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GEOSTATS: Variogram Modeling

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  • Mohammad J Abedini
    Hi everybody That would be a great help if experts out there shed some light on me regarding the following questions. Basically, I am trying to use the
    Message 1 of 7 , Jan 8, 1997
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      Hi everybody

      That would be a great help if experts out there shed some light on me
      regarding the following questions.

      Basically, I am trying to use the available tools such as GSLIB and its
      manual and ... to model the variogram for my continuous elevation data
      sets. Either I am reading too much from those tools or missing an
      important link.

      1. On page 377 of the book written by Isaaks and Srivastava, it reads:

      "The most useful guideline for choosing weight coefficients is to remember
      that their sum must equal the sill of the sample variogram"

      The above condition is not satisfied by neither of the variogram shown on
      page 27(Figure II.5) and page 28 (Figure II.6) of GSLIB manual. If these
      variogram are generic, that is fine otherwise what is the cause of the
      discrepancy?

      2. On page 383 of the same book at the bottom of the page, it reads:

      "To Summarize, the geometric anisotropy requires some foresight in
      modeling the directional sample variograms. All the directional variogram
      models must have identical sill values. Each nested ..........
      with the same coefficient."

      By the last sentence, I am assuming they mean the contribution coefficient
      of each structure (i.e., cc value). First of all, why that should be the
      case?
      If cc should have the same value, does it mean that parameter optimization
      of nested structure for the second direction should be subjected to the
      values obtained in the other direction. Let me give an example.
      Suppose we identify the axis of anisotropy to be x and y axis. In the
      x-direction, we realized that a combination of exponential and Gaussian
      structure will provide the best fit to sample variogram in that direction.
      Clearly enough, we have four parameters to optimize in that direction.
      As soon as we switch to y-direction, the number of parameters will drop to
      two and this less degree of freedom will detoriorate the goodness of fit
      dramatically in that direction while relaxing the equality of cc from one
      direction to another will give rise to a comparable goodness of fit. Am I
      on the right track or some normalization of range should be made at some
      stage. Besides all these things, I cannot understand why the ranges for
      corresponding structure should be divided by each other to define "anis"
      Parameter.

      3- If variogram value start to decrease after reaching the sill, what is
      the physical meaning of that?

      4- Is there any ERRATA being written for GSLIB manual? I need to confirm
      some of the mistakes that I encounter in using the manual.

      With many thanks for your time and help.


      Best of luck.
      Abedini
      \|||/
      (o o)
      +=================oOO==(_)==OOo=================+
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      |_____| |_____|
      |__|__| Mohammad J. Abedini |__|__|
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      |_____| University of Guelph |_____|
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      The world is your exercise-book, the pages on which you do your sums.
      You are also free to write nonsense, or lies, or to tear the pages.

      Richard Bach





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    • Andrew Gill
      My 2cents worth on some of the following posted questions. ... My calculation of the sum of the weight coefficients for Fig 11.5 is 0.22+0.53+0.25=1.00 for
      Message 2 of 7 , Jan 8, 1997
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        My 2cents worth on some of the following posted questions.

        >
        > 1. On page 377 of the book written by Isaaks and Srivastava, it reads:
        >
        > "The most useful guideline for choosing weight coefficients is to remember
        > that their sum must equal the sill of the sample variogram"
        >
        > The above condition is not satisfied by neither of the variogram shown on
        > page 27(Figure II.5) and page 28 (Figure II.6) of GSLIB manual. If these
        > variogram are generic, that is fine otherwise what is the cause of the
        > discrepancy?
        >

        My calculation of the sum of the weight coefficients for Fig 11.5 is
        0.22+0.53+0.25=1.00 for both the NS and EW directions. This appears
        pretty close to the sill values.

        For Fig. II.6 i get 0.40+0.40+0.95+0.9=2.65 for both directions. This
        matches OK the higer sill direction. I presume the question is for the
        other which has a sill around 1.7 or 1.8. The text notes that the
        last spherical structure has a range of 80K and does not practically
        contribute. Take away the weight coefficient of 0.90 and viola you
        get 1.75.

        > 2. On page 383 of the same book at the bottom of the page, it reads:
        >
        > "To Summarize, the geometric anisotropy requires some foresight in
        > modeling the directional sample variograms. All the directional variogram
        > models must have identical sill values. Each nested ..........
        > with the same coefficient."
        >
        > By the last sentence, I am assuming they mean the contribution coefficient
        > of each structure (i.e., cc value). First of all, why that should be the
        > case? etc etc etc [snip]
        >

        Your on you own there. I cheat and assume everythings nice and isotropic.
        Bad boy, I know......

        > 3- If variogram value start to decrease after reaching the sill, what is
        > the physical meaning of that?
        >

        Dont know the technical reason, but I found a similar thing when calculating
        a variogram from output from a numerical model. My dip was due to the
        boundary conditions, same constant head at two ends of an aquifer, so that
        head values at suitably large distances were magically similar and viola
        variogram value goes down.

        > 4- Is there any ERRATA being written for GSLIB manual? I need to confirm
        > some of the mistakes that I encounter in using the manual.
        >

        I hear that a new and improved manual [2nd Edn] is due to be out in January
        by Oxford Uni P for about 45 pounds.

        ** Anyone else heard anything about the release??? **

        > With many thanks for your time and help.
        >
        >
        > Best of luck.
        > Abedini
        > |__|__| Mohammad J. Abedini |__|__|
        > |_____| School of Engineering |_____|
        > |_____| University of Guelph |_____|
        > |_____| Guelph Ont, N1G 2W1 |_____|
        > |__|__| Tel.:(519) 824-4120 ext. 4321 (W) |__|__|
        > |__|__| Tel.:(519) 821-1199 (H) |__|__|
        > |_____| Fax :(519) 836-0227 |_____|
        > |_____| e-mail: mabedini@... |_____|

        Andrew
        --
        --------------------------------------------------------------------------
        Dr Andrew Gill Room:203d

        Department of Applied Mathematics Email: agill@...
        Mathematics Building Phone: 08-3035577
        The University of Adelaide Fax: 08-3033696
        Adelaide, SA, 5005
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      • Geoffrey HENEBRY
        ... Date: Thu, 9 Jan 1997 09:08:12 -0500 (EST) From: Geoffrey HENEBRY To: Geoffrey HENEBRY
        Message 3 of 7 , Jan 9, 1997
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          >From henebry@... Jan 9 09:31:52 1997
          Date: Thu, 9 Jan 1997 09:08:12 -0500 (EST)
          From: Geoffrey HENEBRY <henebry@...>
          To: Geoffrey HENEBRY <henebry@...>
          Subject: Re: GEOSTATS: Variogram Modeling

          On Wed, 8 Jan 1997, Mohammad J Abedini wrote:

          > That would be a great help if experts out there shed some light on me
          > regarding the following questions.
          >
          > 3- If variogram value start to decrease after reaching the sill, what is
          > the physical meaning of that?
          >
          while i may not be a geostatistical expert, i think that this kind
          of behavior indicates spatial nonstationarity. i've seen as much
          frequently in the analysis of satellite imagery.

          tchau, geoff

          ---------------------------------------------------------------
          Geoffrey M. Henebry Assistant Professor
          Dept. of Biological Sciences (201) 648-5053 (voice)
          Rutgers University (201) 648-5518 (fax)
          Newark, NJ 07102 USA henebry@...


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        • roswell marjorie
          While we re at it, are their any errata for the Applied Geostatistics book by Isaaks and Srivastava? I m trying to tackle it on my own, but I was pretty
          Message 4 of 7 , Jan 9, 1997
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            While we're at it, are their any errata for the Applied Geostatistics book
            by Isaaks and Srivastava? I'm trying to tackle it on my own, but I was
            pretty certain of at least one error.

            (I tried reading it on my Carribean trip: I succeeded in completing only
            the review chapters...) (I'll get through it eventually. Does anyone give
            you any "credit" if you learn this stuff on your own?)

            Margie


            On Wed, 8 Jan 1997, Mohammad J Abedini wrote:

            > of the book written by Isaaks and Srivastava, it reads:
            >
            > 4- Is there any ERRATA being written for GSLIB manual? I need to confirm
            > some of the mistakes that I encounter in using the manual.



            _________________________________________________________
            Marjorie Roswell, Spatial Analyst
            UMBC Center for Health Program Development and Management
            1000 Hilltop Circle Fx: (410)455-6850
            Baltimore, MD 21250 E: roswell@...
            Ph: (410)455-6802 http://umbc.edu/~roswell/mipage.html
            _________________________________________________________



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          • Dubois Gregoire
            ... So as mentioned by Prof. Roberto Bruno, this decrease is often refered as hole effect . Here is an attempt to show a semivariogram with this hole
            Message 5 of 7 , Jan 9, 1997
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              Mohammad J Abedini wrote:
              >

              >
              > 3- If variogram value start to decrease after reaching the sill, what is
              > the physical meaning of that?
              >


              So as mentioned by Prof. Roberto Bruno, this decrease is
              often refered as "hole effect" .

              Here is an attempt to show a semivariogram with this "hole effect"


              (gamma)
              |
              | B
              | + + +
              | A + + C +
              | + + + +
              | +
              |+
              |______________________
              A B C Distance



              This suggests you that the samples separated by the distance C
              are more similar than those separated by the distance B, even if
              the distance C is greather that B. So you are in a non stationnary
              as you have a repetition of your structure (the behaviour in A
              is similar in C)

              This case appears if you have repetitions of the spatial pattern
              you are analysing. As an example, it is often the case when analysing
              chemical components in sedimetary rocks (because of the possible repeated
              periodic layers) or in environmental pollution if the pollutant is
              contaminating soils through rainfall. In radioecology, the field I'm
              involved in, we refer to this phenomenon by "hot spots".

              This is another reason why a semivariogram should be calculated at
              different lags and/or on the ranks in order to try to model first
              the very general spatial pattern of your variable.


              Hope this helps,

              Gregoire


              --
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              Joint Research Centre Fax. 39-332-78.54.66
              Environment Inst. TP 321 Email: gregoire.dubois@...
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            • Roberto Bruno
              ... I beg your pardon because I won t answer your questions (only some comment on the third one), but this is not a problem, surely the list will assure you
              Message 6 of 7 , Jan 9, 1997
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                mabedini@... wrote:
                >
                > Hi everybody
                >
                > That would be a great help if experts out there shed some light on me
                > regarding the following questions.
                >
                > Basically, I am trying to use the available tools such as GSLIB and its
                > manual and ... to model the variogram for my continuous elevation data
                > sets.

                I beg your pardon because I won't answer your questions (only some comment on the third
                one), but this is not a problem, surely the list will assure you many contributions such
                as that of Andrew GILL. But I was thinking when reading your message: ...

                you are dealing with elevation data, didn't you find any problem in assuming the
                stationarity necessary (in a broad sense) for using variograms?
                In my experience (about 15 years of geostat) the usefullness (if not the validity) of
                the stationarity hypothesys depends on the working scale, mainly when dealing with
                elevations. And in practice, when you have to make any cartography for example, you need
                a more general tool, able to treat the non stationary case, which includes the
                stationary one.


                > 3- If variogram value start to decrease after reaching the sill, what is
                > the physical meaning of that?

                You can refer to the classical "hole effect" explained in every good geostat text.
                A comment also on the answer of A.Gill case: make attention of calculating variograms
                for large lags, for example considering points near the boundaries, because there is
                always the theoretical need in modeling the variogram of not overcoming the half of your
                field.

                Best regards.

                Roberto BRUNO
                Dept. of Chemical, Mining and Environmental Engineering
                Faculty of Engineering, Univ. of Bologna "Alma Mater Studiorum"
                Tel.(+39-51) 6443393, Fax (+39-51) 6443392
                E-mail SCIENMIN2@...
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              • John Kern
                Additional comments: When the variogram starts to decrease from some sill value this is an indication of periodicities in the data. If periodic behavior is
                Message 7 of 7 , Jan 9, 1997
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                  Additional comments: When the variogram starts to decrease from some sill
                  value this is an indication of periodicities in the data. If periodic
                  behavior is high frequency, then you see the oscilatory behavior in the
                  variogram associated with a hole effect model. If the periodic behavior is
                  low frequency relative to your study area (sampling window) then it is seen
                  as just a decreasing variogram at long lags. This is a good indicator that
                  there are large scale trends (relative to the sampling window) in the data
                  which could be modeled as linear or quadratic or more complex response
                  surfaces with correlated residuals.


                  On anisotopy, Borgman and Li 1994 Math Geology give a nonparametric
                  technique for estimating the axes of anisotropy. The procedure allows one
                  to then use a linear transformation of the lag domain into a single
                  direction in which the model fitting is conducted. This allows the use of
                  all lag pairs in the model fitting procedure and ensures constant sill in
                  all directions and the same set of nested structures in each direction
                  provided you have enough data to detect multiple structures. In my
                  experience it takes a lot of irregularly spaced data to estimate the shape
                  parameters of a single structure let alone more than one structure. Good
                  luck, John Kern

                  >My 2cents worth on some of the following posted questions.
                  >
                  >>
                  >> 1. On page 377 of the book written by Isaaks and Srivastava, it reads:
                  >>
                  >> "The most useful guideline for choosing weight coefficients is to remember
                  >> that their sum must equal the sill of the sample variogram"
                  >>
                  >> The above condition is not satisfied by neither of the variogram shown on
                  >> page 27(Figure II.5) and page 28 (Figure II.6) of GSLIB manual. If these
                  >> variogram are generic, that is fine otherwise what is the cause of the
                  >> discrepancy?
                  >>
                  >
                  >My calculation of the sum of the weight coefficients for Fig 11.5 is
                  >0.22+0.53+0.25=1.00 for both the NS and EW directions. This appears
                  >pretty close to the sill values.
                  >
                  >For Fig. II.6 i get 0.40+0.40+0.95+0.9=2.65 for both directions. This
                  >matches OK the higer sill direction. I presume the question is for the
                  >other which has a sill around 1.7 or 1.8. The text notes that the
                  >last spherical structure has a range of 80K and does not practically
                  >contribute. Take away the weight coefficient of 0.90 and viola you
                  >get 1.75.
                  >
                  >> 2. On page 383 of the same book at the bottom of the page, it reads:
                  >>
                  >> "To Summarize, the geometric anisotropy requires some foresight in
                  >> modeling the directional sample variograms. All the directional variogram
                  >> models must have identical sill values. Each nested ..........
                  >> with the same coefficient."
                  >>
                  >> By the last sentence, I am assuming they mean the contribution coefficient
                  >> of each structure (i.e., cc value). First of all, why that should be the
                  >> case? etc etc etc [snip]
                  >>
                  >
                  >Your on you own there. I cheat and assume everythings nice and isotropic.
                  >Bad boy, I know......
                  >
                  >> 3- If variogram value start to decrease after reaching the sill, what is
                  >> the physical meaning of that?
                  >>
                  >
                  >Dont know the technical reason, but I found a similar thing when calculating
                  >a variogram from output from a numerical model. My dip was due to the
                  >boundary conditions, same constant head at two ends of an aquifer, so that
                  >head values at suitably large distances were magically similar and viola
                  >variogram value goes down.
                  >
                  >> 4- Is there any ERRATA being written for GSLIB manual? I need to confirm
                  >> some of the mistakes that I encounter in using the manual.
                  >>
                  >
                  >I hear that a new and improved manual [2nd Edn] is due to be out in January
                  >by Oxford Uni P for about 45 pounds.
                  >
                  >** Anyone else heard anything about the release??? **
                  >
                  >> With many thanks for your time and help.
                  >>
                  >>
                  >> Best of luck.
                  >> Abedini
                  >> |__|__| Mohammad J. Abedini |__|__|
                  >> |_____| School of Engineering |_____|
                  >> |_____| University of Guelph |_____|
                  >> |_____| Guelph Ont, N1G 2W1 |_____|
                  >> |__|__| Tel.:(519) 824-4120 ext. 4321 (W) |__|__|
                  >> |__|__| Tel.:(519) 821-1199 (H) |__|__|
                  >> |_____| Fax :(519) 836-0227 |_____|
                  >> |_____| e-mail: mabedini@... |_____|
                  >
                  >Andrew
                  >--
                  >--------------------------------------------------------------------------
                  >Dr Andrew Gill Room:203d
                  >
                  >Department of Applied Mathematics Email: agill@...
                  >Mathematics Building Phone: 08-3035577
                  >The University of Adelaide Fax: 08-3033696
                  >Adelaide, SA, 5005
                  >--------------------------------------------------------------------------
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                  >*As a general service to list users, please remember to post a summary
                  >of any useful responses to your questions.
                  >

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