GEOSTATS: gaussian and spherical variograms
- A further comment appropro Gregoire's comment of 22 Sept in
repsonse to Oliveira's comment of 19 September.
It is well-known (although perhaps not written down anywhere)
that it is best to use at least a small nugget with a Gaussian
model otherwise the kriging coefficient matrix is likely to
be ill-conditioned, making the results very unstable (numerically).
As for MLE estimation of variogram parameters this always
requires distributional assumptions that may not be valid or
at least very reasonable.
Any valid model (conditionally negative definite for a variogram)
will result in a unique solution for the kriging equations, i.e.
the coefficient matrix is invertible. It doesn't matter
whether the model "appears" to fit the sample variogram or
not. Obviously that doesn't mean that the kriging results
will be "good", fitting variograms is partly science and
partly an art. The good and the bad part of kriging is that
the weights in the kriging estimator do not directly depend on
the data and do not depend on any distributional assumptions,
I say directly since few if any would apply kriging without
making some effort to match the variogram model to the data.
Note that some software does exactly that however, SURFER
uses a linear variogram model and the slope of the model has no
effect on the kriging weights (only on the kriging variances).
Some other software that supposedly incorporates kriging but
does not allow the user to input a model does the same thing.
The question of whether the results (from kriging) are "good" is
not a statistical nor a mathematical question.
Donald E. Myers
Department of Mathematics
University of Arizona
Tucson, AZ 85721
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