A key step in sparsifying signals is the choice of a sparsity-promoting dictionary. There are two basic approaches to design such a dictionary: the analytic approach and the learning-based approach. Although the analytic approach enjoys the advantage of high efficiency, it lacks adaptivity to various data patterns. On the other hand, the learning-based approach can adaptively sparsify different data sets but has a heavier computational complexity and involves no prior-constraint pattern information for particular data. We have developed a double-sparsity dictionary (DSD) for seismic data to combine the benefits of both approaches. We have evaluated two models to learn the DSD: the synthesis model and the analysis model. The synthesis model learns DSD in the data domain, and the analysis model learns DSD in the model domain. We tested the analysis model and proposed to use the seislet transform and data-driven tight frame (DDTF) as the base transform and adaptive dictionary, respectively, in the DSD framework. The DDTF obtains an extra structure regularization by learning dictionaries, whereas the seislet transform obtains a compensation for the transformation error caused by slope dependency. The DSD aims to provide a sparser representation than the individual transform and dictionary and, therefore, can help to achieve better performance in denoising applications. Although for the purpose of compression, the DSD is less sparse than the seislet transform, it outperforms the seislet and DDTF in distinguishing the signal and noise. Two simulated synthetic examples and three field data examples confirm a better denoising performance of the proposed approach.

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