Planetary surface dating from crater size-frequency distribution measurements: Poisson timing analysis

The predictions of crater chronology models have customarily been evaluated by dividing a crater population into discrete diameter intervals, plotting the crater density for each, and finding a best-fit model isochron, with the uncertainty in the procedure being assessed using 1/√n estimates, where n is the number of craters in an interval. This approach yields an approximate evaluation of the model predictions. The approximation is good until n becomes small, hence the often-posed question: what is the minimum number of craters for an adequate prediction? This work introduces an approach for exact evaluation of a crater chronology model using Poisson statistics and Bayesian inference, expressing the result as a likelihood function with an intrinsic uncertainty. We demonstrate that even in the case of no craters at all, a meaningful likelihood function can be obtained. Thus there is no required minimum count: there is only varying uncertainty, which can be well described. We recommend that the Poisson timing analysis should be preferred over binning/best-fit approaches. Additionally, we introduce a new notation to make it consistently clear that crater chronology model calibration errors are inseparable from stated crater model ages and their associated statistical errors.