The variability of the rainfall rate as a function of area

Distributions of drop sizes can be expressed as DSD =  N t  × PSD, where N t is the total number of drops in a sample and PSD is the frequency distribution of drop diameters ( D ). Their discovery permitted remote sensing techniques for rainfall estimation using radars and satellites measuring over large domains of several kilometers. Because these techniques depend heavily on higher moments of the PSD, there has been a bias toward attributing the variability of the intrinsic rainfall rates R over areas ( σ R ) to the variability of the PSDs. While this variability does increase up to a point with increasing domain dimension L , the variability of the rainfall rate R also depends upon the variability in the total number of drops N t . We show that while the importance of PSDs looms large for small domains used in past studies, it is the variability of N t that dominates the variability of R as L increases to 1 km and beyond. The PSDs contribute to the variability of R through the relative dispersion of χ  =  D 3 V t , where V t is the terminal fall speed of drops of diameter D . However, the variability of χ is inherently limited because drop sizes and fall speeds are physically limited. In contrast, it is shown that the variance of N t continuously increases as the domain expands for physical reasons explained below. Over domains larger than around 1 km, it is shown that N t dominates the variance of the rainfall rate with increasing L regardless of the PSD.