U-Th-Pb discordia regression

Abstract. The actinide elements U and Th undergo radioactive decay to three isotopes of Pb, forming the basis of three coupled geochronometers. The 206Pb / 238U and 207Pb / 235U decay systems are routinely combined to improve accuracy. Joint consideration with the 208Pb / 232Th decay system is less common. This paper aims to change this. Adding 208Pb / 232Th to the mix is particularly useful for discordant samples containing variable amounts of non-radiogenic (common) Pb. The paper presents a maximum likelihood algorithm for joint isochron regression of the 206Pb / 238Pb, 207Pb / 235Pb, and 208Pb / 232Th chronometers. Given a set of cogenetic samples, the algorithm estimates the common Pb composition and concordia intercept age. U-Th-Pb data can be visualised on a conventional Wetherill or Tera-Wasserburg concordia diagram, or on a 208Pb / 232Th vs. 206Pb / 238U concordia diagram. Alternatively, the results of the new discordia regression algorithm can also be visualised as a 208Pbc / 206Pb vs. 238U / 206Pb or 208Pbc / 207Pb vs. 238U / 207Pb isochron, where 208Pbc represents the common 208Pb component. For detrital minerals, it is generally not possible to assume a shared common Pb composition and concordia intercept age. In this case the U-Th-Pb discordia regression method must be modified by tying it to a mantle evolution model. Thus also detrital common Pb correction can be formulated in a maximum likelihood sense. The new method was applied to a published monazite dataset with a Th / U-ratio of ∼ 10, resulting in a significant radiogenic 208Pb component. Therefore the case study represents a worst case scenario for the new algorithm. Nevertheless, it manages to fit the data very well. The method should work even better in low-Th phases such as carbonates. The degree to which the dispersion of the data around the isochron line matches the analytical uncertainties can be assessed using the mean square of the weighted deviates (MSWD) statistic. A modified four parameter version of the regression algorithm quantifies this overdispersion, providing potentially valuable geological insight into the processes that control isotopic closure. All the parameters in the discordia regression method (including the age and the overdispersion parameter) are strictly positive quantities that exhibit skewed error distributions near zero. This skewness can be accounted for using the profile log-likelihood method, or by recasting the regression algorithm in terms of logarithmic quantities. Both approaches yield realistic asymmetric confidence intervals for the model parameters. The new algorithm is flexible enough that it can accommodate disequilibrium corrections and inter-sample error correlations when these are provided by the user. All the methods presented in this paper have been added to the IsoplotR software package. This will hopefully encourage geochronologists to take full advantage of the entire U-Th-Pb decay system.