Pocket Geostatistics for Field Geologists

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POCKET GEOSTATISTICS FOR FIELD GEOLOGISTS

Jean-Michel M. Rendu

Xewmont Services Limited

Danbury, Connecticut

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Introduction

Plost programs can process up to approximately

120 data points. The processing time varies

from a few minutes to a number of hours, depend-. .Increasingly, geologists use geostatistical

ing on the number of data points and the pro-

methods to evaluate drilling programs. The

gram being used. Both storage limitations and

complexity of the methods varies with the processing time should be taken into account

nature of the available sample theto decide whether or not the pocket geostatisti-

objective of the statistical- nalysis and thecal package should be used. In most instances

geologist's experience. Occasionally, estimat-analysis of data sets containing more than 50

ing the average sample grade is all that isdata points will be considered impractical.

required. In other instances a semivariogram

calculation may be considered. If sufficient

data is available, the geologist may wish to

estimate blocks or panels of complex shape,

using the kriging method of evaluation. Histor-

ically, the use of geostatistics by geologists

has been limited by a number of factors, most

significant being the difficulty in accessing

and using computer programs needed to complete

the necessary studies.

Geologists are often working in remote areas

from which electronic communication is unreli-

able, if not impossible. Even when access to

computer programs is technically possible, the

costs incurred in accessing and using these

programs are often considered prohibitive. In

many instances using the programs requires an

understanding of computer systems far beyond

that which a field geologist can justifiably

be interested in acquiring and maintaining.

These problems are resolved in part by

developing better software, and using hardware

which can be installed in exploration offices.

h recent trend is towards user-friendly inter-

Areas of Application

The pocket geostatistical software was

developed for the complete geostatistical

study of two dimensional data sets. It is

directly applicable to the detailed analysis

of vein or stratified deposits, where the

variables of interest are the deposit thickness,

the average grade between foot-wall and hang-

ing-wall, and possibly the metal accumulation.The accumulation is a measure of the quantity

of metal per area unit, and is defined as the

product of thickness by grade. The system is

also applicable to early analysis of large

deposits, when the 'information obtained from

the drill holes can be reduced to a thickness

mineralized and the corresponding metal accumula-

tion. The system has also been used successfully

to study blast hole assays, as well as geo-

chemical data.

Possible direct and indirect applications of

the pocket geostatistical software include the

following:

active programs installed on mini or micro com-

puters. This approach is likely to result in a Statistical Analysis:significantly improved working environment for

many geologists. Xevertheless, there will

remain situations where access to even these

d o n scaled geostatistical systems will be

impossible because of cost and limited trans-

portability.

In an effort to demonstrate thac these last

constraints to the use of geostatistics can be

removed, a complete suite of fully interactive

~eostatistical rograms has been developed for

a 'nand held programmsble calculator. Even

though new limitations are introduced by the

pocket size of the computer, these programs

have been used successfully on a number of

deposits. The questions which can be answeredby using pocket geostatistics, and the func-

tions which have been made available to answer

these questions are described below.

Hardware Configuration and Limitations

The pocket geostatistical package was devel-

oped on a Hewlett Packard HP41CV hand held com-

puter, to which were added a card reader, an

extended function module and two extended

memory modules. This hardware configuration was

chosen to produce a system which can be carried

in a coat pocket. If desired, a Hewlett Packard

printer-plotter can be added to the calculator,

with no significant reduction in portability.A11 ?roSrams have been rdritten to operate

either with or without a princer. The ruse of

rechargeable batteries is reconmended to supply

a reliable source of energy.

- Calculation of average grade with

confidence limits, for normally or

lognormally distributed values.- Calculation of proportion and average

grade of samples above cutoff grade.- Analysis of frequency distribution,

with possible determination of outliers

or mixed geologic or statistical en-

vironments.

Semivariogram:- Semivariogram calculation in user

specified directions, for normally or

lognormally distributed values.- Determination of trends, drifts, and

directions of ore chutes.- Determination of distance of influence

and optimization of drill hole location

and spacing.- Determination of nugget effect and

assessment of sampling and assaying

procedures.- Semivariogram modelling, and development

of a mathematical model to be used Sv

rhe reserve calculating programs.

?.eserve Evaluation:

- Evaluation of the averase value o i

blocks located on a regular 2rid, usin;

linear krigin;.- Evaluation of the average value of

panels of irresular shapes, usin? Linea-

krizin:.



- Estimation of the precision with which

reserves are estimated.- Estimation of the influence of drillhole location and spacing on the pre-

cision with which the reserves are

estimated.- Optimization of drill hole location.

and spacing.- Calculation of optimal kriging weights

to improve routine evaluation of

blocks using suboptimal weightedaveraging techniques.

Influence of Selectivity on Tonnages and

Grades: - Estimation of the variability in grade

which could be ex~ected f a specified

level of mining selectivity was

achieved.- Estimation oi the tonnage of material

and average grade which could be minedabove a given cut off grade, if a

specified level of mining selectivity

was achieved.

As is usually the case with geostatisticalanalysis of geological data, the programs do

not always give a final answer to the questions

which a geologist may have. Hore commonly, theprograms should be used to assist in the

analysis of the data, and to supply quantita-

tive information which must be interpreted,

and added to the geologist's experience and

knowledge to nake a rational decision.

Available Geostatistical Functions

The normal procedure for data analysis con-

sists of creating a drill hole data base,

statistically analyzing the drill hole values,

developing a semivariogram model and estimating

reserves, either for the deposit as a whole,

within specific panels or withia blocks on a

regular grid.

The geostatistical functions which are avail-

able to perform these analyses can be classified

as the following:- Xormal and lognormal statistical

analysis.- Semivariogram calculation and

modelling.

- alock or panel evaluation.- Calculation of grade - tonnage

relationship.

In addition, necessary support functions

are supplied, including data base management,

and optional printer and plotter displays.The pocket geostatistical system is composed

of 13 main programs, which are stored on

approximately 100 magnetic cards.

Data Base Xanagernent:

Programs are supplied for the creation,listing, and updating of drill hole data

bases. The file names are specified by the

user, as well as the number of samples and the

easting, northing and value of each sample.

If areas of irregular shape are estimated

using the panel kriging program, a polygon

data base must be created by the user contain-ing the easting and northing of the vertices

of the polygonal area defining the panel, The

name of the polygon file is user specified.When blocks are estimated by kriging, a

block file is automaticaliy created, containing

for each block the number of samples used to

estimate the block, the estimated value andthe variance of estimation.

All files can be listed and edited. Statis-

tics can be calculated on both drill hole files

and block files. File backups can be saved

on magnetic cards.

Statistical Analysis:

Normal and lognormal statistics can be

calculated on either drill hole files or

block files. If a normal option is specified,

the following statistics are obtained:- Number of samples.- Mean,variance and standard deviation.- Upper and lower confidence limits for

the mean,- Histogram, frequency distribution and

cumulative frequency distribution.

Iflognormal statistics are requested, a

third parameter additive constant can be speci-

fied. The following statistics are obtained:- Number of samples.- Logarithmic mean and variance.- Sichel's "tl' estimate of the mean.- Upper and lower confidence limits for

the mean.- Histogram, frequency distribution, and

cumulative frequency distribution.

The histogram is obtained by classifying thesample values into grade categories. The grade

limits are calculated on a linear of logarithmicscale, depending on the option specified. The

sample average grade is calculated within each

grade class, which can be used to analyze the

influence of cut off grades. When a printer is

available, che above statistics are printed and

a histogram plot is produced.

Calculation of Experimental Semivariograms:

Absolute, relative or logarithmic semivario-

grams can be calculated on drill hole files. In

addition to the name of the file to be processed,

the user must specify the azimuth of the direct-

ion to be considered, a tolerance angle, and

distance classes within which sample pairs willbe grouped. Within each distance interval the

following statistics are calculated:- Number of pairs of samples.- Average distance between samples.- Semivariogram value.

Other statistics which can be calculated at

the user's request include for each distance

interval:- Relative semivariogram.- Average grade of samples.- Average drift between samples.- Average direction of sample pairs.

If a printer is available, all statistics

will be printed and the ~emi~ariogramsill be

plotted.

Xodelling of Semivariogram:

A two diinensional semivariogram aodel can be

specified as the sum of three submodels:



- A nugget effect.- Two spherical submodels, each one hav-

ing its own sill, range, direction,

and ratio of anisotropy.

Once the model has been specified, the

value of the semivariogram can be calculated

in any direction, for any distance. If a

printer is available, the semivariogram will be

plotted. The scale of the plot can be chosen

to facilitate visual coniparison of the experi-

mental and modelled semivariograms.

Estimation of Block Values by Kriging:

A kriging program is available to estimate

blocks on a regular grid. The following para-

meters must be specified by the user:- Name of drill hole file.- hlinimum and maximum easting of the

area to be estimated.- Minimum and maximum northing of the

area to be estimated.- Block size in both the east-west and

north-south directions.- Semivariogram models. This can be the

sum of a nugget effect and two aniso-

tropic spherical submodels.- Xaximum search distance. An aniso-

tropic search can be used.- Xaximum number of samples to be con-

sidered for block evaluation.

In addition, the user must specify the

number of p'oints which should be considered

within each block for the discrete calculation

of block variances and covariances.

An output file is created containing the

number of samples used to estimate each block,the estimated block value and the estimated

variance. The file can be listed, printed or

edited. It can also be accessed by the statis-

tics program to obtain tonnages and average

grades within specified grade intervals.

If desired,additional statistics can be

listed after completion of the kriging program:- Dispersion variance of the true block

values, calculated from the semi-

variogram.- Covariances between the last estimated

block and the samples used to estimate

the block.- Sample values and optimal weight given

to each sample.- Lagrange multiplier.

Estimation of Panels of Irregular Shape:

A kriging program is available to estimate

the average value of panels of irregular shape

from sample values stored in a drill hole file.

The number of samples which can be used to esti-

mate a panel is limited, and analysis of an

entire deposit may require its division into a

number of panels to be estimated independently.

Each panel is specified by a polygon file which

must be created before using the panel kriging

program. The following parameters are request-

ed by the program:- Xame of the drill hole file.- Xame of the polygon file.- Semivariogram model.- A parameter specifying whether all the

samples on the drill hole file should

be considered, or only the samples

within the panel.

In addition, the user must specify a block

size to be used in the east-west and north-

south directions for discrete calculations

of polygon variance and covariances.

The displayed or printed output of the panel

kriging program is similar to that of the block

kriging program.

Calculation of the Grade - Tonnage Relation-

ship:

Provided simple but often satisfied as-

sumptions are made concerning the statistical

properties of selective mining units, it is

possible to estimate what will be the effect

on the reserves of changing mine selectivity

or cutoff grade. A program is available to

assist in this estimation. To specify the

properties of the deposit, the following para-

meters must be specified:- Total tonnage mineralized, assuming no

selectivity.- Average grade of entire deposit.- Whether the block values are normally

or lognormally distributed. If a

distribution is specified, a third

parameter can be added.- Dispersion variance of the selective

mining units.

The dispersion variance is a function of the

block size or the degree of selectivity with

which the deposit will be mined. It can be

estimated using the block kriging program.

Given the above parameters, the following

statistics can be calculated for any userspecified cut off grade:- Tonnage of ore material.- Average grade of ore material.- Metal content of ore material.

Conclusion

Clearly the use of a pocket calculator can-

not be advocated as the ultimate solution to

the geostatistical evaluation of mineral

deposits. The software presented here was

developed in an attempt to alleviate some of

the frustration often expressed by geologists

in their constant effort to use modern tech-

nology in an environment where access to this

technology is often only too limited. Earlyin the development of this software it was

decided that the mathematical integrity of

all calculations should not be compromised.

The cost of this decision is found in the

slow speed of execution of some of the programs.

However, the results obtained are at least as

accurate as those expected from more powerful

software - hardware combinations. Constant

evolution in hardware technology and the develop-

ment of faster processors should result in the

timely elimination of all constraints to the

field utilization of geostatistics.

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Pocket Geostatistics for Field Geologists