# Real-Time Rain Rate Evaluation via Satellite Downlink Signal Attenuation Measurement
## 3 Earth-Space Radio Links in the Presence of Rain  
### Effects of Rain on Earth-Space Radio Links  
At frequencies higher than 5 GHz (as those considered in the NEFOCAST project), the electromagnetic wave that propagates from satellite to the Earth interacts with atmospheric gases and hydrometeors [25]. It has been shown that the attenuation due to the presence of liquid hydrometeors is the predominant mechanism that produces the degradation of the signal [26]. However, the effects of hydrometeors in the mixing phase (such as wet snow, graupel and melting hydrometeors) cannot be neglected. The interaction between electromagnetic waves and rain produces an attenuation of the propagating signal due to the combination of two mechanisms: Scattering in all the directions and absorption, causing a total, path-cumulated attenuation of the signal reaching the receiver. The absorption and scattering produced by liquid particles (such as raindrops) depends on the frequency and the polarization state of the incident wave, and on the characteristics of the precipitation; namely, the size and shape of the raindrops. Different theories have been developed to model the scattering phenomena by raindrops. The theory of Rayleigh-Gans scattering is applicable to small (\(D<<\lambda\), where \(D\) is the diameter of the drops), non-absorbing and spherical drops, while the Mie theory is applicable to spherical raindrops with \(D\cong\lambda\). Finally, the scattering amplitudes of non-spherical drops cannot be computed analytically. However, several numerical techniques are available for calculating the scattering properties, such as the T-matrix method [27]. Therefore, the size distribution of particles plays a central role in the estimation of the total attenuation undergone by the signal that propagates along the radio link and due to rain. The DSD can be modelled by different theoretical parametrizations (see [28] and the references therein) or can be measured by disdometers. Knowing the DSD, either measured or simulated, the specific attenuation \(k\) (dB/km) due to the presence of precipitation can be obtained as  
\[k=4.343\times 10^{-3}\ \int_{0}^{\infty}\sigma_{E}(q,D,\lambda,T,h)\ N(D)\ \mathrm{d}D, \tag{1}\]  
where \(\sigma_{E}\) is the extinction cross section (namely, the area that when multiplied by the power density of incident wave, gives the power loss due to absorption and scattering) that depends on the polarization (\(q\) is a label that stands for \"vertical\", \"horizontal\", or \"circular\"), wavelength (\(\lambda\)) of the incident waves, diameter (\(D\)), type of the particles (\(h\) is a label that stands for \"rain drop\", \"hail\", \"snow flake\", \"graupel\", or \"particle in a mixing phase\") and on the temperature (\(T\)), while \(N(D)\) is the DSD. The ITU-R P.838-3 recommendation [29] provides, for frequencies ranging from 1 to 1000 GHz, the coefficients of the relation \(k\) vs. rain rate \(R\) given in the form of an approximate power law, for horizontal and vertical polarizations. Figure 5 shows clearly that, for the operational frequencies adopted in the NEFOCAST project, attenuation due to severe precipitation (\(R>100\) mm/h) ranges from 3.6 to 11.3 dB/km, depending on frequency and polarization. Similar results can be obtained considering measured DSDs (see Section 4.4). The advantage of using measured DSDs is that, from a climatological point of view, they are more representative of the precipitation characteristics of the geographical area where they have been sampled; however, they can be affected by measurement errors. In addition, it is possible to characterize the error due to the approximation of the \(R\)-\(k\) relation with respect to the variability of the DSD, which is not known from the ITU-R P.838-3 recommendation [29]. Figure 6 shows the geometry of the Earth-space radio-link adopted during the NEFOCAST project, where \(\gamma\) is the elevation angle (about 40\({}^{\circ}\) in this case).