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AI-GEOSTATS: Automated Variogram modelling

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  • m.pawley@auckland.ac.nz
    Hi all, I have a couple of questions for the list. I understand that most theoretical variograms are fit by eye, and I was interested in gauging the usefulness
    Message 1 of 7 , Apr 4, 2004
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      Hi all,
      I have a couple of questions for the list.

      I understand that most theoretical variograms are fit by eye, and I was interested in
      gauging the usefulness of automated (purely data-driven) estimation for theoretical variograms.

      i.e. Would it be useful to practitioners to be able to fit to be able to fit something like a
      'constrained spline' as the theoretical variogram function to give your kriging results?
      (the spline could be constrained to be positive-semi-definite)


      1. Is this something that has been examined in detail in the past?
      2. If not - would it be something that geostatisticians would find useful?


      Any thoughts and references on this matter would be most welcome.

      Many thanks in advance,
      Matthew Pawley



      [Non-text portions of this message have been removed]
    • Pierre Goovaerts
      Hello, The issue of automatic versus manual modeling of semivariogram has been the subject of much debate in the past. In my graduate class, I used to ask the
      Message 2 of 7 , Apr 4, 2004
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        Hello,

        The issue of automatic versus manual modeling of semivariogram
        has been the subject of much debate in the past.
        In my graduate class, I used to ask the students to model their
        experimental semivariograms first manually (i.e. bye eye), then using
        non-linear regression. The resulting models were then used in kriging
        and cross-validation allowed them to assess the prediction
        performances of both types of models. Most were surprised to find out
        that manually fitted semivariograms could lead to more accurate
        predictions than automatically fitted ones. The take-home lesson
        was that the modeling of the semivariogram is usually a preliminary step
        towards prediction or simulation, and influence partially their results.

        Automatic semivariogram modeling is useful to model complex anisotropies
        as long as the experimental semivariograms are reasonably well defined and
        also when multiple semivariograms need to be modeled (i.e. indicator
        kriging). In addition, working now for a software R&D company and
        developing new applications of geostatistics to health science, I have
        to keep in mind that most users migth not have the necessary background to
        compute and model semivariograms. The challenge is then to find a
        procedure to achieve meaningful fits without asking much from the user...

        The issue of automatic versus manual modeling is particularly important
        when data are sparse, making the semivariogram erratic... Then the
        modeling procedure is more than a mere exercice of fitting a curve to
        experimental values. It aims at creating a model for the spatial
        variability of the phenomenon under study and it relies greatly
        on ancillary information (e.g. magnitude of nugget effect, directions of
        anisotropy) typically provided by expert knowledge.


        Pierre

        <><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><>

        Dr. Pierre Goovaerts
        President of PGeostat, LLC
        Chief Scientist with Biomedware Inc.
        710 Ridgemont Lane
        Ann Arbor, Michigan, 48103-1535, U.S.A.

        E-mail: goovaert@...
        Phone: (734) 668-9900
        Fax: (734) 668-7788
        http://alumni.engin.umich.edu/~goovaert/

        <><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><>

        On Mon, 5 Apr 2004 m.pawley@... wrote:

        > Hi all,
        > I have a couple of questions for the list.
        >
        > I understand that most theoretical variograms are fit by eye, and I was interested in
        > gauging the usefulness of automated (purely data-driven) estimation for theoretical variograms.
        >
        > i.e. Would it be useful to practitioners to be able to fit to be able to fit something like a
        > 'constrained spline' as the theoretical variogram function to give your kriging results?
        > (the spline could be constrained to be positive-semi-definite)
        >
        >
        > 1. Is this something that has been examined in detail in the past?
        > 2. If not - would it be something that geostatisticians would find useful?
        >
        >
        > Any thoughts and references on this matter would be most welcome.
        >
        > Many thanks in advance,
        > Matthew Pawley
        >
        >

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      • Edward Isaaks
        Hello, I d like to add a few comments to Pierre s remarks on automatic variogram modeling. Variogram modeling is traditionally done by manually modeling
        Message 3 of 7 , Apr 4, 2004
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          Hello,

          I'd like to add a few comments to Pierre's remarks on automatic variogram
          modeling. Variogram modeling is traditionally done by manually modeling
          individual directional sample variograms and then somehow interpolating
          between all the directional models to obtain the full model in either 2 or 3
          dimensional space. Remember, that the variogram model must be able to return
          Gamma(h) for any separation vector. For example, in 3D space this means the
          model must be able to return Gamma(h) for any lag distance h given any
          azimuth alpha, and any dip angle phi.

          Most geostatisticians find the job of combining more than 4 or 5 directional
          models into one full 3D positive definite model a bit more than they can
          handle. I actually witnessed an enthusiastic geostatistician calculate and
          model something like 50 directional sample variograms. He then laid out 50
          hard copies of the directional models on several large drafting and map
          tables and spent several days trying to put them all together to form one
          consistent model in 3D space. Finally, in disgust he selected 5 directional
          models that more or less corresponded with his prior intuition -- modeled
          them and discarded the remainder. The orientations and ratios of the
          anisotropies (there were several - one for each structure) represented by
          the 50 directional models were more than he could make sense of. The result
          was an over simplified model.

          Such problems are common to the mining industry. Many mining data sets have
          plenty of samples, sometimes as many as 50,000 or more. It turns out that
          mother nature has generally also done a pretty good job of messing up the
          spatial continuity of the regionalized variable(s). So it is not uncommon to
          find that as many as three nested structures may be required to capture the
          messed up spatial continuity with each structure characterized by a unique
          anisotropy ratio and orientation.
          Experience has shown that it is definitely an advantage to be able to model
          all 50 directional sample variograms simultaneously using an automatic
          fitter. Computers are very good at searching out and fitting data in 4
          dimensional space (length, width, depth, and Gamma). The automatically fit
          model can be a very accurate representation of the underlying spatial
          continuity, particularly when the anisotropy ratios are relatively severe.
          But of course, the model must be critically reviewed and judged acceptable
          in the final analysis.

          Many mining companies use automatic variogram modeling routinely; Placer,
          Amec, BHP, Barrick, Newmont, CVRD, and Hecla Mining to name a few. So I
          think it can be said from experience that in general, the advantages of
          automatic variogram modeling outweigh the disadvantages. The proof is in the
          pudding.

          More information on automatic variogram modeling can be found at
          www.isaaks.com

          ----- Original Message -----
          From: "Pierre Goovaerts" <goovaert@...>
          To: <m.pawley@...>
          Cc: <ai-geostats@...>
          Sent: Sunday, April 04, 2004 6:51 PM
          Subject: Re: AI-GEOSTATS: Automated Variogram modelling


          > Hello,
          >
          > The issue of automatic versus manual modeling of semivariogram
          > has been the subject of much debate in the past.
          > In my graduate class, I used to ask the students to model their
          > experimental semivariograms first manually (i.e. bye eye), then using
          > non-linear regression. The resulting models were then used in kriging
          > and cross-validation allowed them to assess the prediction
          > performances of both types of models. Most were surprised to find out
          > that manually fitted semivariograms could lead to more accurate
          > predictions than automatically fitted ones. The take-home lesson
          > was that the modeling of the semivariogram is usually a preliminary step
          > towards prediction or simulation, and influence partially their results.
          >
          > Automatic semivariogram modeling is useful to model complex anisotropies
          > as long as the experimental semivariograms are reasonably well defined and
          > also when multiple semivariograms need to be modeled (i.e. indicator
          > kriging). In addition, working now for a software R&D company and
          > developing new applications of geostatistics to health science, I have
          > to keep in mind that most users migth not have the necessary background to
          > compute and model semivariograms. The challenge is then to find a
          > procedure to achieve meaningful fits without asking much from the user...
          >
          > The issue of automatic versus manual modeling is particularly important
          > when data are sparse, making the semivariogram erratic... Then the
          > modeling procedure is more than a mere exercice of fitting a curve to
          > experimental values. It aims at creating a model for the spatial
          > variability of the phenomenon under study and it relies greatly
          > on ancillary information (e.g. magnitude of nugget effect, directions of
          > anisotropy) typically provided by expert knowledge.
          >
          >
          > Pierre
          >
          >
          <><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><>
          <><>
          >
          > Dr. Pierre Goovaerts
          > President of PGeostat, LLC
          > Chief Scientist with Biomedware Inc.
          > 710 Ridgemont Lane
          > Ann Arbor, Michigan, 48103-1535, U.S.A.
          >
          > E-mail: goovaert@...
          > Phone: (734) 668-9900
          > Fax: (734) 668-7788
          > http://alumni.engin.umich.edu/~goovaert/
          >
          >
          <><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><>
          <><>
          >
          > On Mon, 5 Apr 2004 m.pawley@... wrote:
          >
          > > Hi all,
          > > I have a couple of questions for the list.
          > >
          > > I understand that most theoretical variograms are fit by eye, and I was
          interested in
          > > gauging the usefulness of automated (purely data-driven) estimation for
          theoretical variograms.
          > >
          > > i.e. Would it be useful to practitioners to be able to fit to be able to
          fit something like a
          > > 'constrained spline' as the theoretical variogram function to give your
          kriging results?
          > > (the spline could be constrained to be positive-semi-definite)
          > >
          > >
          > > 1. Is this something that has been examined in detail in the past?
          > > 2. If not - would it be something that geostatisticians would find
          useful?
          > >
          > >
          > > Any thoughts and references on this matter would be most welcome.
          > >
          > > Many thanks in advance,
          > > Matthew Pawley
          > >
          > >
          >
          > --
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        • Isobel Clark
          Matthew I have been praying for someone to come up with automatic semi-variogram fitting for over 30 years. Then I could retire ;-) One problem is that all
          Message 4 of 7 , Apr 5, 2004
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            Matthew

            I have been praying for someone to come up with
            automatic semi-variogram fitting for over 30 years.
            Then I could retire ;-)

            One problem is that all points are not equal on the
            graph. Each point is an estimate of a variance (which
            has a skewed distribution) and is based on different
            numbers of pairs.

            Another problem is that the early (short distance)
            points are more important than longer distances, since
            these are the ones which affect cross validation in
            particular and kriging results in general.

            Noel Cressie had a good go at a 'goodness of fit
            statistic' in the early 90s and I use that as a
            relative fitness measure but not as an optimal fitting
            method -- since it does not give the same fits I would
            get by 'eye balling'.

            If you can produce an algorithm that includes all
            these factors plus the subconscious ones, then we will
            all be grateful.

            Isobel
            http://uk.geocities.com/drisobelclark





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          • Syed Abdul Rahman Shibli
            Just to echo Pierre and Ed s remarks, I think it is easy to be trapped in analysis paralysis when it comes to variogram modelling. It s also easy to lose
            Message 5 of 7 , Apr 5, 2004
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              Just to echo Pierre and Ed's remarks, I think it is easy to be trapped in "analysis paralysis" when it comes to variogram modelling. It's also easy to lose sight of the goals of the interpolation process, and also the inherent limitations of working with experimental variograms. In some problem domains, interpolation results transcend the static and move into the dynamic realm via a transfer function such as a reservoir simulator.

              I've seen some geoscientists (in petroleum) spending an inordinate amount of time fine tuning horizontal variogram models based on five or six appraisal wells and very few pairs at each lag distance. My usual response to such cases is to suggest to the person to come up with a (qualitative) variogram model that made sense to them as geoscientists instead of basing it on well information.

              Some studies spend months coming up with a detailed matrix permeability model in a fractured carbonate reservoir when most well test data point to super K production and is not matrix dominated. This does not help much when one is trying to determine such parameters as injection recovery efficiency. It might only help in getting a first pass volumetric estimate, but unfortunately (or fortuantely) shareholders are normally interested in reserves and not what is in place.

              Just my two cents worth.

              Cheers,

              Syed

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            • Soeren Nymand Lophaven
              Matthew, Variogram modelling is a common way to estimate the parameters of a geostatistical model. Automating the modelling is associated with difficulties as
              Message 6 of 7 , Apr 5, 2004
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                Matthew,

                Variogram modelling is a common way to estimate the parameters of a
                geostatistical model. Automating the modelling is associated with
                difficulties as described very well by Isobel. Some of these can be
                overcome by estimating the parameters using Maximum Likelihood, which is
                the usual way of estimating parameters in statistical models. From my
                point of view the natural way to automate parameter estimation in
                geostatistical models is to use Maximum Likelihood estimation.

                Best regards / Venlig hilsen

                Soren Lophaven
                ******************************************************************************
                Master of Science in Engineering | Ph.D. student
                Informatics and Mathematical Modelling | Building 321, Room 011
                Technical University of Denmark | 2800 kgs. Lyngby, Denmark
                E-mail: snl@... | http://www.imm.dtu.dk/~snl
                Telephone: +45 45253419 |
                ******************************************************************************



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              • sanguinetti.henri
                Pierre and Isobel comments are very interesting. In the particular cases of mining companies, they are even more relevant. Effectively we get many data in
                Message 7 of 7 , Apr 6, 2004
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                  Pierre and Isobel comments are very interesting.
                  In the particular cases of mining companies, they are even more relevant.
                  Effectively we get many data in mining, but we have to remember that the
                  experimental variograms are dependant of the sampling pattern, ie drilling
                  grid, drilling orientation, sampling along drill hole ..etc..
                  Consequently each direction of calculation should be "weighed" carefully and
                  using what is known about the geology.
                  Of course it's long and cumbersome. But if you hate cooking bread you should
                  not be a baker.
                  "the proof in the pudding"..yes, be careful that the mining companies are
                  not as well.

                  Henri



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