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GEOSTATS: Kriging Instability

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  • L Scott Baggett
    Greetings, I m working with a dataset of irregularly sampled sea-surface temperatures following seasonal and gross E-W and N-S trend removal. The estimated
    Message 1 of 2 , Jul 1, 1998
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      Greetings,

      I'm working with a dataset of irregularly sampled sea-surface temperatures
      following seasonal and gross E-W and N-S trend removal. The estimated
      variogram is a linear combination of admissible isotropic variograms as
      functions of lag space and time. The kriging step involves estimation of
      about 250 gridded points through almost 50 years.

      I've found that theoretical efforts to define a kriging neighborhood are
      overshadowed by numerical instabilities in the solution of the (ordinary)
      linear kriging equation G*b=g, where G contains the variogram estimates of
      the observed differences, g is a vector containing variogram estimates of
      the observed - predicted location, and b is the vector of kriging weights
      (and of course single LaGrange multiplier).

      Even with iterative refinement, I've found that kriging neighborhoods of
      about 10 observations are the the largest I can use before G becomes
      ill-conditioned. My only measure of reliability is the condition number
      obtained by taking the ratio of the largest to smallest singular values in
      an SVD. My intuition is that this is a more common problem than has been
      addressed in the literature. I have not yet received a copy of McCarn and
      Carr (1992) and am in hopes this helps. In the meantime, my questions to
      the group are as follows;

      1. Is there a way to estimate numerical precision of the kriging weights
      using the condition number, or something else for that matter? I've seen
      this done using condition numbers calculated from norms of the inverses
      but I question that approach since the inverse is inaccurate in
      ill-conditioned cases. My ultimate goal is to identify and eliminate
      imprecise kriging weights.

      2. Is there a more optimal technique than gaussian elimination with
      partial pivoting combined with iterative improvement? Can someone
      recommend a package or subroutine? I typically cannot use a "canned"
      package because of the spatiotemporal nature of the problem but am open to
      any suggestion.

      3. How have others dealt with this problem? I would be most interested in
      hearing of other experiences with kriging instability. Hopefully, there
      are ways I have not thought of in getting around this.

      Thank you for your comments. I will post the responses.

      L. Scott Baggett
      Rice University
      Statistics Department, MS138
      6100 Main Street
      Houston, TX 77005-1892





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    • Donald Myers
      I am wondering about your modeling of the spatial temporal variograms, this may be the problem. a. Did you attempt to use a metric in space time? b. In general
      Message 2 of 2 , Jul 1, 1998
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        I am wondering about your modeling of the spatial temporal variograms, this
        may be the problem.

        a. Did you attempt to use a metric in space time?

        b. In general using a zonal anisotropy will not work, i.e., writing the
        space-time variogram as the sum of a spatial variogram and a temporal
        variogram. See Myers and Journel, Math Geology circa 1990.

        c. One could use the product of two covariances and then convert to a
        variogram, see De Cesare, Myers and Posa in the proceedings of the 1996
        Geostatistics meeting in Wollongong.

        d. A better choice however is not only the product but also a sum of
        covariances but one has to be careful about the coefficients to ensure the
        conditional negative definiteness of the resulting variogram. De Cesare,
        Posa and I have a paper we will be presenting at the conference in Valencia
        in November that uses this model.

        e. Posa and Journel have a paper in Math Geology on ill-conditioning, i.e.,
        the conditioning number of the coefficient matrix. Also see some work of
        Narcowich and Ward (Texas A & M) on radial basis functions. For the
        connection see a couple of papers of mine.

        Donald E. Myers

        http://www.arizona.edu/~donaldm


        At 03:56 PM 7/1/98 -0500, you wrote:
        >Greetings,
        >
        >I'm working with a dataset of irregularly sampled sea-surface temperatures
        >following seasonal and gross E-W and N-S trend removal. The estimated
        >variogram is a linear combination of admissible isotropic variograms as
        >functions of lag space and time. The kriging step involves estimation of
        >about 250 gridded points through almost 50 years.
        >
        >I've found that theoretical efforts to define a kriging neighborhood are
        >overshadowed by numerical instabilities in the solution of the (ordinary)
        >linear kriging equation G*b=g, where G contains the variogram estimates of
        >the observed differences, g is a vector containing variogram estimates of
        >the observed - predicted location, and b is the vector of kriging weights
        >(and of course single LaGrange multiplier).
        >
        >Even with iterative refinement, I've found that kriging neighborhoods of
        >about 10 observations are the the largest I can use before G becomes
        >ill-conditioned. My only measure of reliability is the condition number
        >obtained by taking the ratio of the largest to smallest singular values in
        >an SVD. My intuition is that this is a more common problem than has been
        >addressed in the literature. I have not yet received a copy of McCarn and
        >Carr (1992) and am in hopes this helps. In the meantime, my questions to
        >the group are as follows;
        >
        >1. Is there a way to estimate numerical precision of the kriging weights
        >using the condition number, or something else for that matter? I've seen
        >this done using condition numbers calculated from norms of the inverses
        >but I question that approach since the inverse is inaccurate in
        >ill-conditioned cases. My ultimate goal is to identify and eliminate
        >imprecise kriging weights.
        >
        >2. Is there a more optimal technique than gaussian elimination with
        >partial pivoting combined with iterative improvement? Can someone
        >recommend a package or subroutine? I typically cannot use a "canned"
        >package because of the spatiotemporal nature of the problem but am open to
        >any suggestion.
        >
        >3. How have others dealt with this problem? I would be most interested in
        >hearing of other experiences with kriging instability. Hopefully, there
        >are ways I have not thought of in getting around this.
        >
        >Thank you for your comments. I will post the responses.
        >
        >L. Scott Baggett
        >Rice University
        >Statistics Department, MS138
        >6100 Main Street
        >Houston, TX 77005-1892
        >
        >
        >
        >
        >
        >--
        >*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!
        >

        --
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