GEOSTATISTICSFOR ENGINEERS ANDEARTH SCIENTISTS

GEOSTATISTICS FOR ENGINEERS AND EARTH SCI ENTISTS

by

Ricardo A. Olea

~.

Kansas Geological Survey The University of Kansas Lawrence, Kansas USA

Springer-Science+Business Media, LLC

Library of Congress Cataloging-ln-Publieatlon Data

Olea, R. A. (Ricardo A.) Geostatistlscs for engineers and earth scientists / by Ricardo A.

Olea. p. cm.

Includes bibliographical references and index. ISBN 978-1-4613-7271-4 ISBN 978-1-4615-5001-3 (eBook) DOI 10.1007/978-1-4615-5001-3

1. Geology--Statistical methods. 2. Kriging. I. Title. QE33.2.S82054 1999 55O'.72-dc21 99-24689

CIP

CopyrIaht C 1999 by Springer Science+Business Media New York. Third Printing 2003. Originally published by K1uwer Academic Publishers in 1999 Softcover reprint ofthe hardcover 1st edition 1999

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, mechanical, photo­copying, recording, or otherwise, without the prior written permission of the publisher, Springer Science+Business Media, LLC.

Printed on acid-free paper.

To lucila

CONTENTSList of Mathematical Definitions x

List of Theorems xii

List of Corollaries xiii

List of Lemmas xiv

Preface xvii

Chapter 1: Introduction 1

Chapter 2: Simple Kriging 7Properties of Linear Combinations of Variates 8Assumptions and Definitions 11The Estimation Variance 11Normal Equations 14Minimum Mean Square Error 15Algorithm 16

EXERCISE 2.1 ••••.•••••••.•.•.••.••.••.•.•••.•••...•.••••••..•..•••.••. 18Properties 20

EXERCISE 2.2 ••.••.•••.•••••.••.•..•..••.•.••.•••..•...••• " ••••••. " .. 27

Chapter 3: Normalization 31Comparing Two Distributions 31

EXERCISE 3.1 •••.••.••••••••••••.•..•.•.•••••••.•••.•••..•..•••••••••.. 32Normal Score Transformation 33Simple Kriging of Normal Scores 35

EXERCISE 3.2 ••••.••.•••..•..•••••.••.•..••....•...••..•..••.•.••..••.. 35

Chapter 4: Ordinary Kriging 39Assumptions 39Important Relationships 40The Estimator 43The Estimation Variance 43The Optimization Problem 44Minimum Mean Square Error 46Algorithm for Intrinsic Random Functions 47

viii Ceostatistics for Engineers & Earth Scientists

Second Order Stationary Ordinary Kriging 48EXERCISE 4.1 ••••••.•.•.•.....•..•..•.•.....••..•.•••........•.••.••••• 52

Properties 53Relating Simple and Ordinary Kriging 59Search Neighborhood 62Quasi-Stationary Estimator 63

Chapter 5: The Semlvarlogram 67The Semivariogram of the Random Function 68The Experimental Semivariogram 70

EXERCISE 5.1 ••.•.••••.••••.••.•.•.••.••....••....•••••.•.•••••..•.•.•• 74Anisotropy and Drift 75Semivariogram Models 76Additivity 80Parameter Estimation by Trial and Error 81Automatic Parameter Fitting 83

EXERCISE 5.2 •••.•••••••..••••••.••••..••.•...••••••••••••••••••••.•••• 86Support 89Direct Applications 89

Chapter 6: Universal Kriging 91The Estimator 91Assumptions 92Unbiasedness 94Estimation Variance 96Optimization 98Minimum Mean Square Error 100Algorithm for Intrinsically Stationary Residuals 102Second Order Stationary Universal Kriging 103Practice " , , 107

EXERCISE 6.1 •••.•.•.••.•••••••.•.•.•.•••.•..•.•.••.•..••••...••.•..• 108

Chapter 7: Crossvalldatlon 115Alternative Evaluation Method 115

EXERCISE 7.1 •••.••••....•••...•.•••...•.••.•.•.•.•....•..•.•.•.•.•.• 117Diagnostic Statistics 119

EXERCISE 7.2 ••.•••.•....••....•.•••••.•••••.•.••••.•.•.•••••••.••••• 121

Chapter 8: Drift and Residuals 129Assumptions 129Unbiasedness 130Estimation Variance 132Optimal Estimator 133Minimum Estimation Variance 135Algorithmic Summary 136Residuals 137

EXERCISE 8.1 ••••.•.••.•.•......•••••.•••..••••.•.•.••.••.•....•...•• 137

Contents ix

Chapter 9: Stochastic Simulation 141Sequential Gaussian Simulation 143

EXERCISE 9.1 ••.•.•••.••.•.•........•.....•.......•...•..•.........•. 147Simulated Annealing 147

EXERCISE 9.2 ...•....•••.•.......•.•.•••..•.•.•..•.•..•..••.........• 152Advantages and Disadvantages of Simulated Annealing 154Lower-Upper (LU) Decomposition 155The Turning Bands Method 158

Chapter 10: Reliability 163Kriging Under Normality of Errors 164

EXERCISE 10.1 •••.•.•.....•.••.•...•.••...•.•.....•..........•..•.••. 165Indicator Kriging " 167

EXERCISE 10.2 ..••.•...••........•....••••..••.....•.•...•....•.•.•.. 171Stochastic Simulation 173

EXERCISE 10.3 •••.•••••••••.••••.••••...•.•.•.•....••.....•.••..•.•.• 175Comparisons 177

Chapter 11: Cumulative Distribution Estimators 179Simulation E-type Estimator 179

EXERCISE 11.1 •••••..••••...•..••......•....•.•......•.........•..... 180Indicator E-type Estimator 182Loss Functions 184

Chapter 12: Block Kriging 187The Estimator 188Assumptions 188Estimation Error 191Normal Equations 193Covariance Modeling 197

EXERCISE 12.J •••.•..•.•.......•....••..•.•...••....•...•..•....•.... 198EXERCISE J2.2 •••••••••..•....•••.•..•.•••.••.••.....•.....•.•....... 202

Remarks 204EXERCISE 12.3 .•.•.....••....•.•..•.•..........••••.....••.•..•...... 206

Chapter 13: Ordinary Cokriglng 209The Estimator 209Assumptions 210Unbiasedness 212Estimation Error 213Optimization 217Minimum Mean Square Error 219Algorithm 220Structural Analysis 221

EXERCISE 13.1 •..•...•.•..•....•.............•••.•.....•......•...... 226Regionalized Compositions 234

Ceostatlstlcs for Engineers & Earth Scientists

Chapter 14: Regionalized Classification 237Typification 238Ward's Method 239Discriminant Analysis 243Allocation by Extension 247

EXERCISE 14.1 ••••••.•••.•.••••..•...••...•••.••.•.•...•••••.••••••.• 248

References 261

Appendix A: West Lyons Field Sampling 267

Appendix B: High Plains Aquifer Sampling 269

Appendix C: UNCF Sampling 279

Appendix 0: Dakota Aquifer Sampling 281

Author Index 289

Subject Index 291

List of Mathematical Definitions

1.1-Random function 32.1-Simple kriging estimator 112.2-R.esidual 122.3-Positive definite function 142.4-Simple kriging covariance matrix 162.5-Vector of unknowns in simple kriging 162.6-Simple kriging covariance vector 172.7-Vector of residuals in simple kriging 17401-0rdinary kriging estimator for the intrinsic and second order

stationary case 0. 0. 0 0. 0 0.. 0 0.. 0. 0. 43402-Negative definite function. 000 000 00000 044403- Lagrangian function for ordinary kriging . 000 0 0 000 0444.4-Semivariogram matrix in ordinary kriging 0 474.5-Vector of unknowns for intrinsic ordinary kriging 0 0 47406-Semivariogram vector in ordinary kriging 00.00 0. 00.0 0474.7-0rdinary kriging sampling vector 484.8-Covariance matrix in ordinary kriging 0 0. 0. 504.9-Vector of unknowns for second order stationary ordinary kriging 514.1 0-Covariance vector in ordinary kriging 0 0 0. 0 514. 11-Ordinary kriging estimator for the quasi-intrinsic and

quasi-stationary case 0 0 635.1-Semivariogram estimator 0 0 70So2-Spherical semivariogram model 765.3-Exponential semivariogram model 0 78

Mathematical Definitions, Theorems, Corollaries, and Lemmas xi

5.4-Gaussian semivariogram model 785.5-Power semivariogram model 795.6-Cubic semivariogram model 795.7-Pentaspherical semivariogram model 795.8-Sine hole effect semivariogram model 795.9-Pure nugget effect semivariogram model 806.1-Universal kriging estimator 916.2-Drift 926.3-Polynomial drift model 936.4-Lagrange function for universal kriging 986.5-Semivariogram matrix in universal kriging 1026.6-Vector of unknowns for intrinsic universal kriging 1026.7-Semivariogram vector in universal kriging 1026.8-Universal kriging covariance matrix 1056.9-Vector of unknowns for stationary universal kriging 1066.10-Universal kriging covariance vector 1066.11-Sampling vector in universal kriging 1068.1- Drift estimator 1298.2-Lagrangian function 1338.3-Drift vector of monomials 1368.4-R.esidual estimator 1379.1-LU decomposition of covariance matrix 15610.1-Indicator 16712.1-Block average 18712.2-Block kriging estimator 18812.3-Point-to-block covariance 18912.4-Block-to-block covariance 18912.5-Lagrangian function for block kriging 19312.6-Point-to-point covariance matrix 19612.7-Block kriging vector of unknowns 19612.8-Point-to-block covariance vector 19612.9-Block kriging sampling vector '" 19613.1-Vectorial random function 21013.2-Cokriging weight matrix 21013.3-0rdinary cokriging estimator 21013.4-Cokriging vector of means 21013.5-Covariance of a vectorial random function in ordinary

cokriging 21013.6-0rdinary cokriging estimation variance 21313.7-Trace of a square matrix 21313.8-Lagrangian function for ordinary cokriging 21713.9-Covariance matrix in ordinary cokriging 22013.1 0-Ordinary cokriging covariance vector 22013.11-Ordinary cokriging vector of unknowns 22013.12-Positive semidefinite matrix 222

xii Ceostatistics for Engineers & Earth Scientists

13. 13-Cross-semivariogram 22213.14- Linear coregionalization model 22213. 15-Additive log-ratio transformation 23413. 16-Additive generalized logistic back transformation 23414.1-Error sum of squares in cluster analysis 24014.2-Within-group error sum of squares in cluster analysis 24014.3-Mahalanobis'distance 244

List of Theorems

2.1- Estimation variance for simple kriging 132.2- Normal equations and nonnegative estimation variance for

simple kriging " 142.3-Minimum mean square error for simple kriging 162.4- Unbiased simple kriging estimator 202.5- Simple kriging exact interpolation 212.6- Orthogonality of estimates and errors 224.1- Estimation variance for ordinary kriging " 434.2- Normal equations and nonnegative estimation variance for

intrinsic ordinary kriging 454.3- Minimum mean square error for intrinsic ordinary kriging 464.4- Normal equations for second order stationary ordinary kriging 494.5- Minimum mean square error for second order stationary

ordinary kriging 504.6- Difference between simple and ordinary kriging weights 604.7- Difference between simple and ordinary kriging estimation

variance 615.1-Estimation variance for assigning one variate to another 686.1- Unbiasedness conditions for the universal kriging estimator 956.2- Estimation variance for universal kriging 976.3-Normal equations and nonnegative estimation variance for

intrinsic universal kriging 986.4- Minimum mean square error for intrinsic universal kriging 1016.5- Normal equations for second order stationary universal

kriging 1036.6- Minimum mean square error for second order stationary

universal kriging 1058.1- Unbiasedness conditions for drift estimator 1318.2- Estimation variance for drift estimator 1338.3- Normal equations for drift estimation 1348.4- Minimum mean square error in drift estimation 1369.1- Drawing from multivariate distributions 1439.2-Normal distribution of errors 143

Mathematical Definitions, Theorems, Corollaries, and Lemmas xiii

9.3-Expectation of product of lower triangular matrix andvector of random numbers drawn for a standard normal

distribution 15610.1-Expected value of indicator .,., ., ., . ., .. .,., .. ., . ., 1691 1.1-Conditional expectation estimator from stochastic

realizations 17911.2-Conditional expectation estimator from indicators 18311.3-0ptimal estimator for linear, asymmetric loss function 18411 .4-Optimal estimator for quadratic loss function 18612.1-Estimation variance for block kriging ., . ., . ., .. .,.,., .. .,.,.,.,., 19212.2-Normal equations for block kriging .. .,., . .,.,.,.,.,.,., .. .,.,., .,19312.3-Minimum mean square error for block kriging 19512.4-Equivalence between a block weight and linear average

weights 20513.1-Unbiasedness conditions for ordinary cokriging 21213.2-Estimation variance for ordinary cokriging 21613.3-Normal equations for ordinary cokriging 21813.4-Minimum mean square error for ordinary cokriging 21913.5-Inequality for the terms in minor determinant of order 2 22313.6-Insufficiency to determine positive semidefiniteness 22413.7-Cross-semivariogram in terms of cross-covariances 22514.1-Probability of belonging to a group 24314.2-Probability of belonging to a group for heteroscedastic normal

group distributions 24414.3-Probability of belonging to a group for homoscedastic normal

group distributions 245

List of Corollaries

2.1-Variance in an exact interpolation 232.2-Singular matrix produced by duplicated sampling sites 232.3-Insensitivity of translation of Cartesian system of the normal

equations for simple kriging .. ., .. ., . ., . ., . ., .. ., . ., .,. 242.4- Insensitivity of estimate to multiplication of covariance by

a factor for simple kriging 242.5- Change in estimation variance by multiplication of covariance

by a factor 242.6-Independence of estimation variance and data configuration 242.7-Lack of direct dependence of estimation variance on data 262.8- Simple kriging weight using a sampling comprising independent

variates 262.9-Simple kriging estimation in the absence of spatial correlation 262.1 O-The mean regarded as an additional observation 272.11- Simple kriging estimate when sum of weights is equal to 1 27

xiv Ceostatistics for Engineers & Earth Scientists

4.1- Equivalence between semivariogram and covariance function 414.2- Insensitivity of translation of Cartesian system of the normal

equations for ordinary kriging 554.3- Insensitivity of estimate to multiplication of covariance by a

factor for ordinary kriging 554.4- Lagrange multiplier using a sampling comprising independent

variates 564.S-0rdinary kriging weight using a sampling comprising

independent variates 564.6-0rdinary kriging estimation variance using a sampling

comprising independent variates 574.7-0rdinary kriging weight in the absence of spatial correlation 584.8-0rdinary kriging estimation variance in the absence of spatial

correlation 584.9-0rdinary kriging estimator in the absence of spatial

correlation 58S. 1-Semivariogram beyond the range 69S.2-Covariance equivalence to semivariogram 7011 .1-Optimal estimator for linear, symmetric loss function 18513.1-Transpose of covariance of the vectorial random function 21213.2-Inequality for the terms in matrix of order 2 22313.3-Cross-semivariogram in terms of symmetric cross-covariances 226

List of Lemmas

2.1- Expected value of variate multiplied by constant 82.2- Expected value of linear combination of random variables 92.3- Expected value of the square of a linear combination of

random variables 92.4- Variance of linear combination of random variables 102.S-Expected value of a residual 122.6-Covariance of variate and its residual 122.7- Combination of covariances for optimal weights in simple

kriging 154.1- Expected value of difference of two variates 404.2- Equivalence of variance of difference of two variates and the

expected value of its square difference 414.3-Equivalence among semivariogram, variance, and covariance 414.4-Triangular relationship for semivariogram 424.5- Combination of semivariograms for optimal weights in ordinary

kriging 464.6- Unbiasedness condition for ordinary kriging estimator 544.7- Matrix form of Lagrange multiplier 594.8- Matrix form of ordinary kriging vector of unknowns 59

Mathematical Definitions, Theorems, Corollaries, and Lemmas

4.9-Matrix form of ordinary kriging estimation variance .... "",.,.',., 606.1-Expected value of universal kriging estimator .. "" .. ,." ,.,.946.2-Estimation error for universal kriging. , , . , , , .. , , . , , . , 966.3-Combination of semivariograms for optimal weights in

universal kriging ' .. ' .. '., " , ,." ". 1008.1-Expected value of the drift estimator , , , , , . , . , 1308.2-Drift error , , .. , .. ,." ,., .. ,., " 1328.3-Combination of covariances for optimal weights in drift

estimation "., .. ,., , ' " , , ,., 13512.1-Point-to-block covariance for quasi-stationary point random

function , .. , , , .. ,. , , , , ,. , ,., , .. , ,18912.2-Block average of second order quasi-stationary point random

function ' ,. , , , , , , .19012.3-Expected value of difference between point variate and

block average " ,., .. " .. , , ,., ,19112.4-Covariance of difference between point variate and block

average , .. , ' , " , , ,. 19112.5- Combination of covariances for optimal weights in block

kriging , , .. , .. , , . , , , . , .. 19413.1-0rdinary cokriging estimation variance, ,."., , .. , ,21413.2-Covariance of combinations of vectorial random variables, ,., .. 21413.3-Expected value of vectorial error , .. , , , 21513.4-Covariance of vectorial differences .. ,., , ,.".".,., , 21513.5-Vectorial combination of covariances for optimal weights

in ordinary cokriging .. , .. " , , , .. ,., ,., .. 219

PREFACEThis textbook is an outgrowth of notes that I prepared for engineering stu­dents attending my graduate-level course taught at The University of Kansasin the United States. Geostatistics builds upon other disciplines, a command ofwhich is basic for a ready understanding of its concepts. Prerequisite to enroll­ment in my class is the successful completion of the equivalent of (American)college-level courses in matrix algebra, probability, and statistics. Rather thanincluding a lengthy introductory review of such subjects here, I have decidedto require from the readers the same level of understanding expected from mystudents. Tutorial deficits or memory lapses should be remedied by referring tothe abundant literature on these topics.

I am indebted to Yun-Shen Yu for inviting me to lecture at his School ofEngineering. For their comments, I thank my students, who have encouragedme to publish the class notes. I am most grateful for remarks by Frederik P.Agterberg and George Christakos which contributed to the improvement of anearly draft of this text. Special thanks are due to Andre Journel, who introducedme to geostatistics in my native Santiago de Chile almost 30 years ago and whowas an insightful and demanding reviewer. Many of the suggestions of mycolleagues were incorporated into the final text, but I reserved the right, andassume the blame, for disagreeing in a few instances.

Preparation of this textbook could not have been possible without supportfrom the Kansas Geological Survey, in particular Jo Anne DeGraffenreid forthe careful editing of the final manuscript and preparation of the camera-readycopy; Patricia Acker for her assistance in the preparation of all final illustrations;Geoffrey Bohling for his meticulous review of the camera-ready COPYi and DanaAdkins-Heljeson for the preparation of a site on the Internet containing all colorillustrations and data used in the exercises.

Ricardo A. Olealawrence, Kansas