Isochrons are usually fitted by “York regression”, which uses a weighted least-squares approach that accounts for correlated uncertainties in both variables. Despite its tremendous popularity in modern geochronology, the York algorithm has two important limitations that reduce its utility in several applications. First, it does not provide a satisfactory mechanism to deal with so-called “errorchrons”, i.e. datasets that are overdispersed with respect to the analytical uncertainties. Second, York regression is not readily amenable to anchoring, in which either the slope or the intercept of the isochron is fixed based on some external information. Anchored isochrons can be very useful in cases where the data are insufficiently spread out to constrain both the radiogenic and non-radiogenic isotopic composition.

This paper addresses both of these issues by extending a maximum likelihood algorithm that was first proposed by Titterington and Halliday (1979). The new algorithm offers the ability to attribute any excess dispersion to either the inherited component (“model 3a”) or diachronous closure of the isotopic system (“model 3b”). It provides an opportunity to anchor isochrons to either a fixed non-radiogenic composition or a fixed age. Last but not least, it allows the user to attach meaningful analytical uncertainty to the anchor. The new method has been implemented in IsoplotR for immediate use in

Isochrons are mixing lines between radiogenic and inherited isotopic endmembers. They are an essential component of radiometric geochronology and exist in several forms. Sections

Equations (

Although this paper will use the York parameters

The most accurate and precise results are obtained from samples that are evenly spread along the isochron line, spanning the entire range of values from the inherited to the radiogenic endmember. Unfortunately, this condition is not always fulfilled. It is not uncommon for most or all aliquots in a sample to cluster together at one point along the isochron, making it difficult to accurately estimate the endmember compositions. For example, no precise isochron ages can be obtained from samples whose radiogenic daughter component is dwarfed by the inherited daughter component. Conversely, the composition of the inherited component cannot be precisely estimated in extremely radiogenic samples. Finally, when the data cluster together in the middle, then neither endmember component can be reliably determined (Fig.

Sometimes, such poorly constrained isochrons can be fixed using external information. For example, if the composition of the inherited component is known through some independent means (e.g. by analysing a cogenetic mineral that is naturally poor in

There is currently no formally documented way to anchor York regression. A commonly used “hack” is to add an extra data point with infinite precision representing either the inherited or radiogenic endmember component. However, this hack does not provide a satisfactory mechanism to assign uncertainty to the anchor. This paper solves that problem. Section

Section

Section

The method of maximum likelihood is a standard statistical technique to estimate the parameters of a probability distribution from a set of measurements. In the case of two-dimensional isochron regression, the parameters are the intercept

This overconstrained problem can be solved by minimizing the paired differences (“residuals”) between the true

The degree to which the residuals

Inflate the analytical uncertainties by a factor

Ignore the analytical uncertainties and replace York regression with orthogonal regression or a similar technique. This approach will not be discussed further in this paper.

Quantify the dispersion as an additional free parameter. There are two options for doing so.

Model 3a isochron regression can be formalized by modifying Eq. (

Three specific cases of Eq. (

This procedure can easily be adapted to model 3 regression by adding

For inverse isochrons, model 3b regression does not actually provide any useful information. Unlike conventional isochrons, whose chronological information is contained in the slope, the chronological information of inverse isochrons is contained in their horizontal intercept. This information can be unlocked by flipping the axes of the isochron diagram around, inverting the isochron, and treating the

The same trick can be used to estimate the slope uncertainty of a conventional isochron. A pragmatic way to avoid the slow convergence rate of model 3b regression is to invert the isochron, flip the dependent and independent variables around, invert the isochron a second time, and carry out a model 3a regression on the transformed data. The resulting

Model 3 regression of two synthetic Ar–Ar datasets.

Anchored isochron regression requires just a trivial modification of the maximum likelihood algorithm. It suffices to treat the anchored parameter as data. For example, the slope and intercept of a model 1 isochron can be anchored by maximizing

When assigning uncertainty to statistical parameters, it is important to clearly define the meaning of this uncertainty. In the case of anchored isochron regression, the uncertainty of the intercept or slope can carry two meanings. For example, when the intercept of an isochron is anchored at a value

The data are underlain by a single isochron whose intercept is only approximately known, with a most likely value of

The data were drawn from a family of isochron lines whose intercepts follow a normal distribution with mean

These two different approaches can be implemented by replacing the log-likelihood functions of Eq. (

Figure

Model 3 partitions the analytical and geological uncertainty between the standard errors of

It is, of course, also possible to treat the dispersion parameter as an unknown by maximizing

Anchored isochrons for the data of Fig.

The

Unlike inverse

There is no need for flipped isochron regression of

The

It is relatively straightforward to generalize the concept of model 3 regression to total

Most published

The semitotal

Figure

Model 3a semitotal

All the algorithms presented in this paper have been implemented in the free and open geochronological toolbox IsoplotR

In the GUI, anchored regression is available from the “isochron” menu (so not from the “concordia” menu for

Finally, carry out model 3b isochron regression anchored to a radiogenic composition that was set at

This paper builds on previous work by

The maximum likelihood formulation of isochron regression can also be used to anchor isochrons to either the inherited or the radiogenic component and to assign geologically meaningful uncertainty to the anchor. In reality it is possible that some samples are affected by both mechanisms so that different aliquots of the same sample differ in their initial ratios as well as the timing of their isotopic closure. Unfortunately, it is not possible to simultaneously capture both types of dispersion using the algorithms of this paper.

The difference between model 1 and model 3 regression represents two contrasting views of geological reality. The model 1 approach assumes that the isotopic composition of minerals represents discrete components recorded at distinct events. In contrast, model 3 isochrons represent a “fuzzier” reality, in which the initial composition or timing of isotopic closure is allowed to vary within a rock. Under the latter model, the concept of an errorchron no longer makes sense.

All uncertainties in this paper are reported as 95 % confidence intervals except if noted otherwise.

Assignment of the isotopic ratios for

The synthetic data in Figs.

Generate

Generate

Turn the resulting 10 pairs of

Use each of the

Use the Poisson values obtained in the previous step to form 10 pairs of

Add some excess dispersion by shrinking the uncertainties by 20 %.

Whether a dataset is overdispersed with respect to the analytical uncertainties can be formally assessed using the chi-square statistic and test. To this end, calculate the

Section

IsoplotR is available from CRAN (

This paper only uses synthetic data, which were generated using the procedures outlined in Appendix B.

The author is a member of the editorial board of

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

The author would like to thank reviewers Donald Davis and John Rudge for suggesting to expand the discussion of model 3 regression and for checking the equations. This work was completed during a sabbatical stay at the Institute of Geology and Geophysics, Chinese Academy of Sciences, which was supported by the CAS President's International Fellowship Initiative (PIFI). The author would like to thank Xian-Hua Li of IGG-CAS and Yang Li of Peking University for hosting him at IGG-CAS. He would also like to thank IsoplotR users Guilhem Hoareau (CNRS, France), Shitou Wu (IGG-CAS, China), and Stijn Glorie (Adelaide, Australia) for discussing the utility of anchored isochrons and moving this feature up the priority list for IsoplotR.

This research has been supported by the Natural Environment Research Council (grant no. NE/T001518/1).

This paper was edited by Noah M McLean and reviewed by Donald Davis and John Rudge.