# Global Formulation of the Synthetic Storm Technique Oriented to Satellite Link-Budget Design
## 3 Model of the Average Exponent of the Global SST  
Following the observations and calculations reported in [37; 38], and based on the rain attenuation time series \(A_{SST}(t)\) obtained from the rain rate time series of the database recorded at the sites listed in Table 2, we have derived the following model of the average value of the parameter \(m\), approximately valid in the range \(20^{\circ}\leq\theta\leq 90^{\circ}\) and in the range  
\begin{table}
\begin{tabular}{l c c c c c c} \hline \hline \multirow{2}{*}{**Site**} & \multirow{2}{*}{**Lat. N (\({}^{\circ}\))**} & \multirow{2}{*}{**Long. E**} & \multirow{2}{*}{**Altitude \(H_{S}\)**} & **Precipitation** & \multirow{2}{*}{**Years of Rain Rate**} & **Wind Speed at** \\  & & & **(\({}^{\circ}\))** & & **Height \(H_{R}\)** & **Measurements \(n_{Y}\)** & **700 Mbar \(v\)** \\  & & & & **(km)** & **(m/s)** \\ \hline Spino d’Adda (Italy) & 45.4 & 9.5 & 84 & 3.341 & 8 & 10.6 \\ Gera Lario (Italy) & 46.2 & 9.4 & 210 & 3.483 & 5 & 8.2 \\ Fucino (Italy) & 42.0 & 13.6 & 680 & 2.905 & 5 & 10.4 \\ Madrid (Spain) & 40.4 & 356.3 & 630 & 3.001 & 8 & 10.9 \\ Prague & 50.0 & 14.5 & 250 & 3.051 & 5 & 12.6 \\ Czech Republic & & & & & & \\ Tampa (Florida) & 28.1 & 277.6 & 50 & 4.528 & 4 & 9.6 \\ Norman (Oklahoma) & 35.2 & 262.6 & 420 & 4.145 & 4 & 12.4 \\ White Sands & & & & & & \\ (New Mexico) & & & & & & \\ Vancouver & & & & & & \\ (British Columbia) & & & & & & \\ Fairbanks (Alaska) & 64.9 & 212.2 & 184 & 2.688 & 3 & 12.4 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Geographical coordinates, altitude (m), rain height \(H_{R}\) (km) [42], number of years \(n_{Y}\) of continuous rain rate measurements and wind speed \(v\) (m/s) at 700 mbar altitude [40], except for Spino d’Adda whose rain storm speed was estimated from radar measurements [1].  
\(10\leq f\leq 100\) GHz. In the following, we have further distinguished three ranges of the elevation angle. The base of the logarithm is 10, the angle is in degrees, and frequency is in GHz.  
**Range (a)**: \(70^{\circ}\leq\theta\leq 90^{\circ}\), \(m=1\). No other action is required.  
**Range (b)**: \(30<\theta<70^{\circ}\).  
1. Calculate the value at 10 GHz, \(m_{f10}(\theta)\) \[m_{f10}(\theta)=2.34\times 10^{-4}\theta^{2}-2.21\times 10^{-2}\theta+1.38\] (4)  
2. Calculate the value at 100 GHz \(m_{f100}(\theta)\) \[m_{f100}(\theta)=1.22\times 10^{-4}\theta^{2}-1.15\times 10^{-2}\theta+1.2\] (5)  
3. Calculate the normalized average value \(m_{n}(f)\) see Figure 1  
\[\begin{array}{ll}m_{n}(f)=&-7.07\times\left(\log f\right)^{4}+44.73\times \left(\log f\right)^{2}-104.57\times\left(\log f\right)^{2}\\ &+107.69\times\log f-40.77\end{array} \tag{6}\]  
4. Calculate, finally, the denormalized value \(m(f,\theta)\) to use in Equation (2) \[m(f,\theta)=m_{n}(f)\Big{[}m_{f100}(\theta)-m_{f10}(\theta)\Big{]}+m_{f10}(\theta)\] (7)  
**Range (c)**: \(20^{\circ}\leq\theta<30^{\circ}\).  
1. Calculate the probability (%) \[p=\left[0.1+0.005\times\left(f-20\right)\right]\] (8) This probability ranges between \(p=0.1\) at \(f=20\) GHz, and \(p=0.5\) at \(f=100\) GHz.  
2. For \(p\leq\left[0.1+0.005\times\left(f-20\right)\right]\), calculate \(m(f,\theta)\) \[m(f,\theta)=d_{2}\big{[}\log(p)\big{]}^{2}+d_{1}\log(p)+d_{0}\] (9) with \[d_{2}=\frac{m_{1}\times\log(0.8)+0.99\times\left(b+2\right)-2-a}{4\times\log (0.8)+2(a^{2}-b^{2})+a^{2}b-ab^{2}}\] \[d_{1}=\frac{m_{1}\big{(}b^{2}-a^{2}\big{)}+0.99\times\left(4-b^{2}\right)+a ^{2}-4}{4\times\log(0.8)+2(a^{2}-b^{2})+a^{2}b-ab^{2}}\] \[d_{0}=\frac{m_{1}ab(a-b)-1.98\times b(b+2)+4a+2a^{2}}{4\times\log(0.8)+2(a^{ 2}-b^{2})+a^{2}b-ab^{2}}\] \[m_{1}=2.30\times\left(1.71\times 10^{-3}f+8.46\times 10^{-2}\right) \times\log(\theta)-9.11\times 10^{-3}f+0.643\] \[a=\log\{0.8\times\left[0.1+0.005\times\left(f-20\right)\right]\}\] \[b=\log[0.1+0.005\times\left(f-20\right)]\]  
3. For \(p>\left[0.1+0.005\times\left(f-20\right)\right]\), no correction is required.