**1. Introduction**

The seismic exploration method is widely used in resource exploration and other fields, and the seismic source, as an important part of the seismic data acquisition, directly affects the exploration effect of the seismic exploration. With the rapid development of vibroseis technology, vibroseis is gradually replacing the traditional dynamite source for seismic data acquisition due to its advantages of safety, environmental protection, high quality, economy, and efficiency. However, limited by its own structure and exploration environment, the single vibroseis has the problems of small output force and limited exploration depth. In order to meet the demand of seismic exploration, the source array based on vibroseis technology is paid more and more attention. In the seismic data acquisition of source array, there are generally more serious coherent noises, such as multiple refraction waves and surface waves of strong energy, strong acoustic interference, and surface direct waves with wide frequency bands, which greatly affect the quality of the seismic data. The removal of coherent noise is essential for the exploration technique of the source array.

How to better remove the seismic noise and improve the SNR of the seismic data has been a hot research topic in the seismic exploration field [1–3]. For coherent noise, scholars of seismic exploration have presented many seismic denoising approaches to suppress it. The traditional noise removal methods mainly suppress coherent noise according to the difference between effective signal and coherent noise in apparent velocity and frequency, such as f-k filter [4,5], K-L transform [6,7], Radon transform [8,9], radial channel transform [10,11], empirical mode decomposition (EMD) [12,13], and so on. Wavelet transform [14,15] is an effective tool for seismic data processing, which has great advantages in one-dimensional data processing, though this advantage cannot be simply extended to two-dimensional or three-dimensional data. Therefore, multi-scale geometric analysis methods such as Ridgelet transform [16,17], Curvelet transform [18,19], Contourlet transform [20,21] and Shearlet

**Citation:** Yu, M.; Gong, X.; Wan, X. Seismic Coherent Noise Removal of Source Array in the NSST Domain. *Appl. Sci.* **2022**, *12*, 10846. https:// doi.org/10.3390/app122110846

Academic Editors: Guofeng Liu, Xiaohong Meng and Zhifu Zhang

Received: 7 September 2022 Accepted: 25 October 2022 Published: 26 October 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

transform [22] have been proposed successively. At present, multi-scale geometric analysis has been widely used in seismic data processing and achieved certain effect.

G. Easley, K. Guo, and D. Labate [23] proposed the Shearlet transform based on the basic theory of wavelet transform and the multi-scale analysis theory. Shearlet transform not only has good localization characteristics, scaling characteristics, direction sensitivity, and near-optimal sparse representation performance, but also has a simple mathematical structure and the transformed coefficients have a one-to-one correspondence with the image points. Therefore, compared with other multi-scale geometric analysis methods, Shearlet transform has a more delicate representation of the scale direction and the physical significance of the transformed coefficients is clearer, which is more conducive to the processing of the coefficients. In the process of the conventional Shearlet transform, the subsampling operation is carried out, which leads to spectrum aliasing in the low frequency and high frequency sub-bands decomposed by Shearlet transform. The spectrum aliasing will cause the information of the same direction to appear in several different sub-bands at the same time, which will weaken the direction selectivity. In addition, the subsampling operation makes the Shearlet coefficient low redundancy, resulting in the lack of the translation invariance. If it is used for image denoising, there will be an obvious ringing effect. In order to overcome the shortcomings mentioned above and enhance the directional selectivity and the translation invariance of Shearlet transform, the NSST came into being. NSST is a non-orthogonal transform. The NSST abandons the subsampling operation and combines the non-subsampled Laplacian pyramid transform (NSLP) with the nonsubsampled directional filter. After the NSST, the size of each direction sub-band at each scale is the same as the original image, which greatly improves the image redundancy. The enhancement of the image coefficient redundancy allows the NSST to have the translation invariance. The NSST has been successfully applied in many fields.

In the field of image processing, the NSST has been widely used. Guo, et al. (2013) proposed a new multi-focus image fusion algorithm based on the NSST [24]. Karami et al. (2016) adopted the NSST and the fully constrained least squares unmixing (FCLSU) to the denoising of hyperspectral images [25]. Priya et al. (2017) used the NSST based on the multislice fusion to the edge enhancement of liver CT images [26]. Qu et al. (2018) presented an image enhancement method based on the NSST and the directional information measurement [27]. Li et al. (2019) performed the enhancement of hyperspectral remote sensing images based on improved fuzzy contrast in the NSST domain [28]. Ramakrishnan et al. (2020) applied the NSST to the fusion of multiple exposure images [29]. Shen et al. (2021) made a change detection in SAR images based on the improved NSST and the multi-scale feature fusion CNN [30]. The NSST also has been increasingly applied in the seismic exploration field in recent years. Liang et al. (2018) presented a denoising method of microseismic data noise by the NSST based on singular value decomposition [31]. Sang et al. (2018) applied the NSST to petroleum seismic exploration denoising [32]. Sang et al. (2020) proposed a seismic random noise attenuation based on PCC classification in the NSST Domain [33].

The existing traditional denoising methods have achieved certain effects in the process of attenuating coherent noise but they also have some problems. For example, some methods will produce some new interference and some will lead to uneven energy after attenuating, even damaging the effective signal. Based on the coherent noise characteristics of the vibrator seismic data and the multi-scale and multi-directional properties of Shearlet transform, coherent noise removal of the source array in NSST domain is proposed in this article. The method is applied in both the synthetic seismic data and the filed seismic data. After processing with this method, the coherent noise in the seismic data is greatly removed and the effective signal information is greatly protected. The analysis of the results demonstrates the effectiveness and practicability of the proposed method on seismic coherent noise removal.
