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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