# Model for the Prediction of Rain Attenuation Affecting Free Space Optical Links
## 2 Predicting Rain Attenuation Impairing FSO Links
### Path Attenuation  
The prediction of the path attenuation induced on FSO links starts from the whole set of synthetic rain fields generated by MultiEXCELL, which only requires the local CCDF of the rain rate, also known as \(P(R)\)[19], as the input.  
As a reference, we have considered the site of Hyderabad, Pakistan (latitude = 25.38\({}^{\circ}\) N, longitude = 68.37\({}^{\circ}\) E), which is characterized by limited precipitation through the year, besides the absence of fog. The input local \(P(R)\) was estimated using the prediction model included in Annex 1 of recommendation ITU-R P.837-6 [28]. The yearly \(P(R)\) curve is reported in Figure 2.  
Figure 3 shows the zoom on a sample rain rate map (out of the whole set consisting of 1088 maps) generated by MultiEXCELL, whose dimension is 230 km \(\times\) 230 km with a spatial resolution equal to 250 m \(\times\) 250 m. For each map, the total rain attenuation \(A\) (in dB) affecting the FSO link is calculated as:  
\[A=\int\limits_{L}\gamma(l)dl=\int\limits_{L}kR(l)^{\alpha}dl=\sum\limits_{i=1} ^{N}kR_{i}{}^{\alpha}\Delta l, \tag{2}\]  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Shape Parameter (\(\mu\))** & \(k\) & \(\alpha\) \\ \hline \(-3\) & 4.0684 & 0.2077 \\ \(-2\) & 2.2838 & 0.4050 \\ \(-1\) & 1.5921 & 0.5506 \\ 0 & 1.2924 & 0.6436 \\ 1 & 1.1394 & 0.7057 \\ 2 & 1.0505 & 0.7497 \\ 3 & 0.9938 & 0.7823 \\ 4 & 0.9551 & 0.8074 \\ 5 & 0.9273 & 0.8273 \\ 6 & 0.9065 & 0.8435 \\ 7 & 0.8905 & 0.8569 \\ 8 & 0.8779 & 0.8682 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Coefficients in (1) for different values of the DSD shape parameter (\(\mu\)).  
Figure 1: Specific rain attenuation (dB/km) as a function of the rain rate (mm/h) for different values of the shape parameter \(\mu\) of the Gamma DSD.  
where \(L\) is the length of the link; \(k\) and \(\alpha\) are the power-law coefficients in (1) and listed in Table 1; and \(R(l)\) is the rain rate at position \(l\) along the path, associated with a given pixel of the rain map. The right-end term of (2) provides the discretized version of the expression to calculate the rain attenuation \(A\), in which \(R_{i}\) is the rain intensity associated with the \(i\)-th pixel and \(\Delta l\) = 250 m. In order to increase the statistical significance of the results and to exploit the large rain maps generated by MultiEXCELL as much as possible, several attenuation values are obtained by moving the hypothetical FSO link across the synthetic rain map. As a result, each rain map is turned into an attenuation map, as shown, for example, in Figure 4 (horizontal link, \(L\) = 5 km, \(\mu\) = 1).  
Figure 4: Zoom on a sample attenuation map (color bar indicating attenuation in dB) obtained from the rain maps in Figure 3 using (2) (5-km FSO link, power-law coefficients corresponding to \(\mu\) = 1 in Table 1).  
Figure 3: Zoom on a sample rain map generated for Hyderabad, Pakistan, using MultiEXCELL (colorbar indicating the rain rate in mm/h).  
Figure 2: Yearly \(P(R)\) for Hyderabad, Pakistan, estimated using recommendation ITU-R P.837-6.  
As a final step, the rain attenuation CCDF, also typically referred to as \(P(A)\), is calculated by gathering all the \(A\) values associated with the set of synthetic rain maps generated by MultiEXCELL for the site of interest. Figure 5 reports the attenuation statistics for a link with \(L=5\) km and for different \(\mu\) values. Results confirm the strong impact of the DSD on rain attenuation, as well as the high \(A\) values affecting FSO links: in the \"best case\", \(A\) exceeds 45 dB for the exceedance probability \(P\approx 0.01\%\) of the yearly time (approximately \(1\) h/year).