# Numerical Modeling of Scholte Wave in Acoustic-Elastic Coupled TTI Anisotropic Media
## Appendix B stability condition  
In this appendix, we derive the stability conditions of the 3D TTI acoustic-elastic coupled equation. Equation (1) is expressed as:  
\[\frac{\partial\text{u}}{\partial\text{t}}=\text{Au}\;, \tag{100}\]  
where \(\text{u}=\left(\text{v}_{\text{x}},\text{v}_{\text{y}},\text{v}_{\text{z}}, \text{P}_{\text{s}},\tau_{\text{x}\text{x}}^{\text{s}},\tau_{\text{y}\text{y}}^ {\text{s}},\tau_{\text{zz}}^{\text{s}},\tau_{\text{x}\text{y}}^{\text{s}},\tau_ {\text{x}\text{z}}^{\text{s}},\tau_{\text{yz}}^{\text{s}}\right)^{\text{T}}\), A is determined by the matrices L and \(\text{D}^{\text{e}\text{e}}\) (the coefficient matrix of TTI acoustic-elastic coupled equation):  
\[\text{A}= \begin{bmatrix}0&\text{A}_{1}\\ \text{A}_{2}&0\end{bmatrix} , \tag{101}\]  
\[\text{A}_{1}=\text{L}= \begin{bmatrix}-\frac{\partial}{\partial x}&\frac{ \partial}{\partial x}&0&0&\frac{\partial}{\partial y}&\frac{\partial}{ \partial x}&0\\ -\frac{\partial}{\partial y}&0&\frac{\partial}{\partial y}&0&\frac{\partial}{ \partial x}&0&\frac{\partial}{\partial x}\\ -\frac{\partial}{\partial z}&0&0&\frac{\partial}{\partial z}&0&\frac{ \partial}{\partial x}&\frac{\partial}{\partial y}\end{bmatrix} , \tag{102}\]  
\[\text{L}= \begin{bmatrix}D_{12}^{\text{e}\alpha}\frac{\partial}{\partial x}+D_ {16}^{\text{e}\alpha}\frac{\partial}{\partial z}&D_{13}^{\text{e}\alpha}\frac {\partial}{\partial y}&D_{14}^{\text{e}\alpha}\frac{\partial}{\partial z}+D_ {16}^{\text{e}\alpha}\frac{\partial}{\partial x}\\ D_{22}^{\text{e}\alpha}\frac{\partial}{\partial x}+D_{26}^{\text{e}\alpha} \frac{\partial}{\partial z}&D_{23}^{\text{e}\alpha}\frac{\partial}{\partial y }&D_{24}^{\text{e}\alpha}\frac{\partial}{\partial z}+D_{26}^{\text{e}\alpha} \frac{\partial}{\partial x}\\ D_{32}^{\text{e}\alpha}\frac{\partial}{\partial x}+D_{36}^{\text{e}\alpha} \frac{\partial}{\partial z}&D_{33}^{\text{e}\alpha}\frac{\partial}{\partial y }&D_{34}^{\text{e}\alpha}\frac{\partial}{\partial z}+D_{36}^{\text{e}\alpha} \frac{\partial}{\partial x}\\ D_{42}^{\text{e}\alpha}\frac{\partial}{\partial x}+D_{46}^{\text{e}\alpha} \frac{\partial}{\partial z}&D_{43}^{\text{e}\alpha}\frac{\partial}{\partial y }&D_{44}^{\text{e}\alpha}\frac{\partial}{\partial z}+D_{46}^{\text{e}\alpha} \frac{\partial}{\partial x}\\ D_{55}^{\text{e}\alpha}\frac{\partial}{\partial y}&D_{55}^{\text{e}\alpha} \frac{\partial}{\partial x}+D_{57}^{\text{e}\alpha}\frac{\partial}{\partial z} &D_{57}^{\text{e}\alpha}\frac{\partial}{\partial z}\\ D_{62}^{\text{e}\alpha}\frac{\partial}{\partial x}+D_{66}^{\text{e}\alpha} \frac{\partial}{\partial z}&D_{63}^{\text{e}\alpha}\frac{\partial}{\partial y }&D_{64}^{\text{e}\alpha}\frac{\partial}{\partial z}+D_{66}^{\text{e}\alpha} \frac{\partial}{\partial x}\\ D_{75}^{\text{e}\alpha}\frac{\partial}{\partial y}&D_{75}^{\text{e}\alpha} \frac{\partial}{\partial x}+D_{77}^{\text{e}\alpha}\frac{\partial}{\partial z }&D_{77}^{\text{e}\alpha}\frac{\partial}{\partial y}\end{bmatrix} . \tag{103}\]  
Taking the temporal derivative of Equation (100) leads to the following equation:  
\[\frac{\partial^{2}\text{u}}{\partial\text{t}^{2}}=\text{A}^{2}\text{u}. \tag{104}\]  
The time-frequency transformation of A\({}^{2}\) is performed to obtain the matrix \(\hat{A}^{2}\). Taking the eigenvalue \(\lambda_{i}\) of the matrix \(\hat{A}^{2}\), we can obtain:  
\[\lambda=\frac{1}{\rho}\text{GK}, \tag{105}\]  
where \(\lambda=\left(\lambda_{1},\lambda_{2},\dots,\lambda_{10}\right)^{\text{T}}\) is eigenvalue vector, \(\text{K}=\left(k_{x}^{2},k_{y}^{2},k_{z}^{2},k_{x}k_{z}\right)^{\text{T}}\) is operators in the frequency-wave number domain, G is the coefficient matrix as:  
\[\text{G}=