REFINING MORAINE AGE ESTIMATES FROM COSMOGENIC EXPOSURE DATES USING THE MAXIMUM LIKELIHOOD METHOD

We suggest a procedure, based on the maximum likelihood method, for estimating the ages of moraines from cosmogenic exposure dates. The procedure described here was originally applied to radiocarbon ages in stratigraphic context by Sharon (2001, Radiocarbon, v. 43, p. 345-354; his eqn. 4).

The maximum likelihood method uses stratigraphic information to refine estimates of moraine ages. In the case of two nested moraines, exposure dates from the inner, younger moraine provide information on the age of the outer, older moraine. To apply the maximum likelihood method, a likelihood function is evaluated at each possible value of the age of the outer moraine. The maximum likelihood estimate of the age of the outer moraine corresponds to the apex of this function.

Under some circumstances, the maximum likelihood method may provide a better estimate of a moraine's age than the mean of the dates from that moraine. To examine this hypothesis, Monte Carlo methods were used to generate synthetic data sets from two nested moraines. The variables in the Monte Carlo simulation were the difference in age between the two moraines and the number of dates available from each moraine. The age of the outer moraine was held constant at 20 ka. All of the dispersion of the dates on each moraine was assumed to be due to measurement error, which was taken to be 5% of the true age of each moraine. For each data set, the age of the outer moraine was estimated using the maximum likelihood method and the mean of the dates from that moraine. The two estimates were then compared to the true age of the outer moraine.

In general, the maximum likelihood method is a better predictor of the age of the outer moraine than the mean of the dates from that moraine in cases where the moraines are close in age (~1 ka or less for pairs of moraines about 20 ka old, assuming a measurement error of 5%) and more dates are available from the inner moraine than the outer. However, the maximum likelihood estimate of the age of the outer moraine depends partly on the difference in age between the moraines. Since the difference in age between moraines is usually not known, rigorous application of the maximum likelihood method is difficult. It can still be applied, on an ad hoc basis, where there is substantial overlap between the probability distribution functions of the dates from the inner and outer moraines.