CHAPTER 2: The
Semi-Variogram
LOG-NORMALITY
I should like, now, to turn to another problem which is often discussed in the literature
when samples are expected to follow a log-normal distribution. Whilst the
construction of the experimental semi-variogram and the estimation procedures
produced by geostatistics do not depend on what distribution the samples
follow, there are one or two ‘side-effects’ which become apparent when dealing
with log-normal samples. As every schoolboy knows, the standard deviation of a
log-normal distribution is directly proportional to its mean. Consequently the
sample variance -- and hence the sill of the semi-variogram -- is proportional
to the square of the mean of the samples. If experimental semi-variograms are
constructed on different sets of samples within a deposit, this ‘proportional
effect’ can have a radical effect on the individual experimental
semi-variograms. Examples in the literature usually concern cases where, in
order to construct experimental semi-variograms in different directions, it has
been necessary to use different sets of, e.g. borehole data in each. As an
example of ‘proportional effect’ consider the following situation. In Cornish
tin lodes the assay values are usually assumed to follow a log-normal
distribution. Such veins are developed by means of horizontal drives
approximately 100ft apart. These drives are sampled every 10ft by taking chip
samples from the roof. In the example under consideration, nine levels have
been developed, from 600ft below surface to 1400ft (6-14 levels).
Semi-variograms were calculated for each level separately. For simplicity, Fig
2.17 shows only three of these experimental semi-variograms, for levels 6, 10
and 12. The other six lie scattered between levels 6 and 12. Figure 2.18 shows
a graph of the average assay value along each drive versus the standard
deviation of the samples along that drive.
Fig 2.17. Example of supposed zonal anisotropy
-- cassiterite vein.
Fig 2.18. Illustration of the proportional
effect -- cassiterite.
The
averages vary between 35lb/ton (pounds of SnO2
per ton of ore) and 80lb/ton, and the standard deviations vary between
35 and 110lb/ton. The relationship is virtually perfect between the two, with a
calculated correlation coefficient of over 0.85. Since the sill of the
semi-variogram is roughly equal to the calculated sample variance, it is easy
to see that the experimental semi-variogram for level 6 will be (and is) the
lowest, with a sill of about 1200(lb/ton)˛; level 10 will be in the middle with a
sill of about 5000; and 12 will be the highest with a sill of 12000(lb/ton)˛. The question is, can we make an overall
semi-variogram for this deposit when the individual experimental
semi-variograms vary by an order of magnitude from area to area.
The published authorities state that the valid way to
combine these semi-variograms is to ‘correct’ each one for the proportional
effect. That is, to divide the individual experimental semi-variograms by the
square of the average of the samples which went into its calculation. This
produces a ‘relative semi-variogram’ -- implying that all values given by the
semi-variogram are now ‘relative to the local mean’. Applying this method to
the above example results in nine experimental relative semi-variograms which
vary in sill between about 1.00 and 1.80. Notice that these values now have no
units. To be converted into meaningful figures they must be multiplied by the
square of the local mean. We can now (supposedly) combine these semi-variograms
into one for the deposit as a whole, and fit a model to it. If we do so we must
remember that in all our estimation procedures etc. we have to ‘uncorrect’ the
values calculated from the semi-variogram -- estimation variances, standard
errors and so on.
This process of correcting experimental semi-variograms for
the proportional effect is widely advocated as the ‘right thing to do’. No one
seems to have bothered to test whether it actually works in practice. In the one case, that
described above, where I have been able to investigate in depth and compare
what happens if you use the ‘relative’ semi-variogram, I found that correction
by the local mean gave completely erroneous results. Therefore, I would not recommend this procedure, but rather
that you should combine the original experimental semi-variograms and try to
fit a model to the ‘uncorrected’ data. In the study mentioned above this was
found to give the correct values at all times.