## 4 Fade Duration Experimental Results  
Figures 4 and 5 show \(P_{S,A}(D)\) and \(P_{S,B}(D)\) for some thresholds in the zenith paths at Spino d'Adda and Tampa. \(D\geq 1\) min, the time resolution of \(R(t)\) and \(A(t)\). For the other sites listed in Table 1, their probability distributions are reported in Appendix A. Notice that the blue line gives \(P_{S,A}(D)\) and \(P_{S,B}(D)\) for \(S=0\) dB, hence the probability distributions of rainfall intervals of the single rain events (thunderstorms, etc.) From Figures 4 and 5 and Appendix A, we can observe the following features:  
1. As threshold increases, fade duration largely decreases, as physically expected. 2. A fade duration \(D\) is exceeded with very different probability at the different sites; or, at the same probability, \(D\) is diverse. 3. The probability distributions tend to overlap at the largest thresholds. In other words, the same fade duration is also substantially found at higher thresholds, as shown in Figure 1.  
Figure 3: Annual probability distribution (%) \(P(A)\) of exceeding the value indicated in abscissa–80 GHz, circular polarization, zenith paths–at the indicated sites. Spino d’Adda: continuous blue line; Gera Lario: continuous black line; Fucino: continuous red line; Madrid: continuous green line; Prague: continuous magenta line; Tampa: dashed red line; Norman: dashed magenta line; White Sands: dashed green line; Vancouver: dashed blue line. The two processes are, of course, not independent. Figure 6 shows the median value-value exceeded with probability 0.5--in process B, \(D_{B}\) (min), versus the median value in process A, \(D_{A}\) (min), for the same threshold, and all sites. Recall that the median value is a statistical parameter more useful than the mean value when a stochastic variable varies in a large range, as fade duration does. We can see that \(D_{B}<D_{A}\), always. Figure 4: Spino d’Adda. (**a**) Probability distribution \(P_{S,A}(D)\) of exceeding the value indicated in abscissa at the following thresholds: 0 dB, blue; 3 dB, green; 6 dB, cyan; 10 dB, magenta; 20 dB, red; 29\(\sim\)40 dB, cyan and black; (**b**) \(P_{S,B}(D)\), same legend. Figure 5: Tampa. (**a**) Probability distribution \(P_{S,A}(D)\) of exceeding the value indicated in abscissa at the following thresholds: 0 dB, blue; 3 dB, green; 6 dB, cyan; 10 dB, magenta; 20 dB, red; 29\(\sim\)40 dB, cyan and black; (**b**) \(P_{S,B}(D)\), same legend. Remember that \(D_{B}\) refers to the number of occurrences and \(D_{A}\) to the fraction of time. For example, in Spino d'Adda, we find that when \(D_{A}=100\) min then \(D_{B}=25\) min; therefore, at a particular threshold (it is \(S\approx 3\) dB, Figure 4a, green line), 50% of the outages are due to fade durations longer than 25 min, while 50% of the outage time is due to fade durations longer than 100 min, events that occur with probability 0.2 (Figure 4b, green line). As observed above, sites belong to quite different climatic regions, and this is clearly observable in Figures 1-6 and Appendix A. Now, it is interesting to see how many occurrences are found in the sites. Figure 7 shows the average annual number of outage occurrences versus attenuation threshold. It is interesting to notice that peak values can be quite sharp, mostly below 10 dB, especially in Gera Lario. Tampa, Norman and White Sands show less pronounced peaks. These features are not directly obtainable from the \(P(A)^{\prime}s\) shown in Figure 3, but only from the histograms giving \(P_{S,A}(D)\) and \(P_{S,B}(D)\). Notice the quite different behavior of Gera Lario, very likely due to the particular geographical location among high mountains at the Northern Lake Como, with more frequent rain events than at Spino d'Adda, just 84 km away. Also of interest is the cumulative number of occurrences versus threshold \(S\) at equal fade duration exceeded, i.e., the actual cumulative number of occurrences which yield \(P_{S,B}(\tau\leq D)\).