Seismic wavelet estimation and deconvolution are essential for high-resolution seismic processing. Because of the influence of absorption and scattering, the frequency and phase of the seismic wavelet change with time during wave propagation, leading to a time-varying seismic wavelet. To obtain reflectivity coefficients with more accurate relative amplitudes, we should compute a nonstationary deconvolution of this seismogram, which might be difficult to solve. We have extended sparse spike deconvolution via Toeplitz-sparse matrix factorization to a nonstationary sparse spike deconvolution approach with anelastic attenuation. We do this by separating our model into subproblems in each of which the wavelet estimation problem is solved by the classic sparse optimization algorithms. We find numerical examples that illustrate the parameter setting, noisy seismogram, and the estimation error of the [Formula: see text] value to validate the effectiveness of our extended approach. More importantly, taking advantage of the high accuracy of the estimated [Formula: see text] value, we obtain better performance than with the stationary Toeplitz-sparse spike deconvolution approach in real seismic data.

Blind sparse-spike deconvolution is a widely used method to estimate seismic wavelets and sparse reflectivity in the shape of spikes based on the convolution model. To increase the vertical resolution and lateral continuity of the estimated reflectivity, we further improve the sparse-spike deconvolution by introducing the atomic norm minimization and structural regularization which, respectively, improve the vertical resolution and lateral continuity of the estimated reflectivity. Specifically, we use the atomic norm minimization to estimate the reflector locations ,which are further used as position constraints in the sparse-spike deconvolution. By doing this, we can vertically separate highly thin layers through the sparse deconvolution. In addition, the seismic structural orientations are estimated from the seismic image to construct a structure-guided regularization in the deconvolution to preserve the lateral continuity of reflectivities. Our improvements are suitable for most types of sparse-spike deconvolution approaches. The sparse-spike deconvolution method with a Toeplitz-sparse matrix factorization (TSMF) is used as an example to demonstrate the effectiveness of our improvements. Synthetic and real examples show that our methods perform better than TSMF in estimating reflectivity of thin layers and preserving lateral continuities.

Seismic denoising is an essential step for seismic data processing. Conventionally, dictionary learning methods for seismic denoising always assume the representation coefficients to be sparse and the dictionary to be normalized or a tight frame. Current dictionary learning methods need to update the dictionary and the coefficients in an alternating iterative process. However, the dictionary obtained from dictionary learning method often needs to be recalculated for each input data. Moreover, the performance of dictionary learning for seismic noise removal is related to the parameter selection and the prior constraints of dictionary and representation coefficients. Recently, deep learning demonstrates promising performance in data prediction and classification. Following the architecture of dictionary learning algorithms strictly, we propose a novel and interpretable deep unfolding dictionary learning method for seismic denoising by unfolding the iterative algorithm of dictionary learning into a deep neural network. The proposed architecture of deep unfolding dictionary learning contains two main parts, the first is to update the dictionary and representation coefficients using least-square inversion and the second is to apply a deep neural network to learn the prior representation of dictionary and representation coefficients, respectively. Numerical synthetic and field examples show the effectiveness of our proposed method. More importantly,the proposed method for seismic denoising obtains the dictionary for each seismic data adaptively and is suitable for seismic data with different noise levels.

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