## 3 Numerical Experiments
### Field Example  
We tested on field seismic data for elastic parameter inversion to further verify the superiority and generality of M2S-MFCRGRU. The field seismic dataset comprises dimensions of 1501 (time) \(\times\) 219 (crossline) \(\times\) 219 (inline), with a time sampling interval of 2 ms. The target layer is the turbidite sedimentary layer, which ranges from 2244 to 2494 ms with 125 time-sampling points. There are only six available well logs in the field seismic dataset, and the log data include the corresponding \(\mathrm{Vp}\) and density. Figure 7 shows the field seismic data. Figure 7a is a time slice, which shows the locations of the crosswell profile and six wells (w1-w6). The seismic data of the crosswell profile and the locations of the six wells are shown in Figure 7b. In order to address the scarcity of label field data and achieve comprehensive training of the network, we utilize data augmentation and transfer learning in the training stage. By incorporating these strategies, accurate and reliable inversion results are obtained. In the pre-training stage, we employ the locally weighted method to enhance the training dataset by interpolating the logging data. This process enables us to obtain interpolated volume for \(\mathrm{Vp}\) and density. Subsequently, we generate synthetic seismic data based on the convolution model, resulting in a total of 12,100 traces. Then, along the crosswell profile, we select and designate 185 traces as the pre-training dataset. Figure 8 shows the interpolated \(\mathrm{Vp}\), interpolated density, and synthetic seismic data of the crosswell profile, respectively. Excluding the 6 wells, we utilize 143 traces (80% of the dataset) to train the network, while the remaining 36 traces (20% of the dataset) are dedicated to validation. The pre-training epoch is set to 2500. In the fine-tuning stage, we select twenty-four seismic data traces from the vicinity of each of the six wells to create the fine-tuning dataset. This dataset comprises a total of 144 seismic data traces. Based on the high likelihood that the interpolated data near the wells closely correspond to the label data, this approach ensures accurate and reliable results. The locations of the selected seismic traces are depicted in Figure 9a. The interpolated seismic data of the crosswell profile are shown in Figure 9b. It is imperative to note that the six wells are deliberately excluded from the dataset. This is to ensure a fair evaluation and avoid potential data leakage. Subsequently, we partition the dataset into two distinct subsets, one for training (115 traces) and the other for validation (29 traces). The pre-trained model is used as the initial model for fine-tuning. In this way, the efficient initialization expedites the network convergence and effectively leverages its knowledge of similar features. The fine-training epoch is 2000. The overall fine-tuning procedure with transfer learning for elastic parameter inversion is summarized in Algorithm 1. Figure 7: The field seismic data. (**a**) A time slice. The pink line is the location of the crosswell profile and the red diamonds are the locations of six wells (w1–w6). (**b**) The seismic data of the crosswell profile (dash-dot vertical lines indicate the location of wells). **Input:** Selected interpolated seismic data, label Vp, label density, training epoch \(N\), early stopping patience \(P\);  
```
1:Initial model: the pre-trained model;
2:for epoch= 1:N do
3: Train and update the model parameters with early stopping criterion;
4: Save the best network parameters;
5:endfor
6:Using the saved network to predict elastic parameters;
7:Inverted Vp and density elastic parameters. ```  
**Algorithm 1** The fine-tuning procedure with transfer learning for elastic parameter inversion. Figure 8: The datasets used in the pre-training stage. (**a**) Interpolated Vp. (**b**) Interpolated density. (**c**) Synthetic seismic data using the convolution model for the crosswell profile. Figure 9: The datasets used in the fine-tuning stage.