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GEOSTATS: 1) stationary mean, 2) correl. dimension

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  • Peter Bossew
    Dear all out there, I have got 2 (maybe very basic) questions, the first one about the requirement of stationary mean in semivariance analysis, the 2. about
    Message 1 of 1 , Jul 25, 2000
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      Dear all out there,

      I have got 2 (maybe very basic) questions, the first one about the
      requirement of stationary mean in semivariance analysis, the 2. about the
      correlation dimension of fractal analysis.

      1.
      before performing a semivariance analysis, any secular trend must be
      removed from the data in order to meet the requirement of stationarity of
      the mean. Question: how do you decide at which scale fluctuations are
      actually trends ? Stationarity of the empirical data means regional mean =
      ยต(x) = const. over x, but over which range of x the regional mean has to
      be taken ?
      As an example, I add (see zipped attachment <examp1.doc>) the spatial
      distribution of count rates over a monazite (a radioactive mineral)
      containing beach of Brazil. The dots represent the sampling locations. The
      picture has been produced using 'naive' kriging with Surfer Software, i.e.
      using the default settings for the variogram and assuming isotropy, just
      in order to grossly visualize the distribution. I would say that there is
      an obvious trend, represented by the maximum between easting 300 and 450
      m; but is the maximum at ca. 160 m also part of the trend ? Doesn't the
      big maximum in fact consist of 3 maxima at ca. 340, 370 and 410 m,
      respectively, which should be modelled by the trend surface ? (Apart from
      the problem, how to model such a trend structure.)

      2.
      It seems to me that the correlation dimension is quite a useful tool to
      assess the topologic structure of the spatial distribution of a variable;
      or could be if used properly. I use this kind of fractal dimension because
      it is (as I think) the easiest to calculate: D :<=> AM(n(r)) ~ r^D, where
      the left hand side denotes the number of points within distance r from a
      fixed point x, averaged over all points x. D is then easily calculated by
      log regression.
      Now the question: there is always an 'edge effect' to D produced by the
      fact that the sampling area is inevitably limited in space. An infinite
      complete regular quadratic sampling grid, e.g., has D = 2, but the same
      grid with finite extension has D < 2, because the points at the border
      have naturally less neighbours than points within the grid and therefore
      contribute to AM(n(r)) by lowering D and thus inferring a fractal
      patchiness of the structure which is clearly an artefact. For this reason,
      D depends heavily on the overall size (extent) of the grid (number of
      sampling points), regardless of its structure, which makes this quantity
      somewhat questionable, I think. Does somebody know how to deal with this
      problem ?

      Thank you very much & regards, PB

      -------------------------------------------------------------------------------------
      Peter Bossew
      Georg Sigl-Gasse 13/11
      A-1090 Vienna, Austria
      ph. +43-1-3177627
      e-mail: p.bossew@...
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