## 4 Result and Discussion  
Because the J07 Model and R&S08 Model have few parameters and are widely used, this paper selects these two models as a reference for subsequent calculations. The J93 Model, proposed by [PERSON] [8], was developed using 11 strong-motion datasets, with \(I_{a}\) ranging from 0.2 to 10.0 m/s. As the dataset expanded, [PERSON] subsequently refined the J93 Model into the J98 Model [36] and J07 Model [35]. Then, [PERSON] and [PERSON] [22], [PERSON] and [PERSON] [37], and [PERSON] [38] applied and validated the J93, J98, and J07 Models, obtaining positive verification results. In this study, the J07 Model is utilized to process the database, with results presented in Equation (10). \[\log u=0.510\log I_{a}-4.195\log\Big{(}\frac{a_{y}}{PGA}\Big{)}-1.668\pm 0.540 \tag{10}\]  
In Equation (10) (Model 1), the standard deviations of \(I_{a},a_{y}/PGA,\) and \(\log(u)\) are 0.031, 0.065, and 0.540, respectively, with a fitting degree (\(R^{2}\)) of 0.866. Comparing Model 1 with the J07 Model reveals that the fitting coefficients are of similar magnitudes, indicating close datasets. Furthermore, Model 1's standard deviation of \(\log(u)\) is lower than that of the J07 Model. Regression analysis was conducted to examine the variation in the residual values of permanent displacement (\(\log u\)) in Equation (10) relative to the fitting parameters (Figure 3). The residual distribution plot reflects the model's goodness of fit. The curves in the Figure 3 illustrate the distribution and variation patterns of the residual points. The more evenly distributed the residual points, the more random the model's errors, indicating a better fit. Figure 3a shows that, in logarithmic coordinates, residuals are concentrated within the range [\(-\)2.0,1.5]. The closer \(\log(u)\) is to zero, the more concentrated the data points, suggesting an approximate normal distribution of permanent displacements relative to the \(\log(I_{a})\). However, Figure 3b reveals that residuals of \(\log(a_{y}/PGA)\) display a nonlinear distribution, implying the potential need for a higher-order term of the yield acceleration ratio in the model. By substituting the data parameters into Equation (10), predicted displacement values were obtained and compared to the calculated displacements, as illustrated in Figure 4. When calculated displacements are small (\(<\)10\({}^{-2}\) cm), Model 1 may overestimate landslide deformation (Figure 4). Conversely, as calculated displacements increase, predicted values may fall below actual values, exhibiting a nonlinear trend. The results of Model 1 (Figure 3a) demonstrate that \(I_{a}\) is a reliable fitting parameter within the empirical model. Using the R&S08 Model as a basis, higher-order terms of \(a_{y}/PGA\) were introduced into Model 1, yielding the fitting results shown in Equation (11). \[ \begin{array}{c}\log u=3.631-22.525\Big{(}\frac{a_{y}}{PGA}\Big{)}+47.591 \Big{(}\frac{a_{y}}{PGA}\Big{)}^{2}-47.840\Big{(}\frac{a_{y}}{PGA}\Big{)}^{3} \\ +15.282\Big{(}\frac{a_{y}}{PGA}\Big{)}^{4}+0.905\log(I_{a})-0.894\log(PGA)\pm 0.410\end{array}  \tag{11}\]  
In Equation (11) (Model 2), the standard deviation of \(\log(u)\) and \(R^{2}\) are 0.410 and 0.923, respectively, reflecting lower deviation and better fit compared to Model 1. These results indicate a significant enhancement in Model 2's accuracy. In Model 2, the residual distribution of permanent displacement with respect to each parameter is presented in Figure 5. Figure 5 shows that the addition of higher-order \(a_{y}/PGA\) terms effectively  
Figure 4: The comparison of predicted displacement and calculated displacement of Model 1. Figure 3: The distribution of residual points of \(\log(u)\) in Model 1 with fitting parameters. (**a**) Distribution with \(\log\) (\(I_{a}\)); (**b**) distribution with \(\log\) (\(a_{y}/PGA\)). mitigates nonlinear effects, with residuals approximately and evenly distributing across the positive and negative regions of \(\log(u)\).