## 2 Rain Rate Statistics  
Rain is generated when water vapor in the atmosphere condenses until it is heavy enough to fall to earth. There are two types of rain: the first is stratiform rain, with almost no vertical wind, covering an extended area and relatively homogenous on a horizontal level. The second is convective rain, with a vertical wind causing up and down draft and high rainfall rates. Rain will affect polarization of signals, as raindrops are not spherical and attenuate horizontally polarized waves more than vertically polarized waves; this must be considered when encoding signals using polarization. In predicting rainfall, one wishes to obtain the probability of rainfall and rain rate. Rainfall can be predicted by partial measurement, by statistical models, and by a combination of these [7; 8; 9; 10; 11; 12; 13; 14; 15; 16]. Empirical models for the probability of rain were developed based on experimental data, valid only in specific climates or frequency ranges.  
Numerical prediction methods have been developed, for example, MultiEXCELL [26] constructs rain cells and collects them to simulate the effects and rate of rainfall. There too, the rain cells' probability of occurrence is analytically derived from the local statistics.  
As detailed in the previous section, ITU-R.Rec.837-7 provides a method for prediction of rainfall rate with a one-minute integration time that is exceeded, for a given average annual probability of exceedance, for a given location. This is based on total monthly data on rainfall generated from two databases, over land and water, as well as monthly mean surface temperature data from ITU-R P.1510 [27]. The land database includes 50 years of data from the GPCC Climatology (V2015) database and, over water, 36 years of data from the European Centre of Medium-range Weather Forecast (ECMWF) Era Interim data. The recommendation provides digital maps of the annual rainfall rate data exceeded for 0.01% of an average year. Some studies display discrepancies between ITU-R results and measurements in certain climates, e.g., in the tropics [28] or South Korea [29]. In our paper, we assume the existence of the rain, so we deal with the symbol error rate in cases of rain.  
It is generally accepted that rain attenuation follows a lognormal distribution [30], although, in tropical regions, the lognormal model is not adequate, and a gamma distribution--or a mixture of log normal and gamma--is a better fit [31]. Our model treats the rain rate as lognormal since the lognormal hypothesis has been shown to match well with measured data [18; 19; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34; 35]. We utilize the lognormal distribution for the sake of definiteness; the model can easily be extended to the alternative distribution.  
The probability density function for rain rate in this model follows a lognormal distribution:  
\[f(r_{r})=\frac{1}{{{{(2\pi)}^{1/2}}{r_{r}}{\sigma_{r}}}}\exp\!\left[{-\frac{1}{2} \left({\frac{{\ln r_{r}-m_{r}}}{{\sigma_{r}}}}\right)^{2}}\right] \tag{1}\]  
where \(r_{r}\) is the rain rate and \(m_{r}\) and \(\sigma^{2}_{r}\) are the mean and variance of the transformed variable. The mean and variance are derived from the mean and variance of the initial variables of rain rate, \(m_{R}\), and \(\sigma^{2}_{R}\), as follows:  
\[m_{r}=\ln m_{R}-\frac{1}{2}\ln\!\left({1+\frac{{\sigma_{R}^{2}}}{{m_{R}^{2}}}}\right) \tag{2}\]  
\[\sigma_{r}=\ln\!\left({1+\frac{{\sigma_{R}^{2}}}{{m_{R}^{2}}}}\right). \tag{3}\]  
The function that predicts the attenuation of the RF signal as a function of rain rate is described in the following section. Table 1 shows the large variation in values of parameters that are typical for certain regions [25].