“Geostatistical
estimation and the lognormal distribution”, Geocongress,
Pretoria RSA, June 1998
Geostatistical
estimation and the lognormal distribution
Dr. Isobel Clark, Geostokos Limited, Alloa, Scotland
It is common knowledge
that ordinary geostatistical methods do not deal well with highly skewed sample
data. In recent times “distribution free” methods have
been advocated to avoid this problem --- the most popular in practice, at this
time, being the multi-indicator methods. As with all techniques, these have
their strengths and weaknesses. One of the most obvious of the drawbacks to a
multi-indicator approach is the necessity to model many semi-variogram graphs
and to carry out many simultaneous kriging or co-kriging estimations. Where the
distribution of sample values is reasonably simple and stable, it would seem
more practical to use the known features of the distribution and associated
methodology.
In this paper we consider the simplest
non-Normal case --- that of lognormal kriging. If the values within a deposit are known to be stationary and lognormal, then the
logarithms of these values should be
We illustrate this paper with a case study on a
Wits type reef deposit which is a simulation based on a real set of data.
Figure 1 shows the histogram of sample values taken at random within an area of
the reef. It can be seen that the data is very well
behaved when logarithms are taken.
Semi-variogram calculation and modelling was carried out on this data set. Cross validation confirmed
the semi-variogram model. A kriging exercise was also
undertaken to estimate average values within square panels over the
study area. The results of this exercise are:
·
an estimate for the average logarithmic
value within the panel 
*;
·
a
standard error for this estimate sok;
·
the within panel variance for logarithmic variances s@0/A
or 
(A,A).
Figure 2 shows the simplified histograms of
original sample values and kriged panel values. Because this is a simulation,
we are also able to show the corresponding histogram of the “true” panel
values. It should be remembered that, in this context,
original sample value means the logarithmic transform of the values. The kriged
panel value is the optimal weighted average estimator produced by ordinary
kriging on the logarithms. The “true”
panel value is the average of the logarithms of all known values (from
simulation) within the panels. It can easily be seen that the kriging estimator
and the panel values are pretty close — even though the “true” values are
produced from about 25 times as much information. All three distributions show
the same mean, whilst both “true” panels and kriged estimators show the reduced
standard deviation (spread) expected when considering panels rather than ‘point’
data. The difference in standard deviations between kriged estimate and true
value is, theoretically, calculated from the classic
volume/variance effect. The difference in variances must be included in the backtransformation to ‘raw’ sample values. Figure 3 shows
the effect of simply anti-logging the kriging estimate of the logarithmic mean
--- in effect, assuming that the variances of estimator and actual value are
equal. The variance of the “true” panel values is given
by:
where
Ctot
represents the total sill or ‘point’ variance, s@. The variance
of the estimators is:
that
is, the total variance less the variance of values amongst the samples in the
weighted average estimator. In practice, this can be
calculated by using the equivalent expression:
where
 is the langrangian multiplier produced by the ordinary kriging
process. If simple kriging is used, the longer form is more appropriate
(obviously). However, adding one-half of this variance difference to the
estimator before anti-logging does not result in sensible
results.
The backtransformed estimates are consistently lower than the “true” panel
values. The larger the panel, the larger the difference
between backtransform and true average (see Figure 4). The reason for this becomes clear if we
simply plot histograms of the average logarithm in the panel and the logarithm
of the “true” value, as in Figure 5. It can be seen from this that, when we
consider panel averages, it is not only the variance which
changes. The logarithmic mean also changes --- in order to preserve the overall
average in the ‘raw’ values. To maintain unbiassedness
in our backtransformed estimates, we must incorporate the change in the
logarithmic average as well as the change in the variances. This may be expressed as follows:
that
is, the correct backtransformed estimator for a panel average is the anti-log
(exponentiation) of:
Figure 6 shows the comparison between kriged
estimates backtransformed using this expression and “true” panel values.
Figure 1: histogram of original sample
values (logarithmic scale) and best fit Normal
distribution
Figure 2: comparison of sample, true panel
and kriged Values
Figure 3: comparison of sample, true
panel and anti-logged kriging estimates
Figure 4: average of true panel values
(logged) versus average of logarithms in panel
Figure 5: histogram of true panel
averages versus histogram of average logarithm in panel
Figure 6: true panel values and backtransformed kriged panel values