# Numerical Modeling of Scholte Wave in Acoustic-Elastic Coupled TTI Anisotropic Media
## Appendix A  
In Appendix A, we first introduce the definitions of the Thomsen parameters and explain how they play a crucial role in our algorithm. We then derive the acoustic-elastic coupled equation in a 3D TTI media. By combining these two parts through the Bond transformation, we can obtain our results. **Thomsen Parameter**  
The specific definitions of the five Thomsen parameters are as follows:  
\[V_{P0}=\sqrt{\frac{c_{33}}{\rho}},\;V_{S0}=\sqrt{\frac{c_{55}}{\rho}}, \tag{1}\]  
\[\varepsilon=\frac{c_{33}-c_{11}}{2c_{33}}, \tag{2}\]  
\[\delta=\frac{(c_{13}+c_{44})^{2}-(c_{33}-c_{44})^{2}}{2c_{33}(c_{33}-c_{44})^ {2}}, \tag{3}\]  
\[\gamma=\frac{c_{66}-c_{55}}{2c_{55}}, \tag{4}\]  
where \(\rho\) represents density, \(V_{P0}\) is the vertical P-wave velocity, \(V_{S0}\) is the vertical S-wave velocity, \(\varepsilon\), \(\delta\) and \(\gamma\) represents the dimensionless anisotropic constants, and \(c_{ij}\) represents the elastic coefficient of VTI media. **acoustic-elastic coupled equation in a 3D TTI media**  
In Appendix A, we deduce the TTI acoustic-elastic coupled equation used for the calculations in this paper. The traditional elastic velocity-stress equation in TTI media (only considering the polarization Angle \(\theta\)) can be expressed as:  
\[\rho\frac{\partial v_{x}}{\partial t}=\frac{\partial\tau_{xx}}{\partial x}+ \frac{\partial\tau_{xy}}{\partial y}+\frac{\partial\tau_{xz}}{\partial z}, \tag{5}\]  
\[\rho\frac{\partial v_{y}}{\partial t}=\frac{\partial\tau_{xy}}{\partial x}+ \frac{\partial\tau_{yy}}{\partial y}+\frac{\partial\tau_{yz}}{\partial z}, \tag{6}\]  
\[\rho\frac{\partial v_{z}}{\partial t}=\frac{\partial\tau_{xz}}{\partial x}+ \frac{\partial\tau_{yz}}{\partial y}+\frac{\partial\tau_{zz}}{\partial z}, \tag{7}\]  
and  
\[\frac{\partial\tau_{xx}}{\partial t}=D_{11}\frac{\partial v_{x}}{\partial x}+ D_{12}\frac{\partial v_{y}}{\partial y}+D_{13}\frac{\partial v_{z}}{\partial z}+D_{15} \bigg{(}\frac{\partial v_{x}}{\partial z}+\frac{\partial v_{z}}{\partial x} \bigg{)}, \tag{8}\]  
\[\frac{\partial\tau_{yy}}{\partial t}=D_{12}\frac{\partial v_{x}}{\partial x}+D_{ 22}\frac{\partial v_{y}}{\partial y}+D_{23}\frac{\partial v_{z}}{\partial z}+ D_{25}\bigg{(}\frac{\partial v_{x}}{\partial z}+\frac{\partial v_{z}}{ \partial x}\bigg{)}, \tag{9}\]\[\frac{\partial\tau_{zz}}{\partial t}=D_{13}\frac{\partial v_{x}}{ \partial x}+D_{23}\frac{\partial v_{y}}{\partial y}+D_{33}\frac{\partial v_{z}}{ \partial z}+D_{35}\bigg{(}\frac{\partial v_{x}}{\partial z}+\frac{\partial v_{z }}{\partial x}\bigg{)}, \tag{10}\] \[\varepsilon=\frac{c_{33}-c_{11}}{2c_{33}}, \tag{2}\]0 (11) \[\varepsilon=\frac{c_{33}-c_{11}}{2c_{33}}, \tag{2}\]1 (12) \[\varepsilon=\frac{c_{33}-c_{11}}{2c_{33}}, \tag{2}\]2  
where \(\rho\) is density, \(t\) is time, \(\rm V=\big{(}v_{x},v_{y},v_{z}\big{)}^{T}\) represents velocity vector of particle, \(\rm\tau=\big{(}\tau_{xx},\tau_{yy},\tau_{zz},\tau_{xy},\tau_{zz},\tau_{yz} \big{)}^{T}\) is stress. The stress tensor \(\rm T\) can be written as:  
\[\varepsilon=\frac{c_{33}-c_{11}}{2c_{33}}, \tag{2}\]3  
where \(\rm I\) is the 3-by-3 identity matrix. The quantity \(\rm T\) is the stress tensor given by  
\[\rm T= \begin{bmatrix}\tau_{xx}&\tau_{xy}&\tau_{xz}\\ \tau_{xy}&\tau_{yy}&\tau_{yz}\\ \tau_{xz}&\tau_{yz}&\tau_{zz}\end{bmatrix} . \tag{15}\]  
The quantity \(P_{s}\) is the synthesized pressure, and \(\tau^{s}\) is the deviatoric stress tensor. The synthesized pressure \(P_{s}\) can be written as follows:  
\[P_{s}=-\frac{1}{3}\rm tr(T). \tag{16}\]  
Substituting Equations (12) and (13) into the elastic velocity-stress equation, we can obtain the TTI acoustic-elastic coupled equation:  
\[ \begin{array}{c}\partial\frac{\partial}{\partial t}\rm V=L\phi,\\ \frac{\partial}{\partial t}\rm\Phi=D^{\alpha\epsilon}L^{\rm T}\rm V+f,\end{array}