| description abstract | In this paper, the delta function is introduced to describe the observed rainfall-rate distribution including the mixture of discrete and continuous parts. Thus the observed rainfall-rate distribution can be expressed as a simple mathematical formula. Also the probability density function (pdf) of frontal rainfall in the region of the United Kingdom is investigated. It is found that the pdf during the period of 26 January?25 February 1990 can be fitted well by the lognormal distribution, and there is a linear correlation between the mean and the standard deviation of rainfall-rate pdf during this period. The GARP Atlantic Tropical Experiment (GATE) data for pixel sizes of 4.0 and 40.0 km are also analyzed in this paper. It is also found that both mean and standard deviation decrease with the increment of the observing pixel size. Based on the rainfall-rate distribution formula; the research results of Kedem et al. regarding modeling the rain-rate pdf as lognormal, gamma, and inverse Gaussian; the correlation between the mean and variance of rainfall-rate pdf; and the specified values of mean and variance, some conclusions on the threshold method (or area?time integral method) are presented that do not require the assumption that rain rate is homogeneous in time and space. The results show that the area-average rain rate and fractional area are nonlinearly related at low rain-rate thresholds and that there is variation of the regression slope with rainfall-rate threshold, observing pixel size, and rain type, and so on. From these results, it can be concluded that the rainfall-rate threshold, precipitation type, and the observing pixel size are three major factors for the threshold method (or area?time integral method). The three elements have to be considered if the threshold method (or area?time integral method) is applied. Also, it is known that the reasons that the threshold method (or area?time integral method) works well arise from the observed precipitation properties such as the lognormal rainfall-rate pdf, the correlation between the mean and variance, and the specified values of the mean and variance. Such observed precipitation properties originate from the mesoscale dynamics and cloud physics. At the same time, such observed precipitation properties should be constrained by mesoscale dynamics and cloud physics. | |
