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247 Pages·1987·10.154 MB·English
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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

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