where ɛ represents the isotopic enrichment factor. This Rayleigh model (Eq. 3) was developed from a standard Rayleigh model (Mariotti et al., 1981) by Druhan & Maher (2017) to better represent isotope fractionation processes in water mixtures subject to different travel times – such as the case for a stream water sample in a catchment, where travel times will differ between different flow pathways within the catchment. Further, MSR was assumed to be equally likely at both end-members as well as the point of mixing; this was in order to conserve pre-MSR end-member fractions to post-MSR conditions. MSR was finally quantified as the percentage of reduced sulfur concentration, i.e., MSR = 100(1 − fred ). Estimates of end-member parameter values (δdep ,δrock , and the sulfur concentration of atmospheric deposition, cdep ) as well as the enrichment factor (ɛ) were derived from a synthesis of regional observations and previously reported literature values.
To better quantify the median and average Monte Carlo outputs when MSR-values were close to 0%, we have modified the analysis methodology presented in Fischer et al. (2022) where a theoretical model output below 0% was truncated (since MSR cannot physically be lower than 0%). This truncation becomes problematic for the current study because we consider average values relatively close to 0% and get both positive and negative random errors of which only the negative random errors will be truncated following the rule. This leads to an overestimation of MSR-values derived from the resulting ensemble statistics, and to keep the output unbiased we omitted the truncation rule.
Results
Regional data synthesis