# Seismic Random Noise Attenuation via Low-Rank Tensor Network
## 2 Materials and Methods  
### Preliminaries  
We first provide the theoretical foundation required to build LRTNet. We start with the definition of a t-product, and then develop the concepts of t-SVD and the tensor nuclear norm (TNN) with the t-product.  
**Definition 1** (t-product [41; 42]).: _The t-product between tensor \(\mathcal{A}\in\mathbb{R}^{n_{1}\times n_{2}\times n_{3}}\) and tensor \(\mathcal{B}\in\mathbb{R}^{n_{2}\times n_{4}\times n_{3}}\) is defined as_  
\[\mathcal{A}*\mathcal{B}=\text{fold}(\text{circ}(\mathcal{A})\text{unfold}( \mathcal{B})), \tag{1}\]  
_where \(\text{fold}(\cdot)\) and \(\text{unfold}(\cdot)\) are determined as_  
\[\text{unfold}(\mathcal{A})=\left[\begin{array}{c}\mathcal{A}^{(1)}\\ \mathcal{A}^{(2)}\\ \vdots\\ \mathcal{A}^{(n_{3})}\end{array}\right],\qquad\text{fold}(\text{unfold}( \mathcal{A}))=\mathcal{A}, \tag{2}\]  
_and \(\text{circ}(\cdot)\) is defined as an \(n_{1}n_{3}\times n_{2}n_{3}\) matrix_  
\[\text{circ}(\mathcal{A})=\left[\begin{array}{cccc}\mathcal{A}^{(1)}& \mathcal{A}^{(n_{3})}&\cdots&\mathcal{A}^{(2)}\\ \mathcal{A}^{(2)}&\mathcal{A}^{(1)}&\cdots&\mathcal{A}^{(3)}\\ \vdots&\vdots&\ddots&\vdots\\ \mathcal{A}^{(n_{3})}&\mathcal{A}^{(n_{3}-1)}&\cdots&\mathcal{A}^{(1)}\end{array} \right]. \tag{3}\]  
It is clear that t-product \(\mathcal{A}*\mathcal{B}\) returns a tensor of size \(n_{1}\times n_{4}\times n_{3}\) and can be interpreted as a convolution-like operator between two tensors [42]. Leveraging the convolution theorem, the t-product can be computed through matrix multiplications involving two frequency slices in the Fourier domain [41; 43]:  
\[\widetilde{\mathcal{T}}^{(k)}=\widetilde{\mathcal{A}}^{(k)}\widetilde{ \mathcal{B}}^{(k)}. \tag{4}\]  
It is worth noting that a significant advantage of the Fourier transform is its capability to improve the low-rank properties of the seismic data tensor [44; 45]. Building on the definition and the convolutional properties of the t-product, a framework for t-SVD can be developed as  
**Definition 2** (t-SVD [42]).: _Given an \(n_{1}\times n_{2}\times n_{3}\) tensor \(\mathcal{A}\), \(\mathcal{A}\) can be factored as_  
\[\mathcal{A}=\mathcal{U}*\mathcal{S}*\mathcal{V}^{T}, \tag{5}\]  
_where \(\mathcal{U}\) and \(\mathcal{V}\) are orthogonal \(n_{1}\times n_{1}\times n_{3}\) and \(n_{2}\times n_{2}\times n_{3}\), respectively, and \(\mathcal{S}\) is an \(n_{1}\ \times\ n_{2}\ \times\ n_{3}\) tensor with each front slice being a diagonal matrix._  
It is evident that t-SVD closely resembles that of matrix SVD. In fact, it is simple to interpret that each front slice of t-SVD is equivalent to each frontal slice of matrix SVD in the Fourier domain [42]. With the definition of t-SVD, the tensor tubal rank and the TNN can be defined as:  
**Definition 3** (Tensor Tubal Rank [41; 43]).: _Given tensor \(\mathcal{A}\in\mathbb{R}^{n_{1}\times n_{2}\times n_{3}}\), we can deconstruct \(\mathcal{A}\) as \(\mathcal{A}=\mathcal{U}*\mathcal{S}*\mathcal{V}^{T}\). The tensor tubal rank of \(\mathcal{A}\), denoted as \(\text{rank}_{t}(\mathcal{A})\), is defined by the number of non-zero tubes of \(\mathcal{S}\), as_  
\[\text{rank}_{t}(\mathcal{A})=\#\{i,\mathcal{S}(i,i,:)\
eq 0\}. \tag{6}\]  
**Definition 4** (Tensor Nuclear Norm [46]).: _The TNN of \(\mathcal{A}\) is defined as_  
\[\|\mathcal{A}\|_{\otimes}\coloneqq\langle\mathcal{S},\mathcal{I}\rangle=\sum_ {r}^{i=1}\mathcal{S}(i,i,1), \tag{7}\]  
_where \(\mathcal{I}\) is the identity tensor, the whole initial frontal slice is an identity matrix and the remaining frontal slices are zeros, and \(r=\text{rank}_{t}(\mathcal{A})\)._