## 3 Methodology
### Scale Decomposition  
In the field of FWI, scale decomposition is a strategy designed to mitigate the nonlinearity of the inverse problem by progressively incorporating information from low to high frequencies. The core concept involves breaking down complex data or models into components of varying frequencies or scales. This approach is especially critical for seismic data, as it facilitates more effective inversion by isolating information across different scales, thereby enhancing model accuracy and convergence. In the dual-PIRNN FWI network architecture, scale conversion enables information obtained through forward modeling in the original-scale branch to be utilized in the downscaled branch. To illustrate this, we use the Marmousi velocity model as an example, dividing it into two scales and verifying the correctness of information conversion between them. One model is designated as the original-scale model, and the other as the downscaled model (Figure 4). In this example, the original-scale model has a grid size of \(100\times 100\) with grid spacing \(dx=dz=10\). The source function is a Ricker wavelet with a sampling rate of \(f_{s}=4000\) Hz. The number of time points is \(NT=1600\). For the downscaled model, the grid spacing \(dx\) and \(dz\) are half that of the original-scale model. Consequently, the number of time points \(NT\) required for forward modeling is also half that of the original scale. The source function for the downscaled model uses a sampling rate of \(f_{s}=2000\) Hz. We employ both surface seismic and VSP observation systems for data acquisition. Additional parameter settings are detailed in Table 1.  
In numerical simulations, we employ a Ricker wavelet as the source, with its discrete-time form given by (12):  
\[R[n]=(1-2\pi^{2}f_{p}^{2}(n\Delta t)^{2})e^{-\pi^{2}f_{p}^{2}(n\Delta t)^{2}}. \tag{12}\]  
In this context, \(f_{p}\) is the peak frequency of the Ricker wavelet, \(\Delta t\) is the sampling interval of the Ricker wavelet, and the sampling interval \(\Delta t\) determines the sampling rate of the Ricker wavelet, \(f_{s}=\frac{1}{\Delta t}\). Using the parameter settings in the Table 1. we will \begin{table}
\begin{tabular}{l c c} \hline \hline
**Parameters** & **Original Scale** & **DownScale** \\ \hline Grid size & nz = nx = 100 & nz = nx = 100 \\ Grid spacing & dz = dx = 10 & dz = dx = 5 \\ Number of shots & 11 & 11 \\ Numbers of receivers & 100 & 100 \\ Source sampling rate & 4000 Hz & 2000 Hz \\ Central frequency & 40 Hz & 40 Hz \\ Time step & 0.5 ms & 0.5 ms \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters for original and downscaled model. Figure 3: SGFD RNN operator. demonstrate the intermediate steps of scale conversion according to (9) and (10), using the Marmousi true velocity model and the initial velocity model, as shown in Figure 5.  
We perform forward modeling using the SGFD RNN operator on both the true and initial models shown in Figure 5 under two scale parameter settings specified in Table 1. The VSP shot gathers data for the first shot and is illustrated in Figure 6. According to the variables in (9) and (10), the four-shot-gather records--(a), (b), (c), and (d)--correspond to \(d_{obs}\), \(d_{cal}\), \(d_{Dobs}\) and \(d_{Dcal}\), respectively. As observed from the figure, the shot gathers records at different scales and exhibits structural similarities, enabling the transformation of information from the original scale to the downscaled domain. In practical scenarios, the shot-gather records in (a) and (b) can be directly obtained at the original scale using the SGFD RNN operator, while the data in (d) can be acquired using the operator at the downscaled level, with velocity model parameters converted from the original-scale initial-velocity model. During the inversion process, the data corresponding to (c), representing the actual record at the downscaled level, remain unknown.