| A signature of non-linear processes in the soil is the non-normal distribution of soil
properties. A common non-normal distribution is the log-normal, in which the variable Z can
be transformed to a variable with a normal distribution by Y = log e(Z). Log-normal
variables are common in soil geochemistry and hydrology. It is standard practice in
geostatistics to use the log-transformation for such variables before spatial modelling and
prediction, and there are procedures to back-transform predictions of Y to the original scale
of measurement Z. This is important because values on the original scale are commonly
required either for scientific purposes or for practical applications such as the assessment of
potential contaminant concentrations in soil.
One of the strengths of geostatistics is that geostatistical prediction returns a prediction
error variance. Furthermore, this variance can be computed before a survey is undertaken, for
a range of possible different sampling networks, since it depends only on the disposition of
sample sites, and the variogram model of spatial dependence. This allows the most efficient
network to be selected: one which will provide estimates of sufficient precision (where the
prediction error variances are within acceptable bounds) without over-sampling. In
log-normal kriging the prediction error variance depends not only on the variogram and the
sampling array, but also on the conditional mean value of the variable, which is not known
until after sampling. This means that the usual pre-survey quality measures which
can be computed to guide the planning of geostatistical surveys are not available
for log-normal variables. Given that many critical variables, such as contaminant
concentrations, are often log-normally distributed, this is a serious gap in the capablity of
geostatistics to facilitate rational sampling design for environmental management and
monitoring.
In this paper I propose and demonstrate some quality measures that can be computed,
pre-survey, for different sampling networks for log-normal variables. I propose measures
that are suitable both for point kriging, when predictions are obtained for the same
sample support as the observations, and for block kriging when the predictions are
upscaled to a region, such as a management unit or a regular block. The quality
measures are dimensionless prediction intervals, scaled relative to the unknown median
value of the target conditional distribution. In the point kriging case the distribution
of interest is the conditional distribution of the variable at the target location. In
the block kriging case the distribution is that of the spatial median value over the
block.
These quality measures are illustrated in a case study on the spatial variability of some
critical metals and metalloids in the soil of eastern England. Target values of the quality
measures are defined in terms of the requirements that land managers or regulators may
specify for uncertainty in final predictions, and the sampling effort required to meet these is
computed. |