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Re: AI-GEOSTATS: Kriging with External Drift

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  • Heinz Burger
    Hello Michael! What you are describing is kriging in the presence of a drift, e.g. kriging of a (single) nonstationary RV. You may proceed as described in
    Message 1 of 13 , Dec 6, 2001
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      Hello Michael!
      What you are describing is kriging in the presence of a drift,
      e.g. kriging of a (single) nonstationary RV. You may proceed as
      described in your email or use universal kriging (UK) for that.

      External drift refers to a second variable which may be used
      in order to improve estimation e.g.
      1st variable: rainfall data measured at a limited number of stations and
      2nd variable: digital elevation data which must be known at each
      grid point to be estimated.
      See Goovaerts' paper for this example:
      http://www.geovista.psu.edu/sites/geocomp99/Gc99/023/gc_023.htm

      For UK see any standard textbook on geostats.

      Regards,
      Heinz Burger

      *****************************************************
      Visit IAMG2002: http://www.fu-berlin.de/iamg2002
      *****************************************************
      Dr. Heinz Burger
      Freie Universitaet Berlin
      - Geoinformatik -
      Malteserstr. 74-100
      12249 BERLIN, Germany
      Tel. (49) 30-838-70561 Fax: (49) 30-838-70723
      mailto: hburger@...-berlin.de
      Web-Seite: http://userpage.fu-berlin.de/~hburger/hb
      ****************************************************



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    • Michael Dennis
      RE: AI-GEOSTATS: Kriging with External DriftThanks for clearing that up for me. I get confused with the terminology sometimes. I ll see if I can get a copy
      Message 2 of 13 , Dec 6, 2001
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        RE: AI-GEOSTATS: Kriging with External DriftThanks for clearing that up for
        me. I get confused with the terminology sometimes. 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. How is this
        shape function computed? Isn't it just a polynomial trend? Or is it
        something more complex than that? I know the bottom row of the matrix is
        computed from the "drift" but I'm interested in how that bottom value is
        derived.

        On another note : I've run into a couple of references on Bayesian Kriging
        which I get the impression is similar to KED. Can anyone tell me what the
        differences are between Bayesian and KED and why you would choose one over
        the other?

        Thanks,

        Mike

        -----Original Message-----
        From: WARR Benjamin [mailto:benjamin.warr@...]
        Sent: Thursday, December 06, 2001 3:59 AM
        To: 'Michael Dennis'
        Subject: RE: AI-GEOSTATS: Kriging with External Drift


        Michael,

        I suggest a book by Hans Wackernagel, An Introduction to Multivariate
        Geostatistics, Springer Verlag, 2nd Edition 1999.

        He describes the technique thoroughly and clearly. Briefly, what you
        describe is Universal Kriging (UK). KED uses a shape function, provided by
        exhaustive secondary data to model a deterministic 'drift' or underlying
        trend, whereas when doing UK you have to model the trend yourself using a
        polynomial.



        Benjamin Warr

        Research Associate
        Centre for the Management of Environmental Resource(CMER)
        INSEAD
        Boulevard de Constance,
        77305 Fontainebleau Cedex,
        France



        > -----Original Message-----
        > From: Michael Dennis [mailto:Mike.D@...]
        > Sent: 06 December 2001 04:13
        > To: AI-Geostats Mailing List
        > Subject: AI-GEOSTATS: Kriging with External Drift
        >
        >
        > Hello all,
        >
        > I'm a bit of a rookie with GeoStatistics. I am interested in
        > Kriging with
        > External Drift but I am having a hard time finding
        > information that tells
        > you how it works in laymans terms (without just firing a
        > matrix at you and
        > leaving you to deduce what it means).
        >
        > I don't think this is right but I'll give it a shot. Does it work as
        > follows :
        >
        > 1) Compute a trend for the drift variable
        > 2) Remove the trend computed in 1) from the main variable
        > 3) Grid the residuals from step 2)
        > 4) Add back the trend from step 1)
        >
        > 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?
        >
        > Also if you have any good sources of information on Kriging
        > with external
        > drift could you pass them on to me?
        >
        > Thanks,
        >
        > Mike
        >
        >
        > --
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        [Non-text portions of this message have been removed]
      • Michael Dennis
        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
        Message 3 of 13 , Dec 6, 2001
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          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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        • 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 4 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 5 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 6 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 7 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 8 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 9 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 10 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 11 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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