## 3 Results
### Impact on Small Islands  
To further explore the impact of LDA over the sea, we consider the performance of WRF, both with and without lightning data assimilation, over six Italian small islands in the central Mediterranean Sea, with available rain gauges. Rain gauge data come from the Italian Department of Civil Protection (DPC) precipitation database, which accounts for more than 3000 rain gauges over Italy [8]. The selected islands are Elba, Giglio, Lipari, Montecristo, Pantelleria, and Ponza, and their locations are shown in Figure 1.  
The performance of the 3 h precipitation forecast of the model is evaluated using RMSE (Root Mean Square Error) and the Pearson correlation coefficient (\(r\)). To calculate the WRF rainfall corresponding to a rain gauge, the nearest-neighborhood method is used. More specifically, we consider the four grid points surrounding the rain gauge and we select the one among them with a predicted value in better agreement with  
Figure 5: Taylor diagram for CTRL (red triangles) and LIGHT (blue squares) forecasts for land and sea. CS (CL) is for the CTRL forecast over the sea (over land) and LS (LL) is for the LIGHT forecast over the sea (over land). the observation. This method allows a spatial error of the predicted precipitation field of \(\Delta x\sqrt{2}\simeq 4.2\) km, where \(\Delta x=3\) km is the WRF grid spacing. The nearest-neighborhood method is often used in the literature [61; 87]. This method gives higher values of the model scores but tolerates a spatial error of the precipitation forecast (4.2 km in our case). Statistics are presented for the 3 h precipitation forecast for the whole period. For clearer visualization, daily rainfall forecast time series at Lipari and Elba islands are also shown. For the daily (24 h) case, precipitation is obtained as the summation of the eight 3 h rainfall values belonging to the same day. Table 3 shows the RMSE and correlation coefficient for the six islands for the WRF 3 h rainfall forecast. An improvement of the RMSE and \(r\) is seen for all islands except Ponza. The improvement given by LDA is larger for Elba and Lipari, while it is smaller for Giglio, showing a notable variability from one island to another. An inspection of the forecast for Ponza shows that the LIGHT forecast overestimates the precipitation on 11 November. In particular, about 55 mm/day (20 mm/day) is forecast by LIGHT (CTRL), while the observation is 10 mm/day. This determines a worse performance of the LIGHT forecast compared to CTRL for Ponza. The time series of observed and forecast daily precipitation for CTRL and LIGHT simulations are shown in Figure 6, and the corresponding statistics are shown in Table 4. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Elba**} & \multicolumn{2}{c}{**Giglio**} & \multicolumn{2}{c}{**Lipari**} & \multicolumn{2}{c}{**Montercisto**} & \multicolumn{2}{c}{**Pantelleria**} & \multicolumn{2}{c}{**Ponza**} \\ \hline  & C & L & C & L & C & L & C & L & C & L & C & L \\ \hline RMSE (mm) & 1.76 & 1.36 & 4.36 & 4.22 & 2.34 & 1.50 & 2.26 & 2.15 & 1.39 & 1.32 & 2.46 & 2.60 \\ \hline r & 0.70 & 0.83 & 0.45 & 0.49 & 0.58 & 0.78 & 0.40 & 0.47 & 0.49 & 0.54 & 0.66 & 0.60 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: RMSE (Root Mean Square Error) and Pearson correlation coefficients for six islands in the central Mediterranean basin around Italy. The statistics are computed considering all the 3 h rainfall forecasts for November 2019 (C is for CTRL; L is for LIGHT). Figure 6: Time series of daily rainfall for Lipari (**a**) and Elba (**b**) islands. Observations are in blue, CTRL forecasts in red, and LIGHT forecasts in green. A comparison with 3 h results reveals that daily analysis produces higher RMSE values and larger correlation coefficients. Considering the time series for Lipari (Figure 6a), it is noted that LIGHT produces fewer false alarms than CTRL.