GEOSTATS: adaptive interpolation
- A few thoughts
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
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
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.
Department of Mathematics
University of Arizona
*To post a message to the list, send it to ai-geostats@....
*As a general service to list users, please remember to post a summary
of any useful responses to your questions.
*To unsubscribe, send email to majordomo@... with no subject and
"unsubscribe ai-geostats" in the message body.
DO NOT SEND Subscribe/Unsubscribe requests to the list!