Introduction

Random noise attenuation is an important task in seismic data processing (Huang et al., 2015; Yang et al., 2015a; Gulunay, 2000; Qu et al., 2015; Liu et al., 2011; Neelamani et al., 2006; Gan et al., 2015b; Yuan et al., 2012; Liu et al., 2012). Among different random noise attenuation approaches, the transform domain thresholding approach is one of the most widely-known approaches (Chen et al., 2016). The principle of this type of approach is simple: seismic reflection has coherent structure and can be sparsely represented while random noise are spreading through the whole transform domain, thus random noise can be removed by applying a simple thresholding operator in the transformed domain. One of the most well-known transforms might be the Fourier transform, or

transform. linear events are transformed into narrow strings in the

domain. The basic assumption of using

transform based thresholding is that the data is built of monochromatic plane waves. To consolidate the assumption, windowed

transform is usually used to ensure the linear property of local seismic events. Another emerging sparsity-promoting transform is Radon transform. According to the shape of the integral operator in the Radon transform, Radon transform can be divided into different types. Among the most popular types are linear Radon transform (also known as slant stack), parabolic Radon transform, hyperbolic Radon transform, and polynomial Radon transform (Xue et al., 2014,2016a). As the standard Radon transform operator is not unitary, least-squares and high-resolution versions of Radon transform are often used to ensure a practical application of the Radon transform. The basic assumption behind the Radon transform based thresholding is that the shape of seismic events follows the shape of the integral operator that is used in the Radon transform. The curvelet transform is becoming more and more popular in the field of exploration geophysics because of its multi-dimensional and multi-scale properties (Neelamani et al., 2008; Hennenfent and Herrmann, 2006; Neelamani et al., 2010; Liu et al., 2016). The curvelet transform can give a very sparse representation of a seismic wavefield which has advantages for interpolation and denoising of these data (Hermann et al., 2007). Therefore it has become more and more popular in exploration geophysics in recent years. Because the curvelet transform takes both the direction and scale into account, it can get sparser representation for complex data than many other alternatives. The curvelet transform does not have any a prior knowledge of the seismic data, which is designed for a general image processing task. Recently popular shearlet transform is also a natural extension of wavelet transform to accommodate the fact that multivariate functions are typically governed by anisotropic features such as edges in images. All of the aforementioned sparsity-promoting transforms are parameter-dependent, rather than data dependent. For example, the good compression performance using curvelet transform depends on the input parameters, such as the scales and the directions, while the seislet transform only depends on the local slope that is estimated directly from the data.

Fomel and Liu (2010) propose a wavelet-like transform that is tailed especially to seismic data, called seislet transform. The seislet transform follows the lifting scheme (Sweldens, 1995) in constructing the second generation wavelet transform. The only difference between the seislet transform and the general second-generation wavelet transform is that the prediction operator in the seislet transform framework is defined as the prediction between different traces according to local slope. The key component of the seislet transform is the estimation of local slope of seismic data using plane-wave destruction algorithm. The seislet transform is a breakthrough in the development of fixed basis sparsity promoting transforms because it can be data adaptive as long as the slope estimation is up to a certain degree of accuracy. The seislet transform has been successfully used in random noise attenuation, seismic data interpolation, stacking without estimating the normal-moveout velocity (Fomel and Liu, 2010), and separation of simultaneous-source seismic data. As the seislet transform compresses the seismic data along local structures, thus seislet thresholding (Fomel and Liu, 2010) can be considered as the simplest structural filtering approach. Different from images from the digital signal processing field, seismic images have well-constructed geological structures (e.g. coherent along the spatial dimension), and thus are suitable for structural filtering in order to enhance useful signals and removing noise (Hale, 2011; Gan et al., 2015a; Xue et al., 2016b; Yang et al., 2015b). However, structural filtering usually requires a precise local slope estimation, which can be demanding in complex seismic profiles because of the difficulty in calculating conflicting dips.

Because of the close dependence on slope estimation, the seislet transform can be limited to practical applications for some specific datasets when slope estimation is not acceptable, such as dip-conflicting profiles and highly noisy profiles. In order to solve this problem, several alternatives to the plane-wave destruction based seislet transform have been proposed. Liu and Fomel (2010) proposed the offset continuation based seislet transform. In the offset continuation based seislet transform, the offset continuation operator is used as a replacement of plane-wave destruction operator for the prediction between different traces. The offset continuation operator can only be used to predict traces along the offset direction because of the sole continuation direction but cannot be used to predict traces along the midpoint direction, thus it cannot be used in image domain (common offset gathers). Liu and Liu (2013) introduced the velocity dependent seislet transform. The velocity dependent seislet transform uses normal-moveout based velocity analysis to obtain velocity spectra and transforms the normal-moveout velocity to local slope which is required by the plane-wave destruction based seislet transform. However, the velocity dependent seislet transform can only be applied in common midpoint gathers since one cannot apply NMO-based velocity analysis in common offset gathers. The problem of estimating correct local slope in common offset gathers, or stacked image, is still an unsolved problem.

Empirical mode decomposition (EMD) based dip filter was first proposed by Chen and Ma (2014). The EMD based dip filter is a data-driven adaptive dip filter. EMD based dip filter utilizes EMD in each frequency slices to separate different spatial oscillating (wavenumber) components that corresponds to different dip components. Because of the adaptive property, the only parameter one need to define is the number of dip components. Chen and Ma (2014) utilized EMD based dip filter to improve the denoising performance of - predictive filtering. - EMD filtering can also be understood as EMD based dip filter considering that random noise, steeply dipping coherent events and ground rolls can be removed by applying a high-cut EMD based dip filter, as mentioned in (Chen and Ma, 2014).

In this paper, I propose an effective way to solve the conflicting-dip problem of many structural filtering by separating different dips of the seismic image first, and then applying the structural filtering to each dip-separated gather secondly. I take the seislet thresholding as an example to show the philosophy of the proposed methodology. When dip conflicts exist, the seislet transform cannot get a good compression performance. For this case, I first apply an EMD based dip filter to separate the seismic data into different dip components such that no conflicting dips exist in each component. Then I apply the traditional seislet thresholding to each separated component to remove random noise and finally I combine all the denoised components together to obtain the output data. The proposed multi-step strategy is very efficient and convenient to implement because no local window is needed for the processing and the EMD based dip filter is data adaptive, without the need to select many parameters that required by other dip filters. The proposed approach is a general framework that can deal with the troubles in any structural filtering approaches by separating the dip components first and then filtering secondly. Since almost all the structural filtering approaches need the calculation of structural information, such as the local slope, and thus will encounter the similar problem as the seislet transform in estimating local slope. Although some novel transforms, such as shearlets, may have the potential to resolving conflicting dips without dip calculation, these transforms are relatively new to seismic data processing and their behaviors in seismic data processing still remains to be investigated. Thus, I only compare the proposed approach with the most widely used curvelet transform.

I organize the paper as follows: I first give short reviews on seislet domain thresholding and point out its dip dependence problem, then I review the adaptive EMD based dip filter and propose the multi-step random noise attenuation by cascading EMD based dip filter and seislet thresholding and give detailed demonstration on how to implement the proposed framework, finally I use both synthetic and field data examples to demonstrate the performance of the proposed approach. The contribution of this paper can be summarized into two aspects. First, the paper relieves the slope dependence of the seislet transform in sparsifying seismic data by applying the seislet transform to EMD based dip-separated seismic images and thus slope estimation can be much more precise. Second, the paper proposes a general dip-separated image filtering framework that can be extended to many other methods which depend on slope information, by utilizing the adaptive property of EMD in separating different dip components.

2020-02-28