## 3 Attenuation, Phase Shift, and Time Delay Due to Rainfall in Zenith Paths  
When the Synthetic Storm Technique is applied to zenith paths, as those in GeoSurf constellations, the rain attenuation time series lasts exactly as the rain-rate time series, as discussed in [17], as it can be read in Equations (18) and (29) of [17]. The vertical (zenith) structure of precipitation is modelled with two layers of precipitation of different depths. Starting from the ground there is rain (hydrometeors in the form of raindrops with a water temperature of 20 \({}^{\circ}\)C called layer A), followed by a melting layer (melting hydrometeors at 0 \({}^{\circ}\)C) called layer B above the rainfall layer and reaching a height of 0 \({}^{\circ}\)C. The vertical rain rate \(R\) (mm/h, averaged in 1 min) in layer A is assumed to be uniform and given by that which is measured at the ground, i.e., by the rain gauge, \(R_{A}=R\). By assuming basic physical hypotheses, calculations show that the vertical precipitation rate in the melting layer, termed \"apparent rain rate\", is also uniform and given by \(R_{B}=3.134\times R_{A}\)[20]. The height of the rainfall above sea level to be used in the simulation is the latest estimate of the ITU-R [21]. The layer B depth is 0.400 km, regardless of the latitude and longitude. Notice that the ITU-R assumes that the depth of the melting layer is 0.360 km, while in the Synthetic Storm Technique, its value is 0.400 km, fixed many years before the ITU-R [20]. Table 2 reports the local precipitation heights for the indicated sites. Now, according to Equation (29) of [17], the rain attenuation \(A(t)\) (dB) in the zenith path is given by the time-series:  
\[A(t)=k_{A}R(t)^{\alpha_{A}}\times(H_{R}-0.4-H_{s})+k_{B}(3.134\times R(t))^{ \alpha_{B}}\times 0.4 \tag{8}\]  
In Equation (1), \(k\) and \(\alpha\) are the constants that give the specific rain attenuation \(Y=kR^{\alpha}\)-\(R\) 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. Similarly, the phase shift \(\Phi(t)\) (\({}^{\circ}\)) in the zenith path is given by the time-series:  
\[\Phi(t)=h_{A}R(t)^{\beta_{A}}\times(H_{R}-0.4-H_{s})+h_{B}(3.134\times R(t))^ {\beta_{B}}\times 0.4 \tag{9}\]  
The constants \(k\), \(\alpha\), \(h\), and \(\beta\) are given in [22]. The time delay \(T\) (ps) is given by:  
\[T(t)=\frac{1000}{360\psi}\Phi(t) \tag{10}\] \begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Site** & **Altitude \(H_{s}\)** & **Precipitation Height \(H_{R}\), ITU-R** & **Local Precipitation Height** & **Local Height of Layer A** \\ \hline Spino d’Adda (Italy) & 0.084 & 3.341 & 3.257 & 2.857 \\ Madrid (Spain) & 0.630 & 3.001 & 2.371 & 1.971 \\ Prague (Czech Republic) & 0.250 & 3.051 & 2.801 & 2.401 \\ Tampa (Florida) & 0.050 & 4.528 & 4.478 & 4.078 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Local precipitation height (km) at the indicated sites to be used in the Synthetic Storm Technique. Notice that the depth of Layer B is 0.400 km, which, if added to the local precipitation height of layer A, gives the local precipitation height. For example, at the latitude and longitude of Spino d’Adda, the ITU-R gives 3.341 km, which becomes 3.257 km after subtracting its altitude of 0.084 km. In Equation (10), \(\psi\) is measured in GHz. See [16] for the application of the time delay due to rainfall in a narrow-band radio links in deep-space communications. Now, because the relationship between \(A(t)\) or \(\Phi(t)\) and \(R(t)\) is monotonic, the on-site probability distribution \(P(R)\) is transformed directly into the \(P(A)\), \(P(\Phi)\) of the zenith path through the variable transform given by Equation (9). Figure 4 of [3] shows the annual average probability distribution \(P(A)\) for the sites listed in Table 1. Figure 3 shows the annual probability distribution \(P(R)\). The different climatic conditions are clearly evident.