Requirements for accurate estimation of fractal parameters for self-affine roughness profiles using the line scaling method

A new concept of feature size range of a roughness profile is introduced in the paper. It is shown that this feature size range plays an important role in estimating the fractal dimension, D, accurately using the divider method. Discussions are given to indicate the difficulty of using both the divider and the box methods in estimating D accurately for self-affine profiles. The line scaling method's capability in quantifying roughness of natural rock joint profiles, which may be self-affine, is explored. Fractional Brownian profiles (self-affine profiles) with and without global trends were generated using known values of D, input standard deviation, σ, and global trend angles. For different values of the input parameter of the line scaling method (step size a₀), D and another associated fractal parameter C were calculated for the aforementioned profiles. Suitable ranges for a₀ were estimated to obtain computed D within ±10% of the D used for the generation. Minimum and maximum feature sizes of the profiles were defined and calculated. The feature size range was found to increase with increasing D and σ, in addition to being dependent on the total horizontal length of the profile and the total number of data points in the profile. The suitable range for a₀ was found to depend on both D and σ, and then, in turn, on the feature size range, indicating the importance of calculating feature size range for roughness profiles to obtain accurate estimates for the fractal parameters. Procedures are given to estimate the suitable a₀ range for a given natural rock joint profile to use with the line scaling method in estimating fractal parameters within ±10% error. Results indicate the importance of removal of global trends of roughness profiles to obtain accurate estimates for the fractal parameters. The parameters C and D are recommended to use with the line scaling method in quantifying stationary roughness. In addition, one or more parameters should be used to quantify the non-stationary part of roughness, if it exists. The estimated C was found to depend on both D and σ and seems to have potential to capture the scale effect of roughness profiles.