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[ai-geostats] Automatic variogram modeling software

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  • Edward Isaaks
    Re automatic semi variogram modeling. Here are a few things to be aware of when looking for automatic semi variogram modeling software. 1. Generally, one is
    Message 1 of 1 , Mar 1, 2006
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      Re automatic semi variogram modeling.


      Here are a few things to be aware of when looking for automatic semi
      variogram modeling software.

      1. Generally, one is faced with the problem of modeling several directional
      sample variograms calculated from a particular data set.
      2. However, modeling the directional sample variograms from a two
      dimensional data set is relatively simple. If a 2D anisotropy is suspected,
      one should calculate multiple directional sample variogram on azimuths of at
      least 30 degree intervals. Four directional sample variograms at 90 degree
      intervals are generally insufficient for accurately modeling an anisotropy.
      The task at hand is to "fit" a model to all of the directional sample
      variograms. Thus, any automatic fitting software that only models one
      directional sample variogram at a time is likely not very useful.
      3. The problem is considerably more difficult when working with 3
      dimensional data. For example, experience suggests that in general, one
      cannot obtain a reasonable sample of the inherent 3 dimensional anisotropic
      spatial continuity with less than 19 directional sample variograms, e.g. 30
      degree increments on the azimuth combined with 30 degree increments on dip.
      (Recall, that the turning bands algorithm is based on the "icosahedron
      approximation" which requires a minimum of 15 lines to simulate an isotropic
      covariance in 3D space). The task at hand is to "fit" a model to all 19
      directional sample variograms. To appreciate the difficulty, consider
      fitting a simple variogram model with two structures to the 19 directional
      sample variograms. One must determine values for 15 parameters, e.g.,
      1. The nugget.
      2. The coefficient for the first structure.
      3. The coefficient for the second structure.
      4. The range along the major axis of the anisotropy model associated with
      the first structure.
      5. The range along the semi-major axis of the anisotropy model associated
      with the first structure.
      6. The range along the minor axis of the anisotropy model associated with
      first structure.
      7. The range along the major axis of the anisotropy model associated with
      the second structure.
      8. The range along the semi-major axis of the anisotropy model associated
      with the second structure.
      9. The range along the minor axis of the anisotropy model associated with
      second structure.
      10. The rotation angle around the Z axis of the anisotropy model for the
      first structure.
      11. The rotation angle around the rotated X axis of the anisotropy model for
      the first structure.
      12. The rotation angle around the rotated Y axis of the anisotropy model for
      the first structure.
      13. The rotation angle around the Z axis of the anisotropy model for the
      second structure.
      14. The rotation angle around the rotated X axis of the anisotropy model for
      the second structure.
      15. The rotation angle around the rotated Y axis of the anisotropy model for
      the second structure.

      Thus, although the problem can be stated quite simply in terms of least
      squares, e.g.;

      Find the set of 15 parameters that;

      Minimize [ Sum_i w_i * (VariogramModel(15 parameters, lag_i) -
      sample_variogram_point_i)^2 ] where "Sum_i" is the sum over all sample
      variogram points from all 19 directional sample variograms. Note that w_i
      may be a set of weights such as "number of pairs" etc.

      The solution is not straightforward. However, the software product
      "SAGE2001" solves the problem above quite quickly by determining the optimum
      set of 15 parameters. The result is a weighted least squares fit to all
      directional sample variogram points simultaneously.

      SAGE2001 is used throughout the mining industry by many companies including
      Newmont, Barrick, Placer, AMEC, and so on.

      Unfortunately, SAGE2001 development costs were steep. Thus, it is not
      freeware or shareware.
      But, SAGE2001 can be downloaded at www.isaaks.com for free and is fully
      functional with no restrictions for 20 days. So if you have a difficult
      modeling problem - feel free to use it.
      Edward Isaaks
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