## 2 The Synthetic Storm Technique Applied to Zenith Paths  
Satellite communication links at centimeter and millimeter wave frequencies are faded by rainfall. For a reliable link-budget design, we need to know, at the very least, the annual probability distribution function, \(P(A)\), of rain attenuation \(A\) (dB) measured/predicted in the up- or down-links to a satellite. Instead of long and expensive measurements of beacon attenuation, prediction models are used for estimating \(P(A)\) from locally, measured or \begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**Site** & **Latitude N (\({}^{\circ}\))** & **Longitude E (\({}^{\circ}\))** & **Altitude \(H_{\text{S}}\) (km)** & **Precipitation** & **Rain Rate Data** \\  & & & & **Height \(H_{\text{R}}\) (km)** & **Bank (Years)** \\ \hline Spino d’Adda (Italy) & 45.4 & 9.5 & 0.084 & 3.341 & 8 \\ Gera Lario (Italy) & 46.2 & 9.4 & 0.210 & 3.483 & 5 \\ Fucino (Italy) & 42.0 & 13.6 & 0.680 & 2.905 & 5 \\ Madrid (Spain) & 40.4 & 356.3 & 0.630 & 3.001 & 8 \\ Prague (Czech Republic) & 50.0 & 14.5 & 0.250 & 3.051 & 5 \\ Tampa (Florida) & 28.1 & 277.6 & 0.050 & 4.528 & 4 \\ Norman (Oklahoma) & 35.2 & 262.6 & 0.420 & 4.145 & 4 \\ White Sands (New Mexico) & 32.5 & 253.4 & 1463 & 4.744 & 5 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Geographical coordinates, altitude (km), rain height \(H_{\text{R}}\) (km), and number of years of continuous rain rate measurements at the indicated sites. estimated, annual probability distributions, \(P(R)\), of 1 min rain rate \(R\). The Synthetic Storm Technique [3] is a powerful and accurate tool, as is now recognized, that can produce all the necessary statistics of rain attenuation, not only \(P(A)\), but also fade durations and rate of change of attenuation because it provides reliable rain attenuation time series \(A(t)\). From knowing the rain-rate time series \(R(t)\) (mm/h, averaged in 1 min intervals), recorded at a site, the SST can generate \(A(t)\), at any frequency and polarization, and for any slant path above about 10\({}^{\circ}\). Because it reproduces reliable \(A(t)\), it has been used in the last 30 years for many purposes by several researchers [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]. When the Synthetic Storm Technique is applied to zenith paths, the rain attenuation time series lasts exactly as the rain-rate time series, as discussed in ref. [3], Equations (18) and (29). The vertical (zenith) structure of precipitation is modeled with two layers of precipitation of different depths, a model taken from ref. [21]. Starting from the ground, there is rain (hydrometeors in the form of raindrops, water temperature of 20 \({}^{\circ}\)C, called layer A), followed by a melting layer, i.e., melting hydrometeors at 0 \({}^{\circ}\)C, layer B. The vertical rain rate \(R\) (mm/h, averaged in 1 min) in layer A is assumed to be uniform and given by that measured at ground, i.e., by the rain gauge, \(R_{A}=R\). By assuming simple physical hypotheses, calculations showed that the vertical precipitation rate in the melting layer, termed \"apparent rain rate\", is also uniform and given by \(R_{B}=3.134\times R_{A}\)[21]. The height of the precipitation (rain and melting layer, \(H_{R}\)) above sea level to be used in the simulation is the latest estimate of the ITU-R [22]. The layer B depth is 0.40 km, regardless of the latitude [3, 21]. Now, according to Equation (29) of ref. [3], rain attenuation \(A(t)\) (dB) in the zenith path is given by:  
\[A(t)=k_{A}R(t)^{\alpha_{A}}\times(H_{R}-0.4-H_{s})+k_{B}(3.134\times R(t))^{ \alpha_{B}}\times 0.4 \tag{1}\]  
In Equation (1), \(k\) and \(\alpha\) are the constants that give the specific rain attenuation \(Y=kR^{a}\)--\(R\) measured in mm/h and \(Y\) in dB per 1 km--in the two layers from the rain rate \(R\) measured at the ground, calculated at 20 \({}^{\circ}\)C in layer A and at 0 \({}^{\circ}\)C in layer B [3], values given in ref. [23].