# Accurate Pseudo-Spectral Acoustic Wave Modelling with Time Dispersion Elimination
## 3 Numerical Examples
(**c**) and (**d**) are an enlarged display of (**a**) and (**b**), respectively. Figure 4: Comparison of analytical solution (solid blue line) with selected trace (red dashed line) for different records with a 25 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 the record trace using the finite difference method with temporal dispersion elimination. (**d**) Comparison with the record trace using the staggered grid-based pseudo-spectral method with temporal dispersion elimination. Figure 5: Comparison of 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 verify that our method can be applied to complex media, the Marmousi synthetic model was used, as shown in Figure 8. Figure 9 shows a comparison of records using the finite difference method with a 40 Hz domain frequency wavelet. The spatial grid sampling interval was 12 m in both the \(x\) and \(z\) directions. The temporal sampling rate was 1 ms. Figure 9a,c are records without temporal dispersion elimination. Figure 9b,d are records with temporal dispersion elimination using Equation (8). Figure 10 is results using the staggered grid-based pseudo-spectral method. At first glance, the spatial dispersion reduction using the staggered grid-based pseudo-spectral method can be seen when comparing Figures 9 and 10. To conduct a detailed investigation into temporal dispersion elimination, comparisons of reference solutions with selected traces for different records in the Marmousi model were implemented, as shown in Figure 11. The reference solution was obtained with the staggered grid-based pseudo-spectral method with a 10 times smaller temporal sampling rate than in Figures 9 and 10 to minimize the temporal dispersion. The strong spatial numerical dispersion energy obviously affected the effectiveness of the temporal dispersion elimination operator (8), as demonstrated in Figure 11a, as there exist both spatial and temporal dispersion residuals when compared with the reference solution. As a comparison, since almost no spatial numerical dispersion exists in the record using the staggered grid-based pseudo-spectral method, the temporal dispersion elimination operator (8) almost fully removed the temporal dispersion, resulting in a record trace free from both spatial and temporal numerical dispersions, which fits very well with the reference solution as listed in Figure 11b. The numerical results from the Marmousi model clearly show that the temporal dispersion elimination process can be easily contaminated by spatial numerical dispersion in complex wavefields. To obtain an accurate acoustic simulation result, the temporal and spatial numerical dispersions should be simultaneously handled, as proposed in our example. For the complex Marmousi model, we ran the wavefield simulation for 3057 temporal discretization steps, with both the conventional finite difference and proposed methods, and the computational times were 117.36 s and 145.71 s, respectively. The computational cost with our present method increased only by  
Figure 6: Comparison of 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. approximately twenty percent. This comparison implies that the proposed method is able to produce higher simulation accuracy with tolerable computational cost.