250
Journal
of the
South
African
Institute
of Mining
and Metallurgy
January,
1969
THE COMPUTER MAPPING AND ASSESSMENT OF
BOREHOLE AND SAMPLING DATA FOR STABLE
MINERALS, P ARTICULARL Y AS APPLIED IN
COAL MINING
By R. B. MacGillivray, * B.S.c. (Eng.) (Rand) (Member),
D. M. Hawkins,t B.Sc. (Hons.) (Rand), F.S.S. (Visitor) and
M. Berjak,t B.Sc. (Hons.) (Rand) (Visitor)
SYNOPSIS
The paper describes a method of producing iso line plans using a mathematical technique for
interpolating between boreholes, and its computer application. The iso line plans are used for
assessing the value parameters of an ore body. The computer programme also produces means and
standard deviations to supplement these iso line plans.
The mathematical techniques tested for the production of a satisfactory fit of the data were
the polynomial, Fourier and moving average technique.
The iso line plans are used for assessing the quality of the ore and other salient features such as
depth to the floor of the ore body at different points, with a view to deciding how best the ore may
be mined.
The plans are plotted by the computer to any scale required.
INTRODUCTION
To assess the economic exploitability of an ore body many tedious and repetitive
calculations must be carried out. Observations of borehole or development samples
have to be processed yielding measures of ore reserve and quality in each sampled
area, and also measures of the worth of the ore body as a whole and of variation in
quality from section to section of the ore body.
The observations made on the ore (which we shall term 'value parameters')
may embrace a variety of characteristics of the ore. Obvious value parameters are
those directly related to the economics of mining, viz. assay values, ore body thickness
and depth below surface. Other value parameters however may be almost as important.
In the case of coal for example a purchasing contract may specify limits of acceptability on other value parameters such as calorific value, ash content or the percentage
of volatile matter. In this case, a detailed study of these value parameters will also
be crucial. This detailed study must include an assessment, not only of the body as a
whole, but also the regional variation in value of the parameter. An obvious and
satisfactory way of displaying this variation is by drawing iso lines of constant
value parameter on a map of the ore body and supplementing this with the
means and standard deviations of the value parameters taken over all boreholes.
Such plans, considered together, show up the zones of significance. The relatively
stereotyped nature of the operations involved in this process makes it amenable to
computer solution. In order to produce iso line plans on the computer we require
some mathematical formula for producing a 'valuation surface'-a
mathematical
*Assistant Technical Manager, Rand Mines Limited.
tDepartment of Statistics, University of the Witwatersrand.
:j:Operations Research Department, Rand Mines, Limited.
---
January, 1969
Journal of the South African Institute of Mining and Metallurgy
251
The computer mapping and assessment of borehole and sampling data for stable minerals, particularly as applied
In mining-R. B. MacGillivray, D. M. Hawkin.' and M. Berjac
function for interpolating between boreholes. This paper deals with one such mathematical technique and its computer implementation. Originally it was developed
for the valuation of coal fields and for this historical reason examples cited are of
colliery data. Both the technique and the implementation are of potentially much
wider application.
TECHNIQUES
EXAMINED
A number of mathematical models for valuation surfaces are available and four
have been considered for computer implementation.
The first is the 'triangle' method of linear interpolation. In this technique which
is used frequently by mining engineers for manual surface fitting, the area is broken
up into small triangles defined by boreholes. The value at any point within a particular
triangle is estimated by linear interpolation of the values at the three boreholes
defining the apices of the triangle. There are several objections to this model. One
is that the decomposition of the area into triangles is subjective and that hence
results are not reproducible. Another (and much more serious) objection from the
statistical point of view is the implicit assumption that the value parameters are
measured with perfect accuracy and that the cores obtained from the boreholes are
representative of the areas nearby. In practice obviously neither assumption is
justified thus no attempt was made to implement this technique.
A second approach which suggests itself is the use of a polynomial. The valuation
surface is obtained as a polynomial in x and y co-ordinates. 1 he value of this polynomial at any given point gives the predicted value of the value parameter at that
point. The coefficients of the polynomial may be found by means of a traditional
least squares fit based on the borehole values given or by the more modern minave
and minimax fitting techniques. This technique has been implemented and tried in
practical cases but results are, on the whole, disappointing. Polynomials of order 6
and less were found to be inadequate for representing data of the complexity quite
commonly found, with the result that valuable information was simply 'smoothed
out'. Further disconcerting properties of these least squares polynomials are that
they are unduly influenced by boreholes near the periphery of the area considered,
and that they must, by their very nature, become unstable at the edges.
A third, related approach involves the use of finite Fourier series rather than
polynomials. These series have several advantages over the polynomial series. They
are considerably more flexible and can give much better representations of complex
surfaces. Further, they are better behaved at the edges. On the other hand, they are
inclined to show the sharp discontinuities characteristic of Fourier series, and once
again are not capable of representing very complex surfaces. On available evidence,
the Fourier series, while better than polynomials, are often simply inadequate for
accurate assessment of the borehole data.
In Figs. 1 and 2 comparative iso line plans of seam widths produced by a 6th
order polynomial, and a 6th order finite Fourier series are shown. The positions and
values (seam widths) of the boreholes are shown in Fig. 3 for comparison. The multiple
correlation, which is a measure of the accuracy of the valuation surface in representing the data, is considerably higher for the Fourier than the polynomial surfaces,
though both techniques suffer from comparison with the moving average technique
discussed below. For example, it is clear that both have smoothed out some useful
information-a
serious shortcoming.
~
~---
------
252
Journal of the South African Institute of Mining and Metallurgy
January, 1969
The computermapping and assessmentof boreholeand samplingdatafor stable minerals,particularly as applied
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Journal of the South African Institute of Mining and Metallurgy
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Thecomputer mapping and as.<essmentof borehole and sampling data for stable minerals, particularly as applied
in mining-R. B. MacGillivray, D. M. Hawkins and M. Berjac
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,
254
Journal of the South African Institute
of Mining and Metallurgy
January, 1969
The computer mapping and assessment of borehole and sampling data for stable minerals, particularly as applzed
in mining-R. R. MacGillivray, D. M. Hawkins and M. Berjac
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-- --
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January, 1969
Journal of the South African Institute of Mining and Metallurgy
255
The computer mapping and assessment of borehole and sampling data for stable minerals, particularly as applied
in mining-R. B. MacGil/ivray, D. M. Hawkins and M. Berjac
The fourth and most satisfactory technique to date is the weighted moving
average technique, pioneered in this country by Krigel. In this technique, the value
over any area is estimated by a weighted average of nearby borehole values, the
weights decreasing with distance. In terms of this very general definition, the 'triangle'
technique is a form of weighted moving average using particularly simple (though
subjective) weights. Krige's moving average was designed for the valuation of South
African gold mines, where the basic data consist of not borehole, but development
and stope samples. The sampling is done at regular intervals over the irregular grid
of reef drives and raises, and the stope sampling along stope faces from time to time
as these advance. Krige's technique breaks the ore body down into regular squares,
and produces estimates of total reserve within each square from data in that and/or
the squares in its vicinity. The technique makes efficient use of the regularity in the
data, but also relies heavily on that regularity. The intensive sampling on gold mines
is motivated by the extreme variability for which gold is notorious. Many other
minerals have much more stable properties, and in such cases the regular grid of
samples will generally be replaced by a much more haphazard set of borehole readings.
In this case, Krige's technique in the form used on the gold mines breaks down.
Accordingly a more general moving average technique is proposed. This moving
average does not depend for its success on any regularity in borehole data, and is
accordingly well suited to valuation of the more stable minerals.
MOVING
AVERAGE
TECHNIQUE
DEVELOPED
Suppose the observations consist of a set of n measurements ri of the value
parameter at co-ordinates Xi, Yi with i = I to n. Suppose that we wish to derive an
estimate r of the value parameter at some arbitrary point with co-ordinates X, y.
The estimate proposed is:
n
r =
i
ri
~I
(ak
+ dik)
~I
(ak
+ dik)
I
n
i
where di is the distance from Xi, Yi to X, Y and a and k are parameters whose physical
significance is discussed below.
r is a weighted moving average of all the measurements. The functional form of
the weights is particularly simple-the
weights assigned to the measurements
~educing exponentially as the distance from the borehole to the point predicted
Increases.
This predictor r does not suffer from any of the drawbacks to the other techniques
mentioned above (though of course it has some peculiar to itself). The valuation
surface makes allowance for errors of measurement; it is stable at the edges and is
not unduly affected by peripheral observations; the surface is smooth but capable
of representing data of any complexity; the use of the predictor does not require
any systematic placing of observations within the ore field.
This technique has been proposed by Sharapov2 and is in use in the United
States of America in the special case a = O. The more general case a > 0 has not
been found in the literature.
As yet no general mathematical formula has been derived which would enable
predictions of optimum values for a and k to be made, but values have been suggested
256
Journal of the South African Institute
of Mining and Metallurgy
January, 1969
The computer mapping and assessment of borehole and sampling data for stable minerals, particularly as applied
in mining-R. B. MacGil/ivray, D. M. Hawkins and M. Berjac
by trial and error. To gain some insight into the physical significance of a and k, let
us consider the effect of varying their values.
If k = 0, then all points carry the same weight. This yields a perfectly flat
valuation surface, the prediction at any point being simply the average of the n
measurements. As we increase k, so we increase the rate at which the weighting factor
falls off with increasing distance. Thus, if the ore is of an unstable type then the
measurements at distant boreholes will provide little information on the value at
the point being predicted. In this case, we will use a large value of k to give low
weight to such distant boreholes. If, on the other hand, the ore is a stable one, then
we will not want the weights to fall off as sharply with increasing distance, and a
smaller value of k will be used. Thus k is determined by the large scale stability of
the ore.
The value of a on the other hand is determined by the magnitude of errors of
measurement and oflocal variability of the ore. Imagine two boreholes close together.
If a = 0 then a prediction close to one of them will mirror the actual borehole measurement. If a > 0 then this will not occur-any prediction, even at the location of a
borehole will not ignore the evidence of other nearby boreholes but will take these
borehole measurements into account. Thus a has the effect of a smoothing factor.
By making a > 0, we allow for the fact that our borehole measurements contain
errors by always bringing into use the information from other nearby boreholes.
Conceptually the use of the factor a is rather similar to moving all boreholes a fixed
distance away from the point being predicted.
If all measurements are highly accurate, and all boreholes are closely representative of the immediately continguous area, then we require very little smoothing
of the data, and should use a value of a close to zero. If there are larger errors or if
there is a large measure of highly localized variability in the ore, then rather more
smoothing is required, and a larger value of a should be used.
So much for the nature of a and k. The choice of actual value for either parameter is a difficult problem. It is hoped that mathematical geological deposition
models will enable optimum values for a and k to be found, but in the meantime these
models are not at hand and the problem must be tackled heuristically. This analysis
has been partially completed for coal data. On the computer, part of the problem
is easily solved by practical
considerations.
Only k values of 0, 2, 4, 6
...
are
economically feasible-any other values of k would be prodigal of computer time.
Initial work suggested the use of k = 2, but recently there have been indications
that a value k = 4 is, in many respects, better, and experimentation on k is
continuing.
A tentative value for a is 200 ft. This choice has been made after a study of the
plans produced using different values of a. Fig. 4, shows an iso line plan produced
by the moving average technique using k = 2 and a = 200 ft. The data are those
used to illustrate the polynomial and finite Fourier techniques. If a is decreased,
then a vast amount of spurious fine detail appears-if a is increased then even broad
features of the plan are smoothed out.
Some theoretical analysis and experience in using the technique have led to the
following comments and conclusions:
(a) The technique works best when the boreholes are drilled on a regular gridthis is to be expected of almost any interpolation scheme.
-----
1969
January,
Journal of the South African Institute
of Mining and Metallurgy
257
The computer mapping and assessment of borehole and sampling data for stable minerals, particularly
in mining-R.
B. MacGillivray, D. M. Hawkins and M. Berjac
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n..-'
,
258
Journal of the South African Institute
of Mining and Metallurgy
January, 1969
The computer mapping and assessment of borehole and sampling data for stable minerals, particularly as applied
in mining-R. B. MacGil/ivray, D. M. Hawkins and M. Berjac
(b) In a 'data void', i.e. an area which has no boreholes near it, the prediction
settles down to the average of the value parameter over all boreholes.
This is a reasonable prediction in the absence of any information.
(c) Discontinuities such as dykes or faults, which can ruin a polynomial or
Fourier fit, do not have an undue effect on the moving average. Near such
a disturbance, some symptomatic crowding of iso lines may occur, but away
from the discontinuity the predictor is unaffected.
(d) The valuation surfaces produced may be (and have been) used to pinpoint
areas of interest at which further observations on the ore body should be
made, and also to assist in the selection of shaft sites and mining layout.
IMPLEMENTA nON
As mentioned earlier, this technique was developed for use in a suite of computer
programmes for the assessment of ore fields. It has been embodied in a FORTRAN
IV programme written for use on the University of the Witwatersrand IBM System
360 Model 50 computer. The programme is a flexible one, and can handle up to 10
distinct value parameters on anyone run. It provides the option of using the moving
average technique or a polynomial or finite Fourier series of any order up to 6. (This
upper limit can readily be extended.) The programme performs many functionssome of them optional. Fairly comprehensive checks are made on the input data
and data errors found are listed out on the printer. Fig. 5 shows an actual error
listing. Each type of error is assigned a severity-for example, a borehole whose
co-ordinates lie outside the prespecified range raises an error severity of 2. The user
specifies the error severity that he is prepared to tolerate, and if an error of higher
severity is encountered, the programme completes the error cheking, then terminates.
VAllATI(~ SEAM A RAWev le.7B.'
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Fig. 5-Error
listing
January, 1969
Journal of the South African Institute
of Mining and Metallurgy
259
The computer mapping and assessment of borehole and sampling data for stable minerals, particularlY as applied
in mining-R. B. MacGi/liyray, D. M. lIawkins and M. Berjac
Certain basic statistics of the value parameters are computed-maxima,
minima,
means, standard deviations and intercorrelations and these are listed out as in Fig. 6.
Other information available on request includes an analysis of the degree of
association between the actual borehole data and the valuation surface. Fig. 7 shows
part of a listing of actual observations, the values predicted by the fitted surface,
and the errors of prediction. Fig. 8 shows a prediction error histogram. This is
invaluable to the statistician studying the properties of the valuation surface. Also
available is some other information of a more specialist nature for use in developing
mathematical geological models.
The runs are relatively inexpensive. On the University installation, there is an
overhead (almost constant) of about R5 for reading in and checking the data. Thereafter the cost of the plans is:
(a) 6th Order polynomial: R3 per square foot of plan.
(b) 6th Order Fourier series: R4.50 per square foot of plan.
(c) Moving average: R4 per square foot per 100 boreholes.
These figures are approximate, and are in any event marginally dependent on a
number of other factors.
[HE
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--"
Journal of the South African Institute of Mining and Metallurgy
260
January, 1969
The computer mapping and assessment of borehole and sampling data jar stable mmerals, particularly as applied
in mining-R. B. l\lacGil/ivray,D. M, Hawk'nsand M. Berjac
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January, 1969
The computer
Journal of the South African Institute of Mining and Metallurgy
mapping and assessment
in mining-R.
of borehole and sampling data for stable minerals, particularly
B. MacGillivray,
D. M. Hawkins and M. Berjac
261
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To date the suite contains two other programmes. The first, also written in
FORTRAN IV is used for automatic selection of a mining width in cases where
several strata are found in the ore body. All possible widths are studied, and the most
satisfactory according to a given criterion is selected. The other programme, written
in PLjl is specific to coal fields. It is used to investigate the association between
-------
262
Journal of the South African Institute
of Mining and Metallurgy
January, 1969
Thecomputermappingand assessmentof boreholeand samplingdatafor stable minerals,particularly as applied
in mining-R. B. MacGillivray. D. M. Hawkins and M. Berjac
common value parameters of coal. A second PL/I programme under development
will be used for comprehensive data checking and for the prediction of missing
information according to regression techniques. Another planned extension is the
shifting of the iso line planning from the line printer to a digital incremental plotter.
Other facilities and programmes may be added as the need for them arises.
APPLICATION
OF THE PROGRAMME
TO THE COAL
MINING
SITUATION
There could be four main reasons for opening a coal mine in South Africa today.
These are-to supply a power station, to supply an oil from coal plant, to supply the
commercial market and to supply special metallurgical coals. All four of these markets
may have different quality requirements. Any previous prospecting in a particular
coal field may not have been undertaken specifically for anyone of these quality
requirements. Under the circumstances, certain value parameters may be missing
from the previous borehole information. If the opportunity should arise for opening
of a captive colliery to supply a power station for example, the mining engineer may
be more interested in the raw coal than washed coal values. Some of these may be
missing from these previous borehole analyses, especially if certain areas of the coal
seam in question contain high quality coal. Using regression techniques and the
information that is given for the previous boreholes, many of these values can be
reasonably well estimated. Some of the values which show high correlation and
therefore lend themselves to prediction are calorific value from ash percentage, washed
ash percentage from raw ash percentage and percentage yield from raw ash percentage.
Table I shows a typical borehole analysis of a coal seam. It will be noticed that while
all the washed coal values are recorded only the raw ash percentage is given. In some
cases, the raw coal values may have been entirely left out or if the coal is of poor
quality the washed coal values may have been excluded.
To produce the required plans and information the borehole data shown in
Table I are punched on to cards. These are fed into the first PL/l programme which
produces the regression curves mentioned above. From these regression curves,
missing information is then predicted. This is currently done by hand, but when the
second PL/l programme is complete, it will take over this function. The borehole
cards are then fed into the seam selection programme.
If so requested, this programme will attempt to select an optimum mining
width; otherwise it will regard the entire seam fed in as the mining width. If manual
selection is desired, this may be achieved by removing the data cards corresponding
to bands in the seam which it is feasible to exclude from a mining point of view.
Exclusion of these bands could have the object of improving quality or mining
conditions.
For the selected mining width the programme calculates the average of each
value parameter at the boreholes. A summary is printed showing these borehole
values, and is transferred to punched cards. Finally, these summary cards are fed
into the surface fitting programme for the calculation of the required iso value lines.
The print outs produced by the computer, can be used as they are. However,
it has been found that transferring the computer information to plans showing known
important geological or surface features is more satisfactory. Fig. 9 shows an iso
value plan prepared in this fashion. Any other desired set of iso line plans may of
course be plotted and so far the technique has been used to plot 'iso-ash', 'iso calorific
value', 'iso volatile' and 'iso width' lines as well as floor contours, middling contours
and depth below surface contour lines.
January, 1969
Journal of the South African Institute
of Mining and Metallurgy
263
The computer mapping and assessment of borehole and sampling data for stable minerals, particularly as applied
in mining-R. B. MacGillivray, D. M. Hawkins and M. Berjac
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264
Journal of the South African Institute of Mining and Metallurgy
January, 1969
The computer mapping and assessmentof borehole and samplingdatafor stable minerals,particularly as applied
in mining-R. B. MacGi/livray, D. M. Hawkins and M. Berjac
AVERAGE
MOIIING
TECHNIOLE
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PRINT
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OUT
January, 1969
Journal of the South African Institute of Mining and Metallurgy
265
The computermappingand assessmentof boreholeand samplingdatafor stable minerals,particularly as applied
in mining-R. B. MacGlllivray, D. M. Hawkins and M. Berjac
CONCLUSIONS
A technique has been developed for fitting a reasonable valuation surface to
irregular borehole or development data, and has been embodied successfully in a
computer programme. This programme has been used effectively in the valuation
of coal fields, and it has been found that the technique, apart from fulfilling the
function for which it was designed, has cast some light on the problem of siting shafts
and further boreholes. Both the technique and the implementation are under further
development.
The authors wish to thank the head of the Coal Division of Rand Mines Limited
for permission to publish this paper.
REFERENCES
1. KRIGE, D. G. 'Two-dimensional
weighted moving average trend surfaces for ore valuation.'
Proc. Symposium on Mathematical Statistics and Computer Application in Ore Valuation, S.A.
Min. Metal/., 1966, p.p. 13-79.
2. SHARAPOV,I. 'Contribution to the theory of the consistency of ore deposits.' Transactions of the
Scientific Coal Institute Perm. Fasc. 4., 1962.
NOTICE~
I
/
The Institute has received requests for copies of 'Uranium in South Africa',
Vols. 1 and 2, but is unable to supply, as all copies have been sold out. If any members
have copies they wish to dispose of it would be appreciated if they would contact
the Secretary, p.a. Box 1183, Johannesburg. Telephone 834-1271.
-
---------