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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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    • Rautman, Christopher A
      Michael, Kriging with External Drift, as implemented in the GSLIB set of programs (Deutsch & Journel) is distinct from Universal Kriging, in that the External
      Message 2 of 13 , Dec 6, 2001
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        Michael,

        Kriging with External Drift, as implemented in the GSLIB set of programs
        (Deutsch & Journel) is distinct from Universal Kriging, in that the
        "External Drift" is intended to be defined by a secondary variable that you,
        the user, "believe" incorporates some information of relevance to the
        primary variable that you are working with.

        The "External Drift" could also be a kriged array of values based on kriging
        some sparse secondary variable (onto a regular grid) that you also feel
        contains information of relevance. You can also specify a "trend"
        mathematically, but the original intent was to incorporate a secondary
        variable "relevant" to estimation of the first.

        An example would be: you have sparse porosity measurements in an oil field,
        but you have a regular array of 3-D seismic amplitudes covering that same
        area. Your external drift variable would be seismic amplitude and your
        primary variable would be porosity. Obviously the two are not precisely the
        same, but "hopefully" they are related.

        It is up to you, the user, to specify external drift terms that "make sense"
        physically. Obviously, this is a matter of interpretation, and you are the
        one responsible for justifying your choices.

        I suggest you read the sections on Kriging with External Drift beginning on
        page 70 and 96 of the Deutsch and Journel (1998) book, "GSLIB Geostatistical
        Software Library and User's Guide," by Oxford University Press. I'm sure
        there are other papers out there on this methodology, but if you are talking
        about the GSLIB program, then it's best to go to the documentation
        associated with that program.

        Best regards,

        Chris

        Christopher A. Rautman, Ph. D., P.G.
        Underground Storage Technology Department
        Sandia National Laboratories
        P. O. Box 5800; MS-0706
        Albuquerque, New Mexico 87185-0706
        505-844-2109; fax: 505-844-4426



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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 3 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 4 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 5 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 6 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
                >
                >
                >
                > --
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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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