###### Abstract  
For sites located in different climatic regions, we estimated the relationship between the annual average probability distributionof exceeding a fixed rain attenuation, and the carrier frequency in the range 16 to 100 GHz, in the zenith paths of GeoSurf satellite constellations. In these constellations rain attenuation is independent of the altitude and number of satellites. Rain attenuation iss calculated with the Synthetic Storm Technique, a reliable prediction method, by using on-site measured rain-rate time series. A suitably defined outage probability factor shows that the outage probability, for fixed power margin, tends to saturate as frequency increases. In wideband radio-links, such as in spread spectrum design, there is very likely a long-term distortion due to the in-band outage probability. The results are oriented to design systems faded by rain attenuation whose value is also the total power margin available due to a mixture of coding and hardware technology, whose combination is not of concern here.  
GeoSurf; in-band outage probability; outage probability; rain attenuation; satellite radio-links; Synthetic Storm Technique; wideband radio-links; zenith path +
Footnote †: journal: 18 August 2022  
rain attenuation \(3\leq A\leq 30\) dBs and carrier frequency \(16\leq f\leq 100\) GHz and studied how \(P(A)\) changes for a fixed value of \(A\). However, we anticipate that the results can be applied also to larger rain attenuation because the modelling of a suitable defined outage probability factor is insensitive to rain attenuation, as shown below.  
Notice that, in our study, it does not matter how the power margin of a radio-link is obtained: it can be a combination of coding and hardware, which can vary very rapidly as technology develops. What counts is the total value, which is here supposed to compensate for rain attenuation.  
For illustrating the general characteristics of the relationship \(P(A)\) versus \(f\), we report the results concerning the sites listed in Table 1, located in different climatic regions. The rain attenuation expected in GeoSurf radio-links is independent of the particular GeoSurf design (e.g., altitude and number of satellites), because all paths to/from a satellite of the constellation are always vertical (zenith). \(A\) was calculated with the Synthetic Storm Technique (SST) [3]. We considered these specific sites because of the availability of the rain-rate time series \(R(t)\) (mm/h), with rain rate averaged in 1 min intervals, continuously recorded on-site for several years, although in zenith paths the SST requires only to know the probability distribution of rain rate \(P(R)\) (see Equation (29) of ref. [3]). In other words, calculations and models reported below can be derived for any site where on-site measured--best choice--or estimated \(P(R)\) is available.  
Since the 1960's, many experiments and studies have been conducted--and reported in a huge numbers of papers--to establish the impact of rain attenuation on the outage probability of fixed (i.e., GEO) and mobile satellite (i.e., MEO and LEO) communications. Because of the saturation of the lower frequency bands currently used, as hardware and transmitter performance improves, satellite communication systems tend to use higher frequencies and larger bandwidths, including THz frequencies [4], for GEO, MEO, and LEO constellations. Unfortunately, rain attenuation, besides clear-sky fading, is still the most serious physical and unavoidable source of outage; therefore, studies, as the one reported in the present paper, or in ref. [5], are always welcome.  
After this introductory section, Section 2 reports and discusses the probability distributions of rain attenuation at the reference sites in zenith paths obtained with the SST, Section 3 defines and reports a suitable outage probability factor at the reference sites, Section 4 models the outage probability factor, Section 5 defines the in-band outage probability factor, and finally Section 6 concludes.