《Table 2Re-Os data for the synthetic samples.》

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《Monte Carlo sampling for error propagation in linear regression and applications in isochron geochronology》

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An additional advantage of using the Monte Carlo based method is that the resulting distribution of age and initial isotopic ratios can be adjusted to integrate with geological information and produce improved chronological constraints.We demonstrate this by using a synthetic example consisting of 12 samples.Their187Re/188Os and187Os/188Os ratios and associated uncertainties including error correlations（Table 2）are used to determine their age and initial isotopic ratio.Results obtained from the Monte Carlo method and the algorithm of the Isoplot program are presented in Fig.5.The ages and initial isotopic ratios from the two methods are essentially the same（Isoplot age=540±2 Ma，initial187Os/188Os=0.600±0.013；Monte Carlo age=540±6，initial187Os/188Os=0.600±0.063），but uncertainties from the Isoplot program are significantly smaller as discussed above.If there is evidence that these samples are younger than 541 Ma，i.e.，based on independent geological constraints，it is reasonable to discard regression results that are older than 541 Ma from the final distribution（Fig.5）.By doing so，the final distribution is altered，and skewed to younger ages and higher initial isotopic ratios（Fig.6）.If we consider quantiles to interpret uncertainties for this distribution，the age estimate changes to 539à6t2Ma and the initial isotopic composition is 0:616à0t0::035026at the 95%percentile level.Similarly，if the initial isotopic ratio can be independently constrained，this information can also be integrated into the Monte Carlo method.This approach is analogous to a common practice in isochron dating，where a sample or a mineral containing low or negligible parent isotope is selected together with samples bearing high parent isotope for isochron dating（e.g.，using matrix and garnet with low and high176Lu/177Hf ratios，respectively for Lu-Hf dating；using plagioclase and pyroxene with low and high147Sm/144Nd ratios，respectively for Sm-Nd dating），through which the y-intercept of the isochron is‘‘fixed”by the sample（e.g.，matrix and plagioclase）plotting near or at the y-intercept.It is possible that the independent constrained geological information would also have uncertainties or follow a certain distribution，these also can be considered in our Monte Carlo method.