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, σ, 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 s (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 σ 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, K s and CD , it seems necessary to have at least 10 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 s and CD , the non-stationarity of profiles should be removed, if it exists. (c) The parameter combinations D and K s , (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 Hanning smoothing techniques seem suitable to use with the spectral technique to obtain accurate estimates for D, K s and CD . (e) To obtain accurate estimates for D, K s , and CD , it is necessary to use a suitable bandwidth for the Parzen window and a suitable number of interactions 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, K s and CD .