Topographic power spectra of cratered terrains: Theory and application to the Moon

Impact cratering produces characteristic variations in the topographic power spectral density (PSD) of cratered terrains, which are controlled by the size-frequency distribution of craters and the spectral content (shape) of individual features. These variations are investigated here in two parallel approaches. First, a cratered terrain model, based on Monte Carlo emplacement of craters and benchmarked by an analytical formulation of the one-dimensional PSD, is employed to generate topographic surfaces at a range of size-frequency power law exponents and shape dependencies. For self-similar craters, the slope of the PSD, β, varies inversely with that of the production function, α, leveling off to 0 at high α (surface topography dominated by the smallest craters) and maintaining a roughly constant value (β ∼ 2) at low α (surface topography dominated by the largest craters). The effects of size-dependent shape parameters and various crater emplacement rules are also considered. Second, we compare the model-derived predictions for the behavior of the PSD with values of β calculated along transects from the Lunar Orbiter Laser Altimeter (LOLA). At small scales (∼115 m to 1 km) model predictions agree well with the PSD slope over the observed range of lunar size-frequency distributions. Differences between global PSD slopes at subkilometer and kilometer scales reflect a scale separation in roughness consistent with prior observations using a variety of surface roughness parameters. Understanding the statistical markers left by the impact cratering process on the lunar surface is useful for distinguishing between competing geological processes on planetary surfaces throughout the solar system.