Numerical error analysis in Ar-40/Ar-39 dating

Calculation of an Ar-40/Ar-39 age involves several sources of systematic (e xternal) and statistic (mostly instrumental) errors that should be propagat ed into the final result for a proper statistical assessment of the age unc ertainty and the overall resolution of the Ar-40/Ar-39 dating technique. Er ror propagation is usually carried out by linearized error expansion techni ques that weight the relative variance contribution of each input parameter by the squared partial derivative of the age function relative to this var iable. Computation of partial derivatives through the working Ar-40/Ar-39 e quations is tedious and error-prone, however. As a result, several data red uction schemes using different levels of approximation are implemented in v arious laboratories, some of which ignore certain sources of error while ot hers use simplified error equations, thus making direct comparison of publi shed age and error estimates sometimes inaccurate. Based on the general num erical approach outlined by Roddick (1987) [Roddick, J.C., 1987. Generalize d numerical error analysis with applications to geochronology and thermodyn amics. Geochim. Cosmochim. Acta 51, 2129-2135], a complete Ar-40/Ar-39 nume rical error analysis (NEA) is proposed that includes up to 28 possible sour ces of error. The NEA code of Roddick (1987) is recast into a more rigorous central finite-difference (CFD) scheme, and applied to three non-ideal, wo rked Ar-40/Ar-39 examples to test underpinning assumptions of the linearize d error propagation by extending the error analysis to higher-order terms o f the Taylor expansion of the age equation. Close to very close agreement b etween the analytic and numerical solutions suggests that the linearized er ror expansion technique is justified for Ar-40/Ar-39 error propagation, des pite strong nonlinearity in the related equations. in one pathological inst ance, nonlinearity is flagged by significant (15%) departure from the algeb raic solution. The linearized age error estimate is still found to be accep tably close to the (exact) NEA estimate, provided however that covariance b etween Ar-40* and Ar-39(K) is precisely accounted for. As most Ar-40/Ar-39 datasets will be invariably corrupted by large covariance corrections, a fu ll-fledged error analysis as made possible through NEA is clearly desirable in most situations. The demonstrated flexibility of the numeric approach s hould be profitably extended to other areas of isotope geochemistry involvi ng complex calculation codes such as treated here. (C) 2000 Elsevier Scienc e B.V. All rights reserved.