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[ai-geostats] Definition of standardized variograms

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  • Gregoire Dubois
    Dear list, While playing around with different software, I encounter different definitions for standardized variograms. Surfer (which is using the terminology
    Message 1 of 3 , Apr 5 6:48 AM
      Definition of standardize variograms

      Dear list,

      While playing around with different software, I encounter different definitions for standardized variograms.

      Surfer (which is using the terminology of Variowin), uses the term "standardized semivariogram"  for variograms obtained by dividing the semivariance by the lag variance, while GS+ uses the total variance. While the function obtained in GS+ is only a matter of rescaling variograms, allowing so various variograms to be compared, those proposed in Surfer have the same pupose as the local, pairwise and/or general relative variograms (see Isaaks & Srivastava, page 163-170), that is to reduce the influence of local means. Interestingly enough, one may note that very few software propose relative variograms while I, very personally, consider these functions as essential for detecting spatial structures of many environmental variables.

      I have thus here two questions about the use of standardized/relative variogram:

      1) What is the correct terminology or definition for standardized variograms?  (I personally do not like very much the use of "standardized" when the standardisation is only applied to each lag...)

      2) The general relative variogram (lag divided by the mean of the lag) has properties that are very similar to the "standardized" variogram (lag divided by the variance of the lag) but both functions differ. How shall one decide what to use and what are the relative properties of these functions?

      Thank you in advance for any feedback.

      Gregoire

      PS: a few points here good be added to Tom Mueller's FAQ on Geostatistical Software Conventions.

      __________________________________________
      Gregoire Dubois (Ph.D.)
      JRC - European Commission
      IES - Emissions and Health Unit
      Radioactivity Environmental Monitoring group
      TP 441, Via Fermi 1
      21020 Ispra (VA)
      ITALY
       
      Tel. +39 (0)332 78 6360
      Fax. +39 (0)332 78 5466
      Email: gregoire.dubois@...
      WWW: http://www.ai-geostats.org
      WWW: http://rem.jrc.cec.eu.int
       
      "The views expressed are purely those of the writer and may not in any circumstances be regarded as stating an official position of the European Commission."

    • Isobel Clark
      Gregoire Michel David coined the term relative semi-variogram back in the 70s for what I think you mean by general relative -- that is, each lag is divided
      Message 2 of 3 , Apr 5 12:55 PM
        Gregoire

        Michel David coined the term "relative semi-variogram"
        back in the 70s for what I think you mean by general
        relative -- that is, each lag is divided by the square
        of the mean of the samples used at that lag.

        Gary Raymond proposed the pairwise relative soon
        after. I used the type you are describing where the
        whole semi-variogram is divided by the same
        mean-squared in my 1979 paper (Does Geostatistics
        Work) because I was analysing a line of samples where
        all samples are used at every lag.

        The term "standardised" in general statistics usually
        means dividing through by the variance or standard
        deviation (not a mean). This is the first time I have
        seen it in context with a semi-variogram. Seen with no
        other information, I would have taken this to imply
        standardised to total sill of 1. This would mean
        dividing by the variance, not the mean-squared.

        Relative semi-variograms help you avoid the
        proportional effect if you are trying to calculate a
        semi-variogram on positively skewed data. Noel Cressie
        wrote a paper in Mathematical Geology (early 90s?)
        which showed that the David relative semi-variogram
        was topologically equivalent to using logarithms. You
        data does not have to be lognormal to do this.

        Computationally, taking logarithms is faster and more
        stable than relative semi-variograms. Probably why
        most people don't bother. Gary Raymond provides
        software for the pair-wise and Geostat Systems will
        have relative semi-variograms. Don't know of any free
        stuff.

        Isobel
        http://geoecosse.bizland.com
      • Pierre Goovaerts
        Hi Gregoire, I agree with you regarding the merits of the standardized semivariogram as implemented in variowin software. In one of my last studies, the
        Message 3 of 3 , Apr 5 2:50 PM
          Hi Gregoire,

          I agree with you regarding the merits of the standardized semivariogram as implemented
          in variowin software. In one of my last studies, the rescaling by the lag variance helped
          correcting the preferential sampling of wells with high arsenic levels, leading to a
          susbtantial decrease in random fluctuations of the experimental semivariograms.
          While the general relative semivariogram approximates the lag variance by the square
          of the lag mean, the standardized semivariogram uses the actual lag variance, hence
          makes less assumptions.
          Regarding the terminology, I guess we should used a term like "lag-standardized"
          to distinguish the global and lag-specific standardization or rescaling of semivariogram
          values.

          Cheers,

          Pierre



          -----Original Message-----
          From: Gregoire Dubois [mailto:gregoire.dubois@...]
          Sent: Tue 4/5/2005 9:48 AM
          To: ai-geostats@...
          Cc: mueller@...
          Subject: [ai-geostats] Definition of standardize variograms



          Dear list,

          While playing around with different software, I encounter different definitions for standardized variograms.

          Surfer (which is using the terminology of Variowin), uses the term "standardized semivariogram" for variograms obtained by dividing the semivariance by the lag variance, while GS+ uses the total variance. While the function obtained in GS+ is only a matter of rescaling variograms, allowing so various variograms to be compared, those proposed in Surfer have the same pupose as the local, pairwise and/or general relative variograms (see Isaaks & Srivastava, page 163-170), that is to reduce the influence of local means. Interestingly enough, one may note that very few software propose relative variograms while I, very personally, consider these functions as essential for detecting spatial structures of many environmental variables.

          I have thus here two questions about the use of standardized/relative variogram:

          1) What is the correct terminology or definition for standardized variograms? (I personally do not like very much the use of "standardized" when the standardisation is only applied to each lag...)

          2) The general relative variogram (lag divided by the mean of the lag) has properties that are very similar to the "standardized" variogram (lag divided by the variance of the lag) but both functions differ. How shall one decide what to use and what are the relative properties of these functions?

          Thank you in advance for any feedback.

          Gregoire

          PS: a few points here good be added to Tom Mueller's FAQ on Geostatistical Software Conventions.

          __________________________________________
          Gregoire Dubois (Ph.D.)
          JRC - European Commission
          IES - Emissions and Health Unit
          Radioactivity Environmental Monitoring group
          TP 441, Via Fermi 1
          21020 Ispra (VA)
          ITALY

          Tel. +39 (0)332 78 6360
          Fax. +39 (0)332 78 5466
          Email: gregoire.dubois@... <mailto:gregoire.dubois@...>
          WWW: http://www.ai-geostats.org <http://www.ai-geostats.org>
          WWW: http://rem.jrc.cec.eu.int <http://rem.jrc.cec.eu.int>

          "The views expressed are purely those of the writer and may not in any circumstances be regarded as stating an official position of the European Commission."
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