| Geostatistical models of soil variation have been used to considerable effect to facilitate
efficient and powerful prediction of soil properties at unsampled sites or over partially
sampled regions. Geostatistical models can also be used to investigate the scaling behaviour
of soil process models, to design sampling strategies and to account for spatial dependence in
the random effects of linear mixed models for spatial variables. However, most
geostatistical models (variograms) are selected for reasons of mathematical convenience (in
particular, to ensure positive definiteness of the corresponding variables). They
assume some underlying spatial mathematical operator which may give a good
description of observed variation of the soil, but which may not relate in any clear
way to the processes that we know give rise to that observed variation in the real
world.
In this paper I shall argue that soil scientists should pay closer attention to the underlying
operators in geostatistical models, with a view to identifying, where ever possible, operators
that reflect our knowledge of processes in the soil. I shall illustrate how this can be done in
the case of two problems.
The first exemplar problem is the definition of operators to represent statistically
processes in which the soil landscape is divided into discrete domains. This may occur at
disparate scales from the landscape (outcrops, catchments, fields with different landuse)
to the soil core (aggregates, rhizospheres). The operators that underly standard
geostatistical models of soil variation typically describe continuous variation, and
so do not offer any way to incorporate information on processes which occur in
discrete domains. I shall present the Poisson Voronoi Tessellation as an alternative
spatial operator, examine its corresponding variogram, and apply these to some real
data.
The second exemplar problem arises from different operators that are equifinal with
respect to the variograms of the corresponding spatial variables. Conventional geostatistical
models are based on two-point statistics, and so only capture those features of spatial
variation in n observations that are represented by the 1
2(n2 - n) pair-wise covariances. It is
known that two-point geostatistics will not identify all the structure in a random spatial field.
For example, connnectivity of regions with distinctive values of a variable, potentially very
influential on scaling behaviour, are not described. I shall show how multipoint geostatistics
can distinguish realizations of spatial processes with identical variogram functions, and
explore the implications for the statistical description of scaling behaviour of process models. |