Requirements for accurate quantification of self-affine roughness using the variogram method

Both stationary and non-stationary fractional Brownian profiles (self-affine profiles) with known values of fractal dimension, D, input standard deviation, σ, and data density, d, were generated. For different values of the input parameter of the variogram method (lag distance, h), D and another associated fractal parameter K_v were calculated for the aforementioned profiles. It was found that σ has no effect on calculated D. The estimated K_v was found to increase with D, σ and d according to the equation K_v = 2.0 × 10^-5d^0.35σ^1.95D^14.5. The parameter K_v seems to have potential to capture the scale effect of roughness profiles. Suitable ranges for h were estimated to obtain computed D within ±10% of the D used for the generation and also to satisfy a power functional relation between the variograrn and h. Results indicated the importance of removal of non-stationarity of profiles to obtain accurate estimates for the fractal parameters. It was found that at least two parameters are required to quantify stationary roughness; the parameters D and K_v are suggested for use with the variograrn method. In addition, one or more parameters should be used to quantify the non-stationary part of roughness, if it exists; at the basic level, the mean inclination/declination angle of the surface in the direction considered can be used to represent the non-stationarity. A new concept of feature size range of a roughness profile is introduced in this paper. The feature size range depends on d, D and σ. The suitable h range to use with the variogram method to produce accurate fractal parameter values for a roughness profile was found to depend on both d and D. It is shown that the feature size range of a roughness profile plays an important role in obtaining accurate roughness parameter values with both the divider and the variogram methods. The minimum suitable h was found to increase with decreasing d and increasing D. Procedures are given in this paper to estimate a suitable h range for a given natural rock joint profile to use with the variogram method to estimate D and K_v accurately for the profile.