# Real-Time Rain Rate Evaluation via Satellite Downlink Signal Attenuation Measurement
## 4 Direct Method for the Estimation of the Rain Rate
### Rain Rate Estimation Algorithm  
The NEFOCAST rain rate estimation algorithm relies on the availability of the instantaneous value of the ratio \(\eta=E_{s}/N_{0}\) between the received energy-per-symbol \(E_{s}\) and the one-sided power spectral density of the additive white Gaussian noise \(N_{0}\). The latter is the sum of the noise picked up by the receive antenna (expressed by the noise temperature \(T_{A}\)) and the noise originated from the receiver electronics (represented by the noise temperature \(T_{RX}\)). The presence of rain induces a specific additional path attenuation \(A\) (with \(A>1\)) which has a twofold effect: (i) it attenuates all RF waveforms (both the useful signal and the cosmic noise) coming from the outer space; (ii) it increases the atmospheric noise picked up by the receive antenna [54]. More specifically, denoting as \(E_{s}\)\({}^{(Clear\;Sky)}\) the energy \(E_{s}\) received in \"clear sky\" conditions, i.e., in the absence of rain (\(A=1\)), \(\eta\) can be expressed as  
\[\eta=\frac{\frac{E\left({}^{Clear\;Sky}\right)}{A}}{k_{B}\left[\underbrace{ \frac{T_{C}}{A_{atm}\cdot A}+T_{m}\bigg{(}1-\frac{1}{A_{atm}\cdot A}\bigg{)}+T _{G}}_{T_{A}}+T_{RX}\right]} \tag{7}\]  
where the meaning of parameters is specified in Table 3 along with their values. In detail, \(T_{m}\) is the mean thermodynamic temperature of the meteorological formation, such as clouds and rain, which \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(R=\alpha\;k^{\beta}\) & \(\alpha\) & \(\beta\) & **NMAE** & **NB** & **RMSE** & **CC** \\ \hline Return link 14,216.6 MHz, & & & & & & \\ H polarization. & 17.910 & 0.836 & 17.9\% & \(-4.3\%\) & 0.90 & 0.983 \\ Forward link 11,345.5 MHz, & & & & & \\ V polarization. & 28.140 & 0.798 & 20.6\% & \(-4.4\%\) & 1.07 & 0.976 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters \(\alpha\) and \(\beta\) of the \(R\)-\(k\) algorithms and statistics indices between measured and estimated rain rates, in the Ku-band. constitute an absorbent, and consequently emissive, medium (see [54], page 184). According to [55; 56], a value of 275 K can be assumed for \(T_{m}\); however, we also numerically verified that taking values in the range 260-280 K (as formerly specified in [57]) has no appreciable effects on the values of the rain rate estimates. \(T_{\text{C}}\) and \(T_{\text{G}}\) are the noise temperatures of the cosmos and of the ground, respectively [54]), while \(T_{RX}\) is the noise temperature of a commercial-grade receive system whose noise figure, referred to 290 K, is 0.2 dB. Finally, \(A_{atm}\) is the gaseous attenuation of the atmosphere [54], which is given by the zenith attenuation for the specified link frequency (available from graphs) divided by the sine of the elevation angle of the satellite above the horizon. After some manipulation, we get  
\[A=\frac{\left[\frac{T_{C}}{A_{atm}}+T_{m}\Big{(}1-\frac{1}{A_{atm}}\Big{)}+T_{ G}+T_{RX}\right]\cdot\frac{\eta^{(Clear\,Sy)}}{\eta}+\frac{T_{m}-T_{C}}{A_{ atm}}}{T_{m}+T_{G}+T_{RX}}=\Xi(\eta) \tag{8}\]  
where \(\eta^{(Clear\,Sy)}\) is the clear-sky value of \(\eta\), which can be obtained letting \(A=1\) in (7). Also, we notice that \(\Xi(\eta^{(Clear\,Sy)})=1\). By using (8), the instantaneous value of \(\eta\) inferred from the received signal (denoted as \(\hat{\eta}\)) can be mapped into an estimate of the attenuation \(\hat{A}\), i.e., \(\hat{A}=\Xi(\hat{\eta})\). Let us now assume a link geometry as depicted in Figure 6a, wherein the rain height \(h_{B}\) is obtained either from a short-term forecast via WRF model or from an archive of historical data such as the MERRA.