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[ai-geostats] regularization

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  • samuel verstraete
    Hi, I have a 3D data set that has been sampled by a private company. They lacked a complete knowledge of geostatistics so there is no sampling strategy
    Message 1 of 3 , Oct 26, 2004
      Hi,

      I have a 3D data set that has been sampled by a private company. They
      lacked a complete knowledge of geostatistics so there is no sampling
      "strategy" involved. Another thing is that the support of the samples is
      strongly fluctuating. Horizontally the sampling support is constant and
      can be considered as a point (about 70cm^2 compared to a few hectares)
      Vertically the sampling support is not stable and rather "huge" in
      comparison with the vertical scale... (sampling can be 0.10 to 1 meter
      and maximum depth would be 5 to 6 meter or even less)

      I've read in the literature that there is a possibility to correct for
      such a things, through regularization. But none of the literature seems
      to discuss the possibility that the samples themself do not always have
      the same support, as stated before samples can have a support that is 10
      times bigger than the smallest sample.

      Question is... Is there any other literature that discusses this matter
      and even more importantly is there any software out there that can take
      this sampling support into consideration when I'm calculating the
      variogram or when I start with estimation/simulation of the field.


      Thanks in advance,

      --
      Samuel Verstraete
      Ghent University
      Faculty of Bioscience Engineering
      Dept. of Soil Management and Soil Care
      Coupure Links 653, B-9000 Gent, Belgium
    • Pierre Goovaerts
      Hi Samuel, I have dealt with similar problems when analyzing the spatial distribution of dioxin and other heavy metals in river sediments. Core lengths can
      Message 2 of 3 , Oct 26, 2004
        Hi Samuel,

        I have dealt with similar problems when analyzing the spatial
        distribution of dioxin and other heavy metals in river sediments.
        Core lengths can strongly fluctuate from one sampling point to the
        next. The empirical approach I used was to weigh each sample
        proportionally to its length both in the computation of semivariograms
        (use of weighted semivariogram estimators) and in the kriging
        procedure (rescaling of kriging weights to account for core length).
        There was no publication on this approach and reports are confidential.
        These days I would use a less empirical approach and capitalize on the
        analogy with the treatment of cancer rates, where the reliability of rates
        is a function of the population size. You could still use weighted
        semivariogram estimator, but use a "kriging with measurement error"
        approach, whereby an error variance term (here inversely proportional
        to the length of the core) is added to the diagonal elemnts of the
        kriging matrix.

        Here is just a suggestion but I am sure that some mining geostaticians
        will come up with a more elegant solution.. I also think that Jayme
        Gomez presented a paper on this issue (and the downscaling or
        disaggregation problem in general) at the last geostat congress in
        Banff, but since I only caught the last part of his presentation I
        might be wrong.

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

        Dr. Pierre Goovaerts
        President of PGeostat, LLC
        Chief Scientist with Biomedware Inc.
        710 Ridgemont Lane
        Ann Arbor, Michigan, 48103-1535, U.S.A.

        E-mail: goovaert@...
        Phone: (734) 668-9900
        Fax: (734) 668-7788
        http://alumni.engin.umich.edu/~goovaert/

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

        On Tue, 26 Oct 2004, samuel verstraete wrote:

        > Hi,
        >
        > I have a 3D data set that has been sampled by a private company. They
        > lacked a complete knowledge of geostatistics so there is no sampling
        > "strategy" involved. Another thing is that the support of the samples is
        > strongly fluctuating. Horizontally the sampling support is constant and
        > can be considered as a point (about 70cm^2 compared to a few hectares)
        > Vertically the sampling support is not stable and rather "huge" in
        > comparison with the vertical scale... (sampling can be 0.10 to 1 meter
        > and maximum depth would be 5 to 6 meter or even less)
        >
        > I've read in the literature that there is a possibility to correct for
        > such a things, through regularization. But none of the literature seems
        > to discuss the possibility that the samples themself do not always have
        > the same support, as stated before samples can have a support that is 10
        > times bigger than the smallest sample.
        >
        > Question is... Is there any other literature that discusses this matter
        > and even more importantly is there any software out there that can take
        > this sampling support into consideration when I'm calculating the
        > variogram or when I start with estimation/simulation of the field.
        >
        >
        > Thanks in advance,
        >
        > --
        > Samuel Verstraete
        > Ghent University
        > Faculty of Bioscience Engineering
        > Dept. of Soil Management and Soil Care
        > Coupure Links 653, B-9000 Gent, Belgium
        >
        >
        >
        >
      • Edzer J. Pebesma
        Two relevant publications mentioned by Carol Gotway during her geoENV keynote (I happen to have them both on my desk right now, but haven t read them yet):
        Message 3 of 3 , Oct 26, 2004
          Two relevant publications mentioned by Carol Gotway during her
          geoENV keynote (I happen to have them both on my desk right
          now, but haven't read them yet):

          Gotway & Young, 2002, Combining incompatible spatial data,
          JASA (I don't have the issue and page numbers) -- this paper
          deals with block kriging when the data are observed on blocks
          with varying size

          A Mockus, 1998, Estimating dependencies from spatial averages.
          Journal of computational and graphical statistics 7:4, 501-513. --
          this paper explains how to estimate the point support variogram
          (probably up to the nugget) from blocks of varying size.

          If you find software for either of the issues in these papers, please let
          the list know.
          --
          Edzer


          samuel verstraete wrote:

          >Hi,
          >
          >I have a 3D data set that has been sampled by a private company. They
          >lacked a complete knowledge of geostatistics so there is no sampling
          >"strategy" involved. Another thing is that the support of the samples is
          >strongly fluctuating. Horizontally the sampling support is constant and
          >can be considered as a point (about 70cm^2 compared to a few hectares)
          >Vertically the sampling support is not stable and rather "huge" in
          >comparison with the vertical scale... (sampling can be 0.10 to 1 meter
          >and maximum depth would be 5 to 6 meter or even less)
          >
          >I've read in the literature that there is a possibility to correct for
          >such a things, through regularization. But none of the literature seems
          >to discuss the possibility that the samples themself do not always have
          >the same support, as stated before samples can have a support that is 10
          >times bigger than the smallest sample.
          >
          >Question is... Is there any other literature that discusses this matter
          >and even more importantly is there any software out there that can take
          >this sampling support into consideration when I'm calculating the
          >variogram or when I start with estimation/simulation of the field.
          >
          >
          >Thanks in advance,
          >
          >
          >
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