## AI-GEOSTATS: Gaussian semivariogram model

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• Dear list Last week I asked the following question regarding the gaussian semivariogram model: I have experienced that the gaussian semivariogram model
Message 1 of 3 , May 7 8:02 AM
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Dear list

Last week I asked the following question regarding the gaussian
semivariogram model:

I have experienced that the gaussian semivariogram model sometimes lead
to a covariance matrix which is not positive definite. I am aware that
the parabolic behavior of the function near the origin could give these
kinds of problems, but I dont think this is the whole story. Do you about

and got some nice and helpful answers. Thanks to Pierre Goovaerts, Donald
Myers, Sean McKenna and Benjamin Warr for providing these answers, which
are given below:

********************************************************************
Pierre Goovaerts wrote:

Problems with the Gaussian semivariogram typically
arise when no nugget effect is specified and
some observations are very close to each other,
leading to covariances matrice with very similar rows.
Wackernagel's book "multivariate geostatistics"
or the recent book by Chiles and Delfiner.

*******************************************************************
Donald Myers wrote:

Theoretically this can not happen (because the gaussian variogram is a
valid model) BUT:

The problem is that the graph of the gaussian model is almost horizontal
for some distance near the origin and if there is no nugget term then
the computed values (for multiple pairs of locations) is either zero or
almost zero. When you have a lot of zeros or entries that are almost
zero in the covariance matrix, i.e., in the coefficient matrix for the
kriging equations, that matrix will not be invertible. The solution is
to incorporate a small nugget term into the variogram. What I said above
is at least related to your observation that the gaussian variogram is
nearly parabolic in shape near the origin.

Note that even though the gaussian covariance is positive definite and
will result in positive definite matrices, if you are using the
variogram form in ordinary or universal kriging then the coefficient
matrix is NOT postive definite although the coefficient matrix is
invertible. See a paper by D. Posa and A. Journel in Math. Geology ,
early 1990's. This distinction is not related to the point I made above.

*******************************************************************
Sean McKenna wrote:

Soren, try Ababou et al., 1994, On the Condition Number of Covariance
Matrices in Kriging, Estimation and Simulation of Random Fields,
Mathematical Geology, 26 (1), pp. 99-133.

******************************************************************
Benjamin Warr wrote:

the addition of a miniscule nugget variance to a variogram model that
inclludes a Gaussian model can rectify this problem, by introducing a
discontinuity at the origin,

Best regards / Venlig hilsen

SÃ¸ren Lophaven
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