# The Characteristics of Seismic Rotations in VTI Medium
## 2 Theoretical Foundations  
Ground motion includes not only three components of linear displacements but also three components of rotation in three-dimensional Cartesian coordinates (Figure 1). The rotation tensor, the curl of the displacement field, is expressed as:  
\[\overset{\rightarrow}{\omega}=\frac{1}{2}\
abla\times\overset{\rightarrow}{ \mathbf{u}} \tag{1}\]  
where \(\overset{\rightarrow}{\omega}\) is the rotation tensor and \(\overset{\rightarrow}{\mathbf{u}}\) is the displacement tensor.  
In a two-dimensional plane, the seismic wave fields can be presented as two translational components, \(X\) and \(Z\) components, and a rotational component, \(\omega_{y}\). In order to  
Figure 1: The translations and rotations.  
correlate with the general seismic velocity observation, the rotation rate vector \(R_{y}\) can be expressed as:  
\[R_{y}=\frac{1}{2}(\frac{\partial\mathrm{v}_{x}}{\partial z}-\frac{\partial v_{z} }{\partial x}) \tag{2}\]  
where \(\mathrm{v}_{x}\) and \(v_{z}\) are the \(X\) and \(Z\) components of the velocity, respectively; \(x\) and \(z\) are the coordinates on the Cartesian system.  
Using the staggered-grid finite-difference method [15], under the assumptions of linear elasticity and small deformation, the one-order velocity-stress equations of the elastic waves in 2D VTI medium can be expressed as:  
\[\left\{\begin{array}{l}\frac{\partial\sigma_{xx}}{\partial x}+\frac{ \partial\sigma_{zz}}{\partial z}=\rho\frac{\partial v_{z}}{\partial t}\\ \frac{\partial\sigma_{zz}}{\partial x}+\frac{\partial\sigma_{zz}}{\partial z }=\rho\frac{\partial v_{z}}{\partial t}\end{array}\right. \tag{3}\]  
\[\left\{\begin{array}{l}\frac{\partial\sigma_{xx}}{\partial t}=c_{11}\frac{ \partial v_{x}}{\partial x}+c_{13}\frac{\partial v_{z}}{\partial z}\\ \frac{\partial v_{z}}{\partial t}=c_{13}\frac{\partial v_{x}}{\partial x}+c_{3 3}\frac{\partial v_{z}}{\partial z}\\ \frac{\partial\sigma_{zz}}{\partial t}=c_{44}(\frac{\partial v_{z}}{\partial x }+\frac{\partial v_{z}}{\partial z})\end{array}\right. \tag{4}\]  
where \(\sigma_{xx}\), \(\sigma_{zx}\) and \(\sigma_{zz}\) are three stress components, \(t\) is time, \(\rho\) is density; \(c_{11}\), \(c_{13}\), \(c_{33}\) and \(c_{44}\) are elastic coefficients, which can be calculated with [14]:  
\[\begin{split} c_{11}&=\rho\mathrm{v}_{p}^{2}(1+2\varepsilon )\\ c_{33}&=\rho\mathrm{v}_{p}^{2}\\ c_{44}&=\rho\mathrm{v}_{s}^{2}\\ c_{13}&=\rho\sqrt{\left[(1+2\delta)\mathrm{v}_{p}^{2}- \mathrm{v}_{s}^{2}\right]\left(\mathrm{v}_{p}^{2}-\mathrm{v}_{s}^{2}\right)}- \rho\mathrm{v}_{s}^{2}\end{split} \tag{5}\]  
where \(\mathrm{v}_{p}\) and \(\mathrm{v}_{s}\) are the velocity of the \(P\) and \(S\) waves, respectively, and \(\varepsilon\) and \(\delta\) are the anisotropic parameters. We define different VTI models by giving different Thomsen parameters to study the influence of anisotropic parameters on the translational and rotational components. When \(\varepsilon=0\) and \(\delta=0\), Equation (3) corresponds to an isotropic medium.  
Based on Equations (2)-(5), we simulate the seismic waves in discrete models with the grids at second-order time and twelfth-order space differential approximations. In addition, we use the Ricker wavelet with a 60 Hz central frequency to simulate the explosion source, the radial concentrated force source, the vertical concentrated force source, and the shear source, respectively. Furthermore, we utilize the splitting form of perfectly matched absorbing layer boundary condition (SPML) [16] to weaken the boundary reflections.