## 3 Numerical Experiments
### Synthetic Example
The MSE values of the inverted Vp results are 0.045136 (FCRN), 0.119093 (MFCRN), 0.034037 (MFCRGRU), and 0.017040 (M2S-MFCRGRU), respectively. The MSE values of the inverted density results are 0.005289 (FCRN), 0.008109 (MFCRN), 0.002761 (MFCRGRU), and 0.002460 (M2S-MFCRGRU), respectively. Noise is inevitable in field seismic data. In this case, different levels of Gaussian white noise are added to synthetic seismic data to evaluate the anti-noise performance of M2S-MFCRGRU. Specifically, signal-to-noise ratios (SNR) of 5 dB, 10 dB, 15 dB, and 20 dB Gaussian white noise are added to clean seismic data. We calculate the MSEs between the inversion results and label data in Table 3. As SNR decre \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multicolumn{2}{c}{**Vp**} & \multicolumn{2}{c}{**Density**} \\ \cline{2-5}  & **MSE \(\downarrow\)** & **PCC \(\uparrow\)** & **MSE \(\downarrow\)** & **PCC \(\uparrow\)** \\ \hline FCRN & 0.022604 & 0.9892 & 0.002810 & 0.9945 \\ MFCRN & 0.021935 & 0.9887 & 0.002595 & 0.9940 \\ MFCRGRU & **0.010300** & **0.9949** & 0.001163 & 0.9972 \\ M2S-MFCRGRU & 0.010717 & 0.9945 & **0.001124** & **0.9973** \\ \hline \hline \end{tabular}
\end{table} 
Table 2: MSEs and PCCs of inversion results for different methods on the Marmousi2 model. \(\downarrow\) means the lower the better and \(\uparrow\) means the higher the better. The best values are in bold. Figure 5: A comparison of the inversion results of FCRN (blue line), MFCRN (green line), MFCRGRU (pink line), and M2S-MFCRGRU (red line) in the 8105 th trace with label values (black line) on the Marmousi2 model. (**a**) Vp. (**b**) Density. method achieves the best inversion performance. It shows that M2S-MFCRGRU has the lowest MSE in all cases. Therefore, although there is a marginal difference in the inversion performance between M2S-MFCRGRU and MFCRGRU in clean seismic data, M2S-MFCRGRU has superior anti-noise performance. This means that M2S-MFCRGRU, using spatial knowledge from seismic data, has the best anti-noise ability and can effectively inverse elastic parameters even with significant noise in the seismic data. We find that FCRN has the second smallest MSE, right after the proposed M2S-MFCRGRU in the high-noise scenario, which shows the importance of GRU and M2S ingredients for the multi-task inversion network. Here, to present a visual comparison, the results from adding Gaussian white noise to 10 dB SNR are presented. The inversion result profiles are shown in Figure 6. Using the Vp inversion results as an example, the outcomes of the M2S-MFCRGRU model demonstrate fewer hanging lines artifacts, enhanced lateral continuity, and superior vertical resolution when compared to the results obtained from the other three models. Comparing the inversion profiles with the label data, M2S-MFCRGRU can deal with the Gaussian white noise well and can maintain the original stratigraphic information of the Marmousi2 model. It shows that the inversion results of M2S-MFCRGRU have the richest details, highest resolution, and least hanging lines. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**MSE \(\downarrow\)** & **SNR (dB)** & **FCRN** & **MFCRN** & **MFCRGRU** & **M2S-MFCRGRU** \\ \hline \multirow{4}{*}{Vp} & 5 & 0.095741 & 0.260494 & 0.195359 & **0.046389** \\  & 10 & 0.040110 & 0.057738 & 0.050924 & **0.018357** \\  & 15 & 0.027561 & 0.030131 & 0.020300 & **0.012805** \\  & 20 & 0.024147 & 0.024260 & 0.013302 & **0.011364** \\ \hline \multirow{4}{*}{Density} & 5 & 0.012879 & 0.018144 & 0.016199 & **0.004482** \\  & 10 & 0.003921 & 0.005268 & 0.004449 & **0.001842** \\ \cline{1-1}  & 15 & 0.003092 & 0.003027 & 0.002056 & **0.001326** \\ \cline{1-1}  & 20 & 0.002887 & 0.002707 & 0.001440 & **0.001182** \\ \hline \hline \end{tabular}
\end{table} 
Table 3: MSEs of inversion results for different methods at different SNR levels on the Marmousi2 model. \(\downarrow\) means the lower the better. The best values are in bold.