the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Short communication: Inverse isochron regression for Re–Os, K–Ca and other chronometers
Yang Li
Conventional Re–Os isochrons are based on mass spectrometric estimates of ${}^{\mathrm{187}}\mathrm{Re}{/}^{\mathrm{188}}\mathrm{Os}$ and ${}^{\mathrm{187}}\mathrm{Os}{/}^{\mathrm{188}}\mathrm{Os}$, which often exhibit
strong error correlations that may obscure potentially important geological complexity. Using an approach that is widely accepted in ${}^{\mathrm{40}}\mathrm{Ar}{/}^{\mathrm{39}}\mathrm{Ar}$ and U–Pb geochronology, we here show that these error correlations are greatly reduced by applying a simple change of variables, using ^{187}Os as a common denominator. Plotting
${}^{\mathrm{188}}\mathrm{Os}{/}^{\mathrm{187}}\mathrm{Os}$ vs. ${}^{\mathrm{187}}\mathrm{Re}{/}^{\mathrm{187}}\mathrm{Os}$ produces an
“inverse isochron”, defining a binary mixing line between an inherited
Os component whose ${}^{\mathrm{188}}\mathrm{Os}{/}^{\mathrm{187}}\mathrm{Os}$ ratio is given by the
vertical intercept, and the radiogenic ${}^{\mathrm{187}}\mathrm{Re}{/}^{\mathrm{187}}\mathrm{Os}$ 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
semisynthetic 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.
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The ${[}^{\mathrm{187}}\mathrm{Os}{/}^{\mathrm{188}}\mathrm{Os}]$ budget of a ^{187}Rebearing rock or mineral can be divided into an inherited component and a radiogenic component:
where λ_{187} is the decay constant of ^{187}Re ($=\mathrm{1.666}\pm \mathrm{0.017}$ yr^{−11}, Smoliar et al., 1996) and t is the time elapsed since isotopic closure. Equation (1) forms the equation of a line:
where x = $\left[{}^{\mathrm{187}}\mathrm{Re}{/}^{\mathrm{188}}\mathrm{Os}\right]$, y = $\left[{}^{\mathrm{187}}\mathrm{Os}{/}^{\mathrm{188}}\mathrm{Os}\right]$, a = $[{}^{\mathrm{187}}\mathrm{Os}{/}^{\mathrm{188}}\mathrm{Os}{]}_{i}$ and b = (exp [λ_{187}t]−1). Both the independent variable (x) and the dependent variable (y) are measured quantities that 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 drawback of the conventional isochron definition of Eq. (1) is that the rarest isotope, ^{188}Os, which is associated with the largest mass spectrometer uncertainties, appears in the denominator of both x and y. This has the potential to produce strong error correlations (Stein et al., 2000). For example, consider the following hypothetical (independent) abundance estimates and their standard errors:
Then, using the methods of Pearson (1896), the ratio correlation between ${[}^{\mathrm{187}}\mathrm{Os}{/}^{\mathrm{188}}\mathrm{Os}]$ and ${[}^{\mathrm{187}}\mathrm{Re}{/}^{\mathrm{188}}\mathrm{Os}]$ is
The strong error correlation between the two variables on the isochron diagram is 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) (Fig. 1a), which represents a mixture of three samples. 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 interpretation of this fit is not so simple:

The error ellipses exhibit a tremendous range of sizes. The plot is dominated by the least precise measurement (i.e. aliquot 14), and the remaining aliquots are barely visible.

The error ellipses are nearly perfectly aligned with the isochron, which makes it difficult to distinguish between geological and analytical sources of correlation.

The isochron fit exhibits a mean squared weighted deviation (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.
All three of these problems can be solved by a simple change of variables:
which defines an “inverse” isochron line:
where ${x}^{\prime}={[}^{\mathrm{187}}\mathrm{Re}{/}^{\mathrm{187}}\mathrm{Os}]$, ${y}^{\prime}={[}^{\mathrm{188}}\mathrm{Os}{/}^{\mathrm{187}}\mathrm{Os}]$, ${a}^{\prime}={{[}^{\mathrm{188}}\mathrm{Os}{/}^{\mathrm{187}}\mathrm{Os}]}_{i}$ and ${b}^{\prime}={{[}^{\mathrm{188}}\mathrm{Os}{/}^{\mathrm{187}}\mathrm{Os}]}_{i}\left(\mathrm{exp}\left[{\mathit{\lambda}}_{\mathrm{187}}t\right]\mathrm{1}\right)$.
Equation (4) defines a mixing line between the nonradiogenic ${[}^{\mathrm{188}}\mathrm{Os}{/}^{\mathrm{187}}\mathrm{Os}]$ ratio (which marks the vertical intercept) and the radiogenic ${[}^{\mathrm{187}}\mathrm{Re}{/}^{\mathrm{187}}\mathrm{Os}]$ ratio (which marks 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
Plotting the Morelli et al. (2007) dataset on an inverse isochron diagram provides a much clearer picture of it (Fig. 1b and c):

Although the error ellipses still exhibit a range of sizes, reflecting the heteroscedasticity of the data, the imprecise measurements no longer dominate the plot to the extent where they obscure the precise ones.

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 geological and analytical sources of correlation.

The overdispersion is clearly visible and can be attributed to aliquots 1, 12 and 14, whose error ellipses exhibit the smallest overlap with the bestfit line. Most of the geochronologically valuable information is contained in the highly radiogenic aliquots (7–11), which tightly cluster near the ${[}^{\mathrm{187}}\mathrm{Re}{/}^{\mathrm{187}}\mathrm{Os}]$ 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 ${[}^{\mathrm{187}}\mathrm{Os}{/}^{\mathrm{188}}\mathrm{Os}]$ ratio, however, is poorly constrained.
Strong error correlations are commonly observed in other conventional isochron systems, where they may arise from a number of mechanisms including poor counting statistics (previous sections), blank correction (e.g. Vermeesch, 2015; Connelly et al., 2017) or fractionation (e.g. Ludwig, 1980). Inverse isochron ratios, in which the radiogenic daughter isotope is used as a common denominator, are commonplace in ${}^{\mathrm{40}}\mathrm{Ar}{/}^{\mathrm{39}}\mathrm{Ar}$ (Turner, 1971) and U–Pb (Tera and Wasserburg, 1972) geochronology. They are equally applicable to other dating methods, such as Rb–Sr (${[}^{\mathrm{87}}\mathrm{Sr}{/}^{\mathrm{86}}\mathrm{Sr}]$ vs. ${[}^{\mathrm{87}}\mathrm{Rb}{/}^{\mathrm{86}}\mathrm{Sr}]$), Sm–Nd (${[}^{\mathrm{144}}\mathrm{Nd}{/}^{\mathrm{143}}\mathrm{Nd}]$ vs. ${[}^{\mathrm{147}}\mathrm{Sm}{/}^{\mathrm{143}}\mathrm{Nd}]$), Lu–Hf (${[}^{\mathrm{177}}\mathrm{Hf}{/}^{\mathrm{176}}\mathrm{Hf}]$ vs. ${[}^{\mathrm{176}}\mathrm{Lu}{/}^{\mathrm{176}}\mathrm{Hf}]$) and K–Ca (${[}^{\mathrm{44}}\mathrm{Ca}{/}^{\mathrm{40}}\mathrm{Ca}]$ vs. ${[}^{\mathrm{40}}\mathrm{K}{/}^{\mathrm{40}}\mathrm{Ca}]$).
In the case of K–Ca dating, the inverse approach offers similar benefits to those for the Re–Os method because ^{44}Ca is typically 100 times less abundant than ^{40}Ca, thus making the conventional isochron plot prone to strong error correlations. Note that some K–Ca studies use ^{42}Ca as a normalising isotope, which is even less abundant ^{44}Ca, and therefore further aggravates the problem. 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.
Given a data table of conventional isochron ratios (x and y in Eq. 2), it is possible to calculate the inverse ratios (x^{′} and y^{′} in Eq. 5), their uncertainties (s[x^{′}] and s[y^{′}]) and error correlations (${\mathit{\rho}}_{{x}^{\prime}{y}^{\prime}}$) 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.
Dalrymple et al. (1988) assert that conventional and inverse isochron regression are mathematically equivalent in the context of ${}^{\mathrm{40}}\mathrm{Ar}{/}^{\mathrm{39}}\mathrm{Ar}$ geochronology. This is indeed the case when the analytical uncertainties of the parent–daughter ratios are relatively small (<5 %, say), as is the case for the Re–Os example of Fig. 1. However, the equivalence breaks down when the analytical uncertainties are large, or when the data are significantly dispersed around the bestfitting isochron line. In those cases, the conventional and inverse isochrons can yield substantially different age estimates. This is because isotopic ratios are strictly positive quantities with skewed error distributions, and the weighted least squares algorithm of York et al. (2004) does not take this skewness into account.
A full theoretical discussion of this phenomenon falls outside the scope of our short communication. Instead, we will compare and contrast the accuracy of conventional and inverse isochrons using a semisynthetic dataset based on 30 K–Ca ion microprobe measurements published by Harrison et al. (2010):

Let x_{i} be the ith ${}^{\mathrm{40}}\mathrm{K}{/}^{\mathrm{44}}\mathrm{Ca}$ ratio measurement, and let σ[x_{i}], σ[y_{i}], ρ[x_{i},y_{i}] be the standard errors and error correlation of the corresponding ${}^{\mathrm{40}}\mathrm{K}{/}^{\mathrm{44}}\mathrm{Ca}$ and ${}^{\mathrm{40}}\mathrm{Ca}{/}^{\mathrm{44}}\mathrm{Ca}$ ratios.

Collect n pairs of log ratios {ln [X_{i}],ln [Y_{i}]} from a bivariate normal distribution with means {ln [x_{i}],ln [y_{i}]} and covariance matrix Σ_{i} where
$$\begin{array}{}\text{(8)}& {y}_{i}={y}_{o}+\mathrm{0.895}{x}_{i}\left(\mathrm{exp}\right[{\mathit{\lambda}}_{\mathrm{40}}t]\mathrm{1}),\end{array}$$in which y_{o}=66 is the initial ${}^{\mathrm{40}}\mathrm{Ca}{/}^{\mathrm{44}}\mathrm{Ca}$ ratio, t=800 Ma is the true K–Ca age, and
$$\begin{array}{rl}{\mathrm{\Sigma}}_{i}=& \left[\begin{array}{cc}\frac{\mathrm{1}}{{x}_{i}}& \mathrm{0}\\ \mathrm{0}& \frac{\mathrm{1}}{{y}_{i}}\end{array}\right]\\ & \left[\begin{array}{cc}\mathit{\sigma}[{x}_{i}{]}^{\mathrm{2}}& \mathit{\rho}[{x}_{i},{y}_{i}]\mathit{\sigma}\left[{x}_{i}\right]\mathit{\sigma}\left[{y}_{i}\right]\\ \mathit{\rho}[{x}_{i},{y}_{i}]\mathit{\sigma}\left[{x}_{i}\right]\mathit{\sigma}\left[{y}_{i}\right]& \mathit{\sigma}[{x}_{i}{]}^{\mathrm{2}}\end{array}\right]\\ & \left[\begin{array}{cc}\frac{\mathrm{1}}{{x}_{i}}& \mathrm{0}\\ \mathrm{0}& \frac{\mathrm{1}}{{y}_{i}}.\end{array}\right]\end{array}$$ 
The semisynthetic dataset is then given by {X_{i},Y_{i}} (for $\mathrm{1}\le i\le n$) with covariance matrices ${\mathrm{\Sigma}}_{i}^{\prime}$ that are computed as follows:
$${\mathrm{\Sigma}}_{i}^{\prime}=\left[\begin{array}{cc}{X}_{i}& \mathrm{0}\\ \mathrm{0}& {Y}_{i}\end{array}\right]{\mathrm{\Sigma}}_{i}\left[\begin{array}{cc}{X}_{i}& \mathrm{0}\\ \mathrm{0}& {Y}_{i}.\end{array}\right]$$
The logarithmic transformation is necessary to account for the inevitable skewness of the error distributions. Even though the semisynthetic dataset is defined in terms of the conventional isochron equation (Eq. 8), Fig. 2 shows that it is the inverse isochron that most accurately estimates the age. We therefore recommend that inverse isochrons replace conventional isochrons in Re–Os and K–Ca geochronology. The difference between the conventional and inverse isochron ages may serve as a measure of robustness for the results.
IsoplotR
Inverse isochrons have been added to all the relevant chronometers in
the IsoplotR
toolbox for radiometric geochronology
(Vermeesch, 2018). This functionality can be used either from the
graphical user interface (which can be accessed both online and
offline, Fig. 3a) or from the command line, using
the R programming language and application programming
interface (Fig. 3b and c). IsoplotR
automatically executes the ratio conversion of Eq. (7) in the background, so the user can supply their data as conventional ratios and still plot them on an inverse isochron diagram.
Conventional isochrons are straightline regressions between two ratios $D/d$ and $P/d$, where P and D are the parent and daughter nuclides, and d is a nonradiogenic 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 in Re–Os and K–Ca geochronology, which uses ^{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, causing strong error correlations. Besides this “spurious” source of correlated uncertainties (sensu Pearson, 1896), additional sources of covariance may include blank corrections, calibrations and fractionation effects that apply to both variables in the isochron regression.
The error 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 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 isochron line but inversely proportional to its horizontal intercept.
Published datasets (which are usually tabulated in a conventional
isochron format) can be reevaluated by transforming them to inverse
isochron ratios using Eq. (7), either
explicitly or internally within IsoplotR
. The two isochron
formulations produce identical results (Dalrymple et al., 1988) if the
relative uncertainties of the ratio measurements are reasonably small
(<5 %, say). In the presence of larger uncertainties, inverse
isochrons produce the most accurate results. We therefore recommend
that inverse isochrons are used instead of conventional isochrons for
Re–Os and K–Ca geochronology and any other datasets exhibiting
strong error correlations.
IsoplotR
is free software released under the GPL3 license. The package and its source code are available from https://cran.rproject.org/package=IsoplotR (last access: 26 July 2021, Vermeesch, 2021).
PV wrote the software and the paper. YL formulated the research question and contributed to the writing of the paper.
Pieter Vermeesch is an associate editor of Geochronology.
Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
We thank David Selby for feedback on an early version of the manuscript. Donald Davis, Ryan Ickert and an anonymous reviewer are thanked for their constructive reviews, which prompted us to develop the semisynthetic K–Ca model.
This research has been supported by the National Key Research and Development Program of China (grant no. 2018YFA0702600), the National Natural Science Foundation of China (grant no. 42022022) and the UK's Natural Environment Research Council (standard grant no. NE/T001518/1).
This paper was edited by Marissa Tremblay and reviewed by Donald Davis and one anonymous referee.
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 Abstract
 Introduction: the conventional Re–Os isochron
 The inverse Re–Os isochron
 Application to other chronometers
 A semisynthetic test of accuracy

Implementation in
IsoplotR
 Conclusions
 Code and data availability
 Author contributions
 Competing interests
 Disclaimer
 Acknowledgements
 Financial support
 Review statement
 References
The requested paper has a corresponding corrigendum published. Please read the corrigendum first before downloading the article.
 Article
(1974 KB)  Fulltext XML
 Abstract
 Introduction: the conventional Re–Os isochron
 The inverse Re–Os isochron
 Application to other chronometers
 A semisynthetic test of accuracy

Implementation in
IsoplotR
 Conclusions
 Code and data availability
 Author contributions
 Competing interests
 Disclaimer
 Acknowledgements
 Financial support
 Review statement
 References