Non-Gaussian and non-symmetric spatial dependence

Many
natural processes lead to spatially distributed variables which exhibit
considerable spatial variability and heterogeneous structures on different
spatial scales. The spatial organization of high and low values is often
different due to the specific character of the generating process. The
geostatistical tools used for their description, interpolation or simulation
are based on second order statistics, variograms and covariance functions.
These were specifically developed for Gaussian processes. However natural
structures often exhibit non-Gaussian features both in their distributions and
in their dependence. One of these is the different spatial dependence of high
and low values. This kind of asymmetrical dependence is a clear sign of a non-Gaussian
organization of the structure. Asymmetrical dependence is a feature which is
independent of the distribution and thus can be related to the dependence
described using copulas. Copulas offer a
comprehensive description of spatial variability offering a framework to
describe and to model asymmetrical behaviour. A third order statistical
function can describe asymmetrical behaviour. Theoretical models allow the
description of fields with asymmetrical dependence. Examples of processes leading
to asymmetrical spatial fields are discussed. Further examples from observed spatial
fields including topographical surfaces, rainfall and groundwater quality
fields illustrate the methodology. Consequences of the models with respect to
spatial scaling are also discussed.