## 5 Conclusions  
The probability of overtopping during the construction period of a dam is influenced by the uncertainty of construction progress, making it more challenging to calculate. The traditional approach is to assign a certain probability distribution to the construction progress. However, the construction progress, being subject to human factors, is uncertain but lacks strict randomness. Therefore, the conventional probabilistic approach has certain limitations in its applicability. Although the traditional INPR method only requires parameter value ranges without precise probability distributions for the parameters, this method still exhibits significant deviations when applied to the calculation of overtopping probability in dams. This study analyzes the causes of deviations and couples the flood frequency with the stress-strength interference of traditional INPR methods, proposing an improved INPR method.  
Taking the Qianping Reservoir as an example, the overtopping probabilities for three construction schedules in different flood seasons were calculated. By comparing the calculation results of different methods, the rationality of the improved method was verified. The calculation results indicated that accelerating the construction progress can reduce the probability of overtopping during the construction period of dam. However, it is difficult to ensure the construction progress, which consequently increases the probability of overtopping. To decrease the probability of overtopping during the construction period to the desired level, two aspects can be considered: (a) develop a faster construction schedule that aligns with the actual engineering conditions, providing more flexibility for the construction progress and (b) strengthen construction progress management and resource supply to reduce uncertainty in the construction progress. This method provides a new approach for the calculation of dam overtopping probability during the construction period of dams and can also be applied in the fields of dam structural design, progress optimization, risk management, and other areas.  
X.G.: Writing--Review and Editing and Methodology. Z.L.: Supervision and Conceptualization. W.G.: Investigation and Funding Acquisition. F.M.: Validation. Y.Z.: Data Curation. H.S.: Formal Analysis. Y.J. and J.W.: Software. All authors have read and agreed to the published version of the manuscript.  
This research was supported by the National Natural Science Foundation of China (Grant 52179144, U2243244, and 51679222), China Postdoctoral Science Foundation (Grant No. 2023M731259), Natural Science Foundation of Henan Province (Grant No. 232300421067), and Program for Science & Technology Innovation Talents in Universities of Henan Province (HASTIT) (Grant No. 22 HASTIT011).  
Not applicable.  
The raw data supporting the conclusions of this article will be made available by the authors on request.  
Author [PERSON] was employed by the Yellow River Engineering Consulting Co., Ltd. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.