Probability density plot for fission-track grain-age samples

Abstract

This paper outlines the statistical basis, construction and interpretation of probability-density (PD) plots as applied to the analysis of mixed distributions of fission-track (FT) grain ages. Such distributions can be viewed as being composed of a mixture of elemental component distributions, with each component characterized by a unique average FT age. The original PD plot of Hurford et al. is shown to have a clear statistical basis, as defined by the Gaussian-kernel method of density estimation. This background is used to develop and justify an improved version of the PD plot. Three modifications are recommended: (1) as originally suggested by Galbraith, the PD plot should be constructed using the transform variable z, which is approximately proportional to the logarithm of FT age. By using z i to represent the FT grain ages τ i , each component distribution within a mixed FTGA sample becomes approximately Gaussian-distributed. This approximation is shown to work well for FT dating of zircon grains, which generally have relatively high uranium content and high track densities, and less so for FT dating of apatite grains; (2) estimation of the PD plot is optimized by setting the width of the Gaussian-kernel equal to α( SE( z ̂ i) , where α is a scaling factor with an optimal value of ∼ 0.6 and SE( z ̂ i) is the standard error of the FT estimate for z for the ith grain. This arrangement ensures the best compromise between resolution and smoothness for the final PD plot; (3) probability density is estimated as a function of z but it is best presented as a function of FT age τ. This objective is accomplished by transforming the z coordinates of the PD plot to τ and plotting τ on a logarithmically-scaled axis, which ensures that the original scaling of the PD plot is preserved. With these modifications, the component distributions in a PD plot will appear as symmetric Gaussian-shaped peaks and the area beneath each peak will be proportional to the relative size of the component. Several examples are given that illustrate the general concepts behind the PD plot and the advantages of the recommended modifications.