**1. Introduction**

Recently, multicomponent seismology has gained considerable attention due to providing more seismic wavefield information for imaging complex structures and predicting reservoirs [1]. Generally, three-component (3C) geophones simultaneously record two horizontal components and one vertical component of the incident seismic wave. Ground roll is coherent noise with characteristics of strong energy, low frequency, low velocity and elliptical polarization, which shield the seismic reflection wave from the middle-deep layer in near offset. Ground roll attenuation is a key step in seismic data processing. An adaptive matched filter is conventionally used to process each component separately for attenuating ground roll. This method consists of a prediction step and a robust subtraction step. Ground roll is predicted by data-driven interferometry method [2] or model-driven semi-analytic modeling [3] methods. Then, an optimization subtraction operation is used to adaptively separate the ground roll [2]. Filter methods in transform domain are also commonly used, such as the frequency-wavenumber (F-K) filter, Radon transform [4], Curvelet-domain filter [5] and time-frequency-domain filter [6], to attenuate the ground roll based on the different characteristics of frequency, velocity and polarization between the ground roll and seismic reflection waves. However, when the acquisition geometry is under sampled in space, the aliasing problem can decrease the effect of these methods. In land multicomponent seismic exploration, 3C seismic data are recorded and processed as a vector, which provides not only a robust analysis of each individual component but also valuable information about the coherency components [7,8].

**Citation:** Xiao, L.; Zhang, Z.; Gao, J. Ground Roll Attenuation of Multicomponent Seismic Data with the Noise-Assisted Multivariate Empirical Mode Decomposition (NA-MEMD) Method. *Appl. Sci.* **2022**, *12*, 2429. https://doi.org/10.3390/ app12052429

Academic Editor: Andrea Paglietti

Received: 14 January 2022 Accepted: 23 February 2022 Published: 25 February 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/).

Empirical mode decomposition (EMD) proposed by Huang et al. [9] decomposes a signal into a series of intrinsic mode functions (IMFs). Each IMF has a relatively local constant frequency ranging from high frequency to low frequency. The EMD method does not require predetermined base functions and can adaptively separate nonlinear and non-stationary signals. EMD and its extensions have good performance on random and ground roll attenuation [10–12]. Chen et al. [13] used complete ensemble empirical mode decomposition (CEEMD) to improve the standard EMD method and obtained better results when they were applied to attenuate the ground roll.

The standard EMD method is conventionally conducted on each component separately, which results in misaligned IMFs for multicomponent data. The ground roll attenuation based on the standard EMD method damages the vector coherence of seismic data. Combining multiple components with a higher-dimensional signal could circumvent the mode alignment problem caused by the standard EMD method [14]. Rehman and Mandic [15] proposed a multivariate empirical mode decomposition (MEMD) method to process multicomponent signals and obtained the IMFs of aligned frequency range. These IMFs match well in the frequency scale and number properties for each component of the multicomponent data. In order to solve the mode mixing effect in standard EMD, extended EMD methods [13,16] have been proposed. Rehman and Mandic [17] proposed noise-assisted MEMD (NA-MEMD) to improve MEMD by adding extra components containing independent white noise to the original multivariate signal. This method helps overcome the mode-mixing problem existing in the extracted IMFs.

In this paper, we propose an alternative ground roll attenuation method for multicomponent seismic data in time domain. First, we start this article with a description of the MEMD and NA-MEMD methods. Then, we demonstrate the principle of the proposed method for ground roll attenuation of multicomponent seismic data. Finally, the synthetic and field seismic data tests demonstrate that the NA-MEMD method can effectively attenuate the ground roll. Compared to the F-K filter method, the proposed method preserves more low-frequency content of the seismic reflection wave and coherency information between components.

### **2. MEMD Method**

The EMD adaptively decomposes the original signal into a finite set of signals, called intrinsic mode functions, abbreviated IMFs [9]. Each IMF represents different vibrational modes embedded in the data, and also has a localized frequency content by preventing frequency spreading because of asymmetric waveforms. Obtaining the IMFs with a frequency range from high to low frequency is an iterative sifting procedure, as follows:


This two-step iterative process is repeated until the stopping criterion is satisfied. The resulting residue of this iteration can be regarded as the first IMF with the lowest frequency content. In standard EMD, the local mean is computed by taking an average of the upper and lower envelopes, and IMFs in turn are obtained between the local maxima and minima. For multicomponent signals, the local maximum and minimum values are not directly defined by the EMD. Rehman and Mandic [15] proposed to generate multiple *n*-dimensional envelopes by taking signal projections along different directions in *n*-dimensional spaces. Cubic spline interpolations in different directions are adopted to form multiple local envelops, and then these values are averaged. Later, the MEMD proposed was used to process the nonlinear *n*-dimensional time-series signals. Let the sequence {*V*(*t*)} = {*v*1(*t*), *v*2(*t*),..., *vn*(*t*)} represent a multivariate signal with *n* components, and *Xθ<sup>k</sup>* = *xk* <sup>1</sup>, *<sup>x</sup><sup>k</sup>* <sup>2</sup>,..., *<sup>x</sup><sup>k</sup> n* denote a set of direction vectors along the directions given by angles

*θ<sup>k</sup>* = *θk* <sup>1</sup>, *<sup>θ</sup><sup>k</sup>* <sup>2</sup>,..., *<sup>θ</sup><sup>k</sup>* (*n*−1) on an (*n* − 1)-dimensional sphere. The main steps of the MEMD method are as follows,


Through a series of MEMD processes, like the EMD decomposition algorithm, the *n*-dimensional multi-signal {*V*(*t*)} = {*v*1(*t*), *v*2(*t*),..., *vn*(*t*)} is decomposed into a series of IMF components and a residue *<sup>r</sup>*(*t*) <sup>=</sup> *<sup>V</sup>*(*t*) <sup>−</sup> *<sup>q</sup>* ∑ *i*=1 *di*(*t*), where *q* denotes the number of

IMF components.

The ability of MEMD which aligns to the intrinsic mode functions is demonstrated in Figure 1. We tested the MEMD method on synthetic data using a three-component signal s = [X, Y, Z] whose components are mixtures of a 16 Hz sinusoid common to all data channels, a 64 Hz and 4 Hz tone in the X component, a 32 Hz tone in the Y component and a 4 Hz tone in the Z components. We observed that all IMFs are three-dimensional and scale-aligned; the 16 Hz tone present in all data channels is localized in a single IMF3, while the 64 Hz, 32 Hz, and 4 Hz tones are localized in IMF1, IMF3, and IMF4, respectively. Such a strict mode alignment cannot be achieved when applying the standard EMD channel wise. Figure 2 shows that the different frequency scale modes exhibit the same IMF.

**Figure 2.** IMFs of the synthetic signal in Figure 1 obtained by standard EMD.
