The 3-D seismic data reconstruction can be understood as an underdetermined inverse problem, and thus, some additional constraints need to be provided to achieve reasonable results. A prevalent scheme in 3-D seismic data reconstruction is to compute the best low-rank approximation of a formulated Hankel matrix by rank-reduction methods with a rank constraint. However, the predefined Hankel structure is easily damaged by the low-rank approximation, which leads to harming its recovery performance. In this article, we present a structured tensor completion (STC) framework to simultaneously exploit both the Hankel structure and the low-tubal-rank constraint to further enhance the performance. Unfortunately, under the assumption of elementwise sampling used by existing methods, STC is intractable to be solved since Hankel constraints cannot be expressed as linear tensor equations. Instead, tubal sampling is proposed to describe the missing trace behavior more accurately and further build a bridge between tensor and matrix completion (MC) to overcome the solving issue in two aspects: through the bridge from tensor to MC, STC can be solved efficiently using MC from random samplings of each frontal slice in the Fourier domain. Through the bridge from matrix to tensor completion, various tensor models within the framework can be developed from noise-specific MC to meet the need for data reconstruction in changeable noise environments. Moreover, alternating-minimization and alternating-direction methods of multipliers are developed to solve the proposed STC. The superior performance of STC is demonstrated in both synthetic and field seismic data.

中文翻译：

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管道采样：在3D地震数据重建中桥接张量和矩阵完成

3-D地震数据重建可以理解为欠定的反问题，因此，需要提供一些其他约束才能获得合理的结果。3-D地震数据重建中的一种流行方案是通过具有秩约束的秩减小方法来计算公式化汉克矩阵的最佳低秩逼近。但是，预定义的汉克尔结构很容易受到低秩近似的破坏，从而损害了其恢复性能。在本文中，我们提出了一个结构化的张量完成（STC）框架，以同时利用汉克尔结构和低管形约束来进一步提高性能。不幸的是，假设现有方法使用的是元素抽样，由于汉克尔约束不能表示为线性张量方程，因此STC难以解决。取而代之的是，建议进行输卵管采样以更准确地描述缺失的迹线行为，并进一步在张量和矩阵完成（MC）之间建立桥梁，以克服两个方面的求解问题：通过从张量到MC的桥梁，可以使用MC来自傅里叶域中每个额叶切片的随机采样。通过从矩阵到张量完成的桥梁，可以从特定于噪声的MC中开发框架中的各种张量模型，以满足在可变噪声环境中重建数据的需求。此外，发展了乘数的交替最小化和交替方向方法来解决所提出的STC。