## 3 Numerical Example  
### Data Organization  
We choose two different velocity models and generate three different frequency band-slice data sets as the data basis for this experiment to verify the generalization of the network in different situations. Namely, the randomly intercepted velocity sub-model (Figure 2a), the Marmous II model (Figure 2b), and the four-layer layered model (Figure 2c). Each velocity model was configured with a uniform grid size of 101 \(\times\) 128, maintaining a grid spacing of 8 m in both horizontal and vertical directions. The finite difference method is used to perform forward modeling on each velocity model, and the Ricker wavelet with a dominant frequency of 20 Hz is selected as the source function. During the forward modeling process, the total recording time is set to 1.2 s, the sampling interval is 0.001 s, and a perfectly matched layer boundary condition of 60 grid point thickness is set. We construct the labeled dataset with full-band seismic records. We generate three different frequency band-slice data as the sample data sets. For the first sample data set, we apply a band-pass filter between 15 Hz and 25 Hz to the full-band data to simulate the band-slice seismic records. Subsequently, we adjust the amplitude in different parts of the first sample data set to generate second and third sample data sets. A total of 1280 seismic records are received, and we randomly select 70% of the entire data set as the training set and the remaining 30% as the test set. Our computing device is NVIDIA GeForce RTX 3080, and our neural network framework is TensorFlow-GPU 2.6.0 with 10 GB of video memory. We built two network models: the BE-Transformer network and the CNN network. During the training process, the Adam algorithm is used to optimize the objective function, and the activation function is Sigmoid. The network models were trained with the same hyperparameters to ensure a fair comparison. The entire training process was carried out for 600 iterations, the learning rate was set to 0.001, the number of training data in each batch was 32, and the inputs and outputs of both networks were 1200 \(\times\) 128. This setting provides a basis for subsequent model evaluation and comparison. In deep learning, loss function is an important concept used to measure the difference between the model predictions and the real data. The predictive accuracy and generalization of the model are improved by minimizing this difference. The loss function is Mean Square Error (MSE), and the formula is:  
\[\text{MSE\_Loss}=\frac{1}{n}\sum_{i=1}^{n}(x_{i}-f(x_{i}))^{2} \tag{6}\]  
In the formula, \(x_{i}\) is the real data, \(f(x_{i})\) is the prediction result of the neural network, and n is the size of the network batch. The loss function curve during the BE-Transformer network training process is shown in Figure 3. The loss curve decreases rapidly at the beginning of the training and gradually stabilizes as the number of iterations increases. Figure 2: Velocity model (**a**) Marmousi II sub-model; (**b**) Marmousi II model; (**c**) Four-layer velocity model. We used the Pearson correlation coefficient as the evaluation index when evaluating the accuracy of the prediction model. The Pearson correlation coefficient measures the linear relationship between two variables, with values between \(-1\) and \(1\). Here, we use it to evaluate the similarity between the predicted output and the expected data. Its expression is:  
\[\mathrm{R}^{2}=\frac{(\sum_{i=1}^{n}(x_{i}-\overline{x})(y_{i}-\overline{y}))^{ 2}}{\sum_{i=1}^{n}(x_{i}-\overline{x})^{2}\sum_{i=1}^{n}(y_{i}-\overline{y})^{ 2}} \tag{7}\]  
In the formula, \(x_{i}\) and \(y_{i}\) (\(i=1,2,\ldots,n\)) are the predicted output and expected data, respectively, and \(\overline{x}\) and \(\overline{y}\) are the mean values of the predicted output and expected data, respectively. Figure 4 shows the Pearson correlation coefficient diagram of the BE-Transformer training process.