Accuracy of the spectral method in estimating fractal/spectral parameters for self-affine roughness profiles

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Abstract

Accurate quantification of roughness is important in modeling strength, deformability and fluid flow behaviors of rock joints. Self-affine fractals seem to have the potential to represent rock joint roughness profiles. Both stationary and non-stationary fractional Brownian profiles (self-affine profiles) with known values of fractal dimension, D, and input standard deviation, a, were generated at different generation levels. A few smoothing techniques were used with the spectral method to calculate D, and two other spectral parameters K, (a proportionality constant; see the text for the details) and CD (the cross-over dimension of the profile) for the fractional Brownian profiles. The effects of smoothing, generation level of the profile, seed value used in the generation, non-stationarity of the profile and a on the accuracy of the calculated D were examined using the spectral method. The following conclusions were obtained: (a) To obtain accurate estimates for D, Ks and CD, it seems necessary to have at least W data points per unit length for a profile having a total length of 100 units (this is equivalent to a generation level of 10). (b) For accurate estimation of D, K, and CD, the non-stationarity of profiles should be removed, if it exists, (c) The parameter combinations D and K, (which has the potential to capture scale effects), and D and CD are recommended for quantification of stationary roughness; in addition, extra parameters are required to quantify the non-stationarity. (d) Both the Parzen and Banning smoothing techniques seem suitable to use with the spectral technique to obtain accurate estimates for D, Ks and CD. (e) To obtain accurate estimates for D, Ks, and CD, it is necessary to use a suitable bandwidth for the Parzen window and a suitable number of interations for the Hanning window; this paper provides guidelines to choose these suitable values, (f) Seed value has negligible effect on the accuracy of estimated D, Ks and CD.

Original languageEnglish (US)
Pages (from-to)789-804
Number of pages16
JournalInternational Journal of Rock Mechanics and Mining Sciences
Volume34
Issue number5
StatePublished - 1997

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smoothing
Fractals
roughness
Surface roughness
Seed
Rocks
seed
scale effect
Fractal dimension
Formability
rock
fluid flow
Flow of fluids
Bandwidth
modeling
method
parameter
effect

ASJC Scopus subject areas

  • Geotechnical Engineering and Engineering Geology
  • Earth and Planetary Sciences(all)
  • Engineering(all)
  • Environmental Science(all)

Cite this

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title = "Accuracy of the spectral method in estimating fractal/spectral parameters for self-affine roughness profiles",
abstract = "Accurate quantification of roughness is important in modeling strength, deformability and fluid flow behaviors of rock joints. Self-affine fractals seem to have the potential to represent rock joint roughness profiles. Both stationary and non-stationary fractional Brownian profiles (self-affine profiles) with known values of fractal dimension, D, and input standard deviation, a, were generated at different generation levels. A few smoothing techniques were used with the spectral method to calculate D, and two other spectral parameters K, (a proportionality constant; see the text for the details) and CD (the cross-over dimension of the profile) for the fractional Brownian profiles. The effects of smoothing, generation level of the profile, seed value used in the generation, non-stationarity of the profile and a on the accuracy of the calculated D were examined using the spectral method. The following conclusions were obtained: (a) To obtain accurate estimates for D, Ks and CD, it seems necessary to have at least W data points per unit length for a profile having a total length of 100 units (this is equivalent to a generation level of 10). (b) For accurate estimation of D, K, and CD, the non-stationarity of profiles should be removed, if it exists, (c) The parameter combinations D and K, (which has the potential to capture scale effects), and D and CD are recommended for quantification of stationary roughness; in addition, extra parameters are required to quantify the non-stationarity. (d) Both the Parzen and Banning smoothing techniques seem suitable to use with the spectral technique to obtain accurate estimates for D, Ks and CD. (e) To obtain accurate estimates for D, Ks, and CD, it is necessary to use a suitable bandwidth for the Parzen window and a suitable number of interations for the Hanning window; this paper provides guidelines to choose these suitable values, (f) Seed value has negligible effect on the accuracy of estimated D, Ks and CD.",
author = "T. Shirono and Pinnaduwa Kulatilake",
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T1 - Accuracy of the spectral method in estimating fractal/spectral parameters for self-affine roughness profiles

AU - Shirono, T.

AU - Kulatilake, Pinnaduwa

PY - 1997

Y1 - 1997

N2 - Accurate quantification of roughness is important in modeling strength, deformability and fluid flow behaviors of rock joints. Self-affine fractals seem to have the potential to represent rock joint roughness profiles. Both stationary and non-stationary fractional Brownian profiles (self-affine profiles) with known values of fractal dimension, D, and input standard deviation, a, were generated at different generation levels. A few smoothing techniques were used with the spectral method to calculate D, and two other spectral parameters K, (a proportionality constant; see the text for the details) and CD (the cross-over dimension of the profile) for the fractional Brownian profiles. The effects of smoothing, generation level of the profile, seed value used in the generation, non-stationarity of the profile and a on the accuracy of the calculated D were examined using the spectral method. The following conclusions were obtained: (a) To obtain accurate estimates for D, Ks and CD, it seems necessary to have at least W data points per unit length for a profile having a total length of 100 units (this is equivalent to a generation level of 10). (b) For accurate estimation of D, K, and CD, the non-stationarity of profiles should be removed, if it exists, (c) The parameter combinations D and K, (which has the potential to capture scale effects), and D and CD are recommended for quantification of stationary roughness; in addition, extra parameters are required to quantify the non-stationarity. (d) Both the Parzen and Banning smoothing techniques seem suitable to use with the spectral technique to obtain accurate estimates for D, Ks and CD. (e) To obtain accurate estimates for D, Ks, and CD, it is necessary to use a suitable bandwidth for the Parzen window and a suitable number of interations for the Hanning window; this paper provides guidelines to choose these suitable values, (f) Seed value has negligible effect on the accuracy of estimated D, Ks and CD.

AB - Accurate quantification of roughness is important in modeling strength, deformability and fluid flow behaviors of rock joints. Self-affine fractals seem to have the potential to represent rock joint roughness profiles. Both stationary and non-stationary fractional Brownian profiles (self-affine profiles) with known values of fractal dimension, D, and input standard deviation, a, were generated at different generation levels. A few smoothing techniques were used with the spectral method to calculate D, and two other spectral parameters K, (a proportionality constant; see the text for the details) and CD (the cross-over dimension of the profile) for the fractional Brownian profiles. The effects of smoothing, generation level of the profile, seed value used in the generation, non-stationarity of the profile and a on the accuracy of the calculated D were examined using the spectral method. The following conclusions were obtained: (a) To obtain accurate estimates for D, Ks and CD, it seems necessary to have at least W data points per unit length for a profile having a total length of 100 units (this is equivalent to a generation level of 10). (b) For accurate estimation of D, K, and CD, the non-stationarity of profiles should be removed, if it exists, (c) The parameter combinations D and K, (which has the potential to capture scale effects), and D and CD are recommended for quantification of stationary roughness; in addition, extra parameters are required to quantify the non-stationarity. (d) Both the Parzen and Banning smoothing techniques seem suitable to use with the spectral technique to obtain accurate estimates for D, Ks and CD. (e) To obtain accurate estimates for D, Ks, and CD, it is necessary to use a suitable bandwidth for the Parzen window and a suitable number of interations for the Hanning window; this paper provides guidelines to choose these suitable values, (f) Seed value has negligible effect on the accuracy of estimated D, Ks and CD.

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