Robust Singular Spectrum Analysis for Erratic Noise Attenuation

1 Robust Singular Spectrum Analysis for Erratic Noise Attenuation Ke Chen*, University of Alberta, Edmonton, Canada [email protected] and Mauricio D. Sacchi, University of Alberta, Edmonton, Canada [email protected] Summary The Singular Spectrum Analysis (SSA) method, also known as Cadzow filtering, adopts the truncated singular value decomposition (TSVD) or fast approximations to TSVD for rank-reduction. SSA is efficient for attenuating Gaussian noise but it cannot eliminate erratic noise (non-Gaussian). We propose a robust SSA method for simultaneously removing Gaussian and non-Gaussian noise. A robust low rank approximation is used in the newly proposed method. Iteratively reweighted least squares (IRLS) is adopted to estimate the approximated robust rank reduction that is required by the SSA method. Synthetic and real data examples are used to illustrate the performance of the proposed method. Introduction Recently, several reduced-rank filtering techniques have been developed for random seismic noise suppression, e.g., f-xy eigenimage analysis (Trickett, 2003) and singular spectrum analysis method (Cadzow filtering) (Sacchi, 2009; Trickett, 2008). Rank-reduction methods have also been developed for simultaneous data completion and random noise attenuation (Oropeza and Sacchi, 2011; Trickett, 2010; Kreimer and Sacchi, 2012; Gao et al., 2013). These rank-reduction methods have two main advantages: first, they are easy and natural to be applied on multidimensional data; second, they preserve the signal. In the SSA method, the seismic data consisting of a superposition of plane waves is transformed to the frequency-space domain. SSA embeds each frequency slice into a Hankel matrix. The rank of this matrix should be equal to the number of distinct dips in the data. Additive incoherent noise in the data will increase the rank of the Hankel matrix. Thus, the denoising problem is posed as a matrix rank-reduction problem. Then, the anti-diagonal elements of the rank-reduced matrix are averaged to recover the signal in frequency domain. In general the TSVD or fast approximations to TSVD are applied in the SSA method. However, the TSVD approximates a matrix by one of a lower rank in a least squares sense. The latter leads to a suboptimal performance of the SSA method when the data are contaminated with erratic (non-Gaussian) noise. Erratic noise is often contained in seismic data in the form of noise bursts, incoherent signals arising from improper geophone coupling and source generated noise. Trickett (2012) proposed a robust rank-reduction filtering method by iteratively applying Cadzow filtering on the reweighted combination of observed and reconstructed data. In this abstract, we use an M-estimator (Huber, 1981) to compute the reduced rank approximation of the noisy Hankel matrix.