A technique to estimate snow depletion curves from timeseries data using the beta distribution

Snow depletion curves, required as input to many snowmelt models, relate the time-varying snow-covered fraction, S(t), to the cumulative melt energy received, M(t). In hydrologic models, snow water equivalent (SWE, particularly the mean value) is generally the most important variable in mass accounting; however, S(t) is more easily observed (by remote sensing, for example). In atmospheric models, S(t) exerts a strong control on land–atmosphere coupling. Generally, the coarse resolution of model grids or response units requires a sub model relating S(t) to areal-average SWE. Depletion curves often express snow-covered fraction directly as a function of cumulative melt, in the form S(M). Under simplifying assumptions, S(M) can be derived from the end-of-winter frequency distribution of snow water equivalent (SWE) (Martinec 1972, Liston 1999). Usually, however, this distribution is unknown, and S(M) must be inferred from sparse observational data of S(t) and estimated M(t). We make the simplifying assumption that all disappearance of snow is due to melt, and define the following variables: Z(x) = SWE [cm] at end of winter (start of melt), a random variable in space, described by a probability density function f(Z); M(t) = Cumulative potential melt energy, expressed as melt depth [cm], monotonically increasing in time; and S(t) = areal fraction snow covered [dimensionless], monotonically decreasing in time. Assuming that M(t) is uniform in space, snow cover depletion S(t) as a function of M(t) is equal to the complement of cumulative distribution function (cdf) for end-of-winter Z(x)