# Dynamic Inversion Method for Concrete Gravity Dam on Soft Rock Foundation
## 4 Engineering Examples
### Comparison and Discussion  
Dam section #7 is taken as a representative example in this study. The observed displacements of the plumb lines IP6, IP10, and PL2 in section #7 are shown in Figures 4-6. Observations were recorded monthly from 26 June 2018 to 26 December 2023, for a total of 66 measurements. In the improved particle swarm algorithm, the population size is taken as 30; the maximum number of iterations is 30; the inertia weight maximum and minimum values are taken as \(\omega_{\mathrm{max}}=0.8\), \(\omega_{\mathrm{min}}=0.2\); the learning factor \(c_{1}=c_{2}=1.5\), \(e_{1}=0.8\), \(e_{2}=0.4\). According to the preliminary calculations, the ranges of values of the mechanical parameters are set as \(E\in[17\,\mathrm{GPa},25\,\mathrm{GPa}]\) for the dam body and \(E\in[2\,\mathrm{GPa},6\,\mathrm{GPa}]\) for the dam foundation. The inversion results are 20.74 GPa for the dam body and 4.67 GPa for the dam foundation. The fitted displacements based on the calibration numerical model are shown in Figures 4-6. Figure 4: The measured and fitted displacements of IP6 (static inversion analysis). Figure 3: Two-dimensional finite element model of dam section 7#. The general consistency between measured and inverted displacement trends across most monitoring points can be observed from the figures. However, a larger discrepancy is noted at measurement points especially near the dam foundation. Based on the fitted results of the static parameter inversion, it is initially inferred that parameter weakening occurs in the dam foundation, while the dam body remains in an elastic working condition. Therefore, the elastic modulus of the dam body is determined to be 20.74 GPa based on the static inversion results. The dynamic inversion analysis incorporated field data from three positioned monitoring points (IP6, IP10, and PL2) in dam section 7#. The calculation results reveal a clear temporal degradation pattern in the mechanical properties of the dam foundation, which was quantitatively characterized through curve fitting. Figure 7 presents the time-dependent reduction in the elastic modulus of the dam foundation. The time-varying functions of material parameters, such as those of geotechnical bodies and concrete, are typically modeled using exponential or logarithmic functions [29; 30]. The time-dependent reduction in the elastic modulus of the dam foundation is fitted as the equation \(y=1.075\cdot\mathscr{E}(-0.114\cdot x)+3.829\). The time-dependent elastic moduli obtained from the dynamic inversion analysis are sequentially implemented in the finite element program to compute theoretical displacements. The fitted displacements compared against field measurements are shown in Figures 8-10. Figure 5: The measured and fitted displacements of IP10 (static inversion analysis). Figure 6: The measured and fitted displacements of PL2 (static inversion analysis). Figure 8: The measured and fitted displacements of IP6 (dynamic inversion analysis). Figure 7: The time-dependent reduction in the elastic modulus of the dam foundation (2018–2023). Figure 9: The measured and fitted displacements of IP10 (dynamic inversion analysis). The measured and simulated displacement time series exhibit strong agreement across all monitoring points (Figures 8-10), demonstrating the effectiveness of the dynamic inversion analysis. In addition, two statistical metrics, the Mean Absolute Error (MAE) and Root Mean Square Error (RMSE), are employed to quantitatively evaluate the model performance, as defined in Equations (13) and (14):  
\[RMSE=\sqrt{\frac{1}{N}{\sum_{i=1}^{N}(\hat{x}_{i}-x_{i})^{2}}} \tag{13}\]  
\[MAE=\frac{1}{N}{\sum_{i=1}^{N}|\hat{x}_{i}-x_{i}|} \tag{14}\]  
where \(N\) is the number of sequences; \(\hat{x}_{i}\) and \(x_{i}\) are the fitted and measured values of the displacement, respectively. The results are shown in Tables 1 and 2.