## 1 GeoSurf Satellite Constellations at Millimeter Wavelengths  
GeoSurf constellations [1] share most of the advantages of current GEO (Geostationary), MEO (Medium Earth Orbit) and LEO (Low Earth Orbit) satellite constellations, without suffering most of the drawbacks of these designs, because GeoSurf uses zenith radio propagation paths worldwide; therefore, it is as if the site were at the equator and connecting to a GEO satellite at the zenith. In [2], we studied the tropospheric attenuation of GeoSurf zenith paths and compared it to that of the slant paths to GEO, MEO and LEO satellites. In [3], we studied the annual average probability distribution \(P(A)\) of exceeding rain attenuation \(A\) (dB), and showed how it depends on carrier frequency at sites located in different climatic regions (sites also considered in the present paper). As experimental results, we considered the rain attenuation time series \(A(t)\) simulated with the Synthetic Storm Technique (SST) [4] from on-site rain rate time series \(R(t)\) recorded for several years. Because these studies refer to channels with negligible linear (amplitude and phase) distortions--i.e., narrow-band channels-in [5], we simulated the slowly time-varying transfer-function and linear distortions in ultra-wideband radio links in GeoSurf constellations working at millimeter wavelengths, namely 80 GHz in 10-GHz bandwidth channels. We think that these channels can be used in future worldwide internet, adopting spread spectrum modulation and code division multiple access (CDMA) [6, 7, 8, 9, 10], with BPSK and QPSK modulation. In fact, CDMA can provide a large processing gain, it is robust against frequency interference and it does not require multiple access coordination (all characteristics which, together with the further advantages of the GeoSurf constellations concerning radio propagation, mentioned in Table 1 of [1], make it an effective choice to provide high bit rates to scattered users). Continuing our development of GeoSurf satellite constellations design, in [11], we studied the interference caused by amplitude and phase distortions induced by rain (simulations with the SST) in ultra-wideband communication systems designed for using amplitude modulation. We have shown that the theoretical capacity loss of these channels is small; therefore, they can be used at millimeter waves. In the present paper, our aim is to study fade durations at millimeter waves by simulating rain attenuation time series \(A(t)\) at 80 GHz, with the SST, from long-term on-site recordings of rain rate time series \(R(t)\). Fade duration is the second most important link parameter required for designing a satellite link faded by rain because it gives the interval \(A(t)\) exceeds an attenuation threshold \(S\) (dB) (usually, the link power margin compared to clear sky). The first is the probability of exceeding the link power margin, with no information on its duration. Therefore, fade durations have been measured, at lower frequency bands, in several satellite links, as reported in more recent references [12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27], in slant paths not larger than about \(60^{\circ}\), as in the ITU-R Recommendation [28]. No measurements or predictions are, however, available for zenith paths, even when low-latitude sites are considered [16]. Now, the SST is a powerful tool that can simulate reliable time series \(A(t)\) in slant paths in which simulations of fade duration statistics were validated years ago [29] and more recently [22]. Figure 1 shows \(A(t)\) at 80 GHz, circular polarization, simulated in the zenith path at Spino d'Adda (Table 1). A threshold \(S\) (dB) can be crossed more than once, with different intervals \(\tau\) (min) which define the stochastic variable \"fade duration\". Since \(S\) is usually the power margin of the radio link, any time \(A(t)>S\), the communication link fades and the communication system experiences an outage. We can count the number of these occurrences and calculate the outage probability.