Tensor completion based on nuclear norm minimization for 5D seismic data reconstruction

Abstract

Prestack seismic data are multidimensional signals that can be described as a low-rank fourth-order tensor in the f requency − space domain. Tensor completion strategies can be used to recover unrecorded observations and to improve the signal-to-noise ratio of prestack volumes. Additionally, tensor completion can be posed as an inverse problem and solved using convex optimization algorithms. The objective function for this problem contains a data misfit term and a term that serves to minimize the rank of the tensor. The alternating direction method of multipliers offers automatic rank determination and it is used to obtain a reconstructed seismic volume. The proposed method converges to a good approximation of the rank of the tensor given the input data. We present synthetic examples to illustrate the behaviour of the algorithm in terms of trade-off parameters that control the quality of the reconstruction. We further illustrate the performance of the algorithm in a land data survey from Alberta, Canada.

5D seismic data completion and denoising using a novel class of tensor decompositions

“…The TNN derived from tSVD differs significantly from the sum of nuclear norms approach used by Gandy et al (2011) and Kreimer et al (2013), where the summation is over the nuclear norms of all the different mode unfoldings of the tensor. Our approach computes the TNN, as derived from the tSVD, by first applying the Fourier transform recursively along the different dimensions and then summing the nuclear norms of the frontal slices of the resulting tensor.…”

“…Even though video data and seismic data are very different data types, we expect that retaining the multidimensional structure of the data should also help with recovery in the seismic case. Nevertheless, it is important to note that because we do not compare our method with that in Kreimer et al (2013), we cannot claim that our approach is superior to theirs. Extensive comparison of our approach with those of Kreimer et al (2013) is a subject of future work and will be undertaken in the future.…”

“…Nevertheless, it is important to note that because we do not compare our method with that in Kreimer et al (2013), we cannot claim that our approach is superior to theirs. Extensive comparison of our approach with those of Kreimer et al (2013) is a subject of future work and will be undertaken in the future. In Silva and Herrmann (2013), the authors use the HT decomposition for data completion with a novel algorithm that exploits the manifold structure of tensors with fixed HT rank.…”

“…All of these decompositions have been applied to the problem of 5D seismic data reconstruction with good performance on real and synthetic data. Seismic data completion using tensor rank minimization under HOSVD is proposed by Kreimer and Sacchi (2012b) and Kreimer et al (2013). Although the performance was shown to be good, the proposed algorithm requires preselection of the truncation ranks along each dimension for effective interpolation.…”

“…If this is not the case, then it may require ground truth and model selection for each section of the data. Kreimer et al (2013) propose seismic data completion based on nuclear norm minimization. This approach flattens the data into various modes, which may be suboptimal because it destroys the multidimensional structure of the data.…”