Fitting precipitation particle size–velocity data to mixed joint probability density function with an expectation maximization algorithm

Abstract

This paper proposes an estimation method of joint size and terminal velocity distribution on the basis of sampling data of precipitation particles containing multiple types. Assuming that the velocity follows the normal distribution and the size follows the gamma distribution, the method searches a locally maximum logarithmic likelihood within a realistic parameter range using the expectation–maximization algorithm. Several test populations were prepared with a realistic number of elements, and then the method was evaluated by retrieving the populations from their sample. The results showed that the original parameters were successfully estimated in most cases of the test population containing some of liquids, graupels, and rimed and unrimed aggregates. The original number of elements was also estimated with an adjustment of the number of elements in a manner such that each of their minority fractions exceeded a threshold. Applied to the two-dimensional disdrometer observation data, the method was helpful to discard frequently observed erroneous data with unrealistically large fall velocity.