# Optimizing Finite-Difference Operator in Seismic Wave Numerical Modeling
## 3 Numerical Simulation
### Model Test  
We choose the \(N=12\) finite-difference coefficients of the second derivative \(c_{tt}\) for wave field simulation. Take [PERSON]'s [16] data as a baseline, which are denoted as'source'. They proposed a Chebyshev auto-convolution combined window. The four finite-difference coefficients of the second derivative are illustrated in the Table 5.  
Take the wave field simulation of the homogeneous medium first. With the earthquake hypocenter obtained from Equation (15), we choose the snapshot at 150 ms, as shown in Figure 5. Since the homogeneous medium is isotropic, the entire wave field can be known by looking at the graph in only one direction. Therefore, we stitched the four figures together to facilitate the comparison. The operator with less waviness suppressed the numerical dispersion better, and the parameters calculated are better, too. It is observed that PSO is quite similar to the source, while KH is the best to have less wave. KH has significantly suppressed numerical dispersion.  
We take the simulation on the Marmousi model shown in Figure 1. Figure 6 gives four wave field snapshots taken at 75 ms intervals.  
Due to the complexity of the model and the small difference between the optimized parameters, the obtained wave field information is mixed and difficult to distinguish. It is not obvious which one is better. Therefore, we make the single traces figure from the receiver to show the differences. A portion of the trace after the arrival of the main waver is captured and shown in Figure 7. To see the differences more clearly, we zoomed in on the waveform after the arrival of the main frequency. CDPSO has a better suppression of numerical dispersion in the later stages, although the suppression is not as good as the source data in the early stage. The overall trend of PSO is the same as source data, while KH is slightly different from the source in the late stage. It is observed from the wave of about 625 that both KH and PSO have a better suppression effect than the source. KH is much better than CDPSO and slightly better than PSO. The differences caused by parameter optimization become less noticeable as the complexity of the model increases.  
Figure 5: Impulse response of CDPSO, KH, PSO, and source in homogeneous medium at 150 ms.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline  & \(\mathbf{c_{0}}\) & \(\mathbf{c_{1}}\) & \(\mathbf{c_{2}}\) & \(\mathbf{c_{3}}\) & \(\mathbf{c_{4}}\) & \(\mathbf{c_{5}}\) & \(\mathbf{c_{6}}\) \\ \hline Source & \(-3.138521049\) & \(1.854196962\) & \(-0.368305121\) & \(0.110295386\) & \(-0.034584857\) & \(0.009637126\) & \(-0.001978972\) \\ PSO & \(-3.126061573\) & \(1.841954323\) & \(-0.3577484\) & \(0.102288844\) & \(-0.029463646\) & \(0.007056116\) & \(-0.001056452\) \\ CDPSO & \(-2.995218501\) & \(1.722195119\) & \(-0.266916073\) & \(0.047045099\) & \(-0.004391552\) & \(2.83\times 10^{-7}\) & \(-0.000323625\) \\ KH & \(-3.11468956\) & \(1.831319621\) & \(-0.349228563\) & \(0.096527374\) & \(-0.026282834\) & \(0.005735015\) & \(-0.000726135\) \\ \hline \end{tabular}
\end{table}
Table 5: Finite-difference coefficients of the second derivative optimized by PSO, CDPSO, and KH (\(N=12\)).  
In a nutshell, the KH shows the best performance among the three algorithms.  
Figure 6: The Chebyshev auto-convolution windows of CDPSO, KH, and PSO.  
Figure 7: The trace of PSO, CDPSO, KH, and source within 300 ms.