Unifying the U–Pb and Th–Pb methods: joint isochron regression and common Pb correction

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. Co-measured 208Pb ∕232Th 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, this total-Pb/U-Th 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 plot. Alternatively, the results of the new discordia regression algorithm can also be visualised as a 208Pbc ∕206Pb vs. 238U ∕206Pb or 208Pbc ∕207Pb vs. 235U ∕206Pb isochron, where 208Pbc represents the common 208Pb component. In its most general form, the total-Pb/U-Th algorithm accounts for the uncertainties of all isotopic ratios involved, including the 232Th ∕238U ratio, as well as the systematic uncertainties associated with the decay constants and the 238U ∕235U ratio. However, numerical stability is greatly improved when the dependency on the 232Th ∕238U-ratio uncertainty is dropped. For detrital minerals, it is generally not safe to assume a shared common Pb composition and concordia intercept age. In this case, the total-Pb/U-Th regression method must be modified by tying it to a terrestrial Pb evolution model. Thus, also detrital common Pb correction can be formulated in a maximum likelihood sense. The new method was applied to three published datasets, including low Th∕U carbonates, high Th∕U allanites and overdispersed monazites. The carbonate example illustrates how the total-Pb/U-Th method achieves a more precise common Pb correction than a conventional 207Pb-based approach does. The allanite sample shows the significant gain in both precision and accuracy that is made when the Th–Pb decay system is jointly considered with the U–Pb system. Finally, the monazite example is used to illustrate how the total-Pb/U-Th regression algorithm can be modified to include an overdispersion parameter. 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 intersample 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.