## 2 Method
The symbols \"+\" and \"-\" represent the upper boundary and the lower boundary of a layer (e.g., in the (i + 1) \(th\) layer, \"+\" represents the upper boundary (\(z_{\rm{i+1}}=0\)), whereas \"-\" represents the low boundary (\(z_{\rm{i+1}}=h_{\rm{r}}\)). For the linearly elastic rock layer, the 2-D wave equation can be written as:  
\[\rho_{r}\frac{\partial^{2}u_{x}}{\partial t^{2}}=\frac{\partial\sigma_{x}}{ \partial x}+\frac{\partial\tau_{xz}}{\partial z} \tag{1}\]  
\[\rho_{r}\frac{\partial^{2}u_{z}}{\partial t^{2}}=\frac{\partial\tau_{xz}}{ \partial x}+\frac{\partial\sigma_{z}}{\partial z} \tag{2}\]  
\[\sigma_{z}=\lambda_{r}\frac{\partial u_{x}}{\partial x}+(\lambda_{r}+2\mu_{r}) \frac{\partial u_{z}}{\partial z} \tag{3}\]  
\[\tau_{xz}=\mu_{r}(\frac{\partial u_{z}}{\partial x}+\frac{\partial u_{x}}{ \partial z}) \tag{4}\]  
where \(u_{\rm{i}}\) denotes the particle displacement and \(\sigma_{\rm{ij}}\) denotes the stress, \(\lambda_{\rm{r}}\) and \(\mu_{\rm{r}}\) are Lame constants for rock and \(\lambda_{\rm{r}}=\rho_{\rm{r}}\) (\(V_{\rm{p}}\)\({}^{2}\))-2\(\rho_{\rm{r}}\) (\(V_{\rm{s}}\)\({}^{2}\)), \(\mu_{\rm{r}}=\rho_{\rm{r}}\) (\(V_{\rm{s}}\)\({}^{2}\)). For the Kelvin viscoelastic filled layer, the 2-D wave equation can be written as:  
\[\rho_{f}\frac{\partial^{2}u_{x}}{\partial t^{2}}=\frac{\partial\sigma_{x}}{ \partial x}+\frac{\partial\tau_{xz}}{\partial z} \tag{5}\]  
\[\rho_{f}\frac{\partial^{2}u_{z}}{\partial t^{2}}=\frac{\partial\tau_{xz}}{ \partial x}+\frac{\partial\sigma_{z}}{\partial z} \tag{6}\]  
\[\frac{\partial\sigma_{z}}{\partial t}=\lambda_{p}\frac{\partial^{2}u_{x}}{ \partial t\partial x}+(\lambda_{p}+2\mu_{p})\frac{\partial^{2}u_{z}}{\partial t \partial z}+\lambda_{f}{{}^{\prime}}\frac{\partial^{3}u_{x}}{\partial x \partial t^{2}}+(\lambda_{f}{{}^{\prime}}+2\mu_{f}{{}^{\prime}})\frac{ \partial^{3}u_{z}}{\partial z\partial t^{2}} \tag{7}\]  
\[\frac{\partial\tau_{xz}}{\partial t}=\mu_{p}\frac{\partial^{2}u_{z}}{ \partial t\partial x}+\mu_{p}\frac{\partial^{2}u_{x}}{\partial t\partial z}+ \mu_{f}{{}^{\prime}}\frac{\partial^{3}u_{z}}{\partial x\partial t^{2}}+\mu_{f }{{}^{\prime}}\frac{\partial^{3}u_{x}}{\partial z\partial t^{2}} \tag{8}\]  
where \(\lambda_{\rm{p}}\) and \(\mu_{\rm{p}}\) are the purely elastic Lame constants of the filled layer when the viscosity is not considered, \(\lambda_{\rm{p}}=\rho_{\rm{f}}\) (\(V_{\rm{p}}\)\({}^{2}\))-2\(\rho_{\rm{f}}\) (\(V_{\rm{s}}\)\({}^{2}\)), \(\mu_{\rm{p}}=\rho_{\rm{f}}\) (\(V_{\rm{s}}\)\({}^{2}\)) and \(\lambda_{\rm{f}}{{}^{\prime}}\) and \(\mu_{\rm{f}}{{}^{\prime}}\) denote the viscoelastic modulus for the filled medium. Equations (1)-(8) are transformed into frequency domains by Fourier transform. For the rock layer,  
Figure 1:0  
Figure 1:1  
Figure 1:2  
Figure 1:3  
where, variables with the symbol \"\(\sim\)\" over them denotes the Fourier transform. For the filled layer,  
Figure 1:4  
Figure 1:5  
Figure 1:6  
Figure 1:7where \(\omega\) is angular frequency and \(i=\sqrt{-1}\). The quality factors related to energy attenuation can be defined as [40]:  
\[ \begin{array}{l}Q_{p}=\frac{\lambda_{p}+2\mu_{p}}{\lambda_{f}+2\mu_{f}{{}^{ \prime}}}\frac{1}{\omega}\\ Q_{s}=\frac{\mu_{p}}{\mu_{f}{{}^{\prime}}}\frac{1}{\omega}\end{array}  \tag{17}\]  
When \(Q_{\rm p}{\rightarrow}\infty\) and \(Q_{\rm s}{\rightarrow}\infty\), Equations (13)-(16) represent the pure elastic case without viscosity energy loss. When \(Q_{\rm p}{\rightarrow}0\) and \(Q_{\rm s}{\rightarrow}0\), the filled medium has a very large viscosity. Based on Equations (15)-(17), we can obtain corresponding complex equivalent Lame constants and an equivalent wave velocity of the filled medium as follows:  
\[