Characterizing fault roughness—Are faults rougher at long or short wavelengths?

Changes in fault roughness with scale, “scaling,” is the topic of this report; changes are considered using a general power law relation between some measure of surface height, H, and another of length, L, H=kLⁿ, where k is a constant and n is an exponent that characterizes the scaling. Extensive profile measurements of natural fault surfaces show that the ratio of average surface height to profile length decreases with scale. Average height is defined using the root mean squared height, Rq. For this analysis, fault surfaces are smoother at long wavelengths (have smaller average height to profile length ratios) than they are at shorter wavelengths. These and other statistical properties of natural fault surfaces hold for more than five orders of magnitude, a huge range from tens of micrometers to 10 meters. However, a different roughness metric, the average height (amplitude) that is specifically associated with a wavelength shows the opposite sense of scaling. The ratio of average amplitude to wavelength increases with wavelength. Thus, the same fault surface can be deemed rougher at long wavelength, or smoother, depending on the chosen metric. This apparent contradiction is a curiosity of the statistics of rough surfaces that have scaling exponents that relate profile length to Rq between 0.5 and 1, as most natural faults do. To add context, the implied roughness scaling for reference synthetic surfaces is determined. These span the natural range of scaling exponents and have moderate to strong point to point amplitude correlation. The potential payoff of expanded descriptions of natural fault roughness and of reference surfaces are improved constraints on physical mechanisms that generate and modify roughness during shear.