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GEOSTATS: Re: Variogram modeling-Summary

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  • Mohammad J Abedini
    With apology for the late summary. I was waiting to clear up some part of question 3 in my mind but it seems it requires more thought. A big thanks should go
    Message 1 of 1 , Jan 17, 1997
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      With apology for the late summary. I was waiting to clear up some part of
      question 3 in my mind but it seems it requires more thought. A big thanks
      should go to the following persons for taking their time and providing the
      response.

      Andrew Gill <agill@...>
      Christian Bauer <cbauer@...-aachen.de>
      Roberto Bruno <scienmin2@...>
      Geoffrey HENEBRY <henebry@...>
      roswell marjorie <roswell@...>
      Dubois Gregoire <gregoire.dubois@...>
      John Kern <johnkern@...>
      Paulo Lopes B. Paraizo <pparaizo@...>

      As some of the responses raised some question along with their comments, I
      will try to comment on their comment in a separate message. Needless to say,
      Some directs their response to my private folder so that I found it useful
      for everybody to be aware of the kind of response that I received.

      *****************************************************************************
      From: Andrew Gill <agill@...>

      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??? **
      ******************************************************************************
      From: Christian Bauer <cbauer@...-aachen.de>

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

      I am sorry, but I just have an idea to answer this question because
      it appeared in my studies as well. This phenomen is mentioned as 'Whole-
      effect' in my german literature. It means that the similarity in space
      increases with distance. e.g take a two-river-valley:
      ..................................................................
      ____________ __________________ ______________
      \___________/ \_____/

      Thie is a profile with sample points above. Those pairs wich cover
      the plateaus and river-beds will be more similar then thos covering
      the bankment. This will lead to smaller variogram-values after the sill is
      reached. Hope that helps,
      ***************************************************************************
      From: Roberto Bruno <scienmin2@...>

      > 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.
      *******************************************************************************
      From: Geoffrey HENEBRY <henebry@...>

      > 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.
      ******************************************************************************
      From: roswell marjorie <roswell@...>

      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.
      ***************************************************************************
      From: Dubois Gregoire <gregoire.dubois@...>
      >
      > 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.
      ****************************************************************************
      From: John Kern <johnkern@...>

      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
      *****************************************************************************
      From: "Paulo Lopes B. Paraizo" <pparaizo@...>

      >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?

      I agree with A. Gill here. There is no discrepancy in this calculations.
      The Gslib models the zonal anisotropy (page 25)by a geometric one with a
      very big anisotropy ratio, and that will make the two directional variograms
      on figure II.6 get to the same sill in a very large distance.
      >
      >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.
      >

      I dont know if it is exactly your point, but as i understand, the book is
      mentioning the geometrical anisotropy, which assumes that the only difference
      in the phenomena is the range for different directions. This hypotesis should
      be assumed for your case, since you want to model it as a geometrical aniso
      tropy. If you deteriorate your goodness of fit, maybe you could try a more
      complex model, like mixing geometrical and zonal anisotropy.


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


      I think you have already had enough. This is non-stationarity, and hole-effects
      is one kind of non-stationarity.

      >
      >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 dont know







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