The theory of cratering statistics: Significant corrections to existing methods

Crater counts provide a rich source of information about the history of our solar system, including age estimates of many geologic terrains. In this work we present substantial improvements to existing statistical methods. Existing methods have only included counting error in age estimates. We have developed mathematical methods to calculate age Probability Density Functions (PDFs) that incorporate the error of identifying craters, measuring diameters, selecting the minimum diameter, and the chronology system. We derive more accurate methods for measuring the slopes of log-log plots of crater size-frequency distributions (SFDs), and we show with synthetic modeling and analytical derivations how existing methods are biased. To measure the full shape of the SFD, we present the Slanted Average Shifted Histogram (SASH), a more accurate method for approximating the SFD from a limited number of observed craters. As examples, we show: (1) how the age PDFs of two counts of Canala Crater become much more consistent when all sources of error are included, (2) a practical guide to the age calculation, (3) a visualization of the math of the age calculation, and (4) how the SASH model reveals that the cratered plains of Mimas, Enceladus, Tethys, Dione, and Rhea display equilibrium saturation. With our methods, future researchers will be able to much more accurately estimate the PDF of age and understand the shapes of crater SFDs.