Unifying the U–Pb and Th–Pb methods: joint isochron regression and common Pb correction
Authors: Vermeesch Pieter 
Published: May 2020

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. 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.