Modeling fission track annealing in apatite: An assessment of uncertainties

Abstract

Abstract If a model is to be used as the basis of an interpretive scheme then it is essential to have some idea of the uncertainties associated with the model predictions. Equations describing the annealing of induced fission tracks in Durango apatite can be applied to variable temperature thermal histories over geologic time scales, but the calculation of confidence limits on the predicted value is not straightforward. The basis of the modeling method is the concept of equivalent time, t 0 , which allows a pair of variable temperature annealing steps, each of time Δ t and temperatures T 1 and T 2 respectively, to be expressed as a single isothermal annealing step of length (Δ t + t 0 ) at temperature T 2 . The numerical approximation is very stable, roundoff errors being insignificant if the time step is less than ≈ 2% of the model time. Error in the predicted value due to uncertainties in the regression coefficients of the model equations is difficult to ascertain analytically and we adopt a Monte Carlo technique to obtain an estimate of this error. Two cases are considered: a comparison with variable temperature annealing experiments in the laboratory and prediction of fission track parameters for simple thermal histories over geologic time scales. Monte Carlo simulations suggest that for variable temperature annealing over an eight hour period, the 2σ error in the predicted value of mean track length is 1-1.5 μm; for thermal histories of 100 Ma (cooling from 100 to 0°C, heating from 0 to 100°C and isothermal heating at 50°C), the 2σ errors for the predicted model ages range from 8 to 26 Ma. A comparison between three different models that have been used to describe the annealing of fission tracks in apatite reveals that there is close agreement between the results from the fanning Arrhenius model and the first order model for thermal histories at > 95°C. This coincidence of results could explain the apparent success of the first order model as an interpretive tool in the past.