Double sparsity dictionary for seismic noise attenuation

Discussion

The proposed DSD method attempts to create an optimal combination of the fixed-basis transform and learning-based dictionary, in order to enhance the performance of each individual type of transform and to overcome the disadvantages of each individual transform. The seislet transform used in this paper is data-adaptive but depends on the estimation of local slopes of seismic events. An incorrect estimation of local slopes may make the transform domain less sparse. The DDTF approach used in this paper may not be optimally efficient when large overlap of filtering windows and large number of iterations are used, because DDTF is sensitive to small-scale random noise and tends to learn the property of random noise when small window overlap or small number of iterations are used. The proposed hybrid framework can relieve the dependency of the seislet transform on slope estimation because the thresholding used for denoising is no longer solely relying on the sparsity of the seislet domain and instead is relying on the DSD domain. Because of the initial seislet transform, the transform domain becomes more compactly supported and distinguishes signal and noise better. DDTF applied after the seislet transform becomes less sensitive to noise and can achieve better performance when learning the behavior of useful signals. As a result, the window overlap and iteration number can both be smaller.

The proposed DSD approach follows the main idea of the multi-scale dictionary learning algorithm proposed by Ophir et al. (2011). The difference between our method and the previous approach is that Ophir et al. (2011) use K-SVD in the wavelet domain, while our method uses DDTF in the seislet domain. We use different tools but keep the same strategy. The seislet transform is more suitable than wavelet transform for seismic data. The DDTF is much faster than K-SVD for the dictionary learning stage. Ophir et al. (2011) split the data (actually split the subband coefficients) into overlapping patches, and then apply K-SVD on each patch. Analogously, we split the seislet subband data into overlapping patches, and apply DDTF on each patch. There is a control on how much overlapping is used. In general, it is advantageous to apply the maximally overlapping patches (i.e., only shift one column/row between the neighbor patches) for the training. This creates the "richness"/"redundancy" in the training data that generates a shift-invariance for the dictionary. Such shift-invariant wavelets are helpful for denoising, which has been demonstrated in Coifman and Donoho (1995). In the DDTF, for each patch, the training filer is of tight frame or orthogonal, but the full transform matrix $\mathbf{W}$ is redundant (each column is a training filter on patches).

The double sparsity is more a concept than a specific type of transform. The seislet transform and DDTF used in this paper can be replaced by other combinations of a fixed-basis transform and a learning-based dictionary to achieve an overall performance which is better than that of each of the individual transforms. The DSD approach is also not limited to the application to random noise attenuation. Other possible applications include seismic data interpolation and regularization, simultaneous-source deblending, and sparsity-based seismic inversion.

Both the noisy synthetic data and the first two field data examples shown in this paper we simulated by adding Gaussian white noise. The advantage of using simulated data is that we know the true answer and can numerically calculate SNR in order to better compare and understand the properties of each transform-domain thresholding-based denoising approach. Although the Gaussian white noise is not always appropriate for simulating real random noise in field data, we use it here for simplicity. Changing the Gaussian white noise to colored noise and then performing similar tests is straightforward.

The fair comparison of different denoising approaches is always a difficult problem. In this paper, we only focus on the coefficient percentage for transform domain thresholding and fix other parameters, which mainly include the patch size of the tight frames and the number of iterations to update the tight frames.

Double sparsity dictionary for seismic noise attenuation