1. Adaptive can mean several different things. The kriging

estimator/predictor is already "adaptive" in several senses.

a. It "adapts" to the spatial pattern of the data locations by

incorporating the spatial correlation for pairs of data locations. (The

data locations need not be on a grid).

b. It "adapts" to the spatial relationship of the location to be estimated

to the data locations used in the estimation

c. By using a search neighborhood it makes several adaptations. Note

however that using search neighborhoods may introduce discontinuities in

the interpolated surface.

One of the supposed justifications for using a search neighborhood is

that for most variogram models the data locations outside the search

neighborhood will have small or zero weights even if included (this

obviously depends on the choice of the search neighborhood and the

variogram model and its parameters) hence there is no substantial change

by not including them.

One of the presumed advantages of using a search neighborhood is that the

constant mean assumption can be slightly weakened, i.e., the mean is only

assumed constant (or nearly constant) within each search neighborhood

(separately). See Journel et al (Math Geology) on "Why do we need a trend

model"?

d. By allowing the variogram model to be "adapted" to the data, the

kriging estimator incorporates several aspects of the data including a

directional effect in the spatial correlation.

Most of the above are what might be called pre-adaptations, i.e, they do

not occur during the actual kriging.

2. In order that the kriging variance be non-negative (remember the

kriging variance is the minimized estimation and mimimizing non-negative

valued functions is very different from mimimizing functions that can take

on both postive and negative values) and that the kriging system have a

unique solution it is (essentially) necessary that the variogram be

conditionally negative definite (or that the covariance be positive

definite). If the variogram is allowed to change from one search

neighborhood to another there are two potential problems, (i) do the

multiple forms collectively determine a valid model, (ii) if the

neighborhoods overlap then which one does one use in the intersection (for

consistency they should in fact match on the intersection). One of the

practical reasons that we commonly use only a small number of standard

models for variograms is that it is difficult to check whether a given

function is a valid variogram.

3. It is well-known that the kriging estimator can be re-written in the

so-called "dual" form, which shows the relationship to radial basis

function interpolation. However there is a significant practical

difference. In the usual form of the kriging estimator the coefficients

change with the location to be estimated whereas in the dual form the

coefficients do not depend on the location to be estimated. That is, the

radial basis form produces an interpolating function whereas the kriging

estimator only implicitly determines the interpolating function (by giving

its values one at a time for each location). Hence it is not so simple to

use a search neighborhood since in that case one must actually confront the

piecing together functions defined on separate (but usually overlapping)

regions and hence the discontinuities will become very apparent. (See

Myers, "Interpolation with Positive Definite Functions", Sciences de la

Terre, 1987).

4. If the function to be interpolated is really so variable that one would

have to consider a locally varying variogram then maybe it is too

complicated to be represented by second order properties.

5. Finally one must remember that geostatistics is essentially based on the

idea that the data is a (non-random) sample from one realization of a

random function. The variogram or covariance function is a characteristic

of the random function, not of the data. The sample variogram (or other

variogram estimator) is a statistic based on the one realization, some form

of ergodicity and stationarity is needed to use this spatial statistic to

estimate/model the variogram which is an ensemble statistic. Ideally one

should have data from multiple realizations at each data location, i.e.,

replications. In some instances where one has spatial temporal data, it

might be assumed that there is no temporal dependence and hence data

collected at multiple time points for a given location might be treated as

replications. This would allow local fitting/estimation of variograms and

without invoking (spatial) stationarity.

This doesn't mean that it might not be possible to have a locally varying

variogram but it does mean that one should be careful both in the modeling

step and in using it for kriging because there are hidden implications.

Donald Myers

Department of Mathematics

University of Arizona

myers@...

(520) 621-6859

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

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