Random noise attenuation plays an indispensable role in seismic data processing. The useful signal that is smeared in the ambient random noise is often neglected and thus may cause fake discontinuity of seismic events and artifact in final migrated image. Enhancing the useful signal while preserving edge properties of the seismic profiles by attenuating random noise can help reduce interpretation difficulties and misleading risks for oil & gas detection. There has been a bunch of random noise attenuation approaches.

The widely used $f-x$ deconvolution (Canales, 1984) can achieve good result for linear events but may fail in handling complex or hyperbolic events. The way to deal with this linear-events dependence limitation is to use local $f-x$ deconvolution with small windows. However, different window sizes will result in different denoising effects and the window size is actually data dependent. Thus, localized $f-x$ deconvolution with small processing windows is often hard to be implemented in practice. In addition to the local $f-x$ deconvolution with small windows, forward-backward prediction method (Wang, 1999) is also effective, but is still limited to linear events. A mean or median filter (Liu et al., 2009; Chen, 2014; Bonar and Sacchi, 2012) is often used to attenuate specific types of random noise. For example, a mean filter is only effective to attenuate highly Gaussian white noise, and a median filter (Chen et al., 2014c; Chen, 2014) can only remove spike-like random noise with excellent performance. An eigen-image based approach (Bekara and van der Baan, 2007), or sometimes referred as global singular value decomposition (SVD), is effective for horizontal-events seismic profiles, but can not be adapted to complex profile. Otherwise, much useful dipping energy will be removed. An enhanced version of this method turns global SVD to local SVD (Bekara and van der Baan, 2007), where a dip steering process is performed in each local processing window to make the local events flat. The problem for local SVD is that only one slope component for each processing window is allowed, and also the size of each processing window is often difficult to select. Matrix completion via $f-x$ domain multichannel singular spectrum analysis (MSSA) can handle complex dipping events well by extracting the first several eigen components after SVD for each frequency slice. The $f-x$ MSSA approach is based on a pre-known rank of the seismic data. However, for complex seismic data, the rank is hard to select, and for curved events, the rank tends to be high and thus will involve a serious rank-mixing problem. One widely used denoising approach that has relative ease for controlling parameters is spectral decomposition. Chen and Fomel (2014) proposed a post-processing strategy in order to guarantee no coherent signal is lost in the removed noise section.

Spectral decomposition of seismic data into different components is often used in random noise attenuation because useful energy and random noise usually reside in different spectral bands. Once signal and random noise are separated in different spectral scales, the random noise can be effectively attenuated by simply removing the large scale components. Existing spectral decomposition schemes include the Fourier transform, wavelet transform (Mallat, 2009; Zhang and Ulrych, 2003), curvelet transform (Candès et al., 2006), seislet transform (Fomel and Liu, 2010; Chen et al., 2014a), and matching pursuit method (Wang, 2007,2010). Each of them has some special properties.

Huang et al. (1998) proposed a new signal decomposition method called Empirical Mode Decomposition (EMD). The original aim of EMD is to stabilize a complex signal, that is, to decompose a signal into a series of Intrinsic Mode Functions (IMF). Each IMF has a locally constant frequency. The frequency of each IMF is decreasing according to the sequence in which each IMF is separated out. EMD is a breakthrough in the analysis of linear and stable spectrum, because it can adaptively separate a nonlinear and non-stationary signal, which is the feature of seismic data, into different frequency ranges. The random noise can be removed by removing the first IMF, which corresponds to the highest oscillatory components. EMD can also be utilized in each frequency slice in the $f-x$ domain to preserve horizontal energy, while leaving the dipping energy dealt with by other specific denoising approaches (Chen et al., 2014b,2015).

Fomel (2013) proposed a novel spectral decomposition scheme termed spectral decomposition using regularized non-stationary autoregression (SDRNAR), which aims at decomposing a seismic signal into sub-signals with smoothly variable frequency and smoothly variable amplitude. This method differs from EMD in that it can more explicitly control the frequency and amplitude of different components and their smoothness than EMD.

The motivation for Fomel (2013) to propose this method is to provide a faster and more precise way to implement spectral analysis. Inspired by the application of EMD to random noise attenuation in the signal-processing field, in this paper, we apply SDRNAR to random noise attenuation in the $t-x$ domain. Instead of removing the first IMF, we remove the residual of SDRNAR, because the random noise is thought to be the highest oscillatory and thus unpredictably distributed. The uniqueness of the proposed denoising approach is that all the spectral information as indicated by Fomel (2013) can be achieved during the denoising process. Compared with EMD, SDRNAR can separate the seismic data into different frequency component with much less frequency-component mixture. We show that SDRNAR can separate the useful signal components and noise effectively while EMD can not. We compared the proposed denoising approach with the well-known $f$-$x$ deconvolution and mean filter according to their denoising performance. Results show that, the proposed approach can preserve much more useful energy than $f$-$x$ deconvolution and mean filter.

We organize the paper as follows: we first introduce the basis of regression analysis, then we review the principle of SDRNAR and propose the application of SDRNAR to random noise attenuation, finally we use both synthetic and field data examples to demonstrate the performance of the proposed approach and make a comparison with the alternatives.