Numerical error analysis in 40Ar/39Ar dating

Calculation of an 40Ar/39Ar age involves several sources of systematic (external) and statistic (mostly instrumental) errors that should be propagated into the final result for a proper statistical assessment of the age uncertainty and the overall resolution of the 40Ar/39Ar dating technique. Error propagation is usually carried out by linearized error expansion techniques that weight the relative variance contribution of each input parameter by the squared partial derivative of the age function relative to this variable. Computation of partial derivatives through the working 40Ar/39Ar equations is tedious and error-prone, however. As a result, several data reduction schemes using different levels of approximation are implemented in various laboratories, some of which ignore certain sources of error while others use simplified error equations, thus making direct comparison of published age and error estimates sometimes inaccurate. Based on the general numerical approach outlined by Roddick (1987) [Roddick, J.C., 1987. Generalized numerical error analysis with applications to geochronology and thermodynamics. Geochim. Cosmochim. Acta 51, 2129-2135], a complete 40Ar/39Ar numerical error analysis (NEA) is proposed that includes up to 28 possible sources 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, worked 40Ar/39Ar examples to test underpinning assumptions of the linearized error propagation by extending the error analysis to higher-order terms of the Taylor expansion of the age equation. Close to very close agreement between the analytic and numerical solutions suggests that the linearized error expansion technique is justified for 40Ar/39Ar error propagation, despite strong nonlinearity in the related equations. in one pathological instance, nonlinearity is flagged by significant (15%) departure from the algebraic solution. The linearized age error estimate is still found to be acceptably close to the (exact) NEA estimate, provided however that covariance between 40Ar* and 39Ar(K) is precisely accounted for. As most 40Ar/39Ar datasets will be invariably corrupted by large covariance corrections, a full-fledged error analysis as made possible through NEA is clearly desirable in most situations. The demonstrated flexibility of the numeric approach should be profitably extended to other areas of isotope geochemistry involving complex calculation codes such as treated here. (C) 2000 Elsevier Science B.V. All rights reserved.