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Re: AI-GEOSTATS: AI-GEOSTATS

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  • Pierre Goovaerts
    Hi, One of the primary objectives of stochastic simulation is to reproduce patterns of spatial variability and in sequential simulation it is ensured by using
    Message 1 of 6 , Nov 30, 2000
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      Hi,

      One of the primary objectives of stochastic simulation
      is to reproduce patterns of spatial variability and
      in sequential simulation it is ensured by using
      previously simulated values to derive probability
      distributions to be sampled randomly. As a consequence, you
      should make sure that the number of selected points
      (data and simulated values) and the size of the search window
      is large enough to allow one to incorporate information
      up to the range of spatial correlation.
      Of course, this may become impractical as the number of simulated
      values increases, hence the concept of multiple-grid simulation
      implemented in the new version of Gslib and that I strongly
      recommend to use.

      Here is the description that I give in my book, page 379.

      "The use of a search neighborhood limits reproduction of the input
      covariance model to the radius of that neighborhood. Another obstacle
      to reproduction of long-range structures is the screening of distant
      data by too many data closer to the location being simulated. The
      multiple-grid concept (G\'omez-Hern\'andez, 1991; Tran, 1994) allows
      one to reproduce long-range correlation structures without having to
      consider large search neighborhoods with too many conditioning data.
      For example, a two-step simulation of a square grid 500X500
      could proceed as follows:

      1. The attribute values are first simulated on a coarse grid (e.g.,
      25x25) using a large search neighborhood so as to reproduce
      long-range correlation structures. Because the grid is coarse, each
      neighborhood contains few data, which reduces the screening effect.

      2. Once the coarse grid has been completed, the simulation continues
      on the finer grid 500X500 using a smaller search neighborhood
      so as to reproduce short-range correlation structures. The
      previously simulated values on the coarse grid are
      used as data for the simulation on the fine grid.

      A random path is followed within each grid.

      The procedure can be generalized to any number of intermediate grids;
      this number depends on the number of structures with different ranges
      final grid spacing.

      Pierre
      <><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><>

      ________ ________
      | \ / | Pierre Goovaerts
      |_ \ / _| Assistant professor
      __|________\/________|__ Dept of Civil & Environmental Engineering
      | | The University of Michigan
      | M I C H I G A N | EWRE Building, Room 117
      |________________________| Ann Arbor, Michigan, 48109-2125, U.S.A
      _| |_\ /_| |_
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      On Tue, 21 Nov 2000, WARR Benjamin wrote:

      > Hi All,
      >
      > When performing SIS we have a choice of the max. no. of data nodes and
      > simulated nodes to use . Is there a general rule defining the number of
      > simulated nodes ? A ratio between the two, beyond which we are really
      > risking artefact creation ? Or any work which highlights the effect of
      > using either a dense set of simulated nodes as opposed to a sparse set. I
      > have thought about the issue and other than an effect on the time taken to
      > simulate the full domain I can't see why a choice of the number of simulated
      > nodes will alter the realisations to a great extent. Any conficting
      > thoughts.
      >
      > Benjamin Warr
      > Research Associate to Prof. Ayres,
      > PhD Student of Geostatistics for Natural Resource Evaluation at Reading
      > University, Soil Science.
      >
      > Postal Address:
      > Centre for the Management of Environmental Resources (CMER)
      > INSEAD
      > Boulevard de Constance,
      > 77305 Fontainebleau Cedex,
      > France
      >
      > Tel: 33 (0)1 60 72 40 00 ext. 4926
      > Fax: 33 (0)1 60 74 55 64
      > e-mail: benjamin.warr@...
      >
      >



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    • Isobel Clark
      ... Sounds really fast, which is an advantage but is t this outweighed by the fact that you have to follow the same path every time? I understood from the
      Message 2 of 6 , Dec 1, 2000
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        > Multiple simulations following the same path,
        > co-simulation and
        > indicator simulations are all implemented.
        Sounds really fast, which is an advantage but is't
        this outweighed by the fact that you have to follow
        the same path every time? I understood from the
        literature that one should vary the path every time
        for a valid siulation.

        Since we are all putting out adverts here, maybe I
        should mention that EcoSSe does sequential gaussian
        simulation and that we speed up the simulation by (a)
        dividing the whole region into subregions and
        randomising within the subregion and (b) writing out
        intermediate grid files for "re-input" for people who
        want to get progressively finer. It's not free, of
        course, ($US1,000) but it is constantly being updated
        to include features the users suggest.

        SGS is not in the demo, but you can check it out
        anyway at
        http://uk.geocities.com/drisobelclark/Ecosse_download.html

        We'll be covering SGS etc in Volume 2, Practical
        Geostatistics 2001 but don't hold your breath as it is
        likely to be mid-year before we get it finished. In
        the meantime, look out for the 350 page "Practical
        eostatistocs 2000: Answers to the Exercises" due out
        this month. Much much more than just "Q1:
        42"..........

        Isobel Clark



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      • Edzer J. Pebesma
        ... A far superior implementation of the concept of multiple grid simulation is found in gstat, found at http://www.geog.uu.nl/gstat/ Suppose you want to
        Message 3 of 6 , Dec 1, 2000
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          Pierre Goovaerts wrote:
          >
          > The procedure can be generalized to any number of intermediate grids;
          > this number depends on the number of structures with different ranges
          > [and] final grid spacing.
          >

          A far superior implementation of the concept of multiple grid simulation
          is found in gstat, found at http://www.geog.uu.nl/gstat/

          Suppose you want to simulate a field with 100 x 100 cells. Gstat then
          starts with the coarsest 2-power grid, which has a cell spacing of
          64 x 64; this grid is placed randomly in the field. After simulating
          these four (or one) cells, following a random path, the grid is refined
          to a grid with 32 cell spacing; after this grid is simulated a 16 cell
          spaced grid is followed, ... etc...; after the grid with a 2-cell
          spacing is followed, the remaining cells are simulated on the
          original field with 1-cell spacing.

          If this sounds confusing to you, see the figure on
          http://www.geog.uu.nl/gstat/manual/node9.html

          Now how can this be done efficiently without adjusting the neighbourhood
          size after each grid refinement? Gstat uses a very efficient
          neighbourhood search algorithm (based on quadtrees, see
          http://www.geog.uu.nl/gstat/manual/node8.html and for the algorithm
          http://www.cs.umd.edu/~brabec/quadtree/index.html : Bucket PR Quadtree)
          that does not call for a neighbourhood size in terms of spatial
          distance, but only in terms of number of nearest points.
          This algorithm selects the nearest n observations at the
          start of the simulations (when data are very sparse) approximately
          as fast as it does at the end, when simulated data are abundant and
          dense.

          The advantages of this algorithm are twofold:
          - no worries about the number of grids to simulate and their densities,
          as this is done recursively;
          - no worries about how to decrease the neighbourhood search radius and
          speed of neighbourhood selection, only define the maximum number of
          nearest points in the neighbourhood.

          Multiple simulations following the same path, co-simulation and
          indicator simulations are all implemented.

          Gstat is GPL'd, so anyone can copy this stuff, to GSLIB or whatever.
          --
          Edzer

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        • Edzer J. Pebesma
          ... For maximum pure-ness: yes. (But then we should also use global neigbourhoods all the time :-) You can of course do this, if you ve got the time and
          Message 4 of 6 , Dec 1, 2000
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            Isobel Clark wrote:
            >
            > > Multiple simulations following the same path,

            > Sounds really fast, which is an advantage but is't
            > this outweighed by the fact that you have to follow
            > the same path every time? I understood from the
            > literature that one should vary the path every time
            > for a valid siulation.

            For maximum pure-ness: yes. (But then we should also use
            global neigbourhoods all the time :-)

            You can of course do this, if you've got the time and
            computing power.

            When you're doing a large Monte Carlo experiment using
            simulated random fields, and the simulation time of these fields
            is a crucial issue, the choice may be between using a
            small sample (of say 100) more `pure' fields versus a large
            sample (of say 1000) slightly correlated fields. In such a
            case, I would probably vote for the second option. [I did a
            little benchmark on 1000 simulations for 3000 cells; the gain
            in speed was about a factor 10 using a single random path.
            This factor will be more for larger fields.]

            Except for the theoretical correlation induced by following
            a single random path, has anyone ever done some computation on how
            large this correlation is in practice?
            --
            Edzer

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