U-Pb geochronology: Systematic development of mixing equations and application of Monte Carlo numerical simulation to the error propagation in the Concordia diagram

Abstract

Geochronological U-Pb information is obtained in the Concordia diagram through two successive stages: 1. (1) the computation of the ratios of radiogenic Pb to uranium isotopic abundances, their uncertainties and correlations; 2. (2) the computation of the ages and their uncertainties on the basis of a linear model, by the determination of the intercepts of the best-fit line with the Concordia curve. The systematic development of equations related to binary and ternary mixtures describing the analysed Pb enables us easy calculation of the atomic abundances of radiogenic Pb isotopes and of common 206Pb. As many parameters are involved, a classical computation of error propagation is tedious. Here, we show how a Monte Carlo-type iterative computation, in which all the parameters vary randomly within their confidence intervals, advantageously resolves the problem. This method implicitly and globally takes into account the totality of the uncertainties and correlations of all the parameters. It enables us the determination and visualization of confidence ellipses around experimental points and allows the discussion of final errors and correlations according to the precision of each initial measurement. The same approach can be used for the determination of the ages and their uncertainties, by iterative computation of the principal axes related to sets of points randomly selected from the confidence ellipses as well as the calculation of ages corresponding to the intercepts of those principal axes with the Concordia curve. The results of this computation are discussed and compared with previous methods of simple and weighted linear regression. The main advantages of the use of this approach are the following: 1. (1) ages and uncertainties are obtained by basic statistics on the distribution of the sets of calculated ages; 2. (2) weighting, always arbitrarily chosen, is here implicitly taken into account. This implicit weighting is more robust than previous methods employed and it corresponds to a stricter application of the postulate of experimental points alignment in the Concordia diagram.