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RE: AI-GEOSTATS: Kriging with External Drift

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  • Michael Dennis
    So are you saying that the external drift variable in the matrix is just the magnitude of the the drift variable at that point? ie : 3 point kriging s = drift
    Message 1 of 13 , Dec 6, 2001
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      So are you saying that the external drift variable in the matrix is just the
      magnitude of the the drift variable at that point?

      ie :

      3 point kriging

      s = drift variable

      [ k11 k12 k13 1 s1 ] [ l1 ] [ k01 ]
      [ k21 k22 k23 1 s2 ] [ l2 ] [ k02 ]
      [ k31 k32 k33 1 s3 ] [ l3 ] = [ k03 ]
      [ 1 1 1 0 0 ] [ u0 ] [ 1 ]
      [ s1 s2 s3 0 0 ] [ u1 ] [ s0 ]


      So in this 3 point Kriging case I just plug in the magnitude of my drift
      variable in for s0, s1, s2,and s3?

      And if you substituted a drift with a magnitude which was computed based
      upon 1st order polynomial you would get the same results from this matrix as
      you would by removing the 1st order polynomial trend and kriging the
      residuals and adding the 1st order polynomial trend back in?

      It is all starting to make sense now (if I'm correct in what I'm saying
      above). Thanks so much for your help!

      Mike

      -----Original Message-----
      From: ai-geostats-list@... [mailto:ai-geostats-list@...]On
      Behalf Of sshibli@...
      Sent: Thursday, December 06, 2001 11:59 AM
      To: Michael Dennis
      Cc: AI-Geostats Mailing List
      Subject: Re: AI-GEOSTATS: Kriging with External Drift



      On Thursday, December 6, 2001, at 04:12 AM, Michael Dennis wrote:
      >
      > I don't think this is right but if you can explain to me how the Drift
      > is
      > actually applied in laymans terms it would be greatly appreciated. Also
      > when you do kriging with external drift do you have to model a
      > variogram or
      > can a reasonable one be computed automatically, if so how would you
      > compute
      > it?

      Kriging with an external drift is just an extension of universal kriging.
      UK assumes that one knows the shape of the trend but not its
      magnitude (or coefficients). For example a linear drift could be modeled
      by Mean = a + bX + cY where X and Y are the coordinates of the data.
      And so on and so forth for higher order polynomial trends.

      In KED, the trend shape is not defined analytically; rather, it is
      assumed that
      it is defined explicitly at all locations based on some densely sampled
      secondary variable. However, such a secondary variable must be
      smoothly varying in space, and also it must be available at all locations
      of the primary data and the locations being estimated.

      As in UK, the magnitude of the trend is unimportant, it is the shape
      that we're interested in. An external drift that varies linearly with X
      and
      Y would be equivalent to UK with an analytical trend of the same
      order polynomial, i.e. 1.

      Regards,

      Syed
      Maersk Copenhagen


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    • Andreas Hartmann
      Hi, a little off the original topic, but one question that puzzles me is the computation of the semivariogramm for Kriging with external drift. If I understand
      Message 2 of 13 , Dec 7, 2001
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        Hi,

        a little off the original topic, but one question that puzzles me is the
        computation of the semivariogramm for Kriging with external drift.

        If I understand correctly, a relation of type
        m*(x) = a1(x) + a2(x)*s(x)
        is assumed between the mean of the primary variable at location x
        (drift) and secondary variable (s). Now, the a's are dependent on the
        location and are not computed explicitly. But in the kriging system the
        residual covarince is needed. To compute the residuals, I would need to
        calculate the equation explicitly, right?

        How do I compute the residual semivariogramm when I don't know the
        residuals? Or have I misunderstood something in the concept of external
        drift?

        Best regards
        Andreas

        --
        Andreas Hartmann
        RWTH Aachen, Angewandte Geophysik
        Lochnerstr. 4-20
        52056 Aachen, Germany
        (+49) (-0) 241 8094835
        mailto:Andreas@...-aachen.de
        http://www.geophysik.rwth-aachen.de




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      • Isobel Clark
        ... you got it. Plus the constraint that the drift components in the samples have to balance with the drift component at the point being estimated. Isobel
        Message 3 of 13 , Dec 7, 2001
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          > So are you saying that the external drift variable
          > in the matrix is just the
          > magnitude of the the drift variable at that point?
          you got it.

          Plus the constraint that the drift components in the
          samples have to balance with the drift component at
          the point being estimated.

          Isobel

          ________________________________________________________________
          Nokia 5510 looks weird sounds great.
          Go to http://uk.promotions.yahoo.com/nokia/ discover and win it!
          The competition ends 16 th of December 2001.

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        • sshibli@mac.com
          ... There is a nice section (Chapter 5.4) in Cressie s textbook (Statistics for Spatial Data) that discusses the potential bias of estimating semivariograms
          Message 4 of 13 , Dec 7, 2001
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            On Friday, December 7, 2001, at 10:05 AM, Andreas Hartmann wrote:
            >
            > How do I compute the residual semivariogramm when I don't know the
            > residuals? Or have I misunderstood something in the concept of external
            > drift?

            There is a nice section (Chapter 5.4) in Cressie's textbook (Statistics
            for
            Spatial Data) that discusses the potential bias of estimating
            semivariograms from residuals and Matheron's formulation of
            IRF-K kriging as an alternative means to krige in the presence of
            a trend. The iterative fitting of the covariance is usually
            non-graphical,
            which is a drawback in itself, and Fig. 5.1 in the same section shows
            Cressie's bold attempt at showing the pitfalls of doing such an
            "automatic" fit. I have not used IRF-K kriging very much in practice,
            for the above reasons. Better to live with the devil that I know, i.e.
            my -- albeit biased -- residual variogram.

            Regards,

            Syed



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          • Nicholas Lewin-Koh
            Hi, Just to add to that Gotway and cressie I can t remember the exact citation did a large simulation study and showed that the bias from using the residual
            Message 5 of 13 , Dec 8, 2001
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              Hi,
              Just to add to that Gotway and cressie I can't remember the exact citation
              did a large simulation study and showed that the bias from using the
              residual variogram is very small in most cases.

              Nicholas

              On Fri, 7 Dec 2001 sshibli@... wrote:

              >
              > On Friday, December 7, 2001, at 10:05 AM, Andreas Hartmann wrote:
              > >
              > > How do I compute the residual semivariogramm when I don't know the
              > > residuals? Or have I misunderstood something in the concept of external
              > > drift?
              >
              > There is a nice section (Chapter 5.4) in Cressie's textbook (Statistics
              > for
              > Spatial Data) that discusses the potential bias of estimating
              > semivariograms from residuals and Matheron's formulation of
              > IRF-K kriging as an alternative means to krige in the presence of
              > a trend. The iterative fitting of the covariance is usually
              > non-graphical,
              > which is a drawback in itself, and Fig. 5.1 in the same section shows
              > Cressie's bold attempt at showing the pitfalls of doing such an
              > "automatic" fit. I have not used IRF-K kriging very much in practice,
              > for the above reasons. Better to live with the devil that I know, i.e.
              > my -- albeit biased -- residual variogram.
              >
              > Regards,
              >
              > Syed
              >
              >
              >
              > --
              > * To post a message to the list, send it to ai-geostats@...
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              >

              CH3
              |
              N Nicholas Lewin-Koh
              / \ Dept of Statistics
              N----C C==O Program in Ecology and Evolutionary Biology
              || || | Iowa State University
              || || | Ames, IA 50011
              CH C N--CH3 http://www.public.iastate.edu/~nlewin
              \ / \ / nlewin@...
              N C
              | || Currently
              CH3 O Graphics Lab
              School of Computing
              National University of Singapore
              The Real Part of Coffee kohnicho@...


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