Message Dear all,atone 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,GregoirePS: 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 CommissionIES - Emissions and Health UnitRadioactivity Environmental Monitoring groupTP 441, Via Fermi 121020 Ispra (VA)ITALYTel. +39 (0)332 78 6360Fax. +39 (0)332 78 5466Email: gregoire.dubois@...- 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

>

> 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 Message 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

SyedDear 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

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 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

> Radioactivity Environmental Monitoring group

> 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