## 1 Introduction  
In designing satellite links above 10 GHz--the frequency beyond which the attenuation due to rainfall is no longer negligible--the traditional approach is to consider the average (annual or worst month) probability distribution \(P(A)\)--namely, the fraction of time concerning the reference interval--of exceeding rain attenuation \(A\) (dB) [1; 2; 3; 4; 5]. In this design, \(P(A)\) represents the outage probability and \(A\) (dB) represents the required power margin necessary to ensure that the link continues to work, i.e., to deliver data with the minimum tolerated probability of symbol error. For example, if the average annual probability is \(P(A)=10^{-4}\) (i.e., \(10^{-2}\) %) then the outage total time is \(365\times 24\times 60\times 10^{-4}=53.56\) min. In [6], in slant paths to geostationary satellites, we have shown that in data channels--i.e., in channels that do not require tight real--time communication--this approach is too pessimistic because it predicts too large a power margin (overdesign). If the data volume up-- or downloaded during rainfall, with a constant probability of symbol error, is more valuable than the instantaneous symbol rate--e.g., in remote sensing and internet of things (IoT) using satellites--the method discussed in [6] can avoid overdesign. In recent years, as far as we know, no other methods with the same interesting potentiality have been proposed. This method should be useful when: (a) data must be up-- or downloaded when it is raining (e.g., because of a satellite pass, or other technical or service constraints); (b) real--time communication is not required. The method is characterized by a single parameter, namely the mean efficiency \(0\leq\eta\leq 1\) of the link, which is suitably defined in [6] and recalled in Section 2. Theoretically, the method can transfer the same data volume as if the link were in clear-sky conditions,and it is directly applicable to BPSK (Bipolar Phase-Shift Keying), QPSK (Quadrature Phase--Shift Keying) and to the more general QAM (Quadrature Amplitude Modulation) schemes. Moreover, as shown in [6], it implements [PERSON]'s channel capacity. In synthesis, according to the theory discussed in [6], a small power margin--regardless of instantaneous rain attenuation-and a small bandwidth expansion allow the delivery of an average symbol rate equal to the symbol rate obtainable when \(A=0\) (i.e., clear--sky conditions) therefore transferring, in a given interval, the same data volume as if there were no rainfall. In [6], we have compared its theoretical performance with several Adaptive Coding and Modulation (ACM) techniques [7; 8; 9]. We have shown that, even theoretically, these latter techniques cannot achieve the maximum efficiency [10; 11; 12; 13; 14; 15] as, on the contrary, our method theoretically can. In the present paper, we extend the theory proposed in [6] (see Section 4) and apply it to the links of GeoSurf satellite constellations, recently proposed in [16], working at millimetre waves, which are the new frontier of satellite frequencies. The GeoSurf constellations share most of the advantages of GEO (Geostationary), MEO (Medium Earth Orbit) and LEO (Low Earth Orbit) satellite constellations, without suffering many of their drawbacks (see Table 1 of [16]) because the propagation paths are vertical (at the local zenith) at any latitude. No measurements or predictions are, however, available for zenith paths. Therefore, we consider as experimental results the rain attenuation time series \(A(t)\) simulated in zenith paths with the Synthetic Storm Technique (SST) (see Equation (29) of [17]) from rain-rate time series \(R(t)\), recorded on-site for several contiguous years in the sites listed in Table 1. The SST is a powerful tool [18; 19; 20; 21; 1, 1]. The sites are in different climatic regions; therefore, the findings show how the design would change worldwide.