Distribution and covariance function of elevations on a cratered planetary surface

Abstract

We derive the distribution and covariance function of elevations on a cratered planetary surface from a representation of the surface as the ‘moving average’ of a random point process. It is assumed that an initially plane surface is excavated by primary impact craters with an inverse-power law size distribution. Crater rim height and rim-to-floor depth are assumed to be power functions of crater diameter. Crater shapes studied include rimless cylinders and paraboloidal bowls, and paraboloidal bowls with power-law external rims and ejecta blanket. The inverse-power law diameter distribution induces a positively skewed ‘stable law’ elevation distribution, with heavy inverse-power law tails whose exponent (for small craters) is two smaller than the crater diameter distribution exponent. The covariance function (equivalently, power spectral density) is shown to be a power-law at moderate distances, whose exponent also depends on the parameters of the cratering process. Observations of lunar elevations and elevation spectral densities on a meter scale agree well with theory.