Introduction

The attenuation of random noise is an important subject in seismic data processing. The enhanced seismic signals with higher signal-to-noise ratio (SNR) can help interpreters to make more accurate decisions. There are generally four different categories of random noise attenuation approaches that exist in the exploration geophysics literatures. The first is based on the predictive property of useful signals in small spatial-temporal windows, such as $f-x$ deconvolution (Chen and Ma, 2014; Canales, 1984), $f-x-y$ prediction filtering (Wang, 2002,1999), $t-x$ prediction error filtering (Abma and Claerbout, 1995), and $f-x$ non-stationary polynomial fitting (Liu et al., 2011). Yang et al. (2015) proposed a novel trace-by-trace random noise attenuation approach based on the predictable spectral components property of useful seismic reflections using regularized non-stationary autoregression (Fomel, 2013). This approach does not require the spatial coherency assumption and has the potential to be widely used to denoise microseismic signal, the SNR of which is very low. The second is based on the statistical property of seismic profile, such as the mean filter (Bonar and Sacchi, 2012), median filter (Chen et al., 2014b; Liu, 2013; Chen, 2014). The third is based on extracting the principal components of seismic data, such as multichannel singular spectrum analysis (MSSA) (Chen et al., 2015; Oropeza and Sacchi, 2011) and EMD based approaches (Chen et al., 2014c). This type of approach is also related with those rank-reduction based approaches, eigenimage-based approaches. The fourth is based on a transformed domain thresholding strategy (Fomel and Liu, 2010; Chen et al., 2014a; Neelamani et al., 2008). The transform operator can be a fixed-basis sparsity-promoting transform, and can also be an adaptively-learned dictionary, while the fixed-basis transform enjoys better efficiency and learning-based dictionary enjoys better adaptivity. Apart from those aforementioned one-step noise attenuation, Chen and Fomel (2015) proposed a two-step approach for retrieving the lost useful signals from the noise section, which can be seen as the residual random noise attenuation.

Freire and Ulrych (1988) proposed to carry out rank reduction of seismic images in the $t-x$ domain via singular value decomposition (SVD). They applied SVD to the seismic data matrix and extracted the first singular value in order to remove random noise based on the assumption that the seismic data with only horizontal events have a rank of one. We refer to this method as the global SVD (GSVD), because this SVD method does not require to be implemented in small windows. The GSVD does not require regularly sampled data because no convolutional operator is used. As long as the seismic profiles are composed with horizontal events, GSVD can obtain a good denoising performance. However, the seismic profiles do not necessarily meet the requirement that there are no dipping events. For those complex profiles, GSVD cannot be applied. Freire and Ulrych (1988) also demonstrated that the GSVD is actually equivalent to Karhunen-Loeve (KL) (or principal component transformation) approach (Jones and Levy, 1987). Because of the horizontal-energy selective property of GSVD, Milton et al. (2009) proposed to use GSVD filtering to remove dipping interference, such as ground-rolls.

The $f-x$ domain rank reduction approaches are independent from dip, and therefore, do not require flattening. They require an eigen-decomposition of the spectral matrix of data, which is connected with Cadzow filtering (Cadzow, 1988) and multichannel singular spectrum analysis (MSSA) (Vautard et al., 1992; Oropeza and Sacchi, 2011). Although MSSA and Cadzow filtering are equivalent, they come from different signal analysis subfields. Cadzow filtering was proposed to denoise images, whereas MSSA was proposed to decompose time series arising in the study of dynamical systems (Oropeza and Sacchi, 2010). Cadzow filtering and MSSA are also referred to as the $f-x$ SVD (FXSVD). For FXSVD, the eigen-decomposition of spectral matrix is usually done by applying an SVD to a pre-constructed (block) Hankel matrix. However, this type of methods depends on the low-rank property of seismic data, in other words, the seismic profile should contain few linear events (2D) or few plane-wave components (3D). The low-rank and linear-events assumptions of the FXSVD are not always met, which enforces the FXSVD to be implemented in local processing windows. The division of local windows and decision of rank value, however, are sometimes an empirical and user-unfriendly process.

In this paper, we propose a novel SVD approach for enhancing useful reflections by removing random noise. The novel SVD is implemented obeying the structural information. We first flatten the seismic reflections according to the local slopes by trace prediction. In the flattened domain, we apply a GSVD to horizontal events in order to remove random noise. The flattened traces are then transformed back to the original structural shape by stacking the traces in the flattened domain. We refer to the proposed novel SVD denoising approach as structure-oriented SVD (SOSVD). The SOSVD does not require local processing windows as LSVD and FXSVD do, and it can distinguish signal and noise better than GSVD. Flattened events make the largest singular value components mainly correspond to useful events, which are much easier to be separated. However, GSVD may not be applied to flattened events.

We organize the paper as follows: we first introduce the well-known GSVD, LSVD along with dip steering, and the theoretical aspects of SOSVD. A brief review of the basic theory of the commonly used $f-x$ deconvolution is also provided in the appendix. We then apply three different SVDs and $f-x$ deconvolution onto three different synthetic examples (with increasing complexity) and one field data example that comes from the North Sea, and compare their performances. The comparisons show that the SOSVD can obtain much better performance in removing noise and preserving useful signals.