26 Feb 2021
26 Feb 2021
Short communication: Inverse isochron regression for Re–Os, K–Ca and other chronometers
 ^{1}State Key Laboratory of Lithospheric Evolution, Institute of Geology and Geophysics, Chinese Academy of Sciences, Beijing 100029
 ^{2}Department of Earth Sciences, University College London, Gower Street, London WC1E 6BT
 ^{1}State Key Laboratory of Lithospheric Evolution, Institute of Geology and Geophysics, Chinese Academy of Sciences, Beijing 100029
 ^{2}Department of Earth Sciences, University College London, Gower Street, London WC1E 6BT
Abstract. Conventional Re–Os isochrons are based on mass spectrometric estimates of ^{187}Re / ^{188}Os and ^{187}Os / ^{188}Os. ^{188}Os is usually far less abundant, and is therefore measured less precisely, than ^{187}Os and ^{187}Re. This causes strong error correlations between the two isochron ratios, which may obscure potentially important geological complexity. Using an approach that is widely accepted in ^{40}Ar / ^{39}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 ^{188}Os / ^{187}Os vs. ^{187}Re / ^{187}Os produces an inverse isochron
, defining a binary mixing line between an inherited Oscomponent whose ^{188}Os / ^{187}Osratio is given by the vertical intercept, and the radiogenic ^{187}Re / ^{187}Osratio, 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. 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.
Yang Li and Pieter Vermeesch
Status: final response (author comments only)

RC1: 'Comment on gchron20217', Donald Davis, 08 Mar 2021
At first glance, this manuscript seems to point out fairly obvious truths about presentation of isochron data but on reflection I have concluded that it is of significant interest to the community.
For historical reasons, isotopic ratios for singledecay radioactive systems where the sample contains significant proportions of nonradiogenic daughter isotope have usually been expressed on a 2D graph where parent and daughter isotopes appear as numerators along the X and Y axes, respectively, and another isotope of the daughter element appears as a common denominator. Analyses of undisturbed coeval samples with a common initial daughter isotopic system should form a linear array on this diagram in which the slope is a function of age and the Y axis intercept gives the initial isotopic composition of the daughter element. As the authors point out, an alternative way to express the data is to plot the parent over daughter isotope on the X axis and a nonradiogenic isotope of the daughter element over the radiogenic isotope on the Y axis. This is usually called an inverse isotope diagram and it has advantages, probably in most cases, but certainly when errors are large and strongly correlated.
One advantage of the inverse plot is that data clearly lie along a mixing line between a pure radiogenic component along the X axis (which could be expressed as age instead of parent over daughter ratio) and a pure initial ratio of the daughter element (the ratio one would get for a sample with parent/daughter equal to zero). The traditional isochron plot is a mixing line too but the radiogenic end member lies at infinity so the age must be calculated from the slope. This is much less intuitive, so I find it hard to see any circumstances in which the traditional isochron plot is better. Therefore, I think that the authors should suggest that the community avoid it completely.
It should be pointed out that the approximate symbol for the correlation coefficient in equation 4 derives from the fact that it is first order approximation to a Taylor expansion, as shown in Vermeesch (2018). Regarding this, it might be interesting if the authors could explore the limitations. For example, if the same data are correctly regressed in both traditional and inverse plots, one should get exactly the same answer for age and initial ratio but because of the firstorder approximation, this should only be true for data with relatively small errors. For example, the age and initial 187Os/188Os ratio given at the top of Fig 1 presumably is the Morelli solution from the normal isochron. The solution from the inverse isochron is not given (it should be) but the 188Os/187Os intercept of about 1.45 on the bottom inverse plot does not correspond to the reciprocal of the initial ratio found from the normal isochron, which is 1.64. It might be interesting to explore just how large errors have to be to produce significant discrepancies and whether or not overdispersion plays a role. That begs the question of which plot gives the closest approximation. One might be able to test this with synthetic data. Even without this, I think that the note is worth publishing but having it would be a new contribution.
Line 32: “are manifested” should be “is manifested”.
Don Davis
 AC1: 'Reply on RC1', Pieter Vermeesch, 07 Apr 2021

CC1: 'Comment on gchron20217', Ryan Ickert, 02 Apr 2021
Community Comment by Ryan Ickert
This is a short manuscript describing how a simple change of variables can be used in isochron plots to reduce the magnitude of the uncertainty correlation. In cases where the uncertainty correlation is very high, the authors argue that these correlations obscure the data structure on conventional plots, making it difficult to use the plots to make decisions about how to use the data, and that the change of variables can be used to display the data in a better manner. The authors provide a set of three equations that can be used to transform the uncertainties and correlations from x/z vs y/z to x/y vs z/y and, apply these to two previously published datasets.
I agree with reviewer Davis that this approach is not new to isotope geochemistry. For example, the highly cited work by Williams (1998; Reviews in Economic Geology v7 p135) explains that one of “(t)he benefits of the TeraWasserburg plot (is)…the errors in measuring ^{207}Pb/^{206}Pb and ^{238}U/^{206}Pb are much more weakly correlated than those in measuring ^{207}Pb/^{235}U and ^{206}Pb/^{238}U…”. And in the popular textbook by McDougall and Harrison (1999; p 113) they state that “…a potential drawback of the conventional isochron plot is that, in general, the isotope measured with the poorest precision, ^{36}Ar, is common to both axes. A result is that the errors associated with both axes are highly correlated and may give rise to misleading linear correlations if the errors or the correlation coefficient are incorrectly estimated.” They later go on “These problems are largely circumvented by an alternate form of isochron analysis in which ^{36}Ar/^{40}Ar is plotted against ^{39}Ar/^{40}Ar…”.
Although the authors of this manuscript appear to be aware that their work is not new (line 66) they don’t make clear what differentiates this contribution from others. The manuscript would be improved if it were better able to highlight a novel contribution.
I agree that plotting highly correlated ellipses is rather annoying, and in rare cases can produce plots that obscure relationships between data – as do many other geochronologists, which is why, as discussed above, many use TeraWasserburg diagrams and Ar isotope inverse isochrons.
The argument that it more easily allows outlier identification is not particularly compelling: The ReOs example in Fig 1 C is unconvincing – the outliers they “identify” on the plot are not clear, at least to me, and anyways that result is muddled somewhat by the fact that they have mixed samples of likely different ages on the same diagram (as described in the original paper). A better way to identify data that have undue weight on the MSWD is to simply inspect the variance normalized residuals and look for the largest values.
On the other hand, it would be nice to have the equation for the change of variables published somewhere clear and convenient, with a brief discussion of the (real) source(s) of correlation in geochronological data, and dispelling some of the myths about the change of variables. For the latter, there is a persistent belief in some workers that ages determined by one regression are better or more precise than using an inverse or vice versa (e.g., Connelly et al., 2017). This would be trivial for the authors to include, by producing regression analysis on both the isochron and it’s inverse and demonstrating substantive equivalence. This is complicated somewhat by the fact that the two regressions become significantly distinct with extremely large uncertainties (as they state on line 100) and also with highly overdispersed data, but it is easy to carve out that as an exception.
As written, the manuscript gives a misleading impression about the origin of correlations in isotopic and geochronological data. While poor counting statistics on denominator isotopes may be important in some Ar isotope datasets, most uncertainty correlations in real, published datasets are due to other factors, such as fractionation corrections, interelement calibration, and blank corrections. See for example, Connelly et al. (2017 GCA V201 p345363), Ludwig (1980 EPSL v46) and the isotope geochemistry textbook by Dickin (2005). This manuscript would be improved if this variety were better covered, however briefly.
I agree with Reviewer Davis that if the manuscript is edited to reflect changes along these lines, it would be a much better candidate for publication.
Minor elements:
Line 1516: It is not stated that the other variables are the present day abundances (or relative abundances, as in the next equation in the text) of 187Os and 187Re and can be directly measured. It’s probably worth making this explicit rather than implying it – for the nonspecialist this might be unnecessarily confusing because everything else is defined.
Line 15: It’s probably a good idea to list the sigma level/coverage factor/confidence limit/credibility interval for the uncertainty.
Line 19 (equation 2): The statement preceding the equation implies that normalizing to 188Os contributes to the equation no longer being underdetermined. This is not correct, of course, and I don’t think that’s what the authors intend to mean, so it should be clarified. There’s really no reason to start at equation 1 – this isn’t a lesson in the basics of isochrons – and for the sake of brevity it would probably be more useful to start with equation 2 and eliminate reference to equation 1 altogether.
Line 27: (Here and elsewhere). The word spurious is defined as “not being what it purports to be; false or fake” or “(of a line of reasoning) apparently but not actually valid”. This adjective does not apply to these correlations, which, as the authors themselves demonstrate on line 29, are real, valid correlations. Perhaps the authors mean to suggest that they are annoying, distracting, or otherwise unwanted, but they are not at all spurious. A spurious correlation is something more akin to the classic “pirates are causing global warming” example (and many others, cf. https://www.tylervigen.com/spuriouscorrelations). This word should not be used in the manuscript to describe any of the correlations, which are all real.
Line 2534: While this section is technically correct, it is misleading because many (probably most?) correlations in isochrontype plots that use more than one element (this is particularly true for measurements by isotope dilution like RbSr, SmNd, KCa, ReOs etc.) are not due to the common denominator effect, but a common correction factor. This is explained in textbooks, such as Dickin (2005, 2^{nd} edition page 36).
Line 3543: This section is misleading, being predicated on a naïve reading of the isochron and the data, and ignores the original interpretation by Morelli et al. No one “at first glance” would read this plot as having an isochron age of 287 Ma – this is something that would be calculated. Such a calculation would almost inevitably be accompanied by an uncertainty and an MSWD, and the overdispersion would then be detected by inspection of the MSWD and pvalue, regardless of whether the data are plotted. Consulting the original paper regarding the source of the overdispersion, reveals that these 16 analyses are an aggregate of 3 different samples. Individually, each of the three samples yield isochrons with little to no overdispersion. The original paper interprets the overdispersion evident upon aggregation as meaning that sample 20A has a different age than the other two. This information should be included in the manuscript and the “outlier identification” and original interpretation of the data should be discussed together.
Unlike points 1 and 3, point 2 is significant and should be the focus – highly correlated analyses are difficult to plot well and present a barrier to effective data communication.
Line 7073: The authors seem to be unaware that low abundance denominators are not the only reason for correlations. Blank corrections can be a significant source of correlation (Connelly et al. 2017 GCA V201 p345363), as can fractionation corrections (Ludwig, 1980 EPSL v46). It might be worth pointing this out here.
Equation 8: I’ve checked the equation and it appears to be correct.
Section 4: This whole section seems superfluous. A statement at the end of the manuscript stating that “these calculations are implemented in Isoplot R” is sufficient. The paragraph and screen grab are unnecessary.
Line 99: What does “mathematically equivalent” mean? For example, using the York et al. (2004) MLE algorithm on the Morelli dataset, the results from the conventional and inverse isochrons are 287.26+/ 1.70 Ma and 287.16 +/ 1.73 Ma. These agree within uncertainty but I don’t think that 287.26 and 287.16 are “mathematically equivalent”. They are not even consistent to within numerical precision (I assume, but do not know, that the difference is due to the Taylor series approximation)
Line 100: Like the previous reviewer, I think it would be fantastic and useful if the authors could expand on this, though I appreciate it may be out of scope. I would suggest that they also include mention that the regression and inverse will be also distinct if the data are highly overdispersed.
Conflict of interest: I am married to Associate Editor Marissa Tremblay
 AC2: 'Reply on CC1', Pieter Vermeesch, 09 Apr 2021

RC2: 'Comment on gchron20217', Anonymous Referee #2, 13 Apr 2021
The authors present an update to their IsoplotR software package enabling the calculation of inverse isochrons, and extoll some of the virtues thereof. While inverse isochrons themselves are hardly new, and have even been applied to the ReOs system before, I have no objection to a paper reminding the community of the utility of inverse isochrons (or really more broadly, the importance of fully accounting for covariances between variables during isochron calculations)  and in any event, the new opensource software enabling the more convenient calculation of such isochrons for multiple isotopic systems is certainly welcome and publicationworthy.
I have not checked in detail of the correctness of all the equations herein, but given the established nature of the inverse isochron approach in general, and the capabilities of the authors, I do not expect any problems.

AC3: 'Reply on RC2', Pieter Vermeesch, 14 Apr 2021
We are pleased that the reviewer found our contribution useful. (S)he claims that inverse isochrons have previously been applied to the ReOs method but, unfortunately, does not provide any actual examples of this. We looked but did not find any inverse ReOs, KCa, RbSr, SmNd or LuHf isochrons in the geological literature. If the reviewer could help us find such examples, then we would be more than happy to include them into the revised manuscript.

AC3: 'Reply on RC2', Pieter Vermeesch, 14 Apr 2021
Yang Li and Pieter Vermeesch
Yang Li and Pieter Vermeesch
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