# Accurate Pseudo-Spectral Acoustic Wave Modelling with Time Dispersion Elimination
## 3 Numerical Examples
Figure 8: Marmousi model: (**a**) velocity model, (**b**) density model. Figure 7: Comparison of the analytical solution (solid blue line) with selected trace (red dashed line) for different records with a 40 Hz domain frequency wavelet. (**a**) Comparison with the record trace using the finite difference (FD) method without temporal dispersion elimination. (**b**) Comparison with the record trace using the staggered grid-based pseudo-spectral (PS) method without temporal dispersion elimination. (**c**) Comparison with record trace using the finite difference method with temporal dispersion elimination. (**d**) Comparison with record trace using the staggered grid-based pseudo-spectral method with temporal dispersion elimination. Figure 3:0 Comparison of the Marmousi model records using the staggered grid-based pseudo-spectral method with a 40 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 3:1 Comparison of the Marmousi model records using the finite difference method with a 40 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. To illustrate the effectiveness of our proposed method for seismic imaging, reverse time migration (RTM) with different numerical modeling methods was implemented, as shown in Figure 3:2 To reduce computational cost, only a partial model, as in Figure 8, was imaged. The calculation parameter for RTM remained unchanged, as in the numerical simulation examples of Figures 9 and 10. Figure 12a,b are imaging results using the finite difference method without and with temporal dispersion elimination, respectively. Figure 13a,b are partial enlarged displays for Figure 12a,b. Similar to the numerical simulation examples of Figure 9, the RTM stack image profile based on the finite difference method with a 40 Hz wavelet contains strong artifacts due to the spatial numerical dispersion, as demonstrated in Figures 12 and 13. In addition, due to the spatial numerical dispersion, the effectiveness of the temporal dispersion elimination process using operator (8) decreased and became almost invisible when comparing Figure 13a,b. Figure 12c,d are imaging results using the staggered grid-based pseudo-spectral method without and with temporal dispersion elimination, respectively. In comparison with the finite difference-based method, as displayed in Figure 12a,b, the imaging artifacts mainly produced by spatial numerical dispersion are significantly reduced in both image profiles with and without temporal dispersion elimination, as illustrated in Figure 12c,d. The reduction in spatial numerical dispersion with the staggered grid-based pseudo-spectral method also highlights the contribution of the temporal dispersion elimination process using operator (8) when comparing Figure 13c,d. The temporal dispersion elimination reduces wavelet oscillation and enhances the coherence of the seismic event in the deep target geological region, as marked by the red arrows. The imaging comparisons clearly demonstrate that the temporal and spatial numerical dispersions should be simultaneously handled to achieve accurate acoustic simulation and imaging results, otherwise, the temporal dispersion elimination process could be contaminated by spatial numerical dispersion energy. Figure 3:3 Comparison of the reference solution (solid red line) with a selected trace without temporal dispersion elimination (solid blue line) and a trace with temporal dispersion elimination (dashed black line). (**a**) Comparison with record trace using the finite difference (FD) method. (**b**) Comparison with record trace using the staggered grid-based pseudo-spectral (PS) method.