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AI-GEOSTATS: Variogram behaviour

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  • WARR Benjamin
    ... With regards Syed Abdul Rahmans comment above: For lags smaller than the first experimental value we do not have information about the variogram structure
    Message 1 of 4 , Mar 16, 2001
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      >I would (dangerously) suggest that in this early lag
      >period, choice of a variogram such as power law, spherical, exponential,
      >or Gaussian would be nit picking. Unless the variogram clearly shows
      >such definitive behavior, of course.

      With regards Syed Abdul Rahmans comment above: For lags smaller than the
      first experimental value we do not have information about the variogram
      structure (see discussion in Cressie's book). And unfortunately it is the
      behaviour of the variogram at the shortest distances which exerts the
      greatest influence. I would suggest that it is always worth reflecting long
      and hard about the likely structure of the variogram towards the origin
      given the variable under study. For example, optical satellite imagery
      should be nugget with gaussian, reflecting both the short range variability
      smaller than the satellite resolution and noise (nugget) and gaussian
      reflecting the convolution of such imagery.

      Benjamin Warr
      Research Associate to Professor Ayres,
      Pedometrician

      Postal Address:
      Centre for the Management of Environmental Resources (CMER)
      INSEAD
      Boulevard de Constance,
      77305 Fontainebleau Cedex,
      France

      Tel: 33 (0)1 60 72 4456
      Fax: 33 (0)1 60 74 55 64
      e-mail: benjamin.warr@...


      -----Original Message-----
      From: Syed Abdul Rahman Shibli [mailto:sshibli@...]
      Sent: Friday, March 16, 2001 7:10 AM
      To: Sara Kustron; ai-geostats
      Subject: Re: AI-GEOSTATS: non-ergodic covariance


      on 16/03/01 2:24, Sara Kustron at skustron@... wrote:

      > It appears that this technique is computationally inaccessible to us
      > non-programmers at this point in time. Could it be argued that though
      > theoretically questionable non-ergodic covariance has some practical value
      > in that it successfully cleans up variograms? I apologize if this offends
      > "purists!"

      It will be foolhardy to be a geostatistical "purist" in this day
      and age. :) Note that one would most likely assume local stationarity within
      a search neigborhood in performing OK estimations, i.e. early lag
      behavior only. I would (dangerously) suggest that in this early lag
      period, choice of a variogram such as power law, spherical, exponential,
      or Gaussian would be nit picking. Unless the variogram clearly shows
      such definitive behavior, of course.

      Syed



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    • colin
      Variogram behaviourHi, ... While I can sympathise with Syed s comments about it being hard to be a purist (it certainly is in petroleum geostats) I think the
      Message 2 of 4 , Mar 16, 2001
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        Variogram behaviourHi,
        >I would (dangerously) suggest that in this early lag
        >period, choice of a variogram such as power law, spherical, exponential,
        >or Gaussian would be nit picking. Unless the variogram clearly shows
        >such definitive behavior, of course.

        While I can sympathise with Syed's comments about it being hard to be a purist (it certainly is in petroleum geostats) I think the most important word in Syed's comment is 'dangerously'. Least we think that there is some kind of relativism about which type variogram to use it is worth looking carefully at the example that Marco gave. There is a huge difference between the behaviour of the two variograms in question (try doing a simulation with them!). It is true that with few samples , the usual variogram might give poor results - but some reasoning about the likely nature of the data under study as suggested by Benjamin is a good way forward. One may use things like 'non-ergodic' variograms - or calculating variograms on gaussian transformed data - or removing outliers before calculating variograms as a way of understanding structure - all of which are accessible to non-programmers but these are merely tools in data exploration. This can help you to decide how to fit a model to the noisy classical variogram - but cannot replace the classical variogram if doing kriging or simulation.

        Colin Daly



        ----- Original Message -----
        From: WARR Benjamin
        To: Ai-Geostats@Unil. Ch (E-mail)
        Sent: Friday, March 16, 2001 8:29 AM
        Subject: AI-GEOSTATS: Variogram behaviour


        >I would (dangerously) suggest that in this early lag
        >period, choice of a variogram such as power law, spherical, exponential,
        >or Gaussian would be nit picking. Unless the variogram clearly shows
        >such definitive behavior, of course.

        With regards Syed Abdul Rahmans comment above: For lags smaller than the first experimental value we do not have information about the variogram structure (see discussion in Cressie's book). And unfortunately it is the behaviour of the variogram at the shortest distances which exerts the greatest influence. I would suggest that it is always worth reflecting long and hard about the likely structure of the variogram towards the origin given the variable under study. For example, optical satellite imagery should be nugget with gaussian, reflecting both the short range variability smaller than the satellite resolution and noise (nugget) and gaussian reflecting the convolution of such imagery.

        Benjamin Warr
        Research Associate to Professor Ayres,
        Pedometrician

        Postal Address:
        Centre for the Management of Environmental Resources (CMER)
        INSEAD
        Boulevard de Constance,
        77305 Fontainebleau Cedex,
        France

        Tel: 33 (0)1 60 72 4456
        Fax: 33 (0)1 60 74 55 64
        e-mail: benjamin.warr@...



        -----Original Message-----
        From: Syed Abdul Rahman Shibli [mailto:sshibli@...]
        Sent: Friday, March 16, 2001 7:10 AM
        To: Sara Kustron; ai-geostats
        Subject: Re: AI-GEOSTATS: non-ergodic covariance



        on 16/03/01 2:24, Sara Kustron at skustron@... wrote:

        > It appears that this technique is computationally inaccessible to us
        > non-programmers at this point in time. Could it be argued that though
        > theoretically questionable non-ergodic covariance has some practical value
        > in that it successfully cleans up variograms? I apologize if this offends
        > "purists!"

        It will be foolhardy to be a geostatistical "purist" in this day
        and age. :) Note that one would most likely assume local stationarity within
        a search neigborhood in performing OK estimations, i.e. early lag
        behavior only. I would (dangerously) suggest that in this early lag
        period, choice of a variogram such as power law, spherical, exponential,
        or Gaussian would be nit picking. Unless the variogram clearly shows
        such definitive behavior, of course.

        Syed




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      • Digby Millikan
        Hello, I have always lead the belief that it is worth while cleaning up a variogram, but with OK a poor variogram model for normally distributed data will give
        Message 3 of 4 , Mar 16, 2001
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          Hello,
          I have always lead the belief that it is worth while cleaning up a variogram, but with OK a poor variogram
          model for normally distributed data will give superior results than inverse distance methods, though with
          lognormally distributed data fitting of a model to a poor variogram can result in highly erroneous estimates,
          as for example the percentage error in your sill will result in an equal percentage error in your estimate.
          Diverging from the topic a bit a method I preferred for lognormal data with poor variograms is to use inverse
          distance or ordinary kriging with a topcut calculated as the topcut which will give you an arithmetic mean
          equal to the sichel t estimator (an estimate of the true mean of a lognormal population). That way overestimation
          due to lognormality is avoided without resorting to lognormal kriging. I beleive their may also be a method for
          estimating the mean of a multiguassian distribution.
          Regards Digby Millikan.


          [Non-text portions of this message have been removed]
        • Isobel Clark
          ... Actually, a percentage error in your semi-variogram sill will be an exponenial error in your final estimates. However, as with Normal distributions
          Message 4 of 4 , Mar 16, 2001
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            > ........... though with
            > lognormally distributed data fitting of a model to a
            > poor variogram can result in highly erroneous
            > estimates,
            > as for example the percentage error in your sill
            > will result in an equal percentage error in your
            > estimate.
            Actually, a percentage error in your semi-variogram
            sill will be an exponenial error in your final
            estimates.

            However, as with Normal distributions everywhere, your
            total sill can be easily checked against your
            estimated population variance.

            > Diverging from the topic a bit a method I preferred
            > for lognormal data with poor variograms is to use
            > inverse distance or ordinary kriging with a topcut
            > calculated as the topcut which will give you an
            > arithmetic mean
            > equal to the sichel t estimator (an estimate of the
            > true mean of a lognormal population). That way
            > overestimation
            > due to lognormality is avoided without resorting to
            > lognormal kriging.
            For the last 30 years I have been advising people not
            to apply top cuts because this is an avoidance of the
            problem not a solution to it.

            You can kill most exploration projects dead in one go
            with this type of arbitrary rule. Of course, that is
            always the safe way to go. If the project dies at
            exploration stage, no-one ever knows what it would
            really have been.

            > I beleive their may also be a method for
            > estimating the mean of a multiguassian distribution.
            I am not sure what you mean by this. Do you mean mixed
            Normals?

            Isobel Clark


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