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AI-GEOSTATS: Kriging with External Drift

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  • Michael Dennis
    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
    Message 1 of 13 , Dec 5, 2001
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      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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    • William V Harper
      Mike, What you are dealing with is universal kriging, i.e., kriging in the presence of a drift. The data set that took away my geostatistical virginity was of
      Message 2 of 13 , Dec 5, 2001
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        Mike,

        What you are dealing with is universal kriging, i.e., kriging in the presence of a drift. The data set that took away my geostatistical virginity was of this same type - the Wolfcamp aquifer data. Since the
        original 1985 geostat pub on this by myself and Jeff Furr, many folks have analyzed this data in numerous ways. The data is downloadable from http://uk.geocities.com/drisobelclark/briefcase.html#PG2000_demo.
        Analyze it with any software that allows universal kriging (e.g., a teaching version of the software on this same page).

        In essence, estimate the trend but keep in mind that universal kriging may not need as high of a trend when fitting local search areas as a global regression trend over your entire data set. Thus while a
        quadratic regression may "best" fit your full data set, you might only need a linear trend when doing universal kriging since you are doing this in each smaller local search area.

        With the appropriate software you are in essence taking out the trend, and fitting the semi-variogram to the residuals. But when you then do the universal kriging after deciding on a reasonable semi-variogram, it
        will be done on the original data then using both the semi-variogram and trend you decided upon.

        See Practical Geostatistics 2000 (ISBN 0-9703317-0-3 or 0-9703317-2-X) for additional details on how one might approach this.

        I hope I am not violating any AI-GEOSTATS bylaws here. I am trying to give an answer that works for me without getting into any hot water.

        Bill

        Michael Dennis wrote:

        > 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
        >
        > --
        > * To post a message to the list, send it to ai-geostats@...
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        --
        William V Harper, Mathematical Sciences, Otterbein College
        Towers Hall 136, Westerville OH 43081-2006 USA
        614-823-1417 Fax 614-823-3201
        Faculty page: http://go.to/billharper
        For the best in geostatistics: http://go.to/geostatistics
        Coming eventually:
        http://www.wvharper.com (currently points to geostat site)



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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 3 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 4 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
            >
            >
            > --
            > * To post a message to the list, send it to ai-geostats@...
            > * As a general service to the users, please remember to post
            > a summary of any useful responses to your questions.
            > * To unsubscribe, send an email to majordomo@... with no
            > subject and "unsubscribe ai-geostats" followed by "end" on
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            > Subscribe/Unsubscribe requests to the list
            > * Support to the list is provided at http://www.ai-geostats.org
            >



            [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 5 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 6 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 7 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.
                  >
                  >
                  > --
                  > * To post a message to the list, send it to
                  > ai-geostats@...
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                  > questions.
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                  http://www.ai-geostats.org

                  ________________________________________________________________
                  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 8 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 9 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 10 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 11 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

                          ________________________________________________________________
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                          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 12 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 13 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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