## 2 Methods
### The 3D Decoupled Nonconversion Elastic Wave Equation in the Cylindrical Coordinate System  
In a 3D homogeneous medium, the time-domain elastic wave equation in Cartesian coordinates [21] can be expressed as:  
\[\rho\frac{\partial^{2}\mathbf{u}}{\partial t^{2}}=(\lambda+\mu)\
abla\
abla\cdot\bm {u}+\mu\
abla^{2}\mathbf{u}-\
abla\lambda\
abla\cdot\mathbf{u}+\
abla\mu\times(\
abla \times\mathbf{u})+2(\
abla\mu\cdot\
abla)\mathbf{u}, \tag{1}\]  
where \(\mathbf{u}\) is the displacement of the elastic wavefield, \(\rho\) denotes the density, \(t\) is the time, and \(\mu,\lambda\) represent Lame constants, respectively. [PERSON] [22] multiplied the above equation by \(\
abla\cdot\) or \(\
abla\times\) and brought the nonconvexity condition into it, obtaining the nonconvexity elastic wave equation as follows :  
\[\frac{\partial^{2}\mathbf{u}}{\partial t^{2}}=\
abla(\mathbf{v}_{p}^{2}\
abla\cdot\bm {u})-\mathbf{v}_{S}^{2}\
abla\times\
abla\times\mathbf{u}, \tag{2}\]  
where \(\mathbf{v}_{S}\), \(\mathbf{v}_{p}\) denote the \(P\)-velocity and \(S\)-velocity, respectively. According to the Helmholtz theory, we can decouple the \(S\)-wavefield and the \(P\)-wavefield and obtain Equation (3),  
\[\left\{\begin{array}{l}\mathbf{u}=\mathbf{u}^{P}+\mathbf{u}^{S},\\ \frac{\partial^{2}\mathbf{u}^{P}}{\partial t^{2}}=\
abla\big{(}\mathbf{v}_{p}^{2} \
abla\cdot\mathbf{u}\big{)},\\ \frac{\partial^{2}\mathbf{u}^{S}}{\partial t^{2}}=\mathbf{v}_{S}^{2}\
abla\times\
abla \times\mathbf{u},\end{array}\right. \tag{3}\]  
where \(\mathbf{u}^{P}\), \(\mathbf{u}^{S}\) are the displacement of \(P\)-wavefield and \(S\)-wavefield, respectively. By introducing the stress field and defining the particle vibration velocity, the first-order velocity-stress equation for the decoupled nonconvexity elastic wave in cylinder coordinates can be expressed as:  
\[\left\{\begin{array}{l}v_{r}=\mathbf{v}_{r}^{p}+\mathbf{v}_{r}^{S},\frac{\partial \mathbf{v}_{r}^{P}}{\partial t_{r}}=\frac{\partial\mathbf{\tau}^{P}}{\partial r}, \frac{\partial\mathbf{v}_{r}^{S}}{\partial t_{r}}=\frac{\partial\mathbf{\tau}_{zr}^{ S}}{\partial z_{z}}-\frac{\partial\mathbf{\tau}_{r\theta}^{S}}{\partial t \theta},\\ v_{\theta}=\mathbf{v}_{\theta}^{p}+\mathbf{v}_{\theta}^{S},\frac{\partial\mathbf{v}_{ \theta}^{S}}{\partial t_{r}}=\frac{\partial\mathbf{\tau}^{P}}{r\partial\theta}, \frac{\partial\mathbf{v}_{\theta}^{S}}{\partial t_{r}}=\frac{\partial\mathbf{\tau}_{ r\theta}^{S}}{\partial r}-\frac{\partial\mathbf{\tau}_{\theta}^{S}}{\partial z },\\ v_{z}=\mathbf{v}_{z}^{P}+\mathbf{v}_{z}^{S},\frac{\partial\mathbf{v}_{z}^{P}}{\partial t _{r}}=\frac{\partial\mathbf{\tau}^{P}}{\partial z},\frac{\partial\mathbf{v}_{z}^{S}}{ \partial t_{r}}=\frac{\partial\mathbf{\tau}_{\theta z}^{S}}{r\partial\theta}- \frac{\partial\big{(}\mathbf{\tau}_{zr}^{S}\big{)}}{r\partial r},\\ \frac{\partial\mathbf{\tau}^{P}}{\partial t}=\mathbf{v}_{p}^{2}\bigg{(}\frac{\partial \mathbf{v}_{r}}{\partial r}+\frac{\mathbf{v}_{r}}{r}+\frac{\mathbf{1}}{r}\frac{\partial \mathbf{v}_{\theta}}{\partial\theta}+\frac{\partial\mathbf{v}_{z}}{\partial z}\bigg{)},\\ \frac{\partial\mathbf{\tau}^{S}_{r\theta}}{\partial t}=\mathbf{v}_{S}^{2}\bigg{(}\frac{ \partial(\mathbf{v}_{p})}{r\partial r}-\frac{\mathbf{1}}{r}\frac{\partial\mathbf{v}_{r}}{ r\partial\theta}\bigg{)},\\ \frac{\partial\mathbf{\tau}^{S}_{\theta z}}{\partial t}=\mathbf{v}_{S}^{2}\bigg{(}\frac{ \partial(\mathbf{v}_{z})}{r\partial\theta}-\frac{\partial\mathbf{v}_{\theta}}{ \partial z}\bigg{)},\\ \frac{\partial\mathbf{\tau}^{S}_{zr}}{\partial t}=\mathbf{v}_{S}^{2}\bigg{(}\frac{ \partial(\mathbf{v}_{r})}{\partial z}-\frac{\partial\mathbf{v}_{z}}{\partial r}\bigg{)},\end{array}\right. \tag{4}\]  
where \(\mathbf{\tau}^{S}_{ij}\), \(\mathbf{\tau}^{p}_{ij}(i,j\in[r,\theta,z])\) are the stress field component of the \(P\)-wavefield and \(S\)-wavefield, respectively. \(\mathbf{v}_{i}(i\in[r,\theta,z])\) denotes the particle vibration velocity component, and \(r,\theta,z\) are the spatial directions in the cylindrical coordinate system.