## 3 Outage Probability Factor  
We modeled the relationship between the outage probability \(P(A)\) due to rain attenuation \(A\) as follows. For a site, let us consider a fixed value \(A\) for all frequencies and start by considering, as reference value, the probability \(P_{f_{o}}(A)\) exceeded at a reference frequency \(f_{o}\), 16 GHz in our modelling. The reference \(P(A)\), for a given \(A\), is therefore that shown in Figure 1 for each site.  
We can relate \(P(A)\) exceeded at frequencies \(f\geq 16\) GHz to the reference probability exceeded at 16 GHz, by writing:  
\[P(A,f)=P(A,f_{o})\rho(A,f) \tag{2}\]  
Now, as \(f\geq f_{o}\) it follows that the probability factor \(\rho(A,f)\geq 1\) because the same attenuation is exceeded at frequency \(f>f_{o}\) with a larger probability.  
Figures 5 and 6 show how \(\rho(A,f)\) changes as a function of frequency, for fixed values of \(A\) ranging approximately from 3 to about 30 dB in 0.2 dB steps, for some sites. Each curve shown in Figures 5 and 6 corresponds, of course, to a fixed value \(A\). The red curve shows the average values \(\rho_{m}(f)\), a parameter independent, of course, of attenuation.  
Figure 4: Annual average probability \(P(A)\) (%) of exceeding the rain attenuation \(A\) (dB) indicated in abscissa, at the indicated sites: Spino d’Adda (SF, blue continuous line); Gera Lario (GL, black continuous line); Fucino (FC, cyan continuous line); Madrid (MD, green continuous line); Prague (PR, magenta continuous line); Tampa (TM, cyan dashed line); Norman (NR, magenta dashed line); White Sands (WS, green dashed line), at 80 GHz in zenith paths. When the power margin in the zenith radio-links of the GeoSurf satellite constellations is equal to \(A\) (dB), \(P(A)\) represents the outage probability.  
Because of the different climatic rainfall regions, each site shows quite different values. However--as the average values (red curves) show--as frequency increases, the curves tend to saturate. The saturation values depend on the site considered, but this occurs approximately at about 100 GHz. Notice, however, that the spread of the curves shown in Figures 5 and 6 is not very large, therefore indicating a small sensitivity to the maximum value (30 dBs) of the rain attenuation considered.  
In conclusion, if a radio-link has a fixed power margin \(A\), as assumed in Figures 5 and 6, as the carrier frequency increases the outage probability also increases across a large bandwidth centered on the carrier frequency \(f\), as next section shows.  
Figure 5: Probability factor \(\rho(A,f)\) as function of frequency, for fixed values of \(A\) ranging approximately from 3 to about 30 dB in 0.2 dB steps (blue continuous curves): (**a**) for Spino d’Adda; (**b**) Prague. The red curve is the average value \(\rho_{m}(f)\), a parameter independent of attenuation. Notice the different ordinate range.  
Figure 6: Probability factor \(\rho(A,f)\) as function of frequency, for fixed values of \(A\) ranging approximately from 3 to about 30 dB in 0.2 dB steps (blue continuous curves): (**a**) Tampa; (**b**) White Sands. The red curve is the average value \(\rho_{m}(f)\), a parameter independent of attenuation. Notice the different ordinate range.