Propagating errors in decay equations: Examples from the Re-Os isotopic system

Statistical evaluation of radiogenic isotope data commonly makes use of the isochron method to determine closure age and initial isotopic composition which can be related to the source region from which the rocks or minerals were derived. Isochron regression algorithms also yield estimates of uncertainties in age and initial isotopic composition. However, geochemists frequently require an estimate of uncertainties associated with the calculation of initial isotopic composition and model age for single samples. This is often the case with Re-Os isotopic data for small sample suites that may not be isochronous. Here we describe two methods of propagating errors associated with Re-Os isotopic measurements in order to estimate uncertainties associated with both of these geologically important parameters; however, these methods are equally applicable to other isotopic systems. The first result is a set of analytical formulae that provide error estimates on both variables, even for the most general case where all dependent variables contain error, and all pairs of variables are correlated. This numerical approach leads to equations that can be easily and efficiently evaluated. A second Monte Carlo procedure was initially implemented to check the accuracy of the analytical formulae, although in the cases tested here it has also proved to be efficient and may even be practical for routine use. The advantage of error analysis of this type is that we can assign a level of confidence and thus significance to calculated initial isotopic compositions and model ages, especially for Archean rocks.