CHAPTER 6: Practice
Because this text was intended to be a ‘first
introduction’ to the subject of Geostatistics, the examples and
situations discussed have tended to be rather simplistic. There are many ore
deposits --- and other applications --- which can be tackled with the methods
described. However, there are at least as many others which cannot because of
the presence of one or more complicating factors. In this chapter, I should
like to mention briefly some of these problems, and perhaps indicate how they
might be tackled. The order in which the problems are presented bears no
relationship to their relative importance.
1.
CONSTRUCTION OF SEMI-VARIOGRAMS USING IRREGULAR DATA
All the discussion on the construction of experimental
semi-variograms in Chapter 2 was based on samples which were spaced more or
less regularly within the deposit. Some grids had missing samples, but this
presents no problems. In a situation where the samples are not regularly
spaced, approximations must be introduced into the calculation. Suppose we wish
to calculate the experimental semi-variogram value for a distance h in a specified
direction (say, north-east). The chance of finding any pairs at exactly this
separation with irregular sampling is quite small. We therefore place a
‘tolerance’ on each specification. We look for samples more-or-less
distance h apart
(within dh) and more-or-less north-east
(within ±dq) ---
see Fig. 6.1 for illustration. The size of the tolerances depends greatly on
the structure of the deposit. This is rather a circular argument, since we do
not know the structure until we construct the semi-variogram. If the deposit is
anisotropic, the semi-variogram will be more sensitive to the tolerance placed
on the search angle. A good practice is to try several dq
values, and a narrow range of dh values. dh should always be small
relative to the sample spacing. As a rule of thumb, you could try dq=5, 0,
0, 5 degrees and dh equal to 10% of the average sample
spacing.
Fig. 6.1. Search area defined by tolerances on
angle and distance between pair in experimental semi-variogram.
2.
SAMPLING ERRORS
This is a field which is skated over in most geostatistical
treatises, and I intend to emulate my predecessors in this. Random errors introduced during
sampling will contribute to the nugget effect in the semi-variogram, i.e. they
will show as an increase in the ‘unpredictable’ component of the
value. Similarly, core loss will also contribute to the nugget effect. Other
possible contributions of ‘error’ --- apart from the mineralisation itself --- are analytical errors in valuation, subsampling and so on. However, the contribution of these errors
should not be over-emphasised. In my Cornish tin example, we carried out a
special sampling scheme to determine the ‘size’ of the operator
error in the vanning assay. This turned out to
be 3% of the total nugget effect. The other 97% was due to the random nature of
the mineralisation.
Systematic errors, be they
sampling, analytical or whatever, will not be picked up by
Geostatistics and will be transferred to any estimates produced.
3.
TRENDS
We have seen in Chapter 2 how to detect the presence of a
significant trend within the deposit. If we construct the experimental
semi-variogram assuming no trend, then the neglected component will show in the
graph. If it is a periodic trend, it will show as a regular rise and fall in
the semi-variogram. If it is a polynomial type of trend, it will show in the
addition of a parabolic component to the ‘true’ semi-variogram.
There are cases where a trend may exist but can be safely ignored, as in the
silver example in Chapter 2. However, there are others in which this is not the
case, e.g. the rainfall example. Kriging, as such, cannot be used in the
presence of a strong trend. It will give erroneous and biased results. Some
technique such as Universal Kriging, or the newer
Generalised Covariances must be applied if the
user is determined to use Geostatistics. My experience in the application of
Geostatistics in the presence of trend is voluntarily non-existent.
4. ANISOTROPY
This is perhaps the easiest ‘problem’ to tackle.
Most frequently, the form of the anisotropy is different ranges of influence in
different directions. For example, a porphyry
molybdenum may have a range of influence of 70m vertically through the deposit,
and one of 350m in all horizontal directions. This is very simply tackled by
changing the units of measurement in one direction so that the ranges appear to be equal. In the
example cited, we might change all horizontal measurements to multiples of 5m
instead of 1. This gives a horizontal range of 70 units. Then, when estimating
block values, it must be remembered that all horizontal distances must be
expressed in units of 5m. Diagonal distances must be corrected accordingly. For
example, a block defined as 50 by 50 by 20m, will be estimated as if it were 10
units by 10 units by 20 units. Similarly, the distance between samples must be
corrected to this new ‘co-ordinate’ system. The simplest way to tackle
this (inside a computer) is to correct all measurements before embarking on the
estimation. The final estimates and standard errors will be as they should be,
and need no ‘uncorrection’.
5.
IRREGULAR SHAPED STOPES OR BLOCKS
In the estimation problems discussed, all panels and blocks
were rectangular in shape. The auxiliary functions are only relevant to such
panels, and so cannot be used with, say, stope
panels such as that shown in Fig. 6.2.
Fig. 6.2. Estimation of irregularly shaped stope.
These
shapes can only be tackled using numerical approximations and a computer. The
principle of the approximation is the same as that applied in calculating the
three-dimensional F
function in Chapter 3. Instead of considering all of the infinite number of
points within the stope, we use a
finite grid of points to represent it. The number of points is in question, but
general agreement seems to lie in the range 64 to 100. This then means that
values such as 
(S,A) are
average semi-variograms between ‘the sample and each point on the grid in
the stope’. A computer program will evaluate
the model semi-variogram value between each pair of points and then produce the
average semi-variogram value required. This principle also applies to irregular
shapes in three dimensions.
6.
THREE-DIMENSIONAL KRIGING
This leads us on nicely to one of my hobby horses --- the
performance of kriging estimation in three dimensions. This problem is most
often encountered in the planning of open pits from borehole results. The
‘standard’ technique is to make an approximation such as described
in Section 5 above. A suggested alternative, which uses less computer time, is
as follows:
i.
slice the deposit into benches;
ii.
represent each block as a panel at
a level midway up the bench;
iii.
approximate this block by a grid of
points in two dimensions;
iv.
take each borehole intersection
with the bench and call this a ‘point’ midway up the bench;
v.
put all the slices back and krige.
The semi-variogram supposedly used in the kriging is that of
‘bench composites’, i.e. sections of length equal to the height of
the bench. This technique is adequate if all boreholes are complete, and if
they start and stop at more or less the same level. However, there are other
situations which it does not represent adequately. Figure 6.3 illustrates some
of these. All of these boreholes would be averaged over the bench, positioned
halfway up the bench and allocated a length equal to the height of the bench.
Fig. 6.3. Some problems with the
simplistic approach to three-dimensional kriging procedures
Borehole A has core loss part way down the bench; borehole B is inclined
so that the composite may be substantially longer than supposed; borehole C
stops before it reaches the bottom of the bench and borehole D does not start
until halfway through. All of these situations can be tackled by taking a truly
three-dimensional approach to the problem. Computer packages are now available
on the market for the above described method, the point approximation method,
and the three-dimensional approach advocated in my own papers.
7. BIAS
ON
After a mine has been estimated on
a block-by-block basis, it is usual to construct the so-called
‘production’ grade/tonnage curve. Unfortunately, even with the best
estimates possible (kriging) this grade/tonnage curve will still be biased.
There are two contributory factors to this bias. The first factor is that the
selection criterion (cutoff value) is being applied to the estimate of the
block grade. No matter how accurate that estimate, it will not exactly equal
the true value of the block. Thus, if a block is estimated to be just below the
cutoff value, there is a finite probability that the true value is above
cutoff. However, this block will be sent as waste. On the other hand there will
be blocks estimated as being above cutoff which are actually below cutoff.
These will be sent as ore. Thus, we will have payable blocks sent to waste and unpayable ore sent to the mill.
This will result in production figures which will differ from those predicted
by the supposed grade/tonnage curve calculated on the block estimates. The
major difference will be a lowering in the grade of ore milled.
The second bias in the grade/tonnage curve is one introduced by the
volume-variance relationship. The estimates of the block values will not
necessarily have the same variance as the actual block values. You may remember
we discussed this matter in connection with the Cornish tin example in Chapter
3. There the estimator was the average grade of two 125-ft strips of ground,
whilst the panel was 125 by 100ft. The variance of these two quantities will
not be the same. In most situations --- except for point kriging --- the variance
of the estimator will be larger than that of the actual blocks. Thus the
grade/tonnage curve based on the block estimates will be biased towards lower
tonnage and an over-optimistic average grade. This problem is currently being
investigated by the staff at Fountainebleau under
the title ‘Disjunctive Kriging’. A simple, empirically justified
technique to correct this bias is currently under investigation at the Royal
School of Mines.
SUMMARY
This summary is more for the book as a whole than for this
chapter particularly. I have endeavoured to give a perhaps over-simplistic
presentation of the theory and practice of the estimation technique known as
kriging. Readers who find this approach tedious are referred to the definitive
(and more mathematical) works mentioned in the bibliography. I have also
endeavoured to detail the practical difficulties which arise in applying the
technique and to suggest some ways of overcoming them.