A pre-trained dictionary learning framework combined with projections onto convex sets for seismic data reconstruction and denoising

The dictionary learning approach has proven effective in seismic data denoising and interpolation. Its core advantage lies in the ability to continuously update the initial dictionary, thereby adapting to the complex structural characteristics of seismic data. However, many existing implementations rely on predefined transforms (e.g., discrete cosine transform) for dictionary initialization. These fixed, data-agnostic bases often fail to fully capture the unique features of seismic signals, which may compromise the sparsity and fidelity of signal representation. Such a limitation can significantly degrade the performance of tasks requiring high-precision reconstruction or noise attenuation. To address this issue, we propose an innovative dictionary learning framework based on a variational sparse representation model. Specifically, this framework first extracts small data patches from arbitrary locations in seismic data, and then constructs a pre-training dataset using a windowing algorithm to preserve fine-grained data features. This process yields an initial dictionary that inherently encodes the intrinsic characteristics of the input seismic data. Subsequently, the initial dictionary is separately refined and updated through the K-singular value decomposition (K-SVD) and sequential generalization of K-means (SGK) algorithms, resulting in an optimized dictionary with more accurate and data-adaptive features. In addition, we integrate a multi-iteration projections onto convex sets algorithm to compensate for missing data features, ultimately achieving high-precision seismic data interpolation and noise attenuation. Numerical experiments demonstrate that the proposed methods (variational SGK and variational K-SVD) outperform the conventional K-SVD and SGK algorithms in both interpolation accuracy and denoising performance.