## 2 Theory and Method
### Network Structure  
The Transformer model was proposed by the Google team in June 2017 by [PERSON] et al. in the paper \"Attention Is All You Need\" [28], and the proposal of this model has had a significant impact in the field of Natural Language Processing (NLP). Unlike traditional recurrent neural networks, the Transformer model abandons the sequential structure and, instead, adopts the mechanism called self-attention, which allows the model to be trained in parallel and improves the training speed. In addition, the self-attention mechanism also allows the model to consider global information when processing the input data, further improving the model's ability to understand and generate text [29; 30; 31]. The BiGRU is an improved recurrent neural network structure that combines the bidirectional recurrent neural network (RNN) and the gated recurrent unit (GRU) [32; 33]. It aims to solve the gradient vanishing or gradient explosion problems that are prone to occur in traditional RNNs when processing long sequences. It consists of two independent GRU layers, one processing the sequence along the forward direction and the other processing the reverse direction. The bidirectional structure allows BiGRU to capture long-distance dependencies in sequences, thus improving the model's performance. To achieve the acquisition and reconstruction of signals in any arbitrary lack of bands of seismic wavefields, the paper proposes the BiGRU-Extended Transformer (hereafter referred to as BE-Transformer) network.  
The study adopts a hybrid architecture of the extended version of Transformer and BiGRU. The introduction of multi-head attention (MHA) allows the model to simultaneously focus on different parts of the input sequence in different heads, thereby enhancing the model's expressiveness and ability to capture complex dependencies. The introduction of Scaled Dot-Product Attention allows the model to learn dependencies in the input sequence based on the similarity between various positions through scaled dot-product operations and weighted sum operations, thereby better capturing semantic information. Position Encoding introduces position information into the embedding vector of sequence data, allowing the model to maintain long sequence dependencies while considering the order information of words or characters in the sequence. BiGRU is introduced to help the model further refine local information and waveform details in seismic images. With such a setting, the network can process local and global features of time series data simultaneously and improve flexibility and prediction performance, thereby more accurately establishing the mapping relationship between frequency band-slice data and full frequency band data.  
When processing large-scale data, the inner product operation relied on by the attention mechanism in the Transformer model may cause numerical instability. Introducing the scaling factor allows us to adjust the scale of the inner product result to finely control the inner product operation, which improves the model's performance when dealing with large-scale data, enhances the modeling capability of the model, and reduces the training difficulty. The regularization technique is introduced into the model, and the data are linearly transformed through the Add Norm layer to eliminate the dimensional differences between the eigenvalues, making the model more stable and reliable. The Dropout layer randomly inactivates some neurons to reduce the model's dependence on training data, thereby alleviating the overfitting problem and improving the network's generalization.  
In summary, this deep learning model can construct a feature-rich and multi-scale domain, which enhances their capabilities when dealing with complex tasks. The network we used is shown in Figure 1.  
Figure 1: Network processing architecture diagram.