Multidimensional seismic data reconstruction and denoising can be achieved by assuming noiseless and complete data as low-rank matrices or tensors in the frequency-space domain. We have adopted a simple and effective approach to interpolate prestack seismic data that explores the low-rank property of multidimensional signals. The orientation-dependent tensor decomposition represents an alternative to multilinear algebraic schemes. Our method does not need to perform any explicit matricization, only requiring calculation of the so-called covariance matrix for one of the spatial dimensions. The elements of such a matrix are the inner products between the lower dimensional tensors in a convenient direction. The eigenvalue decomposition of the covariance matrix provides the eigenvectors for the reduced-rank approximation of the data tensor. This approximation is used for recovery and denoising, iteratively replacing the missing values. Synthetic and field data examples illustrate the method’s effectiveness for denoising and interpolating 4D and 5D seismic data with randomly missing traces.

中文翻译：

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叠前地震数据的重构和方向相关张量分解的去噪

通过将无噪声和完整的数据假定为频空域中的低秩矩阵或张量，可以实现多维地震数据的重建和去噪。我们采用了一种简单有效的方法来插补叠前地震数据，以探索多维信号的低秩特性。依赖于方向的张量分解表示多线性代数方案的替代方案。我们的方法不需要执行任何明确的矩阵化，仅需要为空间维度之一计算所谓的协方差矩阵。这种矩阵的元素是较低维张量之间在方便方向上的内积。协方差矩阵的特征值分解为数据张量的降秩近似提供了特征向量。该近似值用于恢复和去噪，以迭代方式替换丢失的值。合成和现场数据示例说明了该方法对带有随机丢失迹线的4D和5D地震数据进行去噪和插值的有效性。