GEOSTATS: multiscales, anisotropy, non-stationarity and trend
- Re: questions from S. Dufresne
It is probably important to distinguish between a theoretical model and the dataand to ask exactly what is meant by "multiscale" vs "multiple sills"
First of all, when the data is from only one realization it is not possible to test for non-stationarity. However the data can be used to look for evidence of
1. If the data locations are in two space, plot data values against the
horizontal position coordinate (also fit the least squares line), if
the least squares line has non-zero slope this could be interpreted
as evidence of non-stationarity. (Trend in the data), similarly
plot data values against the y coordinate.
2. Does the sample variogram exhibit quadratic or higher order growth?
There are no valid variogram models with these properties and one
explanation of such behavior in the sample variogram is non-stationarity
Non-stationarity (of the model) and trend (a characteristic of the data)
usually have a directional effect (a single point source or sink might
exhibit radial non-stationarity however. Using only data to determine
non-stationarity and anisotropy may make it difficult to distinguish
between a directional effect in the non-stationarity and an anisotropy
in the variogram but these two are definitely not the same and they do
not necessarily go together.
Multiple scale/multiple sills
A nested variogram model with different ranges of dependence (for example as noted in P. Gooavert's comments on simulation) does not shown multiple
"plateaus" in the graph hence one should be a little careful about referring to
multiple sills. For example, Pierre's nested model would not[D[D[D[Dshow plateaus at 100m and at 1000m (he didn't say that it would).
It is illusory to say that the combination of a cardinal sine model and a linear model (power =1) shows many sills, since by the inclusion of the linear
model you are assured that the variogram is unbounded and has NO sill.
A comment about the scale of the problem having an effect is important,
ordinarily one would only use the sample variogram for lag distances up to
half the distance across the region (after that distance the number of pairs
decrease rapidly and also the possible pairings are severely limited geographically). Hence if one or more of the scales of dependence exceed this distance they
may be hard to detect. Also if the order of the non-stationarity is low, the
non-stationarity may not show or have much effect on the sample variogram.
Obviously the physical phenomenon is important in examining these characteristics. Note also that the pattern of sample locations may induce an apparent
non-stationarity or anisotropy.
See "To Be or Not to Be-Stationary", Math. Geology 1988.
Department of Mathematics
University of Arizona
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