Geostatistics with Applications.In

Earth SciencesSecond Ed it ion

Geostatistics with Applications•In

Earth SciencesSecond Edition

By

D.O. SarmaFormerly at Nationa l Geophysical Research Institute

(Council of Scientific and Industrial Research)Hyderabad, India

~ Springer

tP

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Preface to the Second Edition

Geostatistics is expanding very fast: concept- and technique-wise. Keepingin view the importance of the subject, it was thought approp riate to bring outthe second edition of this book. In this process, Chapter I has been expandedincorporating more detail s on sampling and sampling designs. In Chapter 2,a section on simulation has been introduced with emphasis on Monte-Carlosimulation with worked out examples. In Chapter 5, a procedure to computevariogram in the case of irregular grid has been outlined. Minor modific ationshave been made in all other chapters. A new chapter on Introduction toAdvanced Geostat istics has been introduced with discussions on unive rsalkriging , disjunctive kriging, cond itional simulation and median polish kriging.Review Questions are given at the end of each chapter to facilitate a bette runderstanding of the subject by the student/practitioner. The software codesare put in a CD for convenience of the students/practitoner of geostatistics.A few add itions have been made in the bibl iog raphy making it moreexhaustive. This contains references to the concepts and method s presented,in-depth treatment of related topics and possible extensions . My gratefulthanks are due to Dr. B.S. Saini , Principal , Guru Nanak Engg . College,Hyderab ad for very helpful support. I hope that this edition will be a welcom eone.

August 2008 D.O. Sarma

Preface to the First Edition

This book has been designed to serve as a text book for post graduatestudents and research workers in earth sciences who require a backgroundof and a feel for Statistics and the Theory of Regionalised Variables . Thebook is titled 'Geostatistics with Applications in Earth Sciences ' . Althoughthe word geostatistics is used throughout Europe signalling the Theory ofRegionalised Variables as propounded by Prof. George Matheron and hiscolleagues at the Centre de Geostatistique, Fontainebleau, France, still it wasconsidered necessary to include in this book some important classical statisticalmethods which are essential for modelling the processes concerning earthresource systems for optimum appraisal. Thus , Chapters I to 4 deal with theclassical statistical methods including a discu ssion on Box-Jenkins modelsof Time Series Analysis and Chapters 5 to 8 deal with a discussion on theTheory of Regionalised Variables and restricted upto Kriging (StationarityCase) . Chapter 9 deals exclusively with the software developed for some ofthe problems. Practical application of these methods in earth sciences isexplained at every stage .

In all, it is hoped that this book would serve as a practical guide togeostatistics. The units of measurement used in the examples cited in the textare the real ones. No attempt has been made to convert non-metric units intometric units .

I wish to express my grateful thanks to Dr. Harsh K. Gupta , Secretary,Dept. of Ocean Development, Govt. of India and former Director, NationalGeophysical Research Institute for the facilities provided to me in thecompletion of this project and for the Foreword; to Dr. Hari Narain , FormerDirector, National Geophysical Research Institute, Hyderabad, Former Vice­Chancellor, Banaras Hindu University, Varanasi, Former Surveyor-General,Survey of India and Member, Advisory Council , Directorate General ofHydrocarbons, GOI for the Preface. The Chairman and Managing Directorof Bharat Gold Mine s Ltd., Kolar gold field s, Karnataka, the Chairman andManaging Director of Chitradurga Copper Corporation , Chitradurga,Karnataka, the Chairman and Managing Director of Hutti Gold Mine s Ltd. ,Hutti , Karnataka, the Chairman and Managing Director of Hindustan Zinc

x Preface to the First Edition

Ltd., Udaipur , Rajasthan , and the Director-General of the Geological Surveyof India have provided with the necessary assay data for stochastic andgeostatistical modelling studies carried out by me at the National Geophys icalResearch Institute. I express my grateful thank s to all these autho rities.Acknowledgements are due to my colleagues, Mr. N.H. Prasada Rao and Mr.1.B. Selva raj for their help in the finali sation of the software programs listedin this volume. Mr. G.R.K. Rao and Mr. C. Shyam Sunder have done anextremely good job in text processing. Mr. M. Jayarama Rao, Mr. O. PrasadaRao of the Maps & Drawings section of NGRI have given their support intracing the figures listed in the text. Any shortcomings are due to me.

November 200 I D.O. Sarma

Contents

Preface to the Second Edition VII

Pref ace to the First Edition IX

Some Important Symbols Used in the Text xv

1. Statistical Methods in Earth Sciences Il.l Introduction I

1.1.1 Sampling, Data Collection and Sample Design I1.1.2 Sample Design and the Various Steps 21.1.3 Criteria for Selecting/Drawing a Sample 31.1.4 Characteristics of a Good Sample Design 31.1.5 Different Types of Sample Design 31.1.6 Analy sis Aspects 5

1.2 Hypothesi s 101.3 Quantification and Prediction in Earth Sciences II1.4 The Concept of Random Variable 121.5 Probability 121.6 Frequency Function, Joint Frequency Function ,

Continuous Frequency Function and Joint ContinuousFrequency Function 13

Review Questions 15

2. Univa riate Statistical Methods, Frequency Analysis and Simulation 162.1 Univari ate Statistical Method s 162.2 Frequency Analysis 162.3 Graphical Representation of Frequency Distribution 182.4 Arithmetic Representation of Empirical Distributions 29

2.4.1 Measure of Central Location 292.4.2 Measures of Dispersion 302.4.3 Skewness and Kurtosis 30

2.5 Correlation and Regression 312.5.1 Correlation Coefficient 322.5.2 Regre ssion 33

xii Contents

2.6 Simulation 352.6.1 Introduction 352.6.2 Advantages of Simulation 362.6.3 Limitations of Simulation Techniques 362.6.4 Generation of Random Numbers 372.6.5 Monte-Carlo Simulation 37

2.7 Some Applications of Simulation 372.7.1 Applications to Inventory Control 372.7.2 Applications to Gold Mineralisation 40

Review Questions 41

3. Some Statistical Distributions 423.1 The Normal Distribution 42

3.1.1 Salient Features of Normal Distribution andNormal Probability Law 42

3.1.2 Confidence Limits for the Mean (Z) 443.2 The Lognormal Distribution and Properties 45

3.2.1 Estimates for the Mean 453.2.2 Confidence Limits for the Mean 47

3.3 The Chi-square (X2) Test 493.4 Applications 50

3.4.1 Bauxite Example: Distribution of Fep3 Element 513.4.2 Gold Ore Distribution 523.4.3 Copper Example 55

3.5 Case of Rejection of Normal Distribution 60Review Questions 61

4. Stochastic Modelling (Time Series Analysis) and Forecasting 624.1 Introduction 62

4.1.1 Stochastic Processes 624.1.2 The Autocorrelation Function (acl) 63

4.2 Stochastic Modelling (Time Series Analysis) 644.2.1 Physical Significance in Relation to Estimation

of Blocks of Ore 654.2.2 Estimation of Parameters of A.R Proce ss of Order p

~R~] ~

4.2 .3 Moving Average Process [MA(q)] 674.2.4 Auto-regressive and Moving Average Proces s of

Order p, q [ARMA(p, q)] 674.3 Applications 68

4.3.1 Bauxite Example (Fep3) 684.3.2 Gold Mineralisation 694.3.3 Copper Example 71

4.4 Spectral Analysis (Frequency Domain) 714.4 .1 Spectrum, Discrete Fourier Transform (DFT) and

Fast Fourier Transform (FFT) 71

Contents xiii

4.4 .2 Maximum Entropy Method 744.4.3 Spectral Density and Entropy 74

Review Questions 77

5. Concepts of Regionalised Variable s and Variogram Modelling 785.1 Introduction 785.2 Stationarity and Intrinsic Hypothesis 79

5.2 .1 Stationarity 795.2.2 Intrin sic Hypothesis 805.2.3 Stationarity in Actu al Practice 80

5.3 Variogram 805.3.1 Properties of Variogram 815.3 .2 Ani sotropies 825.3.3 Some Practical Points on Variograms 845.3.4 Presence of a Drift 845.3.5 Proportional Effect 855.3.6 Other Features 85

5.4 Commonly Used Variogram Models 865.5 Change of Support, Regul arisation and Estimation Variance 875.6 Examples of Variogram Computation 895.7 Examples of Variograms in Other Fields of Earth Sciences 925.8 Computation of Variogram in the Case of Irregular Grid 94Review Questions 94

6. Regularised Models , Volume-Variance Relationshipsand Economics 95

6.1 Introduction 956.2 Different Situations 976.3 Steps to Be Followed for the Deconvolution Probl em 1006.4 Example: F C203 Element Values for Core Length of

L = 0.5 m 1006.5 Dispersion vs Block size 105

6.5.1 Example: Lode 0 : gold field I 1056.5.2 Example: Lode Z: gold field 2 106

6.6 The Within Variation in Core Length L: Diffe rent Case s 1076.7 Distributions Based on Core Sample Stati stics and Derived

Ones for Point Samples 1086.8 Case of Lognormal Distribution and Blocks of Size V III

6.8.1 Distributi ons Based on Point Samples and DerivedStati stics for Blocks 112

6.8.2 Grade Tonnage Computations Based on Log-transformed Statisti cs and Economic Implications 11 3

Review Questi ons 115

xiv Contents

7. The Concepts of Dispersion, Extension and EstimationVariances 118

7.1 Variance of Dispersion 1187.1.1 Variance of Point Samples within Volume V 1187.1.2 Variance of v within V 120

7.2 Extension Variance 1217.3 Estimation Variance 123Review Questions 124

8. Kriging Variance and Kriging Procedure 1258.1 Towards Kriging Variance 1258.2 Kriging Procedure 1278.3 Kriging System and Kriging Variance in Terms

of y Notation 1308.4 Simple Kriging (also known as Linear Kriging with

Known Expectation) 1308.5 Examples 131

8.5.1 Punctual Kriging 1318.5.2 Block Kriging 133

Review Questions 138

9. Introduction to Advanced Geostatistics 1399.1 Introduction 1399.2 Non-stationary Geostatistics 140

9.2.1 Universal Kriging 1409.2.2 Disjunctive Kriging 1419.2.3 Disjunctive Kriging Estimator 142

9.3 Estimation Based on Conditional Simulation 1439.4 Kriging Nonstationary Data-the Median Polish Method 144

9.4.1 Median Polish Kriging 1449.4.2 Example 145

Review Questions ISO

10. Computer Software lSIPrograms: NORMAL.FOR, LN.FOR, AR.FOR,

MAI.FOR, VGRAM.FOR, ORDKRIG.FOR

Bibliography 195

Index 203

Some Important Symbols(lsed in the Text

y(h)

a

(J

g

y

y

semi-variogram between two points separated bydistance h

semi-variogram between two cores each of length Lseparated by distance h

experimental semi-variogram based on point samples

expe rimental semi-variogram based on core sample s

range of influence of a semi-variogram

sill of a semi-variogram

nugget effect

slope of the linear semi-variogram

sample mean

population mean

standard deviation of x

population std. deviation

mean value of the observations/grades

logarithm of the variable

mean value of the logarithms of observations

standard deviation of the logarithms of observations

standa rd normal deviate in the context of confidence

limits for the mean Zstandard normal deviate in the context of confidence

limits for the mean ( x) of logarithms of data.

average grade above cutoff-E

standard error

xvi Some Important Symbols Used in the Text

cr~Re.V

R.V

RF

E{Z(x)}

C(h) ,cr(h)

A(

V(x)

v(x)

cry,' (v/V)

!J.P{Z = Zi}

!J.kr(x, y)

M/t)

A(z)

M

Pk

<1>1 ' <1>2' ... <l>k

<l>kk

SF(f)

UE(f)

UM

Z

cr2(0/L )

kriging variance

Regionalised Variable

Random Variable

Random Function

expectation of Z(x)

stationary covariance function

weights assigned to various samples in the context ofKriging

domain V centered at x

smaller domain v centered at x

dispersion variance

Lagrangian parameter

probability of Z taking value zi

kth moment about the mean

correlation coefficient between x and y

moment generating function

denotes lognormal frequency function

in the context of lognormal theory, denotes populationmean

in the context of moving average process, denotes theparameter

autocorrelation coefficient at lag k

auto-regressive coefficients

partial autocorrelation coefficient at lag k

spectral density estimates by FFT method

spectral density by maximum entropy method

updated variance

in the proper context, denotes Regionalised/RandomVariable

the value of the regionalised variable at each datapoint Xi

sill value of the variogram with core samples of lengthL as samples

within variation in core of length L

standard deviation of point samples

standard deviation of samples with volume v

i(o/v)y(x; ,V)

y ( v ,v)

y(V,V)

Vj

Some Important Symbols Used in the Text xvii

sample variance of point samples in volume V

average variogram between Xi and the volume V

average value of the variogram between any two pointsX and x' sweeping independently throughout thevolume v

average value of the variogram between any two pointsy and y' sweeping independently throughout the volumeV

variance of v in V

Extension Variance. Error committed when the gradeof a sample of volume v is extended to the grade ofvolume V

covariance between neighbourhood samples i and j.

for all j