Loading ...
Sorry, an error occurred while loading the content.

Re: [ai-geostats] Fractals & Semivariance

Expand Messages
  • Syed Abdul Rahman Shibli
    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 1 of 4 , Jul 19, 2004
    • 0 Attachment
      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
    Your message has been successfully submitted and would be delivered to recipients shortly.