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Rank reduction strategies can be employed to attenuate noise and for prestack seismic data regularization. We present a fast version of Cadzow reduced-rank reconstruction method. Cadzow reconstruction is implemented by embedding 4D spatial data into a level-four block Toeplitz matrix. Rank reduction of this matrix via the Lanczos bidiagonalization algorithm is used to recover missing observations and to attenuate random noise. The computational cost of the Lanczos bidiagonalization is dominated by the cost of multiplying a level-four block Toeplitz matrix by a vector. This is efficiently implemented via the 4D fast Fourier transform. The proposed algorithm significantly decreases the computational cost of rank-reduction methods for multidimensional seismic data denoising and reconstruction. Synthetic and field prestack data examples are used to examine the effectiveness of the proposed method.

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Irregular seismic data causes problems with multi-trace processing algorithms and degrades processing quality. We introduce the Projection onto Convex Sets (POCS) based image restoration method into the seismic data reconstruction field to interpolate irregularly missing traces. For entire dead traces, we transfer the POCS iteration reconstruction process from the time to frequency domain to save computational cost because forward and reverse Fourier time transforms are not needed. In each iteration, the selection threshold parameter is important for reconstruction efficiency. In this paper, we designed two types of threshold models to reconstruct irregularly missing seismic data. The experimental results show that an exponential threshold can greatly reduce iterations and improve reconstruction efficiency compared to a linear threshold for the same reconstruction result. We also analyze the antinoise and anti-alias ability of the POCS reconstruction method. Finally, theoretical model tests and real data examples indicate that the proposed method is efficient and applicable.

A reconstruction method known as Projection Onto Convex Sets (POCS) is an effective, uncomplicated and robust method for the recovery of irregularly missing seismic traces. However, slow convergence of the POCS reconstruction method could jeopardize its computational appeal. For this reason, we investigate the performance of the POCS reconstruction method in terms of different threshold schedules and present a new data driven threshold that leads to an efficient implementation of the POCS method. In particular, we show that high quality solutions can be obtained in a few iterations. In addition, we address an important issue with the classical implementations of POCS reconstruction in that they cannot interpolate regularly missing data. To solve this problem, we introduce a masking operator that is based on a dominant dip scanning method into the POCS iteration. At the end, we present a variant of the POCS method that permits de‐noising seismic volumes during the reconstruction stage. This is achieved by defining a weighted trace re‐insertion strategy that alleviates the influence of noisy traces in the final reconstruction of the seismic volume. We show the effectiveness of the proposed method using synthetic and field data.

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Tensors, also called multilinear arrays, have been receiving attention from the seismic processing community. Tensors permit us to generalize processing methodologies to multidimensional structures that depend on more than 2D. Recent studies on seismic data reconstruction via tensor completion have led to new and interesting results. For instance, fully sampled noise-free multidimensional seismic data can be represented by a low-rank tensor. Missing traces and random noise increase the rank of the tensor. Hence, multidimensional prestack seismic data denoising and reconstruction can be tackled with tools that have been studied in the field of tensor completion. We have investigated and applied the recently proposed parallel matrix factorization (PMF) method to solve the 5D seismic data reconstruction problem. We have evaluated the efficiency of the PMF method in comparison with our previously reported algorithms that used singular value decomposition to solve the tensor completion problem for prestack seismic data. We examined the performance of PMF with synthetic data sets and with a field data set from a heavy oil survey in the Western Canadian Sedimentary Basin.