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.

中文翻译：

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使用低秩张量优化的改进地震数据完成算法：降低成本和优化数据方向

由于设备故障，近端和远端偏移处的震源和接收器位置有限以及缺少交叉线数据，地震数据通常是不完整的。地震数据包含冗余，因为它们被重复记录在相同或相邻的地下区域中，从而导致数据具有低秩结构。为了恢复丢失的数据，可以将数据组织成多维数组或张量，并应用张量完成方法。通过分析张量方向的影响并利用多维傅立叶变换的共轭对称性，可以提高基于张量奇异值分解（tSVD）的低秩数据重建的有效性和效率。实际上，这些结果可以推广到任何阶数张量。将tSVD的奇异值与矩阵的奇异值相关联可以简化分析，结果表明，最方形的方向为低秩重构提供了最佳的数据结构。在tSVD的第一步之后，进行多维傅立叶变换，张量的额叶切片形成共轭对。对于每对，可以用便宜得多的共轭计算代替奇异值分解，从而可以更快地计算tSVD。在我们改进的tSVD算法中使用共轭对称性可以将内部循环的运行时间减少35％–50％。我们考虑了维京人Graben地区和澳大利亚西北大陆架的合成和真实地震数据集，它们以高维张量排列。我们将基于tSVD的重建与传统方法进行了比较，投影到凸集上并进行多通道奇异频谱分析，我们发现基于tSVD的方法具有相似或更好的准确性，并且效率更高，与运行时的收敛速度比传统方法快一个数量级。此外，我们验证了与其他方向相比，最方形的方向可使这些示例的恢复率提高10％–20％。