Distribution and covariance function of elevations on a cratered planetary surface

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.This work summarizes and extends Bellcomm Technical Reports TR-68-340-3, 4, 5, which were supported by the National Aeronautics and Space Administration, Contract NASw-417.Now at Johns Hopkins University, Baltimore, Md.