## 2 Method
\right]e^{-ik_{xi}x} \tag{28}\]  
\[\widetilde{\tau}_{xz}=\left[ \begin{array}{c}2\mu_{i}k_{x}k_{zpi}e^{ik_{xi}x^{ 2}}U_{pi}-2\mu_{i}k_{xi}k_{zpi}e^{-ik_{xi}x^{2}}D_{pi}\\ -\mu_{i}(k_{xi}^{2}-k_{zsi}^{2})e^{ik_{xi}x^{2}}U_{si}-\mu_{i}(k_{xi}^{2}-k_{zsi }^{2})e^{-ik_{xi}x^{2}}D_{si}\end{array} \right]e^{-ik_{xi}x} \tag{29}\]  
Equations (26)-(29) can be written as the matrix form:  
\[\left[ \begin{array}{c}\widetilde{u}_{x}\\ \widetilde{u}_{z}\\ \widetilde{\sigma}_{z}\\ \widetilde{\tau}_{xz}\end{array} \right]_{i}=\left[ \begin{array}{c}-ik_{xi}e^ {ik_{xi}x^{2}}-ik_{xi}e^{-ik_{xi}x^{2}}-ik_{xi}e^{ik_{xi}x^{2}}&ik_{zxi}e^{-ik_ {xi}x^{2}}\\ ik_{zpi}e^{ik_{xi}x^{2}}-ik_{zpi}e^{-ik_{xi}x^{2}}&-ik_{xi}e^{ik_{xi}x^{2}}-ik_{xi }e^{-ik_{xi}x^{2}}\\ (-\frac{\lambda_{i}\omega^{2}}{V_{pi}^{2}}-2\mu_{i}k_{zpi}^{2})e^{ik_{xi}x^{2} }(-\frac{\lambda_{i}\omega^{2}}{V_{pi}^{2}}-2\mu_{i}k_{zpi}^{2})e^{-ik_{xi}x^{2 }}&2\mu_{i}k_{xi}k_{z\pi}e^{ik_{xi}x^{2}}&-2\mu_{i}k_{xi}k_{z\pi}e^{-ik_{xi}x^{ 2}}\\ 2\mu_{i}k_{xi}k_{zpi}e^{ik_{xi}x^{2}}&-2\mu_{i}k_{xi}k_{zpi}e^{-ik_{xi}x^{2}}&- \mu_{i}(k_{xi}^{2}-k_{zsi}^{2})e^{ik_{xi}x^{2}}-\mu_{i}(k_{xi}^{2}-k_{zsi}^{2} )e^{-ik_{xi}x^{2}}\end{array} \right] \tag{30}\]  
For the rock layer,  
\[P_{i}=P_{r}=\left[ \begin{array}{cccc}-ik_{x}^{r}&-ik_{x}^{r}&-ik_{zs}^{r}&ik_ {zs}^{r}\\ ik_{zp}^{r}&-ik_{zp}^{r}&-ik_{x}^{r}&-ik_{x}^{r}\\ (-\frac{\lambda_{i}\omega^{2}}{V_{pi}^{2}}-2\mu_{r}k_{zp}^{r}\,{}^{2})&(-\frac {\lambda_{i}\omega^{2}}{V_{pi}^{2}}-2\mu_{r}k_{zp}^{r}\,{}^{2})&2\mu_{r}k_{x}^{r }k_{zs}^{r}&-2\mu_{r}k_{x}^{r}k_{zs}^{r}\\ 2\mu_{r}k_{x}^{r}k_{zp}^{r}&-2\mu_{r}k_{x}^{r}k_{zp}^{r}&-\mu_{r}(k_{x}^{r}{}^{2 }-k_{zs}^{r}\,{}^{2})&-\mu_{r}(k_{x}^{r}{}^{2}-k_{zs}^{r}\,{}^{2})\end{array} \right]\]  
and  
\[Q_{i}=Q_{r}=\left[ \begin{array}{cccc}e^{ik_{x}^{r}z}&0&0&0\\ 0&e^{-ik_{x}^{r}z}&0&0\\ 0&0&e^{ik_{x}^{r}z}&0\\ 0&0&0&e^{-ik_{z}^{r}z}\end{array} \right]\]  
the subscripts '\(i\)' denote the layer number. For the filled layer,  
\[P_{i}=P_{f}=\left[ \begin{array}{cccc}-ik_{x}^{f}&-ik_{x}^{f}&-ik_{zs}^{f}&ik_ {zs}^{f}\\ ik_{zp}^{f}&-ik_{zp}^{f}&-ik_{z}^{f}&-ik_{x}^{r}\\ (-\frac{\lambda_{f}\omega^{2}}{V_{pi}^{2}}-2\mu_{f}k_{zp}^{f}\,{}^{2})&(-\frac {\lambda_{f}\omega^{2}}{V_{pi}^{2}}-2\mu_{f}k_{zp}^{f}\,{}^{2})&2\mu_{f}k_{x}^{ f}k_{zs}^{f}&-2\mu_{f}k_{x}^{f}k_{zs}^{f}\\ 2\mu_{f}k_{x}^{f}k_{zp}^{f}&-2\mu_{f}k_{x}^{f}k_{zp}^{f}&-\mu_{f}(k_{x}^{f}{}^{2 }-k_{zs}^{f}\,{}^{2})&-\mu_{f}(k_{x}^{r}{}^{2}-k_{zs}^{f}\,{}^{2})\end{array} \right]\]and  
\[Q_{i}=Q_{f}=\left[ \begin{array}{cccc}e^{\frac{ik^{f}_{x}f^{2}}{2 }}&0&0&0\\ 0&e^{-ik^{f}_{x}{p^{2}}}&0&0\\ 0&0&e^{ik^{f}_{x}{q^{2}}}&0\\ 0&0&0&e^{-ik^{f}_{x}{q^{2}}}\end{array} \right]\]  
Equation (30) establishes the relation between stress and displacement and the potential amplitudes in each layer. Note that the matrix \(Q_{i}\) is related to z coordinate. As shown in Figure 1b, two local coordinate systems (\(x_{\text{f}}\)-\(o_{\text{f}}\)-\(z_{\text{f}}\) and \(x_{\text{i + 1}}\)-\(o_{\text{i+ 1}}\)-\(z_{\text{i+ 1}}\)) are adopted, respectively. For the upper boundary of a layer, \(z=0\) and \(Q_{i}\) can be expressed as \(Q_{i}^{+}\). For the low boundary of a layer, \(z=h_{\text{i}}\) and \(Q_{i}\) can be expressed as \(Q_{i}^{-}\). Therefore, we can form:  
\[Q_{r}^{+}=Q_{f}^{+}=\left[ \begin{array}{cccc}1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\end{array} \right]\]  
\[Q_{r}^{-}=\left[ \begin{array}{cccc}e^{ik^{f}_{x}{p_{\text{f}}}}&0&0&0\\ 0&e^{-ik^{f}_{x}{p_{\text{f}}}}&0&0\\ 0&0&e^{ik^{f}_{x}{q_{\text{f}}}}&0\\ 0&0&0&e^{-ik^{f}_{x}{q_{\text{f}}}}\end{array} \right]\]  
and  
\[Q_{f}^{-}=\left[ \begin{array}{cccc}e^{ik^{f}_{x}{p_{\text{f}}}h_{f}}&0&0&0\\ 0&e^{-ik^{f}_{x}{p_{\text{f}}}}&0&0\\ 0&0&e^{ik^{f}_{x}{q_{\text{f}}}}&0\\ 0&0&0&e^{-ik^{f}_{x}{q_{\text{f}}}}\end{array} \right]\]  
The relation of the potential amplitudes between two adjacent rock layers can be derived from the sandwiched element as shown in Figure 1b. As mentioned before, the stress and the displacement at the interface between the rock layer and the filled layer are continuous.