## 3 Methodology
### Generation and Evaluation of Ensemble Flood Forecasting  
The EFF is generated based on the following \"EPF + HM\" schemes: (A) ECMWF-driven XAJ, denoted as ECMWF + XAJ; (B) EPFs (ECMWF and NCEP) driving XAJ, referred to as GE + XAJ; (C) ECMWF driving multiple models (XAJ and GR4J), denoted as ECMWF + MHM; and (D) EPFs driving multiple hydrological models (XAJ and GR4J), known as GE + MHM. Scheme (D) represents the multi-ensemble, multi-model combination [27]. These four schemes generate ensemble members numbering 50, 70, 100, and 140, respectively. Additionally, ensemble forecast products corrected by the GPP method are used for flood ensemble forecasting in these four schemes. The results are evaluated and compared before and after post-processing. Figure 3: Structure of the GR4J model ([PERSON], 2003 [51]). This paper evaluates the performance of EFF using the ensemble mean and the 90% confidence interval (the difference between the 95 th and 5 th percentiles). The ensemble mean reflects the central tendency, while the 90% confidence interval quantifies the range, variability, and uncertainty within the ensemble. This dual assessment allows for a comprehensive evaluation of the ensemble's accuracy in predicting flood events and in assessing the associated uncertainty. Both deterministic and probabilistic indicators are used. The deterministic indicators include the correlation coefficient (\(R\)) and the Relative Mean Error (\(RME\)), which measure the average deviation between the forecast mean and actual observations. Probabilistic indicators include the Continuous Ranked Probability Skill Score (\(CRPSS\)) and quantile histograms. To evaluate EFF uncertainty, three indicators are employed: Average Relative Interval Length (\(ARIL\)), the percentage of observations enclosed by the confidence interval (\(PCI\)), as well as the percentage of observations enclosed by the Unit Confidence Interval (\(PUICI\)) [56]. (1) \(R\) measures the correlation between forecasted and observed values, serving as an indicator of forecast reliability. It is calculated as follows:  
\[R=\frac{\sum_{i=1}^{n}{(O_{i}-\overline{O})(F_{i}-\overline{F})}}{\sqrt{\sum_{ i=1}^{n}{(O_{i}-\overline{O})}^{2}}\sqrt{\sum_{i=1}^{n}{(F_{i}-\overline{F})}^{2}}} \tag{5}\]  
where \(O_{i}\) and \(F_{i}\) represent observed and ensemble forecasted precipitation, respectively, while \(\overline{O}\) and \(\overline{F}\) are the daily mean values of observed and forecasted precipitation over an n-day time series. \(R\) values range from [0, 1], with values closer to 1 indicating more reliable forecasts. (2) \(RME\) reflects the relative deviation between forecast values and the observed mean, assessing the accuracy of the forecasts. It is expressed as follows:  
\[RME=\frac{\sum_{i=1}^{n}{(F_{i}-\overline{O})}}{\sum_{i=1}^{n}{Q_{i}}}\times 1 00\% \tag{6}\]  
\(RME\) values range from [\(-100\%\), \(100\%\)], with values closer to 0 indicating smaller errors between the ensemble mean and observations. (3) \(CRPSS\) evaluates the performance of the quantitative precipitation forecast (QPF) ensemble distribution relative to the climatological distribution. It is defined for a variable n at time t, estimated from the ensemble forecast probability density function as follows:  
\[CRPS_{n}(t)=\int\limits_{-\infty}^{+\infty}{(F_{n,t}(y)-I(y\geq X))^{2}}d_{y} \tag{7}\]  
The continuous skill score, \(CRPSS\), is defined as follows:  
\[CRPSS_{n}=1-\frac{CRPS_{n}}{CRPS_{n,ref}} \tag{8}\]  
where \(CRPS_{n,ref}\) is the \(CRPS\) obtained using a reference forecast. Higher \(CRPSS\) values approaching 1 indicate better ensemble forecast performance. (4) Quantile histograms are used to check if the ensemble forecast values and observed values come from the same probability distribution. A flat, horizontal line shape indicates moderate dispersion within the ensemble; a concave shape with high sides and a low middle suggests under-dispersion; a convex shape with low sides and a high middle indicates over-dispersion.