Abstract
Abstract. The standard classical statistics approach to isochron calculation assumes that the distribution of uncertainties on the data arising from isotopic analysis is strictly Gaussian. This effectively excludes datasets that have more scatter from consideration, even though many appear to have age significance. A new approach to isochron calculations is developed in order to circumvent this problem, requiring only that the central part of the uncertainty distribution of the data defines a “spine” in the trend of the data. This central spine can be Gaussian but this is not a requirement. This approach significantly increases the range of datasets from which age information can be extracted but also provides seamless integration with well-behaved datasets and thus all legacy age determinations. The approach is built on the robust statistics of Huber (1981) but using the data uncertainties for the scale of data scatter around the spine rather than a scale derived from the scatter itself, ignoring the data uncertainties. This robust data fitting reliably determines the position of the spine when applied to data with outliers but converges on the classical statistics approach for datasets without outliers. The spine width is determined by a robust measure, the normalised median absolute deviation of the distances of the data points to the centre of the spine, divided by the uncertainties on the distances. A test is provided to ascertain that there is a spine in the data, requiring that the spine width is consistent with the uncertainties expected for Gaussian-distributed data. An iteratively reweighted least squares algorithm is presented to calculate the position of the robust line and its uncertainty, accompanied by an implementation in Python.
Highlights
The ability to fit a straight line through a body of isotope ratio data in order to form an isochron is the cornerstone of many geochronological methods
This work was motivated by the belief that many isotopic datasets contain meaningful age information that cannot be identified using classical statistical methods and may be discarded or discounted
The age information is contained in a linear spine in the data, but the data contain scatter that is inconsistent with a Gaussian uncertainty distribution, having fatter tails than Gaussian
Summary
The ability to fit a straight line through a body of isotope ratio data in order to form an isochron is the cornerstone of many geochronological methods In detail, this is a non-trivial task, since uncertainties are usually associated with all variables, and these are often correlated, precluding simple “least squares” line-fitting techniques. This approach, referred to here as YORK, became entrenched in the geochemical community, in the last two decades as the essential component of the very widely used software, ISOPLOT, e.g. Our primary focus here will be on general-purpose isochron calculations that determine the age of an “event” that established the isotopic compositions of samples in a dataset This involves what are called model 1 and 2 calculations in ISOPLOT – as described below. Approaches that try to extract detail within events, including ISOPLOT model 3 calculations, are not considered (but see, e.g. Vermeesch, 2018)
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