Error propagation

The errors in fc and t₁ are estimated by a Monte Carlo simulation routine in which the respective equations are solved for a large number of random starting compositions, whose x, y and z are compatible with the observed ²⁰⁶Pb/²³⁸U, ²⁰⁷Pb/²³⁵U and ²⁰⁸Pb/²³²Th ratios, their corresponding errors and the correlation of errors in ²⁰⁶Pb/²³⁸U and ²⁰⁷Pb/²³⁵U. ComPbCorr uses a table of normally distributed, bivariate random numbers as input, which are modified to suit the observed values for each sample. The table is stored in a hidden worksheet within the *.xls file. Do not modify or delete this sheet.

Error propagation expressions for the corrected isotopic compositions are derived by partially differentiating the expressions giving radiogenic lead composition, and adding the error contributions in quadrature (e.g. Ludwig 1980). The error in corrected ²⁰⁶Pb/²³⁸U depends only on the errors in the observed ratio and the error in the fraction of common lead.

On the other hand, the corrected ²⁰⁷Pb/²³⁵U ratio depends on both of the uncorrected U/Pb isotopic ratios, the errors in the corrected ratio will therefore also be influenced by correlation between errors in ²⁰⁶Pb/²³⁸U and ²⁰⁷Pb/²³⁵U (disregarding covariances other than between ²⁰⁶Pb/²³⁸U and ²⁰⁷Pb/²³⁵U):

In contrast, the ²⁰⁸Pb/²³²Th and ²⁰⁶Pb/²³⁸U ratios do not contain a common divisor and the errors in this pair of ratios are therefore considered as uncorrelated:

The correlation coefficient between errors in the corrected ²⁰⁶Pb/²³⁸Pb and ²⁰⁷Pb/²³⁵U is given by:

which indicates that the error correlation is reduced by the correction procedure.

Given the errors in the corrected ²⁰⁶Pb/²³⁸Pb and ²⁰⁷Pb/²³⁵U ratios and the corresponding error correlation coefficient, the error in corrected ²⁰⁷Pb/²⁰⁶Pb is given by:

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