# The Combined Compact Difference Scheme Applied to Shear-Wave Reverse-Time Migration
## 3 Analysis of CCD and CSCD
### Dispersion Analysis  
In numerical simulations, if the size of the spatial grid is too large, it will cause large solution errors and produce numerical dispersion [41; 42]. Therefore, the analysis of the dispersion relationship is not simply an important method to evaluate the advantages and disadvantages of numerical simulation methods, but it is also an important way to determine the size of the spatial grid [43]. \begin{table}
\begin{tabular}{c c c c c c c c} \hline  & \(\alpha\) & \(\beta\) & \(a\) & \(b\) & \(c\) & **Truncation Error** \\ \hline \(\frac{\partial^{2}u}{\partial x^{2}}\) & CFD & 0 & 0 & 3/2 & \(-3/5\) & \(1/10\) & \(\approx 1.78\times 10^{-3}(O(h^{6}))\) \\ \(\frac{\partial^{2}u}{\partial x^{2}}\) & CD & 2/11 & 0 & 12/11 & 3/11 & 0 & \(\approx-4.2163\times 10^{-4}(O(h^{6}))\) \\ \(\frac{\partial^{2}u}{\partial x^{2}}\) & CCD & & / & & & \(-4.9603\times 10^{-5}(O(h^{6}))\) \\ \(\If we use a simple harmonic solution \(f_{i}^{m}=Aexp[I(ihk-kv^{\prime}n\Delta t)]\), we can obtain the following equations for CCD:  
\[\left\{ \begin{array}{l}I[(a_{1}+a_{1})\cos\beta+1]\beta^{\prime}+[(b_{1}+b_{1} )\cos\beta](-1)(\beta^{\prime\prime})^{2}\\ =\sum\limits_{m=1}^{n}[(c_{m}+c_{m})\cos\beta]\\ 2a_{2}(I\sin\beta)\beta^{\prime}+(1-2b_{2}\cos\beta)(-1)(\beta^{\prime\prime })^{2}=\sum\limits_{m=1}^{n}((d_{m}+d_{m})\cos\beta-1)\end{array} \right. \tag{17}\]  
and the following equations for CSCD:  
\[\left\{ \begin{array}{l}(1+\frac{187}{128}\cos(\beta))\beta^{\prime}+\frac{4 7}{128}\sin(\beta)\beta^{\prime\prime}-\frac{6}{128}\cos(\beta)\beta^{\prime \prime\prime}-\frac{2}{768}\sin(\beta)\beta^{(4)}=\frac{315}{128}\sin(\beta) \\ -\frac{325}{64}\sin(\beta)\beta^{\prime}+(\frac{69}{64}\cos(\beta)+1)\beta^{ \prime\prime}+\frac{11}{96}\sin(\beta)\beta^{\prime\prime\prime}-\frac{1}{192 }\cos(\beta)\beta^{(4)}=10(\cos(\beta)-1)\\ \frac{315}{16}\cos(\beta)\beta^{\prime}-\frac{105}{16}\sin(\beta)\beta^{\prime \prime}+(1+\cos(\beta))\beta^{\prime\prime\prime}+\frac{1}{16}\sin(\beta) \beta^{(4)}=-\frac{315}{16}\sin(\beta)\\ \frac{645}{4}\sin(\beta)\beta^{\prime}-\frac{165}{4}\cos(\beta)\beta^{\prime \prime}-5\sin(\beta)\beta^{\prime\prime\prime}+\frac{1}{4}(\cos(\beta)+1) \beta^{(4)}=-240(\cos(\beta)-1)\end{array} \right. \tag{18}\]  
where \(\beta=hk,\beta^{\prime}=hk^{\prime},\beta^{\prime\prime}=hk^{\prime},\beta^{ \prime\prime\prime}=hk^{(3)},\beta^{(4)}=hk^{(4)}\) Solving the equations of CCD gives,  
\[\beta^{\prime\prime}_{CCD}=\sqrt{\frac{81-48\cos\beta-33\cos 2\beta}{48+40\cos \beta+2\cos 2\beta}} \tag{19}\]  
In this equation, \(\beta=kh,k\) is the wavenumber, \(h\) is the spatial gird size, and \(\beta^{\prime\prime}_{CCD}\) is the second-order modified wavenumber of the CCD scheme. If this is substituted into the two-dimensional shear-wave equation, Equation (20) becomes  
\[\beta^{\prime\prime}_{CCD(x)}=\sqrt{\frac{81-48\cos\beta_{x}-33\cos 2\beta_{x }}{48+40\cos\beta_{x}+2\cos 2\beta_{x}}}\beta^{\prime\prime}_{CCD(z)}=\sqrt{\frac{81-48 \cos\beta_{z}-33\cos 2\beta_{z}}{48+40\cos\beta_{z}+2\cos 2\beta_{z}}} \tag{20}\]  
where \(\beta_{x}=\beta cos\theta,\beta_{z}=\beta sin\theta,\) and \(\theta\) is the angle between the wave's propagation direction and the \(x\)-axis. The modified wavenumber of the corresponding derivatives of CSCD can be solved by substituting the numerical solution into the Equation Group (18) obtained by the simple harmonic solution without writing the corresponding analytical expression. The modified wavenumber solution of the finite difference scheme can then be substituted into the finite difference scheme of the shear-wave equation, yielding. \[ \begin{array}{l}2\cos(kv_{num}\Delta t)=2-(v\Delta t/h)^{2}\Big{[}(1+2\gamma )\big{(}\beta^{\prime\prime}_{x}\big{)}^{2}+\big{(}\beta^{\prime\prime}_{z} \big{)}^{2}\Big{]}\\ +(v\Delta t/h)^{4}\Big{[}\big{(}1+2\gamma\big{)}^{2}\big{(}\beta^{\prime\prime }_{x}\big{)}^{4}+2(1+2\gamma)\big{(}\beta^{\prime\prime}_{x}\beta^{\prime \prime}_{z}\big{)}^{2}+\big{(}\beta^{\prime\prime}_{z}\big{)}^{4}\Big{]}/12\end{array}