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Re: AI-GEOSTATS: Gaussian semivariogram model

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  • Pierre Goovaerts
    Hello, Problems with the Gaussian semivariogram typically arise when no nugget effect is specified and some observations are very close to each other, leading
    Message 1 of 3 , May 1, 2002
      Hello,

      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.

      Pierre
      <><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><>

      ________ ________
      | \ / | Pierre Goovaerts
      |_ \ / _| Assistant professor
      __|________\/________|__ Dept of Civil & Environmental Engineering
      | | The University of Michigan
      | M I C H I G A N | EWRE Building, Room 117
      |________________________| Ann Arbor, Michigan, 48109-2125, U.S.A
      _| |_\ /_| |_
      | |\ /| | E-mail: goovaert@...
      |________| \/ |________| Phone: (734) 936-0141
      Fax: (734) 763-2275
      http://www-personal.engin.umich.edu/~goovaert/

      <><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><>


      On Wed, 1 May 2002, Soeren Nymand Lophaven wrote:

      > Dear list
      >
      > I have experienced that the gaussian semivariogram model sometimes leads
      > 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 ??
      >
      > 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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    • 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 2 of 3 , May 7, 2002
        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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