## 2 Semi-Analytical Solution for the Coupled Track-Ground-Tree Grid System
### Coupling the Trees onto the Ground Surface  
As the primary objective is to investigate the attenuation of vertical components of surface waves, each tree is modeled as an elastic, homogeneous rod vibrating axially along its length. The longitudinal vibration behavior of the \(j\)-th tree is described by the following governing equation:  
\[E_{ij}\frac{\partial^{2}\hat{u}_{ij}(z,\omega)}{\partial z^{2}}+\rho_{ij} \omega^{2}\hat{u}_{ij}(z,\omega)=0 \tag{9}\]  
where \(\hat{u}_{ij}(z,\omega)\) denotes the axial motion of the \(j\)-th one and \(E_{ij}\) and \(\rho_{ij}\) correspond to its elasticity and density. The solution to Equation (9) is as followed:  
\[\hat{u}_{ij}(z,\omega)=C_{1j}e^{i\xi_{j}z}+C_{2j}e^{-i\xi_{j}^{2}z} \tag{10}\]  
The parameters \(C_{1j}\) and \(C_{2j}\) denote unknown constants, while \(\xi_{j}=\omega\sqrt{\rho_{ij}/E_{ij}}\) defines the axial wave number for the \(j\)-indexed tree. It is assumed that the top end of each tree (\(z=-h_{\text{t}}\)) is traction-free, whereas the base (\(z=0\)) is subjected to an external force (\(F_{ij}\)) induced by seismic excitation. These unknowns, \(C_{1j}\) and \(C_{2j}\), are then evaluated by enforcing the boundary conditions, yielding the analytical expression for the axial displacement of the circular cross-section rod:  
\[\hat{u}_{ij}(z,\omega)=\frac{\hat{F}_{ij}}{\text{i}\xi_{j}^{2}E_{tj}A_{tj} \Big{(}1-e^{i2\xi_{j}^{2}h_{ij}}\Big{)}}\left(e^{i2\xi_{j}^{2}h_{ij}}e^{i\xi_{ j}^{2}z}+e^{-i\xi_{j}^{2}z}\right) \tag{11}\]  
Equation (12) further establishes the connection between the longitudinal motion and the associated force at the soil-tree interface (where \(z=0\)), leading to  
\[\hat{u}_{ij}(z=0,\omega)=\frac{1+e^{i2\xi_{j}h_{ij}}}{\text{i}\xi_{j}E_{tj}A_{ tj}\Big{(}1-e^{i2\xi_{j}h_{ij}}\Big{)}}\hat{F}_{ij}=\hat{H}_{ij}\hat{F}_{ij}, \hat{F}_{ij}=\hat{H}_{ij}^{-1}\hat{u}_{ij} \tag{12}\]  
Tree responses triggered by the external excitation \(\hat{F}_{ex}\) generates supplementary soil vibrations. Therefore, the total soil response can be decomposed into the free field (no trees) response induced by the loading and the scattered response resulting from the soil-tree dynamic interaction, which yields  
\[\hat{u}(x,y,\omega)=\hat{u}^{f}(x,y,\omega)+\hat{u}^{s}(x,y,\omega)=\hat{G}_{ zz}^{00}(x,y,\omega)\hat{F}_{ex}-\sum_{j=1}^{J}\hat{G}_{zz}^{00}(x-x_{j},y-y_{j}, \omega)\hat{F}_{ij} \tag{13}\]  
where \(\hat{G}_{zz}^{00}(x,y,\omega)\) corresponds to the vertical displacement of the ground surface resulting from a unit force acting at the surface within the space-frequency domain, which is obtained by performing the inverse Fourier transform of the wavenumber \(k_{x}\) applied to Equation (8), yielding  
\[\hat{G}_{zz}^{00}(x,y,\omega)=\sum_{j}^{R-\text{modes}}\frac{1}{4\text{i}} \text{H}_{0}^{(2)}\left(k_{j}\sqrt{x^{2}+y^{2}}\right)\varphi_{zj}^{0}\varphi_ {zj}^{0} \tag{14}\]  
where \(\text{H}_{0}^{(2)}\) represents the zero-order Hankel function of the second kind. Since the displacements of soil and trees are consistent at each soil-tree contact point, a series of linear equations can be obtained by combining Equations (12) and (13):  
\[\left\{ \begin{array}{l}\dot{u}_{l1}(z=0,\omega)=\hat{u}(x_{1},y_{1},\omega)= \hat{G}^{00}_{zz}(x_{1},y_{1},\omega)\hat{F}_{ex}-\sum\limits_{j=1}^{l}\hat{G}^{ 00}_{zz}(x_{1}-x_{j},y_{1}-y_{j},\omega)\hat{H}_{ij}\hat{u}_{lj}(z=0,\omega)\\ \dot{u}_{l2}(z=0,\omega)=\hat{u}(x_{2},y_{2},\omega)=\hat{G}^{00}_{zz}(x_{2},y_ {2},\omega)\hat{F}_{ex}-\sum\limits_{j=1}^{l}\hat{G}^{00}_{zz}(x_{2}-x_{j},y_ {2}-y_{j},\omega)\hat{H}_{ij}\hat{u}_{lj}(z=0,\omega)\\ \vdots\\ \dot{u}_{lf}(z=0,\omega)=\hat{u}(x_{l},y_{l},\omega)=\hat{G}^{00}_{zz}(x_{l}, y_{l},\omega)\hat{F}_{ex}-\sum\limits_{j=1}^{l}\hat{G}^{00}_{zz}(x_{l}-x_{j},y_ {l}-y_{j},\omega)\hat{H}_{lj}\hat{u}_{lj}(z=0,\omega)\\ \end{array} \right.