# Accurate Pseudo-Spectral Acoustic Wave Modelling with Time Dispersion Elimination
## 3 Numerical Examples  
We first verified our method with a homogeneous model. The velocity was set to be 3000 m/s and the density was assumed constant. The spatial grid sampling interval was 12 m in both \(x\) and \(z\) directions. The temporal sampling rate was 2 ms. To illustrate that the present method can achieve high modeling accuracy in both temporal and spatial dimensions, a comparison with the finite difference of tenth order spatial accuracy and second order temporal accuracy was conducted. Figure 2 shows the comparison of records using the finite difference method with a 25 Hz domain frequency wavelet. Figure 2a,c are records without temporal dispersion elimination. It is clear that, although a relatively low frequency wavelet was adopted, the temporal dispersion energy which travels faster than a true seismic wavefield signal can be observed. Figure 2b,d are records of the temporal dispersion elimination using Equation (8). When comparing the enlarged display in Figure 2c,d, we can see the temporal dispersion was clearly removed using Equation (8). The result with the staggered grid-based pseudo-spectral method is displayed in Figure 3. Since the domain frequency of the wavelet is relatively low in this case, spatial dispersion energy is almost invisible and the record calculated with the staggered grid-based pseudo-spectral method is similar to the finite difference-based method. The temporal dispersion is distinct in Figure 3c and effectively eliminated with Equation (8), as shown in Figure 3d. Figure 4 demonstrates the comparison of the analytical solution with a selected trace for different records. The selected trace is at a distance of 1188 m. Due to temporal dispersion, there exists a phase error in both selected traces of the finite difference and staggered grid-based pseudo-spectral methods. After elimination of the temporal dispersion using Equation (8), both traces matched very well with the analytical solution. To highlight the difference between the finite difference and staggered grid-based pseudo-spectral methods, we increased the main frequency of the wavelet to 40 Hz and repeated the experiment on the same homogeneous model. Due to the increase in frequency, in records using the finite difference method, both spatial dispersion energy which travels more slowly, and temporal dispersion energy which travels faster than a true seismic wavefield signal, were observed, as shown in Figure 5a,c. As it is shown in Figure 5c,d, the application of the temporal dispersion operator Equation (8) obviously attenuates the temporal dispersion energy. However, the spatial dispersion is left untouched. In comparison, the staggered grid-based pseudo-spectral method produced almost no spatial dispersion energy, as demonstrated in Figure 6a,c. By adopting the temporal dispersion operator Equation (8), records that are free from both spatial and temporal dispersion can be obtained, as displayed in Figure 6b,d. This is the main advantage of our proposed method since it can generate an accurate wavefield record without both spatial and temporal dispersion errors at the same time. Figure 7 demonstrates the comparison of the analytical solution with selected traces for different records in the case of a 40 Hz wavelet. Severe temporal dispersion contamination can be observed in the results from both the finite difference and staggered grid-based pseudo-spectral methods, as shown in Figure 7a,b. Figure 3: Comparison of records using the staggered grid-based pseudo-spectral method with a 25 Hz domain frequency wavelet. (**a**) Record without temporal dispersion elimination. (**b**) Record with temporal dispersion elimination using Equation (8). (**c**) and (**d**) are an enlarged display of (**a**) and (**b**), respectively. Figure 2: Comparison of records using the finite difference method with a 25 Hz domain frequency wavelet. (**a**) Record without temporal dispersion elimination. (**b**) Record with temporal dispersion elimination using Equation (8).