## 4 Discussion  
1. According to the data in Table 4, the obtained probabilities are 10 times higher than flood routing calculations, with the highest ratio reaching 104 times, such as in the calculations for schedule 1 of the floods in 2019. This is because the traditional methods assume that the upstream flood levels follow a uniform distribution, neglecting the significant differences in flood frequencies corresponding to different flood levels. From Table 4 and Figure 7, it can be observed that the improved INPR model aligns well with the results obtained from flood routing calculations, demonstrating the rationality and feasibility of the proposed method in this study. Additionally, the improved method in this study considers the uncertainty of construction progress, which is more in line with the actual situation of dam construction progress being influenced by multiple factors and characterized by uncertainty. 2. There is no fixed relationship between the calculated results of the improved method proposed in this study and the results obtained from flood routing calculations. This is because the uncertainty of the construction progress affects the range of elevation values for flood retaining structures during the flood season. When the construction progress can be easily ensured, the elevation of the flood retaining structures tends to be higher than the planned elevation, and the calculated results of the improved method will be smaller than those of the flood routing calculations. When the construction progress cannot be easily ensured, the calculated results of the improved method will be higher than those of the flood routing calculations. For example, in schedule 2 during the flood season of 2019, the planned elevation for the dam crest is 392.0 m, with an elevation range of [389.2, 399.6], and the progress risk is 0.27. It is evident that the construction progress can be easily ensured; thus, the \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Schedule** & **Method** & **Year 2017** & **Year 2018** & **Year 2019** & **Year 2020** \\ \hline \multirow{3}{*}{Schedule 1} & Improved method & 0.0522 & 0.0461 & 0.0015 & - \\  & Traditional method & 0.5909 & 0.5826 & 0.1877 & - \\  & Flood routing calculation & 0.0446 & 0.0446 & 0.0018 & - \\ \hline \multirow{3}{*}{Schedule 2} & Improved method & 0.0522 & 0.0461 & 0.0048 & 0.0004 \\  & Traditional method & 0.5909 & 0.5826 & 0.3070 & 0.0149 \\  & Flood routing calculation & 0.0446 & 0.0446 & 0.0063 & 0.0005 \\ \hline \multirow{3}{*}{Schedule 3} & Improved method & 0.0497 & 0.0072 & - & - \\  & Traditional method & 0.5847 & 0.3552 & - & - \\ \cline{1-1}  & Flood routing calculation & 0.0446 & 0.0063 & - & - \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Overtopping probability of Qianping reservoir. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **Schedule** & **Year 2017** & **Year 2018** & **Year 2019** & **Year 2020** \\ \hline \multirow{3}{*}{Schedule 1} & Planned elevation & 374.4 & 374.4 & 403.0 & - \\  & Interval elevation & [373.2, 374.4] & [374.4, 374.4] & [398.1, 408.0] & - \\  & progress risk & 1.00 & - & 0.49 & - \\ \hline \multirow{3}{*}{Schedule 2} & Planned elevation & 374.4 & 374.4 & 392.0 & 416.0 \\  & Interval elevation & [373.2, 374.4] & [374.4, 374.4] & [389.2, 399.6] & [411.7, 423.1] \\  & progress risk & 1.00 & - & 0.27 & 0.38 \\ \hline \multirow{3}{*}{Schedule 3} & Planned elevation & 374.4 & 392.0 & - & - \\  & Interval elevation & [374.1, 374.4] & [387.6, 394.2] & - & - \\ \cline{1-1}  & progress risk & 1.00 & 0.67 & - & - \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Elevation range \(H\) of water-retaining structures and progress risk during the flood season. calculated result of the improved method is 0.0048, which is smaller than the flood simulation result of 0.0063. 3. Theoretically, a faster construction schedule can reduce the probability of overtopping during the flood season in dam construction.