块Hankel矩阵快速低秩估计及在地震信号中的应用

A Fast Rank reduction Algorithm Based on Block Hankel Hankel Block Matrix vector Multiplication with Applications to Seismic Signal Processing

摘要:

通过利用BHHB矩阵(复数块Hankel矩阵)的结构特点，提出了快速稳定的对BHHB矩阵进行SVD(奇异值分解)分解的方法.该方法首先进行Lanczos二对角化，若是对称BHHB矩阵，则进行三对角化来保持对称性；然后利用Twisted 分解方法对实二对角方阵（或对称三对角矩阵）进行SVD分解.此快速SVD算法的优势在于，Lanczos分解过程中使用了新的BHHB矩阵与向量的快速乘法，该乘法通过1维FFT(快速傅里叶变换)代替多维FFT，在加快计算速度的同时减少了存储量；而后Twisted 分解采用部分SVD而不是整体SVD，从而节约了计算时间.数值试验结果表明，快速SVD算法大大提高了计算效率，减少了存储空间；地震信号的实验结果说明，Cadzow滤波方法比目前常用的预测滤波技术效果更好，结合快速SVD算法后，能够快速有效去除信号中的噪声.因此，块Hankel矩阵的快速SVD分解算法在地震信号处理和其他涉及块Hankel矩阵的实际应用中,尤其是解决大规模问题方面,有很好的发展前景.

Abstract:

Cadzow filtering is a well known denoising technique in the seismic signal processing. This method first transforms the data measured in seismic remote sensing into a complex Block Hankel Hankel Block (BHHB) matrix， then it reduces the noise via singular value decomposition (SVD). Usually, the structure of BHHB matrix is ignored in the SVD computation， so that the computational time and memory storage are high especially when the size of matrix is large. This paper presents a fast and stable SVD algorithm for complex BHHB matrices. The fast SVD algorithm consists of two stages: Lanczos bidiagonalization (or tridiagonalization for symmetric BHHB matrix) and diagonalization using twisted factorization. By exploiting the structure of BHHB matrix, the SVD can be accelerated by a fast matrix vector multiplication based on the 1 D Fast Fourier Transform(FFT). Compared to the multi dimensional FFT implementation, the proposed method requires much less memory with the similar computational cost. Numerical experiments support this claim. Finally, the fast SVD method is applied to some seismic examples with the Cadzow filtering technique to reduce noises. It turns out that the proposed method is better than the prediction filtering and it is cost efficient in the speed and memory usage in seismic data processing, especially for large problems.