Statistical Characterization of Spatial Distribution of Impact Craters: Implications to Present-day Cratering Rate on Mars

Introduction: Counting craters has long been widely used in planetary science to get information about age of materials and events in geological history of planetary bodies. It also has long been understood that the uniform surface age and accumulation crater population requires "random" spatial distribution of craters. Such tests for randomness were applied to martian craters, e.g., in [1]. Here I report on my work on development of practical and statistically robust methods for analysis of spatial distribution of craters. I apply these methods to the recently emplaced craters discovered by M. Malin and co-authors [2] and infer a correction factor for the present-day impact cratering rate. Background: Estimation of ages with crater counts is based on the assumption that the cratering process is well described with a mathematical model of Poisson process [e.g., 3]. This means that each impact occurs purely randomly, "not knowing about" the site and time of previous impacts. The validity of this assumption is a special question. Double asteroids, freshly disrupted comets, secondary impacts violate this assumption. The most important of these factors, secondary cratering, plays no role for the very young surfaces on Mars, because they are younger than any large crater able to produce distant secondary craters. Here I assume that the impacts are random. Obliteration of craters is not random (unless they are removed by other impacts). Under the Poisson process assumption, the observed density of craters in an area gives an unbiased estimate of the average crater retention age. The crater retention age depends on crater size. (Note that the crater retention age can vary over the studied area, and the estimate gives the arithmetic average of it.) What is more important for geological applications, the number of craters gives a confidence interval for the average age: ) 1 ; ( ~ ) ; 1 ( 1 1 + < < − − Γ − Γ N p F N N p F , (1) where N~ = age × rate × area is the expected number of craters, N is the actual number of craters, p is the confidence level, for example, 0.9 or 0.95 or 0.99, and ) ; ( 1 ⋅ ⋅ − Γ F is the inverse cumulative gamma distribution. For a large number of craters, practically, for N > 10, this interval is well approximated by traditionally used N error bars: