## 2 Forward Modeling of Pavement Structures
### Spectral Element Method  
For the axisymmetric wave problem, only the radial and vertical displacements are considered. The wave propagation formula of the continuous medium in the time-spatial domain under the cylindrical coordinate system is  
\[ \begin{array}{l}\frac{\partial\sigma_{r}}{\partial r}+\frac{\partial\tau_{zr} }{\partial z}+\frac{\sigma_{r}-\sigma_{\theta}}{r}=\rho\frac{\partial^{2}u_{ r}}{\partial t^{2}},\\ \frac{\partial\tau_{zr}}{\partial r}+\frac{\partial\sigma_{z}}{\partial z}+ \frac{\tau_{zr}}{r}=\rho\frac{\partial^{2}u_{r}}{\partial t^{2}},\end{array}  \tag{1}\]  
where \(\sigma_{i}\) (\(i=r,\theta,z\)) is the normal stress in the subscript direction, \(\tau_{zr}\) is the shear stress in the \(z\)-\(r\) plane, \(\rho\) represents the mass density of the medium, and \(u_{r}\) and \(u_{z}\) represent the radial and vertical displacements of the medium, respectively. In Helmholtz decomposition, the displacement field of the medium can be expressed as the gradient of scalar potential \(\alpha\) and the curl of vector potential \(\beta\), i.e.,  
\[u=\
abla\alpha+\
abla\times\beta, \tag{2}\]  
where \(\
abla\) represents a partial differential operator. In the axisymmetric wave problem, vector potential \(\beta\) has only one component \(\beta_{\theta}\), so the problem to be solved can be simplified as a scalar potential problem. For simplicity, \(\beta\) is used to represent \(\beta_{\theta}\). As it is an axisymmetric problem, the displacement of the medium in the circumferential direction \(\theta\) is zero, and the radial and vertical displacement can be expressed as follows [10]:  
\[u_{r}=\frac{\partial\alpha}{\partial r}-\frac{\partial\beta}{\partial z},\ \ u_{z}=\frac{\partial\alpha}{\partial z}+\frac{1}{r}\frac{\partial(r\beta)}{ \partial r}. \tag{3}\]  
The relationship between stress and displacement in the cylindrical coordinate system is  
\[ \begin{array}{l}\sigma_{r}=(\lambda+2\mu)\frac{\partial u_{r}}{\partial r}+ \mu\Big{(}\frac{u_{r}}{r}+\frac{\partial u_{z}}{\partial z}\Big{)},\\ \sigma_{\theta}=(\lambda+2\mu)\frac{u_{r}}{r}+\mu\Big{(}\frac{\partial u_{r}}{ \partial r}+\frac{\partial u_{r}}{\partial z}\Big{)},\\ \sigma_{z}=(\lambda+2\mu)\frac{\partial u_{z}}{\partial z}+\mu\Big{(}\frac{ \partial u_{r}}{\partial r}+\frac{u_{r}}{r}\Big{)},\\ \tau_{zr}=\mu\Big{(}\frac{\partial u_{z}}{\partial r}+\frac{\partial u_{r}}{ \partial z}\Big{)},\end{array}  \tag{4}\]  
where \(\lambda\) and \(\mu\) are the Lame constants, and  
\[\lambda=\frac{vE}{(1+\
u)(1-2\
u)},\ \mu=\frac{E}{2(1+\
u)}, \tag{5}\]  
where \(E\) is the [PERSON]'s modulus and \(\
u\) is the [PERSON]'s ratio. Equation (4) is substituted into Equation (1) and the displacement is expressed by the scalar potential in Equation (3). After algebraic simplification, the following equations are satisfied:  
\[(\lambda+2\mu)\bigg{(}\frac{\partial^{2}\alpha}{\partial r^{2}}+\frac{ \partial^{2}\alpha}{\partial z^{2}}+\frac{\partial\alpha}{r\partial r}\bigg{)} =\rho\frac{\partial^{2}\alpha}{\partial t^{2}} \tag{6}\]  
\[\mu\bigg{(}\frac{\partial^{2}\beta}{\partial z^{2}}+\frac{\partial^{2}\beta}{ \partial r^{2}}+\frac{\partial\beta}{\partial r}-\frac{\beta}{r^{2}}\bigg{)} =\rho\frac{\partial^{2}\beta}{\partial t^{2}} \tag{7}\]  
Figure 2: The wave propagation in the cylindrical coordinate system. It is evident that Equations (6) and (7) represent two forms of waves, namely P and SH waves, respectively, the wave velocities of which are \(c_{P}=\sqrt{\frac{(\lambda+2\mu)}{\rho}}\) and \(c_{SH}=\sqrt{\frac{\mu}{\rho}}\), respectively. For isotropic layered media, the two waves can be decoupled and solved separately.