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

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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
Message 1 of 1 , Jul 25, 2000
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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