A comparison of the geostatistical ore reserve estimation method over the conventional methods

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1 A comparison of the geostatistical ore reserve estimation method over the conventional methods Item Type text; Thesis-Reproduction (electronic) Authors Knudsen, H. Peter (Harvey Peter), Publisher The University of Arizona. Rights Copyright is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author. Download date 07/06/ :34:15 Link to Item

2 A COMPARISON OF THE GEOSTATISTICAL ORE RESERVE ESTIMATION METHOD OVER THE CONVENTIONAL METHODS by Harvey Peter Knudsen, Jr. A Thesis Submitted to the Faculty of the DEPARTMENT OF MINING AND GEOLOGICAL ENGINEERING In Partial Fulfillment of the Requirements For the Degree of MASTER OF SCIENCE WITH A MAJOR IN MINING ENGINEERING In the Graduate College THE UNIVERSITY OF ARIZONA

3 STATEMENT BY AUTHOR This thesis has been submitted in partial fulfillment of requirements for an advanced'degree at The University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library. Brief quotations from this thesis are allowable without special permission, provided that accurate acknowledgment of source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in his judgment the proposed use of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author. SIGNED: APPROVAL BY THESIS DIRECTOR This thesis has been approved on the date shown below: Associate Professor of Mining and Geological Engineering

4 ACKNOWLEDGMENTS I wish to express a special thanks to Dr. Y. C. Kim, my thesis advisor, for his guidance and advice in all phases of this research. Dr. Donald E. Myers of the Department of Mathematics is also thanked for his interest and help in enabling me to understand the theoretical basis of geostatistics. Dr. DeVerle P. Harris is thanked for his critique and review of this thesis. My sincere gratitude is expressed to the management of the Cyprus Pima Mining Company for supplying the necessary data for this research. Ed Mueller, Data Processing Director, and Don Williamson, Chief Geologist were especially helpful and I thank them both. I wish to thank Richard Bideaux of Computing Associates, Inc., for his suggestion to include the ore reserve estimation method, BLIP, in this research. This research was supported in part by.the Energy Research and Development Administration, Grand Junction, Colorado, through a contractual study titled, "A Comparative Study of the Geostatistical Ore Reserve Estimation Method Over the Conventional Methods.". iii

5 TABLE OF CONTENTS Page LIST OF ILLUSTRATIONS... LIST OF TABLES ABSTRACT... vi vii viii 1. INTRODUCTION ' 1 Purpose of Study... 3 Conventional Ore Reserve Estimation Methods... 4 Polygon Method Inverse of the Distance Squared Method (IDS) Modification of IDS Method (BLIP) Scope of Study A BRIEF INTRODUCTION TO GEOSTATISTICS Regionalized Variables Variogram Function Variogram Models Variance of Block Grades Extension Variance Estimation Variance Kriging GEOSTATISTICAL ORE RESERVE ESTIMATION OF THE PIMA MINE 24 Pima M i n e Variogram Calculation 25 Kriging Kriging Variance and Its Accuracy PREDICTION ACCURACY OF THE FOUR ORE RESERVE ESTIMATION METHODS Tests of Prediction Accuracy Results of Prediction Accuracy Tests Amount Location Computational Aspects iv

6 V TABLE OF CONTENTS--Continued Page 5. CONCLUSIONS AND. SUGGESTIONS FOR FUTURE RESEARCH Considerations In Model Selection Importance of Geology Suggestions for. Future Research APPENDIX A: FLOWCHART AND PROGRAM LISTING FOR PROGRAM GAMMA APPENDIX B: FLOWCHART AND PROGRAM LISTING FOR PROGRAM KRIG REFERENCES CITED

7 LIST OF ILLUSTRATIONS.;ure Page V Variogram of Maggie Canyon manganese deposit Degrees of continuity expressed by variogram Hypothetical example of geometric anisotropy Histogram of DDH assay values, Pima mine Cumulative frequency distribution plot Experimental horizontal variogram, Pima mine Vertical and horizontal DeWijsian variograms, Pima mine Horizontal DeWijsian variograms showing functional anisotropy ' Kriging variance plot Example of calculations used in comparing predicted block grades to actual block grades Histograms of block grade differences between predicted vs. actual Plan map of differences in block grades--blip Plan map of differences in block grades--geostatistics Plan map of differences in block grades--ids L>. Plan map of differences in block grades--polygon

8 LIST OF TABLES Table 1. Accuracy of kfiging variance Results of accuracy tests (% copper)

9 ABSTRACT A comparison was made of the Geostatistical ore reserve estimation and three conventional ore reserve estimation methods. Conventional methods tested are the polygon method, the inverse, of the distance squared. (IDS) method, and BLIP which is similar to IDS but allowing different weights in different directions. Each of the methods was evaluated on its ability to accurately predict individual block grades and total ore reserves for one bench of the Pima Mine, located near Tucson, Arizona. The test bench contained actual block values (arithmetic mean of blast hole assays) for 542 blocks representing 17 million tons of material. Results at the Pima Mine indicate that the geostatistical method is no better nor worse in predicting block grades and total tonnage than the BLIP or IDS methods. In contrast, the polygon method is clearly inferior. All of the methods would have given better results if geology had been included in the block model. At already producing mines BLIP or IDS may be good choices since they involve substantially less cost in computer time. For prospective or new properties, the geostatistical model is more attractive, due to its ability to construct confidence limits on the estimates. viii

10 CHAPTER 1 INTRODUCTION The importance of accurate ore reserve estimates has always been recognized in the past. Mining companies obviously need accurate ore reserve estimates since the quality of the estimate may directly affect the company s profitability. Accurate ore reserve estimates are also becoming increasingly important to,governmental bodies that must make policy decisions regarding nonrenewable natural resources. The U.S. Energy Research and Development Administration, for example, calculates the domestic uranium ore reserves as part of its program to assess the adequacy of domestic uranium supplies and to develop policies relating to long range nuclear fuel supply. These policies involve; 1) the domestic use of foreign uranium, that is, when or if the embargo on foreign uranium should be lifted, 2) planning the operation of existing enrichment plants and the size and timing of new capacity, and 3) planning and development of nuclear energy system (Nuclear Fuel Resource Evaluation, 1973). Such decisions can involve extremely large expenditures and have international implications. Thus, with today1s growing shortages of natural resources and the increasingly large investments to open new mines, the need for accurate ore reserve estimates by both private industry and government becomes almost critical. Yet many of the ore reserve estimation methods used by practioners today lack a sound scientific basis.

11 Results of sampling and ore reserve estimation research by the U.S. Bureau of Mines (Hazen, 1967), by Krige (1951, 1962) and Sichel (1966) in South Africa, and by Matheron (1963) in France have made significant contributions to the current art of ore reserve estimation. Sichel (1966) and Krige (1951, 1952, 1962) successfully applied statistical methods to ore reserve estimates in the gold fields of South Africa. Their work involved use of the lognormal distribution to describe the distribution of assay grades in the gold deposits and were able to more efficiently calculate the ore reserves from a small number of samples. Hazen (1958, 1967) of the U.S. Bureau of Mines studied the use of classical statistics for analyzing sample and assay data of mineral deposit. The use of classical statistics in ore reserve estimation can be questioned on the basis that the assay values are not random but generally are correlated to some degree, as it is well known to geologists. Research by Matheron focused on the study of data that exhibit spatial correlation, a typical example of which is geologic data. The resulting method developed by Matheron (1963) which is commonly referred to as geostatistics has special appeal to geologists and mining engineers. One reason is that it does explicitly take into account the spatial correlations between samples. Another reason is that it makes better use of available data and provides confidence limits for the estimate. The theoretical basis of geostatistics is the Theory of Regionalized Variables developed by Matheron. His earliest publications were in French and not widely known in the United States. His first article

12 in English appeared in Economic Geology in The first really comprehensive summary of the applications of geostatistics published in English was in 1968 by Blais and Carlier. Since 1968 many papers on the theory and applications of geostatistics have been published in English. These papers have stimulated interest in applying geostatistics to ore reserve estimation problems in the United. States. Yet actual applications in the United States seem to be few. Purpose of Study At the majority of open pit copper mines in the U.S., blast hole assays and occasionally face samples are used to distinguish between ore and waste in the actual mining of the deposit. However short range mine planning (monthly to quarterly) is based on the block grades predicted from diamond drill hole assays. These predicted block grades are subject to rather wide variation from the actual grades of the blocks. Thus, a mine planning engineer may face the situation in which a certain area of the pit has no ore even though the predictions indicated otherwise. How critical this problem is to a given mine depends on many factors such as the number of exposed ore faces, the amount of advance stripping, the amount of stockpiled ore, etc. In other words, depending on the particular mining philosophy employed at a mine, this problem can be very critical. Therefore the need is quite important for accurate ore reserve estimates and in particular accurate block grade predictions.

13 There have been many excellent case studies describing the application of geostatistics to the estimation of total ore reserves and individual block grades (David, 1969, 1974; Huijbregts and Segovia, 1973) and the calculation of the possible error of the predictions. There has not been, however, a study comparing the predictions made by the geostatistical method with the actual mined out grades and tonnages at an operating mine.. The purpose of this study was to compare the accuracy of the geostatistical method with the conventional methods. Since the geostatistical method had not been exposed to many of the practitioners in the United States it was hoped that results from this study would contribute to a better understanding of the properties and merits of this method. Conventional Ore Reserve Estimation Methods Among the many conventional ore reserve estimation methods only three methods will be discussed here. They are; 1) the polygon method, 2) the inverse of the distance squared method, IDS for short, 3.) and a modification of the inverse of the distance method, which will be called BLIP in this study. These methods were chosen because they are well known and used by the mining industry. Polygon Method This method has.been widely used in the past for porphyry copper deposits, especially prior to the introduction Of computer in ore reserve calculations.

14 In this method the assay grade of a drill hole is extended halfway to each adjacent hole. This defines the area of influence of the hole. If a block model is used, each block receives the grade of the nearest hole. This method is extremely simple and easily understood. It has the difficulty that when there are few drill holes in an area, each hole can have a rather large zone of influence. To prevent such an influence, common practice has been to choose some arbitrary distance called the radius of influence and to limit the grade assignment of individual blocks to those that lie within the radius of influence. Inverse of the Distance Squared Method (IDS) With the introduction of computers in ore reserve computations, the use of distance weighting techniques became practical. In these techniques the grade assigned to a block is a weighted average of all or nearly all holes surrounding the block. The weights assigned to each hole is a function of the distance between the hole and the center of the block. The weights must sum up to one, thereby forming the linear combination of the grades surrounding the block. In the inverse of the distance squared method (IDS) the weights are inversely proportional to the distance squared. Many other variations of the distance weighting have been devised and used.

15 Modification of IDS Method (BLIP) This is the second distance weighting tested. The basic difference between IDS and this method is that IDS gives equal weight in all directions for a given distance, whereas with BLIP the weight can be varied for different directions. Scope of Study First a geostatistical ore reserve estimation is made for the total ore reserves as well as the individual block grades for one bench of an open pit porphyry copper mine using assay data from diamond drill holes. Computing ore reserves by the geostatistical method involves two distinct phases of the work. Phase one consists of calculating an experimental variogram and determining the theoretical model that best describes the experimental variogram. The estimate of the total ore reserve and its confidence limits are computed using the theoretical variogram. As part of this phase, a computer program is written to calculate variograms from irregularly spaced data. In phase two a computer program is written to do kriging. The individual block grades for the bench are then estimated by kriging. Next, ore reserve estimates of the same bench are obtained by three conventional methods. These estimates as well as the one by the geostatistical method are compared with the actual block grades to determine the accuracy of individual block estimates and the overall estimates for each method. Since the geostatistical method is a relatively new method, a brief review of the basic concepts of geostatistics that are essential

16 to ore reserve estimation is given in Chapter 2. Subsequent chapters discuss the actual geostatistical study and the comparison of the accuracy of the four methods.

17 CHAPTER 2 A BRIEF INTRODUCTION TO GEOSTATISTICS The theoretical basis of geostatistics was developed by Georges Matheron in the late 1950's and early 1960's in order to study data that exhibit spatial correlation to some degree (Matheron, 1963, p. 1247). Such data violate the assumption of random data that are almost always necessary in the treatment by classical statistics. The theory developed by Matheron is known as the theory of regionalized variables. Regionalized Variables - A regionalized phenomenon spreads in space and exhibits a certain spatial structure. The value of this phenomenon at a point x is called a regionalized variable (Matheron, 1971, p. 5). The grade of ore in a channel sample is an example of a regionalized variable. Regionalized variables occur in a given field and have a geometric support. For instance, the field of a channel sample is the ore deposit and its support is the size and shape of the sample. A regionalized Variable is simply a function f(x) of the point x. This function is a very irregular one. The function displays two contradictory aspects; 1) a random aspect and 2) a structured aspect. The random i..aspect is characterized by irregular and unpredictable variations from point to point. The structured aspect is characterized by some degree of spatial correlation of the values of the variable.

18 9 The theory of regionalized variables has two main purposes (Matheron, 1971, p. 5). The first is to express the structural properties of the regionalized variable in some form. The second purpose is to solve this problem of estimation of the regionalized variable from fragmentary sample data. To solve the problem of estimation of the regionalized variable, Matheron introduced a probabilistic interpretation of the regionalized variable in which a regionalized variable is considered to be a realization of a random function (Matheron, 1971, p. 6). Several hypotheses concerning the random function Y(x) have been introduced to define the minimal probabilistic characteristics necessary to solve the estimation problem. The first hypothesis is that the random function is second order stationary. This says that the expectation m is independent of the location x m = E[Y(x)] 2.1 and that the covariogram exists and is independent of the location x but dependent only on the distance h. K(h) = E[Y(x+h) Y(x)]2 - m2 2.2 Note that when h is equal to zero, the covariogram K(o) is equal to the variance of the samples in the deposit. This hypothesis sometimes is too strict and is replaced by the intrinsic hypothesis. Here, only the increments [Y(x+h) - Y(x)] of the random function Y(x) are assumed to be stationary. Both the expectation m(h)

19 E[Y(x+h) - Y(x)] = m(h) and the variance of the increments D2 [Y (x+h) - Y (x) ] = 2y (h) 2.4 are dependent only on the distance h between the samples at point (x) and (x+h). A random function that obeys the hypothesis of second order stationary also obeys the intrinsic hypothesis. In this case, the variogram y(n) and covariogram K(h) are related in accordance with equation 2.5 below: y(h) = K(o) - K(h) 2.5 The variogram 2y(h) in equation 2.4 is used to study regionalized variables obeying either of the above hypotheses. Variogram Function The variogram is a function that expresses the degree of continuity of a regionalized variable. Thus, it is the basic tool of geostatistics. The variogram describes the average squared difference between samples h feet apart and is defined by equation 2.6: 2.6 where 2y(h) = value of the variogram for distance h f(x) = sample value at point x

20 11 f(x+h) = sample value at point x+h h = distance between samples. The variogram is a vector function which generally increases with increasing distance h. Figure 1 is an example of a variogram. On this graph the abscissa value is the distance h and the ordinate the value of y(h) instead of 2y(h). The variogram can express the following structural characteristics of the phenomenon under study (Matheron, 1971, p. 58). 1. Continuity. The continuity of the phenomenon under study is reflected by the rate of growth of the variogram near the origin. Figure 2 illustrates different degrees of continuity expressed by the variogram. A high degree of continuity is reflected by a parabolic growth near the origin. Average continuity is expressed by a regular growth from the origin. A discontinuity at the origin is called a ' nugget effect" and can have real physical meaning or be due to errors of measurement, recording, or sampling. A completely discontinuous variogram indicates a pure random phenomenon. 2. Zone of influence. The variogram gives a concise definition of the notion of zone of influence. Often, beyond a certain distance called the range, the variogram becomes nearly flat indicating that the samples are independent (see Figure 2). A sample has no influence beyond the range of the variogram. A variogram with a range is called a transitive variogram. In some cases the variogram never reaches a limiting value but

21 12 40-n 30- Y(h) n r Feet Figure 1. Variograin of Maggie Canyon manganese deposit. Assay data used in the calculation of this variogram was taken from Hazen (1958, p. 36).

22 13 High degree of continuity. Can be approximated by parabola near the origin. Bed thickness is an example. range Average continuity. Almost linear near origin, Typical of many metal deposits. T Nugget Effect Nugget effect. Discontinuity at the origin and thereafter much like above variogram. y(h) Purely random. Figure 2. Degrees of continuity expressed by variogram.

23 14 instead shows a steady increase with increasing distance. This is an intrinsic variogram and indicates the variogram obeys the intrinsic hypothesis. 3. Anisotropy. Variograms computed in different directions are often different, indicating anisotropy. Figure 3 shows an hypothetical example of anisotropy. The range of the horizontal variogram is 60 feet and the range of vertical variogram is 30 feet. The anisotropy factor is 60/30 or 2. This means that one foot in the vertical direction is equivalent to two feet in the horizontal plane as far as grade variation is concerned. Variogram Models Variograms calculated for mineral deposits have been found to be adequately described by a few theoretical models (David, 1974, p. 59). The most often used models arc the spherical model and the DeWijsian model. The spherical model is a transitive variogram and is expressed by the equation 2.7: Y(h) = Co + C[ - i ^- ] for h < a and y(h) = Co + C for h > a. 2.7 where Cq = nugget value C = sill value minus Co h a = distance = range.

24 Vertical -Variogram Range Horizontal Variogram Range Feet Figure 3. Hypothetical example of geometric anisotropy.

25 16 The DeWijsian variogram is an intrinsic variogram. It is a model of an experimental variogram calculated by using the logarithms of the assay grade rather than the assay grade. The DeWijsian variogram is y(h) = 3a ln(h) + b 2.8 where a = intrinsic coefficient of dispersion h = distance b = constant. The main use of the theoretical model of the experimental variograms lie in the actual calculations required for an ore reserve estimation. Throughout the subsequent sections of this thesis, the theoretical models of the experimental variograms will be referred to as "variograms," for the sake of brevity. Variance of Block Grades The variance of point1 samples o within any size volume v will be denoted by a(-0//vj and it can be calculated by equation 2.9 o ( o/ v) = fv dx /v Y(x~y) dy 2-9 which is condensed notation for a sextuple integral that calculates the average value of the variogram within the volume V when each extremity 1. The term point sample is used to indicate the sample is very small in relation to the ore deposit.

26 of the vector y(x-y) sweep the volume V on its own accord. Explicitly, equation 2.9 can be written as equation 2.10: 17 I X1 x 2 x 3 Yi Y2 a(0 /v) V ^ dxl fo4x2 Jo ^ 3 Jo dyl Jo dy 73 foy(x1-y1, x2-y2, x3-y3) dy The variance a^0/v) depends on the variogram and the size and shape of the volume. The variance of blocks v within a larger block V is calculated using equation 2.11 which is known as Krige's relationship (Matheron, 1963, p. 1254). a(v/v) = a(o/v) " a(o/v) 2-11 The two quantities on the right side of equation 2.11 can be expressed by equation 2.9 in terms of the variogram. This gives equation Or = /y dx Jv y(x-y) dy - f^dx y(x-y) dy 2.12 r 2 (v/v) v 2 "V "V v2 Extension Variance The error made by extending the grade of a sample to another point or to a volume is termed extension variance. It is defined mathematically as q 2 = E[Z(v) - Z(v')]2 2.13

27 18 where Z(v) = the grade of block v that is extended to block v' Z(v ) = the grade of block v'. The calculation of the extension variance of an ore reserve estimate is one of the main purposes of geostatistics. Expanding equation 2.11 and rewriting in terms of variance and covariance gives equation E 0(v/D) + a (v'/d) " 2o(v,v') 2'14 Hie variance is the variance of blocks v within the deposit D. The covariance term ^ is the covariance between block v and block v and is defined in terms of the covariogram by equation a(v,v') = /vdx 4'K(x">,)dy 2-15 Equation 2.14 can now be rewritten in terms of the covariogram to give equation 2.16: oj5 = -1 /dx /K(x-y)dy + - J^, dx /v,k(x-y)dy or in terms of the variogram to give 2.17:

28 19 /y dx /v,y(x-y)dy - ^dx /v r(x-y)dy dx /v. Y(x-y)dy Matheron has shown that equation 2.17 remains valid for any intrinsic random function even if the covariogram does not exist. Estimation Variance The concept of extension variance and estimation variance are the same. However the term estimation variance is used when the extension of several samples to a volume is being made. Equations 2.16 and 2.17 must be modified to calculate the estimation variance of extending N samples to the volume v'. Instead of knowing Z(v) exactly it is instead estimated by Z(v) = h I Y(x.) in i=l The equation for the estimation variance is derived from equation 2.16 by replacing the integrals over the volume v by a discrete summation taken on N samples. This gives equation 2.19 which is the estimation variance in terms of the covariogram. W 1 1 /v 'K(xi"y)dy 2.19

29 the variogram. Equation 2.20 is the estimation variance expressed in terms of E=wr /vy(xi-y)dy-^7 /V'^ 4-Y(x-y)dy- 5f N i k J N 2.20 From equations 2.19 and 2.20, it can be noted that the estimation variance can be decreased by reducing the size of the sampling grid thereby increasing the number of samples. Kriging The previous section discussed how to calculate the probable error of an estimate. This section will discuss how to make the estimate in such a way that it is unbiased and has minimum estimation variance. The procedure used is called kriging and was developed by Matheron (1971, p. 115). 2 The object of kriging is to find the best linear estimator of the grade of a block by taking into account all available samples. Kriging is a weighting procedure. The weights are calculated for each sample in such a manner to minimize the estimation variance subject to the geometrical constraints of the problem. 2. A linear (convex) combination of N samples X^...X^ is de- N

30 21 The derivation of kriging equations for the case of a second order stationary random function with unknown expectation is given in this section. An estimator Z*(v) of the true grade of block Z(v) is expressed as a linear combination of samples X. by equation 2.20 N Z*(V) = I AiX: 2.20 i=l The s are the weights to be determined subj ect to the constraint of equation N V A = i=l 2 The estimation variance E[Z*(v) - Z(v)] modified for the use of weights is as given in equation N N 2.22 N Since the estimator I i=l must be unbiased and must have minimum estimation variance these two conditions become the necessary constraints to the problem of determining the weights. The optimal weights are those that minimize equation 2.22 subject to the constraint of equation This problem expressed as a constrained optimization problem is readily amenable to solution using the method of Lagrange Multipliers (Gupta and Cozzolino, 1974, p. 43).

31 The Lagrangian function of the problem is given as equation 2.23 below. N L ( X 1, X 2,... X n,vi) ( V / D ) ' 2 i ^ 1X i a ( X i, v ) + N N N ik +2-aXi-U 2.23 The use of covariance notation instead of the more explicit integrals is for convenience in writing this and later equations in this section. Taking the partial derivatives of L (equation 2.23) with respect to gives equation % " ' 2a Cxv v) + 2xi (x 1,x1) + 2, l2 Xj (x 1,x.) + 2m 2l24 Repeating this for each and setting the equations equal to zero yields a set of N equations of the form: N Xia(Xi,Xj) + M = (Xi,v) 2-25 The final partial derivative of L is taken with respect to y and yields the original constraint equation 2.21.

32 With this last equation there is now a set of N+l equations containing N+l unknowns as given in matrix notation by equation a(x1,x1) a(x1,x2) a (x2,x1) a(x2,x2) X1 x 2 *(%l,v) *(X2,v) 2.26 XN y 1 The solution to the above simultaneous equation provides the optimal weights and the Lagrange multiplier y. Knowing the weights and y, the estimation variance which is called the kriging variance is calculated by equation K ~ a (v/d) ' ^ Xia(X.,v) N 2.27

33 CHAPTER 3 GEOSTATISTICAL ORE RESERVE ESTIMATION OF THE PIMA MINE The Pima mine is a porphyry copper deposit mined by open pit method and is located near Tucson, Arizona. The Cyprus-Pima Mining Company provided approximately 300 diamond drill hole assays and approximately 10,800 blast hole assays for one bench of the Pima mine for use in this study. The diamond drill hole assays are used to calculate the experimental variogram for the test bench and to assign grades to the blocks, whereas the blast hole assays are used to obtain the true grade of each block which is later compared with the predicted grade from the various estimation methods. The geostatistical ore reserve estimation consists of two phases. Phase one is the calculation of an experimental variogram and determination of the theoretical variogram to be used in phase two. Phase two is prediction of the individual block grades by kriging. Pima Mine A standard block model is used by the Pima mine to describe their orebody. The blocks are one bench high (40') and 100 feet by 100 feet in plan. A copper grade is assigned to each block in the model based on the diamond drill hole assays. The deposit has been developed by diamond drill holes drilled on a more or less regular grid with a spacing of about 200 feet.. 24

34 V ' The test bench contains actual block values for 542 blocks representing 17 million tons of material. The actual block value is defined for this study to be the arithmetic average of the blast hole assays within the block. Typically between 15 and 25 blast holes are drilled in each block. The diamond drill hole assays of the copper grade have a skewed distribution as is shown in Figure 4. The assay values represented by the histogram in Figure 4 were obtained by choosing the assay value of the diamond drill hole nearest to each grid point of a.400 foot square grid point superimposed on the deposit. This was done because the whole deposit has not yet been fully developed by holes on a 200 grid and if the 200 foot grid had been chosen the higher grade portions of the orebody would be given too much weight and a possible bias could result. The cumulative percent frequency distribution of these assays was plotted on logarithmic probability paper to determine if the assay values are lognormally distributed. The nearly straight line shown in Figure 5 indicates the assay values can be adequately described by a lognormal distribution. Yariogram Calculation Diamond drill hole assays of the copper grade were used to calculate the experimental variograms of the Pima mine. The samples used in the calculation were 40 foot composits of the copper grade for the bench under study as well as the composits from the benches above and below the test bench. A Fortran program called Gamma was written to

35 o\ Grade Figure 4. Histogram of DDH assay values, Pima mine.

36 y Grade Cumulative Percent Frequency Figure 5. Cumulative frequency distribution plot.

37 28 calculate the variograms from the irregularly spaced drill holes. Appendix A contains a flowchart and listing of program GAMMA. The first calculations using the actual assay values produced the experimental variogram in Figure 6. This variogram is similar to a transitive variogram except that y(h) never levels off. Instead, y(h) continues to increase with increasing distance. This observation plus the fact that the assay distribution is lognormal suggested that the variogram might be an intrinsic variogram. An experimental variogram was then calculated using the logarithms of the assay values. The result is the average horizontal variogram shown in Figure 7, which is a good DeWijsian variogram. This experimental variogram was modeled by the DeWijsian variogram of equation 2.8 as rewritten here. y(h) = 3a log (h) + C 2.8 The average horizontal variogram at the Pima mine has an alpha value of The vertical variogram is also shown in Figure 7 and has an alpha value of The slopes of the variograms are slightly different, indicating there is anisotropy between the grade variations in the vertical and horizontal directions. This type of anisotropy is known as a functional anisotropy (Carlier, 1964, p. 301) and means that the anisotropy factor is not constant as in the case of geometrical anistropy. Carlier (1964, p. 301) has termed functional anisotropy to be in general unsolvable. Further investigation in the horizontal plane pointed out that a functional anisotropy exists between variograms in the NW and NE

38 29 y (h) Feet Figure 6. Experimental horizontal variogram, Pima mine.

39 30 Vertical Average Horizontal Variogram (a = 0.021) Feet Figure 7. Vertical and horizontal DeWijsian variograms, Pima mine NW -> Feet Figure 8. Horizontal DeWijsian variograms showing functional anisotropy.

40 31 directions as shown in Figure 8. The functional anisotropies encountered may indicate zonation in the deposit or may be due to the possibility that the two basic rock types in the mine have different assay distributions as recent work by the mine s chief geologist has pointed out (Williamson, 1975). The variogram model chosen for the kriging calculations was to use the average horizontal variogram as a three-dimensional isotropic variogram and to add a second term to describe the possible vertical zoning. The model is therefore Y (h) = 3aiso log10(h) + 3azon l o g ^ ( h ^ ) 3'1 where a. ^ = = isotropic coefficient of dispersion a zon h = = zonal coefficient of dispersion = distance invert = vertical component of distance h. The functional anisotropy in the horizontal plane was not included in the above model because there is no apparent way to utilize this informat ion. Kriging Each block on the test bench was assigned a grade by three dimensional kriging of the surrounding diamond drill hole assays. Compos its of the copper assays from the test bench and from the benches above and below the test bench were included in the kriging.

41 32 The kriging variance can be reduced by including more samples in the kriging. However, kriging tends to place the greatest weight on the first ring of samples surrounding the block and very little weight on samples beyond the first ring of samples. This is known as the screen effect. It was decided, therefore, to include only the six nearest holes in the kriging calculations. This was also done to reduce the amount of computer time used in the kriging calculations. Even with only including the 6 nearest holes, each of the holes have 3 composits. Thus there are 18 samples included in the kriging calculations. A computer program called Krig was written to do the kriging for this study. A listing of the program is included in Appendix B. The kriging equations developed in Chapter 2 which utilizes the covariogram cannot be used with an intrinsic variogram such as the one from the Pima mine because the covariogram does not exist in this case. However, Matheron (1971, p. 129) has shown that in the case of an intrinsic random function with a variogram but no covariogram, kriging can still be performed if the covariogram K(h) is replaced with a -y(h). The kriging equations for an intrinsic variogram are given below.

42 and 33 'K = - N fv dx /v Y(x-y)dy + i + I x^cxj^.v) 1 1 The notation y(x,xj) refers to the value of the variogram connecting the two points x^ and Xj. Also y(x^,v) is the average value of the variogram between the point x^ and the volume v. Kriging Variance and Its Accuracy After kriging a block the kriging variance for that block is calculated. This variance can be plotted on a map and areas having a large variance can be outlined to indicate areas of less reliable predictions and possibly needing further development drilling. An example of this is shown in Figure 9 for a small portion of the test bench. The accuracy of the kriging variance computed for each block is determined by taking the difference between the kriged block value and the actual block value and comparing this with the square root of the kriging variance for that block. Statistical theory indicates that 95 times out of a hundred the predicted value should fall within two standard deviations of the actual value, assuming a normal distribution with mean zero and variance equal to the kriging variance. Table 1 shows the results of this test to confirm the accuracy of the kriging variance. For the 542 blocks of the test bench over 95 percent of the blocks had actual values within the two standard deviations of the predicted values, thus confirming the accuracy of the kriging variance that is calculated for each block.

43 o o ;, o o o c < Figure 9. Kriging variance plot, o = Location of diamond drill hole.

44 35 Table 1. Accuracy of kriging variance. Actual Theoretical Blocks within one standard deviation % % Blocks within two standard deviations % % Blocks outside two standard deviations 24 4% 27 5%

45 CHAPTER 4 PREDICTION ACCURACY OF THE FOUR ORE RESERVE ESTIMATION METHODS An ore reserve estimation method should accurately predict the grade and tonnage of ore in a deposit and its approximate location. Predicting the ore s location is the more difficult task due to the extreme variability of the ore grade and the frequently large distances between samples. In this study the prediction accuracy of block grades is chosen as the main criterion for testing the accuracy of the four ore reserve estimates. This testing, however, involves essentially two items; 1) the average grade of the individual ore blocks, and 2) the individual block grade differences. Tests of Prediction Accuracy Let the predicted block grades be Z^*, Z^*,... Z^* and the actual block grades be Z^, Z^,... Z^. The average of differences in block grades is given by 1 n Average of differences F = ^ I 4.1 i=l The above average indicates how well a method predicted the average grade of the blocks for the area under study. A zero difference indicates a perfect prediction (either by chance or reason) whereas a consistent difference in either direction implies a bias. For this 36

46 37 study an average difference in block grade of 0.01% copper represents 1700 tons of copper. Using the same terminology as above, the variance of block differences is computed using equation 4.2. Di = H I, a zi-zi*) - "I The variance of block differences is a measure of how well a method can predict the grade of each individual block estimates. Figure 10 is an example of the calculation of the average and variance of block differences on a small portion of the test bench. Results of Prediction Accuracy Tests The results of the tests described in the previous section are discussed in this section. Amount Each of the four ore reserve estimation methods overestimated the average grade of the 542 blocks of the test bench. These results are shown in Table 2. On the basis of tons of copper, the overestimates ranged from 4,600 tons by ELIP to 11,000 tons by the polygon method. This general overestimation is due in part to the diamond drill hole assays being biased (Williamson, 1975).

47 ACTUAL GRADE OF EACH BLOCK PREDICTED GRADE FROM MODEL Actual Block Predicted Difference Difference Grade Grade Model Grade in Block Squared Average of differences (in block grades) = -56/16 = -3.5 Variance of differences = -^"^26(16-1) ^ = 7*624 Standard deviation of differences = /7~.624 = Figure 10. Example of calculations used in comparing predicted block grades to actual block grades.

48 39 Table 2. Results of accuracy tests.(% copper). Ore Reserve Method Average Difference Variance Standard Deviation BLIP -.03* Geostatistics IDS Polygon ' Negative value of average difference indicates overestimation. The smaller amount of overestimation, shown by BLIP and the geostatistical method is probably due to their utilizing assay values from the test bench as well as the benches above and below. In contrast the IDS method and the polygon method utilize assays only from the test bench and consequently showed a larger overestimation. Examination of the histogram of block grade differences for the four methods in Figure 11 shows that the polygon method made a greater number of large overestimation errors than the other methods. Therefore the polygon method is susceptible to large bias errors. These biases are due to the fact that the distribution of predicted block grades is approximately the same as the distribution of assay values because of the manner of assigning the area of influence. However the actual distribution of block grades is narrower and has less variance. Thus there will be a block with a predicted grade equal to the highest assay grade, although the true block grade is very unlikely to be that high.

49 40 X = -.30 S2 = BLIP S X = -.04 S2 = KRIGING X = S2 = IDS X = -.06, S2 = POLY Figure 11. Histograms of block grade differences between predicted vs. actual.

50 Location A comparison of the variance of block differences in Table 2 shows clearly that the polygon method is the poorest method for predicting the location of the ore. Its variance of block differences is twice as large as the next best method. BLIP gave the best results followed by geostatistics and IDS. Note however that these three methods gave essentially the same results. A hypothesis that the distribution of differences for the three methods have equal variances was tested by applying a test described by Hazen (1967, p. 74). The test is based on the F distribution and tests the assumption that several samples have equal variances. It is similar to the familiar F test for testing if two samples have equal variances. The three variances to be tested are labeled S^2, S^2, and S^2. The number of variances is designated k; the number of items in each sample is designated n^, n^, and n^; and the combined total number of items is designated N. Let (nu-l)s 2 + (n,-l)s 2 + (n -1)S 2 M = (N-k) ln[ ^ 9_] - (N-k) (n^-l)lns^2 - (n2-l)lns22 - (n^-l)^^2

51 which is approximately equal to. Making the substitutions into these formulas yields the value of F to be The book value of Fq gr-(3,4869) is 2.60 therefore the hypothesis that the distributions have equal variances can be accepted. The closeness of results by the three methods is most likely due to the deposit being well developed and the development drilling having been drilled on a more or less regular grid. Plan maps of the test bench showing the differences of block grades for the four methods are shown in Figures 12, 13, 14, and 15. Each method made prediction errors on roughly the same blocks. As can be seen from these figures, the northwest corner of the bench contained the most errors. This is due to inter fingering of homfels and porphyry along the contact between the two rock types. The hornfels are mineralized but the porphyry is essentially barren. Computational Aspects The computer was used in all the ore reserve estimates made with the four methods. With a very large orebody, the computation time required to perform an ore reserve estimate can be quite long and expensive. Thus, it was felt to be of importance to include in this study

52 ++******** + * * ************ ** * ** +*+++*$****** ** *-** *-**** *** ******* **** «- *>** + ****** 4-********** ***** * + + *** ***** ************ ***************** * * * ************ ***** ***+**** * * * **-. **** **** ************** +++* * ************************ * * ** + +** * *+ ************* ***** -**+++-**** * * * ****** *** + ** * + + *********** +** * * ******* ** * ******** *** * *** ** ******** * **** **** **** ******* ****** ***** * ************ -*************-******* _********* ********** * * * * ********* * ******* ********** ******* ******** ************* ********* Figure 12. Plan map of differences in block grades--blip. - Actual block value more than 0.15 lower than predicted value. * Actual block value within 0.15 of predicted value. + Actual block value more than 0.15 greater than predicted value.

53 ** * ********** * * * * + + -^-A ^ 41 ^ ^ ^ - 414* *** ** **** *>******** * >********** *** **+*** ***** ************ ** ************* * ************** ** ********** * * * ** ************* *********** ++** **-****** ** **-*********** +*** ++*** * ***** ****** **** *++++_**** * * * ****** *** + * * **+ *********** + * * + * * ***** ** * ******** **** * * * * ******** * ****** * **** ******* ****** ***** * ************ _********************* ********** ********** * * * * ********* * ******* ********** ******* ******** ************* * * ******* Figure 13. Plan map of differences in block grades--geostatistics. - Actual block value more than 0.15 lower than predicted value. * Actual block value within 0.15 of predicted value. + Actual block value more than 0.15 greater than predicted value.

54 > * * * * * * * * * * * * * * * * ******* * * * * _ **** +++*******+ **** *_*+**+ **** ++********* ***** *+++** _**** **********_* *_** ****_+*$*** *************** ****** +* * * * +** ************* ****** *** * * * * * * * * * * * * * * * * * * _. * + * * * * * + * * * * * * * * * * ***** _.****+_**** _ * * ****** *** + * * * + *********** * * * ***** +-* > ++*-* **** *** +*_***** **** *** **** *** *** ** ** ** + * ******** *_. * **************** * ********** ********** ** * * ********* * ****** ********* ******_ ******** ************* _******** Figure 14. Plan map of differences in block grades--ids. - Actual block value more than 0.15 lower than predicted value. * Actual block value within 0.15 of predicted value. + Actual block value more than 0.15 greater than predicted value.

55 ***++* *********** ** *_** *** * * ** **** *++******** * * * * *+*** **** *********** ***** *+++** ***** ************ ** ** *****+***** * ************** ************* *** ** ***** **** ************* + + * - > * * * * -. * * * * * * * * * * * + * * * + *** -f-f** * ***** ******* ***** **+++***** * * ****** * * * + ** * * * *********** * * + * ***** * +***+* * * * * * *-* -* * * * * * * * * * **** *** **** ******* ****** ***** * ******** *** _********************* ********** ********** ** * * ********* * ******* ********** ******* ******** ************* ********* Figure 15. Plan map of differences in block grades--polygon. - Actual block value more than 0.15 lower than predicted value. * Actual block value within 0.15 of predicted value. + Actual block value more than 0.15 greater than predicted value.

56 47 an order of magnitude comparison of the computational time for each method. The polygon method is the easiest method to program and has the I shortest execution time. IDS is next in complexity and took about twice as long as the polygon method. BLIP is more complex yet and took 50% longer than the IDS method. The execution time of BLIP and IDS are related to the number of holes included in the weighting for each block. Kriging is the most complex method tested and takes the largest amount of computation time.. For the test bench, the execution time was five times as long as the BLIP method. This increase in execution time is due to the amount of time required to solve the n simultaneous equations, where n is the number of holes included in the kriging for a block.

57 CHAPTER 5 CONCLUSIONS AND SUGGESTIONS FOR FUTURE RESEARCH In view of the main purpose of this study, the results for the test bench at the Pima mine indicate that the geostatistical method was no better nor worse in predicting block grades than the BLIP or IDS methods. In.contrast, the polygon method is clearly shown to be inferior to the above three more sophisticated methods. Considerations In Model Selection In general, there is no clear cut choice of which ore reserve estimation method to use on a particular ore deposit. At an already producing mine like the Pima mine having a substantial number of drill holes in a fairly uniform grid, BLIP or IDS may be good choices for the following reasons. 1. They involve substantially less computer cost than the geostatistical method. 2. The accuracy of the method chosen may be determined by applying the simple tests described in Chapter 4, utilizing the records on both diamond drill hole and blast hole assays. For prospective or new properties the geostatistical method is more attractive for the following reasons. 48

58 49 1. It is the only method having the ability to calculate the estimation variance and to construct confidence limits on the estimate. 2. It is capable of pinpointing areas needing more drilling in order to obtain a certain confidence limit, and can calculate the new confidence limit before the holes are drilled. 3. Kriging is a more sophisticated interpolation technique than either BLIP or IDS. Its ability to take the spatial relationships of the samples into account is most apparent in deposits that are very irregularly drilled and have few drill holes, such as is the situation at most prospective properties. Importance of Geology All.of the methods might have given better results.if geology had been included in the block model of the deposit, especially along the contacts between barren and mineralized zones. This would avoid the obvious mistake of extending a high grade assay to a.block in barren ground. This also emphasizes the need for close checking of the results of any computerized ore reserve estimation. Suggestions for Future Research The general field of ore reserve estimation needs more research into ways of further reducing estimation errors and making better use of available data. Specific suggestions for future research are given below. 1. The variogram is the basic tool of geostatistics. There is no method, however, to tell if the variogram is good enough.

59 50 Future research could investigate the problem of how well the experimental variogram represents the true underlying variogram for varying number of samples. 2. The variogram has been shown to be a useful tool to describe the important structural characteristics of the mineralization. Little research has been done to determine what variograms would be produced by the various ore forming processes. Research to determine the variogram models associated with ore forming processes would be useful in finding the correct underlying variogram to describe the experimental variogram calculated from sample data. 3. Functional anisotropy is not well understood. One area of research is to determine if functional anisotrophy has real physical meaning and ways to utilize the functional anisotropy in kriging. 4. Too little of the geological information available at most mines is used.in the ore reserve estimation. Research should be conducted to find ways to better utilize this information in ore reserve estimation and to determine if any new information useful to ore reserve estimation can be obtained from the subsurface drill holes. Information such as the type and degree of alteration, the presence of accessory minerals, and porosity parameters, may be of use in the ore reserve estimation.

60 APPENDIX A FLOWCHART AND PROGRAM LISTING FOR PROGRAM GAMMA \ Program GAMMA is a Fortran IV program written for a CDC 6400 computer. 51

61 52 Read Control Cards / Read Heading Card Ending Card? Yes / Read Input Data / Average^ Variogram No Stop Calculate Mean G Variance Calculate Variogram Yes Print Out Results Print Out Results Stop No //Vveragex Variogram, Yes Add Variogram to Previous Variograms

62 53 1 PROGRAM GAMMA(INPUT,OUTPUT,PUNCH,TAPEl=INPUT,TAPE5=OUTPUT,TAPE8=PU INCH) DIMENSION DATA(100,3),TOT(20r5),TOTVAR(20,5),IFRMT(6),TITLE(81 COMMON VAP(2 0,5),HEAD(8),L0G,CLAS,DLIM,ANG,ICUT,SMEAN,VARI,STO,N 5 COMMON I'PCH, IPC,L,.SPR DIMENSION Z(3) C PROGRAM GAMMA CALCULATES VARIOGRAMS IN ANY DIRECTION FROM DATA C THAT HAS COORDINATES AND ASSAY VALUES,DATA NEED NOT BE ORDERED* 10 C GAMMA IS CURRENTLY DIMENSIONED TO HANDLE UP TO 100 DATA POINTS C AND UP TO 20 CLASS INTERVALS FOR A GIVEN DIRECTION C INPUT CARDS. C CARD ONE. C COL HEADING CARD 15. C CARO TWO. c COL 1 LD LOCATION OF ASSAY VALUE ON DATA CARD (1,2,OR 3) (IE. IS IT FIRST SECOND OR THIRD ON THE DATA CARD) COL 2 NO LOCATION OR NORTHING.(1,2, OR 3) COL' 3 IE LOCATION OF EASTING (1,2, OR OR 3) 20 COL 4 LOG LOG TRANSFORM OF DATA, 1»YES, 0 NO. COL 5 IPC PUNCH CARD OUTPUTw 1=YES, 0=N0. COL 6 ITALLY TOTAL VARIOGRAMS AND AVERAGE. 1 = YES, 0=N0. COL 7 LL WHICH VARIOGRAM VALUES ARE TO BE PLOTTED 0 SECOND MOMENT, 1 = MOMENT CENTER 25 c COL CLAS CLASS INTERVAL TO BE USED TO GROUP DISTANCES COL ANG ANGLE IN DEGREES THAT VARIOGRAM IS TO BE CALCULATED ALONG. c 0. = EAST-WEST 45, «NE-SW. C 90,» NORTH-SOUTH -45. = NW-SE 30 c COL SPR ALLOWABLE ANGLE DEVIATION IN DEGRESS c c CARD THREE COL 1-80 VARIABLE FORMAT CARD. C CARD FOUR C COL TITLE OF VARIOGRAM. 35 C CARD FIVE. DATA CARDS- FOLLOW. C END OF DATA CARDS IS A 7/8/9 CARD.. C PROGRAM WILL. PROCESS- MULTIPLE DATA GROUPS, EACH VARIOGRAM C TO BE CALCULATED MUST CONSIST OF A. TITLE CARD, DATA CARDS.AND AN C EOF CAPO. 40 C C MAJOR VARIABLES USED IN PROGRAM GAMMA C DATA ARRAY CONFINING SAMPLES AND COORDINATES C Z ARRAY USED FOR TEMPORARY STORAGE OF INPUT.DATA C- X DIFFERENCE IN EAST COORDINATES BETWEEN TWO SAMPLES 45 C Y DIFFERENCE IN WEST COORDINATES BETWEEN TWO SAMPLES C G DIFFERENCE IN SAMPLE VALUES C OIS DISTANCE BETWEEN TWO SAMPLES C 05QR 0 SQUARED C DISC OIS TIMES 0 SQUARED 50 C 101(1,1) TOTVARd,!) CUMULATIVE DISTANCE FOR ITH CLASS INTERVAL C TOT (1, 2 ) T O T V A R d, 2). CUMULATIVE DIFFERENCE C TOT (1,3) T O W AR (1,3) CUMULATIVE OSGR C TOT(1,4) TOTVAP(I,4) CUMULATIVE DISTANCE TIMES DIFFERENCE SO. C TOT (1,5) TOTVARd, 5) CUMULATIVE NUMBER OF SAMPLES L * 3. IPCH» 8

63 C IPT» 1 ' - I OUT = 5 R90 a 90,* ZERO ARRAYS DO 5 I a 1, J=l>5 TOT(IfiJ) a VARCI,J) a CONTINUE C READ HEADING AND PARAMETER CARO REA0(IPT,10)HEAD 10 FORMAT(3A10) IF(EOF(IPT).GT.O) GO TO 200. READ (I PI, 20)10, NO, IE, LOG,I TALLY,I PC,LLpCLAS,ANG,SPR 20 FORMAT(711,3X,3F10»0) IF(EOF(IPT).GT.O) GO TO 200 C READ VARIABLE FORMAT CARD REA0(IPT,10) IFRMT IFCEOF(IPT).GT.O) GO TO 200 C**** C SIMPLE LOGIC CHECKS IF(LLoGT.O) L 51 4 IF((LD.LE.3 ),AND.(N0.LE.3)oANO,(IE.LEo3)) GO TO 40, WRITE (10UT,30) 30 FORMAT(* SPECIFIED ORDER OF DATA IS IN ERROR*) GO TO DLIM = 20 o*clas IF(ANG.LE. 90. )G0 TO 70 ANG «ANG WRITE(IOUT,60)ANG 60 FORMAT(* ANG IS TOO LARGE. NEW ANGLE»*,F7.0) C CONVERT ANGLE AND SPREAD TO RADIANS 70 RADI = (ANG-SOR)* RAD2 = (ANG + SPR >* N = 0 C READ DATA CAROS READ(I FT,10) TITLE IF(EOF(IPT).GT.O) GO TO READ(IPT,IFRMT)Z(l),Z(2),ZT 3) IF(EOF(IPT),GT.O) GO TO 85 N a N + l DATA(N,1) = Z(LD) DATA(N,2)» Z(NO) DATA(N,3) = Z (IE) GO TO 80 C CONVERT DATA TO LOGARITHMS IF LOG «1 85 IF(LOG.NE.l) GO TO 91 DO 90 1=1,N DATA(1,1) «AL0G10(DATA(1,1)) 90 CONTINUE C CALCULATE MEAN AND STANDARD DEVIATION 91 SUM = SUM2 = 0.0 DO 100 1=1,N A = DATA(1,1) SUM = SUM * A SUM2» SUM2 * A*A IOC CONTINUE

64 C VAR I «(FLOAT(N)*SUM2-SUM*SUM>/(FLOAT(N >*FLOAT(N-l)) SMEAN» SUM/FLOATtN) STD = SORT(VAR I) START CALCULATION OF VARIOGRAH NI = N-l DO 130 I«1,NI 01 «DATA(I,1) Y1» DATA(I,2) XI = DATA(I,3) II = 1*1 DO 130 J = II,N 02 «DATA(J,1) Y 2 = DATA(J,2) " X2» DATAC Jp 3) X «X1-X2 Y ^ Y1-Y2 0» IF(ABS(X)oGTol) GO TO 110 RAO.= R90 GO TO IF(YeE0«0) GO TO 111 RAO = ATAN(YZX) GO TO RAD = 0. C COMPARE DIRECTION WITH ACCEPTABLE DIRECTIONS. 120 IF((RAD.LT.PADl).OR.(RAD.GT.RAD2)) GO TO 130 DIS * SQPT(Y*Y + X*X) f.0001 IF(OIS,GT.DLIM)GO TO 130 K =* DIS/CLAS * 1 IF(K.GT»20) K=20 DIS» DIS OSOR = 0*0 DISC = DIS*OSOR T0T(K»1) = DIS * TOT(K,1) TOT(K j 2) =Q > T0T(K,2) T0T(K,3) aosor * T0T(k,3) T0T(K,4)» DISO * TOTCK,4) TOT(K, 5) =* T0T(K,5> $ CONTINUE DO 150 I a 1,20 AN» TCI(I,5) IF(AN.EO.O.O) GO TO 150 VAR (1,1) a TOT( I / D /AN VAR(I,2) = TOT(I,2)/AN VAR(I,3) =' TOT(1,3)/(2«0*AN) IF(TOT(I,4).EQ.O) GO TO 149 VAR(I,4) = TOT(1,4)/(TOT(I,l)*2o0) 149 VAR(1,5)» TOT(1,5) 150 CONTINUE C C PRINT OUT RESULTS CALL VAROUT(TITLE) COMBINE.DATA IF AVERAGE VAR IOGR AM IS TO-BE CALCULATED, IF(ITALLY.EQ^O) GO TO 170 DO 160 I = 1,20 DO 160 J=»l, 5 TOT V AR (I, J ) 160 CONTINUE TOTVARCI, J) * TOT(I,J)

65 56 TSUM a SUM * TSUM TSUM2» SUM2 * TSUM2 N N a N N + N 175 C ZERO TOT AND VAR ARRAYS 170 DO 180 I = 1,20 DO 180 J«l,5 TOT(I,J) a VAR(I,J)» 0*0 180 CONTINUE 180 GO TO IF(ITALLYoEOoO) GO TO 194, C CALCULATE AVERAGE VARIOGRAM ' VARI «(FLOAT (NN) 4=TSUM 2-TSUM* TSUM )/( FLOAT (NN )4FL0AT ( N.N-1)) SMEAN * TSUM/FLOAT(NN) 185. STD» SORT(VARI) N a NN DO 190 I a 1,20 AN * T0TVAR(I,5). IF(AN,EOoO.O) GO TO VAR (1,1) = TOTVARt I, D / A N VAR (I,2) = TOTVAR(1,2)/AN VAR (1,3) TOTVAR'C 1,3)/(AN*2*0) IF(T0TVAR(I,4)*E0»0) GO TO 189 VAR(I,4) = TOTVARt1,4)/(TOTVARtI,i)*2*0) VAR(I,5) = AN 190 CONTINUE C PRINT OUT RESULTS.CALL VAROUT(HEAD) 194 WRITE(IOUT,195) FORMAT(* NORMAL END OF JOB*) GO TO WRITE(IOUT,191) 191 FORMAT(* END OF FILE ENCOUNTERED. ERROR IN JOB SETUP,*) 201 STOP 205 END

66 57 1 SUBROUTINE V A R O U T (T I T L E ) c $*$ $$$$$$$$$$$$$ S.$$ss$$$$t$$s$$$$$$$$$$$$$$ss$$$$$$s $$$$$$$$$$$$$$$$$ $$$$$$ C SUBROUTINE VAROUT PRINTS OUT THE RESULTS OF THE VAR IOGRAM CALCULATION. c$$$$$sss$st $.$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$ $ $$$.$$$$$$$$$$$$ $$$$$$ 5 DIMENSION TI T L E (8 )#A (100)#C (11)>D (51)>E (13) COMMON V A R (20,5),HE A D (8),L O G,C LA S,DLIM,ANG,I OUT>SME A N,VARI,STD,NN COMMON IPCH,IPC,L,SPR DATA D /18(1H ), 1HS,1 HE,1H C,IHO^1HN,1HD,1H,1HM,1H0,1HM,1HE,1HN,1HT 1> 20(1H )/ 10 DATA E/1HM,1H0,1HM>IHE^IHN,1HTp1H, 1HC#1HE>1HN#1HT»IKE#1HR/ IF(LoE0«3) GO TO 8 DO 7 I 1,13 -. D(I + 18 ) E(I> 7 CONTINUE 15 8 C(l) = Oo DO 85 1=2,11 C(I) = CLAS&2.0 > CCI-l) 85 CONTINUE ILOG = 2HN0 20 IFtLOG.EOol) ILOG = 3HYES C PRINT OUT PAGE ONE OUTPUT WRI TE(IO UT,10) 10 F0RM"AT<1H1,60X,*VARIQGRAM*) WRITE(I0UT,20)TITLE ' FORMAT(1H0,25X,8A10) W R I T E (IQUT,20)HEAD W R I T E (I0U T,30) ANG, SPR,SMEAN,CLAS,VARI,DLIM,STD,ILOGfNN 30 F O R M A T (1H0,*DIRECT ION = *,F 5.0,5X,*DEV I ATI ON «F 5»0,35X,*MEAN 1= *,E 10e3,/* CLASS SIZE = *, cl 0 e3,5 1X,*VARIANCE = *,E10,3,/* OIS 30 2TANCE LIMIT «*,E10.3,47X*STANDARD DEVIATION a *,E10»3,/* LOGARI 3THMS - *,A5,56X,*N0, OF SAMPLES = *18) IF( IPCeEQeO) GO TO 39 C, PUNCH OUTPUT CARDS W R I T E (I PC H,21) TITLE FORMAT (.8.A10 ) W R I T E (IPCH,23) ANG,SMEAN,VAR I,CLAS,ILOG,NN 23 FORMAT(4E10.3,A10,I10) " 22 FORMAT(6E10,3) 39 W R I T E (IOUT,40) FORMAT(1H0,6X,*0ISTANCE IN FEET*,9X,*N0e OF SAMPLES*,10X>*DIFFEREN 1CE*,8X,*SEC0ND MOMENT*,8 X,*MOMENT CENTER*,4X*AVERAGE DISTANCE*) DO 50 I = 1,20 LOW «t1~1)*clas LUP.* I *CL AS 45 WRITE (I0UT,45 )LOW,LUP,VAR (I, 5), VAR (I, 2), VAR (I, 3), VAR (1,4) j,var (1,1) I F (I PC.E0.1.AND.VAR(I,5),GT,0) W R I T E (IPCH,22) V A R (I,5),V A R (I,2)#V 1AR(I,3),VAR (I,4),V A R (1,1) 45 FORMAT(1H,4X,I 6, * ---- *, I 6,1O X,F8.0,5X,3(11X, E10 3>,10X,F 10* 1) 50 CONTINUE 50 W R I T E (I OUT,10) C PRINT OUT PAGE TWO OUTPUT WRITE(I0UT,20)TITLE W R I T E (IOUT,2 0 )HEAD WRITE(I0UT,100) F OR MA T (1 HO ). C FIND MAX VALUE TEMP «V A R (1,L )

67 58 DO 55 1=2,20 IF(VAR(I,L).GT.TEMP) TEMP «VAR(I,L) CONTINUE UNIT * CLAS/5. DIV = TEMP/50, DO 75 K»1,51 TOP = TEMP 65 BOT» TEMP - DIV DO 65 I» 1,20 IF((VAR(I,L)oGT.TOP>.OR.VAP(X,L)»lE»BOT ) GO TO 65 J» VAP (I, D / U N I T > 1 A (J) = 1KX CONTINUE WRIT (IOUT,60) D(K),TEMP,A 60 FORMAT (1H, 2X, Al, 2X, E10«, 3, **,10061) TEMP = BOT DO 70 1=1, A(I) = 1H 70 CONTINUE 75 CONTINUE WRITE(IOUT,80) 80 FORMAT (1H, 17X, 10(»«$***4$***** ), 4=$*) 80 WRITE(IOUT,9 0 )C 90 FORMAT(1H,13X,11(F6.0,4X)) RETURN END

68 APPENDIX B FLOWCHART AND PROGRAM LISTING FOR PROGRAM KRIG computer. Program Krig is a Fortran IV program written for a DECSYSTEM 10 59

69 Read In All Input Data Calculate Block Variance Do the Following Steps For Each Block Find Nearest Holes Calculate Covariance Between Each Hole and the Block Calculate Covariance Between Each Sample Solve Linear Equations for Weights Calculate Weighted Assay Grade Print Out Results

70 06200 ACCEPT 12*ALFA(1)»ALFA(2) FORHAT(AF) TYPE ' F ORMAT( TYPE NK,NALL,NBC,NEC,NBR,HER*) C READ ON SECOND CONTROL CARD NUMBER OF HOLES IN KRIGXNG, AND C WHICH BLOCKS TO KRIG ACCEPT 16,NK,NALL,NBC,NEC,NBR,HER FORMAT(61) IF(NALL.GT.l) GO TO NBC a NBR = NEC = NC NER a NR C - READ IN THE NAME OF THE INPUT DRILL HOLE FILE TYPE FORMAT!1 TYPE THE NAME OF INPUT DRILL HOLE FILE8) ACCEPT 19,NAME FORMAT(A5) CALL IF I L E U, NAME) I = C~ READ IN THE DRILL HOLES * READ(1,25,END «30)HOLE(I,1),HOL E(I,4),HOLE(I,5),HOLECI, ,HOLE(I,2),HOLE(1,3) FORMAT(A5,lOX,3F5.0,1 OX,2F5»0) C 25 FORMATtA5,4X,3F5»2,2F10 0) " GO TO N «I TYPE 5,N FORMAT! 8 NUMBER OF DRILL HOLES * *15) C CALCULATE THE.BLOCK VARIANCE GM0Y2 * 0, DO 32 I 1, DO 32 J» 1, D C (1,1) = 50. * (1-3)*W4 * W , DC (2,1) -a 50, + -! J-3)*W4 * W DC!3,1) «0, ; DC(1,2) = 5 0 * x D C (2,2) = D C (3,2) = CALL COVAR (DC,GMOY) GM0Y2 = GM0Y2 GMOY CONTINUE BLKVAR = GM0Y2/ C -START THE KRIGGING OPERATIONS YM = YM AX +,5*WI0TH XM = XMIN +,5&WIDTH DO 100 J = NBR,.NER X Y (2) = YM -J*WIDTH DO 100 I «NBC,NEC XYtl > * XM 4- I*WIDTH C FIND NK CLOSEST HOLES DO 40 K = 1, N DIFY = X Y (2 ) - HOLE(K,2) D I F.X = X Y (1) -HOLE (K, 3) D 1 = DIFY*DIFY = DIFX*DIFX DIS(K) = CONTINUE NI = NK K a 1,NK

71 12300 OS(K ) * DISCK) IHQtE(K> = K CONTINUE CALL DSORT (DS,IHOLE,NK) DO 55 K = NI,N IF{DIS(K )9GT»O S (NK)) GO TO OS(NK) = OIS(K) IHOLE(NK) a K CALL DSORT (DS,IHOLE,NK) CONTINUE C--- CREATE SEPARATE SAMPLE INFORMATION FOR EACH COMPOSIT NH «NK D C (1,1) = XYil) D C (2,1) a XY(2) NK3 * NK* D C (3,1) = DO 53 K a 1,NK IK» IHOLE(K) DO 56 II = 1, KI =* K*3-3 II DO 54 L = 1, HOL(KI,L) = HOLE(IK,L) CONTINUE H0L(KI,4) = ELEV(II) H0L(KI,5) * HOLE(IK,11*3) CONTINUE CONTINUE 15000: NN a NMAX «NK NN = NN * IF(NNoGToNMAX) GO TO IF<H0L(NN,5).GT.O.O) GO TO KN s NN * IF(KN.EQ.NMAX) GO TO DO 603 NNN = KN,NMAX DO 603 MM = 1, HOL(NNN 1,MM) * HOL(NNN,MH) CONTINUE NMAX =NMAX GO TO NK3 = NMAX NKH1 a NK3 * NKH2 = NK3 * C CALCULATE COVARIANCE BETWEEN EACH SAMPLE AND THE BLOCK DO 70 K = 1,NK D C (1,2) «HOL(K,3) D C (2,2 ) = HOL(K,2) D C (3,2) a HOL(K,4) CALL COVAR(DC,GMOY) R(K)= GMGY G A (K ) = GMOY A ((K-l )*NKH1 +NKH1) =» loo A ((NKHl-l) *NKH1 * K)» 1, CONTINUE A (NKH1 *NKH1 )» 0» C-- -CALCULATE COVARIANCE BETWEEN THE SAMPLES DO 83 K=1,NK DO 80 L» K,NK DIFY «HOL(K,2)-HOL(L,2) DIFXa H0L(K,3) -HOL(1,3) DIFZ = HOL(K,4) -H0L(L,4)

72 a DIFY*DIFY = SORTC 0 1 } = DI.FX*DIFX «S OR I {02) » SORT(DIFZ*DIFZ) GH GAMMA (01,02,03) A ((K-l) *NKH1 4- L) GH A ((L-1)*NKH1 +K)=GH CONTINUE CONTINUE R(NKH1) * c-- -SOLVE SIMULTANEOUS LINEAR EQUATIONS CALL GEL.G(R, A, NKH1, 1, , IER) C-- CHECK FOR ERROR IN MATRIX OPERATIONS I FdER.NE.0) GO TO c-- B CONTAINS THE SOLUTION A G a 0 a C CU = 0, CALCULATE THE WEIGTHED ASSAY GRADE AND VARIANCE DO 97 K = 1,NK AG * AG * GA(K)*R(K) CU a CU* HOL(K,5)*R(K) CONTINUE VARKG» R(NKHl) * AG - BLKVAR C PRINT OUT RESULTS-- -- * * ICU a CU WRITE(23,103)J,I,ICU,VARKG FORMAT(1 19,315,FlOa3,* 6H0LE 3 LEVEL0) CONTINUE GO TO c - PRINT ERROR MESSAGES TYPE 186,IER,I,J FORMAT!* ERROR IN MATRIX*,15,0 FOR BLOCK I *»,I5, _G0 TO 100 * STOP END SUBROUTINE COVAR (B,GMOY) C SUBROUTINE COVAR DETERMINES THE COVARIANCE BETWEEN A C AND A BLOCK. A DISCRETE SUMMATION IS USED TO APPROXII c THE SEXTUPLE INTEGRAL COMMON ALFA(2),WIDTH DIMENSION 8(3,2) GMOY a 0* W 8 = WIDTH/ W 4 a WI0TH/ I * 1, DO 10 J = 1, D IFX sb {1,2 ) - B (1, 1) * (I-3)*W4 * W DIFY =6(2,2)- 6(2,1) * (J-3)*W4 ti DIFZ '*8(3,2) - 8(3,1) «SORT(DIFY*DIFY) D 2 a SQRT(DIFX*DIFX) o SORT (DIFZ S'DIFZ) GH * GAMMA(01,02,03) GMOY * GMOY * GH CONTINUE GMOY = GHOY/X&o RETURN END SUBROUTINE DSORT (OS,IHOLE,NH) DIMENSION OS(20),IHOLE(20) 64

73 24500 C THIS SUBROUTINE SORTS THE OS ARRAY INTO INCREASING ORDER IF(NH.LT»2) RETURN DO 20 I o 2,NH IP * DO 10 J = Ifil? I F ( D S m. G E. D S C J ) ) GO TO SAVE s DS(I) D S m» DSC J) ISV» IHOLECI) IHOLECI) IHOLECJ) DS(J) 3 SAVE IHOLECJ) * ISV CONTINUE CONTINUE RETURN END SUBROUTINE GELG(R,A,M,N,EPS,IER) C -THIS SUBROUTINE IS FROM THE IBM SSPS PACKAGE OF STATISTICAL c -SUBROUTINES. IT SOLVES SIMULTANEAOUS EQUATIONS c -S I MI LAP. SUBROUTINES MAY BE SUBSTITUTED FOR IT DIMENSION A(1)p R(1) DOUBLE PRECISION A,R,TBfPIV,PIVI,TOL IF(M)23,23, c SEARCH FOR GREATEST ELEMENT IN MATRIX A I ERsO PIVaO NM=N*M DO 3 L=1,MM TBsDABSCACL)) IF{TB-PIV)3,3, PIV=TB Ia L CONTINUE TOL=EPS*PIV 28100,-c. AC I).IS PIVOT ELEMENT. PIV CONTAINS THE ABSOLUTE VALUE OF A c START ELIMINATION LOOP L ST DO 1.7 K*l, M c TEST ON SINGULARITY IF(PIV)23,23, IFCIER )7,5, IF(PIV-T0L)6,6, IERaK-l PIV-I*l./A(I) J»( I - D / M IaI-J*M-K JaJ+l-K I4-K IS ROW I ND E X, J4K COLUMN-INDEX OF PIVOT.ELEMENT c PIVOT ROW REDUCTION AND ROW INTERCHANGE IN RIGHT HAND SIDE I DO 8 L=K,NM,M * TB«PIVI*RCLL) RCLL)-R(L) R (L )atb c IS ELIMINATION TERMINATED IF(K-M)9,18, c

74 30600 C COLUMN INTERCHANGE IN MATRIX A LEND=LST+M-K IF(J)12,12, II* J*J DO 11 l = tstj,leno TB = A ( U Ll=L*II A(L )«ACLL) 11 A (L L )a IB C C ROW INTERCHANGE AND PIVOT ROW.REDUCTION DO 13 L=LST,MM,M *1* TB»PrV.I*A(lL) A(LL)»ACL) A(l )*T c SAVE COLUMN INTERCHANGE INFORMATION A(LST)*U c ELEMENT REDUCTION AND NEXT PIVOT SEARCH PIV*0o LSI*LST* J» DO 16 II=LST,LEND PI VI*-A(II) IS T * J»J* DO 15 L*IST,MM,M LL»L-J A(L)=A(L)+PIVI*A(LL) TB = DA8S(AC L)) IF(T8-PIV)15,15, PIVaTB I = L ' CONTINUE DO 16 L»K,NM,M LL*L+j R(LL)=R(LL)+PIVI*R(L) LST=LST+M END OF ELIMINATION LOOP c BACK SUBSTITUTION AND BACK INTERCHANGE IF(M-l)23,22, ST»MM*M LST»M* DO 21 1*2,M * L ST I IST* IST-*LST L=IST-M L*A(L>* DC 21 J=II,NM,M TB* R (J ) LL 3 J DO 20 K=iST,MM,M L L * LL * TB=TB-A(K)*R(ll) KeJ+L R(J)*R(K) R(K)=TB RETURN c ERROR RETURN MATRIX A

75 IERa-1 RETURN END FUNCTION GAMMA(D 1* 02* 03) COMMON ALFA(2),WIDTH C THIS SUBROUTINE CALCULATES THE VALUE OF THE VAR I OGRAil C FOR A GIVEN DISTANCE AND DIRECTION* H * SORT( D1 *01 + D2*D2 * 03*03) IFIH.LT.IO. ) GO TO 20 GAMMA =ALFA(1)*AL0G10(H) GO TO GAMMA = 0*0 15 H2 o S0RT(D3*D3) IF(H2oLT«10e) GO TO 16 GAMM2 = ALFA(2)*AL0G10(H2) GO TO GAMM2» 0*0 18 GAMMA GAMMA * GAMM2 RETURN e n d

76 REFERENCES CITED Blais, R. A. and Carlier, P. A., 1968, "Applications of Geostatistics in Ore Evaluation." Ore Reserve Estimation and Grade C.I.M.N. Special Vol. No. 9, Montreal, pp Carlier, P. A., 1964, "Contribution Aux Methods d 'estimation des qisement's d 1uranium.'1 Rapport CEA R2332, 410 pp. David, M., 1969, "The Notion of Extension Variance and Its Application to the Grade Estimation of Stratiform Deposits." A Decade of Digital Computing in the Mineral Industry. A.I.M.E., special volume, pp David, M., 1974, "A Course in Geostatistical Ore Reserve Estimation," Ecole Polytechnique, Montreal, Canada,. 303 pp. Gupta, Shiu K. and Cozzolino, John M., 1974, Fundamentals of Operations Research for Management< San Francisco: Holden-Day, Inc. Hazen, S. W., Jr., 1958, "A Comparative Study of Statistical Analysis and Other Methods of Ore Reserves Utilizing Analytical Data From Maggie Canyon Manganese Deposit Artillery Mountains Region, Mohave County." U.S. Bureau of Mines Report of Investigations pp. Hazen, S. W., Jr., 1967, "Some Statistical Techniques for Analyzing Mine and Mineral-Deposit Sample and Assay Data." U.S. Bureau of Mines Bulletin pp. Huijbregts, G. J. and Segovia, R., 1973, "Geostatistics for the Valuation of a Copper Deposit." Eleventh International Symposium on the Application of Computers in the Mineral Industry, Tucson, Arizona, pp. P23-P43. Krige, D. G., 1951, "A Statistical Approach to Some Basic Mine Evaluation Problems on the Witwatersrand.'1 J. Chem. Met. and Min. Soc. South Africa, V. 52, No. 6, pp Krige, D. G., 1952, "A Statistical Analysis of Some of the Borehole Values in the Orange Free State Coldfield." J. Chem, Met. and Min. Soc. South Africa, V. 53, No. 3, pp Krige, D. G., 1962, "Statistical Applications In Mine Valuation." Journal of Inst, of Min. Surveyors of South Africa, V. 12, No. 2 and V. 12, No. 3, pp

77 69 Matheron, G., 1963, "Principles of GeostatisticsEconomic Geology, V. 58, pp Matheron, G., 1971, "The Theory of Regionalized Variables and Its Applications." Les Cahiers du Centre de Morphologic Mathmatique, Fontainebleau, No. 5. Nuclear Fuel Resource Evaluation, 1973, U.S. Atomic Energy Commission, GJ0-105, 25 p. Sichel, H. S., 1966, "The Estimation of Means and Associated Confidence Limits for Small Samples from Lognormal Populations." Symposium on Mathematical Statistics and Computer Applications in Ore Valuation, Johannesburg, The So. African Inst, of Min. and Met. Williamson, Don R., 1975, Chief Geologist at Cypress Pima Mining Co., Tucson, Arizona. Personal communication.

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