## 1 Introduction  
Dams can generate significant economic and social benefits in flood control, power generation, water supply, irrigation, and other aspects [1, 2, 3]. Although the construction technology and management level of dams are improving, with increasingly frequent extreme weather conditions, dams around the world are facing unprecedented risks, such as flooding, seepage in the dam foundation [4], and cracks in the dam body [5]. Among many dam types, earth-rock dams have been widely used due to their good economics and strong applicability [6], accounting for approximately 80% of all dams worldwide [7]. However, their ability to resist flood overtopping is relatively low, and overtopping often leads to dam collapse. Therefore, overtopping is the main cause of dam failures for earth-rock dams [8, 9]. The various indicators of dams during construction have not yet met the design requirements, further reducing their ability to resist floods. Since 1930, half of the failures of earth-rock dams over 30 m and most of the related deaths have been caused by overtopping during construction [10]. Therefore, establishing an accurate and reasonable probability calculation model for overtopping during the construction period of dams has always been a concern for researchers. During the construction period, there is a great deal of uncertainty in the factors affecting the overtopping of dams, and uncertainty analysis typically employs probability methods [11]. [PERSON] and [PERSON] considered the probability of dam overtopping during the construction period under the influence of uncertainty in the upstream inflow and discharge capacity and compared the effects of different lining materials in diversion tunnels on the overtopping probability [12]. [PERSON] et al. used a first-order second moment approach to calculate the influence of uncertainty in the upstream inflow and discharge capacity on the overtopping probability during the construction period of the dam [13]. [PERSON] et al. established a method for calculating the overtopping probability during the construction period of dams considering uncertainty in sediment, hydrology, and hydraulics, particularly for sediment-rich rivers [14]. [PERSON] et al. used the Monte-Carlo method to establish a risk model for diversion during the late stage of dam construction considering hydrological and hydraulic uncertainties and calculated the late-stage diversion overtopping probability for different reservoir operation schemes [15]. The above study focuses on the impact of the uncertainty of upstream inflow (hydrological uncertainty) and discharge capacity (hydraulic uncertainty) on the probability of dam overtopping during the construction period, while the elevation of the top of the dam structures is a fixed value. In actual engineering, the elevation of the top of the dam structures also has uncertainty due to the influence of construction progress uncertainty, which in turn affects the probability of dam overtopping during the construction period. To consider the uncertainty of construction progress, a common method is to assign it a certain probability distribution [16; 17]. However, the random parameters of construction progress are subjective, and there are certain difficulties in parameter calculation [18]. The reason for this is that there are many factors influencing construction progress, such as meteorological factors, geological factors, and human factors. Among them, human factors such as construction organization design and construction management have a greater impact on construction progress. Therefore, although construction progress is uncertain, it does not have strict randomness and cannot be simply described by a random probability distribution [19]. In many engineering projects, it is often difficult to obtain precise probability data for uncertain factors [20], but the magnitude or boundaries of uncertainty are easy to determine [21].