On the scale-dependence of fault surface roughness

Defining roughness as the ratio of height to length, the standard approach to characterize amplitudes of single fault, joint and fracture surfaces is to measure average height as a function of profile length. Empirically, this roughness depends strongly on scale. The ratio is approximately 0.01 at a few mm but 10× smaller at a few tens of meters. Surfaces are rougher at small scales. However, these conclusions are metric-dependent. If instead height is averaged over wavelength, roughness is nearly Brown spatial noise, having almost scale-independent apparent surface height to wavelength ratio. The small deviation from scale-independence is of the opposite sense than found using the standard metric; surfaces are slightly rougher at long wavelengths. Some natural surfaces may be Brownian within the measurement uncertainties. These contradictions are curiosities of surfaces that have Hurst exponents between 0.5 and 1, as natural fault surfaces do. The wavelength-based analysis of roughness and how it changes with scale are straight-forward; a normalized Fourier transform approximately preserves amplitude and its scale dependence in the wavelength domain. Among the conclusions from reconsideration of scale dependence are that the scale dependence is weak and much smaller than that of other fault and shear zone properties. Background and aftershock seismicity, jogs and step-overs indicate strong localization (smoothing) with slip and scale. The lack of strong scale dependence to surface roughness suggests it is not the dominant control on brittle shear zone evolution.