## 4 Numerical Examples  
This study employs both simple uniform and complex SEG models to perform model calculations. For the purposes of clarity, we represent small, medium, and large wavenumbers using three pairs of (\(\mu\), \(\epsilon\)) values for comparative wavefield simulation. Initially, the simple uniform model is utilized for 2D acoustic SGFD modeling with a velocity \(v\) of 2000 m/s, a time step \(\tau\) of 0.001 s, and an operator length \(M\) of 7 for the three methodologies. The specific model size is 3000 m by 3000 m, with a spatial step size \(\Delta x\) and \(\Delta z\) of 10 m each. The Ricker wavelet function \(R(\mathbf{t})\) is expressed as follows:  
\[R(t)=\Big{\{}1-2[\pi f_{0}(t-t_{0})]^{2}\Big{\}}e^{-[\pi f_{0}(t-t_{0})]^{2}} \tag{23}\]  
where,\(f_{0}\) is the dominant frequency of \(R(t)\). The source is in the middle of the model and \(f_{0}\) is 32 Hz, where\(f_{0}\) represents the dominant frequency of the function \(R(t)\), and \(t_{0}=1/f_{0}\). In this model, the source is centrally located. Figure 3: Variation of the stability factor s with respect to \(M\) for various values of (\(\mu\), \(\epsilon\)) in 2D acoustic SGFD modeling: the small wavenumber range in panel (**a**), the medium wavenumber range in panels (**b**), and the large wavenumber range in panels (**c**). Figure 4 presents snapshots of the 2D acoustic SGFD wavefield computed using the TEM, LSM, and TLM(\(L=1\)) methods for various values of (\(\mu\), \(e\)) at \(t=660\) ms, along with a single-channel recording from a coordinate point. The calculations for these three methods employed the 14-point spatial discretization (\(M=7\)). Figure 4: _Cont._  
Figure 4a-c illustrate the numerical artifacts observed using the TEM, LSM, and \(\text{TLM}(L=1)\) for \(\mu=0.120\) and \(\epsilon=0.072\) within a small wavenumber range. These panels reveal that the LSM-based snapshots exhibit significant numerical artifacts, while those derived from the \(\text{TLM}(L=1)\) display fewer artifacts compared to the TEM approach. Moreover, Figure 4h demonstrates that the numerical accuracy of the \(\text{TLM}(L=1)\)-based recordings at the model coordinates point (1000 m, 1370 m) surpasses that of TEM, with LSM showing the least accuracy. Consequently, within the small wavenumber range, the ability of the \(\text{TLM}(L=1)\) method to suppress numerical dispersion is superior, contrasting with the lesser effectiveness of the LSM method. Figure 4a,d,e present wavefield snapshots obtained using TEM, LSM, and TLM for \(\mu=0.620\) and \(\epsilon=0.499\) in the medium wavenumber range. Figure 4i showcases the single-channel recordings at the model coordinates point (1000 m, 1370 m) using these methods, indicating that the \(\text{TLM}(L=1)\) snapshots exhibit minimal numerical artifacts, and their numerical accuracy remains superior to that of LSM and TEM. Figure 4a,f,g display simulated wavefields for \(\mu=1.220\) and \(\epsilon=0.312\) in the large wavenumber range, and Figure 4j depicts the single-channel recordings at the same model coordinates point. It is evident that both \(\text{TLM}(L=1)\) and LSM produce clean wavefield snapshots with overlapping waveforms in single-channel analysis. Therefore,  
Figure 4: Snapshots at 660 ms computed with a 14-point spatial discretization method (\(M=7\)) for varying values of (\(\mu\), \(\epsilon\)). Panels **(a)** illustrate the results using the TEM, while panels **(b,d,f)** depict outcomes using the LSM, and panels (\(\epsilon\),\(\epsilon\),g)** show the results from the \(\text{TLM}(L=1)\). Panels **(h,i,j)** present the seismograms recorded using different simulation methods at coordinates (1000 m, 1370 m). in the large wavenumber range, TLM(\(L\) = 1) and LSM methods demonstrate the strongest and most consistent suppression of numerical dispersion, whereas TEM exhibits more pronounced dispersion.