## 4 Numerical Examples
### Marmousi Model
Figure 14 presents a further comparison of velocity profiles at \(x=300\) m and \(x=700\) m. Evidently, the downtime scale inversion can accurately predict velocities in both shallow and deep regions, with high precision in the highly folded area at \(x=300\) m. The down time-space approach accurately predicts velocities in shallow regions but performs less accurately in deep areas compared to time-scale reduction. Nonetheless, it still captures the velocity variation trends in deep regions. Figure 12: True velocity model and initial model with different Gaussian smoothing. (**a**) True model. (**b**) Initial model, \(\sigma=5\). (**c**) Initial model, \(\sigma=10\). (**d**) Initial model, \(\sigma=20\). Similarly, we employed a less accurate initial model, as shown in Figure 12c, where the Gaussian smoothing parameter \(\sigma\) is set to 10. Following the same parameter settings and inversion process, we obtained the inversion results presented in Figure 15. Figure 15a displays the inversion result under the time-scale reduction strategy. Compared to the results shown in Figure 13a, the inversion results here are satisfactory even with a less accurate initial model. To further evaluate the inversion performance, we compared the velocity profiles of inversion results obtained from initial models with different degrees of smoothing, taking profiles at \(x=300\) m and \(x=700\) m (Figure 16). In Figure 16a, at x = 300 m, the inversion result from the initial model with \(\sigma=5\) slightly outperforms that of \(\sigma=10\) in both shallow and deep regions; however, the overall difference is minimal. Figure 16b shows that at \(x=700\) m, the inversion results from both models are nearly identical. This demonstrates that the current network architecture possesses a certain degree of robustness and adapts well to different initial models. Similarly, comparable results were observed for the inversion under the time -scale reduction strategy, as shown in Figure 16c,d. The quantitative analysis of inversion results under different scenarios is summarized in Table 5, encompassing both the Homogeneous model with different scale-decomposition strategies and the Marmousi model with varying initial models and scale reduction approaches. Notably, for the Marmousi model, the R2 scores, SSIM, and NCC values remain remarkably consistent across different initial models under the same scale-decomposition strategy. This consistency strongly demonstrates the robustness of our dual-PIRNN network architecture to initial model perturbations. To further investigate the impact of initial model errors on inversion results under our current network architecture, we employed a heavily smoothed velocity model with \(\sigma=20\) as shown in Figure 12d as the initial  
Figure 14: Velocity profiles of inversion result at \(\sigma=5\). (a) V profile at x = 300 m. (b) V profile at x = 700 m.  
Figure 13: Inversion results of Marmousi model with the initial model at \(\sigma=5\). (a) Inversion with DownTimeScale. (b) Inversion with DownTimeSpaceScale. model. Following the same inversion procedure, we found that local minima emerged under the time-scale reduction strategy, as illustrated in Figure 17. This indicates that when the initial model is excessively smoothed, the non-uniqueness of the optimization problem becomes more pronounced, leading to inversion errors. For cases with less accurate initial models, additional prior information should be incorporated to better constrain the inversion process. Figure 16: Inversion velocity profiles for different initial velocity models with Gaussian smoothing. (**a**) Velocity profiles at x = 300 m with downtimescale. (**b**) Velocity profiles at x = 700 m with downtimescale. (**c**) Velocity profiles at x = 300 m with downtimescalescale. (**d**) Velocity profiles at x = 700 m with downtimescalescale. The green dashed line represents the Gaussian smoothness of \(\sigma=5\), the red dashed line represents the Gaussian smoothness of \(\sigma=10\), and the black solid line represents the real model.