## GEOSTATS: stationarity

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• Stationarity is a property of the random function, not the data. If one computes a mean on a set of (spatial) data then you are computing a spatial average not
Message 1 of 1 , Apr 22, 1997
Stationarity is a property of the random function, not the
data. If one computes a mean on a set of (spatial) data then
you are computing a spatial average not an ensemble average.
Stationarity pertains to the invariance of the ensemble
mean.

Unfortunately there is no way to compute the ensemble mean
because there are no replications.

See my article in Math Geology, circa 1988.

However there are characteristics that would suggest
non-stationarity such as the quadratic or higher growth
rate of the sample variogram. Plotting data values
against a position coordinate may identify a possible
non-stationarity.

Note that there are several forms of stationarity of
a random function, strong (which depends on the translation
invariance of joint distributions and this form is not
usually used in geostatistics), second order which depends
not only on the mean being constant but also the covariance
being a function of the separation vector, finally there
is intrinsic which relates to the variogram (this is weaker
than second order). If one is working with IRF-k's then
there are even weaker forms of stationarity.

Stationarity is not testable, i.e., a statistical test, because
the test would require stationarity and would require having
data for more than one realization of the random function.

With respect to the idea of using different variogram models
for different directions, that would in general violate the
requirement of having a valid model. The conditional
negative definiteness condition is necessary (and sufficient)
to ensure that the estimation variance is non-negative
and that the kriging system has a unique solution. That is
not to say that there might not be some combinations of
variograms in this manner that are valid but it is very
hard to prove. See Myers and Journel, Math. Geology
circa 1990 for examples of how this fails in even a very
simple case.

Donald E. Myers
Department of Mathematics
University of Arizona
Tucson, AZ
myers@...
http://www.u.arizona.edu/~donaldm
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