A comparison of the geostatistical ore reserveestimation method over the conventional methods

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Authors Knudsen, H. Peter (Harvey Peter), 1945-

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A COMPARISON OF THE GEOSTATISTICAL ORE RESERVE ESTIMATION METHOD OVER THE CONVENTIONAL METHODS

byHarvey Peter Knudsen, Jr.

A Thesis Submitted to the Faculty of theDEPARTMENT OF MINING AND GEOLOGICAL ENGINEERINGIn Partial Fulfillment of the Requirements

For the Degree ofMASTER OF SCIENCE

WITH A MAJOR IN MINING ENGINEERINGIn the Graduate College

THE UNIVERSITY OF ARIZONA

1 9 7 5

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 specialpermission, 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 DIRECTORThis thesis has been approved on the date shown below:

Associate Professor of Mining and Geological Engineering

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

TABLE OF CONTENTS

PageLIST OF ILLUSTRATIONS . . . . . . . . . . . . . . . . . . . . . viLIST OF TABLES......................... .............. . . viiABSTRACT........... viii

1. INTRODUCTION. . . . ........... . . . . ' 1Purpose of Study......... 3Conventional Ore Reserve Estimation Methods ............... 4

Polygon Method. .................... 4Inverse of the Distance Squared Method (IDS). . . . . . 5Modification of IDS Method (BLIP) . . ............. 6

Scope of Study.................... 62. A BRIEF INTRODUCTION TO GEOSTATISTICS . . . . . .......... 8

Regionalized Variables. ............ 8Variogram Function. . . . . . . . . . . . . ............. 10Variogram Models........ 14Variance of Block Grades........................... 16Extension Variance. ............ 17Estimation Variance . . . . . . . . . . ....... 19Kriging.................. 20

3. GEOSTATISTICAL ORE RESERVE ESTIMATION OF THEPIMA MINE 24Pima Mine............ 24Variogram Calculation 25Kriging ........... 31Kriging Variance and Its Accuracy........... 33

4. PREDICTION ACCURACY OF THE FOUR ORE RESERVEESTIMATION METHODS........... 36Tests of Prediction Accuracy........... 36Results of Prediction Accuracy Tests.................... 37

Amount......................... 37Location........... 41

Computational Aspects......................... . . . . . 42iv

VTABLE OF CONTENTS--Continued

Page5. CONCLUSIONS AND. SUGGESTIONS FOR FUTURE RESEARCH........ 48

Considerations In Model Selection ..................... 48Importance of Geology ................................. 49Suggestions for. Future Research .................. 49

APPENDIX A: FLOWCHART AND PROGRAM LISTING FORPROGRAM GAMMA................. 51

APPENDIX B: FLOWCHART AND PROGRAM LISTING FORPROGRAM KRIG . . ........ 59

REFERENCES CITED. ................ 68

LIST OF ILLUSTRATIONS

.;;ure PageV Variogram of Maggie Canyon manganese deposit........... 123. Degrees of continuity expressed by variogram......... 133. Hypothetical example of geometric anisotropy . . . 154. Histogram of DDH assay values, Pima mine ....... . . . 265. Cumulative frequency distribution plot ................. 276. Experimental horizontal variogram, Pima mine . ........... 297. Vertical and horizontal DeWijsian variograms,

Pima mine. ............................. 308.. Horizontal DeWijsian variograms showing functional

anisotropy ............. 309.' Kriging variance plot.................... 3410. Example of calculations used in comparing predicted

block grades to actual block grades. . ............... 3811. Histograms of block grade differences between

predicted vs. actual....... . . . ......... 4012. Plan map of differences in block grades--BLIP. ....... . . 4313. Plan map of differences in block grades--Geostatistics . . . 4414. Plan map of differences in block grades--IDS..... ... 145L>. Plan map of differences in block grades--Polygon . . . . . . 46

LIST OF TABLES

Table1. Accuracy of kfiging variance. . . .2. Results of accuracy tests (% copper)

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

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.

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

in English appeared in Economic Geology in 1963. 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 StudyAt 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.

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 MethodThis method has .been widely used in the past for porphyry copper

deposits, especially prior to the introduction Of computer in ore re­serve calculations.

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.

Modification of IDS Method (BLIP)This is the second distance weighting tested. The basic dif­

ference 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 StudyFirst 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

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.

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 cer­

tain 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 randomi .

■ .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.

9The 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 therandom 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)

10E[Y(x+h) - Y(x)] = m(h) 2.3

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:

The variogram 2y(h) in equation 2.4 is used to study regional­ized variables obeying either of the above hypotheses.

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:

y(h) = K(o) - K(h) 2.5

Variogram Function

2.6

where 2y(h) = value of the variogram for distance h f(x) = sample value at point x

11f(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 isreflected 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

12

40-n

30-

Y(h) 20-

10-

n-------------------1-------------------1------------------ r™50 100 150 200Feet

Figure 1. Variograin of Maggie Canyon manganese deposit.Assay data used in the calculation of this variogram was taken from Hazen (1958, p. 36).

13

High degree of continuity. Can be approximated by parabola near the origin. Bed thickness is an example.

rangeAverage continuity. linear near origin, many metal deposits.

Almost Typical of

TNugget 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.

14instead 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. 2.7

and y(h) = Co + C for h > a

where Cq = nugget valueC = sill value minus Co h = distancea = range.

HorizontalVariogramRange

Vertical-Variogram

Range

Feet

Figure 3. Hypothetical example of geometric anisotropy.

16The 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 vario- grams lie in the actual calculations required for an ore reserve estima­tion. Throughout the subsequent sections of this thesis, thetheoretical models of the experimental variograms will be referred to as"variograms," for the sake of brevity.

Variance of Block GradesThe 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.

17of the vector y(x-y) sweep the volume V on its own accord. Explicitly, equation 2.9 can be written as equation 2.10:

I X 1 x 2 x 3 Yi Y2

a(0/v) V dXl fo4X2 Jo ^ 3 Jo dyl Jo dy22.10

73 foY(x1-y1, x2-y2, x3-y3) dy3

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 2.12.

Or = — /ydx Jv y(x-y) dy - — f^dx y(x-y) dy 2.12r 2(v/v) v2 "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

q2 = E[Z(v) - Z(v')]2 2.13

18where 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 2.14.

°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 blockv’ and is defined in terms of the covariogram by equation 2.15.

a(v,v') = /vdx 4' K(x">,)dy 2-15Equation 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 -2.16

or in terms of the variogram to give 2.17:

19

/y dx /v, Y(x-y)dy - — ■ dx /v r(x-y)dy -2.17

4. 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

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.

Z(v) = h I Y(x.) iN i=l 1

2.18

2.19

W /v 'K(xi"y)dy 1—1

Equation 2.20 is the estimation variance expressed in terms of the variogram.

°E = wr /vY(Xi-y)dy- 7 /V' 4- Y(x-y)dy -2.20

N N5 f i kJ

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.

KrigingThe 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).

2The 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

21The 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

NZ*(V) = I AiX: 2.20

i=l

The ’s are the weights to be determined subj ect to the constraint of equation 2.21.

NV A• = 1 2.21i=l

2The estimation variance E[Z*(v) - Z(v)] modified for the use of weights is as given in equation 2.22.

N N2 . 22

NSince the estimator I must be unbiased and must have

i=lminimum 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 2.21. 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).

The Lagrangian function of the problem is given as equation 2.23below.

NL ( X 1, X2, . . . X n ,vi) ° ( V / D ) ' 2 i ^ 1X i a ( X i ,v) +

2.23N N Nik + 2-aXi-U

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 2.24.

% " ' 2a Cxv v ) + 2xi ° ( x1,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:

NXia(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.

23With this last equation there is now a set of N+l equations containing N+l unknowns as given in matrix notation by equation 2.26.

a(X1,X1) a(X1,X2)

a (x2,x 1) a(X2,X2)

1 1 1 0

X1 *(%l,v)

x 2 *(X2,v)

XN

y 1

2.26

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 2.27.

N°K ~ a(v/D) ' Xia(X.,v) 2.27

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 MineA 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

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 CalculationDiamond drill hole assays of the copper grade were used to cal­

culate 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

o\°

26

20-.

15- -- --

10-

5 -

Grade

Figure 4. Histogram of DDH assay values, Pima mine.

Grade

y

27

0.6-0.5-

0.3-

0.05

5 90 9510 30 50 80Cumulative Percent Frequency

Figure 5. Cumulative frequency distribution plot.

28calculate 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 0 .021.

The vertical variogram is also shown in Figure 7 and has an alpha value of 0.027. 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

29

.09-

.08-

07-

y (h) .06 -

.05-

. 04 —

.03-

1600 20001200400 800Feet

Figure 6. Experimental horizontal variogram, Pima mine.

30

14-Vertical

.12-

.10-

. 08-

Average Horizontal Variogram (a = 0.021)

06-

.04-

0 2 -

100 200 300 500 1000Feet

Figure 7. Vertical and horizontal DeWijsian variograms, Pima mine.

.14-

.12-

10 -

NW ->

.06 -

.04 -

02 -

200 300 500100 1000Feet

Figure 8. Horizontal DeWijsian variograms showing functional anisotropy.

31directions 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. = 0.021 = isotropic coefficient of dispersiona = 0.006 = zonal coefficient of dispersion zonh = distanceinvert = 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.

KrigingEach 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.

32The 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 compo- sits. 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.

33and

'K = -

Nfv dx /vY(x-y)dy + ii + 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 AccuracyAfter 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.

.119 .113 .135 .135 .110 .108 .110 .110

.115 .105 .110 .089o .094 .089 .101 .105

.138 .126 .113 .108 .1130.092 .126 .135

.133 .089;, .103 .126 .128 .108 .115o

.145

.131 .113 .113 .082o .099 .105 .103 .108

.113 o. 080 .124 .108 .105 .1080 .113 .0780

.147 .133 .145 .147 .142 .140 .128 .119

.165 .138 .140 .151 .147 .124 .113 .080o

.163 .087° .110 .142 .131c

.094 .094 .103

.161 .117 .124 .133 .099 < .092 .126 .... .

.124

Figure 9. Kriging variance plot, o = Location of diamond drill hole.

35Table 1. Accuracy of kriging variance.

Actual Theoretical

Blocks within one standard deviation 451 83% 367 68%Blocks within two standard deviations 518 96% 515 95%Blocks outside two standard deviations 24 4% 27 5%

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 AccuracyLet 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 nAverage of differences F = ^ I 4.1i=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

37study 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*) - "I2 4.21-1

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.

AmountEach 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).

ACTUAL GRADE OF PREDICTED GRADEEACH BLOCK FROM MODEL38 31 22 23

59 33 24 22

53 23 23 23

45 30 20 24

61 39 29 31

51 30 30 26

42 25 24 23

47 34 27 25

Actual Block Predicted Difference DifferenceGrade Grade Model Grade in Block Squared38 61 -23 52931 39 - 9 8122 29 - 7 4923 31 - 8 6459 51 8 6433 30 3 924 30 - 6 3622 26 - 4 1653 42 11 12123 25 - 2 423 24 - 1 123 23 0 045 47 - 2 430 34 - 4 1620 27 - 7 4920 25 - 5 25

-56 1068Average of differences (in block grades) = -56/16 = -3.5 Variance of differences = -^"^26(16-1) ^ = 7*624Standard deviation of differences = /7~.624 = 2.7

Figure 10. Example of calculations used in comparing predicted block grades to actual block grades.

39Table 2. Results of accuracy tests .(% copper).

Ore Reserve MethodAverageDifference Variance

StandardDeviation

BLIP -.03* 0.023 0.15Geostatistics -.04 0.025 0.16IDS -.05 0.028 0.17Polygon

...... '-.06 0.057 0.24

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.

40

.25-20-

1S-10-

05-- .77-.30

25-20-

15-10-

05--.84-.30

20-

15-10-

05--.79 -.30

.20-

.15-,10-05-

-1.62-.30

X = -.30 S2 = .023

-.15 0.0BLIP

.11 .30 .94

-.15

X = -.04 S2 = .025

0 . 0KRIGING

.15 .30 1.10

-.15

X = -.05 S2 = .028

0 . 0IDS

80

X = -.06 S2 = .057

-.15 0. 0POLY

.15 .30 .93

Figure 11. Histograms of block grade differences between predicted vs. actual.

LocationA 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 2M = (N-k) ln[ — £ 9_] -

(N-k)

(n^-l)lnS^2 - (n2-l)lnS22 - (n^-l)^^2

which is approximately equal to . Making the substitutionsinto these formulas yields the value of F to be 2.50. 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 AspectsThe 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

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* * * * * * * * * * * * * * * — — * * * * * — * * * + * * * ** * * **-.— * *** — ** * * — * * * * * * * * * * * * * *

+ + + * *— — * * * * * * * * * * * * * * * * * * * * * * * ** * * * — — + + * * — * — * + — * * * * * * * * * * * * *

* * * * * - * * + + + - * * * * * * * * * * * * ** * * + * * * + + * * * * * * * * * * *+** * * * * * * * * *** — * * * * * * * * * * * * ** * * * * ********* * * * * — * * * * ***** * * * * * * * * * * * *

— — * * * * * * * * * * * * * * * * * * - * * * * * * * * * * * * * - * * * * * * *_ * * * * * * * * * * * * * * * * * * ** * * * * * * * * * * * * ** * * * * * * * * * * * * * * * ******** ** * * * * * ** * * * * * * * * * * * *

* * * * * * * * *

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.

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*************** ** *********** * * ** — * * * * * * * * * * * * * — * * * * * * * * * * *

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— — * * * * * * * * * * * * * * * * * * _ * * * * * * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * * * * ** * * * * * * * * * * * * *** * **** * * * * * * * * * ** * * * * * * * * * * * * * *

* * * * * * * * * * * * ** * *******

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.

> * * * * * * * * * * * * * * * — * —+ + + + ——******* * * * — * _—**** +++*******+ **** *_*+**+— **** ++********* *****— *+++** _**** **********_* *_**__****_+*$*** ***************___******——+* * * * +** *************— ******— ***__+ + + * * * * * * * — * * * * * * * — * * * * _ . — *

+ * * * - — + + * — * + * * * * * — * * — * * ****** _.****+_**** _ * * ********* + * * * + — * * * * * * * * * * *

— — — * * — * * * * * *—+-* >— ++*-* ******* — + * _ * * * * *— * * * * — — * * * * * * ****—*** **——**—— ** + —— * ******** — — *_.

— *— * * * * * * * * * * * * * * * * _ _ ** * * * * * * * * * * * * * * * * * * *** * * *** * * * * * * ** * * * * * — — * * * * * * * * ** * * * * * _ * * * *****

* * * * * * * * * * * * *_ * * * * * * * *

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.

* * * + + * * * * * * * * * * * * ** *_**+ + + + + — — — — —*** * * ——** ——**** *++******** * * * * ——*+*** — — **** *********** *****——*+++** ***** ************ **—**—*****+***** * * * * * * * * * * * * * * * — — — * * * * * * * * * * * * *

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* * * * * * * * * * * * ** * * * * * * * *

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.

47an order of magnitude comparison of the computational time for each method.

The polygon method is the easiest method to program and has theIshortest 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.

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 sub­

stantial number of drill holes in a fairly uniform grid, BLIP or IDSmay be good choices for the following reasons.

1. They involve substantially less computer cost than the geo- statistical method.

2. The accuracy of the method chosen may be determined by applyingthe simple tests described in Chapter 4, utilizing the recordson both diamond drill hole and blast hole assays.

For prospective or new properties the geostatistical method is more attractive for the following reasons.

48

49■ 1. It is the only method having the ability to calculate the esti­

mation 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.

50Future 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.

APPENDIX A

FLOWCHART AND PROGRAM LISTING FOR PROGRAM GAMMA\

Program GAMMA is a Fortran IV program written for a CDC 6400 compu­ter.

51

52

Read Control Cards

/ Read Heading Card

EndingCard?

Yes

NoAverage^Variogram Stop/ Read Input Data /

YesCalculate Mean G Variance Print

OutResults

CalculateVariogram

StopPrintOut

Results

//VveragexVariogram,No

YesAdd Variogram to Previous Variograms

53

1 PROGRAM G A M M A ( I N P U T , O U T P U T , P U N C H , T A P E l = I N P U T , T A P E 5 = O U T P U T , T A P E 8 = P UINCH)DIM ENSIO N D A T A ( 1 0 0 , 3 ) , T O T ( 2 0 r 5 ) , T O T V A R ( 2 0 , 5 ) , I F R M T (6),T I T L E (81 COMMON V A P (2 0 , 5 ) , H E A D ( 8 ) , L 0 G , C L A S , D L I M , A N G , I C U T , S M E A N , V A R I , S T O , N

5 COMMON I'PCH, IPC,L,.SPRD I M E NSIO N Z(3)

C PROGRAM GAM MA CALCULATES V AR IOGRA MS IN ANY D I R E CTIO N FROM DATAC THAT HAS C OO RDINA TES AND ASSAY V A L U E S , DATA NEED NOT BE ORD ERED*

10 C GAMMA IS CUR RENTLY DIM ENS I O N E D TO HANDLE UP TO 100 DATA POINTSC AND UP TO 20 CLASS INTER VALS FOR A GIV EN DIR ECTIONC INPUT CARDS.C CARD ONE.C COL 1 - 80. HEADING CA RD

15 . C CARO TWO.c COL 1 LD L OC ATION OF ASS AY VAL UE ON DATA CARD (1, 2,OR 3)c (IE. IS IT FIRST SE COND OR THIRD ON TH E DATA CARD)c COL 2 NO LOCATION OR N O R T H I N G . (1,2, OR 3)c COL' 3 IE LOCATION OF EAS TING (1,2, OR OR 3)20 c COL 4 LOG LOG T R A N SFOR M OF DATA, 1»YES, 0° NO.c COL 5 IPC PUNCH CARD O U T P U T w 1=YES, 0=N0.c COL 6 ITALLY TOTAL VAR I O G R A M S AND A V E R A G E . 1 = YES, 0=N0.c COL 7 LL WHICH V A R I O G R A M VA LUES ARE TO BE PLOTTEDc 0 ■ SEC OND MOMENT, 1 = MO MENT CEN TER25 c COL 11 - 20 CLAS CLASS INTERVAL TO BE USED TO GROUP DI STANCESc COL 21 - 30 ANG ANGLE IN DEGREES THAT V A R IOGRA M IS TO BEc C A L C ULAT ED ALONG.c 0. = EAS T-WEST 45, « NE -SW

. C 90, » N O R T H - S O U T H -45. = NW-SE30 c COL 31 - 40 SPR ALL OWABL E ANGLE DEV I A T I O N IN DEGRESS®c CARD THREEc COL 1 -80 VARIABLE FORMAT CARD.

C CARD FOURC COL 1 - 80. TITLE OF V AR IOGRA M.

35 C CARD FIVE. DATA CARDS- FOL LOW.C END OF DATA CARDS IS A 7/8 /9 CARD. .C PROGRAM W I L L . PROCESS- MULTIPLE DATA GROUPS, EACH VAR IO G R A MC TO BE CAL C U L A T E D MUST CON SIST OF A. TITLE CARD, DATA CARDS .AND ANC EOF CAPO.

40 CC MAJOR VAR IABLE S USED IN PRO GRAM GAMMAC DATA ARRAY C O N F I N I N G SA MPLES AND C O O RDINA TESC Z ARRAY USED FOR TEM PO R A R Y STORAGE OF INP UT.DATAC- X DIFFERENCE IN EAST C O O R D INA TES BETWEEN TWO SAMPLES

45 C Y DIFFERENCE IN WEST C O O R D INA TES BET WEEN TWO SAMPLESC G DIF FERENCE IN SAMPLE VALUESC OIS DIS TANCE B E T WEEN TWO SAMPLESC 05QR 0 SQUAREDC DI SC OIS TIMES 0 SQU ARED

50 C 101 (1,1) T O T V A R d , ! ) C U M U LATI VE DISTANCE FOR ITH CLASS INTERVALC TOT (1 , 2 ) T O T V A R d , 2). CU M U L A T I V E DIFFERENCEC TOT (1,3) T O W AR (1,3) C U M U LATI VE OSGRC T O T (1,4) T O T V A P (I ,4) CUM ULATI VE DIS TANCE TIMES DIFFERENCE SO.C TOT (1,5) T O T V A R d , 5) CUM ULATI VE NUMBER OF SAMPLES

L * 3 .IPCH » 8

54

65

70

75

80

85

90

95

1 100

105

110

IPT » 1 ' -I OUT = 5R90 a 9 0 , * . 0 1 7 4 5 3 2 9 2 8 1 9 9 4 3

C ZERO ARR AYSDO 5 I a 1,20 00 5 J=l>5T O T (IfiJ) a V A R C I , J ) a 0.0

5 - CONTINUEC READ HEADING AND PA RAMETER CAR O

REA 0 ( I P T , 1 0 ) H E A D 10 F O R M A T (3A10)

I F( EOF(IPT).GT.O) GO TO 200 .READ (I PI, 20)10, NO, IE, LOG,I TALLY,I P C , L L p C LA S , A N G , S P R

20 F O R M A T (7 11,3X, 3F10»0)IF( EOF(I PT).GT.O) GO TO 200

C READ VARIABLE FO RMAT CA RDREA0( IPT,10) IFRMT IFCEOF(I PT). GT.O) GO TO 200C * ***

C SIMPLE LOGIC CHECKSIF(LLoGT.O) L 51 4I F ( ( L D . L E .3 ),A N D . ( N 0 . L E . 3 ) o A N O , ( I E . L E o 3 ) ) GO TO 40

, WRITE (10 U T ,30)30 F O R M A T (* SPECIFIE D ORDER OF DATA IS IN ERROR*)

GO TO 201 40 DLIM = 20 o *CLAS

IF(ANG.LE. 90. )G0 TO 70 ANG « ANG - 150.

■ W R I T E ( I O U T , 6 0 ) ANG 60 F O R M A T (* ANG IS TO O LARGE. NEW ANGLE »*,F7.0)C CONVERT ANGLE AND SPREAD TO RADIANS70 RADI = ( A N G - S O R ) * . 0 1 7 4 5 3 2 9 2 5 1 9 9 4 3

RAD2 = (ANG + SPR >*.01745 329 2 51 9943 65 N = 0C READ DATA CAROS

READ(I F T , 10) TITLE I F( EOF(IPT).GT.O) GO TO 185

80 R E A D (I P T , I F R M T )Z( l),Z( 2),ZT 3)IF(EOF(I PT), GT.O) GO TO 85 N a N + lDAT A(N,1) = Z(LD)D A T A ( N , 2) » Z(NO)D A T A ( N , 3) = Z (IE)GO TO 80

C CONVERT DATA TO L OG ARITH MS IF LOG « 185 IF(LOG.NE.l) GO TO 91

DO 90 1 =1 ,ND A T A (1,1) « A L 0 G 1 0 ( D A T A (1,1))

90 CONTINUEC CAL CULAT E MEAN AND STANDARD D E V I A T I O N91 SUM = SUM2 = 0.0

DO 100 1 =1 ,NA = D A T A (1,1)SUM = SUM * A SUM2 » S U M 2 * A*A

IOC CONTINUE

55

115

120

125

130

135

140

145

150

155

160

165

170

VAR I « (F L O A T ( N ) * S U M 2 - S U M * S U M > /(F L O A T (N >*FLOAT(N -l)) SMEAN » SUM /FLOA TtN)STD = SORT(VAR I)

C START C A L C U L A T I O N OF V A R I O G R A HNI = N-l DO 130 I « 1,NI01 « DAT A(I,1)Y1 » DATA(I,2)XI = D A T A ( I , 3)II = 1*1DO 130 J = II,N02 « D A T A ( J , 1)Y 2 = DATA(J,2) "X2 » DATAC Jp 3)X « X1-X2Y ^ Y1-Y2 0 » 01-02IF( ABS(X)oGTol) GO TO 110 RAO .= R90 GO TO 120

110 IF(YeE0«0) GO TO 111 RAO = ATAN(YZX )GO TO 120

111 RAD = 0.C COMPARE DIR ECTIO N WITH ACCEPTAB LE D IR ECTIO NS.120 I F ( ( R A D . L T . P A D l ) . O R . ( R A D . G T . R A D 2 ) ) GO TO 130

DIS * S Q P T (Y*Y + X*X) f .0001 I F ( O I S , G T . D L I M ) G O TO 130 K =* DIS/CLAS * 1 IF( K.GT»20) K=20 DIS » DIS - .0001 OSOR = 0*0 DISC = DIS*OSOR T 0 T ( K » 1 ) = DIS * T O T ( K , 1)T O T (K j 2) =Q > T0T(K,2)T0T (K,3) aOSOR * T0T(k,3)T0T (K,4) » DISO * T O T C K , 4)TOT(K, 5) =* T0T (K,5> •$- 1.0

130 CONTINUEDO 150 I a 1,20 AN » T C I (I,5)IF(AN.EO.O.O) GO TO 150 VAR (1,1) a TOT( I / D / A N V A R (I,2) = T O T (I,2 ) /AN V A R ( I , 3) =' T O T ( 1 , 3 ) / (2«0*AN)IF( TOT(I ,4). EQ.O) GO TO 149V A R ( I , 4) = T O T ( 1 , 4 ) /(TOT(I, l)*2 o0)

149 V A R (1,5) » T O T (1,5)150 CONTINUEC PRINT OUT RESULTS

CALL VAR OUT(TITLE)C C O M B I NE. DATA IF AVERAGE VAR IOGR AM IS TO-BE CAL CULAT ED,

IF ( ITALL Y.EQ ^O) GO TO 170 DO 160 I = 1,20 DO 160 J=»l, 5TOT V AR (I , J ) TOTVARCI, J) * TOT(I,J)

160 CONTINUE

56

TSUM a SUM * TSUM TSUM2 » S U M2 * TSUM2 N N a N N + N

175 C ZERO TOT AND VAR ARRAYS170 DO 180 I = 1,20

DO 180 J«l,5T O T (I, J ) a VAR(I,J) » 0*0

180 CONTINUE 180 GO TO 65

185 IF(ITALL YoEO oO) GO TO 194 ,C CAL CULAT E AVERAGE VAR IOGRA M '

VARI « ( FLOAT (NN) 4= TSUM 2-T SUM* TSUM ) /( FLOAT ( NN ) 4 F L 0AT ( N.N-1) ) SMEAN * T S U M / F L O A T ( N N )

185 . STD » SORT(VARI)N a NNDO 190 I a 1,20 AN * T0T VAR(I,5)

. IF(AN,EOoO.O) GO TO 190 190 VAR (1,1) = TOTVARt I, D / A N

VAR (I,2) = T O T V A R (1, 2 ) / AN VAR (1,3) “ TOTVAR'C 1 , 3 ) / (AN*2*0)IF( T0TVA R ( I , 4 ) * E 0 » 0 ) GO TO 189V A R ( I , 4) = T O T V A R t 1 , 4 ) / ( T O T V A R t I , i ) *2*0)

195 189 V A R ( I , 5) = AN190 CONTINUEC PRINT OUT RESULTS

.CALL V A R O U T ( H E A D )194 W R I T E ( I O U T , 195)

200 195 F O R M A T (* NORMAL END OF JOB*)GO TO 201

200 W R I T E (I O U T , 191)191 F O R M A T (* END OF FILE ENCOUNTE RED. ERROR IN JOB SETUP,*)201 STOP •

205 END

57

1 S U B R O U T I N E V A R O U T (T I T L E )c $*$•$$$$$$$$$$$$$ S. $ $ s s $ $ $ $ t $ $ s $ $ $ $ $ $ $ $ $ $ $ $ $ $ s s $ $ $ $ $ $ s $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $$ $$$$C S U B R O U T I N E V A R O U T P R I N T S OU T THE R E S U L T S OF THE VAR I O G R A M C A L C U L A T I O N .c $ $ $ $ $ s s s $ s t $. $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$■$ $$$.$$$$$$$$$$$$ $$$$$$

5 D I M E N S I O N TI T L E (8 ) # A ( 1 0 0 ) # C ( 1 1 ) > D ( 5 1 ) > E (13)C O M M O N V A R ( 2 0 , 5 ) , HE A D ( 8 ) , L O G , C L A S , D L I M , A N G , I O U T > SME A N , V A R I , S T D , NN C O M M O N I P C H , I P C , L , S P RD AT A D / 1 8 ( 1 H ) , 1 H S ,1 H E , 1 H C , I H O ^ 1 H N , 1 H D , 1 H , 1 H M , 1 H 0 , 1 H M , 1 H E , 1 H N , 1 H T

1> 2 0 ( 1 H )/10 D A T A E / 1 H M , 1 H 0 , 1 H M > I H E ^ I H N , 1 H T p 1H , 1 H C # 1 H E > 1 H N # 1 H T » I K E # 1 H R /

I F ( L o E 0 « 3 ) GO TO 8DO 7 I ■ 1 , 1 3 - .D(I + 18 ) ® E(I>

7 C O N T I N U E15 ■ 8 C (l) = Oo

D O 85 1 = 2 , 1 1C( I) = C L A S & 2 . 0 > C C I - l )

85 C O N T I N U E ILOG = 2 H N 0

20 I F t L O G . E O o l ) I LO G = 3 H Y E SC P R I N T O U T PA G E ONE O U T P U T

W R I T E ( I O U T , 10)10 F 0 R M " A T < 1 H 1 , 6 0 X , * V A R I Q G R A M * )

W R I T E ( I 0 U T , 2 0 ) T I T L E25 20 ' F O R M A T ( 1 H 0 , 2 5 X , 8 A 1 0 )

W R I T E ( I Q U T , 2 0 ) H E A DW R I T E (I 0 U T , 3 0 ) A N G , S P R , S M E A N , C L A S , V A R I , D L I M , S T D , I L O G f N N

30 F O R M A T (1 H 0 , * D I R E C T ION = * , F 5 . 0 , 5 X , * DE V I ATI ON « F 5 » 0 , 3 5 X , * M E A N 1= * , E 1 0 e 3 , /* C L A S S SIZ E = * , cl 0 e3 , 5 1 X , * V A R I A N C E = * , E 1 0 , 3 , / * OI S

30 2 T A N C E L I M I T «■ * , E 1 0 . 3 , 4 7 X * S T A N D A R D D E V I A T I O N a * , E 1 0 » 3 , / * L O G A R I 3 TH M S - * , A 5 , 5 6 X , * N 0 , OF S A M P L E S = *18)IF( I P C e E Q e O ) G O T O 39

C , P U N C H O U T P U T C A R D S W R I T E ( I PC H , 21) T I T L E

3.5 21 F O R M A T (.8.A10 )W R I T E ( I P C H ,2 3) A N G , S M E A N , V A R I , C L A S , I L O G , N N

23 F O R M A T ( 4 E 1 0 . 3 , A 1 0 , I 1 0 ) "22 F O R M A T ( 6 E 1 0 , 3)39 W R I T E (I O U T , 40)

40 . 40 F O R M A T ( 1 H 0 , 6 X , * 0 I S T A N C E IN F E E T * , 9 X , * N 0 e OF S A M P L E S * , 1 0 X > * D I F F E R E N 1 C E * , 8 X , * S E C 0 N D M O M E N T * , 8 X , * M O M E N T C E N T E R * , 4 X * A V E R A G E D I S T A N C E * )DO 50 I = 1 , 2 0 L OW « t 1 ~ 1 )* C L A S LUP .* I * CL AS

45 W R I T E (I 0 U T , 4 5 ) L O W , L U P , V A R ( I, 5), VA R (I, 2), VA R (I, 3), VA R (1,4) j, VA R (1,1) I F ( I PC . E 0 . 1 . A N D . V A R ( I , 5 ) , G T , 0 ) W R I T E (I P C H , 22) V A R ( I , 5 ) , V A R (I , 2 ) # V

1 A R ( I , 3 ) , V A R ( I , 4 ) , V A R (1,1)45 F O R M A T ( 1 H , 4 X, I 6, * ---- * , I 6, 1O X , F 8 . 0 , 5 X , 3 ( 1 1 X , E 1 0 • 3 > , 1 0 X , F 10* 1)50 C O N T I N U E

50 W R I T E (I O U T , 10)C P R I N T O U T P A G E TWO O U T P U T

W R I T E ( I 0 U T , 2 0 ) T I T L E W R I T E (I O U T , 2 0 )HE AD W R I T E ( I 0 U T , 1 0 0 )

55 100 F OR MA T (1 HO ) .C FIN D MAX V A L U E

TE M P « V A R ( 1 , L )

58

DO 55 1=2,20I F ( VAR(I ,L). GT.TE MP) TEMP « VAR(I,L)

60 55 CONTINUEUNIT * CLAS/5.DIV = TEMP/50®

, DO 75 K »1,51 TOP = TEMP

65 BOT » TEMP - DIVDO 65 I » 1,20I F ( ( V A R ( I , L ) o G T . T O P > . O R . V A P ( X , L ) » l E » B O T ) GO TO 65 J » VAP ( I, D / U N I T > 1 A (J ) = 1KX

70 65 CONTINUEW R I T £ (I OU T,60) D ( K ),TEM P,A

60 FOR MAT (1H , 2X, Al, 2X, E10«, 3, ” **,10061)TEMP = BOT DO 70 1=1,100

75 A(I) = 1H70 CONTINUE75 CON TINUE

W R I T E ( IOUT,80)80 FORMAT (1H , 17X, 10( »«$***4$***** ), 4=$*)

80 W R I T E ( I O U T , 9 0 )C90 F O R MAT(1 H , 1 3 X , 1 1( F6.0 ,4X))

RET URNEND

APPENDIX B

FLOWCHART AND PROGRAM LISTING FOR PROGRAM KRIG

Program Krig is a Fortran IV program written for a DECSYSTEM 10 computer.

59

Read In All Input Data

PrintOut

Results

Find Nearest Holes

Calculate Weighted Assay Grade

CalculateBlock

Variance

Solve Linear Equations for Weights

Calculate Covariance Between Each Sample

Do the Following Steps For Each Block

Calculate Covariance Between Each Hole and

the Block

06200 AC CEPT 1 2 * A L F A (1)»A L F A (2)063 00 12 FORHAT(AF)06400 TYPE 14065 00 14 ' F O R M A T (•TYPE N K , N A L L , N B C , N E C , N B R , H E R * )06600 C — READ ON SECOND CONTROL CARD NUM BER OF HOLES IN KRI GXNG, AND067 00 C WHICH BLOCKS TO KRIG068 00 ACCEPT 1 6 , N K , N A L L , N B C , N E C , N B R , H E R069 00 16 F O R M A T (61)070 00 IF(NALL.GT.l) GO TO 2007100 NBC a 107200 NBR = 107300 NEC = NC07400 NER a NR075 00 C— - READ IN THE NAME OF THE INPUT DRILL HOLE FILE07600 20 TYPE 1807700 18 F O R M A T !1 TYPE THE NAME OF INPUT DRILL HOLE F I L E 8 )078 00 ACCEPT 1 9,NAME079 00 19 FORMAT(A5)08000 CALL IF I L E U , NAME)

■ 081 00 I = 008200 C ~ — READ IN THE DRILL HOL ES08300 22 1 * 1 + 108400 R E A D ( 1, 25,END « 3 0 ) H O L E (I, 1 ) , H O L E (I , 4 ) , H O L E (I, 5 ) , HOL ECI,08500 1 , H O L E (I , 2 ) , H O L E (1,3)08600 25 F O R M A T ( A 5 , l O X , 3 F 5.0,1 O X , 2 F 5»0)08 700 C 25 F O R M A T t A 5 , 4 X , 3 F 5 » 2 , 2 F 1 0 ® 0 )088 00 " GO TO 22089 00 30 N « I -1090 00 TYPE 5, N091 00 5 F O R M A T!8 NUMBER OF DRILL H OL ES * *15)092 00 C CAL CULATE THE .B L O C K VARIANCE093 00 GM0Y2 * 0,0094 00 . DO 32 I ® 1,4095 00 DO 32 J » 1,409600 D C (1,1) = 50. * ( 1 - 3 ) *W4 * W8097 00 , „ DC (2,1) -a 50, + -! J-3)*W4 * W 809800 . D C ! 3,1) « 0,0099 00 ; D C (1,2) = 5 0 * x10000 D C (2,2) = 50.10100 D C (3,2) = 0.010200 CALL COVAR (D C , G M O Y )10300 G M 0 Y 2 •= GM0Y2 ♦ GMOY10400 32 CO NTINUE10500 BLKVAR = GM0 Y2/16.10600 C— -START THE KRI G G I N G O PE RATIO NS10700 YM = YM AX + ,5*WI0TH10800 XM = XMIN + ,5& WIDTH10900 DO 100 J = NBR,.NER11000 X Y (2) = YM - J * W I D T H11100 DO 100 I « NBC,NEC11200 XYtl > * XM 4- I* WIDTH11300 C— FIND NK CLOSEST HOL ES -11400 DO 40 K = 1, N11500 DIFY = X Y (2 ) - H O L E ( K , 2)11600 D I F.X = X Y (1) -HOLE (K, 3)11700 D 1 = D I F Y*DIF Y11800 02 = D I F X *DIF X11900 DIS(K) = 0 1 + 0 212000 40 CONTINUE12100 NI = NK ❖ 112200 00 50 K a 1,NK

12300 OS(K ) * DISCK)12400 IHQtE(K> = K ■12500 50 CONTINUE12600 CALL DSORT (DS,IHOLE,NK)12700 DO 55 K = NI,N12800 I F {D I S (K )9 GT »O S (N K )) GO TO 5512900 OS(NK) = OIS(K)13000 IHOLE(NK) a K13100 CALL DSORT (DS,IHOLE,NK)13200 55 CONTINUE13300 C--- CREATE SEP ARATE SAM PLE I NF ORMAT ION FOR EACH C O M P O S I T13400 NH « NK13500 D C (1,1) = XYil)13600 D C (2,1) a XY(2)13700 NK3 * NK*313800 D C (3,1) = 2790®13900 DO 53 K a 1,NK14000 IK » IHOLE(K)14100 DO 56 II = 1,314200 KI =* K*3 -3 II143 00 DO 54 L = 1,314400 H O L (K I , L ) = H O L E (I K , L )14500 54 CONTINUE146 00 H0 L(KI,4) = ELEV(II)14700 H0L (KI,5) * H O L E ( I K , 11*3)14800 56 CONTINUE14 900 53 . CONTINUE1 5 0 0 0 : NN a 015100 NMAX « NK315200 600 NN = NN *115300 IF(NNoGToNMAX) GO TO 65015400 IF< H 0 L ( N N , 5 ) . G T . O . O ) GO TO 60 015500 KN s NN *115600 IF( KN.EQ .NMA X) GO TO 64915700 DO 603 NNN = KN, NMAX15800 DO 603 MM = 1,515900 H O L (NNN— 1 , M M ) * H O L (NNN,MH)16000 603 CONTINUE16100 649 ■ NMAX =NMAX - 116200 GO TO 60016300 650 NK3 = NMAX16400 NKH1 a NK3 * 116500 NKH2 = N K 3 * 216600 C CAL CULAT E COV A R I A N C E BET WEEN EACH SAMPLE AND THE BLOCK16700 75 DO 70 K = 1,NK316800 D C (1,2) « H O L (K , 3)16900 D C (2,2 ) = HOL(K,2)17000 D C (3,2) a HOL (K,4)17100 CALL C O V AR(DC ,GMO Y)17200 R ( K ) = GMGY17300 G A (K ) = GMOY17400 A ( (K-l )*NKH1 +NKH1) =» loO17500 A ( (NKHl-l) *NKH1 * K) ■» 1,017600 70 CONTINUE17700. A (NKH1 *NKH1 ) » 0»017800 C-- -CALCULATE COV A R I A N C E BET WEEN THE SAMPLES17900 74 DO 83 K=1,NK318000 DO 80 L » K ,N K 318100 DIFY « H O L (K , 2) - H O L ( L ,2)18200 DIFXa H0L(K,3) - H O L (1,3)18300 DIFZ = H O L (K ,4) -H0L(L,4)

64

18400 01 a DIF Y*DIF Y18500 01 = SORTC 01} .18600 02 = DI.FX*DIFX18700 02 « S OR I {02)18800 . 03 » SORT(DIF Z*DI FZ)18900 GH ■ GAMMA (01 ,02,03)19000 126 A ( (K-l) *NKH1 4- L) ° GH19100 A ((L - 1 ) *NKH1 + K ) = G H19200 80 CONTINUE19300 83 CONTINUE19400 R(NKH1) * 1.019500 c-- -SOLVE SIM ULTANEOUS LINEAR E Q U A T I O N S --- -19600 CALL GEL.G(R, A, NKH1, 1, 0.000001, IER)19700 C-- • CHECK FOR ERROR IN MATRIX OPE RATIO NS19800 I F d E R . N E .0) GO TO 18519900 c-- B CO NTAINS THE S O L U T I O N ___20000 89 A G a 0 a20100 CU = 0,20200 C— — •CALCULATE THE W E I GTHED ASSAY GRADE AND VARIANCE20300 DO 97 K = 1, NK320400 AG * AG * GA(K)*R(K)20500 CU a CU* HOL ( K , 5 ) * R ( K )20600 97 CONTINUE20700 VARKG » R(NKHl) * AG - BLK VAR20800 C — PRINT OUT R E S U L T S - - — -- *-- --- — — — *----- -—20900 ICU a CU21000 W R I T E ( 2 3 , 1 0 3 ) J , I , I C U , V A R K G21100 103 F O R M A T (1 1 9 ® , 3 1 5 , F l O a 3,* 6H 0LE 3 L E V E L 0 )21200 100 CONTINUE21300 GO TO 19121400 c— - PRINT ERROR MESSAGES21500 185 TYPE 1 8 6 , IER,I,J21600 186 FORMAT!* ERROR IN M A T R I X * , 1 5 , 0 FOR BLOCK I *»,I5,21700 _G0 TO 100 * •21800 191 STOP21900 END ...22000 SUBROUTINE COVAR (B,GMOY)22100 C — SUBROUTINE COVAR DETERMIN ES THE C O V ARIAN CE BETWEEN A22200 C— ■— AND A BLOCK. A DISCRETE S U M MATIO N IS USED TO APPROXII22300 c — THE SEXTUPLE INTEGRAL.22400 COM MON A L F A ( 2 ) , W I D T H22500 • DIM EN S I O N 8(3,2)22600 GMOY a 0*22700 W 8 = W I D T H /8 .2 2 800 W 4 a WI0TH/4.22900 00 10 I * 1,423000 DO 10 J = 1,423100 D IFX sb {1,2 ) - B (1, 1) * (I-3)*W4 * W 823200 DIFY = 6( 2,2)- 6(2,1) * (J-3)*W4 ❖ ti823300 DIFZ '*8(3,2) - 8(3,1)23400 01 « S O R T(DIF Y*DI FY)23500 D 2 a S Q R T (DIF X*DI FX)23600 0 3 o SORT ( D I F Z ’S'DIFZ)23700 GH * G A M M A (01 ,02,03)23300 GMOY * GMOY * GH2 3 900 10 CONTINUE24000 GMOY = GHOY/X&o24100 RETURN24200 END24300 SU BROUTINE DSORT (OS,IHOLE,NH)24400 D I M ENSIO N O S ( 2 0 ) , I H O L E (20)

24500 C THIS S U B ROUTI NE SORTS THE OS ARRAY INTO INCREASING ORD ER24600 IF(NH .LT»2) RETURN24700 DO 20 I o 2,NH24800 IP * 1 - 124900 DO 10 J = Ifil?25000 I F ( D S m . G E . D S C J ) ) GO TO 1025100 SAVE s DS(I)252 00 D S m » DSC J)25300 ISV» IHOLECI)25400 I H O L E C I ) ° IHOLECJ)25500 DS(J) 3 SAVE25600 IHOLECJ) * ISV25700 10 CON TINUE25800 20 CONTINUE25900 RET URN26000 END26100 SUBROUTI NE G E L G ( R , A , M , N , E P S , I E R )26200 C — -T HIS SUBROUTINE IS FROM THE IBM SSPS PACKAGE OF STA TISTI CAL26300 c— - S U BROUT INES . IT SOLVES S I M U L T AN EAOU S EQUATIONS.26400 c— -S I MI LAP. SUB ROUTI NES MAY BE S UB STITU TED FOR IT.26500 DI M ENSIO N A ( 1 ) p R(1)26600 DOUBLE PRE CISIO N A , R , T B f P I V , P I V I , T O L26700 I F ( M ) 2 3 , 23,126800 c26900 c SEA RCH FOR G R E A T E S T ELEMENT IN MATRIX A27000 1 I E R s O27100 PIVaO.27200273 00 NM= N*M27400 DO 3 L=1 ,MM27500 T B s D A B S C A C L ) )27600 I F {T B - P I V )3,3,227700 2 PIV=TB27800 Ia L27900 3 C ON TINUE28000 T O L = E P S * P I V28100 ,-c. AC I).IS PIV OT ELE MENT. PIV CON TAINS THE ABSOLUTE VALUE OF A28200 c28300 ‘ c START ELI M I N A T I O N LOOP28400 L S T 3 128500 DO 1.7 K*l, M28600 c TEST ON S I N G U L A R I T Y28700 IF ( P I V )2 3,23 ,428800 4 IFC IER ) 7,5,728900 5 I F ( P I V - T 0 L ) 6 , 6 , 729000 6 IERa K-l29100 7 PIV-I*l./A(I)29200 J»( I - D / M293 00 IaI-J*M-K29400 JaJ+l-K29500 c I4-K IS R O W — I ND E X , J4K C O L U M N - I N D E X OF PIVOT . ELEMENT29600 c PIV OT ROW RED UC T I O N AND ROW INTERCHANGE IN RIGHT HAND SIDE I29700 DO 8 L = K ,NM,M2 9 800 11*1+129900 TB« PIVI*RCLL)30000 RCLL)-R(L)30100 8 R (L ) aTB30200 c30300 c IS ELIMINAT ION TER MINAT ED30400 I F ( K-M)9 ,18, 1830500 c

30600 C COL UMN INT ERCHANGE IN MATRIX A30700 9 L E N D = L S T + M - K30800 I F ( J)12,12,1030900 10 II* J*J131000 DO 11 l = tSTj,LENO31100 TB = A ( U31200 Ll=L*II31300 A(L )«ACLL)31400 11 A (L L ) a IB31500 C31600 C ROW INTERCHANGE AND PIVOT R O W . R E DU CTIO N31700 12 DO 13 L =L ST,MM ,M31800 11*1*131900 TB»PrV.I*A(lL)32000 A(LL)»ACL)32100 13 A(l )*T832200 c32300 c SAVE C O L UMN INTERCHANGE I N F O RMAT ION .32400 A(LST)*U32500 c32600 c ELEMENT R E D U C T I O N AND NEXT PIV OT SEARCH32700 PIV*0o32800 LSI *LST*132900 J»033000 DO 16 II= LST,L END33100 PI VI*-A(II)33200 IS T * 1133300 J»J*133400 DO 15 L * I S T,MM ,M33500 LL» L-J33600 A(L ) = A ( L ) + P I V I * A ( L L )33700 TB = DA8S(AC L ) )33800 I F ( T 8 - P I V ) 1 5 , 1 5 , 1 433900 14 P I V a TB34000 I = L '34100 15 CON TINUE34200 DO 16 L»K ,NM,M34300 ■ LL*L+j34400 16 R ( L L ) = R ( L L ) + P I V I * R ( L )34500 17 LST =LST+ M34600 c END OF E L I M I N A T I O N LOOP34700 c BACK S U B S T I T U T I O N AND BACK INT ERCHA NGE34800 18 I F (M - l ) 2 3 , 2 2 , 1934900 19 1ST »MM*M35000 LST»M*135100 DO 21 1* 2,M35200 11 * L ST— I35300 IST* IST-*LST35400 L = I ST-M3 5500 L*A (L>*® 535600 DC 21 J=I I,NM, M35700 T B * R (J )35800 LL 3 J35900 DO 20 K = i S T , M M , M36000 L L * LL *136100 20 TB= TB - A ( K ) * R ( l l )36200 KeJ +L36300 R(J)*R(K)36400 21 R(K)=TB36500 22 RE TURN36600 c ERR OR RETURN

MATRIX A

3670036800369003700037100372003730037400375003760037700378003790038000381003820038300384003850038600

23 IERa-1 R ET URN END

F U N CTION G A M M A (D 1* 02* 03)COMMON A L F A ( 2 ), WIDT H

C THIS SUBROUTINE CALCULAT ES THE VAL UE OF THE VAR I OGRAilC— FOR A GIV EN DISTANCE AND DIRECTION*

H * S O R T ( D 1 *01 + D2*D2 * 03*03)IFI H.LT.IO. ) GO TO 20 GAMMA =AL FA(1) *AL0 G10(H )GO TO 15

20 GAMMA = 0*015 H2 o S0RT(D3*D3)

I F ( H 2 o L T « 1 0 e ) GO TO 16GAMM2 = A L F A ( 2)* AL0G 10(H2 )GO TO 18

16 GAMM2 » 0*018 GAMMA ® GAMMA * GAMM2

RETURNe n d

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. 41-68.

Carlier, P. A., 1964, "Contribution Aux Methods d'estimation des qisement's d 1 uranium.'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

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