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

[ai-geostats] Fractals & Semivariance

Expand Messages
  • Gregoire Dubois
    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
    • 0 Attachment
      Message
      Dear all,
       
      at
       
      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
       
       
    • Syed Abdul Rahman Shibli
      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
      • 0 Attachment
        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
      • Gregoire Dubois
        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
        Message 3 of 4 , Jul 19, 2004
        • 0 Attachment
          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

          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
        • 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 4 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.