## Snow accumulation

Annual snow accumulation, a (l, 0), is determined by the local precipitation rate, P(l, 0), and an estimate of the fraction of precipitation to fall as snow, fS(l, 0): a (l, 0) = fS(l, 0)P(l, 0), where spatial locations are denoted by their longitude-latitude co-ordinates (l, 0). Given spatially distributed precipitation rates at a particular time, an arbitrary air temperature cutoff is normally used to dictate whether local precipitation falls as snow, T < Tt, for a threshold temperature Tt. Operationally, Tt is typically taken to be 0°C or 1°C. This is a poor treatment if monthly or mean annual climatology are used to estimate snowfall, as the cutoff temperature prohibits snowfall at mean values slightly above the threshold, and diurnal and synoptic variability will obviously give extensive periods with subfreezing temperatures. Degree-day methodology offers a physically based alternative to estimate the fraction of precipitation to fall as snow, fS. If observational weather data are available at frequent (subdiurnal) time intervals, fS can be estimated from the fraction of time below Tt during a precipitation event.

Using monthly or annual climatology, simple statistical models can be introduced to parameterize temperature variability. For instance, temperature can be assumed to have a Guassian distribution around mean monthly temperatures Tm, with standard deviation sm. Under this assumption, fS is calculated from the integral

Tt fS=smk J

The standard deviation om is derived ideally from hourly temperature data, in order to capture the diurnal temperature cycle. For annual climatology, a sinusoidal temperature distribution is typically assumed to represent the seasonal temperature cycle (Reeh, 1991)

where t is time in days, t is the length of the year (365.24 days), Td is the daily temperature, Tmax is the summer temperature maximum, and T is the annual average air temperature.

January 1 is taken to be t = 0 in Equation (2), although a time lag can be built in to incorporate the effects of seasonal surface temperature lags, a function of local surface heat capacity and the radiative environment. The equivalent integration to Equation

fS=Jt Jexp d 0

where sd represents the standard deviation in daily temperature, calibrated to include both diurnal and synoptic temperature variability. In model implementations, atmospheric temperature lapse rates become important, as monthly or annual temperatures need to be interpolated or extrapolated over the landscape for application in Equations (1)-(3).