## 4 Numerical Examples
### Marmousi Model  
In this experiment, the Marmousi model is employed for inversion using the proposed dual-PIRNN network architecture. The initial Marmousi velocity models are derived from the true Marmousi velocity model by applying varying degrees of Gaussian smoothing. The velocity parameters are treated as trainable parameters within the dual-PIRNN network. The original-scale grid size is \(100\times 100\), with grid spacing \(dz=dx=10\) m. Split-field perfectly matched layer (SPML) absorbing boundaries are implemented around the model domain. The specific velocity models are shown in Figure 12. Figure 12a represents the true Marmousi velocity model at the original scale, while Figure 12b-d display the initial velocity models obtained by applying Gaussian smoothing to Figure 12a with \(\sigma\) values of 5, 10, and 20, respectively. As described in the previous section, the scale-decomposition strategies are categorized into time-scale and spatial-scale reduction. Inversion is performed using the dual-PIRNN FWI network architecture. The inversion parameters and corresponding data scales for the Marmousi model at different scale levels are presented in Table 4. \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Parameters** & **Original Scale** & **DownTime** & **DownTimeSpace** \\ \hline Grid Points & nz = nx = 100 & nz = nx = 100 & nz = nx = 50 \\ Grid Spacing & dz = dx = 10 & dz = dx = 5 & dz = dx = 5 \\ Number of timesteps & 1600 & 800 & 400 \\ Central frequency & 40 Hz & 40 Hz & 40 Hz \\ Source sampling rate & 4000/8000 Hz & 2000 Hz & 2000 Hz \\ Total data scale & \(11\times 100\times 100\times 1400\) & \(11\times 100\times 100\times 800\) & \(11\times 50\times 50\times 400\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Parameters for different scale Marmousi models. Figure 11: Velocity profiles of homogeneous layer model and loss curve. (**a**) V profile at x = 300 m. (b) V profile at x = 700 m.  
Figure 10: Inversion results of homogeneous layer model with different DownScale. (**a**) Inversion with DownTimeScale. (**b**) Inversion with DownTimeSpaceScale. We first conducted inversions on the initial model with \(\sigma=5\), applying both time-scale reduction and combined time-space reduction strategies according to the parameter settings in Table 4. Figure 13a displays the inversion result under the time-scale reduction strategy. This approach accurately identifies several velocity interfaces and structural information in the shallow layers. The resolution of structural details in most deep areas is also relatively high. However, it can be observed in Figure 13a that the inversion effect in the deep regions on the left side of the velocity model is less accurate than on the right. This discrepancy arises because the reflective signals from the velocity structures on the left side of the deep Marmousi model are only partially received by the VSP observation system. In contrast, the reflective signals from the sloped structures on the right are more easily captured, leading to noticeably higher inversion accuracy on the right side. Figure 13b shows the inversion result under the combined time and space-scale reduction strategy. While this approach significantly reduces computational costs by decreasing the number of grid points and time steps (as detailed in Table 3), due to decomposition in both the time and space domains, the resolution is reduced on the coarse grid. This trade-off between computational efficiency and resolution is a key consideration in practical applications. If interpolated back to the original scale, the details could be clearer. However, the overall structure of the Marmousi model is still well-reconstructed.