stationarity. In geostatistics there are two forms of stationarity that are

used or referred to: second order and intrinsic. Both pertain to the random

function and NOT to data. Second order stationarity requires that (1) the

mean be constant (with respect to position) and (2) that the covariance

function be a function of the separation vector rather than of the two

positions. Covariance functions must be positive definite and also bounded.

Intrinsic stationarity (sometimes referred to as the "intrinsic hypothesis)

requires that the first order increments have mean zero and that the

variance of the first order increment be a function of the separation

vector (rather than of the two positions). Variograms must be negative

conditionally definite and do not have to be bounded. Every covariance

corresponds to a variogram but not every variogram corresponds to a

covariance (e.g., a power model variogram). Note that second order

stationarity implies that the variance is a constant whereas intrinsic

stationarity does not require or imply that the variance exists and in

particular that it is finite.

The problem is that in essentially all applications encountered in

geostatistics there are no replications, i.e., the data is considered as a

(non-random) sample from one realization of the random function.

Consequently one can not test for stationarity, i.e., statistical test.

Instead one has to rely on various empirical checks to see whether one or

the other of the two forms of stationarity is reasonable.

One of the simplest is to make a bi-plot of the data values vs the position

coordinates (one coordinate at a time). Analytically this means to fit a

trend surface and ask whether the coefficients are non-zero.

Note that even though a variogram does not have to be bounded it can not

grow at exponential rate, in fact it must grow at less than a quadratic

rate. The usual sample/empirical variogram does not estimate the

variogram, i.e. 0.5Var{Z(x+h)-Z(x)}. Rather it estimates 0.5E{

[Z(x+h)-Z(x)]^2}. If E{Z(x+h) - Z(x)} = 0 then the two are the same. If

this latter expected value is linear (or higher order) in one or more of

the position coordinates then the sample variogram will exhibit more than

quadratic growth. This is evidence of non-stationarity.

The optimal way to estimate the non-constant mean is by kriging but of

course this requires knowing the variogram/covariance (which comes first

the chicken or the egg). In practice one must attempt to estimate/model the

non-constant mean (least squares, median polish, etc), then compute

residuals and use the residuals to estimate/model the variogram/covariance.

There are some problems with using least squares, there will be some bias

in the sample variogram.

There is no substitute for exploratory analysis of the data set both before

and after variogram estimation/modeling.

You might want to look at a paper of mine in Math Geology concerning

stationarity, circa 1988-89, a paper by Journel and Rossi in Math Geology

entitled "when do we need a trend model?" and a paper by two others showing

explicitly the relationship between a trend surface and universal kriging

with a pure nugget model.

Finally note that since the kriging equations (simple, ordinary, universal

and point or block) do not directly depend on the data values, only the

data locations. Hence any valid variogram will produce results but

obviously if the variogram is a poor fit or the stationarity conditions are

badly violated then the results will bear little resemblence to the data.

It is also likely that the results will be of any use.

Donald Myers

myers@...

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

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