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