# 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 inline-formula206Pb ∕238U and inline-formula207Pb ∕235U decay systems are routinely combined to improve accuracy. Joint consideration with the inline-formula208Pb ∕232Th decay system is less common. This paper aims to change this. Co-measured inline-formula208Pb ∕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 inline-formula206Pb ∕238Pb, inline-formula207Pb ∕235Pb and inline-formula208Pb ∕232Th chronometers. Given a set of cogenetic samples, this inline-formula $M8inlinescrollmathmlchem\mathrm{normal total}-\mathrm{normal Pb}/\mathrm{normal U}-\mathrm{normal Th}$ 78pt14ptsvg-formulamathimgbeb17270a234dc51af38023124204551 gchron-2-119-2020-ie00001.svg78pt14ptgchron-2-119-2020-ie00001.png 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 inline-formula208Pb ∕232Th vs. inline-formula206Pb ∕238U plot. Alternatively, the results of the new discordia regression algorithm can also be visualised as a inline-formula208Pbc ∕206Pb vs. inline-formula238U ∕206Pb or inline-formula208Pbc ∕207Pb vs. inline-formula235U ∕206Pb isochron, where inline-formula208Pbc represents the common inline-formula208Pb component. In its most general form, the inline-formula $M17inlinescrollmathmlchem\mathrm{normal total}-\mathrm{normal Pb}/\mathrm{normal U}-\mathrm{normal Th}$ 78pt14ptsvg-formulamathimged5d783723953e4a52c016d04bcaef63 gchron-2-119-2020-ie00002.svg78pt14ptgchron-2-119-2020-ie00002.png algorithm accounts for the uncertainties of all isotopic ratios involved, including the inline-formula232Th ∕238U ratio, as well as the systematic uncertainties associated with the decay constants and the inline-formula238U ∕235U ratio. However, numerical stability is greatly improved when the dependency on the inline-formula232Th ∕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 inline-formula $M21inlinescrollmathmlchem\mathrm{normal total}-\mathrm{normal Pb}/\mathrm{normal U}-\mathrm{normal Th}$ 78pt14ptsvg-formulamathimg3f31ce6d71b0a7f6de7c82bed1f338e8 gchron-2-119-2020-ie00003.svg78pt14ptgchron-2-119-2020-ie00003.png 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 inline-formulaTh∕U carbonates, high inline-formulaTh∕U allanites and overdispersed monazites. The carbonate example illustrates how the inline-formula $M24inlinescrollmathmlchem\mathrm{normal total}-\mathrm{normal Pb}/\mathrm{normal U}-\mathrm{normal Th}$ 78pt14ptsvg-formulamathimge26db333b9040173372a653f288209e0 gchron-2-119-2020-ie00004.svg78pt14ptgchron-2-119-2020-ie00004.png method achieves a more precise common Pb correction than a conventional inline-formula207Pb-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 inline-formula $M26inlinescrollmathmlchem\mathrm{normal total}-\mathrm{normal Pb}/\mathrm{normal U}-\mathrm{normal Th}$ 78pt14ptsvg-formulamathimg39f766a6cf0cb74311bb3de72c6bd90b gchron-2-119-2020-ie00005.svg78pt14ptgchron-2-119-2020-ie00005.png 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.

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Vermeesch, Pieter: Unifying the U–Pb and Th–Pb methods: joint isochron regression and common Pb correction. 2020. Copernicus Publications.

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