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AI-GEOSTATS: Gaussian semivariogram model

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  • Soeren Nymand Lophaven
    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, 2002
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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
      this phenomenon, and where to read more about it ??

      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.
      You can read more about this "pathological" model in Hans
      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

      Søren Lophaven
      Master of Science in Engineering | Ph.D. student
      Informatics and Mathematical Modelling | Building 321, Room 011
      Technical University of Denmark | 2800 kgs. Lyngby, Denmark
      E-mail: snl@... | http://www.imm.dtu.dk/~snl
      Telephone: +45 45253419 |

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