## 3 Results  
### Quantitative Analysis of \(Z\) and \(H\)  
#### 3.1.1 Upstream Water Level Range Z  
Under current technological conditions, it is difficult to accurately forecast hydrological events exceeding one year [30]. Therefore, the upstream water level range \(Z\) needs to be discussed and determined in conjunction with engineering realities and the calculation requirements of INPR models. For the lower limit of the upstream water level range, the minimum value of the annual flood season upstream water level during the construction period is taken, corresponding to the elevation of the once-in-a-year flood in the flood season. Regarding the upper limit of the upstream water level range, in actual engineering, overflow occurs when the upstream water level exceeds the elevation of the top of the damming structure, which constrains the upstream water level. However, INPR methods require knowledge of the maximum height that the upstream water level might reach. Therefore, assuming that the diversion and drainage structures remain consistent with the design, and the elevation of the top of the damming structure is consistently higher than the upstream water level, the upper limit of the range of upstream floodwater levels that can be reached during upstream flooding is denoted as \(Z\).  
The \"Standard for Rank Classification and Flood Protection Criteria of Water and Hydropower Projects (SL 252-2017)\" [31] in China stipulates that the maximum flood standard for the construction period of dams is a 5000 to 10,000-year return period. The corresponding water level of the flood with a return period of 5000 years or 10,000 years is taken as the upper limit of the upstream water level range, with confidence levels of 99.98% and 99.99% for the two intervals. For the interval values of indicators in engineering problems, the Pauta criterion, known as the \"3\(\sigma\)\" criterion, is commonly used to determine if the observed values exceed the limit [32]. For variables following a normal distribution, the confidence level of the data range satisfying the \"3\(\sigma\)\" criterion (\(\mu-3\sigma\), \(\mu+3\sigma\)) is 99.74% [33]. The confidence level of the upper limit of the upstream water level range for the construction period of dams, based on the water level corresponding to the flood with a return period of 5000 or 10,000 years, exceeds 99.74%; hence, it is considered reasonable. In many small and medium-sized basins in China, hydrological stations have relatively short  
Figure 5: Spatial schematic diagram of the coupled model of flood frequency and stress–strength interference: (**a**) \(H_{1}\
eq H_{2}\); (**b**) \(H_{1}=H_{2}\).  
operating periods, limited hydrological data, and a lack of long-term flood observation samples [34]. Therefore, the water level corresponding to the flood with a return period of 5000 years is used as the upper limit of the upstream water level range.  
Based on the upstream flood hydrograph of the Qianping Reservoir dam site, the relationship between the reservoir water level and surface area, the discharge capacity of the diversion and spillway structures, and the upstream water level and its flood frequency during the construction period of flood season of the Qianping Reservoir dam are calculated, as shown in Table 2. According to the data in Table 2, the upstream water level range is \(Z=[344.12,416.67]\).