Unsupervised deep learning with higher-order total-variation regularization for multidimensional seismic data reconstruction

In 3D marine seismic acquisition, the seismic wavefield is not sampled uniformly in the spatial directions. This leads to a seismic wavefield consisting of irregularly and sparsely populated traces with large gaps between consecutive sail lines, especially in the near offsets. The problem of reconstructing the complete seismic wavefield from a subsampled and incomplete wavefield is formulated as an underdetermined inverse problem. We have investigated unsupervised deep learning based on a convolutional neural network for multidimensional wavefield reconstruction of irregularly populated traces defined on a regular grid. Our network is based on an encoder-decoder architecture with an overcomplete latent representation, including appropriate regularization penalties to stabilize the solution. We proposed a combination of penalties, which consists of the [Formula: see text]-norm penalty on the network parameters, and a first- and second-order total-variation penalty on the model. We determined the performance of our method on broadband synthetic data and field data represented by constant-offset gathers from a source-over-cable data set from the Barents Sea. In the field data example, we compare the results to a full production flow from a contractor company, which is based on a 5D Fourier interpolation approach. In this example, our approach displays improved reconstruction of the wavefield with less noise in the sparse near offsets compared with the industry approach, which leads to improved structural definition of the near offsets in the migrated sections.