Explicit solutions for a probabilistic moraine preservation model

Abstract

If a series of glacial advances occurs over the same pathway, the moraines that are now present may constitute an incomplete record of the total history. This is because a given advance can destroy the moraine left by a previous one, if the previous advance was less extensive. Gibbons, Megeath and Pierce (GMP) formulated an elegant stochastic model for this process; the key quantity in their analysis is $\bi P(n\vert N)$ , the probability that n moraines are preserved after N glacial advances. In their paper, GMP derive a recursion formula satisfied by $\bi P(n\vert N)$ , and use this formula to compute values of P for a range of values of n and N. In the present paper, we derive an explicit general answer for $\bi P(n\vert N)$ , and show explicit, exact results for the mean value and standard deviation of n. We use these results to develop more insight into the consequences of the GMP model; for example, to a good approximation, 〈n〉 increases as ln(N). We explain how a Bayesian approach can be used to analyze $\bi P(N\vert n)$ , the probability that there were N advances, given that we now observe n moraines.

Keywords

• glaciation
• moraines
• stochastic model