Short communication: Inverse isochron regression for Re–Os, K–Ca and other chronometers

Conventional Re–Os isochrons are based on mass spectrometric estimates of 187Re/188Os and 187Os/188Os, which often exhibit strong error correlations that may obscure potentially important geological complexity. Using an approach that is widely accepted in 40Ar/39Ar and U–Pb geochronology, we here show that these error correlations are greatly reduced by applying a simple change of variables, using 187Os as a common denominator. Plotting 188Os/187Os vs. 187Re/187Os produces an “inverse isochron”, defining a binary mixing line between an inherited Os component whose 188Os/187Os ratio is given by the vertical intercept, and the radiogenic 187Re/187Os ratio, which corresponds to the horizontal intercept. Inverse isochrons facilitate the identification of outliers and other sources of data dispersion. They can also be applied to other geochronometers such as the K–Ca method and (with less dramatic results) the Rb–Sr, Sm–Nd and Lu–Hf methods. Conventional and inverse isochron ages are similar for precise datasets but may significantly diverge for imprecise ones. A semi-synthetic data simulation indicates that, in the latter case, the inverse isochron age is more accurate. The generalised inverse isochron method has been added to the IsoplotR toolbox for geochronology, which automatically converts conventional isochron ratios into inverse ratios, and vice versa.


Introduction: the conventional Re-Os isochron
The Re-Os method is based on the β-decay of 187 Re to 187 Os: where λ 187 is the decay constant of 187 Re (= 0.01666×0.00017 Gyr −1 , Smoliar et al., 1996) Equation 2 forms the equation of a line: where x = 187 Re/ 188 Os y = 187 Os/ 188 Os , a = 187 Os/ 188 Os i , and b = (exp[λ 187 t] − 1). Both the independent variable (x) and the dependent variable (y) are associated with analytical uncertainty. Therefore, linear regression of the isochron line is typically done by weighted least squares regression with uncertainty in both variables (York et al., 2004).
One problem with the conventional isochron definition of Equation 2 is that the rarest isotope, 188 Os, which is associated 25 with the largest mass spectrometer uncertainties, appears in the denominator of both x and y. This has the potential to produce strong spurious error correlations (Pearson, 1896). For example, consider the following hypothetical (independent) abundance estimates: X ≡ 187 Os = 2, 000 ± 10 fmol; Y ≡ 187 Re = 30, 000 ± 50 fmol and Z ≡ 188 Os = 10 ± 2 fmol then the ( 187 Os/ 188 Os) and ( 187 Re/ 188 Os) isotope ratio estimates exhibit a correlation coefficient of The strong error correlation between the two variables on the isochron diagram are manifested as narrow and steeply inclined error ellipses, which may graphically obscure any geologically significant trend.
As an example, consider the Re-Os dataset of Morelli et al. (2007) (Figure 1a). At first glance, this dataset appears to define an excellent isochron with a clear slope corresponding to an isochron age of 287 Ma. However upon closer inspection, the 35 interpretation of this fit is not so simple: 1. The isochon fit exhibits an MSWD of 2.5, which indicates the presence of a moderate amount of overdispersion of the data with respect to the formal analytical uncertainties. It is not immediately clear which aliquots are responsible for the poor goodness-of-fit.
2. The error ellipses exhibit a tremendous range of sizes. The plot is dominated by the least precise measurement (i.e. 40 aliquot 14), and the remaining aliquots are barely visible.
3. The error ellipses are nearly perfectly aligned with the isochron, which makes it difficult to distinguish between geochronologically significant and statistically spurious sources of correlation.

The inverse Re-Os isochron
All three of these problems can be solved by a simple change of variables: which defines an 'inverse' isochron line: Equation 5 defines a mixing line between the non-radiogenic 188 Os/ 187 Os -ratio (which marks the vertical intercept) and the radiogenic 187 Re/ 187 Os -ratio (which markes the horizontal intercept). By moving the least abundant nuclide to the numerator of the dependent variable, instead of the denominator of both the dependent and the independent variables, the inverse isochron reduces the error correlations. Revisiting the earlier hypothetical example yields an error correlation of:  1. The overdispersion is clearly visible and can be attributed to aliquots 1, 12 and 14. Most of the geochronologically valuable information is contained in the highly radiogenic aliquots 7-11, which tightly cluster near the [ 187 Re/ 187 Os]intercept. Even though the data are overdispersed, the overall composition is very radiogenic and can therefore be used to obtain precise age constraints. The initial [ 187 Os/ 188 Os]-ratio, however, is poorly constrained. 60 2. Although the error ellipses still exhibit a range of sizes, reflecting the heteroscedasticity of the data, the imprecise measurements do no longer dominate the plot to the extent where they obscure the precise ones.
3. The error ellipses are no longer aligned parallel to the isochron line, but are oriented at an angle to it. This makes it easier to see the difference between the geochronological and statistical sources of correlation.

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Inverse isochron ratios, in which the radiogenic daughter isotope is used as a common denominator, are frequently used in 40 Ar/ 39 Ar (Turner, 1971) and U-Pb (Tera and Wasserburg, 1972)  plot prone to strong error correlations (Figure 2). For other chronometers such as Rb-Sr, Sm-Nd and Lu-Hf, whose nonradiogenic isotopes are at least as abundant as the radiogenic daughter isotopes, the benefits of the inverse isochron approach are less obvious. ) and error correlations (ρ x y ) using the following equations: This transformation is perfectly symmetric in the sense that it can also be used to convert inverse isochron ratios to conventional ones. To do this, it suffices to swap x and y for x and y and vice versa.

Implementation in IsoplotR
Inverse isochrons have been added to all the relevant chronometers in the IsoplotR toolbox for radiometric geochronology 80 (Vermeesch, 2018). This functionality can be used either from the graphical user interface (which can be accessed both online and offline, Figure 3a), or from the command line, using the R programming language and application programming interface

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Conventional isochrons are straight line regressions between two ratios D/d and D/P , where P and D are the parent and daughter nuclides, and d is a non-radiogenic isotope of the daughter element. This paper reviewed the phenomenon whereby strong error correlations arise when d is less abundant than D, and is therefore measured less precisely than D. This is the case 6 https://doi.org/10.5194/gchron-2021-7 Preprint. Discussion started: 26 February 2021 c Author(s) 2021. CC BY 4.0 License.
in Re-Os and K-Ca geochronology, which use 188 Os and 44 Ca as normalising isotopes, respectively. These isotopes are tens to hundreds of times less abundant than the radiogenic 187 Os and 40 Ca. Note that some K-Ca studies use 42 Ca as a normalising 90 isotope, which is even less abundant 44 Ca, and therefore further aggravates the problem.
The spurious correlation between the isochron ratio measurements can be so strong (r > 0.99) that it outweighs and obscures the geochronological correlation. This is not only inconvenient from an esthetic point of view, but may also cause numerical problems. It is not uncommon for data tables to either not report error correlations at all, or to report them to only one significant digit. However, the difference between error correlations of r = 0.991 and r = 0.999, say, may have a large effect on the 95 isochron age. All these problems can be solved by recasting the isochron regression into a new form, by plotting d/D vs. P/D.
This produces a different type of linear trend, in which the vertical intercept yields the reciprocal daughter ratio, and the age is not proportional to the slope of the isochorn line, but inversely proportional to its horizontal intercept. The two formulations are mathematically equivalent (Dalrymple et al., 1988) provided that the relative uncertainties of the ratio measurements are reasonably small (< 10%, say).

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Code and data availability. IsoplotR is free software released under the GPL-3 license. The package and its source code are available from https://cran.r-project.org/package=IsoplotR.
Author contributions. PV wrote the software and the paper. YL formulated the research question and contributed to the writing of the paper.
Competing interests. Pieter Vermeesch is an associate editor of Geochronology.