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.

Most water resources are a mixture of different water sources that have traveled via distinct flow paths in the landscape (e.g., streams, lakes, groundwater). A key challenge in hydrology is to infer source contributions to understand the flow paths to a given water body using a source attribution technique. A classic example is the two-component hydrograph separation model to quantify the proportion of groundwater and rainfall in streamflow, often referred to as “pre-event” water vs. “event” water (Burns et al., 2001; Klaus and McDonnell, 2013; Schmieder et al., 2016). Other examples include estimating the proportional contribution of rainfall and snowmelt to groundwater recharge (Beria et al., 2018; Jasechko et al., 2017; Jeelani et al., 2010), fog to the amount of throughfall (Scholl et al., 2011, 2002; Uehara and Kume, 2012), and soil moisture (at varying depths) and groundwater to vegetation water use (Ehleringer and Dawson, 1992; Evaristo et al., 2017; Rothfuss and Javaux, 2017).

The primary goal of such attribution in hydrology is to infer the contribution of different sources to a target water body; the tracer can be an observable compound like a dye, a conservative solute, or even a proxy for chemical composition such as electrical conductivity. The key requirement is that the concentration of the tracer is distinguishable between different sources. The stable isotope compositions of hydrogen and oxygen in water (subsequently referred to as “stable isotopes of water”) are used as tracers in hydrology. Other commonly used tracers include electrical conductivity (Hoeg et al., 2000; Laudon and Slaymaker, 1997; Lopes et al., 2018; Pellerin et al., 2007; Weijs et al., 2013) and conservative geochemical solutes such as chloride and silica (Rice and Hornberger, 1998; Wels et al., 1991).

Classically, attribution analysis is done by assigning an average tracer concentration to each source, typically estimated from time or space averages of observed field data (Maule et al., 1994; Winograd et al., 1998), and then solving a series of linear equations. In order to express uncertainty in the attribution analysis, a tracer-based hydrograph separation approach was first proposed in the work of Genereux (1998) and has subsequently been used in many studies (Genereux et al., 2002; Koutsouris and Lyon, 2018; Zhu et al., 2019). Bayesian mixing approaches offer a useful alternative to classic hydrograph separation, as Bayesian approaches explicitly acknowledge the temporal variability of source tracer concentrations estimated from observed samples (Barbeta and Peñuelas, 2017; Blake et al., 2018). Rather than a single estimate of source contributions, Bayesian approaches yield full probability density functions (PDFs) of the fraction of different sources in the target mixture (Parnell et al., 2010; Stock et al., 2018), hereafter referred to as “mixing ratios”.

Bayesian mixing was first developed in ecology to estimate the proportion of different food sources to animal diets (Parnell et al., 2010; Stock et al., 2018). Hydrological applications of such models are still rare (Blake et al., 2018; Evaristo et al., 2016, 2017; Oerter et al., 2019). In a Bayesian mixing model, a statistical distribution is fitted to both the measured source tracer concentrations and to the measured tracer concentrations from the target (e.g., river, groundwater, vegetation). The distribution of the mixing ratios is then inferred via Bayesian inference. With recent advances in probabilistic programming languages like Stan (Carpenter et al., 2017), Bayesian inference has become a relatively simple task.

However, the key limitation with the above approach is that the source compositions are assumed to come from standard statistical distributions. Typically, the sources are assumed to be drawn from Gaussian distributions, which can be fully characterized by the mean and variance of the data available for each source (Stock et al., 2018). This limits both the potential applicability and the insights that can be gained from tracer information in hydrology because the sample mean and variance may not accurately reflect the statistical properties of the actual source composition, and the Gaussian approach represents an unnecessary simplification in cases in which a large amount of information on source composition is available.

An additional complication in hydrology comes from the fact that observed point-scale samples do not necessarily capture the tracer concentrations in the actual sources, which are distributed heterogeneously in space and whose contribution can be temporally variable depending on the state of the catchment (Harman, 2015). For instance, if we were to characterize the contribution of snowmelt to groundwater, we would need to capture (1) the temporal evolution of the isotopic ratio of snowmelt, which strongly varies in space (Beria et al., 2018; Earman et al., 2006), and (2) the temporal evolution of the area actually covered by snow. This spatially and temporally distributed nature of the sources can be hard to account for in both analytical and Bayesian mixing approaches.

To overcome the limitations of source heterogeneity and the previously discussed restriction to Gaussian distributions, we present a new mixing approach for hydrological applications called HydroMix. This approach does not require a parametric description of observed source or target tracer concentrations. Instead, HydroMix formulates the linear mixing problem in a Bayesian inference framework similar to hydrological rainfall–runoff models (Kavetski et al., 2006a), wherein the mixing ratios of the different sources are treated as model parameters. Multiple model parameters can be inferred in such a setup, allowing for the parameterization of additional hydrologic processes that can modify source tracer concentrations (shown in Sect. 3.5). A more detailed account of the advantages and limitations of this new approach is given in Sect. 5.

In this paper, we first describe the theoretical details of HydroMix for a simple case study with two sources, one mixture and one tracer (Sect. 2). Section 3 presents synthetic and real-world case studies that demonstrate the accuracy, robustness, and flexibility of HydroMix. In the synthetic case study, we use a conceptual hydrologic model to simulate tracer concentrations. We also introduce a composite likelihood function that accounts for the magnitude of the different source fluxes. The real-world case study applies HydroMix in a high-elevation headwater catchment in Switzerland. The results of these applications are presented in Sect. 4 before summarizing the main outcomes, applicability, and limitations of HydroMix in Sect. 5.

A system with

Section 2.1 details the general modeling approach for a simplified system with two sources and one tracer. This is followed by a detailed discussion on the choice of the parameter inference approach used.

For a system with two sources that combine linearly to form a mixture, the
mixing model can be formulated as

The two parameters in this system, the mixing ratio (

Long time series of the tracer concentration in both the sources and mixture are rare.

The effect of seasonality in precipitation can make the inference of

The tracer concentrations in the different sources are generally measured at point scales, whereas the tracer concentration in the target integrates inputs over the entire source area.

By utilizing all the available measurements

Assuming that the residuals can be described with a Gaussian error model
with a mean of zero and constant variance

In the case of linear mixing between two sources, the two model parameters
considered at this stage are the mixing ratio

In order to avoid numerical problems, we use the log-likelihood form of Eq. (8),

Following the general Bayes' equation, the posterior distribution of the
model parameters can be written as

We implement an MCMC sampling algorithm using a Metropolis–Hastings (Hastings, 1970) criterion to infer the posterior distribution of the mixing ratio. For the synthetic case study (Sect. 3.1), we set up 10 parallel MCMC chains to monitor convergence according to the classical Gelman–Rubin convergence criterion (Gelman and Rubin, 1992). Each chain is initiated by assigning a uniform prior distribution for the mixing ratio, and the mixing ratio varies between 0 and 1. For the subsequent case studies, we use importance sampling for the sake of simplicity. The prior distributions of additional model parameters (if applicable) are discussed in the corresponding case study section. Apart from the prior distribution of the model parameters, HydroMix requires tracer concentration of the different sources and of the mixture. The error model variance is not jointly inferred with other model parameters but calculated for each sample parameter set from the residuals according to Eq. (6).

We provide a comprehensive overview of the performance of HydroMix based on a set of synthetic case studies (case studies in Sect. 3.1 and 3.2) and a real-world application to demonstrate the practical relevance for hydrologic applications (case studies in Sect. 3.4 and 3.5). The first case study demonstrates the ability of HydroMix to converge on the correct posterior distribution for synthetically generated data. The second case study uses a synthetic dataset of rain, snow, and groundwater isotopic ratios using a conceptual hydrologic model and compares the results of HydroMix to the actual mixing ratios assumed to generate the dataset. It then weights the source samples and evaluates the effect of weighting on the mixing ratio (case study in Sect. 3.3). In the last two case studies, HydroMix is applied to observed tracer data from an Alpine catchment in the Swiss Alps to infer source mixing ratios and an additional parameter (isotopic lapse rate).

In this example, source concentrations

The sensitivity of HydroMix to the number of samples drawn from

In this case study, we build a conceptual hydrologic model wherein groundwater
is assumed to be recharged directly from rainfall and snowmelt. Stable
isotopes of deuterium (

Synthetic time series are generated for precipitation, the isotopic ratio in
precipitation, and air temperature at a daily time step. For generating the
precipitation time series, the time between two successive precipitation
events is assumed to be a Poisson process with the precipitation intensity
following an exponential distribution (Botter et al., 2007;
Rodriguez-Iturbe et al., 1999). Time series of air temperature and of
isotopic ratios in precipitation are obtained by generating an uncorrelated
Gaussian process with the mean following a sine function (to emulate a
seasonal signal) and with constant variance (Allen et al., 2018; Parton and
Logan, 1981). The separation of precipitation into rainfall (

Only the last 2 years of the model runs are used to obtain the time series
of isotopic ratios in rainfall, snowmelt, and groundwater. These years are
then used to estimate the mixing ratio of snowmelt in groundwater, which is
the fraction of groundwater recharged from snowmelt. Rainfall and snowmelt
samples are the two sources and groundwater samples represent the mixture.
For the HydroMix application, all the modeled rainfall and snowmelt samples
generated using the hydrologic model are used, whereas for groundwater, only
one isotopic ratio per month is used (randomly sampled). The mixing ratios
inferred using HydroMix are compared to the actual recharge ratio obtained
from the hydrologic model as

In Sect. 3.2, rainfall and snowmelt samples are not weighted by the
magnitude of their fluxes while computing the mixing ratios with HydroMix.
As all rainfall and snowmelt samples are used, the weights are implicitly
determined by the number of rainfall and snowmelt events instead of their
magnitudes. This is a general problem in all mixing approaches and has not
been adequately acknowledged in the literature. Ignoring the weights may
lead to biased mixing estimates if the proportional contribution of one of
the components (e.g., rainfall or snowmelt) is low but the number of
samples obtained to represent that component is proportionally much higher
(Varin et al., 2011). For example, in a given
catchment, the amount of total snowfall may be a small proportion of the
annual precipitation, but the number of days when snowmelt occurs may be
comparable to the total number of rainfall days in a year. If this is not
specified a priori, HydroMix may overestimate the proportion of groundwater being
recharged from snowmelt. To account for this, we introduce a weighting
factor in the likelihood function originally formulated in Eq. (8) to make
a new composite likelihood (Varin et al., 2011):

The objective of this case study is to infer the proportional contributions of snow versus rainfall to the groundwater of an Alpine headwater catchment, Vallon de Nant (Switzerland), using stable water isotopes.

Vallon de Nant is a 13.4 km

Vallon de Nant has a typical Alpine climate, with around 1900 mm of annual
precipitation and a mean air temperature of 1.8

Map showing Vallon de Nant along with the locations of meteorologic and hydrologic observations as well as frequent sampling sites. Composite samples of precipitation were collected at the weather stations. Groundwater samples were collected at the groundwater monitoring points and the installed piezometers. The groundwater piezometers were installed by James Thornton from the University of Neuchâtel (Thornton et al., 2018).

Vallon de Nant has been extensively monitored since February 2016. Water
samples are collected from streamflow, rain, snowpack, and groundwater at
different elevations, which are then analyzed for the isotopic ratios in
deuterium (

Summary of the isotopic data (

HydroMix is used to estimate the proportion of snow recharging groundwater (subsequently referred to as the “snow recharge coefficient”). In order to obtain a PDF of the snow recharge coefficient, isotopic ratios in all the water samples from rain, snowpack, and groundwater are used. A uniform prior distribution is assigned to the snow recharge coefficient, which varies between 0 and 1, representing the entire range of possible values.

In any mixing analysis, it may be useful or desirable for users to specify
an additional model parameter that is able to modify the tracer
concentrations based on their process understanding of the system. In the
case of Alpine catchments with large elevation gradients, stable isotopes in
precipitation often exhibit a systematic trend with elevation, becoming more
depleted in heavier isotopes with increasing elevation. This is also known
as the “isotopic lapse rate” (Dansgaard,
1964; Friedman et al., 1964). In typical field campaigns, because of
logistical challenges, precipitation samples are collected only at a few
points in a catchment, with often fewer precipitation samples at high
elevations. This leads to oversampling at lower elevations and undersampling at higher elevations, which can bias mixing estimates. This has
been found to be especially relevant for hydrograph separation in forested
catchments (Cayuela et al., 2019). To
allow a process compensation for this, an additional lapse rate factor is
introduced with which each observed point-scale sample (observed at a given
elevation) is corrected to a reference elevation as follows:

The lapse rate factor is allowed to modify both rainfall and snowpack isotopic ratios to obtain a catchment-averaged isotopic ratio, which is then used in the mixing model. Using this formulation of an isotopic lapse rate makes the following implicit assumptions: (1) precipitation storms on aggregate move from the lower part of the catchment to the upper part of the catchment, thus creating a lapse rate effect, and (2) precipitation falls uniformly over the catchment. It is important to note that the isotopic lapse rate is different from the precipitation lapse rate; i.e., the rate of change of precipitation with elevation is different from the rate of change of the precipitation isotopic ratio with elevation.

Prior distribution of the different model parameters as specified to HydroMix.

Diagnostic plots showing the convergence characteristics of MCMC
chains for five different mixing ratios for the low variance dataset (shown
in Table 3). Panels

Mean and variance of the two sources,

It is important to note that the precipitation isotopic ratio is not only a function of elevation, but also depends on other factors such as the source of moisture origin, cloud condensation temperature, and secondary evaporation. Similarly, strong spatial variability exists in the isotopic ratio of snowmelt water, depending on catchment aspect, snow metamorphism, and wind distribution. This case study is a mere demonstration that HydroMix allows for the inference of additional parameters that can account for various physical processes that may modify isotopic ratios.

The prior distribution of the isotopic lapse rate is specified based on
isotopic data collected across Switzerland under the Global Network of
Isotopes in Precipitation (GNIP) program (IAEA/WMO, 2018).
Using the monthly isotopic values collected between 1966 and 2014,
average lapse rate values are obtained for both

A uniform prior distribution is assigned to the isotopic lapse rate
parameter, with the lower bound specified as 3 times the Swiss lapse
rate for both

The results for the different case studies are discussed in the sections below.

The mean and standard deviations used to generate the low and high variance source distributions for the synthetic case studies are summarized in Table 3. We randomly generated 100 samples from each of the two source distributions and from the target distribution and varied the mixing ratios between 0.05 and 0.95 in 0.05 increments. It should be noted that HydroMix permits using a different number of samples for the sources and for the mixture.

For the low variance case, the mixing ratio inferred with HydroMix with 1000 MCMC simulations closely reproduces the theoretical mean of the mixing ratios used to generate the synthetic data (Fig. 2a). However, for the high variance case, the inferred mixing ratios do not match the true underlying mixing ratios, especially for low and high mixing ratios. This is partly due to the poor identifiability of the sources (given that their distributions are highly overlapping) and partly due to the relatively small sample size of 100. The inferred mean should reproduce the theoretical mean with increasing sample size and we clearly see this for the low variance case in Fig. 2b, where the model performance markedly improves with an increasing number of samples. The performance is measured here in terms of the absolute error between the posterior mixing ratio mean and the true mean summed and averaged over all tested ratios from 0.05 to 0.95. We did not perform inferences for sample sizes larger than 100 as the computational requirement increases exponentially with increasing sample sizes.

The model converges fairly quickly for the low variance case after

Figure 4 shows the variation in the isotopic ratio
of groundwater over the entire 100-year period, showing that the system achieves
a steady-state condition after

Parameters used to generate time series of precipitation, air
temperature, and isotopic ratios in precipitation.

Evolution of the modeled isotopic ratio in groundwater over a
100-year period with

The mixing ratios inferred with HydroMix are very similar regardless of whether snowfall or snowmelt is used across the entire range of recharge efficiencies (Fig. 6). This provides confidence in the use of snowfall samples as a proxy for snowmelt when estimating mixing ratios. However, it is clear from Fig. 6 that an important bias emerges between the estimated mixing ratio from HydroMix and the actual mixing ratio known from the hydrologic model, especially for low mixing ratios.

This bias can be expected to emerge when the source contributions are not weighted according to their fluxes, which to our knowledge has not been explicitly addressed in the hydrological literature. As already discussed in Sect. 3.3, the absence of sample weighting typically induces a bias when there is a large divergence between the number of samples taken over a certain period (e.g., 1 year) to characterize a source and the magnitude of source flux over that period (e.g., 40 snow and 10 rain samples taken to characterize the two sources, for which snow only accounts for a very small portion, e.g., 10 %, of the annual precipitation).

Box plot showing the variability in the isotopic ratio of snowfall and snowmelt as simulated by the hydrologic model. The box plot extends from the 25th to 75th percentile value, with the median value depicted by the orange line. The whiskers extend up to 1.5 times the interquartile range. The black circles are the outliers.

Ratios of snow in groundwater estimated with HydroMix plotted against ratios obtained from the hydrologic model for the last 2 years of simulation. Also shown are the separate results obtained by using samples of either snowmelt or snowfall. The full range of ratios is obtained by varying rainfall and snowmelt recharge efficiencies from 0.05 to 0.95. The numbers of rainfall, snowfall, and snowmelt days are 39, 24, and 107 in the last 2 years of simulation.

After taking into account the magnitude of rainfall and snowmelt events in the composite likelihood function of Eq. (23), it is clear that many of the unweighted biases can be removed (Fig. 7). The most significant improvement is seen at very low mixing ratios for which the divergence between the conceptual model and the mixing model estimates error is reduced by almost 50 %. In this study, we have used a relatively simple normalization-based weighting function (Eq. 25). Testing other weighting functions that have been proposed in the past (Vasdekis et al., 2014) is left for future research.

Ratios of snow in groundwater estimated using HydroMix plotted against ratios obtained from the hydrologic model for both weighted and unweighted mixing scenarios. The full range of ratios is obtained by varying rainfall and snowmelt recharge efficiencies from 0.05 to 0.95. The numbers of rainfall, snowfall, and snowmelt days are 39, 24, and 107 in the last 2 years of simulation.

Using the dataset from an Alpine catchment (Vallon de Nant, Switzerland),
HydroMix estimates that 60 %–62 % of the groundwater is recharged from
snowmelt (using unweighted approach), with the full posterior distributions
shown in Fig. 8a. This estimate is consistent for
both of the isotopic tracers (

Histogram showing the fraction of snow recharging groundwater in
Vallon de Nant using the isotopic ratios in

As can be seen from Fig. 8a, the estimated
distribution of the snow ratio in groundwater is very narrow. This can be
explained by the fact that we assume that the collected precipitation
samples represent the variability actually occurring in the catchment. To
overcome this limitation, we infer an additional parameter called the
isotopic lapse rate that accounts for the spatial heterogeneity in terms of
catchment elevation. As shown in Fig. 9, the
posterior distributions of the isotopic lapse rate (for both

However, an important consequence of additional parameter inference without
providing additional data or constraints is an increase in the degree of
freedom, which can then increase the uncertainty in source contributions.
This effect is seen in Fig. 8b, especially in
contrast with the previous result in Fig. 8a,
where the median mixing ratios of the posterior distributions remain similar
(

Histogram showing the posterior distribution of the isotope lapse
rate parameter in

As with all linear mixing models, the quality of the underlying data determines the accuracy and utility of the results. If the tracer compositions of the different sources are not sufficiently distinct, the uncertainty in the estimated mixing ratios will become very large. This means that if either the underlying data quality is poor or the source contribution dynamics are not well conceptualized, then the uncertainty in the mixing ratios will be too high to be useful.

In cases in which a large number of source samples are available, the computational requirements of HydroMix outweigh the benefit from using it. These are likely cases in which the statistical distribution of the source tracer composition is well understood, and therefore fitting a probability density curve to the source and target samples and then inferring the distribution of the mixing ratio using a probabilistic programming approach is more appropriate (Carpenter et al., 2017; Parnell et al., 2010; Stock et al., 2018). Also, HydroMix might not be an appropriate method in instances in which fitting statistical distributions to source and target compositions reflects a priori knowledge of the system.

A key difference between HydroMix and other Bayesian mixing approaches is that HydroMix parameterizes the error function, whereas other Bayesian approaches parameterize the statistical distribution of source and mixture compositions. Parameterizing source compositions requires large sample sizes, which is seldom the case in tracer hydrology. Error parameterization offers a useful alternative and can also be verified against the posterior error distribution. In the case studies demonstrated in this paper, a normally distributed error model was found to be appropriate. However, error models other than Gaussian can be used by formulating the respective likelihood function.

HydroMix builds the model residuals by comparing all the observed source samples with all the observed samples of the target mixture, assuming that all available source and target samples are independent. Interestingly, the assumption of independence holds even if the source and target samples are taken at the same time, since the target samples result from water that has traveled for a certain amount of time in the catchment and hence is not related to the water entering the catchment. However, if a system has instantaneous mixing, then the source and target samples taken at the same moment in time will necessarily be strongly correlated. In such cases, the assumption of independent samples would not make sense and the method might give spurious results.

Finally, it is noteworthy that adding additional parameters to characterize the source tracer composition increases the degree of freedom of the model, which implies that adding such parameters leads to an increase in the uncertainty of the source contribution estimates unless new information, i.e., new observed data, is added to the model. This means that users who are interested in incorporating additional modification processes by adding parameters should ideally provide additional tracer data able to constrain this process, subject to tracer data being available.

For consistency and simplicity, the case studies and synthetic hydrological examples provided here focused on the contribution of rain and snow in recharging groundwater. However, it is important to emphasize that the opportunities to implement HydroMix extend to all cases in which mixing contributions are of interest and for which it is difficult to build extensive databases of source tracer compositions. Such examples include quantifying the amount of “pre-event” vs. “event” water in streamflow; pre-event water refers to groundwater and event water refers to rainfall or snowmelt. Another interesting use case might be to quantify the proportion of streamflow coming from the different source areas in a catchment to capture the spatial dynamics of streamflow. Other uses include quantifying the amount of fog contributing to throughfall, the proportion of glacial melt vs. snowmelt flowing into a stream, the amount of vegetation water use from soil moisture at different depths vs. groundwater, the interaction between surface water and groundwater at the hyporheic zone (Leslie et al., 2017), and sediment fingerprinting to quantify the spatial origin of river sediments. In all of these cases, understanding source water contributions, both spatially and temporally, will improve the physical understanding of the system.

We develop a new Bayesian modeling framework for the application of tracers in mixing models. The primary application target of this framework is hydrology, but it is by no means limited to this field. HydroMix formulates the linear mixing problem in a Bayesian inference framework that infers the model parameters with a Metropolis–Hastings-based MCMC sampling algorithm based on differences between observed and modeled tracer concentrations in the target mixture using all possible combinations between all source and target concentration samples. This is especially useful in data-scarce environments where fitting probability distribution functions is not feasible. HydroMix also makes the inclusion of additional model parameters to account for source modification processes straightforward. Examples include known spatial or temporal tracer variations (e.g., isotopic lapse rates or evaporative enrichment).

An evaluation of HydroMix with data from different synthetic and field case
studies leads to the following conclusions.

HydroMix gives reliable results for mixing applications with small sample
sizes (

As revealed by our synthetic case study with a conceptual hydrological
model, at low source contributions (i.e.,

The use of composite likelihoods to weight samples by their amounts can significantly reduce the bias in the mixing estimates. At low source proportions, the estimated mixing ratio improves by more than 50 % after accounting for the amount of all the sources. We show this using a simple normalization-based weighting function. Future studies should explore the usage of different weighting functions that have been proposed in the past (Vasdekis et al., 2014).

A synthetic application of HydroMix to understand the amount of snowmelt-induced groundwater recharge revealed that using the snowfall isotopic ratio instead of the snowmelt isotopic ratio leads to similar mixing ratio estimates. This is particularly useful in high mountainous catchments, where sampling snowmelt is logistically difficult.

A real case application of HydroMix in a Swiss Alpine catchment (Vallon de Nant) showed a clear winter bias in groundwater recharge. About 60 %–62 % of the groundwater is recharged from snowmelt (unweighted mixing approach), while snowfall only accounts for 40 %–45 % of the total annual precipitation. This has also been previously suggested elsewhere in the European Alps (Cervi et al., 2015; Penna et al., 2014, 2017; Zappa et al., 2015).

The model code is implemented in Python 2.7 and can be downloaded along with
the dataset from Zenodo at

The paper was written by HB with contributions from all coauthors. HB and BS formulated the conceptual underpinnings of HydroMix. JRL helped in framing the statistical and hydrological tests to evaluate HydroMix. AM and NCC helped in compiling data used for model evaluation and provided critical feedback during model validation.

The authors declare that they have no conflict of interest.

We would also like to thank Lionel Benoit for his inputs on the formulation of the Bayesian mixing model. We thank the four anonymous reviewers and the editors for their constructive feedback that considerably improved the paper.

This research has been supported by the Swiss National Science Foundation (grant no. PP00P2_157611).

This paper was edited by Bethanna Jackson and reviewed by four anonymous referees.