Introduction

Seismic noise attenuation is one of the most significant steps in the whole seismic data processing and imaging workflow. It has great influence to many subsequent processing tasks, such as amplitude-variation-offset inversion, reverse time migration, full waveform inversion, and automatic interpretation for oil&gas detection (Chen, 2018; Qu et al., 2016; Gao et al., 2016; Xue et al., 2016b; Gan et al., 2016; Huang et al., 2016; Xue et al., 2016c; Wang et al., 2015; Zeng et al., 2017; Zhang et al., 2017; Asgedom et al., 2017).

In the past several decades, a large number of algorithms have been developed for seismic noise attenuation. Stacking the seismic data along the spatial directions, e.g., the offset direction, can enhance the energy of spatially coherent useful waveform signals as well as mitigate the spatially incoherent random noise (Wu and Bai, 2018c; Xie et al., 2016; Liu et al., 2009a). One of the commonly used state-of-the-art algorithms is the prediction-based method, including t-x predictive filtering (Abma and Claerbout, 1995), f-x deconvolution (Canales, 1984), the polynomial fitting based approach (Liu et al., 2011), and non-stationary predictive filtering (Liu and Chen, 2013; Liu et al., 2012). This type of methods utilize the predictive property of useful signals along spatial direction to create a regression-like model for distinguishing between signal and noise.

Another type of commonly used methods are based on data decomposition. This type of methods assume that noisy seismic data can be decomposed into different components where signal and noise are separated based on their frequency difference or morphological difference (Huang et al., 2017). Empirical mode decomposition (EMD) (Huang et al., 1998; Chen, 2016) and its improved version, e.g., ensemble empirical mode decomposition (EEMD) (Wu and Huang, 2009), complete ensemble empirical mode decomposition (CEEMD) (Colominas et al., 2012), have been used intensively for reducing the noise in seismic data (Chen et al., 2016a). Variational mode decomposition was proposed by Dragomiretskiy and Zosso (2014) for substituting EMD because of its explicit control on the decomposition performance. It has been utilized for noise attenuation in Liu et al. (2017) and for time-frequency analysis by Liu et al. (2016a). Regularized non-stationary decomposition (Wu et al., 2016; Yang et al., 2014) is another decomposition method which is also based on a solid mathematical model. Singular value decomposition (SVD) can also be used to extract the most spatially coherent components from the multi-dimensional seismic data (Siahsar et al., 2017).

Sparse transform based approaches assume that multi-dimensional seismic data can be compressed in a sparse transformed domain, where the signal is represented by high-amplitude coefficients and the noise is represented by small-amplitude coefficients (Bai and Wu, 2018; Gholami, 2013). Hence, by transforming data to the sparse domain, a soft thresholding can be applied to reject those small-amplitude coefficients that correspond to noise, which is followed by an inverse transform from the thresholded coefficients to time-space domain (Chen, 2017). This type of methods are closely connected with the compressive sensing paradigm (Lorenzi et al., 2016). Widely used sparse transforms are Fourier transform, curvelet transform (Candès et al., 2006; Zu et al., 2017; Herrmann and Hennenfent, 2008; Herrmann et al., 2007; Wang et al., 2011), seislet transform (Chen and Fomel, 2015a; Gan et al., 2015a,b; Fomel and Liu, 2010), shearlet transform (Kong et al., 2016), Radon transform (Foster and Mosher, 1992), and a variety of sparse wavelet transforms (Mousavi and Langston, 2016b; Anvari et al., 2017), e.g., synchrosqueezing (Daubechies et al., 2011; Mousavi and Langston, 2017,2016a; Mousavi et al., 2016) or empirical wavelet transforms (Liu et al., 2016b), etc. Recently, the adaptive dictionary learning has gained a lot of attention in the seismic data processing field (Chen, 2017; Wu and Bai, 2018b). The dictionary learning based sparse representation differs from the traditional sparse transforms in that the basis functions for the sparse transform are adaptively learned from the data itself, instead of being fixed in the traditional transforms.

Rank reduction methods are one of the most effective methods in the seismic data processing community, which includes the Cadzow filtering (Trickett, 2008), singular spectrum analysis (Wu and Bai, 2018a; Vautard et al., 1992; Zhou et al., 2018), damped singular spectrum analysis (Zhang et al., 2016a,b; Chen et al., 2016b), and multi-step singular spectrum analysis (Zhang et al., 2016c). There are two least-squares projection step in the damped singular spectrum analysis method. The first step can be considered as a rank reduction method while the second step can be interpreted as a compensation step for the non-optimal performance of the rank-reduction method, i.e., the approximated signal subspace in the traditional rank-reduction framework is a mixture of both signal and noise subspaces. From a different aspect, Xue et al. (2016a) proposed a rank-increasing method for iteratively estimating the spike-like noise instead of estimating signals in deblending of simultaneous-source data (Zu et al., 2016b; Zhou and Han, 2018; Bai et al., 2018b; Zu et al., 2016a; Bai and Wu, 2017; Wu and Bai, 2018d; Bai et al., 2018a).

Mean and median filters utilize the statistical difference between signal and noise to reject the Gaussian white noise or impulsive noise (Liu, 2013; Liu et al., 2009b). In addition to these classic noise attenuation methods, some advanced denoising methods have been proposed in recent years. Time-frequency peak filtering (Kahoo and Siahkoohi, 2009; Lin et al., 2013,2015) based approaches utilize the high-resolution property of time-frequency transform to distinguish between useful signals and random noise. Instead of developing a standalone denoising strategy, Chen and Fomel (2015b) proposed a two-step denoising approach that tries to solve a long-existing problem in almost all denoising approaches: the signal leakage problem. By initiating a new concept called local orthogonalization, Chen and Fomel (2015b) successfully retrieved the coherent signals from the removed noise section to guarantee no signal leakage in any denoising algorithms.

For all the aforementioned state-of-the-art noise attenuation algorithms, none of them are specifically designed for preserving the strong amplitude-variation details in seismic data. As we know, the amplitude variations in seismic data greatly affect the subsurface oil & gas exploration and production. Hence, the amplitude preserving capability is one of the backbone features we need to keep in mind when designing a new denoising algorithm. In this paper, we are solving the serious problem that is often neglected in traditional seismic data processing by proposing the plane-wave orthogonal polynomial transform method. Here we want to clarify that the amplitude variation we mention here refers to strong amplitude variation, not simply the edge details, or weak signals that are often mentioned in the literature. We first introduce the basic knowledge of the orthogonal polynomial transform (OPT), which is the key component that brings us the amplitude-preserving capability in the proposed framework. We then introduce the theory of plane-wave trace continuation that is used for flattening the seismic events without damaging the amplitude information. We show that both plane-wave trace continuation and OPT can well perserve the amplitude variation details in the seismic data, which accounts for the superb performance of preserving the amplitude details in the real data applications. Considering the strong influence of the slope estimation to the plane-wave flattening, we introduce a robust slope estimation method that can substitute the traditional plane-wave destruction (PWD) (Fomel, 2002) based methods in the presence of strong noise. A group of synthetic, pre-stack and post-stack field seismic data are used for demonstrating the performance of the proposed framework.