## 4 Results and Discussion
### Empirical Relationship with the SPT Blow Counts  
For the sandy silt mixed with silty sand, six sets of key parameters \(E_{50}^{ref}\) of the soil layers were retrieved using the aforementioned method, and an empirical relationship was established between these parameters and the SPT blow counts (\(N\) value) of the corresponding soil layers. The results are presented in Figure 9. The \(R^{2}\) value of the goodness of fit is 0.93, indicating a satisfactory curve fitting effect. It can be concluded that there is a good linear relationship between the compression modulus \(E_{\text{s1-2}}\) and the SPT blow count (\(N\)) of the soil layer, as represented by Equation (7). This linear relationship is consistent with the existing literature [25; 26; 27]. Additionally, since there is a certain proportional relationship between \(E_{50}^{ref}\) and \(E_{\text{s1-2}}\) (in Table 4), the constitutive parameters can be accurately determined. The \(E_{50}^{ref}\) and \(N\) values obtained through step-by-step inversion analysis can serve as an empirical formula for determining the constitutive parameters of the soil. The proposed method for determining the constitutive parameters can be applied to other soil layers, providing a basis for accurate numerical analysis during the design phase.  
\[E_{50}^{\text{ref}}=61.72N+15341.15 \tag{7}\]  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Soil Layer** & \(E_{\text{eod}}^{ref}\) & \(E_{\text{s0}}^{ref}\) & \(E_{\text{s1-2}}^{ref}\) & \(G_{\text{s0}}^{ref}\) & \\  & (**MPa**) & (**MPa**) & (**MPa**) & (**MPa**) & \\ \hline
2-1 Clayey silt & 9.4500 & 9.450 & 37.800 & / & / \\ \hline
2-2 Sandy silt & 12.345 & 15.000 & 81.000 & / & / \\ \hline
2-3 Sandy silt mixed with silty sand 1 & 13.179 & 16.012 & 86.465 & / & / \\ \hline
2-3 Sandy silt mixed with silty sand 2 & 13.251 & 16.100 & 86.940 & / & / \\ \hline
2-3 Sandy silt mixed with silty sand 3 & 13.338 & 16.206 & 87.512 & / & / \\ \hline
2-3 Sandy silt mixed with silty sand 4 & 13.419 & 16.304 & 88.041 & / & / \\ \hline
2-3 Sandy silt mixed with silty sand 5 & 13.507 & 16.411 & 88.619 & / & / \\ \hline
2-3 Sandy silt mixed with silty sand 6 & 13.589 & 16.507 & 89.138 & / & / \\ \hline
3-1 Muddy clay & 4.167 & 5.000 & 43.000 & 96.000 & \(2.3\times 10^{-4}\) \\ \hline
3-2 Silty clay & 6.087 & 7.000 & 40.600 & 96.600 & \(2.7\times 10^{-4}\) \\ \hline
4 Silty clay & 6.087 & 7.000 & 40.600 & 96.600 & \(2.7\times 10^{-4}\) \\ \hline \hline \end{tabular}
\end{table}
Table 6: Optimization results of the constitutive parameters.  
Figure 9: The empirical relationship between the inversion parameter \(E_{50}^{ref}\) and SPT blow counts (sandy silt mixed with silty sand).  
The ground settlement measurements at section A-A were adopted to verify the effectiveness of the proposed SPT-based step-by-step inverse method for the constitutive parameters. As shown in Figure 10, the ground surface settlements predicted by the proposed method agree well with the measurements in the level 4, 3, and 2 excavation stages. The proposed method can not only predict the horizontal displacement of the retaining wall but also the ground settlement at different excavation stages. Overall, the parameter selection method of the constitutive model proposed in this paper is an accurate, practical, and efficient method.