Geostatistics for Engineers and Earth Scientists

309 Pages·1999·16.95 MB·English

by  Ricardo A. Olea (auth.)

GEOSTATISTICS
FOR ENGINEERS AND
EARTH 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, photo
copying, recording, or otherwise, without the prior written permission of the
publisher, Springer Science+Business Media, LLC.
Printed on acid-free paper.
To lucila
CONTENTS
Listof Mathematical Definitions x
List ofTheorems xii
List ofCorollaries xiii
List ofLemmas xiv
Preface xvii
Chapter 1: Introduction 1
Chapter2: Simple Kriging 7
Properties ofLinear Combinations ofVariates 8
Assumptions and Definitions 11
The Estimation Variance 11
Normal Equations 14
Minimum Mean Square Error 15
Algorithm 16
EXERCISE2.1 ••••.•••••••.•.•.••.••.••.•.•••.•••...•.••••••..•..•••.••. 18
Properties 20
EXERCISE2.2 ••.••.•••.•••••.••.•..•..••.•.••.•••..•...•••" ••••••." .. 27
Chapter 3: Normalization 31
Comparing Two Distributions 31
EXERCISE3.1 •••.••.••••••••••••.•..•.•.•••••••.•••.•••..•..•••••••••.. 32
Normal Score Transformation 33
Simple Kriging ofNormal Scores 35
EXERCISE 3.2 ••••.••.•••..•..•••••.••.•..••....•...••..•..••.•.••..••.. 35
Chapter 4: Ordinary Kriging 39
Assumptions 39
Important Relationships 40
The Estimator 43
The Estimation Variance 43
The Optimization Problem 44
Minimum Mean Square Error 46
Algorithm for Intrinsic Random Functions 47
viii Ceostatistics for Engineers&Earth Scientists
Second Order Stationary Ordinary Kriging 48
EXERCISE4.1 ••••••.•.•.•.....•..•..•.•.....••..•.•••........•.••.••••• 52
Properties 53
Relating Simple and Ordinary Kriging 59
Search Neighborhood 62
Quasi-Stationary Estimator 63
Chapter 5: The Semlvarlogram 67
The Semivariogram ofthe Random Function 68
The Experimental Semivariogram 70
EXERCISE 5.1 ••.•.••••.••••.••.•.•.••.••....••....•••••.•.•••••..•.•.•• 74
Anisotropy and Drift 75
Semivariogram Models 76
Additivity 80
Parameter Estimation by Trial and Error 81
Automatic Parameter Fitting 83
EXERCISE 5.2 •••.•••••••..••••••.••••..••.•...••••••••••••••••••••.•••• 86
Support 89
Direct Applications 89
Chapter 6: Universal Kriging 91
The Estimator 91
Assumptions 92
Unbiasedness 94
Estimation Variance 96
Optimization 98
Minimum Mean SquareError 100
Algorithm for Intrinsically Stationary Residuals 102
Second Order Stationary Universal Kriging 103
Practice " , , 107
EXERCISE6.1 •••.•.•.••.•••••••.•.•.•.•••.•..•.•.••.•..••••...••.•..• 108
Chapter 7: Crossvalldatlon 115
Alternative Evaluation Method 115
EXERCISE 7.1 •••.••••....•••...•.•••...•.••.•.•.•.•....•..•.•.•.•.•.• 117
Diagnostic Statistics 119
EXERCISE7.2 ••.•••.•....••....•.•••••.•••••.•.••••.•.•.•••••••.••••• 121
Chapter 8: Drift and Residuals 129
Assumptions 129
Unbiasedness 130
Estimation Variance 132
Optimal Estimator 133
Minimum Estimation Variance 135
Algorithmic Summary 136
Residuals 137
EXERCISE8.1 ••••.•.••.•.•......•••••.•••..••••.•.•.••.••.•....•...•• 137
Contents ix
Chapter 9: Stochastic Simulation 141
Sequential Gaussian Simulation 143
EXERCISE9.1 ••.•.•••.••.•.•........•.....•.......•...•..•.........•. 147
Simulated Annealing 147
EXERCISE9.2 ...•....•••.•.......•.•.•••..•.•.•..•.•..•..••.........• 152
Advantages and Disadvantages ofSimulated Annealing 154
Lower-Upper (LU) Decomposition 155
The Turning Bands Method 158
Chapter 10:Reliability 163
Kriging Under Normality ofErrors 164
EXERCISE 10.1 •••.•.•.....•.••.•...•.••...•.•.....•..........•..•.••. 165
Indicator Kriging " 167
EXERCISE 10.2 ..••.•...••........•....••••..••.....•.•...•....•.•.•.. 171
Stochastic Simulation 173
EXERCISE 10.3 •••.•••••••••.••••.••••...•.•.•.•....••.....•.••..•.•.• 175
Comparisons 177
Chapter 11: Cumulative Distribution Estimators 179
Simulation E-type Estimator 179
EXERCISE 11.1 •••••..••••...•..••......•....•.•......•.........•..... 180
Indicator E-type Estimator 182
Loss Functions 184
Chapter 12: Block Kriging 187
The Estimator 188
Assumptions 188
Estimation Error 191
Normal Equations 193
Covariance Modeling 197
EXERCISE 12.J •••.•..•.•.......•....••..•.•...••....•...•..•....•.... 198
EXERCISE J2.2 •••••••••..•....•••.•..•.•••.••.••.....•.....•.•....... 202
Remarks 204
EXERCISE 12.3 .•.•.....••....•.•..•.•..........••••.....••.•..•...... 206
Chapter 13: Ordinary Cokriglng 209
The Estimator 209
Assumptions 210
Unbiasedness 212
Estimation Error 213
Optimization 217
Minimum Mean Square Error 219
Algorithm 220
Structural Analysis 221
EXERCISE 13.1 •..•...•.•..•....•.............•••.•.....•......•...... 226
Regionalized Compositions 234
Ceostatlstlcs for Engineers &Earth Scientists
Chapter 14: Regionalized Classification 237
Typification 238
Ward's Method 239
Discriminant Analysis 243
Allocation 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 3
2.1-Simple kriging estimator 11
2.2-R.esidual 12
2.3-Positive definite function 14
2.4-Simple kriging covariance matrix 16
2.5-Vector ofunknowns in simple kriging 16
2.6-Simple kriging covariance vector 17
2.7-Vector ofresiduals in simple kriging 17
401-0rdinary kriging estimator for the intrinsic and second order
stationary case 0.0.0 0.0 0.. 0 0..0.0.43
402-Negative definite function. 000 000 00000 044
403-Lagrangian function for ordinary kriging .000 0 0 000 044
4.4-Semivariogram matrix in ordinary kriging 0 47
4.5-Vector ofunknowns for intrinsic ordinary kriging 0 0 47
406-Semivariogram vector in ordinary kriging 00.00 0.00.0 047
4.7-0rdinary kriging sampling vector 48
4.8-Covariance matrix in ordinary kriging 0 0.0.50
4.9-Vector ofunknowns for second order stationary ordinary kriging 51
4.10-Covariance vector in ordinary kriging 0 0 0.0 51
4.11-Ordinary kriging estimator for the quasi-intrinsic and
quasi-stationarycase 0 0 63
5.1-Semivariogram estimator 0 0 70
So2-Spherical semivariogram model 76
5.3-Exponential semivariogram model 0 78
MathematicalDefinitions, Theorems, Corollaries, andLemmas xi
5.4-Gaussian semivariogram model 78
5.5-Power semivariogram model 79
5.6-Cubic semivariogram model 79
5.7-Pentaspherical semivariogram model 79
5.8-Sine holeeffect semivariogram model 79
5.9-Pure nugget effect semivariogram model 80
6.1-Universal krigingestimator 91
6.2-Drift 92
6.3-Polynomial drift model 93
6.4-Lagrange function for universal kriging 98
6.5-Semivariogram matrix in universal kriging 102
6.6-Vector ofunknowns for intrinsic universal kriging 102
6.7-Semivariogramvector in universal kriging 102
6.8-Universal kriging covariance matrix 105
6.9-Vector ofunknowns for stationary universal kriging 106
6.10-Universal kriging covariance vector 106
6.11-Samplingvector in universal kriging 106
8.1-Drift estimator 129
8.2-Lagrangian function 133
8.3-Drift vectorofmonomials 136
8.4-R.esidual estimator 137
9.1-LU decomposition ofcovariance matrix 156
10.1-Indicator 167
12.1-Blockaverage 187
12.2-Block krigingestimator 188
12.3-Point-to-block covariance 189
12.4-Block-to-blockcovariance 189
12.5-Lagrangian function for block kriging 193
12.6-Point-to-point covariance matrix 196
12.7-Blockkriging vector ofunknowns 196
12.8-Point-to-block covariancevector 196
12.9-Block krigingsampling vector '" 196
13.1-Vectorial random function 210
13.2-Cokrigingweight matrix 210
13.3-0rdinarycokriging estimator 210
13.4-Cokrigingvector ofmeans 210
13.5-Covarianceofa vectorial random function in ordinary
cokriging 210
13.6-0rdinarycokriging estimation variance 213
13.7-Traceofa square matrix 213
13.8-Lagrangian function for ordinary cokriging 217
13.9-Covariance matrix in ordinary cokriging 220
13.10-Ordinarycokriging covariancevector 220
13.11-Ordinary cokrigingvector ofunknowns 220
13.12-Positivesemidefinite matrix 222
xii Ceostatistics for Engineers& Earth Scientists
13.13-Cross-semivariogram 222
13.14-Linear coregionalization model 222
13.15-Additive log-ratio transformation 234
13.16-Additive generalized logistic back transformation 234
14.1-Error sum ofsquares in cluster analysis 240
14.2-Within-group error sum ofsquares in cluster analysis 240
14.3-Mahalanobis'distance 244
List of Theorems
2.1-Estimation variance for simple kriging 13
2.2-Normal equations and nonnegative estimation variance for
simple kriging " 14
2.3-Minimum mean square error for simple kriging 16
2.4-Unbiased simple kriging estimator 20
2.5-Simple kriging exact interpolation 21
2.6-Orthogonality ofestimates and errors 22
4.1-Estimation variance for ordinary kriging " 43
4.2-Normal equations and nonnegative estimation variance for
intrinsic ordinary kriging 45
4.3-Minimum mean square error for intrinsic ordinary kriging 46
4.4-Normal equations for second order stationary ordinary kriging 49
4.5-Minimum mean square error for second order stationary
ordinary kriging 50
4.6-Difference between simple and ordinary kriging weights 60
4.7-Difference between simple and ordinary kriging estimation
variance 61
5.1-Estimation variance for assigning one variate to another 68
6.1-Unbiasedness conditions for the universal kriging estimator 95
6.2-Estimation variance for universal kriging 97
6.3-Normal equations and nonnegative estimation variance for
intrinsic universal kriging 98
6.4-Minimum mean square error for intrinsic universal kriging 101
6.5-Normal equations for second order stationary universal
kriging 103
6.6-Minimum mean square error for second order stationary
universal kriging 105
8.1-Unbiasedness conditions for drift estimator 131
8.2-Estimation variance for drift estimator 133
8.3-Normal equations for drift estimation 134
8.4-Minimum mean square error in drift estimation 136
9.1-Drawing from multivariate distributions 143
9.2-Normal distribution oferrors 143