Hello
Syed,
I
was hoping a reply from you :)
I
didn't think about the problematic of anisotropy and the potential use
of ratios of fractal dimensions. It might be worth some further investigation.
The
physical meaning of fractals derived from directional variograms is tricky
indeed.
I was
wondering if the average of all these fractal dimensions would be formally equal
to the fractal dimension derived from omnidirectional variogram.
My
first guess would be yes, but this would depend on the angular tolerance of
the directional variograms. And would the average value of the fractal dimension
have any reasonable physical meaning?
Any
experience with this?
Thanks
again for the kind help.
Gregoire
-----Original Message-----
From: Syed Abdul Rahman Shibli [mailto:[EMAIL PROTECTED]
Sent: 16 July 2004 19:23
To: Gregoire Dubois
Cc: [EMAIL PROTECTED]
Subject: Re: [ai-geostats] Fractals & Semivariance
Not sure how anisotropic "fractal" spatial correlation models would fit in the whole scheme of things. You're essentially assuming a power law model (Brownian motion) to model the spatial correlation, which implicitly assumes a phenomena with an infinite capacity for dispersion, i.e. no range. The ratio of two fractal dimensions is not necessarily the same as the ratio of two ranges in the two directions of maximum and minimum continuity, which is the traditional measure of "anisotropy".
However, practically speaking you can still calculate experimental variograms for two, three, or four separate directions and derive the log-log estimate of the fractal dimension from these separate variograms. I wouldn't know what this will physically mean, except to say that I have a phenomena with different capacities for dispersion in different directions.
Cheers
Syed
Dear all,
at
http://www.umanitoba.ca/faculties/science/botany/labs/ecology/fractals/measuring.html
one can read the following
"The fractal dimension is estimated separately for each profile from the log-log plot of cell count against step size (D = 2 - slope, where 1 <= D <= 2). The average of these values plus one provides an estimate of the surface fractal dimension."
Burrough's method (using the slope of the log-log plot of the semivariogram to calculate the fractal dimension of 1 dimensional transect or profile) could thus be extended to a 2 D case (a surface). Has anyone references discussing the use of Burrough's method when applied to a 2 D case?
Unless one considers the investigated phenomenon completely isotropic, averaging the fractal dimensions derived from the slopes of directional log-log semivariograms may not provide any useful/reliable information.
Has someone on the list any experience with this kind of issue?
Thanks very much for any help.
Best regards,
Gregoire
PS: I know there are other techniques to calculate the fractal dimension of a surface but I'm only interested in those involving the computation of the semivariance.
__________________________________________
Gregoire Dubois (Ph.D.)
JRC - European Commission
IES - Emissions and Health Unit
Radioactivity Environmental Monitoring group
TP 441, Via Fermi 1
21020 Ispra (VA)
ITALY
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