# Sensitivity Analysis of Numerical Coherency Model for Rock Sites
## 3 Sensitivity Study
### Spatial Variability Parameters  
Site-specific coherency has been reported to be highly influenced by spatial variability [1; 22; 23; 24; 22]. We selected the CV and CL, which represent site variability, as sensitivity analysis variables. For the Pinyon Flat site, reference CV and CL profiles were not measured and therefore, are not available. In this study, sensitivity analyses were performed using the range of CV and CL values reported for rock sites [25; 26; 27]. The vertical correlation length (CLv) was fixed at 10 m, as recommended in a previous study [23]. Table 1 lists the selected case matrix.  
Figure 6 presents the results of the sensitivity analysis of plane-wave coherency with respect to the CV. The analysis shows that plane-wave coherency decreases as the CV and separation distance increase. This trend is likely due to increased site heterogeneity and seismic wave scattering caused by the spatial variability of \(V_{s}\) and the greater separation distance. This can be interpreted as a decrease in coherency resulting from increased phase differences and interference among individual seismic wave components as the CV increases. Similar patterns have been observed in soil sites with relatively lower stiffness [23]. This suggests that the CV strongly influences coherency regardless of site conditions. Therefore, careful consideration is required when selecting and applying the CV in the development of numerical coherency models for incoherency SSI analysis.  
Figure 7 shows the results of the sensitivity analysis of plane-wave coherency with respect to the horizontal correlation length (CLh). Given that the CLh has a greater impact on ground motion coherency than the CLv, the analysis was conducted by fixing the CLv at 10 m and varying the CLh from 20 m to 40 m. The results indicate that the influence of the CLh on plane-wave coherency is minimal as the CLh increases.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Case Number** & **CV (\%)** & **CLh (m)** & **Analysis Depth (m)** \\ \hline Case 1: CV & 15, 20, 25, 30 & 20 & 150 \\ Case 2: CLh & 20 & 20, 25, 30, 40 & 150 \\ Case 3: Depth & 20 & 20 & 50, 100, 150, 200 \\ \hline \end{tabular}
\end{table}
Table 1: Spatial variability and numerical model parameters simulated in this study.  
Figure 5: Result of convergence with respect to number of numerical simulations.  
the size of the heterogeneity is small relative to the wavelength (i.e., when the CLh is small), the scattering effect of seismic waves due to heterogeneity is reduced, and the influence of separation distance on plane-wave coherency is limited. Especially in the low frequency range, an increase in the CLh has no effect, as the long wavelength seismic components are less influenced. That is, while a larger CLh increases the scale of individual heterogeneities, it does not significantly affect coherency in cases with long wavelengths, such as rock sites. This suggests that when developing numerical coherency models, applying an excessively high CLh value is unlikely to impact coherency.  
Figure 6: Evaluation of sensitivity analysis regarding effect of CV on plane-wave coherency.  
Figure 7: Evaluation of sensitivity analysis regarding effect of CLh on plane-wave coherency.