## 3 Proposed Method
### Nsect Prediction  
Although wavelet transforms (WT) [47] (e.g., Daubechies wavelet, symlets, and coiflets) have been widely applied in signal processing, they exhibit significant limitations. On the one hand, a WT primarily processes single-channel data and demonstrates an insufficient capability to characterize directional features within two-dimensional or higher-dimensional spaces, making it difficult to effectively capture multi-dimensional signals with singularities and their spatial correlations. On the other hand, conventional multi-scale decomposition methods often employ subsampling operations to adjust the directional responses of filters, yet this approach induces high-frequency aliasing effects, leading to the resolution degradation of raw data and severely compromising the preservation of high-frequency details that are critical for deep learning denoising. To address these technical challenges, this study adopts the non-subsampled contourlet transform (NsCT) [48] to construct a multi-scale decomposition framework for seismic data. As a multi-scale and multi-directional image transformation technique, the NsCT overcomes the dimensional constraints of traditional methods through its directional representation system. Inheriting and extending the multi-scale analytical framework of the shearlet transform [49], the NsCT employs a cascaded structure comprising a non-subsampledpyramid filter bank (NSPFB) and a non-subsampled directional filter bank (NSDFB). This architecture achieves \(K\)-level decomposition, generating a low-frequency (LF) sub-band and multiple high-frequency (HF) sub-bands that retain the original image dimensions, thereby effectively capturing the edge and texture information in the images. For a given seismic image, the contourlet affine system can be formally defined as  
\[\left\{\psi_{j,l,k}=|\text{det}A|^{j/2}\psi\left(S^{l}A^{j}x-k\right):l,j\in Z,k \in Z^{2}\right\}, \tag{10}\]  
where \(\psi\) represents a set of basis functions, and it satisfies the condition that \(\psi\) belongs to the \(L^{2}(R^{2})\) space, where \(j\), \(l\), and \(k\) represent the scale, direction, and translation parameters, respectively. Matrices \(A\) and \(S\) are both \(2\times 2\) invertible matrices, and they satisfy \(|\text{det}S|=1\). Matrix \(A\) is an anisotropic matrix that dominates the scaling characteristics of the contourlet transform; matrix \(S\) is responsible for controlling the direction of the contourlet transform. According to the standard, \(A=A_{0}=[4,0;0,2]\)\(A=A_{1}=[2,0;0,4]\); \(S=S_{0}=[1,1;0,1]\) or \(S=S_{1}=[1,0;1,1]\). Based on these settings, the contourlet transform function can be expressed as follows:  
\[\hat{\psi}_{j,l,m}^{(0)}(x)=2^{j\frac{3}{2}}\psi^{0}(S_{0}^{l}A_{0}^{j}x-k), \tag{11}\]  
\[\hat{\psi}_{j,l,m}^{(1)}(x)=2^{j\frac{3}{2}}\psi^{1}(S_{1}^{l}A_{1}^{j}x-k), \tag{12}\]  
where \(j\leq 0\), \(-2^{j}\leq l\leq 2^{j}-1\), and \(m\in Z^{2}\). NST decomposition consists of two main parts: multi-scale decomposition and multi-direction decomposition. The multi-scale decomposition is achieved through the non-subsampled Laplacian pyramid transform, while the multi-direction decomposition is completed by combining various shear filters. In the seismic data processing of the NST, although the normalization of preprocessing and the thresholding of post-processing bring a certain computational overhead, this overhead is relatively small and is crucial to improving the overall processing effect. Normalization ensures the consistency of the input data in scale, reduces the deviation introduced by data amplitude differences, and, thus, enhances the sensitivity and stability of the NST to the seismic characteristics of different scales. Thresholding effectively suppresses low-intensity noise while retaining key seismic information and further optimizes the denoising performance.