Table Of ContentFurther titles in this series
F.P. AGTERBERG
GEOMATHEMATICS
S. TWOMEY
INTRODUCTION TO THE MATHEMATICS OF INVERSION IN REMOTE
SENSING AND INDIRECT MEASUREMENTS
Developments in Geomathematics 2
GEOSTATISTICAL ORE RESERVE
ESTIMATION
by
MICHEL DAVID
Ecole Poly technique, Universite de Montreal, Montreal, Quebec, Canada
Elsevier Scientific Publishing Company
Amsterdam Oxford New York 1977
ELSEVIER SCIENTIFIC PUBLISHING COMPANY
Molenwerf 1,
P.O. Box 211,1000 AE Amsterdam, The Netherlands
Distributors for the United States and Canada:
ELSEVIER SCIENCE PUBLISHING COMPANY INC.
52, Vanderbilt Avenue
New York, N.Y. 10017
First edition 1977
Second impression 1978
Third impression 1979
Fourth impression 1982
Library of Congress Cataloging in Publication Data
David, Michel, 19^5-
Geostatistical ore reserve estimation.
(Developments in geomathematics ; 2)
Bibliography: p.
Includes index.
1. Mine valuation—Statistical methods. 2. Mines
and mineral resources—Statistical methods. I. Title.
II. Series.
TN272.D37 622M3 76-39808
ISBN o-Wt-41532-7
ISBN 0-444-41609-9 (series)
ISBN 0-444-41532-7 (vol. 2)
© Elsevier Scientific Publishing Company, 1977
All rights reserved. No part of this publication may be reproduced, stored in a retrieval
system or transmitted in any form or by any means, electronic, mechanical, photocopy
ing, recording or otherwise, without the prior written permission of the publishers, Elsevier
Scientific Publishing Company, P.O. Box 330, 1000 AH Amsterdam, The Netherlands
Printed in The Netherlands
PREFACE
Having been associated intimately with the birth and early developments
of what is now widely known as geostatistics it is a privilege to introduce
this comprehensive and practical work of Prof. David to mining engineers,
geologists and statisticians. Prof. Matheron together with his present and
ex-colleagues and students (including Prof. David) deserve the lion's share of
the credit due for the development and establishment of geostatistics.
They encountered no serious problems of acceptance in the latin-speaking
countries of Europe and South America nor in Eastern Europe but at times
had some stormy receptions from the English-speaking mining countries
around the world. Even today, there is still a significant skepticism in certain
circles in the latter countries due mainly to the paucity, until recently, of
authoritative and comprehensive English publications as well as the lack of
fully reported practical case studies.
In recent years the position in English-speaking countries has changed
dramatically due to an increasing number of published papers and the teach
ing of geostatistics courses at many mining universities and at international
symposia. The need for comprehensive reference works is, however, still felt
keenly, and against this background this publication by Prof. David will be
extremely welcome and useful. His emphasis of the practical side and the
large number of practical case studies should particularly assist in illustrating
applications of the theory involved and serve to provide proof of the advan
tages to be gained from the geostatistical approach.
D.G. KRIGE
INTRODUCTION
Peut-etre Vesprit humain va-tHl
etre amene a s'occuper serieusement,
c'est d dire avec precision de Faleatoire?
Andre Malraux
Hotes de passage
This text intends to be a technical one. This means that techniques to
solve identified problems will be presented. As the theory which serves as
a basis for these techniques is very new, and relatively unfamiliar to the
mineral industry, several chapters or sections will be devoted to it. These
two ideas of a technique and a theory have been my guideline in preparing
this course on the geostatistical estimation of mineral resources. The main
target was to stay, as much as possible, close to the practical problems. This
is the reason for the many examples which are intermeshed with the text;
however, in many cases, staying too close to a problem obscures the broader
frame into which a question has to be asked before finding a correct answer.
This is the reason for some theoretical digressions, which may seem to some
as an attempt to try and make things look complicated. Certainly, in a
particular mine, many problems can be solved without a total understanding
of the complete theory. On the other hand, when one considers all the
problems occurring in different mines, one cannot hope to solve them with
out having a good grasp, a synthetic view of the theory of regionalized
variables as developed by G. Matheron in France, the most advanced develop
ments of which have just been published in the Proceedings of a N.A.T.O.
Advanced Study Institute (Guarascio, Huijbregts, David, 1976).
What is presented here are the basic tools, derived from his early theory
(1962), which were found useful for answering questions in mineral reserves
estimation, grade control, sampling optimization and related problems of the
mineral industry. The presentation made here has been derived from ex
perience in teaching this topic in a North American context, or in countries
where mining bears an Anglo-Saxon influence rather than a French one. This
can immediately indicate some problems encountered in the redaction of
this text in English. People who developed geostatistics in French may be
surprised at the pragmatic presentation which is made of this conceptually
perfectly well-grounded mathematical theory. English speaking people will
wonder at the "exotic flavour" still retained in many expressions. Rather
than waiting for the perfect presentation of all this, I have decided to present
this account of my experience, hoping that several other better texts will
bridge all the gaps of this one.
The text has mainly been written for mining engineers and geologists, who
are the people facing the problem of ore reserves and grade control, and who
VII
usually have had little exposure to probability and statistics. This is why the
book starts with the very elementary concepts of descriptive statistics. Many
ore reserve problems can be solved with these very elementary tools and
knowing what they can do, will at the same time show where they fail and
hence what the new theory should answer to be of any help. Next, Chapter
3 presents a review of what ore reserves calculations are so that the link of
Matheron's theory with the industry's problems and its present solutions is
shown. An intuitive presentation of the theory is given in Chapter 4, where
the variogram is introduced. Chapter 5 is a few pages of theory to firmly
ground the model although statisticians will find many unqualified state
ments here. This is not a book for professional statisticians. Better exposes
of the theory will certainly be available very shortly and at present, the
same textbook cannot satisfy practizing engineers and mathematical statis
ticians. Then, from Chapter 6 and on, the application of the theory is
shown; first how to compute variograms (Chapter 6); how to estimate the
variability of blocks (Chapter 7); how to assess the accuracy of an estima
tion method (Chapter 8); and finally how to find the best method of ore
reserve estimation, kriging (Chapters 9 and 10). Chapter 11 discusses
problems of grade-tonnage curves, just as in Chapter 2, showing the solution
obtained and the answers still missing. Chapter 12 is devoted to orebody
modelling and general remarks on mine planning. Chapter 13, which should
really be the zeroth chapter, discusses the problem of sample preparation;
the twelve previous chapters discuss the grade of cores, the numbers which
are manipulated are in fact the grade of a few grams of powder. An account
of Gy's and Ingamells' work in this field is presented. Other chapters present
ing case studies were intended. The case studies have in fact been torn into
pieces and presented throughout the book. In the list of references, complete
ones in English are pointed out. In the examples or more particularly in the
figures relating particular real situations, some numbers are missing; it is not
possible to release grade information for many of the mines considered in
this volume, and it is not possible to spend time reworking examples on
public data for which everybody has lost interest.
This account of a ten-year experience could never have been published
without the help of many people. Andre, Charles, Pierre, Massimo: we
spent more time discussing things other than geostatistics, however, this was
probably the essential support to the whole adventure. I started assembling
notes to present geostatistics at a one-day short course of the 1973 APCOM
meeting in Golden, Colorado. Let me thank Dr. T.B. Johnson who invited
me at that time; a more complete version was produced at McKay School
of Mines to teach a one-week short course in Reno, Nevada. The possi
bility of spending so many years on geostatistics is owed to Dr. R.A. Blais
and Dr. Guy Perrault of Ecole Polytechnique de Montreal. More basic
is the debt I contracted to Dr. Matheron, Dr. Formery and Dr. Serra who
introduced me to probability and geostatistics at the School of Mines
VIII
in Nancy. Dr. D.G. Krige and Michel Dagbert patiently read and edited the
text; suggestions by CO. Ingamells and D.R. Williamson have also been
incorporated into the final version, but the responsibility for errors that
remain is mine. Michel Dagbert, my research associate, contributed in addition
by carrying out most of my usual workload at Ecole Polytechnique,
Universite du Quebec a Chicoutimi and Mineral Exploration Research Insti
tute. Dr. M.W. Davis who masters equally well Shakespeare language and
Matheron's mathematics offered numerous suggestions and corrections and
all my friends of the department of Geological Engineering who at one time
or the other have worked on geostatistical projects provided a constant
encouragement, Guy Daoust, Peter Dowd, Jean-Marc Belisle, Raymonde
LeDti, Raymond Sabourin, Yves Beauchemin and dozens of students. All
these efforts and science would have been useless without the marvelous help
of Valerie Virard and Raymonde Robert who typed the first and second
version of this text. Jose Beauchemin and Andre Lacombe made the draft
ing. The financial help of the National Research Council of Canada (Grant
NRC7035) is gratefully acknowledged as well as the opportunity to use the
drafting facilities of the department of Mineral Engineering at Ecole Poly
technique. The Mineral Exploration Research Institute provided a most wel
come typing help in rush periods. Many illustrations have been borrowed as
well as a few examples, permission to use this material is certainly appreciated.
Finally, all this help to develop an engineering tool would not have been
sufficient without the support of many engineers and geologists from
companies around the world. I wrote this book for them and hope that it
can be a first introduction to this still mysterious geostatistics which has
devoured a few of us.
Montreal, December 31st, 1975. M. DAVID
LIST OF NOTATIONS
A word of caution. It has been known for a long time that geostatisticians
seem to have that capability of changing notations twice or more in the same
page and still understand each other. This text has been written, keeping this
in mind and an effort has been made to keep notation consistent throughout
the volume. Some problems remain and it is not possible to» have a perfectly
standard notation as for instance 7 has been the standard to designate the
median of a lognormal distribution, but it is also the standard to designate a
variogram, similarly a has been commonly used to designate the average of
the logarithims in a lognormal population, it is also the standard in the
equation of a De Wijsian variogram: j(h) = 3a In (h). The cumulative prob
ability of a random variable has been called F(x) forever, so has the variance
of a point sample within a volume! SichePs £-estimator has been in use for
30 years; we also have Student-f and tonnage above a cut-off £(x/). "a" can
be a slope, a range, a weighting coefficient or an accumulation and some
times one has to use all these variables in the same report; some local changes
Eire thus obviously necessary. It is not obvious, however, that it would
necessarily be a good thing to have different symbols for all these variables,
and certainly judgement and common sense can sort them out and tell which
is which, while a totally new system would break too many well-established
habits (. . . who said geostatistics is trying to break habits in ore reserve
estimation!). We hope, however, to come out with a better system in forth
coming editions.
The notations commonly used by Sichel, Gy and Ingamells have been
retained when contributions of these authors Eire discussed. The definitions
are given in each particular section.
About the unit system, it has also been decided not to break habits; this
volume is written in a North American context which might be together with
Yemen, Gambia and Sierra Leone among the last areas to use the British
System; however, to today's North American and most English-speaking
geologists and mining engineers, feet and ounce per ton make more sense
than meters and gram per ton. There is a trend to shift to metric, this is why
we left some examples in metric units but we definitely feel that a 50'
block is something easy to work with while an 18.22 m block requires more
skill! We have thus chosen to report examples in the actual units they were
originally worked out. The notation used is shown below in alphabetical
order and a short list of abbreviations and symbols used is also given.
* superscript implies an estimated value
a isotropic range of a spherical variogram
Subscript ranSe of a spherical variogram in subscripted direction
a,- weighting coefficient in a linear combination
an coefficient of Z{X{) in the estimation of d*
A average value of C(h) (nugget effect), or defined in context
XVII
[A] vector of unknown coefficients in kriging system
A area of influence of sample indexed i
t
a intrinsic coefficient of dispersion (3a = slope of a De Wijsian variogram)
a. average of the logarithms of values in a lognormal population
B a block
standard deviation of the logarithms of values in a lognormal distribution
nugget effect
C + C sill
0
C(h) covariance of a random function
cov(x, y) covariance of quantities X and Y
7 median of a lognormal population
yy((hh)) variogram of a random function
YY((hh)) local value ofy(h)
7v(h) variogram of samples of size V
1(0, h) variogram in direction 6
lij = Jih
y(v, V) = average value of the variogram between v and V (one end of h is in v, the
IIvvvv
other in V)
Jiso(h) isotropic component of a variogram
TzonW zonal component of a variogram
7Rc(h) cross covariogram of variables R and C
JMIW variogram of samples of length /, perpendicular to h
d,D dimensions defined in context
di 1th coefficient of the drift
D diametral variation in direction A
A
[D] the right-hand side vector in the kriging system
5 density of a material
A a direction
Ah, Ad increment of h, 6
E(X) expected value of X
E(Y\X) conditional expectation of Y given X
E(Y\X = conditional expectation of Y given that X = XQ
Ci error of estimation of set indexed /
ff((XX)) density of probability of random variable X
AAxx)) 1th function in the expansion of the drift
F(X) cumulative distribution of random variable X
F(set) variance of point sample within set (set = /, S, V . . .)
G(Z) the Gauss integral (from Z to 00)
distances defined in context
h a vector of length h
^subscript component of h in subscripted direction
h' h corrected for anisotropics
^subscript component of h in subscripted direction
distance between points X,- and Xj
hij
U, i, J integers defined in context
k defined in context
kk((xx)) characteristic function of a set
kk a correction factor
KK Lasky's constants
U 2
K(h) the covariogram of a random function
lL lengths defined in context
y
X a kriging coefficient
m average value of a population
m average grade of a deposit with zero cut-off
0
XVIII
m(x) average grade of tonnage above cut-off x
c c
m(x) expected grade at point x (the drift)
average distance between two points within a surface
jU, lli Lagrange multipliers
n number of samples
N parameter in the negative binomial distribution
N(h) number of pairs of samples, a distance h apart
0 a point sample
co factor in the computation of the mean of a truncated normal distribution
p defined in context
p(x) 8L discrete probability
P a parameter in the negative binomial distribution
P\x} probability of X
P\X | y} conditional probability of X given Y
P{X, Y} joint probability of X and Y
Py(h) the geometric covariogram of set V
q(x) quantity of metal above x
c c
Q a parameter in the negative binomial distribution
Q(x) proportion of quantity of metal above x
c c
R the three-dimensional space
Rn a space in n dimensions
r correlation coefficient within a finite subsample
p correlation coefficient for a population
s, S surfaces defined in context
s standard deviation of a group of samples
Sjp standard error of the mean
s variance of a group of samples
s2(0/i) local variance of point samples within a block indexed i
o2 variance of a population
o\ estimation variance
o2 variance of the mean
m
o(X) = o standard deviation of X
x
°XiXj ~ °ij covariance of grades at point Xf and point Xj
Oyx covariance of grades of set V and sample Xi
t
Oy variance of grade of blocks of size V
02 = o2(B/D) variance of blocks of size JB within deposit D
B
o{ Vj, V/D) covariance of grades of sets V and V within D
2 x 2
ojsj variance due to a nugget effect
ao — O2(0/D) variance of grade of point samples within deposit
Ofty estimation variance of the grade of a block V estimated from the average
of N samples
o2 estimation variance of a surface s
(Jsup variance due to the border effect
o\ the kriging variance
[2] the matrix of in the kriging system
t Sichel's t-estimator
^(i-a,iV) Student t for a risk a and N samples
t(x) total tonnage above cut-off grade x
c c
T total tonnage of a deposit with zero cut-off
0
T(x) proportion of tonnage above cut-off x
c c
T a time interval
6 a direction
