the "equality" of two variograms. If one uses an estimator such as the

sample variogram, one only obtains estimates of the values of the variogram

for a finite number of lags (note that dealing with a possible anisotropy

makes it even more complicated). Moreover the reliability of these

estimates varies, in part because the numbers of pairs will vary. If one is

using the variogram for kriging or simulation then one is most interested

in the behavior of the variogram, i.e., the values for short lags and

unfortunately the short lags usually have the smallest numbers of pairs. If

one uses least squares or maximum likelihood then one must first choose a

model (or models in the case of a nested model) and then one of these is

used to estimate the parameters.

There is an old paper by Davis and Borgman in Mathematical Geology (circa

1980) on the distribution of the sample variogram, they give two results:

(1) beginning with an assumption of multivariate normality (which is not

testable) and an assumed model type then they obtain numerical results for

the distribution , (2) they obtain asymptotic results which are

theoretically interesting but probably not much help in practice.

There is also a paper in Mathematical Geology, circa 1990, on the "true"

numbers of pairs. The problem as is well known is that there is an

interdependence between the pairs used to estimate for one lag and those

used to estimate for another. The author has to assume multivariate

normality to derive the results.

It is known that the kriging estimator is relatively robust with respect to

the variogram, i.e., slight changes in the variogram will result in only

slight changes in the kriging weight vector and hence in general only

slight changes in the kriged values. There are at least two different ways

to quantify the "distance" between two variograms, these correspond to a

notion of continuity. A third one corresponds to differentiability, none of

the three implies the others.

In practice one often uses a search neighborhood in kriging hence it is

only of interest whether the variograms match or are at least close for up

to some maximum lag. One will have very little information about the

variogram for longer lags anyway.

In general statistical tests will require some distributional assumptions

and these are hard to obtain for variograms/variogram estimators. It is an

interesting question to ask, i.e., are the variograms for two different

variables or the same variable for two different regions the same but one

that will be hard to test without making very strong assumptions

(non-testable assumptions).

Finally one might want to consider the question of sample location pattern

design relative to testing the equality of two variograms. I have an old

paper with A.W. Warrick on the design of sampling plans in order to control

the numbers of pairs for each lag. If one assumes isotropy (it is even more

complicated in the case of anisotropy) then the pattern that generates an

equal number of pairs is a spiral, not a very practical result.

Note also that if one assumes normality then the distribution of the

half-squared differences will be Chi-Squared (one can see this effect in

most sample variograms, the VARIO component of GEOEAS will provide

histograms for these distributions). Not a particularly nice distribution

for testing because of the "fat" tails.

Donald E. Myers

Department of Mathematics

University of Arizona

Tucson, AZ 85721

http://www.u.arizona.edu/~donaldm

At 05:02 PM 2/14/00 -0800, you wrote:>Assume that we have two sets of geostatistical data. Is there any

--

>statistical test to determine whether variograms on those two sets are the

>same?

>

>Thanks,

>

>A. Lazarevic

>

>

>

>

>--

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