5D and 4D pre-stack seismic data completion using tensor nuclear norm (TNN)

In this paper we present novel strategies for completion of 5D pre-stack seismic data, viewed as a 5D tensor or as a set of 4D tensors across temporal frequencies. In contrast to existing complexity penalized algorithms for seismic data completion, which employ matrix analogues of tensor decompositions such as HOSVD or use overlapped Schatten norms from different unfoldings (or matricization) of the tensors, our approach uses a recently proposed decomposition called tensor SVD or tSVD for short, proposed in [Kilmer and Martin (2011)]. We show that seismic data exhibits low complexity under tSVD, i.e. is compressible under tSVD representation, and we subsequently propose a new complexity penalized algorithm for pre-stack seismic data completion under missing traces. This complexity measure which we call the Tensor Nuclear Norm (TNN) is motivated by algebraic properties of the tSVD. We test the performance of the proposed algorithms on synthetic and real data and show that missing data can be reliably recovered under heavy down-sampling.