## 3 Proposed Method
### Architecture for FID-N Seismic Denoising Network  
As shown in Figure 1, the proposed FID-N has two modules: a signal shallow feature extraction (SSFE) module and a feedback module that has iterativeness and corrects the input of the system by rerouting the output in each loop. The feedback module contains a two-path information distillation module (IDB) that can distill more useful information based on collecting as much information as possible. The FID-N can be unfolded as a set of temporally ordered iterations ranked from 1 to \(T\). \(T\) represents the number of iterations over which the FID-N unfolds and processes temporal information in a sequence.  
Inspired by [45], our proposed network is optimized with \(L1\) loss. Given the noisy seismic signals \(I^{Noisy}\) and clean seismic signals \(I^{Clear}\), the FID-N loss function is given by  
\[L(\theta)=\frac{1}{T}\sum_{i=1}^{T}W^{t}\Big{\|}I^{noisy}-I^{clear}\Big{\|}_{1}, \tag{1}\]  
where \(W^{t}\) denotes a weight factor that quantifies the importance or relevance of the output generated in the \(t\)-th iteration in the context of the iterative process. This factor can be used to adjust the influence of the \(t\)-th iteration's output on the subsequent computations or final results. \(\theta\) represents the parameter of the FID-N. Following [45], \(W^{t}=1\) for each iteration, representing that the contribution of each output is equal.  
Specifically, double convolution layers (\"Layer-1\" and \"Layer-2\", respectively) are utilized to perform SSFE on noisy seismic signals. The first layer captures the initial features, and the second layer refines them, improving feature extraction. The extracted shallow feature \(F_{t}^{in}\) in the \(t\)-th iteration is calculated by  
\[F_{t}^{in}=H_{1}\Big{(}H_{2}\Big{(}I^{noisy}\Big{)}\Big{)}, \tag{2}\]  
Figure 1: Architectural diagram of our FID-N network. Red and blue arrows denote the feedback and global residual skip connections, respectively.  
where \(H_{1}\) and \(H_{2}\) are convolution operations of Layer-1 and Layer-2, respectively. The shallow features \(F_{t}^{in}\) are then used as the input to the two-path IDB.  
In the two-path IDB, the hidden state of the previous iteration \(F_{t-1}^{out}\) is fed back to the \(t\)-th iteration and the shallow features \(F_{t}^{in}\). \(F_{t}^{out}\) represents the output of the two-path IDB:  
\[F_{t}^{out}=f_{TPIDB}\Big{(}F_{t-1}^{out},F_{t}^{in}\Big{)}, \tag{3}\]  
where \(f_{TPIDB}\) represents the operation of our proposed two-path IDB as well as the actual feedback process. More details of the two-path IDB can be found in Section 3.2. Integrally, to obtain the feature maps \(I_{t}^{output}\) in the \(t\)-th iteration, global residual learning is used in this work, which can be formulated as  
\[I_{t}^{Output}=F_{t}^{G}+F_{t}^{in}, \tag{4}\]  
where \(F_{t}^{G}=H_{3}\big{(}F_{t}^{out}\big{)}\), and \(H_{3}\) denotes the convolution operation of 'Conv 3' in Figure 2.  
Each convolutional layer in our FID-N has 64 filters. This paper obtains, in total, \(T\) denoised seismic data after \(T\) iterations.  
In Figure 1, the SSFE module consists of two convolutional layers with 3 \(\times\) 3 and 5 \(\times\) 5 convolutional kernels, with a stride of 1. Conv 3 is the convolutional layer with 3 \(\times\) 3 convolutional kernels. Table 1 shows the SSFE structure.