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Re: AI-GEOSTATS: Kriging with External Drift

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  • sshibli@mac.com
    ... 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
    Message 1 of 13 , Dec 6, 2001
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      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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    • Isobel Clark
      Michael Maybe we have not made this clear. Universal kriging is a two stage process. (1) Fit a trend (global) or local trends and calculate the residuals. From
      Message 2 of 13 , Dec 6, 2001
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        Michael

        Maybe we have not made this clear.

        Universal kriging is a two stage process.

        (1) Fit a trend (global) or local trends and calculate
        the residuals. From these residuals you obtain a
        de-trended (drift-less) semi-variogram.

        (2) Using the semi-variogram derived in (1) together
        with the established form of the trend, you krige the
        complete value with trend from your original sample
        values -- not from the residuals.

        The only time you use the residuals is to get the
        semi-variogram model.

        People who refer to 'kriging with external drift' seem
        to mean different things and you would need to read
        each case on its own merits. I am sure there are
        people out there who can point you to the best
        references for that terminology.

        Try the kriging game together with our Wolfcamp
        tutorial which is freely available and distributable
        to anyone who wants it.

        Are we getting closer?
        Isobel Clark
        http://uk.geocities.com/drisobelclark/briefcase.html

        --- Michael Dennis <Mike.D@...> wrote: >
        Thanks for the info. I don't mind you plugging your
        > book, that was one of
        > the questions I asked : What is a good reference on
        > this subject.
        >
        > See this is where I start getting confused with
        > terminology. I'm talking
        > about KED (Kriging with External Drift) and then you
        > start talking about
        > Universal Kriging. If I understand correctly they
        > are not the same thing?
        > With Universal Kriging you remove the trend and Krig
        > the residuals but in
        > KED you use the trend(drift) in the actual krig
        > matrix?
        >
        > Mike
        >
        > -----Original Message-----
        > From: Isobel Clark
        > [mailto:drisobelclark@...]
        > Sent: Thursday, December 06, 2001 11:18 AM
        > To: Michael Dennis
        > Cc: wharper@...
        > Subject: RE: AI-GEOSTATS: Kriging with External
        > Drift
        >
        >
        > > I'll see if I can get a
        > > copy of the book you mention to look at to see if
        > it
        > > helps me out.
        > >
        > > You say that KED uses a shape function for the
        > > "Drift" data.
        > 'shape functions' can be anything you like but most
        > people stick to simple polynomials. To use Univeral
        > Kriging the only constraint is that it has to be
        > expressed as a linear function in the coefficients.
        > That is: b0 + b1 * some function + b2 * some other
        > function and so on, where the b's are the
        > coefficients.
        >
        > All this is explained in detail in Chapter 12 of
        > Practical Geostatistics 2000, but we aren't allowed
        > to
        > say that on the open list ;-)
        >
        > You can try it out free (completely) with the
        > kriging
        > game in my briefcase. This shows you the equations
        > and
        > the terms calculated. A full tutorial on Universal
        > Kriging is also available in the briefcase and can
        > be
        > run with the free PG2000 software.
        >
        > Find them all at:
        >
        > http://uk.geocities.com/drisobelclark/briefcase.html
        >
        > Let us know if we can be of further help.
        >
        > Isobel Clark
        >
        >
        ________________________________________________________________
        > 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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        ________________________________________________________________
        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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      • 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 3 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 4 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 5 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 6 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 7 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 8 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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