## 4 Numerical Examples
### Homogeneous Layer Model  
In this experiment, we conduct FWI on a homogeneous layer model and the Marmousi model using the proposed dual-PIRNN network architecture. The Adam optimizer is employed with a learning rate of 40. The time step \(dt\) for finite-difference calculations is set to 0.0005 s, while the specific scale of each model determines the grid spacing and the number of grid points. The forward modeling component of the PIRNN network generates observed shot-gather records. Based on the grid parameters, 11 sources are positioned equidistantly along the top of the model to simulate synthetic shot-gather data. The receivers for the VSP observation system are arranged in a vertical array at the center of the model. The actual model and initial model for the homogeneous layer are shown in Figure 9. The original-scale velocity model has a grid size of \(100\ \times 100\), with grid spacing \(dz=dx=10\). The three layers have velocities of [2500, 3000, 3500] m/s, respectively. The initial model is uniformly shifted by 200 m/s from the actual model, resulting in velocities of [2700, 3200, 3700] m/s. Following the dual-PIRNN network architecture, we employ \begin{table}
\begin{tabular}{l c} \hline
**Parameters** & **Forward Modeling** \\ \hline Finite-Difference parameter & nz = nx = 200, Order = 4/10 \\ PML Width & 20 \\ Width of velocity model & 2000 m \\ Depth of velocity model & 2000 m \\ Central frequency & 20/40 Hz \\ Source Location & Center of model \\ Velocity value & 3000 m/s \\ \hline \end{tabular}
\end{table} 
Table 2: Parameters for forward modeling. two scale-decomposition strategies: one that decomposes only in time and another that decomposes in both time and space. Table 3 presents the inversion parameters and corresponding data scales across different model scales. Inversion tests were conducted using both scale-decomposition strategies within the dual-PIRNN FWI network architecture. The results are presented in Figures 10 and 11. To comprehensively evaluate the inversion performance, we employ three quantitative metrics: R-squared score (R2), Structural Similarity Index (SSIM), and Normalized Cross-Correlation (NCC). The R2 score measures the linear correlation between inverted and true models, SSIM evaluates structural similarity, and NCC assesses overall similarity. Figure 10a shows the inversion results using the time-scale reduction strategy, while Figure 10b displays the results using both time- and spatial-scale reduction. From these results, we observe that the time-scale reduction strategy in Figure 10a produces striped noise in the inversion results. This is attributed to limited source-receiver coverage, leading to poor resolution in certain directions. Additionally, the reduced time-scale results in a lack of high-frequency information in some areas of the model, manifesting as rapid changes and stripe artifacts. Although scale decomposition may lead to some loss of accuracy in the inversion results, a potential improvement is to incorporate a velocity smoothness regularization term during the inversion process. In Figure 10b, the inversion is performed on a smaller model grid by reducing both time and spatial scales, which mitigates local minima and results in a more stable inversion process. Furthermore, due to the reduced number of grid points and time steps required for numerical simulation, as shown by the data scale comparison in Table 3, this approach significantly decreases GPU resource consumption and improves computational efficiency during the inversion process.