## 2 The Smart Rainfall System (SRS)
### Satellite-Earth Link Model  
It is well known that rainfall scatters microwaves, and the \(Ku\) band used for most of DVB-S/S2 transmissions is particularly sensitive to this phenomenon [16; 17]. To describe this phenomenon, the radiation emitted by a geostationary satellite can be modeled as a plane wave at the earth station thanks to the far-field condition [18]. Thus, the electric \(\mathbf{E}\) and the magnetic \(\mathbf{H}\) fields can be locally described as  
\[\mathbf{E}(\mathbf{r}_{rx})=E_{0}(\mathbf{r}_{rx},\mathbf{r}_{tx})\mathrm{e}^{ -j\hat{\mathbf{k}}\cdot(\mathbf{r}_{rx}-\mathbf{r}_{tx})}\mathbf{\hat{p}} \tag{1}\]  
\[\mathbf{H}(\mathbf{r}_{rx})=\frac{1}{\eta_{0}}\hat{\mathbf{k}}\times\mathbf{E} (\mathbf{r}_{rx}) \tag{2}\]  
where \(\mathbf{r}_{rx}\) and \(\mathbf{r}_{tx}\) are the position vectors for the receiving antenna and the satellite, respectively, \(\hat{\mathbf{k}}=(\mathbf{r}_{rx}-\mathbf{r}_{tx})/||\mathbf{r}_{rx}-\mathbf{ r}_{tx}||_{2}\) is the unit vector indicating the direction of propagation, \(\hat{\mathbf{p}}\) is the unit vector for the polarization, \(k=\omega\sqrt{\eta_{0}\epsilon_{0}}\) is the vacuum's wavenumber (with \(\mu_{0}\) and \(\epsilon_{0}\) magnetic permeability and dielectric permittivity of the vacuum, respectively, and \(\omega=2\pi f\) angular frequency), \(\eta_{0}=\sqrt{\mu_{0}/\epsilon_{0}}\) is the vacuum's intrinsic impedance, and \(E_{0}\) is the wave's complex amplitude. The time factor \(e^{j\omega t}\) has been omitted to simplify the notation. The power gathered by the antenna can be formulated as [18]  
\[P_{R}=A_{e}p_{inc}(\mathbf{r}_{rx})=e_{p}e_{l}\frac{\lambda_{0}^{2}}{4\pi}G_{ rx}p_{inc}(\mathbf{r}_{rx})=e_{p}e_{l}\frac{\lambda_{0}^{2}}{8\pi\eta_{0}}G_{ rx}|E_{0}(\mathbf{r}_{rx},\mathbf{r}_{tx})|^{2}e^{-2\alpha_{min}l}=P_{R}^{0}e^{-2 \alpha_{min}l} \tag{3}\]  
where \(A_{e}\) is the antenna's effective area; \(p_{inc}\) is the incident power density; \(e_{p}\) and \(e_{l}\) are the polarization and load mismatch efficiencies, respectively, \(\lambda_{0}\) is the wavelength in the vacuum; \(G_{rx}\) is the antenna's gain; \(\alpha_{min}\) is the rain's attenuation constant; \(P_{R}^{0}\) is the received power in case of clear sky; and \(l\) is the distance, taken along the wave's path, between the receiving antenna's focus and the melting layer, whose altitude \(h_{0}\) is here assumed to coincide with the zero-degree isotherm level. The latter is here estimated by means of a linear extrapolation on the base of environmental temperature measurements gathered by two weather stations located at a small geographic distance from the SRS sensors (this approach relies on the assumption of a linear vertical temperature profile). About the quantity \(l\), the following relation holds,  
\[l=\frac{h_{0}-h_{ant}}{\sin(\vartheta)} \tag{4}\]  
where \(h_{ant}\) and \(\vartheta\) are the receiving antenna's altitude and the elevation angle, respectively. Figure 1 exemplifies the considered configuration. It is important to highlight that \(\alpha_{rain}\) explains the rain contribution only; any other attenuation terms can be included in \(E_{0}\). Moreover, in writing the Equation (3), it is assumed that the rain's effect is constant along \(l\). Passing to a logarithmic representation, the total attenuation due to the meteorological perturbation is given as  
\[L_{rain}^{[\mathrm{dB}]}=P_{R}^{0}\mathrm{,}\mathrm{(dBm)}-P_{R}^{[\mathrm{dBm }]}=20\alpha_{rain}l\mathrm{log}_{10}(e)=\gamma_{rain}l^{[\mathrm{km}]} \tag{5}\]  
where \(P_{R}^{0}\mathrm{,}\mathrm{(dBm)}\) and \(P_{R}^{\mathrm{(dBm)}}\) are \(P_{R}^{0}\) and \(P_{R}\) expressed in dBm, respectively, \(\gamma_{rain}=2\cdot 10^{4}\alpha_{rain}\mathrm{log}_{10}(e)\) is the rainfall specific attenuation given in dB/km, whereas \(l^{[\mathrm{km}]}\) is \(l\) expressed in km.