# Seismic Random Noise Attenuation via Low-Rank Tensor Network
## 2 Materials and Methods
### Optimization Procedure  
We introduce two auxiliary variables, \(\mathcal{Z}=\mathcal{X}\) and \(\mathcal{S}=\mathbb{D}(\mathcal{Z})\), where \(\mathbb{D}(\cdot)=\left[\tau_{x}\mathcal{D}_{x};\tau_{y}\mathcal{D}_{y};\ \tau_{z}\mathcal{D}_{z}\right]\) is the concatenated difference operation, and \(\|\mathcal{S}\|_{1}=\|\mathcal{Z}\|_{TV}\), which is able to utilize the alternating direction method of multipliers (ADMM) framework to resolve the above problem (9). Then, the constrained minimization problem can be expressed as follows:  
\[ \begin{split}&\min_{\mathcal{X},\mathcal{Z},\mathcal{S},\mathcal{N}} \alpha\|\mathcal{X}\|_{\mathbf{w},\mathcal{B}}+\mu\|\mathcal{S}\|_{1}+\gamma \|\mathcal{N}\|_{F}^{2}\\ &\text{s.t.}\quad\mathcal{Y}=\mathcal{X}+\mathcal{N},\ \mathcal{Z}= \mathcal{X},\ \|\mathcal{S}\|_{1}=\|\mathcal{Z}\|_{TV}.\end{split}  \tag{11}\]  
Subsequently, the above problem (11) can be further transformed to find a saddle point of the following disputedLagrangian function:\[L(\mathcal{X},\mathcal{Z}, \mathcal{S}_{r},\mathcal{N},\Gamma_{1}\Gamma_{2},\Lambda)=\alpha\| \mathcal{X}\|_{\mathbf{w},\mathbb{R}}+\mu\|\mathcal{S}\|_{1} \tag{12}\] \[+\gamma\|\mathcal{N}\|_{F}^{2}+\frac{\rho}{2}\Bigg{(}\bigg{\|} \mathcal{X}+\mathcal{N}-\mathcal{Y}+\frac{\Gamma_{1}}{\rho}\bigg{\|}_{F}^{2}\] \[+\bigg{\|}\mathcal{X}-\mathcal{Z}+\frac{\Gamma_{2}}{\rho}\bigg{\|} _{F}^{2}+\bigg{\|}\mathbb{D}(\mathcal{Z})-\mathcal{S}+\frac{\Lambda}{\rho} \bigg{\|}_{F}^{2}\Bigg{)},\]  
where \(\Gamma_{1}\), \(\Gamma_{2}\), and \(\Lambda\) are the Lagrangian multipliers, and \(\rho\) is a positive penalty parameter. Therefore, we can obtain a solution to the original problem (11) after a finite number of iterations according to the ADMM framework. The iteration scheme can be expressed as follows:  
\[\left\{ \begin{aligned} \mathcal{X}^{k+1}&=\mathcal{U} *\mathbb{S}_{\mathbf{w}^{k}}(\mathbf{\Sigma})*\mathcal{V}^{\dagger},\\ \mathcal{Z}^{k+1}&=\mathbb{F}^{-1}\Bigg{(}\frac{ \mathbb{F}(\rho^{k}(\mathcal{X}^{k+1}+\mathbb{D}^{*}(\mathcal{S}^{k}))+\Gamma _{2}^{k}-\mathbb{D}^{*}(\Lambda^{k}))}{\rho^{k}(\mathcal{I}+\mathcal{A})} \Bigg{)},\\ \mathcal{S}^{k+1}&=\mathbb{S}_{\mu/\rho^{k}}( \mathbb{D}(\mathcal{Z}^{k+1})+\Lambda^{k}/\rho^{k}),\\ \mathcal{N}^{k+1}&=(2\gamma+\rho^{k})^{-1}(\rho^{k}( \mathcal{Y}-\mathcal{X}^{k+1}-\Gamma_{1}^{k}/\rho^{k})),\end{aligned} \right. \tag{13}\]  
where \([\mathcal{U},\ \mathbf{\Sigma},\ \mathcal{V}]=\text{t-SVD}(\mathcal{D}^{k})\) denotes the tensor singular value decomposition of \(\mathcal{D}^{k}\)[47] and \(\mathcal{D}^{k}=(\mathcal{Y}+\mathcal{Z}^{k}-\mathcal{N}^{k})/2-(\Gamma_{1} ^{k}+\Gamma_{2}^{k})/2\rho^{k}\), the weight vector \(\mathbf{w}\) in WTNNM is defined as \(w_{i}=\frac{c}{\sqrt{\max{(\sigma_{i}^{2}(\mathcal{D}^{k})-\alpha/\rho_{i} \mathcal{D})+\epsilon}}}\) with a constant \(c\) and a small positive constant \(\epsilon\) according to [14], \(\mathbb{F}(\cdot)\) and \(\mathbb{F}^{-1}(\cdot)\) indicate the 3D Fourier transform and its inverse transform, \(\mathcal{A}=\tau_{x}^{2}|\mathbb{F}(\mathcal{D}_{x})|^{2}+\tau_{y}^{2}| \mathbb{F}(\mathcal{D}_{y})|^{2}+\tau_{z}^{2}|\mathbb{F}(\mathcal{D}_{z})|^{2}\), and it is the same for all iterations, and \(\mathbb{S}_{\rho}(X)=\frac{X}{|X|}\max{(X-\rho,0)}\) is the soft-thresholding operator. For the Lagrange multipliers, we have  
\[\left\{ \begin{aligned} &\Gamma_{1}^{k+1}=\Gamma_{1}^{k}+\rho^{k}( \mathcal{X}^{k+1}+\mathcal{N}^{k+1}-\mathcal{Y}),\\ &\Gamma_{2}^{k+1}=\Gamma_{2}^{k}+\rho^{k}(\mathcal{X}^{k+1}- \mathcal{Z}^{k+1}),\\ &\Lambda^{k+1}=\Lambda^{k}+\rho^{k}(\mathbb{D}(\mathcal{Z}^{k+1 })-\mathcal{S}^{k+1}).\end{aligned} \right. \tag{14}\]
\[\mathcal{L}\Big{(}\mathcal{X}^{(n)},\mathcal{Y}^{(n)};\Theta\Big{)}=\arg\min_{ \Theta}\frac{1}{N}\sum_{n=1}^{N}\left\|H_{\Theta}\Big{(}\mathcal{Y}^{(n)}\Big{)} -\mathcal{X}^{(n)}\right\|^{2}, \tag{16}\]  
where \(\Big{\{}(\mathcal{X}^{(1)},\mathcal{Y}^{(1)}),\cdots,(\mathcal{X}^{(N)}, \mathcal{Y}^{(N)})\Big{\}}\) are the \(N\) pairs of the training data.