HydroMix v1.0: a new Bayesian mixing framework for attributing uncertain hydrological sources

Abstract

Tracers have been used for over half a century in hydrology to quantify water sources with the help of mixing models. In this paper, we build on classic Bayesian methods to quantify uncertainty in mixing ratios. Such methods infer the probability density function (PDF) of the mixing ratios by formulating PDFs for the source and target concentrations and inferring the underlying mixing ratios via Monte Carlo sampling. However, collected hydrological samples are rarely abundant enough to robustly fit a PDF to the source concentrations. Our approach, called HydroMix, solves the linear mixing problem in a Bayesian inference framework wherein the likelihood is formulated for the error between observed and modeled target variables, which corresponds to the parameter inference setup commonly used in hydrological models. To address small sample sizes, every combination of source samples is mixed with every target tracer concentration. Using a series of synthetic case studies, we evaluate the performance of HydroMix using a Markov chain Monte Carlo sampler. We then use HydroMix to show that snowmelt accounts for around 61 % of groundwater recharge in a Swiss Alpine catchment (Vallon de Nant), despite snowfall only accounting for 40 %–45 % of the annual precipitation. Using this example, we then demonstrate the flexibility of this approach to account for uncertainties in source characterization due to different hydrological processes. We also address an important bias in mixing models that arises when there is a large divergence between the number of collected source samples and their flux magnitudes. HydroMix can account for this bias by using composite likelihood functions that effectively weight the relative magnitude of source fluxes. The primary application target of this framework is hydrology, but it is by no means limited to this field.