Carbonate rocks are only a minor component of the continental crust. However, their scientific importance far outweighs their volumetric abundance. Biogenic carbonates and speleothems document the history of life and of Earth's climate and environment. To generate detailed time series of past changes from these carbonate archives, an accurate and precise chronological framework is essential. This framework is anchored in absolute time using radiometric dating methods, with two techniques commonly employed for this purpose. ²³⁰Th/U dating is the method of choice for young samples whose ²³⁰Th, ²³⁴U, and ²³⁸U activities are out of secular equilibrium . ²⁰⁶Pb / ²³⁸U dating is the default method for older rocks, in which the secular equilibrium between ²³⁰Th, ²³⁴U, and ²³⁸U has been restored .

Ironically, the absence of detectable ²³⁴U / ²³⁸U disequilibrium compromises the accuracy of the ²⁰⁶Pb / ²³⁸U method. Any initial excess or deficit of ²³⁴U and ²³⁰Th affects the ²⁰⁶Pb / ²³⁸U ratio and, hence, the age estimate derived therefrom. In clean, detritus-free carbonates, it is often safe to assume the absence of initial ²³⁰Th. This assumption is not valid for ²³⁴U, which can be enriched (or occasionally depleted) relative to ²³⁸U by physiochemical processes such as (1) α-recoil ejection and preferential leaching of ²³⁴U from α-damaged mineral sites and (2) chemical fractionation between preferentially oxidized ²³⁴U⁶⁺ and ²³⁸U⁴⁺ .

Corrections for initial ²³⁴U / ²³⁸U disequilibrium can be done either by assuming a specific initial ²³⁴U / ²³⁸U activity ratio or by inferring the initial ratio from any measured residual ²³⁴U / ²³⁸U disequilibrium . In Sect.  of this paper, we review the second approach using matrix exponentials. We show that initial ²³⁴U / ²³⁸U disequilibrium can bias ²⁰⁶Pb / ²³⁸U dates by up to 4 Myr.

Current disequilibrium correction algorithms use a “Monte Carlo” approach to propagate the errors. In Sect.  we will show that this approach can underestimate the analytical uncertainties of ²⁰⁶Pb / ²³⁸U dates by an order of magnitude when samples are within a few per mil of secular equilibrium (which typically happens before ca. 2 Ma). This observation undermines the results of several published studies .

In Sect.  we introduce a deterministic Bayesian approach to estimate the uncertainties of disequilibrium-corrected ²⁰⁶Pb / ²³⁸U dates. We use this alternative algorithm to show that beyond ca. 2 Ma, disequilibrium-corrected ²⁰⁶Pb / ²³⁸U dates are impractically imprecise, unless highly enriched initial ²³⁴U / ²³⁸U activity ratios can be ruled out a priori. The large uncertainty associated with the ²⁰⁶Pb / ²³⁸U method degrades its ability to constrain reliable chronologies for carbonates whose initial disequilibrium has expired. However, a more accurate approach is available. Following the example of , , , and others, Sect.  makes a case for the little-used ²⁰⁷Pb / ²³⁵U clock as a replacement for the ²⁰⁶Pb / ²³⁸U method.

²⁰⁷Pb / ²³⁵U isochrons are, essentially, immune to the effects of initial disequilibrium (apart from a minor correction for ²³¹Pa, which becomes smaller with increasing age). In Sect.  we present examples from Siberia and Israel to show that the ²⁰⁷Pb / ²³⁵U method is more accurate than the ²⁰⁶Pb / ²³⁸U method, whilst being less precise for young samples. Both the Bayesian uncertainty estimation method and ²⁰⁷Pb / ²³⁵U isochrons have been implemented in the IsoplotR toolbox for radiometric geochronology (Sect. ).

This paper will use the following symbols and notations:

n38, n34, n30, n26 and n06: the number of atoms of ²³⁸U, ²³⁴U, ²³⁰Th, ²²⁶Ra and ²⁰⁶Pb, respectively;

Even though the U–Pb decay systems consist of numerous steps (14 for the ²³⁸U→206Pb chain and 11 for the ²³⁵U→207Pb chain), conventional U–Pb geochronology ignores this complexity, and the method is mathematically treated as a set of simple parent–daughter pairs. This simplification is justified once a state of secular equilibrium is established between all the intermediate daughter products in the decay chains. Such secular equilibrium is practically reached after 1 to 2 Myr. As mentioned in Sect. , any disequilibrium that might exist prior to this secular equilibrium can be used as a chronometer in its own right.

Initial disequilibrium of the U-decay series affects the accuracy of the U–Pb method. For example, ignoring any initial excess ²³⁴U results in an overestimated ²⁰⁶Pb / ²³⁸U age, and ignoring any initial ²³¹Pa deficit results in an underestimated ²⁰⁷Pb / ²³⁵U age. Therefore, initial disequilibrium is one mechanism to produce discordant U–Pb results. Extreme ²³⁴U enrichments have been observed in places such as South Africa ([4/8]i < 12; ), Siberia ([4/8]i < 6; ) and Japan ([4/8]i < 12; ). Using the [4/8]i = 12 value as an upper bound, the maximum effect of initial ²³⁴U / ²³⁸U disequilibrium can be approximated as follows: 1Δ(t)≈1λ38ln⁡1+206Pb238U+(12-1)λ38λ34+(0-1)λ38λ30-1λ38ln⁡1+206Pb238U≈11λ34-1λ30=3.8Myr, where λ38 = 0.155125(83) Gyr⁻¹ , λ30 = 9.1705(16) Myr⁻¹, and λ34 = 2.82206(80) Myr⁻¹ . For old carbonates (> 100 Ma, say), a 4 Myr bias may be inconsequential. However, for young carbonates, the relative effect of initial disequilibrium can result in order-of-magnitude levels of bias. A disequilibrium correction is needed to remove this bias.

If the intermediate daughter is sufficiently long lived and the sample is sufficiently young (t < 5/λ, say) to retain some of its disequilibrium, then the activity ratios can be back-calculated to the time of isotopic closure (assuming subsequent closed-system behaviour). This strategy applies to ²³⁴U / ²³⁸U disequilibrium and, for very young samples, to ²³⁰Th / ²³⁸U disequilibrium. The complex evolution of the U-decay series was first described by and subsequently applied to U–Pb geochronology by , , and . Here we opt for an alternative formulation, using matrix exponentials . For example, the ²³⁸U → ²⁰⁶Pb decay chain can be expressed in matrix form as follows: 2∂∂tn38n34n30n26n06=-λ380000λ38-λ340000λ34-λ300000λ30-λ260000λ260n38n34n30n26n06, where λ26 = 0.4332(19) kyr⁻¹ , and the shortest-lived intermediate daughters (< 1 kyr half-lives) have been omitted. The solution to Eq. () is a so-called matrix exponential: 3n38n34n30n26n06=expm-λ380000λ38-λ340000λ34-λ300000λ30-λ260000λ260tn38n34n30n26n06i, which expresses the present-day amounts of the different isotopes as a function of the initial amounts. An interesting result is obtained by setting t = ∞ in Eq. () to estimate the activity ratio under secular equilibrium: 4[4/8]∞=λ234λ234-λ238=1.000055.

Note that this activity ratio is not exactly equal to unity. This is because 0.0055 % of ²³⁸U is lost during ²³⁴U's mean lifetime of 1/λ34 = 354 kyr. Equation () can also be inverted to express the initial amounts as a function of the present-day amounts: 5n38n34n30n26n06i=expm--λ380000λ38-λ340000λ34-λ300000λ30-λ260000λ260tn38n34n30n26n06.

Equations () and () can be used to construct a concordia diagram in the presence of disequilibrium. If measured activity ratios are used to infer the initial conditions, then the concordia line terminates where those inferred activity ratios reach unrealistic values (e.g. [4/8]i = 500 and [0/8]i = 0; Fig. a). Beyond 10 or so ²³⁴U half-lives, it becomes very difficult to estimate [4/8]i from [4/8]m, and it is even more difficult to quantify the analytical uncertainty of the disequilibrium correction. In Sect.  we review the current “Monte Carlo” approach to uncertainty estimation for initial disequilibrium correction, and in Sect.  we propose an alternative “Bayesian” approach, which offers significant advantages for samples that are close to secular equilibrium.

Existing data processing software for disequilibrium-corrected ²⁰⁶Pb / ²³⁸U geochronology, such as DQPB , estimates the uncertainty of the disequilibrium correction by Monte Carlo simulation. Given a linear array of isotopic data in Tera–Wasserburg space (i.e. n07/n06 vs. n38/n06), paired with an activity ratio measurement [4/8]m with standard error s[4/8]m, this approach works as follows: 1.

Draw a random value [4/8]t from a normal distribution with mean [4/8]m and standard deviation s[4/8]m.

For the purpose of the present study, we have implemented our own version of this algorithm , using R and IsoplotR . The only major difference between our code and DQPB is that it does not sample the [4/8]m distribution randomly but uses a targeted approach to sample [4/8]m as a sequence of regularly spaced normal quantiles. This is faster and produces deterministic results that do not depend on the seed of a random number generator. Figure  summarizes the application of this approach using the “Corchia” dataset of , producing identical results to DQPB. To reflect this equivalence of outcomes, we will refer to our version of the algorithm as a “Monte Carlo” method, despite the fact that it does not actually use a random number generator.

Next, let us apply the same approach to older materials such as sample AV03 (Bolt's Farm, South Africa) of . The uncorrected U–Pb isochron age for this sample is 5.6 ± 0.9 Ma, which is 22 half-lives of ²³⁴U. Consequently, the measured present-day ²³⁴U / ²³⁸U activity ratio is statistically indistinguishable from secular equilibrium, at [4/8]m = 1.0046 ± 0.0063. Despite the lack of measurable disequilibrium, the “Monte Carlo” approach appears to have successfully applied a disequilibrium correction, resulting in a corrected age that is less than half the uncorrected age, with a precision of better than 12 % (Fig. ). How is this possible? The answer lies in the rejected solutions (step 4 of the algorithm), which are marked in black in Fig. b. Ignoring these “physically impossible” initial ratios suppresses the equilibrium solutions and skews the distribution of Monte Carlo solutions towards high [4/8]i values (Fig. c) and young ages (Fig. d).

To demonstrate that the result of Fig. () is wrong, let us replace the measured ²³⁴U / ²³⁸U activity ratio with the equilibrium ratio (Eq. ): [4/8]m=[4/8]∞±0.0063.

Plugging this value into the “Monte Carlo” algorithm yields an impossible result (Fig. ). It has applied a disequilibrium correction without any actual disequilibrium, by ignoring exactly half of the [4/8]t distribution (Fig. b). This was necessary because, for this old sample, essentially any [4/8]t value that is less than the equilibrium ratio would require a negative [4/8]i ratio or a negative isochron age t.

The previous section showed that the “Monte Carlo” algorithm produces incorrect results for samples whose [4/8]t-distributions fall within, say, 3 standard errors (i.e. 3×s[4/8]m) from secular equilibrium. One way to address this issue is for the “Monte Carlo” algorithm to issue a warning when the [4/8]m value is close to [4/8]∞. This is the approach taken by DQPB . In this section we introduce an alternative approach that automatically handles problematic cases, without the need to define a nominal “applicability cutoff”. Given any values of [4/8]m and s[4/8]m, our new “Bayesian” algorithm proceeds as follows: 1.

Define a prior distribution for [4/8]i. In a first instance, we will use a uniform distribution that stretches between a minimum [4/8]i value m = 0 and a maximum [4/8]i value M = 20. However, this uniform distribution can easily be replaced by a more informative prior. One flexible way to capture a diversity of prior information is the logistic normal distribution:6ln⁡[4/8]i-mM-[4/8]i∼N(μ,σ2),where μ and σ are the location and dispersion parameters of the distribution, respectively. An application of this informative prior will be given in Sect. .

Applying this method to the 580 ka Corchia example (Fig. ) yields essentially identical results to the “Monte Carlo” algorithm (Fig. ).

However, when the Bayesian approach is applied to the older Bolt's Farm data (Fig. ), it produces a very different result than the “Monte Carlo” approach (Fig. ). The posterior distributions for Bolt's Farm sample AV03 (shown in Fig. c and d) still have maxima at [4/8]i = 9 and t = 2.6 Ma, just like the “Monte Carlo” distributions (Fig. c and d). But unlike the “Monte Carlo” solution, the result of the Bayesian approach also assigns a significant probability to older ages, including the uncorrected U–Pb date of 5.6 Ma. The similarity of this posterior distribution to the prior distribution reflects the fact that the measured ²³⁴U / ²³⁸U activity ratio contains relatively little information. The resulting uncertainties are large but correctly reflect our ignorance about the true extent of the disequilibrium in this case.

Finally, changing the [4/8]m ratio to the equilibrium value (Fig. ) produces a posterior distribution that is nearly identical to the prior distribution. This means that the likelihood function contains almost no information. In other words, the measured ²³⁴U / ²³⁸U activity ratio does not tell us anything about the initial disequilibrium (except that [4/8]i < 10). If it cannot be ruled out that the sample may have experienced extreme ²³⁴U / ²³⁸U disequilibrium, then it is not possible to undo the effects of this disequilibrium using the modern (measured) ²³⁴U / ²³⁸U activity ratio.

The applicability range of the ²⁰⁶Pb / ²³⁸U method depends on the [4/8]i ratio and on the precision of the [4/8]m measurements. Although sub-per-mil level analytical uncertainties can be routinely achieved for individual [4/8]m measurements, the external reproducibility is likely worse than this in most samples. This is due to a combination of two competing factors. First, [4/8]i is negatively correlated with U concentration . Second, the U concentration must exhibit large variations to form a statistically robust ²⁰⁶Pb / ²³⁸U isochron.

The combination of these two effects has the potential to cause intra-sample variations in [4/8]m, exceeding the analytical uncertainties. The exact magnitude of the dispersion is unknown because most speleothem U–Pb dating studies report only one or a few [4/8]m values per isochron. Here we will assume a conservative value of s[4/8]m=2‰, based on a collection of 12 speleothems analysed by .

Disequilibrium corrections using measured ²³⁴U / ²³⁸U activity ratios are only feasible if those activity ratios are statistically distinguishable from secular equilibrium. To turn these conclusions into quantitative guidelines, let us define “statistically distinguishable” as “at least 3 × s[4/8]m removed from secular equilibrium”. Using this definition, a sample with [4/8]i < 2.7 would become indistinguishable from secular equilibrium after ca. 2 Ma. In other words, it is impossible to correct a 2 Ma sample whose [4/8]i = 2.7, say. The uncorrected ²⁰⁶Pb / ²³⁸U isochron age of such a sample would be 2.6 Ma, corresponding to a bias of 30 %. Table  shows the outcomes of the same exercise for a range of other ages.

Figure  presents a more extensive exploration of the magnitude (panel a) and precision (panel b) of disequilibrium-corrected ²⁰⁶Pb / ²³⁸U geochronology assuming the aforementioned 2 ‰ reproducibility. Alternative versions of this diagram can be generated by modifying the reproducibility value in the R code provided in the Supplement of .

Based on these considerations, we judge carbonate ²⁰⁶Pb / ²³⁸U geochronology to be unreliable beyond ca. 1.5 Ma and impossible beyond ca. 2 Ma unless initial ²³⁴U / ²³⁸U disequilibrium can be confidently ruled out (Fig. ). However, there is a solution to the conundrum of ²³⁴U / ²³⁸U disequilibrium. This solution is the ²⁰⁷Pb / ²³⁵U isochron method .

In the previous section, we showed that the ²⁰⁶Pb / ²³⁸U method's accuracy is hampered by the extreme enrichment (up to double the equilibrium value or more) of ²³⁴U observed in certain groundwaters (e.g. ). This problem can be solved by avoiding ²³⁴U altogether and sidestepping the ²³⁸U–²⁰⁶Pb decay chain in favour of the ²³⁵U–²⁰⁷Pb decay chain .

There are two kinds of ²⁰⁷Pb / ²³⁵U isochrons. The simplest kind plots ²⁰⁴Pb / ²⁰⁷Pb ratios against ²⁰⁴Pb / ²³⁵U ratios, defining the following linear relationship: 7204Pb207Pb=204Pb207Pbi1-235U207Pb(exp⁡[λ35t]-1), where ²⁰⁴Pb is used as a proxy for common Pb. Alternatively, one can also use ²⁰⁸Pb to fulfil this role. This gives rise to a ²⁰⁸Pbi/207Pb vs. ²³⁵U / ²⁰⁷Pb isochron, where ²⁰⁸Pb_i is the non-radiogenic ²⁰⁸Pb component (with the decay products of ²³²Th removed). Equation () is then replaced with 8208Pbi207Pb=208Pb207Pbi1-235U207Pbexp⁡[λ35t]-1, where 9208Pbi207Pb=208Pb207Pb-232Th207Pbexp⁡λ32t+1, in which λ32 = 0.0495(25) Gyr⁻¹ and the ²³²Th / ²⁰⁷Pb ratio can be obtained from the product of the ²³²Th / ²³⁸U, ²³⁸U / ²³⁵U, and ²³²Th / ²⁰⁷Pb ratios. Because Th is insoluble in water, radiogenic ²⁰⁸Pb is often absent from carbonates. Therefore, it is generally safe to assume that 208Pbi/207Pb≈208Pb/207Pb.

The ²³⁵U → ²⁰⁷Pb method has just one long-lived intermediate daughter, ²³¹Pa, which requires a correction. Due to ²³¹Pa's short half-life of 32.65 kyr (λ31 = 21.158(71) Myr⁻¹; ), it is generally not possible to measure any remaining disequilibrium in the Myr time range, where the ²³⁵U → ²⁰⁷Pb method offers a tangible advantage over the ²³⁸U → ²⁰⁶Pb method. Therefore, the [1/5]i value must be assumed.

Pa is chemically similar to Th and insoluble in water. Therefore, ²³¹Pa is always depleted relative to ²³⁵U in carbonates, so whereas [4/8]i can vary anywhere between 0 and 12 , [1/5]i is always less than 1 and can be safely assumed to be zero. In a worst-case scenario, in which one assumes [1/5]i = 0 but the true activity ratio is [1/5]i = 1, this would only bias the ²³⁵Pb / ²³⁵U age by a relatively small amount (Table ).

The degree of potential bias of the ²⁰⁷Pb / ²³⁵U method decreases with increasing age, unlike the ²⁰⁶Pb / ²³⁸U method, whose bias increases with age (Table ). In this sense, the ²⁰⁷Pb / ²³⁵U and ²⁰⁶Pb / ²³⁸U methods are complementary to each other. The ²⁰⁷Pb / ²³⁵U is most accurate for samples older than 1 Ma, whereas the ²⁰⁶Pb / ²³⁸U is more accurate for samples younger than 1 Ma. Note that the latter is similar to the applicability range of the ²³⁰Th / U method, so one could argue that disequilibrium-corrected ²⁰⁶Pb / ²³⁸U dating is of limited use to carbonate U–Pb geochronology (except to infer [4/8]i; ). Although the ²⁰⁷Pb / ²³⁵U method outperforms the ²⁰⁶Pb / ²³⁸U method at ca. 1 Ma in terms of accuracy, its poorer precision means that its potential benefits do not materialize until ca. 2 Ma. In the next section, we will demonstrate this by applying both methods to three different case studies.

Having made a largely theoretical case against ²⁰⁶Pb / ²³⁸U dating and for ²⁰⁷Pb / ²³⁵U dating of old carbonates that are suspected to have experienced initial ²³⁴U / ²³⁸U disequilibrium, we will now compare and contrast the two chronometers using three practical case studies. The first example will demonstrate the accuracy of the ²⁰⁷Pb / ²³⁵U method by showing its consistency with disequilibrium-corrected ²⁰⁶Pb / ²³⁸U dates of young (< 2 Ma) samples.

The second example uses ID-TIMS data to serve two purposes. First, it will show that the ²⁰⁷Pb / ²³⁵U method produces more accurate, more consistent, and more precise results than the ²⁰⁶Pb / ²³⁸U method for older (> 2 Ma) carbonates. Second, it will demonstrate how the Bayesian framework can use prior information to overcome the inaccuracy of the ²⁰⁶Pb / ²³⁸U method.

In the third case study we apply the ²⁰⁷Pb / ²³⁵U method to LA-ICP-MS data using ²⁰⁸Pb as a proxy for common Pb. In addition to highlighting a successful application where the ²⁰⁷Pb / ²³⁵U method produces demonstrably superior results to the ²⁰⁶Pb / ²³⁸U method, this dataset also illustrates limitations of the ²⁰⁷Pb / ²³⁵U approach with a sample that yields a precise ²⁰⁶Pb / ²³⁸U isochron and an unusably imprecise ²⁰⁷Pb / ²³⁵U isochron.

All the methods described in this paper have been implemented in the IsoplotR toolbox for geochronological data processing . The matrix exponential disequilibrium correction method of Sect.  has been part of IsoplotR since version 3.0, whereas the deterministic Bayesian uncertainty estimation routine of Sect.  was introduced in version 5.2. At the time of writing, IsoplotR (version 6.7) supports 12 different U–Pb data formats. Disequilibrium corrected U–Pb isochron regression is available for all these formats, in different forms.

Formats 1–3 contain neither ²⁰⁴Pb nor ²⁰⁸Pb. Therefore, isochron regression for these formats must be done by semitotal-Pb / U regression in Tera–Wasserburg concordia space. Formats 4–6 include ²⁰⁴Pb as a common-Pb tracer. These formats permit the calculation of both ²⁰⁶Pb / ²³⁸U and ²⁰⁷Pb / ²³⁵U isochrons, either jointly (by three-dimensional total-Pb / U isochron regression; ) or separately. To take full advantage of the ²⁰⁷Pb / ²³⁵U method's superior accuracy, it is recommended to use the two-dimensional option. Formats 7 and 8 use ²⁰⁸Pb as a common-Pb tracer. They are also amenable to both ²⁰⁶Pb / ²³⁸U and ²⁰⁷Pb / ²³⁵U isochron regression, either jointly (by total-Pb / U–Th regression; ) or separately. Formats 9–10 and 11–12 are simplified versions of formats 4–6 and 7–8, respectively, which only permit two-dimensional regression. Formats 9 and 11 are meant for ²⁰⁶Pb / ²³⁸U isochron regression, whereas formats 10 and 12 are meant for ²⁰⁷Pb / ²³⁵U isochron regression.

The disequilibrium corrections can be accessed from IsoplotR's GUI (either online or offline) by using the “isochron” function and ticking the “apply disequilibrium correction” check box in the options menu. Alternatively, the same functionality can also be accessed from the command-line API. Bayesian uncertainty estimation is possible using either interface, but visualizing the posterior distributions of the parameter space is currently only possible from the command line.

The Supplement provides all the R code that was used to reproduce the figures in this paper .

In this paper, we presented a critical appraisal of carbonate U–Pb geochronology and proposed three improvements to the technique. First, we introduced a matrix exponential solution to the initial disequilibrium problem, extending the work of . This formulation produces identical results to the conventional solution by but can be written out more succinctly and can more easily be modified to suit other problems. For example, the matrix exponential approach can be adjusted to calculate disequilibrium-corrected U–Th–He ages . Second, we presented a deterministic Bayesian algorithm to quantify the statistical uncertainty associated with the disequilibrium correction. This algorithm was used to demonstrate that, for samples older than ca. 2 Ma, disequilibrium-corrected ²⁰⁶Pb / ²³⁸U geochronology is unreliable. Third, we advocated the use of the ²⁰⁷Pb / ²³⁵U isochron method as a more accurate alternative to the ²⁰⁶Pb / ²³⁸U method.

Although our findings are most relevant to young carbonates, the inaccuracy of the ²⁰⁶Pb / ²³⁸U method equally applies to old samples. Only the relative difference between the ²⁰⁶Pb / ²³⁸U and ²⁰⁷Pb / ²³⁵U ages reduces with time. The absolute difference remains constant at up to 4 Myr (Eq. ). The corresponding systematic uncertainty cannot be removed without making unverifiable assumptions about the initial ²³⁴U / ²³⁸U activity ratio. For samples older than > 100 Ma, say, the systematic error caused by initial disequilibrium is generally smaller than the random errors associated with the isotope ratio measurements. However, given a sufficiently precise set of isochrons, it is theoretically possible to reconstruct the U disequilibrium conditions at the time of isotopic closure from the difference between the ²⁰⁶Pb / ²³⁸U and ²⁰⁷Pb / ²³⁵U clocks.

advocate using the same procedure in Quaternary studies. They propose a two-step procedure, whereby the difference between the ²⁰⁶Pb / ²³⁸U and ²⁰⁷Pb / ²³⁵U dates is used to estimate [4/8]i; and this [4/8]i value is then used to calculate a corrected ²⁰⁶Pb / ²³⁸U age. IsoplotR implements a one-step algorithm that achieves the same goal using the total-Pb / U algorithm of and the total-Pb / U–Th algorithm of . However, we would like to add a note of caution about the usefulness of this joint regression procedure. Beyond ca. 2 Ma, all the age-resolving power of the paired ²⁰⁶Pb / ²³⁸U and ²⁰⁷Pb / ²³⁵U approach resides in the ²⁰⁷Pb / ²³⁵U clock, so the ²⁰⁶Pb / ²³⁸U data add no value.