Sparse representation and inversion have been widely used in the acquisition and processing of geophysical data. In particular, the low-rank representation of seismic signals shows that they can be determined by a few elementary modes with predominantly large singular values. We review global and local low-rank representation for seismic reflectivity models and then apply it to least-squares migration (LSM) in acoustic and viscoacoustic media. In the global singular value decomposition (SVD), the elementary modes determined by singular vectors represent horizontal and vertical stratigraphic segments sorted from low to high wavenumbers, and the corresponding singular values reflect the contribution of these basic modes to form a broadband reflectivity model. In contrast, local SVD for grouped patch matrices can capture nonlocal similarity and thus accurately represent the reflectivity model with fewer ranks than the global SVD method. Taking advantage of this favorable sparsity, we introduce a local low-rank regularization into LSM to estimate subsurface reflectivity models. A two-step algorithm is developed to solve this low-rank constrained inverse problem: the first step is for least-squares data fitting and the second is for weighted nuclear-norm minimization. Numerical experiments for synthetic and field data demonstrate that the low-rank constraint outperforms conventional shaping and total-variation regularizations, and can produce high-quality reflectivity images for complicated structures and low signal-to-noise data.

稀疏表示和反演在地球物理数据的采集和处理中得到了广泛的应用。特别是，地震信号的低阶表示表明它们可以由一些具有主要大奇异值的基本模式来确定。我们回顾了地震反射率模型的全局和局部低阶表示，然后将其应用于声学和粘声介质中的最小二乘偏移（LSM）。在全局奇异值分解（SVD）中，奇异向量确定的基本模式代表从低到高波数排序的水平和垂直地层段，相应的奇异值反映了这些基本模式对形成宽带反射率模型的贡献。相比之下，分组块矩阵的局部 SVD 可以捕获非局部相似性，从而比全局 SVD 方法用更少的秩准确地表示反射率模型。利用这种有利的稀疏性，我们在 LSM 中引入局部低秩正则化来估计地下反射率模型。开发了一个两步算法来解决这个低秩约束逆问题：第一步是最小二乘数据拟合，第二步是加权核范数最小化。合成数据和现场数据的数值实验表明，低秩约束优于传统的整形和全变分正则化，并且可以为复杂结构和低信噪比数据生成高质量的反射率图像。