**1. Introduction**

Noise decreases the signal-to-noise ratio (SNR) of seismic data and affects the quality of subsequent processes [1,2]. There are six types of noise recorded using geophones [3], in which periodic noise is a kind of noise caused by power lines, pump jacks [4], engine operation [5], or other interferences, shown as monochromatic noise or multitoned noise. Sometimes, periodic noise is so strong that seismic records are severely contaminated. However, it is not easy to attenuate periodic noise, since it overlaps with seismic waves in the time domain and the frequency domain.

For periodically monochromatic noise, the conventional method of attenuation is notch filtering, which requires exact knowledge of the frequency of monochromatic noise [6]. Obviously, the notch filtering method can attenuate seismic waves at the cutoff frequency. Model-based approaches [7–10] such as sinusoidal approximation are used to remove power line noise, though this method requires accurate estimation of noise frequency and needs significant computation time [11]. Henley [12] presented a spectral clipping method to detect monochromatic noise automatically; however, it is not applicable to weak periodic noise. Karsli and Dondurur [13] used an improved mean filtering method to attenuate power line harmonic noise without noise frequency estimation; however, this requires knowledge of the rough frequency band.

In recent years, following the application of wide-azimuth, broadband, high-density seismic acquisition technology—including micro-seismic observation [14] and time-lapse monitoring based on fiber sensing—the size of 3D seismic data is increasing, and the frequency band of reflections is becoming wider. Besides monochromatic noises suppression, recognition of multitoned noises and their automatic suppression are urgently needed.

**Citation:** Sun, L.; Qiu, X.; Wang, Y.; Wang, C. Seismic Periodic Noise Attenuation Based on Sparse Representation Using a Noise Dictionary. *Appl. Sci.* **2023**, *13*, 2835. https://doi.org/10.3390/ app13052835

Academic Editor: Lamberto Tronchin

Received: 15 January 2023 Revised: 18 February 2023 Accepted: 19 February 2023 Published: 22 February 2023

**Copyright:** © 2023 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/).

<sup>1</sup> SINOPEC Research Institute of Petroleum Engineering Co., Ltd., Beijing 102206, China

Methods based on sparsity representation, such as S-transform, singular spectrum analysis and empirical mode decomposition, extend realization of automatic denoising [15–17]. Xu et al. [4] proposed a method based on morphological diversity of monochromatic noise and seismic waves, which assumed that the raw data are only composed of two kinds of signals, the monochromatic noise and seismic waves. It is not applicable to seismic data with strong white Gaussian noise or multitoned noise.

In this paper, a new method is proposed to attenuate periodic noise. The proposed method is based on sparse representation using a noise dictionary. The novelty is the construction of a noise dictionary, which can represent periodic noise sparsely. First, the noise dictionary is constructed using ambient noise. Next, the noise dictionary is used to sparsely represent periodic noise. Then, the de-noised data are obtained by subtracting the periodic noise from the raw seismic data. The method is applied to synthetic and field seismic data. The effectiveness of the proposed method is that it can subtract periodic noise from raw seismic data without any notches in the spectrum compared with the notch filtering method.
