# Rainfall Estimation from Tropospheric Attenuation Affecting Satellite Links
## 6 Performance Evaluation  
The calibration of the model's parameters was carried out on a statistical basis, i.e., by maximizing the agreement between the measured and estimated \(P(R)s\). However, it is worth verifying the accuracy in estimating the rain rate applicability when the model is used on an instantaneous basis. To this aim, we have resorted to the following indicators of the rain event:  
* total rainfall accumulation, \(R_{T}\) [mm];
* maximum rain rate, \(R_{M}\) [mm/h]; and,
* average rain rate, \(\overline{R}\) [mm/h].  
For each indicator, we have defined the difference between the daily estimated (denoted with the subscript _est_) and measured (denoted with the subscript _meas_) quantity of interest, for each of the 70 considered rain events:  
\[\epsilon_{r,T}=R_{T,est}-R_{T,meas}\ [\text{mm}], \tag{16}\]  
\[\epsilon_{r,M}=R_{M,est}-R_{M,meas}\ [\text{mm}/\text{h}], \tag{17}\]  
\[\overline{\epsilon}_{r}=\overline{R}_{est}-\overline{R}_{meas}\ [\text{mm}/\text{h}]. \tag{18}\]  
Table 5 lists the associated average (on all the events) mean and RMS values of the residuals.  
The results obtained are satisfactory, as the indicators reveal that the proposed method allows (on average) for estimating the rainfall accumulation with an error in the order of few millimeters. Concerning the detection of peak rain rates, the resulting value of the RMS error is considered to be acceptable, given the difficulty in estimating extreme rain rate values, which might even range between 150 mm/h and 200 mm/h.  
Besides the overall statistical indicators, it is worth further investigating the model accuracy by selecting some sample events. Figure 11 depicts the trend of the rain rate during a typical Spring rain event, which is characterized by moderate rain rates and occasional more intense peaks. The equivalent rain height is calculated while using (10) since the event occurred in April (\(m=4\)): the  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Residual** & **Mean Value** & **RMS Value** \\ \hline \(\epsilon_{r,T}\) & 1.39 mm/h & 5.34 mm/h \\ \(\epsilon_{r,M}\) & \(-\)2.15 mm/h & 11.83 mm/h \\ \(\overline{\epsilon}_{r}\) & 0.04 mm/h & 1.52 mm/h \\ \hline \hline \end{tabular}
\end{table}
Table 5: Average (on all the events) of the mean and Root Mean Square (RMS) values of the residuals in (21)–(23).  
variability of the precipitation is tracked with a significant degree of accuracy, leading to an estimate of the total rainfall accumulation that closely matches the measurements (see Figure 12).  
Figures 13 and 14 refer to a typical event in Autumn (November, \(m=11\)): in this case, the equivalent rain height is calculated while using (8). The model accuracy is lower than in the previous event, but the results are still quite satisfactory.  
Figure 11: Rain rate time series for the event on 26 th April 2017.  
Figure 12: Accumulated rainfall for the event on 26 th April 2017.  
The correct evaluation of the equivalent rain height plays a key role in the prediction of the rain rate, as also shown by the curves in Figure 7. Figure 15 depicts the trend of \(h_{R}\) for the previously mentioned convective and stratiform events, as reported in Figures 11 and 13, respectively, and points out the importance of using local forecast values of \(h_{R}\) with high temporal resolution. The values in Figure 15 indicate the opposite for the two events, although the rain height is generally higher for convective events than for stratiform ones; in this case, using monthly values of \(h_{R}\) (or even worst, a single yearly value) would have led to a much lower prediction accuracy than that shown in Figures 11 and 13.  
Figure 14: Accumulated rainfall for the event on 25 th November 2017.  
Figure 13: Rain rate time series for the event on 25 th November 2017.