# The Combined Compact Difference Scheme Applied to Shear-Wave Reverse-Time Migration
## 3 Analysis of CCD and CSCD
### The Numerical Simulation Accuracy Analysis  
To compare the numerical simulation accuracy of the models, we calculate the simulation error by simulating the two-dimensional plane harmonic initial value problem and then compare the numerical simulation accuracy of the CCD, CSCD, and CFD schemes. The initial value problem of the two-dimensional plane wave can be expressed as  
\[\left\{\begin{array}{c}\frac{\partial^{2}p_{ab}(t,x,z)}{\partial t^{2}}=v^{2 }\frac{\partial^{2}p_{ab}(t,x,z)}{\partial x^{2}}+v^{2}\frac{\partial^{2}p_{ ab}(t,x,z)}{\partial z^{2}}\\ p_{sh}(0,x,z)=\cos\!\left(-\frac{2\pi f_{0}}{v}\cdot\cos\theta\!\cdot\!x-\frac {2\pi f_{0}}{v}\cdot\sin\theta\!\cdot\!z\right)\\ \frac{\partial p_{ab}(0,x,z)}{\partial t}=-2\pi f_{0}\sin\!\left(-\frac{2\pi f _{0}}{v}\cdot\cos\theta\!\cdot\!x-\frac{2\pi f_{0}}{v}\cdot\sin\theta\!\cdot\!z \right)\end{array}\right. \tag{23}\]  
where \(v\) is the plane wave velocity, \(\theta\) is the angle between the propagation direction and the \(x\)-axis, and \(f_{0}\) is the frequency of the simple harmonic plane wave.  
Figure 1: Velocity ratio curves for different numerical simulation methods. The blue curve is the traditional central difference scheme, the green curve is the traditional implicit difference scheme, the red curve is the CCD difference scheme, the blue curve is the CSCD difference scheme, and the black line is the velocity ratio constant of 1: (a) \(\theta=0,\alpha=0.25\); (b) \(\theta=\pi/6,\alpha=0.25\); (c) \(\theta=\pi/3,\alpha=0.25\).  
The analytical solution for the above initial value problem is  
\[p_{sh}(t,x,z)=\cos\left[2\pi f_{0}\left(t-\frac{x}{v/\cos\theta}-\frac{z}{v/\sin \theta}\right)\right]. \tag{24}\]  
To simulate the two-dimensional shear wavefield, we specify the following: \(f_{0}=20\) Hz and \(\theta=\pi/4\), the wave velocity used is 1000 m/s, the length and depth of the model are 2000 m, the length of the vertical and horizontal grids are the same, and the sampling time is 1 s. The relative error of numerical simulation is calculated under different spatial mesh lengths and time steps, and it is defined as  
\[E_{r}(\%)=\left\{\frac{1}{\sum\limits_{i=1}^{N}\sum\limits_{j=1}^{N}\left[p_{ sh}\left(t_{n},x_{i},z_{j}\right)\right]^{2}}\sum\limits_{i=1}^{N}\sum\limits_{j=1}^{N} \left[p_{sh}\left(t_{i,j}\right)-p_{sh}\left(t_{n},x_{i},z_{j}\right)\right]^{2 }\right\}^{\frac{1}{2}}\times 100 \tag{25}\]  
where \(p_{sh}\left(i,j\right)\) is the numerical solution, and \(p_{sh}\left(t_{n},x_{i},z_{j}\right)\) is an analytical solution. We then compare the relative error curves of the CCD, CSCD, CD, and CFD schemes under different spatial steps (10 m, 15 m) and fixed time steps of 1 ms (\(\Delta t=0.001\) s), as shown in Figure 2. It can be seen from Figure 2 that the relative error gradually increases with increasing spatial grid length, time step, and simulation time. When the spatial step is 15 m and the time step is 1 ms, the maximum relative error of the CCD scheme is 8%, which is much smaller when compared with the maximum relative error of the CFD scheme (39%). When the smaller spatial grid size (10 m) is adopted, the accuracy is significantly improved, and the relative error is only 0.8%. The shear-wave simulation result using the CCD scheme has high accuracy and can handle the numerical simulation of the seismic wavefield with long sampling times. The simulation accuracy of the CSCD scheme is even higher than the CCD scheme. For CSCD, when the spatial grid size is 15 m, and the time step size is 1 ms, the maximum relative error is only 0.05%, further reducing to 0.0036% when a small spatial step (10 m) is used.