A fundamental differential equation that links rain attenuation to the rain rate measured at one point, and its applications in slant paths

We have studied how the rain attenuation A (dB) predicted with the Synthetic Storm Technique in a slant path of precipitation path length L is built up by parts ¿A (dB) due to partial precipitation path lengths ¿L . We have found that ¿A/ A can be estimated from ¿L / L independently of the value of A and L, and that this relationship leads to a differential equation whose integral yields the average relationship A [CokAR¿A+(1 ¿ Co)kB(3.134R)¿B]L0.90 with Co and L constants that depend only on latitude in a known way, and k and ¿ are the usual constants that transform rain rate into specific (dB/km) rain attenuation for the two layers A and B of precipitation modelled in the SST. With this expression, which links monotonically rain attenuation of a slant path (an integral) to the rain rate measured at a point by a rain gauge, we show that it is possible to reliably estimate the same cumulative probability distribution of rain attenuation predicted by the SST, without knowing the rain rate time series. The method could be extended to real time predictions.