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AI-GEOSTATS: Observations with a known standard deviation

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  • Soeren Nymand Lophaven
    Dear list I am currently working with spatial interpolation of geophysical data. Each observation is associated with an individual and known standard
    Message 1 of 3 , Jan 30, 2003
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      Dear list

      I am currently working with spatial interpolation of geophysical
      data. Each observation is associated with an individual and known standard
      deviation. How should this infomation be incorporated if I want to use
      ordinary kriging for interpolation ?? My idea was the following:

      When finding the vector of weights (w) by solving the system of linear
      equations A*w=b, I would exchange the zeros in the diagonal of the
      A-matrix with the individual observation variances. Does this sound
      reasonable ??


      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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    • Isobel Clark
      Soeren I presume what you have is a sort of analytical error for each sample? That is, the standard deviation for two samples at the same location around the
      Message 2 of 3 , Jan 30, 2003
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        Soeren

        I presume what you have is a sort of 'analytical
        error' for each sample? That is, the standard
        deviation for two samples at the same location around
        the 'true value' at the same location?

        In this case, you can put the variance down the
        diagonal of your kriging system to obtain optimal
        weights under the uncertainty admitted for your data
        values.

        You would need to be careful that the 'analytical
        variance' was not greater than the nugget effect of
        the semi-variogram model.

        The kriging system would be similar to that obtained
        when the sample is not treated as a 'point', but
        rather as a volume. This results in a lower kriging
        variance than using zero on the diagonal, so to
        compensate you should probably add the complete
        'analytical variance' back on to get realistic
        estimation variances.

        There seems to be a lot of confusion in the books (and
        software) about what happens if you have a significant
        replication variance.

        Isobel Clark
        http://geoecosse.bizland.com/news.html



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      • Colin Daly
        Soeren That works if your matrix is made up of covariance terms rather than variogram terms. However you should use the variance of the error term instead of
        Message 3 of 3 , Mar 1 8:32 AM
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          Soeren

          That works if your matrix is made up of covariance terms rather than
          variogram terms. However you should use the variance of the error term
          instead of the standard deviation.

          So in your notation A_ij = C_ij = D - Gamma_ij

          where Gamma_ij are the variogram values, D is the sill of the variogram
          (or larger than the largest variogram value if your variogram does not have
          a sill) and at the diagonal you use C_ii+K_ii where K_ii are the variance
          error terms.

          This will not interpolate your data! It will filter the noise terms (which
          you say that you know the variance of at each point)

          Regards

          Colin Daly



          ----- Original Message -----
          From: "Soeren Nymand Lophaven" <snl@...>
          To: <ai-geostats@...>
          Sent: Thursday, January 30, 2003 3:22 PM
          Subject: AI-GEOSTATS: Observations with a known standard deviation


          Dear list

          I am currently working with spatial interpolation of geophysical
          data. Each observation is associated with an individual and known standard
          deviation. How should this infomation be incorporated if I want to use
          ordinary kriging for interpolation ?? My idea was the following:

          When finding the vector of weights (w) by solving the system of linear
          equations A*w=b, I would exchange the zeros in the diagonal of the
          A-matrix with the individual observation variances. Does this sound
          reasonable ??


          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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