## 5 Conclusions  
In this work, the proposed method, M2S-MFCRGRU, is designed to simultaneously invert Vp and density by leveraging the advantages of the hybrid network and multi-task learning. The hybrid network captures both local information and time-related information, while multi-task learning effectively exploits the correlation between elastic parameters. Furthermore, the M2S strategy incorporates spatial information from seismic data, thereby enhancing the stability and reliability of the inversion results. Moreover, we utilize data augmentation and transfer learning to overcome the challenge of insufficient field data labels and improve the generalization capability.  
Experimental results on both the synthetic Marmousi2 model and field data demonstrate that the M2S-MFCRGRU method exhibits superior horizontal continuity and vertical resolution. Moreover, when compared to well data in the field example, the proposed method shows great consistent performance. In addition, the anti-noise experiment reveals  
\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}  & **NRMSE \(\downarrow\)** & **PCC \(\uparrow\)** & **NRMSE \(\downarrow\)** & **PCC \(\uparrow\)** \\ \hline FCRN & 0.154176 & 0.7106 & 0.250259 & 0.5180 \\ MFCRN & 0.124208 & 0.7976 & 0.146003 & 0.8130 \\ MFCRGRU & 0.127864 & 0.7577 & 0.128573 & 0.8816 \\ M2S-MFCRGRU & **0.120695** & **0.8036** & **0.094228** & **0.9292** \\ \hline \hline \end{tabular}
\end{table}
Table 5: NRMSEs and PCCs of inversion results for different methods at blind well w5. \(\downarrow\) means the lower the better and \(\uparrow\) means the higher the better. The best values are in bold.  
that M2S-MFCRGRU has the strongest noise immunity, while the blind well test confirms its highest generalization capability. Consequently, despite the limited availability of label data, our proposed M2S-MFCRGRU method yields promising inversion results for elastic parameters.  
Conceptualization, Q.Z., H.R. and B.W.; methodology, Q.Z., H.R. and B.W.; formal analysis, Q.Z. and B.W.; funding acquisition, C.W. and X.Y.; investigation, Q.Z.; data curation, Q.Z.; visualization, B.W.; supervision, C.W., X.Y. and B.W.; writing--original draft preparation, Q.Z. All authors have read and agreed to the published version of the manuscript.  
This research was funded by Key Laboratory of Geophysics, PetroChina.  
Not applicable.  
Data Availability StatementData are available upon request due to restrictions, e.g., privacy or ethical restrictions.  
The authors declare no conflict of interest.  
The following abbreviations are used in this manuscript:  
\begin{tabular}{l l} FCRN & fully convolutional residual network \\ GRU & gated recurrent unit network \\ M2S & multi-trace to single-trace \\ M2S-MFCRGRU & M2S multi-task FCRN and GRU hybrid network \\ Vp & P-wave velocity \\ DL & deep learning \\ ANNs & artificial neural networks \\ DNNs & deep neural networks \\ CNNs & convolutional neural networks \\ FCNs & fully convolutional neural networks \\ RNNs & recurrent neural networks \\ LSTM & long short-term memory \\ BiLSTM & bidirectional LSTM \\ BiGRU & bidirectional GRU \\ DCGRU & deep convolutional GRU \\ BN & batch normalization \\ ReLU & rectified linear unit \\ MSE & mean squared error \\ NRMSE & normalized root mean squared error \\ PCC & Pearson correlation coefficient \\ MFCRN & multi-task FCRN \\ MFCRRU & multi-task FCRN and GRU hybrid network \\ SNR & signal-to-noise ratio \\ \end{tabular}