## 2 Establishment of Prediction Model
### Simplified Tunnel Model Using VM
\begin{bmatrix}\mathbf{K}_{\mathrm{F}}-\mathbf{K}_{\mathrm{F}}^{ \mathrm{G}}\end{bmatrix} -\omega^{2}\begin{pmatrix}\mathbf{M}_{\mathrm{F}}- \mathbf{M}_{\mathrm{F}}^{\mathrm{G}}\end{bmatrix}\right)\mathbf{u}_{\mathrm{F }}=\mathbf{P}+\mathbf{F}_{\mathrm{F}} \tag{37}\]  
Ground system:  
\[\left( \begin{bmatrix}\mathbf{K}_{\mathrm{F}}^{\mathrm{G}}&\mathbf{K}_{ \mathrm{FG}}\\ \mathbf{K}_{\mathrm{GF}}&\mathbf{K}_{\mathrm{G}}\end{bmatrix} -\omega^{2} \begin{bmatrix}\mathbf{M}_{\mathrm{F}}^{\mathrm{G}}\\ \mathbf{M}_{\mathrm{G}}\end{bmatrix} \right) \begin{pmatrix}\mathbf{u}_{ \mathrm{F}}\\ \mathbf{u}_{\mathrm{G}}\end{pmatrix} = \begin{pmatrix}\mathbf{F}_{\mathrm{G}}\\ \mathbf{0}\end{pmatrix}  \tag{38}\]  
where \(\mathbf{F}_{\mathrm{F}}\) and \(\mathbf{F}_{\mathrm{G}}\) represent the interaction force vector between the foundation system and the ground system due to the separation of the total system, satisfying the force balance relationship \(\mathbf{F}_{\mathrm{F}}+\mathbf{F}_{\mathrm{G}}=\mathbf{0}\). To determine \(\mathbf{u}_{\mathrm{F}}\), \(\mathbf{u}_{\mathrm{G}}\) yields need to be eliminated, expressed as follows:  
\[ \begin{split}\mathbf{u}_{\mathrm{F}}=\mathbf{A}(i\omega)\mathbf{F }_{\mathrm{G}}\\ \mathbf{A}(i\omega)=\left(\begin{pmatrix}\mathbf{K}_{\mathrm{F}}^{ \mathrm{G}}-\omega^{2}\mathbf{M}_{\mathrm{F}}^{\mathrm{G}}\end{pmatrix}- \mathbf{K}_{\mathrm{FG}}\begin{pmatrix}\mathbf{K}_{\mathrm{G}}-\omega^{2} \mathbf{M}_{\mathrm{G}}\end{pmatrix}^{-1}\mathbf{K}_{\mathrm{GF}}\right)^{-1} \end{split}  \tag{39}\]  
where \(\mathbf{A}(i\omega)\) represents the relationship between the exciting force and displacement of the interaction nodes between the foundation and the foundation in the layered foundation soil, that is, the dynamic flexibility matrix of the foundation. \(\mathbf{A}(i\omega)\) can be calculated using [PERSON]'s function of the TLM of substratum under a concentrated point load derived in Section 2.1. It should be noted that if the horizontal coordinates of the observation point and the excited point are the same, the displacements at the excited point and along the vertical axis passing through the excited point will diverge. In order to eliminate this singularity, the fundamental solution of the displacement in the frequency-space domain under uniformly distributed circular loads can be used to calculate the displacement. By combining Equation (37) with Equation (39), the impedance matrix \(\mathbf{S}(\mathrm{i}\omega)\) in the ground-foundation system can be expressed as follows:  
\[ \begin{split}&\mathbf{S}(\mathrm{i}\omega)\mathbf{w}_{\mathrm{F}}= \mathbf{P}\\ &\mathbf{S}(\mathrm{i}\omega)=\left(\left(\mathbf{K}_{\mathrm{F}} -\mathbf{K}_{\mathrm{F}}^{\mathrm{G}}\right)-\omega^{2}\left(\mathbf{M}_{ \mathrm{F}}-\mathbf{M}_{\mathrm{F}}^{\mathrm{G}}\right)\right)+\mathbf{A}( \mathrm{i}\omega)^{-1}\end{split}  \tag{40}\]  
By introducing the deformation of Equation (40) into Equation (2), the interaction force between the ground and foundation interaction nodes can be written as follows:  
\[\mathbf{F}_{\mathrm{G}}=-\mathbf{F}_{\mathrm{F}}=\mathbf{A}(\mathrm{i}\omega)^ {-1}\mathbf{S}(\mathrm{i}\omega)^{-1}\mathbf{P} \tag{41}\]  
After the ground-foundation interaction force of each node is obtained, the displacement response of any observation point of foundation soil under \(\mathbf{F}_{\mathrm{F}}\) action can be obtained according to the fundamental displacement solution of layered foundation under unit point load derived in Section 2.1. When the external excitation force \(\mathbf{P}\) is the vertical unit harmonic load, the displacement response of the soil around the foundation derived above is the foundation-subsoil displacement frequency response function. Analysis model of tunnel-foundation soil dynamic interaction  
In the underground rail transit system, the movement of subway trains causes the vibration of the tunnel structure. The dynamic load of the train is transmitted to the entire tunnel structure through the track and then from the tunnel foundation to the surrounding foundation soil.