The complexity of precipitation spatialization methods has been commented on by . When orographic effects are not well understood, complex approaches do not necessarily reproduce local measurements efficiently . In the central Himalayas, various hydrologic and glaciological studies are based on observation networks to produce a precipitation grid. However, few studies provide precipitation fields on an hourly timescale , and precipitation fields on spatial scales lower than 1 km are always obtained using altitude linear lapse rates . However, the considered lapse rates are constant in time and/or uniform in space. The spatial and temporal variability in the precipitation is then not represented in these studies. Moreover, the geostatistical co-kriging method has been applied by for monsoon precipitation interpolation over the Kosi catchment. However, the provided precipitation fields overall underestimate the observations, and this method is shown not to be adequate for the interpolation of solid precipitation. The IDW method is a simple, widely used method to spatialize precipitation in mountainous areas . In the French Alps, combine the IDW method with a multiplicative altitudinal factor. Precipitation at any point M of the mesh extrapolated by the IDW is given by P(M)=∑Sid-1(M,Si)⋅(P(Si)exp⁡(β(zM-zi))∑Sid-1(M,Si). In Eq. (), the altitude effect is represented through the introduction of the altitudinal factor β, defined by as the slope of the linear regression between the altitude of stations (in m a.s.l.) and the logarithm of the seasonal volume of total precipitation expressed in millimetres. This method presents the advantage of using an altitudinal factor which can vary in time and space. The spatial and temporal variability in the precipitation is therefore represented in this method. Moreover, the effect of altitude is independently studied and the controlling parameters have physical meaning.

Several studies based on observations or theoretical approaches have observed that precipitation in the Himalayan Range generally presents a multimodal distribution along elevation. Precipitation is considered to increase with altitude until a first altitudinal threshold located between 1800 and 2500 m, depending on the study, and to decrease above 2500 m. Moreover, the linear correlation of precipitation with altitude is reported to be weak for measurements above 4000 m . The decreasing of precipitation with altitude is characterized through various functions . Nevertheless, the hypothesis of linearity of precipitation (P) with altitude (z) is often made with a constant or time-dependent lapse rate . noted that, in mountainous areas, the hypothesis of a linear relation between P and z is only acceptable over a small spatial extension and for homogeneous weather types. Consequently, we considered altitude lapse rates for precipitation on the seasonal timescale, and we analysed the spatial variability in the relation between P and z.

For this purpose, we chose to regroup the stations into three groups (see Fig. ): (1) stations with elevation ranging from 2078 to 3022 m, following a west–east transect (Group 1); (2) stations with elevation ranging from 2619 to 3570 m following a south-westerly transect (Group 2); and (3) stations with elevation above 3970 m (Group 3). Figure  shows that for Group 1, observed seasonal volumes of precipitation increase globally with altitude at a rate lower than 0.1 km-1; for Group 2, seasonal volumes decrease at a rate around -0.3 km-1; for Group 3, seasonal volumes decrease at a rate lower than 0.2 km-1, with a poor linear trend.

The overlapping of altitude ranges between Group 1 and Group 2 highlights that the relation between precipitation and altitude strongly depends on terrain orientation. The difference in seasonal volumes at the BAL (2575 m a.s.l., 3471 mm year-1) and MER stations (2561 m a.s.l., 2245 mm year-1) (GROUP 1) also results from site effects on precipitation. In summary, β values inferred from local observations mainly express local variability and are not sufficient to establish any explicit relation between precipitation and altitude on the catchment scale. However, for operational purposes, the β factor can be simplified as a multimodal function of altitude within the Dudh Kosi catchment. The β factor is represented as a piecewise linear function of altitude using two altitude thresholds (z1 and z2) and three altitude lapse rates (β1, β2 and β3): β(z)=β1>0ifz≤z1β2<0ifz1<z≤z2β3∼0ifz>z2..

As no deterministic value can be ensured for the five parameters controlling the shape of Eq. (), an ensemble approach was applied (see Sect. ) to estimate parameter sets on the scale of the entire Dudh Kosi River basin that are optimally suitable for both the Tauche and the Kharikhola catchments.

We considered that there was no interannual storage in either of the two subcatchments studied; i.e. the variation in the groundwater content was considered zero from one hydrological year to the other. Consequently, annual simulated discharges were computed as the sum over all grid cells and all time steps of simulated surface flow and simulated subsurface flow. The question of the calibration of flow routing in the catchment was thus avoided.

The ISBA surface scheme simulates interactions between the soil, vegetation and the atmosphere with a sub-hourly time step (SVAT model). The multilayer version of ISBA (ISBA-DIF) uses a diffusive approach : surface and soil water fluxes are propagated from the surface through the soil column. Transport equations for mass and energy are solved using a multilayer vertical discretization of the soil. The explicit snow scheme in ISBA (ISBA-ES) uses a three-layer vertical discretization of snowpack and provides a mass and energy balance for each layer . Snowmelt and snow sublimation are taken into account in balance equations. The separation between runoff over saturated areas (Dunne runoff), infiltration excess runoff (Horton runoff) and infiltration is controlled by the variable infiltration capacity scheme (VIC) . The ISBA code is freely available from the respective authors.

The precipitation phase was estimated depending on hourly air temperature readings. Mixed phases occurred for temperatures between 0 and 2 ∘C, following a linear relation. Other input variables required for ISBA (atmospheric pressure, relative humidity, wind speed, short- and long-wave radiations) were interpolated from measurements at Pyramid station as functions of altitude, using the method proposed by . Short-wave radiation and wind speed are not spatially interpolated and are considered to be equals to the measurements at Pyramid station for the two catchments.

Several products provide parameter sets for physical properties of surfaces on the global scale . However, these products are not accurate enough at the resolution required for this study. The most recent analysis exclusively used knowledge garnered from the literature. To detail the approach, in this study the parametrization was based on in situ measurements. A classification into nine classes of soil/vegetation entities was defined based on Sentinel-2 images at a 10 m resolution , using a supervised classification tool of the QGIS Semi-Automatic Classification Plugin .

In and around the two catchments, 24 reference sites were sampled during field missions. Data collection included soil texture, soil depth and root depth, determined by augering to a maximum depth of 1.2 m. Vegetation height and structure and dominant plant species were also determined. The results were classified into nine surface types. The nine classes and their respective fractions in the Kharikhola and Tauche catchments are presented Table .

Analysis of soil samples showed that soils were mostly sandy (∼ 70 %), with a small proportion of clay (∼ 1 %). Soil depths varied from very thin (∼ 30 cm) at high altitudes to 1.2 m for flat cultivated areas. Forest areas were separated into three classes: dry forests were characterized by high slopes and shallow soils; wet forests presented deep silty soils (1 m), with high trees (7 m); intermediate forests had moderate slopes and relatively deep, sandy soils. Crop areas presented different soil depths depending on their average slope. In addition, values for unmeasured variables (leaf area index (LAI), soil and vegetation albedos, surface emissivity, surface roughness) were taken from the ECOCLIMAP1 classification for ecosystems representative of the study area. ECOCLIMAP1 provides the annual cycle of dynamic vegetation variables, based both on a surface properties classification and on a global climate map . The ECOCLIMAP2 product is derived from ECOCLIMAP1 and provides enhanced descriptions of surfaces. However, ECOCLIMAP2 is only available for Europe and therefore is not used in this study.

The precipitation fields generated using each behavioural parameter set were used as input data within the ISBA surface scheme. The simulations over the Tauche and Kharikhola catchments were run separately over the 1 January 2013 to 31 March 2016 period, on an hourly timescale. The 2013–2014 hydrological year was used as a spin-up period, and the results were observed for the 2014–2015 and 2015–2016 hydrological years. To overcome the issue of calibrating a flow-routing module, the simulated discharge were aggregated on an annual timescale and compared to annual observed discharge at the outlet (Q‾obs).

Figure  presents boxplots obtained for the 712 behavioural parameter sets for the terms of the annual water budget, i.e. liquid and solid precipitation, discharge, and evapotranspiration. The dashed line represents Q‾obs for each catchment. The mean annual volumes of simulated variables were also computed for each parameter set in 2014–2015 and 2015–2016, and the intervals of uncertainty associated with simulated annual volumes are provided. This method highlights the propagation of uncertainties associated with the representation of orographic effects toward the simulated terms of annual water budgets.

Table  presents the mean value, standard deviation and relative standard deviation for all of the ISBA-simulated variables for the Kharikhola and Tauche catchments for 2014–2015 and 2015–2016. The annual actual evapotranspiration accounted for 26 % of annual total precipitation for Kharikhola and 34 % for Tauche. In comparison, evapotranspiration was estimated at about 20, 14 and 53 % of total annual precipitation, respectively, by , and over the entire Dudh Kosi Basin, and estimated it at 36.2 % of annual total precipitation for the upper part of the Langtang Basin.

Annual snowfall volume for Kharikhola was a neglectable fraction of annual total precipitation (∼ 1 %), and it was around 44 % for Tauche. Annual snowfall was estimated at, respectively, 15.6 and 51.4 % of annual total precipitation by (entire Dudh Kosi River basin) and (upper part of the Langtang Basin).

Moreover, this statistical approach shows that the only uncertainties associated with representation of the orographic effect result in significant uncertainties in simulated variables. These uncertainties account for up to 16 % for annual total precipitation, up to 25 % for annual discharge and up to 8 % for annual actual evapotranspiration. Uncertainty in annual snowfall is quantified at 16 % for a high-mountain catchment and up to 32 % for a middle-mountain catchment. These uncertainty intervals are essentially conditioned by model structure and parametrization, and these results indicate that simulated water budgets provided by modelling studies must necessarily be associated with error intervals.

Going further into the simulation results, the hydrological cycle was inverted, in order to use observed discharge to optimize the relation between precipitation and altitude, as presented for mountainous areas by . Precipitation fields were then constrained on the local scale according to simulated discharges. Annual bias in discharge was computed for each catchment as the absolute value of the ratio between the observed and simulated annual discharges minus 1. Figure  presents the scatter plot of the distributions of bias in annual discharge for the Kharikhola and Tauche catchments. The Pareto optima, minimizing bias in annual discharge for both catchments, were computed using the R rPref package . For example, the first 10 Pareto optima were selected among the 712 behavioural parameter sets considered. The values of parameters for the winter and monsoon seasons for the 10 first optimum sets are summarized in Table . For the 10 parameter sets selected, the altitudinal threshold z1 was located between 2010 and 3470 m a.s.l. during the monsoon season and between 2287 m a.s.l. and 3488 m a.s.l. during winter. The second altitudinal threshold z2 was located between 3709 and 6167 m a.s.l. during monsoon and between 3734 and 6466 m a.s.l. during winter. Altitudes found for z1 were globally higher than altitudes proposed in the literature for the second mode of precipitation (between 1800 and 2400 m a.s.l., as described in Sect. ). Since these values were calibrated on the local scale, according to ground-based measurements, they can be considered to accurately represent the local variability encountered in the Tauche and Kharikhola catchments. Moreover, values for an altitudinal threshold of precipitation located above 4000 m a.s.l. were proposed.

Observed precipitation at measuring stations was then interpolated on an hourly timescale over the Dudh Kosi River basin at the 1 km spatial resolution. The method given by Eq. () is applied, using shape parameters for the altitudinal factor selected Table . The average annual volumes of computed total precipitation ranged between 1365 and 1652 mm, and annual snowfall volumes ranged between 89 and 126 mm, on average over the 2014–2015 and 2015–2016 hydrological years. These values are consistent with other products available for the area. In particular, showed that the APHRODITE product underestimates total precipitation over the Dudh Kosi River basin, with annual total precipitation of 1311 mm for the interannual average between 2001 and 2007, and proposed a mean annual total precipitation for the Dudh Kosi Basin of 2114 mm over the 1986–1997 period. The ERA-Interim reanalysis (25 km resolution) provided a mean annual precipitation of 1743 mm over the 2000–2013 period. Different relations between altitude and annual precipitation are then represented. The steeper the slope, the more variation in precipitation. This has to be considered in light of the physical properties of convection at such high altitudes.