Statistical modelling of thermal annealing of fission tracks in apatite

We develop an improved methodology for modelling the relationship between mean track length, temperature, and time in fission track annealing experiments. We consider “fanning Arrhenius” models, in which contours of constant mean length on an Arrhenius plot are straight lines meeting at a common point. Features of our approach are explicit use of subject matter knowledge, treating mean length as the response variable, modelling of the mean-variance relationship with two components of variance, improved modelling of the control sample, and using information from experiments in which no tracks are seen.

This approach overcomes several weaknesses in previous models and provides a robust six parameter model that is widely applicable. Estimation is via direct maximum likelihood which can be implemented using a standard numerical optimisation package. Because the model is highly nonlinear, some reparameterisations are needed to achieve stable estimation and calculation of precisions. Experience suggests that precisions are more convincingly estimated from profile log-likelihood functions than from the information matrix.

We apply our method to the B-5 and Sr fluorapatite data of Crowley et al. (1991) and obtain well-fitting models in both cases. For the B-5 fluorapatite, our model exhibits less fanning than that of Crowley et al. (1991), although fitted mean values above 12 μm are fairly similar. However, predictions can be different, particularly for heavy annealing at geological time scales, where our model is less retentive. In addition, the refined error structure of our model results in tighter prediction errors, and has components of error that are easier to verify or modify. For the Sr fluorapatite, our fitted model for mean lengths does not differ greatly from that of Crowley et al. (1991), but our error structure is quite different.