## 3 Experiments
### Experiment on the Marmousi2 Model  
The impedance of the Marmousi2 model and its corresponding synthetic data are the same as in [23], and are shown in Figure 2. For this model, 101 traces were selected from the synthetic seismic data and impedance data as the training set through isometric sampling, and 1350 traces were randomly selected from the remaining 13,500 traces as the validation set and the whole 13,601 traces comprised the test set. The Adam optimization method was adopted in this paper. The weight decay rate was set to \(1\times 10^{-7}\). The learning rate was set to 0.001. The number of epochs was set to 50, and the batch size was set to 10. The above hyperparameters and network structure were decided from an ablation study by adjusting the convergence of the training set and the validation set. The training of the network was implemented under the PyTorch framework, and a GPU was applied to accelerate the calculation.  
The validation Mean Squared Error (MSE) of the two tasks with different weights is shown in Table 1. The training/validation procedure was executed five times and the average results were output for comparison. The performance of the model for a single task is shown in the first and last rows. When the validation loss is minimal during the 50 epochs, the corresponding \(\sigma_{pre}\) = 0.890 and \(\sigma_{rec}\) = 0.759, then we can calculate that the optimal weight for impedance prediction and seismic data reconstruction is 0.421:0.579 by Equations (5) and (6). Figure 3 shows the variation at different epochs of \(\sigma_{pre},\ \sigma_{rec}\) and the corresponding \(w_{pre},\ w_{rec}\). Under this optimal weight, the validation MSE of the two task is 0.0365 and 0.00054. Compared with results in Table 1, it can be seen that the MSEs of the multi-task network of the two tasks under different weights are smaller than that of the single-task network, and the optimal weight determined by the proposed method makes the MSE smaller for both tasks, which proves the effectiveness of the method. The training and validation loss curves of different tasks under the optimal weight are shown in Figure 4. Figure 5 illustrates the profiles predicted by the single-task network (Figure 5a) and the hard parameter sharing multi-task network (Figure 5b) with the optimal weight for visual comparison. The MSEs of the profiles predicted by the single-task network and the hard parameter sharing multi-task network are 0.0478 and 0.0353, respectively, which proves the advantage of the hard parameter sharing multi-task network from a quantitative perspective.  
Figure 2: Marmousi2 model dataset: (**a**) impedance; (**b**) synthetic seismic data generated by 30 Hz 0\({}^{\circ}\) phase Ricker wavelet.  
To further demonstrate the effectiveness of the proposed method, Figure 6 shows the results of impedance prediction for the 731 st (Figure 6a) and 8991 st (Figure 6b) trace data points. We can see that the impedance predicted by the hard parameter sharing multi-task network (green) matches the true impedance (red) better than that of the single-task network (blue). The blue and green dotted lines in Figure 6 represent residuals between the true impedance and the impedance predicted by the two networks. Table 2 shows the Pearson Correlation Coefficient (PCC) between the predicted value and the truth. Under the optimal weights, the PCC between the predicted value and the ground truth of the 731 st and 8991 st traces of the multi-task network (third column) are both higher. We also tested the tolerance of the two networks to noise. We added six levels of Gaussian noise with a signal-to-noise ratio (SNR) of 0, 5, 15, 25, 35, and 45 dB to the test dataset. The MSEs between the prediction of different SNR data and the true impedance are presented in Table 3.