Pulse Average Requirement for Semi-Fractal Spatial Distributions of Raindrops

In light of mounting evidence that the rain drop number distribution (DND) in radar pulse sampling volumes may significantly deviate from Poisson, on which the classical theory of radar pulse averaging is based, we derive anew the pulse intensity distribution with explicit dependence on rain DND using conditional probability. A bounded cascade model is then used to simulate semi-fractal rain DND in space. In these simulations the variance of the rain DND in the pulse sampling volume depends on the depth level of the cascade used, in addition to the controlling parameters of the cascade. Explicit analytical formula is derived for the DND variance of the entire simulation domain. The local variance of DND, which exerts the greatest influence on the required number of pulses in an average for a given confidence, is determined stochastically therefore not amenable to an analytical expression. It is found that significantly more samples (15% to 80%) may be needed in pulse averages to achieve the same confidence as predicted by the classical theory based on an idealized Poisson distribution. It then follows that the noise errors introduced to the radar reflectivity factor (Z) by this effect, relative to the nominal averaging procedure from classical theory, will be passed on to retrieved rain rate (R) in accordance with the underlying Z-R relationship used to estimate precipitation.