Tracer data have been successfully used for hydrograph separation in glacierized basins. However, in these basins uncertainties of the hydrograph separation are
large and are caused by the spatiotemporal variability in the tracer signatures of water sources, the uncertainty of water sampling, and
the mixing model uncertainty. In this study, we used electrical conductivity (EC) measurements and two isotope signatures (

Glaciers and snowpack store a large amount of fresh water in glacierized basins, thus providing an important water source for downstream human societies and ecosystems (Barnett et al., 2005; Viviroli et al., 2007; He et al., 2014; Penna et al., 2016). Seasonal meltwater and rainfall play significant roles in shaping the magnitude and timing of runoff in these basins (Rahman et al., 2015; Pohl et al., 2017). Quantifying the seasonal contributions of the runoff components (CRCs), including groundwater, snowmelt, glacier melt, and rainfall, to the total runoff is therefore highly necessary for understanding the dynamics of water resources in glacierized basins under the current climate warming (La Frenierre and Mark, 2014; Penna et al., 2014; He et al., 2015).

The traditional end-member mixing approach (abbreviated as EMMA) has been widely used for hydrograph separation in glacierized basins across the world
(Dahlke et al., 2014; Sun et al., 2016a; Pu et al., 2017). For instance, studies in the glacierized catchments of the Italian Alps indicate the successful
application of the EMMA to estimate the proportions of groundwater, snow, and glacier meltwater based on water stable isotopes and electric
conductivity (EC; e.g., Chiogna et al. 2014, Engel et al. 2016, and Penna et al. 2017). Using EMMA, Li et al. (2014) confirmed significant contributions of snow
and glacier melt runoff to total runoff in the Qilian Mountains. Maurya et al. (2011) reported the contribution of glacial ice meltwater
to the total runoff in a Himalayan basin on

However, uncertainties of CRCs quantified by EMMA in glacierized basins are typically high (Klaus and McDonnell, 2013; Rahman et al., 2015) because of the following reasons. (1) The catchment elevation generally extends over a large range, leading to strong spatial variability in climate forcing (precipitation and temperature) and the tracer signatures of water sources. (2) The number of end-member water sources for runoff is typically high, including snow and glacier meltwater. (3) Water sampling in a high-elevation glacierized catchment is difficult due to logistical limitations that result in small sample sizes for the application of EMMA. The uncertainties of CRCs can be categorized into statistical uncertainty and model uncertainty. Statistical uncertainty refers to the spatiotemporal variability of the tracer signatures, sampling uncertainty, and laboratory measurement error (Joerin et al., 2002). Model uncertainty is determined by the assumptions of the EMMA, which might not agree with the reality in the basin (Joerin et al., 2002; Klaus and McDonnell, 2013). For example, the fractionation effect on isotope ratios caused by evaporation during the mixing process can result in significant errors given the constant tracer assumption in the EMMA (Moore and Semmens, 2008).

The Gaussian error propagation technique has been typically applied along with EMMA to estimate the statistical uncertainty for hydrograph separation,
assuming that the uncertainty associated with each source is independent of the uncertainty of other sources (Genereux, 1998; Pu et al., 2013). The
spatiotemporal variability of the tracer signatures is estimated by multiplying the

The Bayesian end-member mixing approach (shortened to Bayesian approach) shows the potential of estimating the proportions of individual components to the mixing variable in a more rigorous, statistical way (Parnell et al., 2010). For hydrograph separation, the tracer signatures of the water sources are first assumed to obey specific prior distributions. Their posterior distributions are then obtained by updating the prior distributions with the likelihood derived from water samples. In the last step, the CRCs to the total runoff are estimated based on the balance of the posterior tracer signatures. The posterior distributions of the CRCs are typically estimated in a Markov chain Monte Carlo (MCMC) procedure. In the Bayesian approach, both the statistical and model uncertainties are represented by the posterior distributions of parameters. The parameter uncertainty is estimated based on likelihood observations using MCMC.

Although the Bayesian approach can be applied in cases when the sample sizes are small (Ward et al., 2010), it has rarely been used for hydrograph separation in glacierized basins. To the authors' knowledge, there have been only four studies, including Brown et al. (2006), that conducted the hydrograph separation using a three-component Bayesian approach in a glacierized basin in the French Pyrenees. Furthermore, Cable et al. (2011) quantified the CRCs to total runoff in a glacierized basin in the North American Rocky Mountains. They used a hierarchical Bayesian framework to incorporate the temporal and spatial variability of the water isotope data into the mixing model. Rodriguez et al. (2016) investigated the effects of tracer measurements and mixing model parameters on the quantification of CRCs in a Chilean glacierized basin using an informative Bayesian framework. Recently, Beria et al. (2020) used a classic Bayesian approach to estimate the uncertainty of CRCs at a Swiss Alpine catchment. However, the performance of the Bayesian approach has not been evaluated in comparison to the EMMA. Moreover, the sensitivity of the Bayesian approach to the water sampling uncertainty associated with the representativeness of the water samples caused by the limited sample site and sample size is still not clear. Benefiting from the prior assumptions of changes in isotope signatures during the mixing process, the Bayesian approach bears the potential to estimate the fractionation effect on isotopic signatures (Moore and Semmens, 2008), which has, however, not been investigated either.

In this study, we compare the EMMA and the Bayesian approach for hydrograph separation in a Central Asian glacierized basin using water isotope and EC measurements. In Central Asia, glacierized catchments provide an important fresh water supply for downstream cities and irrigated agriculture. Quantifying the contributions of multiple runoff components to total runoff is important for understanding the dynamics of water resource availability at the regional scale. However, the uncertainty of the quantification of runoff components in the glacierized catchments is particularly large, as mentioned before. Our research question is twofold. First, how do the EMMA and Bayesian approaches compare with respect to the quantification of CRCs? Second, what are the influences of the different uncertainty sources (including variability of the tracer signatures, sampling uncertainty, and model uncertainty) on the estimated CRCs in the two mixing approaches?

The paper is organized as follows: details on the study basin and water sampling are introduced in Sect.

Located in Kyrgyzstan, Central Asia, the Ala-Archa basin drains an area of 233

Study area of the Ala-Archa basin (derived from the World Topographic Map by © ESRI) and the Golubin glacier, including the locations of the water sampling points.

Two meteorological stations (Fig. 1), i.e., Alplager (at an elevation of 2100

Since July 2013, local station operators have collected weekly stream water samples from the river channel close to the Alplager and
Baitik meteorological sites using pure 50

Glacier meltwater was sampled during the summer field campaigns in each year from 2012 to 2017. Samples of meltwater flowing on the Golubin glacier in the
ablation zone and at the glacier tongue were collected by pure 50

All samples were stored at 4

The hydrograph separation is carried out in each of the three seasons (i.e., cold season, snowmelt season, and glacier melt season). Water samples collected in the period from 2012 to 2017 are split into each of the three seasons for the hydrograph separation. The CRCs estimated by the mixing approaches refers to the mean contributions in each of the three seasons during the period from 2012 to 2017. The mixing approaches applied for the hydrograph separation in each season are summarized in Table 2. Considering that the groundwater and snowmelt samples were rarely collected in the cold season, we used all available groundwater and snowmelt samples from the three seasons for hydrograph separation in the cold season. Tracer signatures of rainfall are assumed to be the same as the measured tracer signatures of precipitation samples in all three seasons.

The main assumptions of EMMA include the following points (Kong and Pang, 2012). (1) The tracer signature of each runoff component is constant during the analyzed
period. (2) The tracer signatures of the runoff components are significantly different to each other. (3) Tracer signatures are conservative in the
mixing process. In the cold and snowmelt seasons, a three-component EMMA method (EMMA

Assuming the uncertainty of each variable is independent of the uncertainty in others, the Gaussian error propagation technique is applied to
estimate the uncertainty of the CRCs (

The Bayesian approaches applied for each season are summarized in Table 2. Similar to the EMMA, we apply a three-component Bayesian approach to all
seasons and, additionally, a four-component Bayesian approach in the glacier melt season. The three-component Bayesian approach has two types. The
Bayesian

The priors of EC values of runoff components are assumed to be normal distributions (Eq. 8a), with mean

The initial value ranges of parameters that need to be estimated in Eqs. (6)–(9) are summarized in Table 3. The posteriors of parameters describing the
spatial variability of tracer signatures in Eqs. (7) and (8b) are first estimated by the mean tracer signatures of runoff components measured at different
spatial locations. Parameters describing the overall variability of tracer signatures in Eqs. (6) and (8a) are then constrained by likelihood
observations of tracer signatures from all water samples at different times and locations. The posterior distributions of CRCs (

Note that the four-component approaches (EMMA

Due to limited accessibility, meltwater samples are typically difficult to collect in high-elevation glacierized areas. Often only a few water samples are available to represent the tracer signatures of meltwater generated from the entire glacierized area. Hence, the representativeness of collected meltwater samples implies an additional uncertainty source in the hydrograph separation.

We thus define three virtual sampling scenarios to evaluate the effects of meltwater sampling on the EMMA and Bayesian mixing approaches. Scenario I is
used to evaluate the effects of the sample size of meltwater in which four groups of meltwater sample are tested. The four sample groups have the same
mean value and SD of

Tracer signatures measured from water samples in three seasons.

CV stands for coefficient of variation.

Mixing approaches applied for hydrograph separation in different seasons.

Parameters used for prior distributions in the Bayesian approaches.

The water sources for runoff, such as rainfall and meltwater, are subject to evaporation before reaching the basin outlet – especially in
summer. However, the isotopic composition of stream water was measured at the basin outlet, and the contributions of runoff components are also quantified
for the total runoff at the basin outlet. After the long routing path from the sampled sites to the basin outlet, the isotopic compositions of
rainfall and meltwater mixing at the basin outlet could be different from those measured at the sampled sites, which is caused by the evaporation fractionation
effect. To consider the changes in the isotope signatures of water sources caused by the fractionation effect during the mixing process, we set up two
modified Bayesian approaches, i.e., Bayesian

Tracer measurements from all the water samples are summarized in Table 1 and Fig. 2 (see also Fig. S1). The mean values in Table 1 indicate that
precipitation is most depleted in heavy water isotopes (

Isotope signatures of water samples from the three seasons in the Ala-Archa basin.

CV values in Table 1 and boxplots in Fig. 3a–f show that the

For each water source, except groundwater, the tracer signatures show a significant seasonality (Table 1 and Fig. 3). In particular, the

Figure 3j–l show the

Table 4 and Fig. 4 compare the CRCs estimated by the mixing approaches. In the cold season (Fig. 4a), the EMMA

Contributions of runoff components (CRCs) estimated by the different mixing approaches (percentage, %). The ranges (%) show the difference between the 95 % and 5 %

Contributions of runoff components (CRCs) to total runoff estimated by different mixing approaches in three seasons. The Bayesian

In the snowmelt season (Fig. 4b and Table 4), the EMMA

When treating glacier melt and snowmelt as two separate end members in the glacier melt seasons (Fig. 4d), the EMMA

The posterior distributions of tracer signatures estimated by the Bayesian

Posterior distributions of tracer signatures estimated by the Bayesian

The Bayesian

Comparison of the posterior distributions of tracer signatures estimated by the Bayesian approaches with Bayesian

Correlation between posterior

Figure 8 shows the sensitivity of the Bayesian

Sensitivity of the CRCs estimates to the sample size (Scenario I), the mean (Scenario II), and standard deviation (Scenario III) of

The changes of

Effects of isotope fractionation on the estimates of CRCs in the Bayesian approach for the three seasons.

The fractionation effect also produced visible changes in the posterior distributions of

Effects of isotope fractionation on the estimated posterior distributions of tracer signatures of water sources in the glacier melt season.

The EMMA estimated similar CRCs but with a larger uncertainty than the Bayesian approaches. The reasons for this are twofold. First, the EMMA estimated the uncertainty ranges of CRCs using the standard deviations (SD) of the measured tracer signatures. SD values are likely overestimated in this study due to the small sample sizes (i.e., low number of water samples) and thus represent the variability of the tracer signatures of the corresponding water sources across the basin insufficiently. Due to the limited accessibility of the sample sites caused by snow cover, the water samples of meltwater and groundwater are often collected sporadically. The small sample size and strong variability in sampled tracer signatures likely led to a large SD value in the measurement. Second, the EMMA assumes that the uncertainty associated with each water source is independent of the uncertainty of other water sources (Eq. 5), which increases the uncertainty ranges for CRCs.

In contrast, by updating the prior probability distributions, the Bayesian approaches estimated a smaller variability of tracer signatures in the posterior distributions when compared to the measured tracer
signatures. The posterior distributions were sampled continuously from the assumed initial value
ranges by the MCMC runs, thus reducing the sharp changes and yielding lower variability for the tracer signatures. Moreover, the uncertainty ranges
for CRCs were quantified using Eqs. (6)–(10) instead of calculating independently as in the EMMA. Additionally, the assumed prior distributions of tracer
signatures and the CRCs take the correlation between the tracer signatures and the dependence between the runoff components into account, thus resulting in smaller uncertainty ranges (Soulsby et al., 2003). For example, the Bayesian approaches that considered the
correlation between

The Gaussian error propagation technique is only capable of considering the uncertainty of CRCs resulting from the variation in the tracer signatures (Uhlenbrook and Hoeg, 2003). The uncertainty of CRCs that originated from the sampling uncertainty of meltwater was then investigated in separate virtual sampling experiments. The EMMA produces large uncertainty ranges and SD values for CRCs in the glacier melt season when the meltwater sample size is rather small. The mean CRC quantified by the EMMA relies more heavily on the mean tracer values of the sampled meltwater, since the mean tracer values have been used directly in Eqs. (1)–(4), compared to the mean CRC estimated by the Bayesian approach.

The EMMA assumes that the tracer signature of each runoff component is constant during the mixing process; thus, it is unable to estimate the uncertainty of CRCs caused by the isotope fractionation effect. The virtual fractionation experiments, using the modified Bayesian approaches, show that the isotope fractionation could increase the contribution of snowmelt by 8 % and reduce the contribution of rainfall by 7 % in the snowmelt season. We assume that the mean CRCs estimated by the Bayesian approaches that consider the isotope fractionation are more plausible – despite the larger uncertainty ranges. Along the flow path from the source areas to the river channel, the isotopic compositions of meltwater and rainfall are likely increased by the evaporation fractionation effect – especially in the warm seasons. The increased isotopic compositions of meltwater and rainfall during the routing process need to be considered in the mixing approaches for hydrograph separation.

In general, the uncertainty of CRCs is visibly caused by the spatiotemporal variability in the tracer signatures, the water sampling uncertainty, and the isotope fractionation during the mixing process. The uncertainty caused by the water sampling of meltwater tends to be smaller than the uncertainty caused by the variations of the tracer signatures in both the EMMA and Bayesian mixing approaches. This is consistent with the findings that the SD values of the tracer measurements of water samples are the main uncertainty sources for the quantification of CRCs (Schmieder et al., 2016, 2018). The Bayesian approach tends to be superior for narrowing the variability of posterior tracer signatures benefitting from the prior assumptions and the consideration of the dependence between tracer signatures and runoff components when compared to EMMA.

The representativeness of the water samples is one of the limitations of this study. The groundwater was only sampled from a single spring located at
an elevation of 2400

We split the entire sampling period (from 2012 to 2017) into three seasons, i.e., cold season, snowmelt season, and glacier melt season, due to the low availability of the water samples in each year. By concentrating water samples in the three seasons, we increased the sample sizes of each runoff component for each season, thus increasing the ability of water samples to represent the spatiotemporal variability of seasonal tracer signatures. We used all available groundwater and snowmelt samples from the three seasons for hydrograph separation in the cold season due to the rather low number of samples collected in the cold season. This likely leads to overestimated contributions from groundwater and snowmelt in the cold season. However, the overestimation of the groundwater contribution is probably small because the tracer signatures of groundwater generally show small seasonal variability. The estimated contributions of snowmelt in the cold season are a bit higher than the contribution modeled by He et al (2018) during winter months of December, January, and February; these are still reasonable when considering that the cold season includes October and November when the snow is more prone to melting.

The assumptions of the mixing approaches lead to another limitation of this study. The EMMA assumes the tracer signatures of water sources are constant during the mixing process, which is a common assumption for the practical application of EMMA. It thus fails to consider the uncertainty originating from the changes of tracer signatures. In the Bayesian approach, we assumed normal prior distributions for the tracer signatures of water sources and Dirichlet prior distribution for the CRCs based on the literature (Cable et al., 2011). To refine the description of the temporal and spatial variability of the CRCs in the Dirichlet distribution, more hydrological data relating to the runoff processes in the basin are required. We acknowledge that the estimated CRCs could be strongly affected by the assumptions of prior distributions. However, testing the effects of the prior assumptions goes beyond the scope of this study. We assume that collecting more water samples from various locations and at different times for each water source could improve the estimation of tracer signature distributions.

This study compared the Bayesian end-member mixing approach with a traditional end-member mixing approach (EMMA) for hydrograph separation in
a glacierized basin. The contributions of runoff components (CRCs) to the total runoff were estimated for three seasons, i.e., cold season, snowmelt, and
glacier melt seasons. The mean CRCs estimated by the two mixing approaches are similar in all the three seasons. The uncertainties of these contributions, caused by the variability of tracer signatures, water sampling uncertainty, and isotope fractionation, were evaluated as follows:

The Bayesian approach generally estimates smaller uncertainty ranges of CRCs in comparison to the EMMA. Benefiting from the prior assumptions of tracer signatures and CRCs, and from the incorporation of the correlation between tracer signatures in the prior distributions, the Bayesian approach reduced the uncertainty. The Bayesian approach jointly quantified the uncertainty ranges of CRCs. In contrast, the EMMA estimated the uncertainty of the contribution of each runoff component independently, thus leading to higher uncertainty ranges.

The estimates of CRCs in EMMA tend to be more sensitive to the sampling uncertainty of meltwater when compared to those in the Bayesian approach. For small sample sizes (e.g., two), EMMA estimated very large uncertainty ranges. The mean of the CRCs quantified by EMMA is also more sensitive to the mean value of the tracer signature of the sampled meltwater than those values estimated by the Bayesian approach.

Ignoring the isotope fractionation during the mixing process likely overestimates the contribution of rainfall and underestimates the contribution of meltwater in the melt seasons. The EMMA currently used is unable to quantify the uncertainty of CRCs caused by the isotope fractionation during the mixing process due to the underlying assumptions.

The supplement related to this article is available online at:

ZH, KUS, and SV conceptualized this research. ZH, KUS, SMW, OK, and AG collected the data. ZH, KUS, and SV developed the methodology that was used. The original draft was compiled by ZH, SV, and DD. All authors contributed to writing the review and the editing of the paper.

The authors declare that they have no conflict of interest.

This paper has been funded by the German Federal Ministry for Science and Education (GlaSCA-V; grant no. 88 501) and the Volkswagen Foundation (GlaSCA; grant nos. 01DK15002A and B).The article-processing charges for this open-access publication were provided by a Helmholtz Association German Research Centre.

This paper was edited by Markus Weiler and reviewed by three anonymous referees.