## 4 Numerical Examples
In summary, simple homogeneous 2D SGFD modeling results indicate that the TLM(\(L\) = 1)-based approach maintains fewer numerical artifacts across the full wavenumber range than the TEM and LSM methods. To further highlight the benefits of the TLM(\(L\) = 1) approach, it was tested on a 2D SEG/EAGE BP salt model by comparing the snapshots and single-channel waveform records. Figure 5 displays the velocity model of the 2D SEG/EAGE BP salt. The BP salt model delineates several critical parameters: acoustic velocities range from 1486 to 4790 m/s, the grid features a consistent mesh size of 20 m in both \(x\) and \(y\) directions, and the dominant frequency of the Ricker wavelet is established at 14 Hz. The source of generated vibrations is fixed at the coordinate point (8000,1). A time step of 1 ms was utilized, with seismograms recorded at the surface of the model. To ensure comparability and simplicity in our analyses, each of the three simulation schemes employed the same SGFD operator length of \(M\) = 7, although the values for (\(\mu\), \(\epsilon\)) varied across the full wavenumber range. According to Figure 6, which displays snapshots at the maximum simulation time of 4 s, the findings are as follows:  
1. In the small wavenumber range, the SGFD based on TLM(\(L\) = 1) demonstrates significantly lesser numerical dispersion compared to the TEM and LSM methods, with TEM showing higher numerical accuracy than LSM (Figure 6a-c). 2. In the medium wavenumber range, the TLM(\(L\) = 1) SGFD still exhibits markedly lower numerical dispersion relative to both TEM and LSM, with LSM achieving better numerical accuracy than TEM (Figure 6a,d,e). 3. In the wide wavenumber range, the numerical accuracy of the TEM-based SGFD with \(M\) = 7 is substantially inferior to those of the LSM- and TLM(\(L\) = 1)-based schemes. Furthermore, wavefield snapshots from LSM and TLM(\(L\) = 1) appear indistinct (Figure 6a,f,g). A comparative analysis was further conducted by modeling single-trace records, which highlighted the performance distinctions among the methods. Figure 7 shows the waveforms at surface positions \(x\) = 5000 and \(x\) = 10,000 for different values of (\(\mu\), \(\epsilon\)). In the small and medium wavenumber range, the TLM(\(L\) = 1)-based scheme consistently exhibits the most robust capability to suppress numerical artifacts, with LSM displaying the least accuracy. Additionally, in the large wavenumber range, the waveforms of the LSM and  
Figure 5: Two-dimensional SEG/EAGE BP salt velocity model. TLM(\(L=1\)) methods overlap, with both methods showing significantly fewer numerical artifacts than the TEM (Figure 7e,f). These results clearly demonstrate that while the TEM-based scheme can effectively suppress numerical artifacts within the small wavenumber range, it is less suitable for broader applications. Conversely, our proposed TLM(\(L=1\)) method shows the strongest suppression of numerical dispersion across all wavenumber ranges, thereby confirming its superior effectiveness in minimizing numerical artifacts throughout the full wavenumber range. Figure 6: Cont. Figure 6: Snapshots from 2D acoustic SGFD modeling using heterogeneous models delineate the performance of three different simulation schemes for different values of (\(\mu\), \(\epsilon\)) with \(M\) = 7: TEM-based (**a**), LSM-based (**b**,**d**,**f**), and TLM(\(L\) = 1)-based (**c**,**e**,**g**). Figure 7: _Cont._