| Two different upscaling methods for the snowmelt process have been developed: numerical upscaling (grid-based distributed model) and stochastic upscaling (pdf method). As a numerical upscaling approach, it is demonstrated that the distributed snowmelt model with short-wave radiation, which was estimated by solar geometry, performs very well with given spatially distributed information. The estimated snow distribution in the ungaged basin, Upper Cosumnes River watershed, is compared to the satellite image and they correspond well. On the other hand, as a stochastic upscaling approach, the Fokker Planck equation (FPE) has been derived and applied to an actual watershed, the Ward Creek test basin. The FPE for the prediction of the probability density of snowmelt state variables has been validated by Monte Carlo simulations in a three-day period. The Fokker Planck equation, the stochastically upscaled equation for the snowmelt process, worked properly under sufficiently diverse sources of heterogeneity. An advantage of the stochastic upscaling is that since this model does not choose any shape of physical domain, it can adapt to any hydrological model for other processes. Additionally, the snow temperature profile is studied using both the field observations and numerical experiments in order to support the assumptions made in the upscaling procedures. |
