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Repeatedly recording seismic data over a period of months or years is one way to identify trapped oil and gas and to monitor CO2 injection in underground storage reservoirs and saline aquifers. This process of recording data over time and then differencing the images assumes the recording of the data over a particular subsurface region is repeatable. In other words, the hope is that one can recover changes in the Earth when the survey parameters are held fixed between data collection times. Unfortunately, perfect experimental repeatability almost never occurs. Acquisition inconsistencies such as changes in weather (currents, wind) for marine seismic data are inevitable, resulting in source and receiver location differences between surveys at the very least. Thus, data processing aimed at improving repeatability between baseline and monitor surveys is extremely useful. One such processing tool is regularization (or binning) that aligns multiple surveys with different source or receiver configurations onto a common grid. Data binned onto a regular grid can be stored in a high-dimensional data structure called a tensor with, for example, x and y receiver coordinates and time as indices of the tensor. Such a higher-order data structure describing a subsection of the Earth often exhibits redundancies which one can exploit to fill in gaps caused by sampling the surveys onto the common grid. In fact, since data gaps and noise increase the rank of the tensor, seeking to recover the original data by reducing the rank (low-rank tensor-based completion) successfully fills in gaps caused by binning. The tensor nuclear norm (TNN) is defined by the tensor singular value decomposition (tSVD) which generalizes the matrix SVD. In this work we complete missing time-lapse data caused by binning using the alternating direction method of multipliers (or ADMM) to minimize the TNN. For a synthetic experiment with three parabolic events in which the time-lapse difference involves an amplitude increase in one of these events between baseline and monitor data sets, the binning and reconstruction algorithm (TNN-ADMM) correctly recovers this time-lapse change. We also apply this workflow of binning and TNN-ADMM reconstruction to a real marine survey from offshore Western Australia in which the binning onto a regular grid results in significant data gaps. The data after reconstruction varies continuously without the large gaps caused by the binning process.

 

Seismic data are often incomplete due to equipment malfunction, limited source and receiver placement at near and far offsets, and missing crossline data. Seismic data contain redundancies because they are repeatedly recorded over the same or adjacent subsurface regions, causing the data to have a low-rank structure. To recover missing data, one can organize the data into a multidimensional array or tensor and apply a tensor completion method. We can increase the effectiveness and efficiency of low-rank data reconstruction based on tensor singular value decomposition (tSVD) by analyzing the effect of tensor orientation and exploiting the conjugate symmetry of the multidimensional Fourier transform. In fact, these results can be generalized to any order tensor. Relating the singular values of the tSVD to those of a matrix leads to a simplified analysis, revealing that the most square orientation gives the best data structure for low-rank reconstruction. After the first step of the tSVD, a multidimensional Fourier transform, frontal slices of the tensor form conjugate pairs. For each pair, a singular value decomposition can be replaced with a much cheaper conjugate calculation, allowing for faster computation of the tSVD. Using conjugate symmetry in our improved tSVD algorithm reduces the runtime of the inner loop by 35%–50%. We consider synthetic and real seismic data sets from the Viking Graben Region and the Northwest Shelf of Australia arranged as high-dimensional tensors. We compare the tSVD-based reconstruction with traditional methods, projection onto convex sets and multichannel singular spectrum analysis, and we see that the tSVD-based method gives similar or better accuracy and is more efficient, converging with runtimes that are an order of magnitude faster than the traditional methods. In addition, we verify that the most square orientation improves recovery for these examples by 10%–20% compared with the other orientations.