# Seismic Random Noise Attenuation via Low-Rank Tensor Network
## 2 Materials and Methods
### Optimization Procedure
The \(l\)-th layer is composed of four modules, and the details of these modules are described as follows:  
(1) Low-Rank Approximation (\(LRA\)) Module: This module recovers the seismic data by LRTA operation based on WTNNM, and the soft-thresholding operator is replaced by the ReLU function. Then, variable \(\mathcal{X}^{l+1}\) is updated as follows:  
\[\mathcal{X}^{l+1}=\mathcal{U}*\text{ReLU}(\mathbf{\Sigma}-\mathbf{w}^{l})* \mathcal{V}^{\dagger}, \tag{17}\]  
where \([\mathcal{U},\ \mathbf{\Sigma},\ \mathcal{V}]=\text{t-SVD}(\mathcal{D}^{l})\) denotes the tensor singular value decomposition of \(\mathcal{D}^{l}\), \(\mathcal{D}^{l}=(\mathcal{Y}+\mathcal{Z}^{l}-\mathcal{N}^{l})/2-(\Gamma_{1} ^{l}+\Gamma_{2}^{l})/2\theta_{\rho_{1}}^{l}\), and \(w_{i}=\frac{\theta_{\rho_{i}}^{l}}{\sqrt{\text{ReLU}(\sigma_{i}^{2}(\mathcal{ D}^{l})-\Gamma_{k}^{l}/\kappa_{1})^{2})+\epsilon}}\). (2) TV Regularization (\(TVR\)) Module: This module introduces the TV regularization to the new denoised seismic data \(\mathcal{X}^{l+1}\) of the previous \(LRA\) module and replaces the soft-thresholding operator by the ReLU function. Then, the output of this module, the updated smooth denoised seismic data \(\mathcal{Z}^{l+1}\), can be expressed as follows:  
\[\left\{ \begin{aligned} &\mathcal{Z}^{l+1}=\mathbb{F}^{-1}\left( \frac{\mathbb{F}\Big{(}\theta_{\rho_{2}}^{l}(\mathcal{X}^{l+1}+\mathbb{D}^{* }(\mathcal{S}^{l}))+\Gamma_{1}^{l}-\mathbb{D}^{*}(\Lambda^{l})\Big{)}}{\theta _{\rho_{2}}^{l}(\mathcal{I}+\mathcal{A}^{l})}\right),\\ &\mathcal{S}^{l+1}=\frac{\mathcal{B}^{l}}{|\mathcal{B}^{l}|}* \text{ReLU}(|\mathcal{B}^{l}|-\frac{\theta_{\mu}^{l}}{\theta_{\rho_{2}}^{l}}),\\ &\Lambda^{l+1}=\Lambda^{l}+(\theta_{\rho_{2}}^{l})(\mathbb{D}( \mathcal{Z}^{l+1})-\mathcal{S}^{l+1}),\end{aligned} \right. \tag{18}\]  
where \(\mathcal{B}^{l}=\mathbb{D}(\mathcal{Z}^{l+1})+\Lambda^{l}/\theta_{\rho_{2}}^{l}\) and \(\mathcal{A}^{l}=(\theta_{x}^{l})^{2}|\mathbb{F}(\mathcal{D}_{x})|^{2}+( \theta_{y}^{l})^{2}|\mathbb{F}(\mathcal{D}_{y})|^{2}+(\theta_{z}^{l})^{2}| \mathbb{F}(\mathcal{D}_{z})|^{2}\). (3) Random Noise Removal (\(RNN\)) Module: This module updates the removed random noise \(\mathcal{N}\) according to the new denoised result \(\mathcal{X}^{l+1}\). This sub-problem is a standard  
Figure 1: The flowchart of LRTNet. least square regression problem with a closed-form solution, and the output of this module, the new removed random noise \(\mathcal{N}^{l+1}\), can be expressed as follows:  
\[\mathcal{N}^{l+1}=\frac{\theta_{\rho_{3}}^{l}(\mathcal{Y}-\mathcal{X}^{l+1})- \Gamma^{l}}{2\theta_{\gamma}^{l}+\theta_{\rho_{3}}^{l}}. \tag{19}\]  
(4) Lagrange Multiplier Updating (_LMU_) Module: This module updates the Lagrange multipliers \(\Gamma_{1}\) and \(\Gamma_{2}\) based on the new denoised result \(\mathcal{X}^{l+1}\), the new removed random noise \(\mathcal{N}^{l+1}\), and the new smooth denoised seismic data \(\mathcal{Z}^{l+1}\) according to the ADMM framework. Then, the outputs of this module, the new Lagrange multipliers \(\Gamma_{1}^{l+1}\) and \(\Gamma_{2}^{l+1}\), can be expressed as follows:  
\[ \begin{cases}\Gamma_{1}^{l+1}=\Gamma_{1}^{l}+(\theta_{\rho_{4}}^{l})(\mathcal{ X}^{l+1}+\mathcal{N}^{l+1}-\mathcal{Y}),\\ \Gamma_{2}^{l+1}=\Gamma_{2}^{l}+(\theta_{\rho_{4}}^{l})(\mathcal{X}^{l+1}- \mathcal{Z}^{l+1}).\end{cases}  \tag{20}\]  
The pseudocode of LRTNet is given in Algorithm 1. ```
0:\(\mathcal{Y}\)
0:\(\mathcal{X}\)  Set \(\mathcal{X},\mathcal{N},\mathcal{Z},\mathcal{S},\Lambda,\Gamma_{1}\), and \(\Gamma_{2}\) to zero tensor. \(\triangleright\) Initial \(P=\{\mathcal{Y},\mathcal{X},\mathcal{N},\mathcal{Z},\mathcal{S},\Lambda, \Gamma_{1},\Gamma_{2}\}\) // layer 1 \(P.\mathcal{X}\gets LRA(P)\). \(\triangleright\) update \(\mathcal{X}\) with LRT Module \(P.\mathcal{Z},P.\mathcal{S},P.\Lambda\gets TVR(P)\). \(\triangleright\) update \(\mathcal{Z},\mathcal{S},and\Lambda\) with TVR Module \(P.\mathcal{N}\gets RNR(P)\). \(\triangleright\) update \(\mathcal{N}\) with RNR Module \(P.\Gamma_{1},P.\Gamma_{2}\leftarrow L\mathcal{M}U(P)\).