Geostatistical Case Studies

247 Pages·1987·10.154 MB·English

by  Jacques Rivoirard (auth.), G. Matheron, M. Armstrong (eds.)

Geostatistical Case Studies
Ge statistical
0
Case Studies
edited by
G. MATHERON
and
M. ARMSTRONG
Centre de Geostatistique,
Fontainebleau, France
D. Reidel Publishing Company
A MEMBER OF THE KLUWER ACADEMIC PUBLISHERS GROUP
Dordrecht / Boston / Lancaster / Tokyo
Library of Congress Cataloging in Publication Data
Geostatistieal case studies.
(Quantitative geology and geostatistics)
Includes index.
\. Mines and mineral resources-Statistical methods. 2. Geology-
Statistical methods. Matheron, G. (Georges) II. Armstrong, M. .
1950- Ill. Series_
TN153.G46 1987 662'.1 86-31518
ISBN -13: 978-94-010-8018-7 e-ISBN :978-94-009-3383-5
001: 10.1007/978-94-009-3383-5
Published by D. Reidel Publishing Company,
P.O. Box 17,3300 AA Dordrecht, Holland.
Sold and distributed in the U.S.A. and Canada
by Kluwer Academic Publishers.
101 Philip Drive, Assinippi Park, Norwell, MA 02061, U.S.A.
In all other countries, sold and distributed
by Kluwer Academic Publishers Group,
P.O. Box 322, 3300 AH Dordrecht, Holland.
All Rights Reserved
© 1987 by D. Reidel Publishing Company, Dordrecht, Holland
Softcover reprint of the hardcover 1st edition 1987
No part of the material protected by this copyright notice may be reproduced or
utilized in any fonn or by any means. electronic or mechanical
induding photocopying, recording or by any information storage and
retrieval system. without written permiSSion from the copyright owner
TABLE OF CONTENTS
Preface Vll
J. RIVOIRARD / Computing Variograms on Uranium Data 1
P. CHAUVET / The Comparison Between the Gamma Logs and the Grades
in the Estimation of a Uranium Deposit 23
P. A. DOWD and D. W. MILTON / Geostatistical Estimation of a Section of
the Perseverance Nickel Deposit 39
G. CAPELLO, M. GUARASCIO, A. LIBERTA, L. SALVATO, and G. SANNA /
Multipurpose Geostatistical Modelling of a Bauxite Ore body in
Sardinia 69
L. MOINARD / Application of Kriging to the Mapping of a Reef from
Wireline Logs and Seismic Data; A Case History 93
A. GALLI and G. MEUNIER / Study of a Gas Reservoir Using the External
Drift Method 105
CH. KA VOURINOS / The Grade-Tonnage Curves for a Zinc Mine in France 121
A. ZAUPA REMACRE / Conditioning by the Panel Grade for Recovery
Estimation of Non-Homogeneous Orebodies 135
D. GUIBAL / Recoverable Reserves Estimation at an Australian Gold
Project 149
H. SANS and J. R. BLAISE / Comparing Estimated Uranium Grades with
Production Figures 169
C. DEMANGE, CH. LAJAUNIE, CH. LANTUEJOUL, and 1. RIVOIRARD /
Global Recoverable Reserves: Testing Various Changes of
Support Models on Uranium Data 187
L. DE CHAMBURJ;, CH. DE FOUQUET, and H. FRAISSE / Calculating Ore
Reserves Subject to Mining Constraints, for a Uranium Deposit 209
Index 247
PREFACE
It is now nearly 25 years since the first textbook on
geostatistics ("Traitj de gjostatistique appliquje" by
G. Matheron) appeared in print in 1962. In that time geostatis
tics has grown from an arcane theory regarded with scepticism by
statisticians and miners alike, to a reputable scientific disci
pline which is routinely used in the geosciences. In the mining
industry, in particularly, comparisons between predicted reserve
estimates and actual production figures have proved its worth.
Few now doubt its usefulness as a statistical tool in the earth
sciences.
Over the past quarter of a century, many geostatistical
case studies have been published but the vast majority of these
are routine applications of kriging. Our objective with this
volume is to present a series of innovative applications of
geostatistics. These range from a careful variographic analysis
on uranium data, through detailed studies on geologically complex
deposits right up to the latest nonlinear methods applied to
deposits with highly skew data distributions. Applications of new
techniques such as the external drift method for combining well
data with seismic information have also been included.
Throughout the volume the accent has been put on how to
apply geostatistics in practice. Notation has been kept to a
mininmum and mathematical details have been relegated to annexes.
We hope that this will encourage readers to put the more sophis
ticated techniques into practice in their own fields.
G. MATHERON
M. ARMSTRONG
COMPUTING VARIOGRAMS ON URANIUM DATA
Jacques RIVOIRARD
Centre de Geostatistique
ECOLE NATIONALE SUPERIEURE DES MINES DE PARIS
35 rue Saint-Honorf, 77305 FONTAINEBLEAU, France
ABSTRACT
The objective of this paper is to show how well known the
structure is for a particular uranium deposit. After presenting
the average vertical variogram for all the holes, some of the
individual variograms will be studied and we will show the
influence of a few very rich holes on the overall variogram. This
turns out to be poorly defined. The first order variogram, which
also has been considered, is curiously similar to the usual
variogram, whereas the structure of the translated logarithm
proves to be better known.
1. PRELIMINARY REMARKS ON COMPUTING VARIOGRAMS
Determining the geostatistical structure is often a difficult
task. Basically the objective is to estimate the spatial
integral:
J
",((h) 1 [z(x+h) - z(x)]2 dx
2lsns_hl Sns
-h
where z(x) is the regionalised variable, S is its field (or a
part of its field that is supposed to be homogeneous), and S-h is
the field translated by -h.
With a regular grid of measurements, the experimental variogram
is just a discretised approximation of this integral. When no
G. Matheron and M. Arm5lrong (eds.), Geostatis/ira! Case Studies, 1-22.
© !<)87 by D. Reidel F/lb/islril/g Olll,,<l"),.
2 J. RIVOIRARD
regular grid exists, estimating the structure may be hasardous.
But even in the most favourable cases (e.g. variograms along
drillholes), computing the structure is not always easy, and it
can be useful to appreciate how well known the structure is. For
example, when holes in several directions are available, the
differences between directional variograms can be due to
anisotropy (independently from different levels of variability,
due to a proportional effect); but they can also be accidental, f
the structure are badly known.
The objective of this paper is to show how well known the
structure is for a particular uranium deposit. After presenting
the average vertical variogram for all the holes, some of the
individual variograms will be studied and we will show the
influence of a few very rich holes on the overall mean variogram.
The variogram obtained after taking translated logarithms and the
first order variogram will also be considered.
2. THE CASE-STUDY
The data come from a regular grid of 37 uncorrelated vertical
holes, drilled in a part of a stockwerk uranium deposit. Each
hole intersects a small but variable number of randomly oriented
mineralised veins (fig. 1). As there is no correlation between
veins from neighbouring holes, the veins cannot be studied
individually. At this stage, this orebody has to be estimated as
a massive deposit which will be mined in a selective way.
The variable under study is the average radiometric grade of each
1.5m sample (in order to preserve confidentiality these values
have been multiplied by an arbitrary coefficient). The mean m is
1.10, while the variance a2 is 12.78, and so the ratio aIm is
3.25 (This is independent of the confidentiality coefficient).
The histogram is very skew (fig. 2). The waste passes represent
62% of the total length of the holes. The distribution can not be
considered as lognormal, for a test of lognormality would only
concern 38\ of the values. It is important to note that the
radiometry values for waste passes are subject to errors. They
are all low but differences between the values are not
significant.
3. THE AVERAGE VARIOGRAM AND THE VARIANCES
The average variogram looks quite good (fig. 3). It increases for
the first 8 lags (up to 12 meters); then it reaches a sill which
drops down slowly a bit further on. Then, at about 45 meters, it
drops suddenly indicating that both ends of the holes are
relatively poor.
COMPUTING VARIOGRAMS ON URANIUM DATA 3
This variogram is the average of all the individual vertical
variograms. For each hole the average value of 1/2 {z.-z.)2 for
all pairs (z. ,z.) of data [i.e. the average of all the1pofnts of
the variogra~, Jzero lag included, weighted by the number of
pairs] is equal to the sample variance of this hole. So the
influence of a given hole on the average variogram depends
directly on its variance (and on the number of data, of course,
but this is constant in our case).
As can be seen on the scatter diagram (fig. 4), the variance of a
hole is roughly related to its mean grade. But above all this
figure shows the extreme diversity of the variance values.
Twenty-one values are quite small (less than 3), three are
enormous (147, 69 and 61), some others are still important (18.6,
15.1,14.4, 12.8, 10.6).
The lack of robustness of the variance to "outliers" is well
known. In this case the variance drops from 12.78 to 7.54 (and
then 5.57, 3.90 and 1.88) as the most variable hole (or the most
variable 2, 3 or 8 holes respectively) are removed. This shows
how imprecisely the sample variance (and hence the sill of the
variogram) is known.
In the same way, the holes with the highest variance have a very
great influence on the average variogram. For example, except for
the value of the sill, the average variogram of the 8 most
variable holes is identical to the overall average vertical
variogram (fig. 5).
It is therefore very instructive to look at the individual
structure of each of these eight holes (it is important to note
that their spatial location is not preferential). For the rest of
the article the drillholes have been numbered in decreasing order
of variability.
4. THE INDIVIDUAL VARIOGRAMS OF THE 8 HOLES
The behaviour of some of the individual variograms such as No 3
(fig. 6) is typical. This hole contains one very large value
(50.4) located ten 1.5m lags from the top and 31 lags from the
bottom (see the sample values in the Appendix). Clearly, for the
first 10 points of the variogram, this extreme value is present
in 2 pairs, whereas, from the 11th to 31st points, it occurs in
only one pair, which explains the sudden decrease in the
variogram between the 10th and 11th points, and also between the
31st and 32nd.
It is easy ,to see that, if the n = 42 samples of the hole were
equal to 0.1, except the 11th which were equal to 50.4, the
4 J. RIVOIRARD
variogram (fig. 7) would be:
- hyperbolic l(h) = (50.4 - 0.1)2 /(n-h) for the first 10 lags,
- then, after dropping to half its initial value, it would be
hyperbolic again l(h) = (50.4 - 0.1)2 /2(n-h) the next 22 lags,
before dropping to zero.
So the distances of the sudden drops (as well as the rate of
increases) depend on the position of these values relative to the
ends of the hole.
Drillholes Nos 6 and 2, which contain some following large
values, surrounded by much lower values, have the same general
shape (fig. 8 and 9) . But also, the succession of these large
values is responsible for the continuity given by the first
points of the variogram.
Hole No 8 is very interesting (fig. 10). There are 2 rich
samples, 17.9 and 11.8, 3 lags apart. In addition to the drops
caused by the disappearance of these values, the variogram shows
a hole effect at the 3rd lag, where the difference between the 2
high values (17.9 - 11.8) reduces their enormous influence on the
variogram. Holes Nos 7 and 5 show the same type of behaviour for
similar reasons (hole effects at the 11th lag on fig. 11, and
from the 2nd to the 4th lags on fig. 12). The sudden drop seen in
fig. 13 for hole No 4 is also due to the position of the maximum
value 25.46. The hole effects at the 24th and 10th lags are due
to differences between relatively large values (25.46 - 9.57, and
then 9.57 - 5.00 and 6.67 - 4.54 respectively).
Lastly, hole No 1 (fig. 14) contains a lot of rich samples. Its
structure is rather complex. Nevertheless, both of the most
sudden drops are due to the disappearance of the value 57.94 from
the pairs. Each of these drops is followed by several decreases
due to the rich neighbouring samples. The hole effect at the 4th
lag is due to the difference 57.94 - 42.79.
5. COMPOSITION OF THE AVERAGE STRUCTURE
Except for the value of the sill (and of the variance), we have
seen that the structure is due to the most variable holes. It is
most surpr1s1ng to see that, in spite of the chaotic individual
variograms (increases interrupted by sudden drops, and hole
effects) the average variogram has a reasonably well defined sill
with a range of about 8 lags.
Of course the most variable hole has the dominant role in the
average variogram. It is responsible for the anomally at the 4th
lag (due to a hole effect): see fig. 3 and 14. If this hole is
removed, the variogram still shows a reasonable degree of
COMPUTING V ARIOGRAMS ON URANIUM DATA 5
continuity, which is mainly due to hole No 2 (see fig. 9, 15 and
16) .
The sudden drop for large values of h is still evident, even
after the three most variable holes have been removed. This is
because the extremities of the holes are relatively poor. But
this decrease disappears after the 8 most variable holes have
been removed (fig. 18). So the extremities are not without
mineralisation (in which case they should have been eliminated),
but this is not as rich as in the 8 holes.
6. INFLUENCE OF THE RICHEST SAMPLES AND HOLES
In cases like this it is the richest samples that can make an
orebody payable. As they are not preferentialy located, removing
the extreme values because they would perturb the variogram could
be dangerous because the residual structure is likely to be
meaningless.
But the influence of a few holes can be so high, that a real
knowledge of the structure can be based only on their stability.
This is the only way to interpret the evolution of the average
variogram, as the richest holes are removed. We have seen that it
is not very stable. The most stable element is undoubtly the
permanence of a sill between the 10th and 30th lags.
This shows how instable second order statistics such as the
variogram and the variance, are. It suggests testing other
related estimators of spatial structure, for instance the first
order variogram:
J
'( 1 (h) = --- ! z (x+h) - z (x)! dx
2!SOS_h! SoS
-h
Unfortunately this turns out to be just as unstable as the
ordinary variogram, and the similarity is so high that the ratio
of these two variograms is nearly constant (fig. 19). This result
is very interesting, for it is a good indication for the mosaic
model (Matheron, 1984), which effectively can give excellent
results for grades distributions in cases like this (Lantuejoul
and al., 1986).
Another way to reduce the influence of the high values without
eliminating them is to take logarithms. Care has then to be taken
not to amplify small differences between low values, which a
gaussian anamorphosis would also do. This danger can be avoided
with the translated logarithm: log(n+z), the translation constant