## 2 Theory and Method
### Deep Learning Algorithms  
Based on the above problems in the reconstruction process of non-extensive passive source distributions, we hope to restore virtual source records that conform to physical rules, have good continuity, weak coherent noise, and do not contain spurious axes. Here, we propose to use the UNet network for imperfect virtual shot records. Since the direct wave energy in the virtual shot records is strong and the reflected wave energy is weak, we introduced attentional gating (AG) [33] to add multiple attentional layers to the UNet, allowing it to focus more of its learning on the reconstruction of the reflected waves in the lower part of the record. The AG, first proposed in the medical field, aims to mimic the human attention mechanism by targeting and focusing on salient features in the data, which can make the model more efficient. It enables local regions to receive special attention, automatically learning to focus on the structural features of the target, suppressing irrelevant regions of the input image, and highlighting features useful for a specific task. Validated on a dataset in the medical domain, the results obtained show that not only can it be integrated into a network model with minimal computational overhead, but it will also improve the flexibility and predictive performance of the network. Here, we adopt a lightweight CBAM attention mechanism [34] to make the network more capable of learning. When the network transforms a graph into a feature map, it can simultaneously generate specific feature map information on the channel and spatially. It also performs adaptive operations with the incoming feature maps and finally outputs the feature maps after the action of the attention mechanism. We drew inspiration from residual thinking in ResNet and added residual connections to the network to achieve optimal performance and reduce training difficulty. The network we used is shown in Figure 1.  
We used virtual shot records with fewer sources and unevenly distributed sources as training datasets, respectively. We used virtual shot records with more sources and randomly distributed in the subsurface as labels. We normalized the virtual shot records with the following equation:  
\[x=\frac{x-x_{min}}{x_{max}-x_{min}}\]  
where \(x\) represents the value of a point in the virtual shot record; \(x_{min}\) represents the minimum value in that record; and \(x_{max}\) represents the maximum value in that record. Here, we did not normalize the data using the extreme values in the whole data. This is to prevent anomalies in signal strength for some records. We also did not normalize the data using the usual per-track extremes. This is to prevent a blind-like imbalance in the processed data due to differences between seismic traces. We use single record normalization, using the maxima and minima of each virtual shot record to normalize, ensuring high contrast and continuity of pixel points horizontally for each record. After preprocessing the data, we can then feed the processed dataset into our network. The optimizer is Adam; the activation function is ReLU except for the last layer, which is sigmoid; and the loss function is MSE:  
\[loss(y_{1},y_{2})=\frac{1}{N}\sum_{i=1}^{N}(y_{2}-y_{1})^{2}\]  
Figure 1: Network structure of improved U-net. where \(N\) represents the sum of the number of pixels in a single seismic record; \(i\) represents the number of elements processed by the neural network; and \(y_{2}\) and \(y_{1}\) represent the predicted and target values (labels) of the network, respectively. In general, there are some difficulties in using passive seismic reconstruction records as a dataset. If the active seismic record is chosen as the label, the valuable low frequency information in the passive source data will be lost. If the more effective virtual shot records are chosen as labels, some noise will inevitably be generated. In this paper, we believe that it is more important to protect the low-frequency signal.