . . .
Further Reading 248
xvi CONTENTS
11 Point Estimation 249
Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
Triangulation . . . . . . . . . . . . . . . . . . . . . . . . 251
Local Sample Mean . . . . . . . . . . . . . . . . . . . . . 256
. . . . . . . . . . . . . . . . .
Inverse Distance Methods 257
Search Neighborhoods . . . . . . . . . . . . . . . . . . . 2 59
Estimation Criteria . . . . . . . . . . . . . . . . . . . . . 260
Case Studies . . . . . . . . . . . . . . . . . . . . . . . . 266
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
Further Reading . . . . . . . . . . . . . . . . . . . . . . 277
12 Ordinary Kriging 278
The Random Function Model and Unbiasedness . . . . . 2 79
The Random Function Model and Error Variance . . . . 2 81
The Lagrange Parameter . . . . . . . . . . . . . . . . . .2 84
Minimization of the Error Variance . . . . . . . . . . . .2 86
Ordinary Kriging Using y or p . . . . . . . . . . . . . .2 89
An Example of Ordinary Kriging . . . . . . . . . . . . .2 90
Ordinary Kriging and the Model of Spatial Continuity . 296
An Intuitive Look at Ordinary Kriging . . . . . . . . . .2 99
Variogram Model Parameters . . . . . . . . . . . . . . .3 01
Comparison of Ordinary Kriging to Other Estimation
Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 313
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
Notes 321
Further Reading . . . . . . . . . . . . . . . . . . . . . . 322
13 Block Kriging 323
The Block Kriging System . . . . . . . . . . . . . . . . .3 24
Block Estimates Versus the Averaging of Point Estimates327
Varying the Grid of Point Locations Within a Block . . 327
A Case Study . . . . . . . . . . . . . . . . . . . . . . . . 330
14 Search Strategy 338
Search Neighborhoods . . . . . . . . . . . . . . . . . . .3 39
Quadrant Search . . . . . . . . . . . . . . . . . . . . . . 344
. . . . . . . . . . . .
Are the Nearby Samples Relevant? 347
Relevance of Nearby Samples and Stationary Models . . 349
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
CONTENTS xvii
15 Cross Validation 351
Cross Validation . . . . . . . . . . . . . . . . . . . . . . 352
. . . . . . . . .
Cross Validation as a Quantitative Tool 352
. . . . . . . . . .
Cross Validation as a Qualitative Tool 359
. . . . . . . .
Cross Validation as a Goal-Oriented Tool 364
16 Modeling the Sample Variogram 369
. . . . . . . . . . .
Restrictions on the Variogram Model 370
. . . . . . . . . . . .
Positive Definite Variogram Models 372
Models in One Direction . . . . . . . . . . . . . . . . . .3 75
. . . . . . . . . . . . . . . . . . . .
Models of Anisotropy 377
. . . . . . . . . . . . . . . . . . . . . .
Matrix Notation 386
Coordinate Transformation by Rotation . . . . . . . . . 3 88
. . . . . . . . . .
The Linear Model of Coregionalization 390
. .
Models For the The Walker Lake Sample Variograms 391
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397
. . . . . . . . . . . . . . . . . . . . . .
Further Reading 398
17 Cokriging 400
The Cokriging System . . . . . . . . . . . . . . . . . . . 4 01
. . . . . . . . . . . . . . . . . . .
A Cokriging Example 405
. . . . . . . . . . . . . . . . . . . . . . . .
A Case Study 407
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
Notes 416
Further Reading . . . . . . . . . . . . . . . . . . . . . . 416
18 Estimating a Distribution 41 7
. . . . . . . . . . . . . . . . .
Cumulative Distributions 418
. . . . . . . . .
The Inadequacy of a Naive Distribution 419
. . . . . . . . . . . .
The Inadequacy of Point Estimates 420
Cumulative Distributions. Counting and Indicators . . . 4 21
. . . . . .
Estimating a Global Cumulative Distribution 424
Estimating Other Parameters of the Global Distribution 428
. . . . . . . . . . . . . .
Estimating Local Distributions 433
. . . . . . . . . . . . . .
Choosing Indicator Thresholds 435
. . . . . . . . . . . . . . . . . . . . . . . .
Case Studies 438
. . . . . . . . . . . . . . . . . . . .
Indicator Variograms 442
. . . . . . . . . . . . . . . .
Order Relation Corrections 447
. . . . . . . . . . . . . . . . . . . . .
Case Study Results 448
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
Notes 456
. . . . . . . . . . . . . . . . . . . . . .
Further Reading 457