The ocean floor on the flanks of mid-ocean ridges is covered by abyssal hills, topographic features elongated perpendicularly to the direction of relative plate motion. These topographic features are interpreted as normal fault blocks and/or volcanic constructions that originated near the ridge axis and were later rafted onto the ridge flanks by seafloor spreading. The purpose of this paper is to quantify the topographic roughness of a profile perpendicular to the strike of a number of normal faults, given a fault population and a mechanical model for the response of the lithosphere to faulting. We obtain expressions for the variation in root-mean-square roughness with profile length and for the power spectral density of a profile given three parameters: a fault density (number of faults per unit length crossed by the profile), an average squared fault scarp height, and a characteristic length of flexure. To keep matters simple, we make a number of assumptions and approximations, namely, that the lithosphere behaves as an elastic plate, that faults have an infinite length and a vertical dip, that the response of topography to a number of faults is simply the sum of the responses to each fault, and that faults have random locations and scarp heights independently chosen from some statistical distribution. The theory predicts that the roughness-length relationship/power spectral density should follow power laws for scales/wavelengths less than a characteristic scale proportional to the length scale of flexure. We compare the predictions of the theory with actual measurements of mid-ocean ridge flank roughness, and find good first-order agreement. In particular, we use independent estimates of fault densities, average squared fault scarp heights, and flexural length scales to predict the topographic roughness of the East Pacific Rise, and we find that normal faulting can explain all the observed topographic roughness. Nevertheless, there are some differences between predictions and observations. These differences are likely to be due to processes other than faulting that create topographic relief (e.g., volcanism) and to spatial correlations of fault scarp heights. Despite these shortcomings, the approach presented here provides a first step in understanding the topographic roughness signal by quantifying the contribution of the geological processes that generate surface relief.
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