## 3 Methodology
### Verification Metrics  
The evaluation of raw and post-processed EPFs was conducted using the basin-averaged value of observed precipitation during the 2014-2018 summer periods as a benchmark. As revealed in [PERSON], [PERSON] [35], relying solely on deterministic or probabilistic criteria is typically insufficient to assess EPF performance. Therefore, both deterministic and probabilistic metrics were adopted; the former was used to reflect the performance of ensemble mean forecasts and the latter was used to reflect the performance of ensemble forecasts. The deterministic metric used was the mean absolute error (\(MAE\)) [36], defined as:  
\[MAE=\frac{1}{n}{\sum\limits_{i=1}^{n}}{|F_{i}-O_{i}|} \tag{4}\]  
where \(F_{i}\) and \(O_{i}\) are the EPFs and observed precipitation time series, respectively. \(\overline{F}\) and \(\overline{O}\) are the average of all daily values for EPFs and observed precipitation, respectively. \(n\) is the length of daily time series. The \(MAE\) was used to assess the average absolute difference between the ensemble mean forecasts and the observations. \(MAE\) ranges from 0 to infinity, and a lower MAE signifies superior accuracy. On the probabilistic front, the continuous ranked probability skill score (\(CRPSS\)) [37] and threat score (TS) [38] were adopted. The \(CRPSS\) is used to assess the performance of the ensemble spread of EPF over climatology, which is defined as the mean squared difference between the distribution of forecasts and corresponding distributions of observations:  
\[CRPS=\int\limits_{-\infty}^{+\infty}{(PF(x)-OF(x))^{2}dx} \tag{5}\]  
\[CRPSS=1-\frac{CRPS}{CRPS^{*}} \tag{6}\]  
where \(PF\) and \(OF\) are the cumulative distribution functions (CDFs) of the forecasts and observations, respectively. \(x\) represents the event to be analyzed. \(CRPS^{*}\) is the value of the reference forecasts. The \(CRPSS\) ranges from negative infinity to 1, and a \(CRPSS\) value greater than 0 indicates skillful forecasts, with 1 being ideal. The most widely used index, TS, was used to assess the performance of ensemble forecasts for four rainfall categories, i.e., light rain, moderate rain, heavy rain, and torrential rain. Table 2 details the category of valley area rainfall [39]. \(TS\) is given by:  
\[TS=\frac{N_{A}}{N_{A}+N_{B}+N_{C}} \tag{7}\]  
Using the definitions of \(N_{A}\), \(N_{B}\), and \(N_{C}\) from the contingency table (Table 3), the \(TS\) can take values between 0 to 1; a \(TS\) value of 1 represents an ideal forecast, while 0 denotes no skill. \begin{table}
\begin{tabular}{c c} \hline
**Category** & **24 h (mm)** \\ \hline Light rain & 0.1-5.9 \\ Moderate rain & 6.0–14.9 \\ Heavy rain & 15–29.9 \\ Torrential rain & \textgreater{}30 \\ \hline \end{tabular}
\end{table} 
Table 2: Category of valley area rainfall. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Observations** & \multicolumn{3}{c}{**Forecast**} \\ \cline{2-3}  & **Yes** & **No** \\ \hline Yes & N\({}_{\text{A}}\) & N\({}_{\text{C}}\) \\ No & N\({}_{\text{B}}\) & N\({}_{\text{D}}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Two-category contingency table. Additionally, the rank histogram, a renowned tool, was also used for ascertaining ensemble forecast reliability. As [PERSON] [40] noted, a uniform rank histogram typifies a reliable forecast system, while deviations signify miscalibration. The associated reliability metric, \(\Delta\), quantifies departures from a flat histogram as:  
\[\Delta=\sum\limits_{i=1}^{m+1}\left|F_{i}-\frac{1}{m+1}\right| \tag{8}\]  
where \(m\) is number of forecasting members, \(\Delta\) ranges from 0 to 1, and a value of 0 symbolizes a flawless score and a balanced rank histogram. For comprehensive insights on the aforementioned metrics, please refer to.