Optimizing time step size for apatite fission track annealing models

Empirical isothermal apatite fission track (AFT) annealing models are used to extract variable temperature histories from measured AFT parameters by forward and inverse modeling techniques. Using one such published annealing model based on Durango apatite data, a method has been developed for optimally discretizing thermal histories into isothermal steps for the evaluation of the annealing model. The equation S=(root log(T-a/T-m)/A(1)+A(2)e(-(Tm-T)/A3))/R(A4) predicts isothermal time step size (S) as a function of maximum interval temperature (T-m in kelvins) and rate of temperature change (R in K m.y.(-1)) with constants A(1)=0.00596, A(2)=0.043, A(3)=34.8, and A(4)=0.978, and with the total annealing temperature (T-a) given as a power function of R, T-a=398.15(R(0.0157)). This equation offers improved computational efficiency and accuracy for the full range of track length reduction in comparison with other methods published. Improved model performance is important particularly for inversion type models which may require the generation of thousands of model temperature histories with large variations in heating and cooling rates. For similar amounts of annealing, integration step sites differ by two orders of magnitude for heating/cooling rates that range between 0.1 and 10 K m.y.(-1), a range that encompasses most sedimentary basins. As an added advantage, users can specify the approximate degree of accuracy for track length calculations by multiplying S by the scaling factor, root 10E, where E is the approximate percent error on calculated track lengths. For E values of 0.1 and 1.0, computed thermal histories are within less than or equal to 0.2 degrees C and less than or equal to 1.0 degrees C, respectively, of the true numerical solution.