# The Combined Compact Difference Scheme Applied to Shear-Wave Reverse-Time Migration
## 2 Principle of RTM and Combined Compact Difference Scheme
### Principle of Combined Compact Difference Scheme  
For the construction of the compact difference scheme, [PERSON] extended [PERSON]'s equation in 1992 and created the compact difference scheme as follows:  
\[ \begin{split} cf_{i-1}^{\prime}+f_{i}^{\prime}+cf_{i+1}^{\prime} =a_{2}\frac{f_{i+2}-f_{i-2}}{4h}+a_{1}\frac{f_{i+1}-f_{i-1}}{2h}\\ cf_{i-1}^{\prime\prime}+f_{i}^{\prime\prime}+cf_{i+1}^{\prime\prime} =a_{2}\frac{f_{i+2}-2f_{i}+f_{i-2}}{4h^{2}}+a_{1}\frac{f_{i+1}-2f_{i}+f_{i-1} }{h^{2}}\end{split}  \tag{2}\]  
In Equation (2), \(h\) is the grid spacing, \(a,c\) are the difference coefficient matrices. \(f\) is the function value of node \(i\). \(f_{i}^{\prime}\) and \(f_{i}^{\prime\prime}\) represent the first- and second-order derivatives of node \(i\), respectively; \(f_{i+1},f_{i+2},f_{i-1},f_{i-2}\) represent the function values of node i successively two nodes forward and two nodes backward; \(f_{i+1}^{\prime},f_{i-1}^{\prime}\) represent the first-order derivatives of node \(i\) successively one node forward and one node backward, respectively; \(f_{i+1}^{\prime\prime},f_{i-1}^{\prime\prime}\) represent the second derivative of i node one node forward and one node backward, respectively. Following [38], the wave equation of a shear wave in a two-dimensional medium is  
\[\frac{\partial^{2}V_{y}}{\partial t^{2}}=(1+2*\gamma)v_{80}^{2}\frac{\partial ^{2}V_{y}}{\partial x^{2}}+v_{80}\frac{\partial^{2}V_{y}}{\partial z^{2}}+ \rho F_{y} \tag{3}\]  
where \(V_{y}(x,z)\) is the displacement component of the shear wave. \(v_{80}(x,z)\) is the shear-wave velocity in the vertical direction, and \(\rho F_{y}\) is the shear-wave source, that is, the concentrated force source excited in the y direction on the surface. If the input medium is isotropic, the value of the anisotropic parameter \(\gamma=0\). Expanding the above shear-wave equation to represent the time of \(n+1\) and \(n-1\), we obtain  
\[V_{y(i,j)}^{\,n+1} =V_{y(i,j)}^{\,n}+\Delta t\left(\frac{\partial V_{y}}{\partial t }\right)_{i,j}^{\,n}+\frac{(\Delta t)^{2}}{2}\left(\frac{\partial^{2}V_{y}}{ \partial t^{2}}\right)_{i,j}^{\,n}+\frac{(\Delta t)^{3}}{6}\left(\frac{ \partial^{3}V_{y}}{\partial t^{3}}\right)_{i,j}^{\,n}+\frac{(\Delta t)^Equation (6) can be used to advance the time step of the shear wavefield in a two-dimensional medium. The equation contains the second and fourth derivatives of the sum of displacement pairs in the _horizontal and vertical_ directions, and the sum of displacement pairs in the _horizontal and vertical_ directions can be differentiated by the finite difference scheme. [PERSON] (1998) and others constructed a combined compact difference (CCD) scheme with higher accuracy as follows:  
\[\left\{\begin{array}{c}a_{1}\big{(}f_{i+1}^{\prime}+f_{i-1}^{\prime}\big{)}+f _  
\[C= \begin{bmatrix}1&0&a_{1}&-a_{2}/h\\ 0&1&-b_{1}h&b_{2}\\ a_{1}&a_{2}/h&1&0\\ b_{1}h&b_{2}&0&1\\ &a_{1}&a_{2}/h\\ &&b_{1}h&b_{2}\\ &&&&\ddots&a_{1}&-a_{2}/h\\ &&&&\ddots&-b_{1}h&b_{2}\\ &&&&\ddots&1&0\\ &&&&\ddots&0&1\end{bmatrix}  \tag{11}\]  
\[D= \begin{bmatrix}0&(-2\sum\limits_{m=1}^{n}d_{m})/h^{2}&-\sum\limits_{m=1}^{n} c_{m}/h&(\sum\limits_{m=1}^{n}d_{m})/h^{2}\\ \sum\limits_{m=1}^{n}c_{m}/h&(\sum\limits_{m=1}^{n}d_{m})/h^{2}&0&(-2\sum \limits_{m=1}^{n}d_{m})/h^{2}\\ &&\sum\limits_{m=1}^{n}c_{m}/h&(\sum\limits_{m=1}^{n}d_{m})/h^{2}\\ &&&&\ddots&-\sum\limits_{m=1}^{n}c_{m}/h&(\sum\limits_{m=1}^{n}d_{m})/h^{2}\\ &&&&\ddots&0&(-2\sum\limits_{m=1}^{n}d_{m})/h^{2}\\ &&&&\ddots&0&(-2\sum\limits_{m=1}^{n}d_{m})/h^{2}\end{bmatrix}  \tag{12}\]  
\[E=