A stochastic snow model.

Abstract

The purpose of this study was to develop a stochastic model of the snowfall, snow accumulation and ablation process. Snow storms occurring in a fixed interval were assumed to be a homogeneous Poisson process with intensity X. The snow storm magnitudes were assumed to be independent and identically distributed random variables. The magnitudes were independent of the number of storms and concentrated at the storm termination epochs. The snow water equivalent from all storms was a compound Poisson process. In the model, storms then occurred as positive jumps whose magnitudes equaled the storm amounts. Between storms, the snowpack ablated at a constant rate. Random variables characterizing this process were defined. The time to the occurrence of the first snowpack, generated by the first storm, was a random variable, the first snow-free period. The snowpack lasted for a random duration, the first snowpack duration. The alternating sequence of snow-free periods followed by snowpacks of random duration continued throughout the fixed interval. The snow-free periods were independent and identically distributed random variables as were the snowpack durations. The sum of each snow-free period and the immediately following snowpack duration formed another sequence of independent and identically distributed random variables, the snow-free, snow cycles. The snow-free, snow cycles represented the interarrival times between epochs of complete ablation, and thus defined a secondary renewal process. This process, called the snow renewal process, gave the number of times the snowpacks ablated in the interval. Distribution functions of the random variables were derived. The snow-free periods were exponentially distributed. The distribution function of the snowpack durations was obtained using some results from queueing theory. The distribution function of the first snow-free, snow cycle was derived by convoluting the density function of the first snowfree period and the first snowpack duration. The distribution of the sum of n snow-free, snow cycles was then the n-fold convolution of the first snow-free, snow cycle with itself. The probability mass function of the snow renewal process was evaluated numerically, from a known relationship with the sum of snow-free, snow cycles. The snowpack ablation rate was considered to be a random variable, constant within a season, but varying between seasons. The snowpack durations and snow-free, snow cycles were conditioned on the ablation rate, then unconditional distributions derived. An application of the model was made in the case where snow storm magnitudes were exponentially distributed. Specific expressions for the distribution functions of the random variables were obtained. These distributions were functions of the Poisson parameter X, the exponential parameter of storm magnitudes, Ne l, and the snowpack ablation rate. The snow model was compared with data from the climatological station at Flagstaff, Arizona. Snow storms were defined as sequences of days receiving 0.01 inch or more of snow water equivalent separated from other storms by one or more dry days. Snow storms occurred approximately as a homogeneous Poisson process. Storm magnitudes were exponentially distributed. Empirical distributions of snowpack ablation rates were obtained as the coefficients of a regression analysis of snowpack ablation. Two methods of estimating the Poisson parameter were used. The theoretical distribution functions were compared with the observed. The method of moments estimate generally gave more satisfactory results than the second estimate.