# Seismic Random Noise Attenuation via Low-Rank Tensor Network
## 1 Introduction  
Seismic exploration is a critical tool for subsurface imaging, widely used not only in oil and gas exploration, but also in applications such as geothermal energy assessment, carbon capture and storage monitoring, and earthquake hazard analysis [1]. The data collected during seismic surveys, however, are often contaminated by random noise, which arises from limitations in acquisition techniques, geological conditions, and environmental factors. This noise impairs subsequent interpretation tasks, such as seismic inversion, seismic facies analysis, and attribute analysis [2, 3, 4]. In this study, we focus on Gaussian random noise, which is statistically characterized by a normal distribution and is distinct from structured noise types such as footprint noise. Gaussian noise is frequency-independent and can significantly degrade the quality of seismic data across all frequency ranges. Therefore, the attenuation of random noise (and the enhancement of effective information) in seismic reflection remains a fundamental and enduring research topic. Over recent decades, numerous methods have been proposed to suppress random noise. Among these, the low-rank approximation (LRA) has received significant attention due to its superior performance [5; 6; 7; 8; 9; 10]. LRA operates on the hypothesis that seismic signals are inherently low-rank, and seeks to reduce the rank affected by random noise in the physical or transformed domain. LRA can be further categorized as low-rank matrix approximation (LRMA) and low-rank tensor approximation (LRTA). LRMA views seismic data as images or matrixes and then applies image denoising techniques. Due to the presence of similar geologic structures in the subsurface, clean seismic profiles exhibit a high degree of structural similarity, which can be effectively represented using LRMA [11]. References [12; 13; 14] stack blocks together with a similar seismic structures in a seismic profile to establish a matrix, and then perform LRA to suppress random noise. The Hankel matrix formulated from the useful seismic signal is low-rank [15], so references [16; 17; 18; 19] map the seismic data to a Hankel matrix or a block Hankel matrix and recover the clean seismic data with LRMA. Despite the promising results, LRMA simply treats the seismic data as matrixes and neglects the spatial similarity, i.e., associations along the inline and crossline dimensions, and frequency association in the high-dimensional space. In contrast, LRTA considers seismic data to be tensors satisfying the low-rank property in high-dimensional space, effectively utilizing their highly structured three-dimensional features [20]. References [21; 22; 23; 24; 25] perform tensor singular value decomposition (t-SVD) on seismic data to mitigate random noise. In [26; 27], the CANDECOMP/PARAFAC (CP) decomposition is applied to the stacked tensor to achieve the optimal approximation. Although both LRMA and LRTA demonstrate great power, they all share the common limitation that their parameters must be manually selected and meticulously tuned based on prior knowledge. Deep learning (DL) also shows great potential in eliminating random noise [28; 29; 30; 31; 32]. Most DL methods use a supervised learning strategy to train an end-to-end model. Typical models include convolutional neural networks (CNNs) [33], the 3D denoising CNN (3D-DnCNN) [34], generative adversarial networks (GANs) [35], and U-Net [36]. Besides supervised learning, there also exists unsupervised learning, for example, autoencoders [37], the robust tensor DL (RTDL) [38], etc. Compared with LRA, DL has the advantage of not relying on prior knowledge. However, the end-to-end model and purely data-driven attribution make it hard to interpret why the features are learned by DL. In light of the aforementioned aspects, we introduce the low-rank tensor network (LRTNet), a novel approach that combines LRTA and DL to address the challenge of seismic random noise attenuation.