Reflectivity inversion methods based on a stationary convolution model are essential for seismic data processing. They compress the seismic wavelet, and by broadening the bandwidth of seismic data, they assist the interpretation of seismic sections. Unfortunately, they do not apply to realistic nonstationary deconvolution cases where the seismic wavelet varies as it propagates in the subsurface. Deep learning techniques have been proposed to solve inverse problems where networks can behave as the regularizer of the inverse problem. Our goal is to adopt a semi-supervised deep learning approach to invert reflectivity when the propagating wavelet is considered unknown and time-variant. To this end, we design a prior-engaged neural network by unrolling an alternating iterative optimization algorithm, where convolutional neural networks are used to solve two sub-problems. One is to invert the reflectivity, and the other is to estimate the time-varying wavelets. Generally, it is well-known that when working with geophysical inverse problems such as the one at hand, one has limited access to labeled data for training the network. We circumvent the problem by training the network via a data-consistency cost function where seismic traces are honored. Reflectivity estimates are also honored at spatial coordinates where true reflectivity series derived from borehole data are available. The cost function also penalizes time-varying wavelets from varying abruptly along the spatial direction. Experiments are conducted to show the effectiveness of the proposed method. We also compared the proposed approach to a nonstationary blind deconvolution algorithm based on regularized inversion. Our findings show that the proposed method improves the vertical resolution of seismic sections with noticeable correlation coefficient improvements over the nonstationary blind deconvolution. In addition, the proposed method is less sensitive to initial estimates of nonstationary wavelets. Moreover, it needs less human intervention when setting parameters than regularized inversion.

We have developed an adaptive singular spectrum analysis (ASSA) method for seismic data denoising and interpolation purposes. Our algorithm iteratively updates the singular-value decomposition (SVD) of current spatial patches using the most recently added spatial sample. The method reduces the computational cost of classic singular spectrum analysis (SSA) by requiring QR decompositions on smaller matrices rather than the factorization of the entire Hankel matrix of the data. A comparison between results obtained by the ASSA and SSA methods, in which the SVD applies to all of the traces at once, proves that the ASSA method is a valid way to cope with spatially varying dips. In addition, a comparison of the ASSA method with the windowed SSA method indicates gains in efficiency and accuracy. Synthetic and real data examples illustrate the effectiveness of our method.

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Seismic deconvolution used for improving the bandwidth of data is inherently nonstationary, mixed phase, and blind. Due to some restricting assumptions imposed by conventional deconvolution methods, they are either stationary or semiblind. A fully nonstationary blind deconvolution method is proposed that is able to simultaneously take into account different sources of nonstationarity and to improve the bandwidth of highly nonstationary seismic data in a fully blind manner. Based on the concept of block convolution and the overlap method, the convolutional model of seismic data is generalized to consider nonstationary cases and to model nonstationary data. This generalized convolutional model is then used for nonstationary blind deconvolution, in which the statistical characteristics of the wavelets are allowed to arbitrarily change in the vertical and horizontal directions. Given a nonstationary seismic record, several time-space-varying wavelets are simultaneously determined with the reflectivity model in an alternating direction algorithm using a variational approach. Numerical tests are presented showing the high performance of our nonstationary blind deconvolution for improving the temporal resolution of data in comparison with their stationary counterparts. The results indicate that in comparison with patched deconvolution, our nonstationary method is more robust and stable for different window sizes and it produces better results with a higher signal-to-noise ratio.

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