Comparing Bayesian and traditional end-member mixing approaches for hydrograph separation in a glacierized basin

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 (δ18O and δ2H) to label the runoff components, including groundwater, snow and glacier meltwater, and rainfall, in a Central Asian glacierized basin. The contributions of runoff components (CRCs) to the total runoff and the corresponding uncertainty were quantified by two mixing approaches, namely a traditional end-member mixing approach (abbreviated as EMMA) and a Bayesian endmember mixing approach. The performance of the two mixing approaches was compared in three seasons that are distinguished as the cold season, snowmelt season, and glacier melt season. The results show the following points. (1) The Bayesian approach generally estimated smaller uncertainty ranges for the CRC when compared to the EMMA. (2) The Bayesian approach tended to be less sensitive to the sampling uncertainties of meltwater than the EMMA. (3) Ignoring the model uncertainty caused by the isotope fractionation likely led to an overestimated rainfall contribution and an underestimated meltwater share in the melt seasons. Our study provides the first comparison of the two end-member mixing approaches for hydrograph separation in glacierized basins and gives insight into the application of tracer-based mixing approaches in similar basins.

standard deviations (Sd) of the measured tracer signatures (Pu et al., 2013;Penna et al., 2016;81 Sun et al., 2016b). Although this approach has been successfully used in various glacierized 82 basins, some recurring issues remain. These include (1)  is also often ignored. 88 The Bayesian end-member mixing approach (abbreviated as Bayesian approach) shows 89 the potential to estimate the proportions of individual components to the mixing variable in a 90 more rigorous statistical way (Parnell et al., 2010). For hydrograph separation, the water tracer 91 signatures of the water sources are first assumed to obey specific prior distributions. Their Although the Bayesian approach can be applied in cases when the sample sizes are 98 small (Ward et al., 2010), it has been rarely used for hydrograph separation in glacierized basins. 99 To the authors' knowledge, there have been only three studies, including Brown et al. (2006),  In this study, we compare TEMMA and the Bayesian approach for hydrograph 111 separation in a Central Asia glacierized basin, using water isotope and EC measurements. The The paper is organized as follows: details on the study basin and water sampling are 117 introduced in Section 2; assumptions of the two mixing approaches are described in Section 3; 118 Section 4 estimates the CRC, as well as the corresponding uncertainties; discussion and 119 conclusion finalize the paper in Sections 5 and 6, respectively.  (Aizen et al., 2000;Aizen et al., 2007). In particular, the snowmelt runoff mainly occurs in the 128 warm period from early March to middle September, and the glacier melt typically generates 129 from the high-elevation areas during July to September (Aizen et al., 1996;He et al., 2018;He 130 et al., 2019). We subsequently defined three runoff generation seasons as follows. Cold season: 131 from October to February, in which the streamflow is fed mainly by groundwater and to a 132 smaller extent by snowmelt and rainfall; Snowmelt season: from March to June, in which the 133 streamflow is fed chiefly by snowmelt and groundwater and additionally by rainfall; Glacier 134 melt season: from July to September, in which the streamflow is fed by significant glacier melt 135 and groundwater, rainfall and snowmelt.

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Two meteorological stations (Fig. 1), i.e., Alplager (at elevation of 2100 m a.s.l.) and 137 Baitik (at elevation of 1580 m a.s.l.), have been set up in the basin since 1960s to collect daily 138 precipitation and temperature data. The Ala-Archa hydrological station has been set up at the 139 same site of the Baitik meteorological station to collect daily average discharge data since 1960s.

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The dynamics of glacier mass balance and snow mass balance in the accumulation zone have  first collected from fixed rain collectors (immediately after the rainfall/snowfall events), and 149 then accumulated in two indoor rain containers over one month. The mixed water in the 150 containers were then sampled for isotopic analysis every month. The indoor rain containers 151 were filled with thin mineral oil layers for monthly precipitation accumulation and stored in 152 cold places. Additionally, two plastic rain collectors PALMEX, specifically designed for 153 isotopic sampling to prevent evaporation, were set up at the elevations of 2580 m a.s.l. and 3300 154 m a.s.l. to collect precipitation in high-elevation areas (Fig. 1). Precipitation samples were 155 collected monthly from these two rain collectors during the period from May to October when 156 the high-elevation areas were accessible.    The mixing approaches applied for the hydrograph separation in each season are summarized 186 in Table 2.  (Genereux, 1998

AB AB A B A B A B A B f A B A B A B A B A B A B
204 1 2 1 2 1 2 2 1 3 1 2 1 3 2 3 2 1 3 1 3 2

AB AB A B A B A B A B f A B A B A B A B A B A B
where the subscripts 1-3 refer to the three runoff components (i.e., groundwater, Assuming the uncertainty of each variable is independent from the uncertainty in others,

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the Gaussian error propagation technique is applied to estimate the uncertainty of the CRC (f1-213 3) using the following equation (Genereux, 1998):   (Table 3). V is a 2*2 unit positive-definite matrix, and '2' stands for the degree of freedom in 248 the Wishart prior distribution.

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The priors of EC values of runoff components and stream water are assumed as normal 250 distributions (Eq. 8a), with mean ɛ and variance τ. Similarly, the spatial variability of the mean 251 EC values of runoff components (ɛ) are assumed to follow a normal distribution with mean θ 252 and variance ω (Eq. 8b). τ, θ and ω are parameters estimated by likelihood observations (Table   253 3).
The mean isotopes (μ 18 O and μ 2 H) and EC (ɛ) of stream water are constrained by a normal distribution with means and precision matrix β and Ω (Eq.9d). β is a parameter vector 265 estimated by likelihood observations (Table 3).

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The value ranges for the parameters need to be estimated in Eqs. 6-9 are summarized in 267 Table 3. The posteriors of parameters describing the spatial variability of water tracers in Eqs.

Effects of water isotope fractionation on hydrograph separation
where, η 18 O and η 2 H are the mean values of the changes in isotopes caused by the fractionation 308 effect, which are parameters need to be estimated. Ω is the inverse of the covariance matrix 309 defined in Eq. 6b. The parameters in Eqs. 6-11 are then re-estimated by the measurements of 310 water tracer signatures using the MCMC procedure. Tracer measurements from all the water samples are summarized in Table 1    with the corresponding mixing approaches ( as one end-member (i.e. meltwater) in the glacier melt season (Fig. 4c), the TEMMA_3 375 estimated the mean contributions of groundwater, meltwater and rainfall of 45%, 46% and 9%, 376 respectively. The Bayesian_3 and Bayesian_3_Cor estimated a lower contribution of 377 groundwater (43-44%) and a higher contribution of rainfall (11%) compared to the TEMMA_3.

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In general, the TEMMA_3 estimated the largest uncertainty for the contributions in all the three 379 seasons, followed by the Bayesian_3. The Bayesian_3_Cor slightly reduced the uncertainty 380 ranges compared to the Bayesian_3 (Table 4).

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When treating glacier melt and snowmelt as two separate end-members in the glacier 382 melt seasons (Fig. 4d), the TEMMA_4 failed to separate the hydrograph in the glacier melt  (Table 4).    The currently used TEMMA is unable to quantify the uncertainty for CRC caused by the isotope 566 fractionation during the mixing process, due to the underlying assumptions. Competing interests.

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The authors declare no conflict of interest.