CHAPTER 2: The Semi-Variogram
COMPLEX MODELS
Now let us try some real semi-variograms, rather than these hand-picked simple
ones. Figure 2.13 shows the experimental semi-variograms for three metals in
another complex base-metal sulphide. The metal of economic importance is the
copper, but the other two metals are of sufficient value to warrant investigation.
The semi-variograms are ‘down-the-hole’ in direction, and contain information
from about 50 boreholes perpendicular to the plane of the ore-body.
Fig 2.13. Experimental
semi-variograms for a complex base-metal sulphide deposit.
My interpretation of the
lead and zinc semi-variograms is pure nugget effect. That is, the ‘model’ is a
horizontal line at a value equal to the sample variance. There appears to be
very little relationship even between neighbouring cores! On the other hand, the
copper semi-variogram appears to be a combination of a nugget effect (constant)
and a parabola. As in the previous example, the parabola implies a polynomial
trend, in this case acting on pairs of samples even at 1m spacing. The nugget
effect implies completely random behaviour. So we have a trend with random
variation; an ideal case for Trend Surface Analysis.
The next example concerns a nickel ore-body disseminated in peridotite, which
has been ‘proved’ by means of about 45 vertical boreholes. The average spacing
between the boreholes was about 60m and they were not regularly spaced, so that
only the ‘down-the-hole’ experimental semi-variograms were calculated.
Altogether
approximately 4000m of core was recovered and assayed in 2m core sections. In this
case the logarithm of the grades was used, rather than the grades; the reason
for this has no relevance here. The experimental semi-variogram is shown in Fig
2.14, and the numerical values are given in Table 2.8.
Fig 2.14. Experimental semi-variogram for a
nickel deposit -- logarithms of grade values.
Table 2.8 Experimental semi-variogram from a disseminated nickel deposit
(logarithm of grade)
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There appears
to be a definite flat sill at about 2.55(log %)˛. However, drawing a straight line through
the first two points, as we did in the silver semi-variogram produces two odd
results. First, the line intersects the semi-variogram axis at 0.40(log%)˛ not at zero. This suggests that there
is a component of each value which is ‘random’ or unpredictable. Samples very
close together still have a reasonably large difference in value. Remembering
that the sill (if it exists) is equal to the sample variance, we can see that
0.40/2.55=0.156 suggests that about 16%
of the variation in the sample values is random and unpredictable. Thus, no
matter how closely we sample, this unpredictability will still exist. The
semi-variogram model will need to be of the form:
where g’(h) is the usual sort of model (e.g. linear).
In effect, the nugget effect is a simple constant raising the whole theoretical
semi-variogram 0.4 units. Thus we now seek a model with a sill of 2.15(log %)˛. Now, we saw in the silver example that
extending the initial straight line slope up to the sill gave a value of
two-thirds of the range of influence, when using a spherical model. In this
case the intersection produces a value of 13m implying a range of influence of
about 20m. On the other hand, the curve does not even approach the sill until
some distance past 45m. Clearly neither of our ideal models will cope with this
sort of situation. Let us look again at the experimental curve. There seems to
be an ‘intermediate’ sill, reached at about 14m and a value on the g-axis of 1.95-0.40=1.55 (to allow
for nugget effect). We seem to have a mixture of two spherical type models, one
with a shortish range and one with a range of about 50m. Let us try out this
tentative model and see how it fits the experimental semi-variogram. We have a
fairly complex model:
Putting these values
into the proposed model produces the following:
For distances (h) between 14 and 50m, the model is
given by:
and when the distance between the two samples is greater
than 50m, the model semi-variogram takes the form:
In order
to compare the theoretical model with the experimental points we must evaluate
the model at various distances, and draw the resulting curve onto the same
graph. For example, for distance h equal to 2m:
and for a distance h equal to 40m:
A set of values was selected for h and the theoretical curve constructed. The values are
shown in Table 2.9, and the resulting model has been plotted in Fig 2.15. The
experimental points are also shown for comparison.
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Fig 2.15. First
attempt to fit a mixture of spherical models to the nickel semi-variogram. |
Table 2.9 First attempt
to fit a mixture of Spherical models to the experimental nickel
semi-variogram (parameters in text)
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The ‘model’ curve fits fairly well to the beginning and end
of the experimental semi-variogram, but does not seem too good in the middle.
The kink in the curve is at far too high a level -- it needs to occur at g=1.95.
We
assumed that this level was equal to C0+C1. What was forgotten is that, even
at short distances, the second spherical component still contributes some value to the model, so that the
value 1.95 should actually be equal to:
In other words, we need to lower the value of C1 and raise the value of C2, and then try the fit again. After a few
tries, I got the following model:
This model is shown in Fig 2.16 alongside the experimental
semi-variogram, and seems to be a relatively good fit. Perhaps the reader would
like to try to improve upon it? Table 2.10 gives the corresponding numerical
values for the model curve.
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Fig 2.16. Final attempt to fit a mixture of spherical
models to the nickel semi-variogram. |
Table 2.10 Final attempt to fit a mixture of Spherical models to the
experimental Nickel semi-variogram (parameters in text)
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Examples of semi-variogram models which are mixtures of
spherical components abound in the geostatistical literature, and seem to be
about the most common type encountered, especially in low concentration
minerals such as cassiterite, copper veins, uranium and so on.