mining geostatistics.pdf

GEOSTATISTICS

.

A.G. JOURNEL AND. .

Ch. J. HUIJBREGTSACADEMIC PRESS @

Mining GeostatisticsA. G. JOURNEL

andCH. J. HUIJBREGTS

Centre de GeostatisriqueFontainebleau, France

ACADEMIC PRESSHarcourt Brace Jovanovich, Publishers

London . San Diego . New York BerkeleyBoston . Sydney . Tokyo . Toronto

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ACADEMIC PRESS LIMITED24-28 Oval Road

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United States edition published byACADEMIC PRESS INC.

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Copyright 1978 byACADEMIC PRESS LIMITEDReprinted with corrections 1981

Third Printing 1984Fourth Printing with corrections 1989

All Rights Reserved

No part of this book may be reproduced in any form by photostat, microfilm,or any other means, without written permission from the publishers

Library of Congress Catalog Card Number: 77-92823ISBN Hardback: 0-12-391050-1ISBN Paperback: 0-12-391056-0

Printed in Great Britain by St Edmundsbury Press LimitedBury St Edmunds, Suffolk

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Forewordby G. MA THERON

The distribution of ore grades within a deposit is of mixed character, beingpartly structured and partly random. On one hand, the mineralizing processhas an overall structure and follows certain laws, either geological ormetallogenic; in particular, zones of rich and poor grades always exist, andthis is possible only if the variability of grades possesses a certain degree ofcontinuity. Depending upon the type of ore deposit, this degree of continuitywill be more or less marked, but it will always exist; mining engineers canindeed be thankful for this fact because, otherwise, no local estimation and,consequently, no selection would be possible. However, even thoughmineralization is never so chaotic as to preclude all forms of forecasting, it isnever regular enough to allow the use of a deterministic forecasting tech-nique . This is why a scientific (at least, simply realistic) estimation mustnecessarily take into account both features - structure and randomness ~inherent in any deposit. Since geologists stress the first of these two aspects,and statisticians stress the second, I proposed, over 15 years ago, the namegeostatistics to designate the field which synthetizes these two features andopens the way to the solution of problems of evaluation of mining deposits.

Practicians are well aware that such problems are numerous and oftenvery complex. They sometimes infer from this that the correspondingmathematical theory is necessarily very complicated. Actually, they shouldset their minds at ease, for this is luckily not the case . The mathematicalformalism necessary for geostatistics is simple and, in some cases, even trite.The user of A. Journel and Ch. Huijbregts' book will come across nothingmore than variances and covariances, vectors and matrices; in short, onlylinear calculations comparable to those which any computer user manipu-lates every day. After the preliminary effort of familiarizing themselves withthe notations, they will note, possibly to their surprise, that they can followall of the authors' steps withou t difficulty.

I might address a symmetric warning to statisticians. Those unfamiliarwith the complexity and specificity of mining problems might find this

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VI FOREWORD

treatise banal and of little interest. I feel that they would be mistaken. At thispoint, a comparison with geophysics could be instructive (besides, the veryterm "geostatistics" was formed after the model of "geophysics" or of"geochemistry' ').

If a theoretical physicist were to read a treatise on geophysics, he wouldrisk getting the disappointing impression that it is but a collection ofparticular cases of the theories of potential and of wave propagation: a bit ofgravitation and magnetism, quite a lot of electromagnetism, acoustics, etc.,nothing very original, really. And yet, without special training, that veryphysicist, be he a Nobel prize winner, would admit to be unable to interpretthe results of a seismic prospection campaign.

In the same way, a specialist in mathematical statistics, having pagedthrough the present book, would not find much more than variances,covariances, optimal linear estimators, etc., and would find no justificationin this for creating a separate field. However, let us submit to this specialistthe following problem, which is quite a standard one for the mining engineer(in geostatistical terminology, it is the problem of the estimation of recover-able reserves).

"We have, at a certain depth, an interesting copper deposit. The geolo-gists who have studied the deposit have arrived at certain conclusions aboutits genesis and structure, which you will find in this report. The availablenumerical data consist of a square grid of drill -holes 200 m apart. The coresections were analysed metre by metre, and, thus, we have a total of some5000 chemical analyses. We are considering open-pit selective mining. Thismeans that, during the mining of our approximately 200 Mm3 deposit, whichhas been mentally divided into 50 000 blocks 20 m x 20 m x 10m, we havethe choice of sending each newly excavated block either to waste , or to theprocessing plant. Taking account of an economic pre-feasibility study, wehave decided that the blocks with a mean ore grade greater than x = 0.4%Cu should be sent to the plant, and that the rest should be set aside with thewaste . This cut-oft grade x is subject to revision, since the market price ofcopper as well as the mining and processing costs can vary. x should beconsidered as a parameter, and the evaluations should be made undervarious hypotheses, i.e., with x =0.3%,0.4%,0.5% , for example.

"Moreover, we must consider the fact that our block by block decision isbased, not on the unknown true block grade but on an estimator, thisestimator being based on information which will be much richer than thatnow available (core drillings on a 200 ill X 200 m grid): at the time ofselection, we shall have the results of the chemical analyses of blast holes ona 5 m x 5 m grid. Based upon this Anal information, we shall build anestimator z 1 for each of our blocks, and we shall send this block to the plantonly if its estimated grade z i is greater or eq ual to the cut-off grade .r.

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FOREWORD VII

"We would like to know, as a function of x, the tonnage T(x)ofthe finallyselected blocks, and their mean grade m (x); this time, of course, we arereferring to the mean value of the true mean grades of the selected blocks,and not of their estimated mean grades z": On top of this, the evaluation,based upon the rather poor information we have now (drilling on a 200-mspacing), will inevitably be erroneous. So we shall also want to know theorders of magnitude of the errors committed in the evaluation of T(x) andm (x), for example through their estimation variances."

In the very same way our Nobel prize physicist above admits being unableto interpret the results of a seismic prospection campaign, I think that astatistician not versed in mining practice would honestly have to admit hisinability to give a realistic solution to the above problem. Actually, this is arather simple and favourable case: only one metal, a well-explored depositwith a regular sampling grid, no missing data points nor preferential orbiased data, etc. One should also note that the mere statement of theproblem itself takes up about a page. A statistician who is not familiar withmining may well be discouraged before he can even get a good idea of theproblem at hand.

In fact , the difficulties are several. First of all, there is a violent contrast inmasses: even though the experimental data are numerous (5000 samples),they represent but a minute part of the total mineralized mass (about SOtonnes out of hundreds of millions). Then there is a volume (or support)effect: the 20 m x 20 m x 10m blocks are not at all distributed (in the spatialsense as well as in the statistical sense) in the same way as the kilogram-sizecore sections. Finally, there is the fact that the criteria to be used for thefuture selection of blocks will not be based on the true grades of the blocks,nor upon the data now at hand, but on an intermediate level of information(the blast-holes). One can, thus, understand why the need to solve suchproblems would instigate, over the years, the creation of a field as specializedas geophysics, with its own terminology and special techniques - a fieldwhich carries the name of geostatistics.

Andre Journel and Charles Huijbregts, who have worked at the Centre deGeostatistique at Fontainebleau since its creation in 1968, are surely amongthe persons who have contributed the most to elevating this field to itspresent level of practical and operational efficiency. They have personallyachieved studies and estimations upon more than 30 ore deposits of the mostdiverse types. The vast experience which they thus accumulated, and thedose and permanent contact they maintained with the everyday realities ofmining, account for the primarily operational and concrete nature of thisbook. Here is a practical treatise , written by practising mining engineers, forthe benefit of other practicians, The authors have deliberately ~ and justlyso - simplified to the utmost the purely mathematical part of their expose,

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viii FOREWORD

while remammg as rigourous as possible. They have concentrated theirefforts upon two main points: first of all, clearly define the problems at hand,and we have seen that this is no mean task; second, show how one should goabout solving them in practice. Practicians will appreciate this doubleapproach: clarifying the objectives and efficient use of the techniques.Moreover, the mathematical formalism is laced with concrete commentswhich make intuitive sense to the practician. Each time a new notion isintroduced, its definition is immediately followed by examples that illustrateits meaning.

This book is structured according to the most typical problems whichoccur in practice, progressing from the simplest to the most complicated:structural analysis, estimation of in situ resources, estimation of recoverablereserves, simulation of deposits. In each instance, several examples ofapplications to actual deposits of the most diverse types are given: thepractician will be led by the hand through the labyrinth and forewarned oftraps and accidents which lurk at each corner.

In Chapter VII, devoted to conditional simulations, the strictly "linear"code of the first six chapters is dropped. The Object of such a simulation is tobuild a numerical model which has the same structure as the real deposit.More precisely, the numerical model must have the same histogram, vario-gram and cross-correlations as the real deposit, and must coincide with theexperimental data values at the locations where they were taken. Theauthors describe the novel technique developed by the Centre de Geostatis-tique. These conditional simulations will be used more and more often tosolve difficult non-linear problems which defy the use of any other technique(e.g., forecasting the fluctuations of ore characteristics at the mill feed,homogenization obtained after judicious ore blending and/or stockpiling,etc.). Here, geostatistics no longer confines itself to predicting once and forall the recoverable reserves, but helps in optimal planning and control of themining operations, at the daily, monthly or yearly scale. The last chaptergives general indications about "non-linear" geostatistics, which is stillunder elaboration.

In summary, A. Journel and Ch. Huijbregts' book constitutes a systematicstate-of-the-art account of geostatistics and traces its progress over the lastten years (progress to which the authors have largely contributed). Thistreatise, intended for practicians, is surely destined to become the referencetext for the field during the 1980's.

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Contents

Foreword by G. Matheron vContents xi

INTRODUCTION 1

I. GEOSTATISTICS AND MINING APPLICATIONS 7Summary 7

LA. The steps involved in proving a deposit 7I.E. The geostatistical Ianguage 10I.e. Some typical problems and their geostatistical

approach 18

II. THE THEORY OF REGIONALIZED VARIABLES 26Summary 26

II.A. Regionalized variables and random functions 27(I.B. Estimation variance 48I1.c. Dispersion variance 61I1.D . Regularization 77I1.E. Calculation of the mean values y 95

III. STRUCTURAL ANALYSIS 148Summary 148

liLA. Nested structures and the nugget effect 148I1LB. Models of variograms 161IILe. The practice of structural analysis 195

IV. CASE STUDIES OF STRUCTURAL ANALYSIS 236Summary 236

IV.A. Fitting models 236IV.B. Hole effect and proportional effect 247IV.C. Coregionalizations 253IV.D. Anisotropies 262

IX

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x CONTENTS

IV.E. Various structural analyses 271IV.F. Grade control during production 293

V. KRIGING AND THE ESTIMATION OF IN SITURESOURCES 303Summary 303

V.A. Theory of kriging 304V.B. The application of kriging 343V.c. Variance of global estimation 410

VI. SELECTION AND ESTIMATiON OF RECOVERABLERESERVES 444Summary 444

VI.A. In situ resources and recoverable reserves 444VI.B. Permanence of distribution 466

VII. SIMULATION OF DEPOSITS 491Summary 491The objectives of simulation . 491VILA. The theory of conditional simulations 494VILB. The practice of simulations 517

VIII. INTRODUCTION TO NON-LINEAR GEOSTATISTICS 555Summary 555Why non-linear geostatistics? 556VIII.A. Kriging in terms of projections 557VIII.B. Disjunctive kriging and its applications 573

Bibliography 581

Index A, Geostatistical concepts 591

Index B. Quoted deposits 595Index C. FORTRAN programs 596

Index D. Notations 597


Introduction

This book is an attempt to synthetize the practical experience gained inmining geostatistics by the researchers from the Centre de MorphologicMathernatique in Fontainebleau, Francet and by mining engineers andgeologists around the world who were kind enough to let us know of theirexperience. This book is designed for students and engineers who wish toapply geostatistics to practical problems, and, for this reason, it has beenbuilt around typical problems that arise in mining applications. SeveralFORTRAN programs are given, and the techniques developed are illus-trated by a large number of case studies. A modest knowledge of integralcalculus, linear algebra, statistics, notions of stochastic processes, variance,covariance and distribution functions is required, as well as a geologicaland mining background. With this background, the elements of mininggeostatistics can be understood and the approximations necessary for anexperimental approach can be accepted.

Definition and basic interpretation of geostatisticsEtymologically, the term "geostatistics" designates the statistical study ofnatural phenomena. G. Matheron (1962) was the first to use this termextensively, and his definition will be retained: "Geostatistics is the appli-cation of the formalism of random functions to the reconnaissance andestimation of natural phenomena".

A natural phenomena can often be characterized by the distributionin space of one or more variables, called "regionalized variables". Thedistribution of grades in the three-dimensional space, for example,

t Namely and particularly 1. Serra, A. Marechal, J. Deraisme, D. Guibal. Special thanks to P.A. Dowd who contributed extensively to the English version from the original French draft.

1

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2 MINING GEOSTATISTICS

characterizes at least some of the aspects of a mineralization. The dis-tribution of altitude in a horizontal space characterizes a topographicalsurface.

Let z (x) be the value of the variable z at the point x. The problem is torepresent the variability of the function z(x) in space (when x varies). Thisrepresentation will then be used to solve such problems as the estimation ofthe value z(xo) at a point Xo at which no data arc available, or to estimatethe proportion of values z (x ) within a given field that are greater than agiven limit (e.g., cut-off grade).

The geostatistical solution consists of interpreting each value z (Xi) (z inlower case) as a particular realization of a random variable Z(Xi) (Z inupper case) at the point Xi. The set of these auto-correlated randomvariables {Z (x), when x varies throughout the deposit or domain D}constitutes a random function. The problem of characterizing the spatialvariability of z (x) then reduces to that of characterizing the correlationsbetween the various random variables Z (Xi), Z (Xi) which constitute therandom function {Z (x), xED}. This fundamental interpretation is justifieda posteriori if it results in coherent and acceptable solutions to the variousproblems encountered in practice.

The adaptability of geostatisticsGeostatistics, and the probabilistic approach in general, is particularlysuited to the study of natural phenomena. Among the many techniques andmathematical models available to the practitioner of geomathematics, aparticular technique or approach will be chosen if, from experience, it isknown to be best suited to the problem concerned. A Fourier analysis willsuit one problem. while, in other cases, geostatistical tools are best suited;in some cases, a mathematical formulation may even be rejected for simplequalitative experience.

In the mining field, geostatistics provides a coherent set of probabilistictechniques, which are available to each person (geologist, mining engineer,metallurgist) involved in the mining project; the proper use of these tech-niques depends on these people. Just as the efficiency of the axe and thesaw depends on the arm of the lumberman, so the efficiency of anygeostatistical study depends on the liaison between the various people anddepartments involved. The proper conclusions of an analysis of spatialvariability will be reached only if there is a close contact with the geologistor, better still, if the analysis is carried out by the geologist himself. Theevaluation of recoverable reserves has no meaning unless technicallyfeasible processes of selection and mining are clearly defined; in other


INTRODUCTION 3

words, the mining engineer must be involved. Outside the mining field,geostatistics applied to forestry requires the ability to interpret spatialcorrelation between the densities of growth of different species of trees;geostatistics applied to bathymetry must take into account all the problemsrelated to the localization of the hydrographic survey ship at sea. Thus, abook on geostatistics cannot be dissociated from its practical context, inthis case mining.

This instrumental nature of geostatistics both opposes and completes theexplanatory aspects of the standard, deterministic , geological approach .For example, even if a metallogenic study is sometimes successful inexplaining the overall genesis of a given deposit , such a deterministicapproach wil1 be clearly insufficient on a smaller scale, such as that of metalconcentrations. In such a case, it is said that the grades are "randomly"distributed within the defined genetic outline.

But nothing is more precise than this "random" distribution. It will differfrom a gold placer deposit to a porphyry-copper deposit, and the randomdistribution of the grades within the deposit will condition its survey and itsselective mining, which represent considerable investments. On the smallerscale of mining, the probabilistic approach can relay the deterministic andgenetic approaches by which the deposit was delineated on a larger scale.Our lumberman is not a botanist; similarly, the mining geostatisticianaccepts the reality imposed by nature, and is content to study theconsequences of this reality on the mining project he must prepare. Aprobabilistic approach can, of course, be guided by genetic considerations,and the best mining geostatistician is the one who is able to combine hisgeological knowledge and his technical mining skill with a good use of theprobabilistic language.

Plan of the bookThe acceptance of geostatistics, and mining geostatistics in particular, restson the coherence and effectiveness of the solutions it provides to thevarious problems encountered in practice. Thus, the primary aim of thisbook is to convince the various people involved in a mining project of thepractical interest of geostatistics, An engineer will not be convinced in thesame manner as a mathematician or a researcher; he will first want to knowhow the techniques have fared in practice. For this reason, before begin-ning any theoretical development, a practical presentation of geostatistics,by way of its mining applications, is given in Chapter 1. This chapter can beread separately from the rest of the book, and is intended as a resume forreaders who are pressed for time, and anxious to make a quick decision onthe usefulness of the proposed techniques. In the remainder of the book,



each new concept is introduced by a practical example drawn from thestudy of a real deposit.

The theoretical structure of the book begins with Chapter It whichpresents the theory of regionalized variables and the main tool of geo-statistics - the variogram - and the two variances of estimation and dis-persion.

Any geostatistical study begins with a structural analysis, which consistsof modelling the in situ spatial variabilities of the parameters studied, e.g.,grades, thicknesses, accumulations. Chapters III and IV show how a struc-tural analysis is carried out in practice from the construction of variogramsto their interpretation and fitting by models. Both chapters insist on the"skilled" nature of the structural analysis. Even though many computerprograms have been written for this purpose, a structural analysis mustnever be carried out blindly; it must integrate the qualitative experience ofthe geologist and carry it through to the subsequent resource-reservesestimation and the eventual planning of the mining project.

It is obvious that a massive porphyry-copper deposit cannot be estimatedin the same way as a sedimentary phosphate deposit. The estimationprocedure must take into account both the structure of the spatial vari-ability peculiar to each deposit, and the particular manner in which thedeposit was surveyed. It is only by taking these peculiarities into accountthat a confidence interval can be ascribed to each estimated value. Krigingtakes these factors into account, and is presented in Chapter V, along withthe geostatistical approach to the estimation of the in situ resources.

The next step is to evaluate the proportion of these in situ resources thatcan be recovered within a given technological and economical framework .This estimation of recoverable reserves must take the nature of the pro-posed selection and mining methods into account. Chapter VI presents thegeostatistical formalization of these problems of selection and estimationof recoverable reserves.

Since geostatistics interprets the regionalized variable z (x) as a partic-ular realization of a random function Z (x), the possibility arises of drawingother realizations of this same random function, in much the same way ascasting a die several times. This new, simulated realization represents anumerical model of the real spatial variability z (x). Within certain limits,the consequences of various planned mining methods can then beexamined by simulating these methods on the numerical model. The tech-niques for simulating deposits and the practical uses of these simulationsare given in Chapter VII.

Each chapter begins with a summary of its contents, so as to allow thereader to sort out items of particular interest. For this purpose, the readercan also make profitable use of the indexes of geostatistical concepts,

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INTRODUCTION 5

quoted deposits and FORTRAN programs. At a first reading, the reader isadvised to omit the various theoretical demonstrations. These demon-strations are necessary for the logical coherence of the book, but not for apreliminary overall understanding of geostatistics.

First notations

For the uninitiated, one of the main difficulties of geostatrsncs is thenotation. These notations are precisely defined as they enter into the work,and are summarized in Index D. However, it may prove helpful at thisstage to introduce the following basic notations.

A deposit is always a regionalized phenomenon in the three-dimen-sional space. To simplify notation, the coordinates (xu, xv, x.,) of a point areusually denoted x. Thus, the integral (extended over the three-dimensionalspace) of the function [(Xu., xv, x.,) is written as

f+OO r f+> r.-co I(x) dx = -co dx, -:x;l dx, -co [(x"' Xli' x.,) dx w The vector with three-dimensional coordinates (h u , hv , h w ) is denoted h,

with a modulus

and direction Cf. Note that a direction in a three-dimensional space isspecified by two angles, its longitude and its latitude.

Lower-case letters, z or y, are reserved for real values as, for example,the grade z(x) at point x or the density y (x) at the same point. Thecorresponding upper-case letters, Z or Y, denote the interpretation of thisreal value as a random variable or function, as, for example, the randomfunction Z(x) with expectation E{Z(x)}. The real value z(x) is said to be aparticular realization of the random variable or function Z (x).

An estimated value is distinguished from a real value. known orunknown, by an asterisk *. Thus, z *(x) represents the estimated value ofz(x), and Z*(x) represents the interpretation of this estimator as a randomvariable or function.

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IGeostatistics and Mining

Applications

SUMMARY

This chapter is intended to be both an introduction to, and an overview of,geostatistics as applied to the mining industry. Each successive problem encoun-tered from the exploration phase to the routine operating phase is presented alongwith its geostatistical approach. The problems of characterizing variability struc-tures of variables of interest within the deposit are solved with the "variogram",Optimal local estimates with corresponding confidence intervals are found by"kriging". The choice of various working techniques (machinery, need for stock-piling or blending) can be facilitated by simulating them on a geostatisticallydetermined numerical model of the deposit.

The main steps in proving a deposit are outlined schematically in section LAalong with some typical evaluation problems arising from such steps.

Section LB introduces the geostatistical terminology from a practical and intui-tive point of view. Such key notions as regionalized variables, variogram, estima-tion and dispersion variances, are presented.

The geostatistical approach to several of the problems encountered in section LAis given in section I.e, which also outlines the sequence of steps in a geostatisticalstudy.

I.A. THE STEPS INVOLVED IN PROVING A DEPOSIT

In this section, we shall consider the various steps involved in proving adeposit from prospection to production. The sequence of these steps andthe critical nature of certain problems will obviously depend on the type ofdeposit.

7



Geological surveys

A preliminary survey of a province or favourable metallogenic environ-ment, by such various means as geological mapping, geochemistry orgeophysics, will usually result in the location of a certain number ofprospects. If one of these prospects locates a mineralized zone, the nextstep is to determine its value and extent by a more detailed survey in-volving, for example, drilling, trenches or even, sometimes, drives. Thelocations of these samples are generally not regular, but are based ongeological considerations such as the need to verify the existence of themineralization and to provide a qualitative description of it (e.g.,mineralogy, tectonics, metallogenic type). The results of such samplingmay be-a proper basis on which to plan further reconnaissance, but they arenot entirely suited for quantitative estimations of tonnage and yardage.The first estimations will tend to be more qualitative than quantitative , aswill be thegeologist's image of the deposit.

Systematic survey on a large grid

The second stage of the survey usually consists of drilling on a large andmore or less regular grid. In general, this grid will cover what is consideredto be the mineralized zone and should be sufficient to evaluate the overallin situ resources (tonnages and mean grades). If promising results areobtained from this second stage, the survey may be extended to some in-filldrilling or by drilling in adjacent zones. At this stage, the geologist andproject engineer are faced with two fundamental quantitative problems.

The first one is how to deduce the required global estimations from theavailable information, which consists of qualitative geological hypothesesand quantitative data from core samples, trenches, pits, etc. In general,estimations of a geometric nature such as mineralized surface and yardagewill make more use of qualitative information than estimations of a"concentration" nature such as mean grades and densities.

The second problem is to express the confidence that can be given tothese estimations. In practice, it is common to classify the in situ resourcesin different categories such as, for example, those recommended by the USBureau of Mines: measured ore, indicated ore and inferred ore. Theboundaries between these categories do not always appear clearly. Thegeologist knows from experience that the precision of an estimation, suchas mean grade, does not only depend on the number of the available assaysbut also on the in situ variability of the grade in the deposit and the mannerin which this variability has been sampled. He also knows that the conceptsof continuity and zone of influence, which are used in the evaluation of


1. GEOSTATISTICS AND MINING APPLICAnONS 9

resources, depend on the particular deposit and the type of are considered.And, finally, he knows that for a given ore the variability of assays of coresamples, cuttings or channel samples can be markedly different.

In this chapter, it will be shown how geostatistics takes these variousconcepts into account by means of the variogram and structural analysis.

Sampling on a small grid

Once the deposit has been deemed economically mineable, the next step isto define the technological and economical framework for its exploitation.

The total in situ resources of a deposit are seldom entirely mineable,both for technological reasons (depth, accessibility) and for economicreasons (mining cost, cut-off grade). To define recoverable reserves withina given technological and economic context, or, conversely, to define thetype of mining which will result in maximum profit from the deposit it isnecessary to have a detailed knowledge of the characteristics of thisdeposit. These characteristics include the spatial distribution of rich- andpoor-grade zones, the variability of thicknesses and overburden, thevarious grade correlations (economic metals and impurities).

To determine these characteristics, it is usual to sample on a small grid,at least over the first production zone. The results of this samplingcampaign are then used to construct block estimations within this zone.These local estimations set the usual problem of determining estimatorsand evaluating the resulting estimation error. There will also be problemsassociated with the use of these local estimations: the evaluation ofrecoverable reserves depends on whether the proposed mining method isblock-caving with selection units of the order of 100 m x 100 ill X 100 m, oropen-pit mining with selection units of the order of 10 m x 10 ill xl 0 m.

From experience, the mining engineer knows that, after mining, theactual product will differ from the estimations of the geologist. In general,he knows that tonnage is underestimated and, more importantly, quality isoverestimated. In practice, this usually results in a more or less empirical"smudge factor" being applied to the estimation of recoverable reserves,which may reduce the estimated quantity of recovered metal from 5 to20%. This is not a very satisfactory procedure, because it does not explainthe reasons for these systematic biases between the geologist's estimationsand actual production; in addition, it is far too imprecise when evaluating amarginally profitable new deposit.

Given the size of present day mining projects and the cost of theirinvestments, the risks of such empirical methods are no longer acceptable,and it is imperative to set and rigorously solve such problems as those givenbelow.


MINING GEOSTATISTICS

(i) The definition of the most precise local estimator and its confidenceinterval, taking account of the peculiarities of the deposit and theore concerned. It is obvious that a gold nugget deposit cannot beevaluated in the same way as a sedimentary phosphate deposit.

(ii] The distinction between in situ resources and mineable reserves. Itshould be possible to model rigorously the effect of the miningmethod on the recovery of the in situ resources.

(iii) The definition of the amount of data (drill cores, exploration drives)necessary for a reliable estimation of these recoverable reserves.

Mine planning

Once the estimations of resources and recoverable reserves have beenmade, the next step is to plan the mining project itself. This involveschoosing mining appliances, type of haulage, requirements of stockpilingand blending, mill planning, etc. Some of these choices are conditional, inall or part, upon the structures of the variabilities of the are characteristics,either in situ or at any of the various stages of mining, haulage, stockpilingand mill feeding. It will be shown later how geostatistics can be used toprovide representations of these variabilities by means of numerical modelsof the, deposit. The effects of various extraction and blending processes canthen"be tested by simulating them on these numerical models.

LB. THE GEOSTATISTICAL LANGUAGE

I.B.1 Regionalized variable and random functionA mineralized phenomenon can be characterized by the spatial distributionof a certain number of measurable quantities called "regionalized vari-ables". Examples are the distribution of grades in the three-dimensionalspace, the distribution of vertical thicknesses of a sedimentary bed in thehorizontal space, and the distribution of market price of a metal in time.

Geostatistical theory is based on the observation that the variabilities ofall regionalized variables have a particular structure. The grades z(x) andz (x + h) of a given metal at points x and x + h are auto-correlated; thisauto-correlation depends on both the vector h (in modulus and direction)which separates the two points, and on the particular mineralizationconsidered. The variability of gold grades in a placer deposit will differfrom that of gold grades in a massive deposit. The independence of the twogrades z(x) and z (x + h) beyond a certain distance h is simply a particularcase of auto-correlation and is treated in the same way.

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r. GEOSTATISTICS AND MINING APPLICATIONS 11

The direct study of the mathematical function z (x) is excluded becauseits spatial variability is usually extremely erratic, with all kinds of dis-continuities and anisotropies. It is equally unacceptable to interpret all thenumerical values z (x), z (x') as independent realizations of the samerandom variable Z, because this interpretation takes no account of thespatial auto-correlation between two neighbouring values z(x) and z(x +h). Both the random and structured aspects of the regionalized variable areexpressed by the probabilistic language of "random functions".

A random function Z(x) can be seen as a set of random variables Z(Xi)defined at each point Xi of the deposit D: Z(x)={Z(Xi)' \;fXiED}. Therandom variables Z (x,) are correlated and this correlation depends on boththe vector h (modulus and direction) separating two points Xi and Xi + hand the nature of the variable considered . At any point Xi, the true gradez(Xj), analysed over a core sample, for example, is interpreted as oneparticular realization of the random variable Z (Xi)' Similarly, the set oftrue grades {Z(Xi), \iXi E D} defining the deposit or the zone D is inter-preted as one particular realization of the random function {Z (Xi), 'rIXi ED}.

The next section will deal with the operational aspect of this formaliza-tion by describing how the spatial variability of the regionalized variablez(x) can be characterized.

1.8.2. The variog ramConsider two numerical values z (x) and z (x + h), at two points x and x + hseparated by the vector h. The variability between these two quantities ischaracterized by the variogram function 2y(x, h), which is defined as theexpectation of the random variable [Z(x)-Z(x+h)J 2 , i.e.,

2)'(x, h) = E{[Z(x)-Z(x +h)J2}. (1.1 )

In all generality, this variogram 2'}'(x, h) is a function of both the point xand the vector h. Thus, the estimation of this variogram requires severalrealizations, [zk(x),zdx+h)], [zk(x),zdx+h)], ... , [Zk"(X),Zk,,(x+h)),of the pair of random variables [Z(x), Z (x + h)]. Now, in practice, at leastin mining applications, only one such realization [z(x), z(x+h)] is avail-able and this is the actual measured couple of values at points X and X +h.To overcome this problem, the intrinsic hypothesis is introduced. This hy-pothesis is that the variograrn function 2y(x, h) depends only on theseparation vector h (modulus and direction) and not on the location x. It isthen possible to estimate the variogram 2y(h) from the available data: anestimator 2'}'*(h) is the arithmetic mean of the squared differences between

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two experimental measures [z (X;), Z(Xi + h)] at any two points separated bythe vector h ; i.e.,

(1.2)

where N(h) is the number of experimental pairs [z (Xi), Z(X; + h)] of dataseparated by the vector h.

Note that the intrinsic hypothesis is simply the hypothesis of second-order stationarity of the differences [2 (x) - 2 (x + h)]. In physical terms,this means that, within the zone D, the structure of the variability betweentwo grades z (x ) and z (x + h) is constant and, thus, independent of x; thiswould be true, for instance, if the mineralization within D were homo-geneous.

The intrinsic hypothesis is not as strong as the hypothesis of stationarityof the random function 2 (x) itself. In practice, the intrinsic hypothesis canbe reduced, for example, by limiting it to a given locality; in such a case, thefunction y(x, h) can be expressed as two terms: )I(x, h) = I(x) . )lo(h),where yo(h) is an intrinsic variability constant over the zone D and [(x)characterizes an intensity of variability which depends on the locality x, d.the concept of proportional effect, treated in section IILB.6.

The variogram as a tool for structural analysis"Structural analysis" is the name given to the procedure of characterizingthe structures of the spatial distribution of the variables considered (e.g.,grades, thicknesses, accumulations). It is the first and indispensable step ofany geostatistical study. The variagram model acts as a quantified summaryof all the available structural information, which is then channelled into thevarious procedures of resource and reserve evaluation. Thus y(h) can beregarded as injecting extensive geological experience into the sequence ofstudies involved in a mining project.

In the proving of any mining project, it is imperative to have goodcommunication between all the departments involved (geology, miningtechniques and economy, processing). It is a waste, for instance, to have ageologist carry out an excellent study of geological structures, only to befollowed by a project engineer evaluating resources by such arbitraryprocedures as polygons of influence or inverse-squared distance weighting.Such evaluations are independent of the deposit and its geology, and makeno distinction, for example, between a gold placer deposit and a porphyry-copper deposit. The variogram model 2y(h) can help in this essentialcommunication between the geologist and the project engineer.


I. GEOSTATISTICS AND MINING APPLICATIONS 13

Continuity

In the definition of the variograrn 2y(h), h represents a vector of modulus Ih Iand direction a. Consider a particular direction a. Beginning at the origin,y(O) = 0, the variogram increases in general with the modulus Ih I. This issimply an expression of the fact that, on average, the difference betweentwo grades taken at two different points increases as the distance Ih Ibetween them increases. The manner in which this variogram increases forsmall values of Ih I characterizes the degree of spatial continuity of thevariable studied. Figure 11.10 shows profiles of four piezometer readings,giving the height of a water table, measured at four different points(denoted by 3, 4,33 and 18), as a function of time. The four correspondingsemi-variograms have markedly different shapes, from which a geostatisti-cian would be able to deduce the degree of continuity of the fourpiezometer profiles.

Anisotropies and zone of influenceIn a given direction a, the variogram may become stable beyond somedistance Ih I= a called the range, cf. Fig. 1.1. Beyond this distance a, themear. square deviation between two quantities z(x) and z(x +h) no longerdepends on the distance Ih I between them and the two quantities are nolonger correlated. The range a gives a precise meaning to the intuitiveconcept of the zone of influence of a sample z (x). However, there is noreason for the range to be the same in all directions a of the space. In Fig.1.1, for instance, the vertical range characterizing the mean verticaldimension of the mineralized lenses differs from the horizontal ranges. Fora given distance Ihl, the horizontal variograrn presents a weaker variabilitythan the vertical variograrn: this expresses the horizontal sedimentarycharacter of the phenomenon considered.

C7- - - - Horizontal

Ihl

FIG. 1.1. Structural anisotropy expressed by the variogram.

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1.8.3. The estimation variance

The variogram defined in (1.1) can be viewed as an estimation variance, thevariance of the error committed when the grade at point x is estimated bythe grade at point x + h. From this elementary estimation variance, 2y(h) =E{[Z (x) - Z (x + h )]2}, geostatistical techniques can be used to deduce thevariance of estimation of a mean grade Zo by another mean grade Zu. Thisestimation variance is expressed as

(1.3)The mean grades Z; and Zv can be defined on any supports, e.g., V mayrepresent a mining block centred on the point y, and v the set of N drillcores centred on points {z, i = 1 to N}. The mean grades are then definedby

Zv = VI(

) f Z(x)dxy v(y) 1 Nand z; =- I Z(x;).N ;=1The notation H V, v), for example, represents the mean value of the

elementary semi-variogram function y(h) when the end-point x of thevector h = (x - x') describes the support V and the other end-point x'independently describes the support v, i.e.,

1 N fHV, v)=NV ,I y(x;-x)dx.,~l v(y)

The general formula (1.3), shows that the estimation variance, i.e., thequality of the estimation, depends on all four of the following.

(i) The relative distances between the block V to be estimated and theinformation v used to estimate it. This is embodied in the termy(V, v).

(ii) The size and geometry of the block V to be estimated. This isembodied in the term y(V, V).

(iii) The quantity and spatial arrangement of the information v, which isembodied in the term -rev, v).

(iv) The degree of continuity of the phenomenon under study, which isconveyed by its characteristic serni-variogram y(h).

Thus, the theoretical formula (1.3) expresses the various concepts thatwe intuitively know should determine the quality of an estimation.

Confidence intervalThe estimation variance itself is not enough to establish a confidenceinterval for the proposed estimate, the distribution of the errors must also



be known. In mining applications however, the standard 95%Gaussian confidence interval [ 2ifEl is used, where (T~ is the estimationvariance.

Experimental observation of a large number of experimental histogramsof estimation errors has shown that the Gaussian distribution tends tounderestimate the proportion of low errors (within O"E/2) and of higherrors (outside the range 30"E)' However, the standard Gaussianconfidence interval r20"E] in general correctly estimates the 95 %experimental confidence interval. Figure II.17 shows the histogram ofestimation errors of 397 mean block, grades from the Chuquicamatadeposit in Chile. The arithmetic mean of the errors is O.Ol % Cu for amean grade of 2% Cu. The experimental variance of the 397 errors isequal to 0.273 (% CU)2, which is close to the geostatistically predictedvalue of 0.261. The predicted 95% confidence interval is, according to theGaussian standard, [2.J(O261)] = [102% CuI. In fact, 96% of theobserved errors fall within this interval.

Change of support (volume)The elementary variogram 2y(h), estimated from experimental data, isdefined on the support of these data, e.g., core samples of given cross-section and length. By means of formula (1.3), the variograrn 2yv(h)defined on any other support V can be deduced from the variogram 21'(h),e.g., V may be the support of the mining block. In fact,

2yv(h)= E{[Zv(x) - Zv(x + h )]2}appears as the variance of estimation of the mean grade of the block Vcentred on point x by the mean grade of the block V centred on the pointx + h. This formalism for the change of support has great practical appli-cations. Since future mining will be carried out on the basis of blocks andnot drill cores, it is of great benefit to be able to evaluate the spatialvariability of the mean grades of these blocks by means of the variogram2Yv(h) defined on the block support.

1.8.4. The dispersion variance

There are two dispersion phenomena well known to the mining engineer.The first is that the dispersion around their mean value of a set of datacollected within a domain V increases with the dimension of V. This is alogical consequence of the existence of spatial correlations: the smaller V,the closer the data and, thus, the closer their values. The second is that thedispersion within a fixed domain V decreases as the support v on which



each datum is defined increases: the mean grades of mining blocks are lessdispersed than the mean grades of core samples.

These two phenomena are expressed in the geostatistical concept ofdispersion variance. Let V be a domain consisting of N units with the samesupport v. If the N grades of these units are known, their variance can becalculated. The dispersion variance of the grades of units v within V,written D 2(v/ V), is simply the probable value of this experimental vari-ance and is calculated by means of the elementary variogram 2y(h)through the formula

D 2(v/V)= i'(V, V)-y(v, v). (IA)Taking the generally increasing character of the variogram into account,

it can be seen that D 2(v/V) increases with the dimension of V and.decreases with the dimensions of v.

This formula can be used, for example, to calculate the dispersionvariance of the mean grades of production units when the size of the units(v) varies or when the interval of time considered varies (i.e., V varies).

1.8.5. CoregionalizationsThe concepts of variogram and various variances, which so far have beendefined for the regionalization of one variable, can be generalized to spatialcoregionalizations of several variables. In a lead-zinc deposit, for example,the regionalizations of the grades in lead, ZI (x), and in zinc Zz(x), arecharacterized by their respective variograms 2YI(h) and 2Y2(h). However,these two variabilities are not independent of each other and we can definea cross-variograrn for lead and zinc:

Once the cross-variograrn is calculated, we can form the matrix

["(1 "(IZ],"(12 "(2

called the coregionalization matrix. This coregionalization matrix can thenbe used to estimate the unknown lead grade Zj(x) from neighbouring leadand zinc data.

Sometimes, two different types of data relating to the same metal can beused in block estimation. One set may be the precise data taken from coresamples, while the other may be imprecise data taken from cuttings andblast-holes. A study of coregionalization of these two types of data willdetermine the proper weights to be assigned to them during evaluation.

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1.8.6. Simulations of depositsThe regionalized variable z (x), which is the subject of the geostatisticalstructural analysis, is interpreted as one possible realization of the randomfunction Z (x). It is possible to generate other realizations zs(x) of thisrandom function Z(x) by the process of simulation. This simulated real-ization zs(x), will differ from point to point from the real realization z (x),but, statistically, will show the same structure of variability (same histo-grams and variograms); moreover, at the experimental data locations Xi, thesimulated values will be equal to the true values, i.e., Zs(Xi) = Z(Xi), "tIx; Edata set. The simulated and actual realization will, thus, have the sameconcentrations of rich and poor values at the same locations. The tworealizations can be viewed as two possible variants of the same geneticphenomenon characterized by the random function Z (x ). The simulatedrealization, however, has the advantage of being known at all points andnot only at the data points Xi.

Various methods of extraction, stoping, blending, stockpiling, etc., canthen be simulated on this numerical model {z.x), xED} of the deposit orzone D. Using feedback, the effect of these simulated processes on recov-ery, fluctuations of ore characteristics, and on the running of the mine and

100

50

Rain - go uge

FIG. 1.2. Locations of the rain gauges.



mill in general can be used to modify certain parameters of the miningproject.

The techniques for the simulation of a regionalization z (x) can beextended to simulations of coregionalizations, i.e., to the simulations of thesimultaneous spatial distributions of the various characteristic variables(thicknesses, grades, density) of a mineralization.

variant I

50

Variant 2

Voria nt 3

FIG 1.3. Three simulated rainfalls.

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Case study, after 1. P. Delhomme and P. Delfiner (1973)This was a study of the regionalization of water heights (measured in raingauges) after rain showers, but it could equally well apply to the regional-ization of the vertical thickness of a sedimentary bed.

Figure 1.2 shows the location of rain gauges in the Kadjerneur basin inTchad (Central Africa). After one particular rainfall, the height of water ineach rain gauge was measured. These data were then used to construct anexperimental semi-variogram to which a linear model was fitted. Figure 1.3shows three simulated rainfalls, each of which has the same linear vario-gram as the real rainfall and the same water heights at the actual rain gaugelocations. At points other than these rain gauge locations, the three simu-lations can differ markedly from each other. This is noticeably evident inthe extreme right-hand corner of the basin, where there are no raingauges.

I.C. SOME TYPICAL PROBLEMS AND THEIR GEOSTATISTICALAPPROACH

As previously mentioned, the initial geological surveys of a deposit resultin essentially qualitative information, and any quantitative information willgenerally be on a non-systematic grid. In general, it is not possible toconstruct sufficiently representative histograms and variograms on thebasis of such information. However, even if it is of a purely qualitativenature (e.g., structural geology) this information can be used during thegeostatistical structural analysis, which will later be carried out on the dataobtained from the first systematic sampling campaign. The geology is usedas a guide in the construction and interpretation of the various experimen-tal variograms. These variograms then channel this geological knowledgeinto the various studies involved in a mining project.

l.C.}. Global estimationOnce the first systematic campaign is completed, it is usual to proceedto estimations of global in situ resources, such as tonnages of are andoverburden, quantities of metals and mean grades. These estimations mustalso include confidence intervals which will determine the point at whichthe in situ resources are sufficiently well estimated to proceed to the nextstage of evaluation, i.e., that of the recoverable reserves.

At this stage of global evaluation of resources, there are no specificgeostatistical methods for determining estimators. For example, amineralized surface may be estimated by interpolation between positive



and negative drill-holes. Mineralized thickness or mean grade may beestimated by the arithmetic mean of the positive data on a pseudo-regulargrid. or by any of the traditional methods of extension of profiles orcross-sections. However, through the estimation variance. geostatistics canquantify the reliability of each of these traditional methods and so providea criterion for choice between them. The estimation variance can also beused to objectively classify resources and to predict the number, locationand distribution of additional information required to improve the qualityof estimations. e.g., to move a given tonnage from the "possible" to the"proved" category, d. A. G. Royle (1977).

I.C.2. Local estimation

Once the deposit has been deemed globally mineable, the next phase islocal, block by block estimation. This local estimation gives an idea of thespatial distribution of the in situ resources. which is necessary for theevaluation of the recoverable reserves. This local estimation is usuallycarried out on the basis of samples obtained from a smaller sampling grid.e.g., through in-fill drilling. The estimation variance of a given panel, d.formula (1.3), can be calculated as a function of the grid size and, thus.indicates the optimal sampling density to be used for the local estimation.

Once this second, denser sampling campaign is completed, the nextproblem is to determine the best possible estimate of each block. Thisamounts to determining the appropriate weight to be assigned to eachdatum value, whether inside or outside the panel. The determination ofthese weights must take into account the nature (core samples, channelsamples) and spatial location of each datum with respect to the block andthe other data. It must also take into account the degree of spatialcontinuity of the variable concerned. expressed in the various features ofthe variograrn, 2y(lhl. a).

For example, suppose that the mean grade Zv of the block V is to beestimated from a given configuration of N data Z; as shown on Fig. IA.The geostatistical procedure of "kriging" considers as estimator Z t of thismean grade, a linear combination of the N data values:

N

zt= L AjZj j~ I

(1.6)

The kriging system then determines the N weights {Ai, i = 1 to N} suchthat the following hold.

(i) The estimation is unbiased. This means that) on average, the errorwill be zero for a large number of such estimations, i.e .

E{Zv -zt} = o.


I. GEOSTATISTICS AND MINING APPLICAnONS 21

(ii) The variance of estimation u~ = E{[Zv - zt ]2} is minimal. Thisminimization of the estimation variance amounts to choosing the(linear) estimator with the smallest standard Gaussian 95%confidence interval (2UE).

Consider the example shown on Fig. 1.4, in which structural isotropy isassumed, i.e., the variability is the same in all directions. Provided that allthe N = 8 data are of the same nature, and all defined on the same support,kriging gives the following more or less intuitive results:

(i) The symmetry relations, "\2 = "\3, As = "\6;(ii) The inequalities, Al ~Ai, Vi '1'-1; A7 :os ;\4, i.e., the datum Z4 screens

the influence of Z7; As~ ;\4, i.e., part' of the influence of Z4 istransferred to the neighbouring data Z5, Z6 and Z7.

v

FIG,IA. Kriging configuration of block V.

I.C.3. Selection and recoverable reservesThe terms "selection" and "recoverable reserves" have little meaningunless they are used within the context of precisely defined technologicalconditions for recovery, From experience, the mining engineer knows thatthe quantity and quality of recovery improves as:

(i) The mining selection unit becomes smaller (although this intuition istrue only for highly selective mining);

(ii) The information available at the time of selection improves, to allowbetter discrimination between rich and poor units.

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These two essential concepts are formalized by geostatistics in thefollowing way.

1. Recovery depends on the support of the selection unitConsider the two histograms shown on Fig. 1.5(a). The first one representsthe distribution of the mean grades of 5 m x 5 m x 5 m units v in the depositD. These units may represent, for example, selective underground miningby small blocks. The second histogram represents the distribution of themean grades of 50 m x 50 m x 50 m units V in the same deposit D, whichmay represent, for example, mining by block-caving. It is obvious that thefirst histogram (v) will be much more dispersed than the second one (V),

(0 )

FIG 1.5. Recoveries estimated from cut-off grades applied to histograms. (a)Influence of the support; (b) influence of the standard of information.

and that the same cut-off grade Zo applied to both histograms will result invery different recoveries. These different dispersions are characterized bythe two dispersion variances D\v/D) and D 2(V/D) with D\vIDD 2( VID). By assuming some hypotheses related to the conservation of thetype of distribution law, it is possible to deduce these two histograms fromthe experimental histogram defined on the support C of core samplelengths. For any given selection unit size, the estimated histogram can thenbe used to determine the recoveries corresponding to different cut-offgrades zoo These recoveries are then used to build the so-called "grade-tonnage" curves.

2. Recovery depends on the standard of available informationThe preceding histograms concerned the real grades of units v and V. Inpractice, however, it seldom happens that the true grades are available at


L GEOSTATISTICS AND MINING APPLICATIONS 23

the time of selection. Selection is made on the basis of the estimated gradesof these units, which will inevitably result in a recovery different from theoptimum recovery based on the true grades.

Figure 1.5(b) shows the histogram of the true grades of the units Vsuperimposed on the histogram of their estimated values. As actual selec-tion will be made on the basis of these estimated values, the proportion ofrecoverable units V must be read from the histogram of the estimatedgrades (dotted area on Fig. 1.5(b)) and not from the histogram of the truegrades (hatched area on Fig. 1.5(a)).

If the estimated grades are obtained by kriging, then the dispersionvariance D 2 ( V* I D) of the histogram of the estimated grades can be cal-culated from a relation known as the "smoothing effect" due to kriging:

(I. 7)

where D 2( VID) is the dispersion variance of the true grades given byformula (1.4), and (T~ is the estimation variance given by the krigingsystem.

Knowing the two dispersion variances, D 2( V* / D) and D 2( VID), theestimates of the two histograms on Fig. 1.5(b) can be constructed and theeffect on recovery of selection, based on estimated grades rather than realones, can be evaluated. A balance can then be made between the profit inimproving the recovery and the cost of increased and better informationfor selection.

The preceding recoveries, estimated from the histograms, take noaccount of the spatial locations of the recovered units. It is weI! known thata rich unit (with a grade greater than the cut-off zo) in a poor zone will notbe mined, and a number of poor units in a rich zone will be mined. Thus, aprecise evaluation of recoverable reserves should take into account all thetechnological parameters of the envisaged mining method, e.g., pitcontours, conditions and cost of accessibility, haulage, etc. The geostatisti-cal solution to this problem is to work on a numerical model of the deposit.This model will have the same spatial distribution of grades, and, thus, canbe considered as a deposit similar to the real one. The simulation of theforeseen mining process on this model will, thus, provide the requiredevaluation of recoveries and "smudge" factor.

I.CA. Working out the mining projectOnce the general framework of the mining method has been defined andthe corresponding recoverable reserves have been estimated, the nextproblem is a more precise determination of the parameters of the mining



project. Some of these parameters depend on the variability of the arecharacteristics. Examples are as follows:

(i) The number and size of simultaneous stapes, and whether or notblending of the production from these stapes is required. Thesechoices will of course affect the trucking equipment.

(ii) The need for stockpiling, for control or blending, and the charac-teristics of the stockpiles.

(iii) The flexibility of the mill, to allow absorption of the fluctuations inquantity and quality of the recovered are.

The choice of such parameters involves a study of the fluctuations overtime of the mined ore after each of the mining phases, trucking andstockpiling. The structure of the in situ variabilities will be partiallydestroyed after mining and haulage, and the modified structures must beknown to determine the blending procedures necessary to meet the millfeed constraints. It may be possible to make such choices from experiencegained on similar already mined deposits, but the analogy can sometimesbe limited.

When there is no such "similar" deposit, the geostatistical approachconsists in building a numerical model of the deposit on which the variousprocesses of mining, trucking and blending can be simulated: theirconsequences on the production and the mill feed can then be observed,and, by feedback, the parameters adopted for the various mining processescan be corrected. These simulations are, of course, only as reliable as thenumerical model of the deposit. This model must faithfully reproduce all thefeatures of the spatial distribution of the characteristic variables (grades,thicknesses, accumulations, but also morphology and tectonics) of the realdeposit. The better the characteristics of variability and morphology of thereal deposit are known, the better the numerical model will be. Modellingdoes not create information, it can only reproduce the available structuralinformation in a simple numerical manner, easy to handle.

l.e.5. Mine planning in the routine phase

Once routine mining of the deposit has begun, the information about thedeposit becomes increasingly more detailed, both qualitatively (e.g., theworking faces can be seen) and quantitatively (data from blast-holes, millfeed grades, etc.), Using these data, the estimations can be constantlyupdated to improve overall operation. These estimations may be the basisfor short- or long-term mine planning. For example, for a given period ofproduction, these estimations may be used to determine hanging-wall and

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L GEOSTATISTICS AND MINING APPLICATIONS 2S

foot-wall outlines in a subhorizontal sedimentary deposit, or to carry outselection that satisfies the various constraints on tonnage and quality.Mine-mill production schedules can be made more reliable by specifyingconfidence intervals for the mean grades of ore sent for milling.


IIThe Theory of Regionalized

Variables

SUMMARYThis fundamental theoretical chapter gives the conceptual basis of geostatistics,then introduces, successively, the three main geostatistical tools: the variogram, theestimation variance and the dispersion variance.

Section II.A gives the interpretation of a regionalized variable z (x) as a realiza-tion of a random function Z (x ), The spatial correlations of this random function,i.e., the variability in space of the variable being studied, are characterized by thecovariance function C(h) or by its equivalent, the semi-variogram function definedby relation (JIA):

y(h) = 1E{[Z(x + h)- Z(X)]2}.Section II.B introduces the distribution function of the error of estimation. If a

normal distribution is assumed, this error is then completely characterized by itsexpectation (or estimation bias) and its variance (or estimation variance). For theimportant class of linear estimators, the estimation variance of any domain V byany other domain v is given by a combination of mean values of the semi-variogramy(h), d. formula (11.27):

(T~(v, V)= 2-y(V, v)- )I(V, V)- Y(v, v).Section II.C introduces a second variance, the dispersion variance D 2(v/ V),

which characterizes the dispersion of the grades of mining units v within a stope V,for example. This dispersion variance is written as a combination of mean values ofthe semi-variogram 'Y(h), cf. formula (11.36):

D 2(v/ V) =)i( V, V) - )I(v, u),Section II.D introduces the important notion of regularization (taking the

average value in a certain support v). If Z (x) is the point grade with the pointserni-variogram vin). the mean value Z,,(x) in the volume v is then characterizedby a regularized semi-variogram y,,(h), which is written as a combination of meanvalues of the point serni-variogram, d. formula (11.41):

'Yv(h}=y(v, Vh)'--Y(V, v}.26


II. THE THEORY OF REGIONALIZED VARIABLES 27

Two sets of information defined on two different supports v and v' and charac-terized by the two regularized semi-variograms Y~ and Yv' can then be compared.

As all the previous geostatistical operations make continuous use of mean valuesy of the point serni-variogram y(h), section II.E is devoted to the practicalcalculation procedures of those mean values y. Either a direct numerical calculationor the use of appropriate auxiliary functions and charts is possible.

11.A. REGIONALIZED VARIABLES AND RANDOM FUNCTIONS

II.A 1. Regionalized variables and their probabilistic representationWhen a variable is distributed in space, it is said to be "regionalized". Sucha variable is usually a characteristic of a certain phenomenon, as metalgrades, for example, are characteristics of a mineralization. Thephenomenon that the regionalized variable is used to represent is called a"regionalization", examples of which are:

(i) the market price of a metal which can be seen as the distribution ofthe variable price in time (one-dimensional space);

(ii) a geological phenomenon such as the thickness of a subhorizontalbed which can be regarded as the distribution in the two-dimen-sional space of the variable thickness.

(iii) a mineralized phenomenon can be characterized by the distributionin the three-dimensional space of variables such as grades, densities,recoveries, granulornetries etc.

Regionalized variables are not restricted to mining, and examples fromother fields include population density in demography, rainfall measure-ments in pluviometry and harvest yields in agronomy. In fact, almost allvariables encountered in the earth sciences can be regarded as regionalizedvariables.

The definition of a regionalized variable (abbreviated to ReV) as avariable distributed in space is purely descriptive and does not involve anyprobabilistic interpretation. From a mathematical point of view, a ReV issimply a function f(x) which takes a value at every point x of coordinates(xu, xv, x w ) in the three-dimensional space. However, more often than not,this function varies so irregularly in space as to preclude any directmathematical study of it. Even a curve as irregular as that of the nickelgrades of core samples shown in Fig. ILl underestimates the true vari-ability of the point grades due to the smoothing effect of taking averagevalues over the volume of the core. However, the profiles of the nickelgrades from three neighbouring vertical bore-holes have a commongeneral behaviour; a relatively slow increase of the grade values followedby a rapid decrease, and, then, at the contact with the bedrock, a further



increase. This behaviour is characteristic of the vertical variability of nickelgrades in the lateritic type deposits of New Caledonia, (cf. Ch. Huijbregtsand A. Journel, 1972).

f.--------~400m --------+-1....---- 20 m --..........,~

0-8 Ni 0-8 Ni 0-8 Ni

s:ii.Q)p

10

20

30

FIG. ILL Vertical variability of Ni grades.

Poorzone

x

FIG. Il.l. Random and structured aspect of a ReV.



In practically all deposits, a characteristic behaviour or structure of thespatial variability of the ReV under study can be discerned behind a locallyerratic aspect. In almost all deposits, there exist zones which are richer thanothers; samples taken in a rich zone will be, On average, richer than thosetaken in a poorer zone, i.e., the value of the ReV I(x) depends on thespatial position x, see Fig. II.2.

Thus a ReV possesses two apparently contradictory characteristics:

0) a local, random, erratic aspect which calls to mind the notion ofrandom variable;

(ii) a general (or average) structured aspect which requires a certainfunctional representation,

A proper formulation must take this double aspect of randomness andstructure into account in such a way as to provide a simple representationof the spatial variability and lead to a consistent and operational approachto the solution of problems. One such formulation is the probabilisticinterpretation as provided by random functions.

The concept of a random functionA random variable (abbreviated to RV) is a variable which takes a certainnumber of numerical values according to a certain probability distribution.For instance, the result of casting an unbiased die can be considered as aRV which can take one of six equally probable values. If one particular castresult is 5, then, by definition, we shall say that this value 5 is a particularrealization of the RV "result of casting the die".

Similarly, let us consider the grade z(xd= 15% Cu at a particular pointx I in a copper deposit. This grade can be considered as a particularrealization of a certain RV Z(Xl) defined at point XI. Thus, the set ofgrades z(x) for all points x inside the deposit, i.e., the regionalized variablez (x), can be considered as a particular realization of the set of RV {Z (x),X E deposit}. This set of RV is called a random function and will be writtenZ (x ).1

This definition of a random function (abbreviated to RF) expresses therandom and structured aspects of a regionalized variable:

(i ) locally, at a point x I, Z (x i) is a random variable;(ii) Z(x) is also a RF in the sense that for each pair of points XI and

Xl + h, the corresponding RV's Z (x I) and Z (x1+ h) are not, in1- Notation In order 10 distinguish clearly between observations and a probabilistic model.

the regionalized variable or the realizations of the RV or RF are written in lower-caseletters, thus: y(x), z (x ), t(x), while the RV or RF themselves are written in upper-caseletters, thus: Vex), Z(x). T(x).



general, independent but are related by a correlation expressing thespatial structure of the initial regionalized variable z (x).

Statistical inferenceThe probabilistic interpretation of a ReV z (x) as a particular realization ofa certain RF Z (x) has an operative sense only when it is possible to infer allor part of the probability law which defines this RF in its entirety.

Obviously, it is not rigorously possible to infer the probability law of aRF Z (x) from a single realization z (x) which is, in addition, limited to afinite number of sample points Xi- It is impossible, for instance, to deter-mine the law of the RV "the result of casting the die" from the singlenumerical result 5 resulting from a single cast of the die, and, in particular,it is not possible to determine whether or not the die is biased. Many castsof the die are necessary to answer such a question. Similarly, many realiza-tions z 1(x), Z2(X), ... , zdx) of the RF Z (x) are required in order to inferthe probability law of Z (x). Since, in practice we shall be limited to a singlerealization {z (Xj)} of the RF at the positions Xj, we appear to have come to adead end and, in order to solve it, certain assumptions are necessary. Theseassumptions involve various degrees of spatial homogeneity and are intro-duced under the general heading of the hypothesis of stationarity.

In practice, even if only over a certain region, the phenomenon understudy can very often be considered as homogeneous, the ReV repeatingitself in space. This homogeneity or repetition provides the equivalent ofmany realizations of the same RF Z (x) and permits a certain amount ofstatistical inference. Two experimental values z (xo) and z (xo+ h) at twodifferent points Xo and Xo + h can, thus, be considered as two differentrealizations of the same RV Z (xo). This type of approach is not peculiar togeostatistics, it is used to infer the distribution law of the RV Z (x) from ahistogram of data values {z(Xj)}, or more simply, to infer the mathematicalexpectation E {Z (x)} from the arithmetic mean of the data.

II.A.2. Moments and stationarity

Consider the RF Z (x). For every set of k points in R" (n -dimensionalspace), x h X2, , xi, called support points, there corresponds a k-component vectorial random variable

This vectorial RV is characterized by the k -variable distribution function

FXIX;:_ .. X~ (ZI, Z2, . , Zk)= Prob {Z(Xl)< ZI, ... , Z(Xk)< zd.



The set of all these distribution functions, for all positive integers k and forevery possible choice of support points in R", constitutes the "spatial law"of the RF Z(x).

In mining applications, the entire spatial law is never required, mainlybecause the first two moments of the law are sufficient to provide anacceptable approximate solution to most of the problems encountered.Moreover, the amount of data generally available is insufficient to infer theentire spatial law. In geostatistics (more precisely, linear geostatistics), onlythe first two moments of the RF are used; in other words, no distinction ismade between two RF's Z1(X) and Z2(X) which have the same first- andsecond-order moments and both functions are considered as comprisingthe same model.

Mathematical expectation, or first-order momentConsider a RV Z(x) at point x. If the distribution function of Z(x) has anexpectation (and we shall suppose that it has), then this expectation isgenerally a function of x, and is written

E{Z(x)} = m(x).

Second-order moments

The three second-order moments considered in geostatistics are as follows.

(i) The variance, or more precisely the "a priori" variance of Z(x).When this variance exists, it is defined as the second-order momentabout the expectation m(x) of the RV Z(x), i.e.,

Var {Z (x)} = E{[Z(x) - m (x )]2}.As with the expectation m (x), the variance is generally a function ofx.

(ii) The covariance. It can be shown that if the two RV's Z(XI) andZ (X2) have variances at the points XI and X2, then they also have acovariance which is a function of the two locations x I and Xl and iswritten as

(iii) The uariogram. The variogram function is defined as the variance ofthe increment [Z (Xl)- Z (X2)], and is written as

2y(XI, X2) = Var {Z(XI)- Z(X2)}'The function Y(X1' X2) is then the "serni-variogram".



The hypothesis of stationarityIt appears from the definitions that the covariance and variogram functionsdepend simultaneously on the two support points Xl and X2. If this is indeedthe case, then many realizations of the pair of RV's {Z(xd, Z(X2)} must beavailable for any statistical inference to be possible.

On the other hand, if these functions depend only on the distancebetween the two support points (i.e., on the vector h = x, - X2 separating Xland X2), then statistical inference becomes possible: each pair of data[z (Xk), z (xd} separated by the distance (Xk - xd, equal to the vector h, canbe considered as a different realization of the pair of RV's {Z(x,}, Z(X2)}'

It is intuitively clear that, in a zone of homogeneous mineralization, thecorrelation that exists between two data values Z(Xk) and Z (XI,,,) does notdepend on their particular positions within the zone but rather on thedistance which separates them.

Strict stationarity A RF is said to be stationary, in the strict sense, whenits spatial law is invariant under translation. More precisely, the twok-component vectorial RV's {Z(x,), ... ,Z(xd} and {Z'(x, +h), ... , Z (x, + h)} are identical in law (have the same k -variable dis-tribution law) whatever the translation vector h.

However, in linear geostatistics, as we are only interested in the pre-viously defined first two moments, it will be enough to assume first that thesemoments exist, and then to limit the stationarity assumptions to them.

Stationarity of order 2 A RF is said to be stationary of order 2, when:0) the mathematical expectation E{Z (x)} exists and does not depend

on the support point x; th us,

E{Z(x)} = m, Ir/x; (11.1 )(ii) for each pair of RV's {Z (x ), Z (x + h)} the covariance exists and

depends on the separation distance h,

C(h)=E{Z(x +h) Z(x)}- m;" 't/x, (II. 2)where h represents a vector of coordinates (h u ) h.; h w ) in the three-dimensional space.

The stationarity of the covariance implies the stationarity of the varianceand the variograrn. The following relations are immediately evident:

Var{Z(x)} = E{[Z(x) - m]2} = C(O), "Ix (11.3)y(h) =!E{[Z (x + h)- Z (x )]2} = C(O)- C(h), Vx. (IL4)



Relation (11.4) indicates that, under the hypothesis of second-orderstationarity, the covariance and the variogram are two equivalent tools forcharacterizing the auto-correlations between two variables Z (x + h) andZ (x) separated by a distance h. We can also define a third tool, thecorrelogram:

(h)= C{h)= 1- y(h)p C(O) C(O) (11.5)

The hypothesis of second-order stationarity assumes the existence of acovariance and, thus, of a finite a priori variance, Var {Z(x)} = C(O). Now,the existence of the variogram function represents a weaker hypothesisthan the existence of the covariance; moreover, there are many physicalphenomena and RF's which have an infinite capacity for dispersion (ct.section II.A.S, Case Study 1), i.e., which have neither an a priori variancenor a covariance, but for which a variogram can be defined. As aconsequence, the second-order stationarity hypothesis can be slightlyreduced when assuming only the existence and stationarity of the vario-gram.

Intrinsic hypothesis A RF Z(x) is said to be intrinsic when:(i) the mathematical expectation exists and does not depend on the

support point x,

E{Z(x)}=m, Vx;(ii) for all vectors h the increment [2 (x + h) - Z (x)] has a finite variance

which does not depend on x,

Var {Z(x +h)- Z(x)} = E{[Z(x + h)- Z(X)]2} = 2)1(h), Vx. (11.6)Thus, second-order stationarity implies the intrinsic hypothesis but the

converse is not true: the intrinsic hypothesis can also be seen as thelimitation of the second-order stationarity to the increments of the RFZ(x).

Quasi -stationarity In practice, the structural function, covariance orvariogram, is only used for limited distances Ih 1:0::::; b. The limit b represents,for example, the diameter of the neighbourhood of estimation (i.e., thezone which contains the information to be used, cf. Fig. II.3). In othercases, b may be the extent of an homogeneous zone and two variablesZ(xd and Z(Xk + h) cannot be considered as coming from the samehomogeneous mineralization if Ih I> b.

In such cases, we can, and indeed we must, be content with a structuralfunction C(x, x + h) or y(x, x + h), which is no more than locally stationary



(for distances Ih I less than the limit b). This limitation of the hypothesis ofsecond-order stationarity (or the intrinsic hypothesis if only the variograrnis assumed) to only those distances Ih I~ b corresponds to an hypothesis ofquasi-stationarity (or to a quasi-intrinsic hypothesis).

In practical terms, we can define sliding neighbourhoods inside of whichthe expectation and the covariance can be considered as stationary and thedata are sufficient to make statistical inference possible.

The hypothesis of quasi-stationarity is really a compromise between thescale of homogeneity of the phenomenon and the amount of available data.In fact, it is always possible to produce stationarity by considerably reduc-ing the dimension b of the zones of quasi-stationarity, but then most ofthese zones would not have any data in them, and, thus, it would beimpossible to infer the quasi-stationary moments in these zones.

From a data grid of 1 to 3 m, a 200-m-Iong hillside would appear as aquasi-stationary model, but would require a non-stationary model(because of the increasing altitude) for a grid of 20 to 30 m, cf. Fig. 11.3. Ifthe data grid is of the order of 200 m, then a certain stationarity againbecomes evident: at the scale of the peneplanar basin the entire hillsideappears as a local variability of the stationary RF "altitude". This exampleshows, incidentally, that it is not possible to devise a theoretical test for thea priori verification of the hypothesis of stationarity using a single realiza-tion of the RF. It is always possible to find a particular realization of astationary RF which would display a systematic and continuous change invalue (apparent non-stationarity) over a limited distance, so that atheoretical test could never refute the hypothesis of stationarity.

x

x

x

z(xo}x x

- I b=IO m I~~I I'I I


Their proof is of no interest in the application of geostatistics; we shallsimply state the properties and comment on them.

Positive definite conditionsLet Z(x) be a stationary RF of expectation m and covariance C(h) orsemi-variogram l'(h). Let Y be any finite linear combination of the type

"y= L AiZ(Xi)i=l

(II.7)

for any weights Ai.This linear combination is a RV and its variance must never be negative,

Var {Y} ~ O. Explicitly, this variance is written

Var {Y}= L L AiAiC(Xi-Xj)~O.i j

(11.8)

The covariance function must be such that it ensures that the previousvariance will always be positive or zero. By definition then, the functionC(h) is said to be "positive definite".

Thus, not any function g(h) can be considered as the covariance of astationary RF: it is imperative that it be positive definite.

Taking account of relation (11.4), expression (11.8) can be written interms of the serni-variogram:

Var {Y} = C(O)I AiL Aj - L I AiA j l' (Xi - x,).iii

(11.9)

In the case where the variance C(O) does not exist, only the intrinsichypothesis is assumed, and the variance of Y is defined on the conditionthat

and the term C(O) is thus eliminated, leaving

Var{Y}= -LLA iA j l' (Xi - X j)i i

(11.10)

The variogram function must be such that Var {Y} is positive or zero, withthe condition that It Aj = O. By definition, -l'(h) is said to be a "conditionalpositive definite" function.



Authorized linear combinations

When only the intrinsic hypothesis is assumed, y(h) exists but C(h) maynot exist; the only linear combinations that have a finite variance are thoseverifying the weights condition L Ai = O. Thus, if only these authorizedlinear combinations are considered (which is the case in linear geostatis-tics), there is no need to calculate the a priori variance C(O), nor to knowthe expectation m =E{Z (x)} or the covariance C(h); it is enough to knowthe serni-variogram.

Note that, for the authorized linear combinations, formula (II.IO) issimply deduced from formula (11.8) by substituting (- y) for C. This rulefor passing from a formula written in terms of covariance to the cor-responding formula written in terms of semi-variogram is genera! in lineargeostatistics.

Properties of the covarianceThe positive definite character of the covariance function C(h) entails thefollowing properties:

C(O) = Var {Z(x)} ~ 0, an a priori variance cannot be negative; }C(h) """ C(-h), the covariance is an even function; (II.ll)

IC(h)1 =:;; C(O), Schwarz's inequality.

Since the degree of correlation between two variables Z (x) and Z (x + h)generally decreases as the distance h between them increases, so, ingeneral, does the covariance function, which decreases from its value at theorigin C(O). Correspondingly, the semivariogram y(h) """ C(O)- C(h)increases from its value at the origin, 1'(0) = 0, see Fig. 11.4.

Absence of correlationVery often, in practice, the correlation between two variables Z (x) andZ (x +h) disappears when the distance h becomes too large:

C(h)~O, when Ihl~co,

and, in practice, we can put C(h) = 0, once Ih1~ a. The distance a beyondwhich C(h) can be considered to be equal to zero is called the "range" andit represents the transition from the state in which a spatial correlationexists (Ih I< a) to the state in which there is absence of correlation (Ih j> a),d. Fig. IIA.



Properties of the uariogramThe definition of the variogram as the variance of increments entails thefollowing properties:

1'(0) = 0, y(h)=y(-h);::.O. (II, 12)In general, but not always, as h increases, the mean quadratic deviation

between the two variables Z (x) and Z (x + h) tends to increase and so y(h)increases from its initial zero value.

r(m)=c{O)C(O) - - - - - - ---- - --~--_----:._----

r (II)

C(II)

C{co)::Oo o

FIG. 11.4. Covariance and semi-variogram.

Transition phenomena Very often, in practice, the serni-variogram stopsincreasing beyond a certain distance and becomes more or less stablearound a limit value ")'(00) called a "sill" value, which is simply the a priorivariance of the RF

")'(00)= Var{Z(x)} = C(O). (11.13)In such cases, the a priori variance exists as does the covariance. Such

variograms, which are characterized by a sill value and a range, are called"transition" models, and correspond to a RF which is not only intrinsic butalso second-order stationary, d. Fig. II.4.

Zone of influence In a transition phenomenon,

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