## 3 Numerical Experiments
### Synthetic Example  
The Marmousi2 model has a complex stratigraphic structure, which includes faults, interfaces, and strong lateral and vertical variations; it is representative of a geophysical model [47]. Synthetic seismic data are generated using the zero-phase Ricker wavelet with a main frequency of 30 Hz. The synthetic seismic data, Vp, and density are shown in Figure 3. The parameters of synthetic data are set as follows: the trace number is 13,601, the sample number per trace is 2800, and the time sampling interval is 1 ms. For the experiments, 101 data pairs of equidistantly sampled traces are utilized for training, 34 pairs for validation, and the complete profile for testing. The training epoch is set to 1000. The inversion results of Vp and density are shown in Figure 4. For both Vp and density inversion results, the red rectangles in inversion profiles show that the inversion details of FCRN and MFCRN are insufficient with obvious hanging lines and rough structures, while the layer interfaces of MFCRGRU and M2S-MFCRGRU are sharp with better horizontal continuity. Using the M2S strategy, the geological structures of the inversion results are clearer and closer to the geological structures of the labeled data. Moreover, when inverting the structure to the gas-charged sand channel indicated by the arrows, the inversion results of M2S-MFCRGRU accurately capture the important details. The entire inversion profiles of M2S-MFCRGRU are very close to label data. Therefore, M2S-MFCRGRU can deal with complex stratigraphic structures, like shallow horizontal layers and steep faults, achieving the best inversion performance. \begin{table}
\begin{tabular}{c c} \hline \hline
**Training Parameter** & **Value** \\ \hline Learning rate & 0.001 \\ Weight decay rate & \(1\times 10^{-7}\) \\ Batch size & \(12\) \\ Optimization algorithm & Adam \\ Network weight initialization & Kaiming initialization \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The parameters used in the training stage. To assess the inversion performance of the proposed method, quantitative evaluations are conducted for all the results. The evaluation results, utilizing MSE and PCC metrics, are summarized in Table 2. From the computation results, the GRU-adds networks that have lower MSE and higher PCC, which means GRU can significantly improve the network learning ability. The operation of adding GRU into FCRN is reasonable and significant. For density inversion, M2S-MFCRGRU has the smallest MSE (0.001124) and the highest PCC (0.9973). For Vp inversion, we find that the difference in quantitative evaluations between MFCRGRU and M2S-MFCRGRU is not significant for the noise-free data. In particular, although it can be seen that MFCRGRU has the best MSE (0.010300) and PCC (0.0049), it is only 0.000417 smaller than the MSE (0.010717) of M2S-MFCRGRU and it is only 0.000417. Figure 4: Inversion results of the Vp and density on the Marmousi2 model without noise. Top row: inverted Vp of FCRN (**a**), MFCRN (**b**), MFCRGRU (**c**), M2S-MFCRGRU (**d**); Bottom row: inverted density of FCRN (**e**), MFCRN (**f**), MFCRGRU (**g**), M2S-MFCRGRU (**h**). Figure 3: Marmousi2 model. (**a**) Synthetic seismic data. (**b**) Vp. (**c**) Density. 0.0004 higher than the PCC (0.9945) of M2S-MFCRGRU. In view of the lateral continuity of the inversion profile, the slight loss in accuracy is acceptable. To further compare the inversion results of elastic parameters, we extracted the 8105 th trace to showcase the differences in the outcomes produced by the aforementioned four networks relative to the labeled values (Figure 5). Figure 5a,b show the comparison between the Vp and density, respectively. The overall trend of the inversion results of M2S-MFCRGRU (the red line) aligns more closely with the label data (the black line) compared to the results from FCRN (the blue line), MFCRN (the green line), and MFCRGRU (the pink line). In addition, we calculate the MSE between the label data and the inversion results of FCRN, MFCRN, MFCRGRU, and M2S-MFCRGRU on the 8105 th trace.