## 3 Results
### Pearson Correlation Coefficient, Anomaly Correlation, and Taylor Diagram
The anomaly correlation (AC) [85] is a measure of the association that operates on pairs of grid-point values in the forecast and observed fields, and it is designed to reward good forecasts of the pattern of the observed field, with less sensitivity to the correct magnitudes of the field variable. The centered difference of the AC was computed:  
\[AC=\frac{\sum_{m=1}^{M}\left(y_{m}^{\prime}-\overline{y^{\prime}}\right)\left( o_{m}^{\prime}-\overline{o^{\prime}}\right)}{\left[\sum_{m=1}^{M}\left(y_{m}^{ \prime}-\overline{y^{\prime}}\right)^{2}\sum_{m=1}^{M}\left(o_{m}^{\prime}- \overline{o^{\prime}}\right)^{2}\right]^{\frac{1}{2}}} \tag{3}\]  
\[y^{\prime}=y-c_{m}\] \[o^{\prime}=o-c_{m}\]  
where \(c_{m}\) is the monthly average of the 3 h precipitation of the IMERG dataset for each grid point, \(y^{\prime}\)_m_ and \(o^{\prime}\)_m_ are the anomalies of the 3 h precipitation of the WRF forecast and IMERG with respect to \(c_{m}\), and \(M\) is the total number of land or sea grid points, depending on whether the AC is computed for land or for sea. Overbars are averages over the \(M\) grid-points. The averages of the AC for CTRL and LIGHT for the whole period for the 3 h precipitation forecast are shown in Table 2.  
Positive values of the AC show the ability of WRF to represent the observed precipitation spatial pattern. LDA improves the WRF performance for both sea and land, with an increase in the AC of 8% and 5%, respectively. The AC is higher for land grid points, showing a better performance of WRF over land. The performance gap between land and sea is reduced when LDA is applied, because LDA has a greater impact over the sea. The AC is positive and shows a good performance of WRF for most of the considered days, both for CTRL and for LIGHT. Values vary considerably among different times, being in some cases low and sometimes negative over land. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{4}{c}{**Correlation Coefficient Threshold**} \\ \hline
**R** & **0.1** & **0.3** & **0.5** & **0.7** \\ \hline CTRL & 24,783 & 16,237 & 5476 & 731 \\ LIGHT & 24,908 & 18,327 & 8067 & 1242 \\ \hline \end{tabular}
\end{table} 
Table 1: Number of grid points out of a total of 27,602 grid points where the CTRL or LIGHT correlation coefficient is greater than a selected threshold. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**AC** & **CTRL** & **LIGHT** & **Variation (\%)** \\ \hline LAND\_3 h & 0.50 & 0.53 & 5\% \\ SEA\_3 h & 0.44 & 0.49 & 8\% \\ \hline \end{tabular}
\end{table} 
Table 2: Average of the anomaly correlation (AC) for the whole period for the 3 h precipitation, for the CTRL forecast, the LIGHT forecast, and the improvement of LIGHT compared to CTRL in percentage (variation). The AC is improved by LDA, as shown by its larger values for LIGHT with respect to CTRL. This improvement is larger until 20 November. From 20 to 30 November, a lower number of convective events occurred over Italy and, therefore, the impact of LDA is lower. The impact of LDA is larger over the sea than over land. This result is partially determined by the lower performance of the WRF forecast over the sea, as confirmed by the average values of the AC over land and sea (Table 2 and Figure 4). Other factors, however, could contribute to this result. The smoothing of the precipitation field given by the regridding applied to the WRF forecast can penalize the impact of LDA over land, where, because of the complex orography of Italy, small-scale thunderstorms develop due to local forcing conditions. These thunderstorms are likely smaller than those developing over the sea, and regridding can impact unevenly the performance of LDA over land and sea. The time series of the AC over land and sea are shown in Figure 4.  
The Taylor diagram was computed for the whole period for land and sea grid points, and it is shown in Figure 5.  
The correlation between IMERG observations and the WRF forecast is improved by LDA both over land and over the sea.