Investigation of Crater “Saturation” Using Spatial Statistics

Abstract

Conventional techniques for investigation of crater populations make use of size–frequency distributions, but typically disregard the statistical properties of the spatial distribution of craters. Cratering of a surface is generally a spatially random process, so the distributions of craters on lightly cratered surfaces can be expected to be random. As crater obliteration becomes important, however, the situation changes. Every random distribution has regions of sparseness and regions of clustering. Craters that form in areas of sparseness obliterate few older craters, so the sparseness is filled in. Craters that form in areas of clustering obliterate more older craters, relatively reducing the clustering. As sparseness is filled in and clustering is reduced, the spatial distribution should progress from one that is random to one that is more uniform than random. We show that the spatial distribution of craters on a lightly cratered lunar maria surface is consistent with randomness, but that the distributions on heavily cratered portions of Rhea and Callisto are markedly more uniform than random. We develop a simple model of crater formation and obliteration, and use it to investigate the statistical behavior of crater spatial distributions as crater “saturation” is approached. We find that the use of spatial distributions to investigate saturation is more robust to model uncertainties than is the use of crater density alone. Using a simple statistical parameter, the “ Z-statistic,” we find that ≳ 25% of the craters formed to date on our study areas on Rhea and Callisto have been obliterated by subsequent cratering. Adjusting the Z-statistic for crater floor area effects, we find evidence that crater saturation has been reached on Rhea.