## 2 Materials and Methods
### INPR Theory
#### 2.3.2 Improved INPR Calculation Method  
If there is probability density information, it should be fully utilized to derive probabilistic reliability [29]. For the overtopping event during the construction period of dams, the probability density of the top elevation of the water-retaining structure under the influence of construction progress is unknown. For the upstream flood level, different flood frequencies (which are the reciprocals of flood return period) correspond to different upstream flood level \(Z\), and there is certain probability density information associated  
Figure 3: Spatial schematic diagram of the stress–strength interference model of overtopping accidents.  
Figure 2: Interval (**a**) and ellipsoid (**b**) models.  
with them. Therefore, it is necessary to make full use of this information to improve the rationality of the calculation results.  
To reflect the influence of flood frequency on the probability of overtopping during the construction period of dams, this study couples the upstream flood frequency with the stress-strength interference model. The upstream flood level is differentiated within the range [\(Z_{1}\), \(Z_{2}\)], denoted as \(\mathrm{d}z\). Then, the flood frequency, failure domain area, and basic variable domain area corresponding to any flood level \(Z\) are \(f(Z)\), \(L_{f}\) (\(Z\))\(\mathrm{d}z\), and (\(H_{2}-H_{1}\))\(\mathrm{d}z\), respectively. \(L_{f}\) (\(Z\)) and \(H_{2}-H_{1}\) represent the failure domain width and the basic variable domain width corresponding to the upstream flood level \(Z\), respectively, as shown in Figure 4.  
The traditional INPR neglects the differences in flood frequency within the interval [\(Z_{1}\), \(Z_{2}\)]. Therefore, flood frequency of the upstream flood level \(f(Z)\) is used to modify the failure domain area and the basic variable domain area, and integration is performed over the interval [\(Z_{1}\), \(Z_{2}\)]. The improved probability of overtopping during the construction period of dams is shown in Equation (4).  
\[P_{f}=\frac{\int_{H_{1}}^{Z_{2}}f(Z)L_{f}(Z)dz}{(H_{2}-H_{1})\int_{Z_{1}}^{Z_ {2}}f(Z)dz}=\frac{V_{F}}{V_{T}}, \tag{4}\]  
The geometric significance of the frequency of floods and the stress-strength interference coupling model is as follows: in the three-dimensional variable space with H, F, and \(Z\) as coordinate axes, the plane \(Z=H\) divides the frequency of floods and the stress-strength interference coupling model into the safe domain and the failure domain, as shown in Figure 5a. The ratio of the volume of the failure domain to the total volume of the basic variable domain is used as the probability of overtopping during the construction period of dams. Specifically, when the elevation of the dam's crest is a constant value, i.e., \(H_{1}=H_{2}\), the spatial diagram of flood frequency of the upstream flood level and the stress-strength interference coupling model is shown in Figure 5b, where the line represents the flood frequency curve of the upstream flood level. Therefore, the probability of the elevation \(H_{1}\) corresponding to the flashover is the area above elevation \(H_{1}\) in the red region, and the corresponding probability of overtopping is shown in Equation (5).  
\[P_{f}=\frac{\int_{H_{1}}^{Z_{2}}f(Z)dz}{\int_{Z_{1}}^{Z_{2}}f(Z)dz}, \tag{5}\]  
Figure 4: Schematic diagram of the coupling model of flood frequency and stress–strength interference.