## [ai-geostats] Fractals & Semivariance

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• Dear all, at http://www.umanitoba.ca/faculties/science/botany/labs/ecology/fractals/m easuring.html one can read the following The fractal dimension is
Message 1 of 4 , Jul 16, 2004
Message
Dear all,

at

"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
TP 441, Via Fermi 1
21020 Ispra (VA)
ITALY

Tel. +39 (0)332 78 6360
Fax. +39 (0)332 78 5466

• 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
Message 2 of 4 , Jul 16, 2004
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
> TP 441, Via Fermi 1
> 21020 Ispra (VA)
> ITALY
>
> Tel. +39 (0)332 78 6360
> Fax. +39 (0)332 78 5466
> Email: gregoire.dubois@...
> WWW: http://www.ai-geostats.org
> WWW: http://rem.jrc.cec.eu.int
>
>
> * By using the ai-geostats mailing list you agree to follow its rules
> ( see http://www.ai-geostats.org/help_ai-geostats.htm )
>
> * To unsubscribe to ai-geostats, send the following in the subject or
> in the body (plain text format) of an email message to sympa@...
>
> Signoff ai-geostats
• Gregoire, To be honest I have never attempted this, although as you said the angular tolerance, bandwidth, and lag tolerance will ultimately determine whether
Message 3 of 4 , Jul 19, 2004
Gregoire,

To be honest I have never attempted this, although as you said the
angular tolerance, bandwidth, and lag tolerance will ultimately
determine whether the directional fractal dimensions can be averaged to
give an "omnidirectional" dimension, D. I would argue that two
directional variograms each with a directional tolerance of about 45
degrees on either side of the azimuth in the two principal directions
would yield an average D similar to an omnidirectional case, but this
will not strictly be true the smaller the tolerances used.

I have used simulated annealing to generate (stochastic) fractal fields
with different dimensions in three directions X, Y, and Z in 3D space,
e.g. assumption of fractional Gussian noise vertically with high Hurst
exponent (persistence) and fractional Brownian motion laterally with
lower Hurst exponent (anti-persistence).

Cheers

Syed

> 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:sshibli@...]
> Sent: 16 July 2004 19:23
> To: Gregoire Dubois
> Cc: ai-geostats@...
> 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
> TP 441, Via Fermi 1
> 21020 Ispra (VA)
> ITALY
>
> Tel. +39 (0)332 78 6360
> Fax. +39 (0)332 78 5466
> Email: gregoire.dubois@...
> WWW: http://www.ai-geostats.org
> WWW: http://rem.jrc.cec.eu.int
>
>
> * By using the ai-geostats mailing list you agree to follow its rules
> ( see http://www.ai-geostats.org/help_ai-geostats.htm )
>
> * To unsubscribe to ai-geostats, send the following in the subject or
> in the body (plain text format) of an email message to sympa@...
>
> Signoff ai-geostats
> * By using the ai-geostats mailing list you agree to follow its rules
> ( see http://www.ai-geostats.org/help_ai-geostats.htm )
>
> * To unsubscribe to ai-geostats, send the following in the subject or
> in the body (plain text format) of an email message to sympa@...
>
> Signoff ai-geostats
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