## 3 Numerical Studies
### Impact of Stochastic Tree Parameters on Attenuation Performance
This is because low-frequency surface waves with longer wavelengths can propagate around the tree grid to farther observation points, reducing the influence of the forest trees on them. Figure 15 shows the ARS at different observation points when the density \(\rho_{\mathrm{ij}}\) of each tree has a range of variation \(\alpha\) = 0.4. Despite the presence of high uncertainty, the designed tree grid can still achieve evident mitigation effects with 8-32 Hz and 63-80 Hz. The fluctuation in attenuation efficiency induced by tree density is more pronounced in the low-frequency range than in the high-frequency range for observation points A, D, and E, which are comparatively close to the forest trees, and this trend is reversed at higher frequencies. Moreover, the uncertainty in attenuation performance due to random variations in tree density is marginally lower than that associated with random variations in tree height. Figure 16 shows the ARS for different observation points by the tree grid over a homogeneous half-space, considering a variation range of \(\alpha=0.4\) for trunk radii \(r_{\text{tg}}\). The results demonstrate that the impact of uncertainty in trunk cross-sectional dimensions on attenuation performance is significantly smaller than that of tree height and density. Taking point A as an example, the randomness in tree radius causes a maximum ARS fluctuation of 1.2 dB in the low-frequency range, while in the high-frequency range, the fluctuation remains within 0.1 dB, indicating good stability in attenuation performance. Figure 14: Average ARS value and \([-2\sigma,2\sigma]\) range at observation points A (**a**), B (**b**), C (**c**), D (**d**), and E (**e**) with simultaneous variation \(\alpha=0.4\) in tree height \(h_{\text{tg}}\). Figure 17 shows the ARS for different observation points by the tree grid over a homogeneous half-space, where simultaneous variations in the tree height \(h_{\text{ij}}\), radius \(r_{\text{ij}}\), and density \(\rho_{\text{ij}}\) of each tree simultaneously exhibit a variation range characterized by \(\alpha=0.4\). The results reveal that the attenuation performance of the tree grid shows greater uncertainty compared to scenarios where randomness is introduced in a single parameter. For point A, the fluctuation amplitude reaches up to 2 dB in the low-frequency range and 1 dB in the high-frequency range. Figure 15: Average ARS value and \([-2\sigma,2\sigma]\) range at observation points A (**a**), B (**b**), C (**c**), D (**d**), and E (**e**) with variation \(\alpha=0.4\) in tree density \(\rho_{\text{ij}}\). In summary, despite the randomness of tree parameters, the tree grid can still maintain effective attenuation performance. Variation in tree height causes more significant fluctuations in attenuation efficiency compared to variations in radius and density. The attenuation performance of the tree grid demonstrates greater uncertainty when the variations in height, radius, and density are simultaneously taken into account. In addition, due to the propagation of low-frequency surface waves around the tree grid toward farther observation points, the performance uncertainty at these points tends to reduce in the low-frequency region while amplifying in the high-frequency domain. Figure 16: Average ARS value and \([-2\sigma,2\sigma]\) range at observation points A (**a**), B (**b**), C (**c**), D (**d**), and E (**e**) with variation \(\alpha=0.4\) in tree radius \(r_{\text{g}}\).