the Bolivian Royal Andes. The Bolivian Royal Andes separate the warm wet Amazon on the East and the dry cold Altiplano on the West. The discharge from the basins flows towards the Tuni reservoir. Water stored at the Tuni reservoir is used for human water supply.

proposed by [22] . The conversion relation between potential evapotranspiration (PET) and actual evapotranspiration (AET) requires a welting point threshold and a field capacity threshold. The wilting point and the field capacity threshold were expressed as a saturation percentage of the soil. The field capacity and the wilting point were assumed as 46% and 20% respectively, as they are typical values suggested for clay soils [23] . In the wetland cells the AET extracts water from the bottom tank. AET from wetland cells is equal to the PET since any water deficit is replaced by water from the saturated zone [24] . The glacier cells do not consider evaporation since sublimation influences the mass balance, but not the hydrologic response of the cell. The discharge from glacier cells is separated into a fast englacial flow from surface, delayed englacial flow and a subglacial flow [25] . Thus, the linear cascade of 3 vertical tanks is also applied, but with a correction factor accounting for different hydraulic conductivity. Figure 2 shows the linear cascade of the three vertical tanks. Due to the lack of vegetation there is no interception on the bare ground cells. It is important to note that the model assumed that land cover does not change during the simulation, i.e., the area covered by glacier remains constant. The dynamic glacier evolution process will be covered by further research. The Supertank model requires two kinds of data: meteorological data and geographical data.

3.1.1. Meteorological data

This data refers to the external meteorological conditions of the basin. It consists of time series data of precipitation, glacier melting and potential evapotranspiration. Precipitation was obtained from the tipping bucket rain gauges. The wind induced errors of the precipitation measurements were corrected according to the World Meteorological Organization [26] . However, such measurements are point ones. The spatial distribution of precipitation was obtained by considering that precipitation increases with elevation at a rate of 0.24 mm m−1 [27] .

Glacier ablation was obtained considering the energy balance method [28] [29] .

(1)

where QM is the energy flux available for ablation (W m−2), SWI is the incoming short wave radiation (W m−2),

Figure 2. Schematization of the vertical cascade of tanks of the Supertank model.

LWI is the incoming long wave radiation (LWR) (W m−2), SWO is the outgoing short wave radiation (W m−2), LWO is the outgoing LWR (W m−2), QH is the sensible heat flux (W m−2), QL is the latent heat flux (W m−2), and Qo are other minor heat fluxes like rain (W m−2). Since the energy from Qo is small compared to the other sources of energy, it was neglected. The turbulent and sensible heat fluxes were estimated according to the bulk aerodynamic approach. Sensible heat flux was calculated according to:

(2)

where Cp is the specific heat air at constant pressure (1010 J/kg K), k is the Von Karman constant (0.41), ρ is the air density (1.29 kg/m3), P is the atmospheric pressure (Pa), u is the wind speed (m/s), Po is the standard atmospheric pressure (101325 Pa), Z is the instrument height (m), Zow is the roughness for wind logarithmic profile (m), Zot is the roughness for temperature logarithmic profile (m), T is the local temperature (K), T0 is the freezing temperature (273.15 K). The latent heat flux was calculated according to:

(3)

where L is the latent heat flux of evaporation or sublimation (kJ/kg), e2 is the vapour pressure at 2 m (Pa), eice is the vapour pressure at melting surface (Pa), Zoe is the roughness parameter (m). The roughness coefficients were assumed as Zoe = Zot = Zow/10; Zow=10 mm [30] .

The total energy available for ablation was converted into its water equivalent by considering the latent heat of fusion (334 kJ/kg) in the case of melting or the latent heat of sublimation (2838 kJ/kg) in the case of sublimation. It is important to differentiate melting from sublimation because melting glacier enters the basin as water; on the other hand, sublimation does not enter the basin and goes directly to the atmosphere. Energy is used for sublimation when latent heat flux is negative [31] . The sign of such flux is defined by the vapour pressure at 2 m and the vapour pressure at melting ice surface. Vapour pressure at 2 m was estimated by [32] :

(4)

The vapour pressure of melting ice was computed by [33] :

(5)

where T0 is the freezing temperature (0˚C) in Kelvin degrees (273.15 K). Thus, the vapour pressure of melting ice is 611.15 Pa. Any meteorological conditions of temperature and relative humidity that result in a vapour pressure lower than 611.15 Pa will produce sublimation.

Potential evapotranspiration was obtained according to the Priestley-Taylor method [34] .

(6)

where Rn is the solar radiation(W m−2), G is the ground flux(W m−2), G is the slope of saturation vapour pressure [kPa ˚C−1], Δ is the psychrometric constant [kPa ˚C−1], γ is the coefficient that relates the aerodynamic evaporation as a percentage of the radiative evaporation (1.74). Ground flux was assumed as percentage of the net radiation [35] . γ is given by:

(7)

where P is the atmospheric pressure (kPa). The value of Δ is given by:

(8)

where T is the air temperature (˚C).

3.1.2. Geographical data

It consists on the data representing the landforms and physical characteristics of each basin. This data is used for the semi distribution of the model. It consists of different raster files. The respective raster files were obtained using a digital elevation model (DEM) from the ASTER mission. The DEM was processed with the TauDEM algorithm implemented in the GIS software MAPWINDOWS [36] .

Other important raster file is the one defining the land cover of the grid cells. This file was obtained by processing remote sensing data from the Advanced Visible and Near Infrared Radiometer type 2 sensor (AVNIR-2) of the Advanced Land Observation Satellite (ALOS). The ALOS AVNIR-2 images were processed using the multispectral image data analysis system software [37] . Figure 3 shows the raster maps used in the present study.

The only geographical data that was not obtained from the rasters is the hydraulic conductivity. Hydraulic conductivity was obtained from field measurements using the minidisk infiltrometer [38] . The assumed hydraulic conductivity was 0.04 cm h−1.

3.2. Modelling

The model was calibrated at the Tuni basin considering the hydrological year 2011-2012. Then, the model was validated at the Tuni basin for the hydrologic year 2012-2013 and at the Huayna West basin for the hydrologic

Figure 3. Rasters used for the hydrological simulation of the basins Tuni and Huayna West. Such rasters are used for the semi-distribution of the model.

year 2011-2012 (Table 2). The performance of the model was evaluated according to the Nash Sutcliffe Efficiency (NSE), root mean square error ratio (RSR), percentage bias (BIAS) and correlation coefficient(CC) [39] - [41] . NSE is calculated by:

(9)

where Qobs,l is the observed discharge [m3 s−1] at time step l, Qsim,l is the simulated discharge at time step l and Qav is the average discharge [m3 s−1].

RSR is calculated by:

(10)

The BIAS is calculated by:

(11)

The CC is calculated by:

(12)

where Qsav is the average simulated discharge (m3 s−1).

In order to have an evaluation of the glacier melt water, two scenarios were modelled and compared: one current scenario including glaciers and one hypothetical scenario without glaciers. For the scenario without glaciers, all the glacier cells were assumed as bare ground cells. The meteorological conditions of both scenarios were the same. The total volume of water discharge from the simulation without glacier (VNG) was estimated:

Table 2. Simulation periods used for simulating the basins Tuni and Huayna West.

(13)

The melt water contribution of both basins during the hydrological year 2011-2012was estimated.

3.3. Temperature uncertainty

The present study considered temperature measurements from a HOBO temperature smart sensor. This sensor has a measurement range between −40˚C and 75˚C with a resolution of 0.02˚C. The accuracy of the sensor varies with temperature [11] and the possible temperature measurement errors are estimated by:

(14)

where tm is the measured temperature (˚C) and te is the error of the measured temperature (˚C). Then, the real but unknown temperature can be expressed by the corrected temperature tc (˚C):

(15)

The uncertainty induced by temperature was addressed by performing a Monte Carlo (MC) analysis. The MC analysis was performed by running the hydrological model s multiple times considering different values of temperature. Then, the total cumulated discharge of each simulation was calculated:

(16)

where Vs is the total water volume from the simulation s, Qi,s is the hourly discharge at time step i of the simulation s and n is the total number of time steps. The melt water contribution from simulation s (MVs) was estimated as the difference between the cumulated discharge with glaciers (Vs) and the cumulated discharge without glaciers (VNG).

(17)

The MC analysis included a convergence analysis considering the variation of the mean and standard deviation as the number of iterations increases [42] [43] .

(18)

(19)

where MMVk and MSDk are the mean and standard deviation of the melt water up to the kth run. Then, the hydrological simulations from the confidence levels of 95% and 5% were selected as the upper and lower confidence estimations respectively. The melt water was then estimated as the difference between the volume from the simulation with glaciers and the volume from the simulation without glaciers.

4. Results

4.1. Hydrological model calibration

The hydrological model Supertank was initially applied to the Tuni basin for the hydrological year 2011-2012. Once it was satisfactorily calibrated it was validated at the Tuni basin for the hydrological year 2012-2013 and at the Huayna West basin for the hydrological year 2011-2012. The Supertank model was calibrated considering the measured temperature. Table 3 shows the parameters of the calibrated model. Table 4 shows the performance of the model. Figure 4 shows the observed and simulated hydrographs for the calibration period. Figure 5 and Figure 6 show the observed and simulated hydrographs at the Tuni basin and the Huayna West basin for the verification periods.

4.2. Temperature errors

Figure 7 shows the maximum possible temperature measurement errors for both basins. In both stations the errors during winter are higher than the errors during summer. Between November and March, the errors at the Condoriri station are almost constant of ±0.21˚C. On the other hand, the errors for those same months at the Huayna West station are quite higher, with more errors higher than ±0.21˚C.

The hydrological model was applied to the basins Tuni and Huayna West considering the possible errors due to temperature measurements. The uncertainties of the temperature errors were addressed by applying a MC analysis. The melt water contribution was estimated considering the temperature measurements and the possible errors. The temperature measurement errors were sampled according to a uniform distribution considering the whole range of possible errors. The average cumulated discharge and standard deviation of the cumulated discharge are stable after 1500 iterations in the case of Tuni (Figure 8) and after 1000 iterations in the case of Huayna West (Figure 9). Thus, 1500 iterations is a reasonable number of simulations that provides reliable results.

4.3. Glacier melt water

Figure 10 shows the monthly melt water contribution for the hydrological year 2011-2012 considering the 5% and 95% confidence intervals for the basins Tuni and Huayna West. Although the winter months have the highest errors, the highest uncertainty in melt water contribution occurs during the summer months (wet season). The yearly melt water contribution from the Tuni and Huayna West basins during the hydrological year 2011-2012 was between 1.37 × 106 m3 and 1.72 × 106 m3 (Table 5). Comparing with the total observed discharge, the total melt water contribution accounts for about 13.45% of the total discharge (5% confidence) and 17.01% of the total discharge (95% confidence).

Table 3. Parameters of the calibrated Supertank model.

Table 4. Performance evaluation of the hydrological model for the basins Tuni and Huayna West.

Figure 4. Observed and simulated discharge of the Tuni basin for the hydrologic year 2011-2012.

4.4. General Discussion

According to the parameterization of the systematic errors, temperatures lower than 0˚C have higher errors than temperatures above 0˚C. For this reason, in both basins the highest temperature errors occur during the dry season months (between April and September) which is coincident with the winter. During the winter months the freezing temperatures below zero occur more often. Figure 11 shows that during winter 40% of the temperature measurements from the Huayna West station are below 0˚C and 30% of the temperature measurements from the Condoriri station are below 0˚C. On the other hand during summer months the occurrence of freezing temperatures is lower than 5%. The measurement errors at the Huayna West station are higher than the errors at the Condoriri station. This is because the Huayna West station is located at higher elevation and has lower temperatures than the Condoriri station. Thus, temperature measurement devices installed at higher elevations are prone to higher errors.

Although temperature errors are higher during winter, uncertainties related to melt water contribution are higher during summer. This is because during winter most of the ablation is due to sublimation [44] which goes to the atmosphere. Thus, the total amount of melt water during winter is minimum.

The melt water contributed from glaciers is higher during the wet season, which is coincident with the austral summer. 87% of Tuni melt water contribution occurs during the wet season (October to March) and 81% of the Huayna West melt water contribution occurs during the wet season. It is important to remember that the seaso-

Figure 5. Observed and simulated discharge of the Tuni basin for the hydro- logic year 2012-2013.

nality of the area is defined not by temperature cycles, but by hygric conditions [45] defined by the seasonal oscillation of the inter-tropical convergence zone (ITCZ). During the austral winter the ITCZ is north and the study area has a clod dry season; but during the austral summer the ITCZ is south. During the summer months the ITCZ oscillates over the Bolivian Amazonia coincident with the eastern inter-tropical flux that brings water vapor from the so called Amazon Ocean [14] . Thus, the wet season coincides with the eastern inter-tropical flux that brings water vapor from the Atlantic [46] . For this reason the relative humidity during summer is much higher than the relative humidity during winter. This higher relative humidity influences the amount of energy used for melting. Melting occurs when the vapor pressure is higher that the vapor pressure at melting surface [31] . Water pressure in turn depends on temperature and relative humidity. The small temperature variation of tropical latitudes leads to small increments of water pressure; thus, temperature increase does not increase the energy used for melting. On the other hand, the higher relative humidity of the wet season will lead to higher increments in the vapor pressure; thus, there is more energy used for melting.

The total melt water contribution for the hydrological year 2011-2012 was between 13.45% and 17.01% of the total discharge. The comparison of such contribution with contributions from other tropical glaciers is quite difficult, because there are few studies to compare and the high variability of melt water contribution. One tropical glacierized basin with a similar degree of glaciation is the Rio Santa in the Cordillera Blanca (Peru) which is about 10% glacierized (in the present study the glacier area is about 10.1% of the total area). Studies from the Rio Santa showed a high variability of the melt water contribution. The melt water contribution from Rio Santa during the hydrological period 1998-1999 was between 12% and 20% [47] ; but the melt water contribution for the hydrological period 2001-2004 was between 30% and 40% [48] . For the same periods and the same river (Rio Santa), other studies [49] estimated the melt water to be between 16.5% and 20.1%. The melt water contribution from the present study is similar to the mentioned ranges (Table 6). Nevertheless, the present estimation seems a low one compared with estimation of the Rio Santa for the period 2001-2004. Such difference may be due to the fact that Rio Santa is located in the inner tropics where the melting is constant throughout the year, while the present study is located in the outer tropics where melting occurs mainly during the wet season [13] .

Unlike other tropical glacier studies that estimated melt water contribution, the melt water contribution from the present study is directly related to the human consumption of a big city (El Alto). Thus, it is also important to express the melt water contribution related to human consumption (Table 7). Considering the population of El Alto (1182319 habitants) and that the estimated water demand of El Alto is about 46.7 l habitant−1 day−1 [50] , the melt water contributed by the Tuni-Huayna West glaciers is equivalent to the yearly demand of 6.81% to 8.55% of the city of El Alto (between 80497 habitants and 101148 habitants respectively).

5. Conclusions

The present study analyzed the melt water contribution from the Glaciers Tuni and Huayna West to their respective basins considering the systematic errors of the temperature measurements. Melt water contribution from the Tuni and Huayna West glaciers was estimated for the hydrologic year 2011-2012.

Figure 6. Observed and simulated discharge of the Huyana West basin for the hydrologic year 2011-2012.

Figure 7. Errors from the temperature measurements from the Condoriri station (a) and the Huayna West station (b).

Figure 8. Convergence of the Monte Carlo analysis applied to the Tuni basin. a) The average melt water volume converges to stability after 1000 iteration. b) The standard deviation converges to stability after 1000 iteration.

Figure 9. Convergence of the Monte Carlo analysis applied to the Huayna West basin. a) The average melt water volume converges to stability after 1500 iteration. b) The standard deviation converges to stability after 1000 iteration.

The hydrological model Supertank was satisfactorily applied to the basins Tuni and Huayna West. Then, the hydrological model was applied considering a scenario without glaciers. The melt water contribution was estimated as the difference between those scenarios.

The temperature measurement errors are higher during freezing temperatures. Thus, the temperature measurements from winter months have higher errors. Temperature measurements from higher elevations (lower temperatures) are expected to have higher errors.

Most of the melt water contribution occurs during the wet season months (October to March). 87% of melt water from Tuni glacier occurs during the wet season and 81% of the melt water contribution from the Huayna West glacier occurs during the wet season.

Although temperature measurement errors are higher during the dry season, the melt water contribution un-

Figure 10. Monthly melt water contribution from the Tuni glacier (a) and the Huayna west glacier (b).

Table 5. Melt water contribution from the glaciers Tuni and Huayna West for the hydrological year 2011-2012 considering confidence level of 5% (CL5) and 95% (CL95). The contribution is expressed in terms of water volume and water percentage from the total discharge.

Figure 11. Probability curve of the temperature measurements from the Condoriri station and the Huayna West station for the winter months (a) and the summer months (b).

Table 6. Comparisson of melt water contribution from the present study with the melt water contribution estimated at the Rio Santa by other studies.

Table 7. Melt water contribution from the glaciers Tuni and Huayna West for the hydrological year 2011-2012 considering confidence levels of 5% and 95%. The melt water volume is expressed in volume, in percentage from the total discharge, percentage of El Alto water demand (EAWD) and equivalent El Alto population (EAP).

certainties due to temperature measurement errors are higher during the wet season. This is because during the dry season the total melting is reduced to its minimum.

The melt water contribution from the Tuni and Huayna West glaciers during the hydrological year 2011-2012 was between 1.37 × 106 m3 (for a 95% confidence) and 1.72 × 106 m3 (for a 5% confidence). Such volumes are equivalent to 13.45% and 17.01% of the total discharge.

The melt water contribution from the Tuni and Huayna West glaciers during the hydrologic year 2011-2012 was equivalent to the yearly water demand of 6.81% to 8.55% of El Alto (Bolivia). Those percentages are equivalent to 80497 habitants and 101148 habitants respectively.

Acknowledgements

This research is developed within the framework of the GRANDE project, financed by SATREPS from JST- JICA. We thank PRAA project for the ALOS data.

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NOTES

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