Low‐rank seismic denoising with optimal rank selection for hankel matrices

Abstract

A B S T R A C TBased on the fact that the Hankel matrix representing clean seismic data is low rank, low-rank approximation methods have been widely utilized for removing noise from seismic data. A common strategy for real seismic data is to perform the low-rank approximations for small local windows where the events can be approximately viewed as linear. This raises a fundamental question of selecting an optimal rank that best captures the number of events for each local window. Gavish and Donoho proposed a method to select the rank when the noise is independent and identically distributed. Gaussian matrix by analysing the statistical performance of the singular values of the Gaussian matrices. However, such statistical performance is not available for noisy Hankel matrices. In this paper, we adopt the same strategy and propose a rule that computes the number of singular values exceed the median singular value by a multiplicative factor. We suggest a multiplicative factor of 3 based on simulations which mimic the theories underlying Gavish and Donoho in the independent and identically distributed Gaussian setting. The proposed optimal rank selection rule can be incorporated into the classical low-rank approximation method and many other recently developed methods such as those by shrinking the singular values. The low-rank approximation methods with optimally selected rank rule can automatically suppress most of the noise while preserving the main features of the seismic data in each window. Experiments on both synthetic and field seismic data demonstrate the superior performance of the proposed rank selection rule for seismic data denoising.

Reply to letter to the editor “Low‐rank seismic denoising with optimal rank selection for Hankel matrices” by S. Trickett

“…We welcome the letter by Trickett (2020) and thank the author for providing the in‐depth comments on our paper (Wang et al ., 2020) and pointing out the very related work (Trickett, 2015), which we were not aware of; we apologize for the omission. Onwards, we will refer to Trickett (2020) as T20, Trickett (2015) as T15, and Wang et al ., (2020) as WZG20, respectively.…”

Seismic signal enhancement based on the low‐rank methods

“…The main requirement of the LR methods is the low rank of the frequency‐domain Hankel matrix. The rank of the Hankel matrix equals the number of distinct dips (Oropeza and Sacchi, 2011; Chen et al ., 2016; Wang et al ., 2020). However, the real seismic data are complicated, where the linear‐events assumption is not met.…”

Letter to the Editor on: Low‐rank seismic denoising with optimal rank selection for Hankel matrices by C. Wang, Z. Zhu, and H. Gu, Geophysical Prospecting 68, 892–909

“…A paper by Wang et al . (2020), which I will refer to as WZG20, discussed how a popular method of random noise attenuation can be made to automatically adapt to changing signal and noise conditions within the data. I delivered a similar paper at the 2015 SEG annual convention entitled `Preserving signal: Automatic rank determination for noise suppression’ (Trickett, 2015), which I will refer to as T15.…”