Attenuation of swell noise in marine streamer data via nonnegative matrix factorizationYoones Vaezi and Nasser Kazemi, Department of Physics, University of Alberta

SUMMARY

High-amplitude swell noise is a common sort of noise in ma-rine seismic surveys. It can obscure the lower amplitude sig-nals of interest and result in erroneous imaging and conse-quently wrong interpretations of the data. Therefore, it isof great importance to attenuate them in the early processingsteps. In this paper we propose a new alternating minimiza-tion technique for swell noise attenuation which is based onnon-negative matrix factorization of the power spectrum of thetime-frequency representation of the signal and noise compo-nents of the data. The key element of our method is that thenoise has sufficiently different time-frequency characteristicswith respect to the signals of interest. The noise componentis trained by a data-driven dictionary learning step. We havesuccessfully applied this technique to a shot gather largely con-taminated by swell noise. The swell noise is reduced signifi-cantly with very little impacts on the useful signals. We sug-gest that this method can be applied to ground roll attenuationas well.

INTRODUCTION

High-amplitude noise attenuation is one of the most time con-suming and challenging tasks in seismic data processing whichdirectly affects the reliability of the interpretations of the finalsubsurface images. These noises are characteristically dividedinto two categories: coherent high-amplitude noises, such asground roll, seismic interferences, etc., and incoherent high-amplitude noises, such as swell noise.

The principal idea behind almost all signal-to-noise ratio (SNR)enhancement methods is to transfer the data to a domain wherethe signal and the noise components are separable. Having re-moved the presumed noise, the data component is transformedback to normal physical t− x space. Therefore, the main chal-lenge is to find such a domain where the signal and the noiseare well separated.

The most common methods for attenuating coherent high am-plitude noise are Radon-based transforms in the τ− p domain,wavelet transforms (Deighan and Watts, 1997), and singularspectrum analysis (SSA) (Nagarajappa, 2012; Chiu, 2013). Theconventional methods for incoherent noise suppression are basedon either prediction error filters in the f − x (Canales, 1984)and t−x (Abma and Claerbout, 1995) domains or reduced rankfiltering, such as SSA (Chen and Sacchi, 2015; Sacchi, 2009;Oropeza and Sacchi, 2011) and eigenimage filtering (Ulrych etal., 1988). Time-frequency filtering has also been used as animportant and versatile approach for seismic data de-noising,for either coherent or incoherent noise (Elboth et al., 2010).

In this paper we propose the application of non-negative matrixfactorization (NNMF) with sparse coding for de-noising (Ci-chocki et al., 2009; Hoyer, 2002). The basic idea is that we can

obtain a meaningful part-based factor decomposition (Lee andSeung, 1999) from a single-channel time series imposing onlythe constraints of non-negativity and sparseness of the data.A nonnegative representation suitable for NNMF scheme canbe provided by transforming the data into the power spectraldomain.

Our approach is a modified version of the workflow developedby Cabras et al. (2010) for processing audio records. Their ap-proach was later used as a tool for single-channel source sepa-ration of seismic signals in a volcanic environment to separatea high convective, relatively transient, seismic source of inter-est from a low convective, relatively continuous, noise (Cabraset al., 2012), and also to remove relatively transient wind noisefrom a relatively continuous volcanic tremor signal (Cabras etal., 2014). A similar approach was followed by Mehmood etal. (2012) in a seismic footstep detection-based system to sep-arate the human footstep signatures from the horse footstepspectral signatures.

In the marine environments, swell noise is an incoherent typeof noise that is characterized by low frequencies (2 - 10 Hz)and high amplitudes that can mask signals of interest (Elboth,2009). It usually affects a number of neighboring traces, andcan be observed in seismic data as vertical stripes. It has an ad-verse effect on seismic data quality and may even lead to tem-porary suspension of acquisition. We have applied our tech-nique to a seismic shot gather from a marine seismic surveythat suffers from significant swell noise (Figure 1a). The resultshows significant reduction of the swell noise.

THEORY

In this section we will explain an alternating minimization tech-nique for separation of the swell noise from the seismic records.The method uses non-negative matrix factorization of the powerspectrum of the time-frequency representation of the signaland noise components of the data. A data-driven dictionarylearning step is also introduced in the context of semi-blindsignal and noise separation optimization. Below we explainthe technique step by step.

Seismic time series can be written as

x = s+n, (1)

where x is the recorded time series, s is the desired signal andn is noise. In this abstract, we only consider swell noise formarine streamer data. Separation of the noise from the sig-nal for single-channel data is an underdetermined problem andis difficult to solve. However, one can take advantage of theredundant based transformation tools like short-time Fouriertransform (STFT) to separate the desired signal from noise. Aswe mentioned earlier, swell noise has low frequency character-istics and its high amplitude contribution can be seen mainly

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Swell noise attenuation

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Figure 1: (a) A marine seismic gather, showing clear swell-noise contamination (vertical stripes). Data courtesy of Fugro. (b) Theresult after swell-noise attenuation. (c) The difference section between (a) and (b). The data within the black boxes are comparedin Figure 2.

between 2−10 Hz in the STFT domain. Unfortunately, the de-sired signal and noise are sharing common image space in theSTFT domain and the noise part cannot be removed via low-cut filters without removing a significant part of the desiredsignal. As an alternative, we propose an optimization methodthat can recover the signal and noise with the smallest possi-ble a priori knowledge. If we assume that the noise and signalare not correlated, then equation 1 can be represented in STFTdomain as

X(t, f ) = S(t, f )+N(t, f ), (2)

where X(t, f ), S(t, f ), and N(t, f ) are STFT representation ofx, s, and n, respectively. Hereafter, for simplicity, we drop thetime and frequency dependency of the signal in the STFT do-main. We can then calculate the β -power spectra of the signalas

|X|2/β = |S|2/β + |N|2/β . (3)

The non-negativity of the power spectra can be used as a greatsource of knowledge for the noise and signal separation. Noiseattenuation can be posed as factorizing the signal into two non-negative parts. Non-negative matrix factorization is a powerfultechnique for such a problem.

Non-negative matrix factorization can be expressed as factor-izing a non-negative m× n matrix X ∈ R+ into two positive-valued rank k≤min(m,n) matrices: D which is a m×k matrix,called dictionary or basis function, and H that is a k×n matrix,called sparse code or coefficients. In other words,

X = DH+E, (4)

where E is the approximation error to be minimized. Follow-

ing the same rules equation 3 can be written as

X =[Dn Ds

][HnHs

]+E, (5)

where N = DnHn and S = DsHs. By solving equation 5 wecan design a Wiener filter that can remove one component orthe other from the original power spectrum representation ofthe recorded data. However, the number of unknowns is big-ger than the number of knowns and the problem is ill-posedand hard to solve. To tackle this problem, we will introduce adictionary learning step that can find the support regions andthe degree of importance of the signal and noise factors in thepower spectrum domain. Hence, we propose to modify equa-tion 5 to

X = Wn�N′+(1−Wn)�S′+E, (6)

where Wn ∈ R+ is some weighting dictionary for the noisecomponent which has values between 0 and 1, � stands forelement-wise multiplication, 1 is a matrix of ones with a propersize, and N′ = D′nH′n and S′ = D′sH′s. Unlike previous con-tributions, we are not going to use strict zeros- and ones-basedmasking matrices for the weighting function. In this way, wecan guarantee preserving the signal in the noisy regions andvice versa. Moreover, it is worth mentioning that because ofthe positivity of the weighting functions the non-negativity ofthe components will remain untouched and we can still benefitfrom non-negative matrix factorization. Later, we will discussin more details how one can build such a dictionary that candefine the support regions and the degree of importance of thepower spectrum coefficients in reconstructing the noise com-ponent. Equation 6 has five types of unknowns. To solve the

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Figure 2: (a), (b), & (c) The wiggle plots of the data corresponding to the windowed areas in Figure 1. The arrows show the tracewhose time-frequency representation is analyzed in Figure 3.

problem we propose to minimize

argmin{N′,S′,Wn}

12||X− [Wn�N′+(1−Wn)�S′]||22 +λ |H|1, (7)

where λ =[λn λs

], |H|1 =

[|Hn|1 |Hs|1

]T and T is trans-pose operator. Equation 7 will be solved by an alternatingminimization technique. We will solve one set of unknownsby fixing the others and then alternate between the solutionsuntil convergence. Our method works trace by trace on the β -power spectrum of the recorded signals. However, the initialdictionary is built based on the averaged version of the powerspectrum of several noisy traces in the gather. In other words,we select the traces with very low SNR in the gather. Next,we transform them into the STFT domain. Finally, we averagethe β -power spectra. Note, we need to have weightings withvalues between zero and one, hence, we divide the weightingmatrix by its maximum value. Initial estimate of the N′ is cho-sen to be the original β -power spectrum of the recorded sig-nal and S′ is initialized by a random matrix with independent,identically distributed samples drawn from a Gaussian distri-bution. Algorithm 1 shows the main steps of our optimizationprocedure.

After calculating Xn = D′nH′n and Xs = D′sH

′s one can build a

Wiener filter to recover the desired signal as

s = IST FT (X(t, f )� Xβ/2s

Xβ/2n + Xβ/2

s

), (8)

and the noise component by

n = IST FT (X(t, f )� Xβ/2n

Xβ/2n + Xβ/2

s

), (9)

where IST FT stands for inverse short-time Fourier transform.

Algorithm 1 Alternating minimization algorithm

1: procedure : INITIALIZE D′s,H′s,Wn,k

2: while {Not converged}3: calculate Z = NNMF(Wn�X,k) = D′nH′

n4: for kn = 1 : Nn

iter

5: normalize D′n => D′n =D′n||D′n||22

6: calculate X = D′nH′n

7: update H′n←H′

n�D′Tn Z

D′Tn Xn+λn

8: calculate X = D′nH′n

9: update D′n← D′n�ZH′Tn +D′n�(1(XnH′Tn �D′n))XnH′Tn +D′n�(1(ZH′Tn �D′n))

10: endfor11: RETURN D′n,H

′n

12: calculate Y = NNMF(X−Wn� X,k) = D′sH′s

13: for ks = 1 : Nsiter

14: normalize D′s => D′s =D′s||D′s||22

15: calculate X = D′nH′n + D′sH

′s

16: update H′s←H′

s�D′Ts Y

D′Ts Xn+λs

17: update H′n←H′

n�D′Tn Y

D′Tn Xn+λn

18: calculate X = D′nH′n + D′sH

′s

19: update D′s← D′s�YH′Ts +D′s�(1(XH′Ts �D′s))XH′Ts +D′s�(1(YH′Ts �D′s))

20: endfor21: RETURN D′n,H

′n,D

′s,H

′s

22: update Wn←Wn +D′nH′n

23: calculate Wn =Wn

max(Wn)

24: endwhile

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Figure 3: (a), (b), & (c) The β−power STFT representations of the trace number 245 at the original gather, de-noised gather, andthe noise section, respectively. The color bar shows the β−power STFT amplitude which is normalized separately for each section.

REAL DATA EXAMPLE

Figure 1a shows a shot gather from a marine seismic surveythat is largely contaminated by swell noise. Elboth et al. (2010)and Bekara and van der Baan (2010) have previously used anidentical shot gather for their de-noising studies. The brack-ets in this figure denote the traces that have been used fortraining the weighting dictionary of the noise (Wn in equa-tion 6). A heuristic value of β = 3 was selected as it pro-vides good NNMF data decomposition in terms of SNR, andalso it seems to produce the best separation results for the cur-rent dataset. Figure 1b shows the resulting section of our pro-posed de-noising technique after 3 outer iterations for weight-ing (Algorithm 1) and Figure 1c depicts the difference sectionbetween the original and the de-noised sections. The amountof swell noise is greatly attenuated while the signals of inter-est, including reflection and refraction arrivals, have been wellpreserved.

To better demonstrate the effectiveness of our suggested methodwe have plotted in Figure 2 the data associated with the win-dows highlighted in Figure 1. Figure 2 shows that the re-flection arrivals of interest that were once obscured by high-amplitude swell noise in the original section can now be easilytraced in the section obtained after noise attenuation.

Figure 3 shows the corresponding β−power STFT representa-tion of the entire sample trace pointed in Figure 2 by the blackarrows (trace number 245). The dominating swell noise at thefrequencies below 12 Hz have been greatly attenuated via ourde-noising approach, leading to a clean β−power STFT repre-

sentation which is most representative of the signal of interest(Figure 3b).

CONCLUSIONS

We have proposed a new alternating minimization techniquefor separation of swell noise from seismic records. It is basedon non-negative matrix factorization of the power spectrum ofthe time-frequency representation of the signal and noise com-ponents of the data. The procedure is data-driven and thusadaptive. We have applied this method to a shot gather froma marine seismic survey that is significantly contaminated byswell noise. The results show that this technique has been suc-cessful in effectively removing the swell noise with very lim-ited influence on the signal of interest. According to the the-ory, this technique is potentially applicable to any sort of noiseprovided that it has sufficiently different time-frequency char-acteristics with respect to the signals of interest and that it canbe trained by a part of the data that is mainly representative ofthe noise. With this in mind, we suggest that this techniquecan be used for ground roll attenuation as well.

ACKNOWLEDGMENTS

The authors would like to thank Thomas Elboth for providingthe data. The authors would also like to thank Mauricio Sac-chi for sharing with us interesting ideas about seismic noisesuppression.

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EDITED REFERENCES Note: This reference list is a copyedited version of the reference list submitted by the author. Reference lists for the 2016

SEG Technical Program Expanded Abstracts have been copyedited so that references provided with the online metadata for each paper will achieve a high degree of linking to cited sources that appear on the Web.

REFERENCES Abma, R., and J. Claerbout, 1995, Lateral prediction for noise attenuation by t-x and f-x techniques:

Geophysics, 60, 1887–1896, http://dx.doi.org/10.1190/1.1443920. Bekara, M., and M. van der Baan, 2010, High-amplitude noise detection by the expectation-maximization

algorithm with application to swell-noise attenuation: Geophysics, 75, no. 3, V39–V49, http://dx.doi.org/10.1190/1.3428749.

Cabras, G., S. Canazza, P. L. Montessoro, and R. Rinaldo, 2010, The restoration of single channel audio recordings based on nonnegative matrix factorization and perceptual suppression rule: Proceedings of the 13th International Conference on Digital Audio Effects, 458–465.

Cabras, G., R. Carniel, and J. Jones, 2012, Non-negative matrix factorization: An application to Erta Àle volcano, Ethiopia: Bollettino di Geofisica Teorica ed Applicata, 53, 231–242, http://dx.doi.org/10.4430/bgta0056.

Cabras, G., R. Carniel, J. Jones, and M. Takeo, 2014, Reducing wind noise in seismic data using non-negative matrix factorization: An application to Villarrica volcano, Chile: Geofísica Internacional, 53, 77–85, http://dx.doi.org/10.1016/S0016-7169(14)71491-6.

Canales, L., 1984, Random noise reduction: 54th Annual International Meeting, SEG, Expanded Abstracts, 525–527, http://dx.doi.org/10.1190/1.1894168.

Chen, K., and M. D. Sacchi, 2015, Robust reduced-rank filtering for erratic seismic noise attenuation: Geophysics, 80, no. 1, V1–V11, http://dx.doi.org/10.1190/geo2014-0116.1.

Chiu, S. K., 2013, Coherent and random noise attenuation via multichannel singular spectrum analysis in the randomized domain: Geophysical Prospecting, 61, 1–9. http://dx.doi.org/10.1111/j.1365-2478.2012.01090.x.

Cichocki, A., R. Zdunek, T. Phan, and S. Amari, 2009, Nonnegative matrix and tensor factorizations: applications to exploratory multi-way data analysis and blind source separation: Wiley, http://dx.doi.org/10.1002/9780470747278.

Deighan, A. J., and D. R. Watts, 1997, Ground-roll suppression using the wavelet transform: Geophysics, 62, 1896–1903, http://dx.doi.org/10.1190/1.1444290.

Elboth, T., B. A. Reif, and Ø. Andreassen, 2009, Flow and swell noise in marine seismic data: Geophysics, 74, no. 2, Q17–Q25, http://dx.doi.org/10.1190/1.3078403.

Elboth, T., I. Vik Presterud, and D. Hermansen, 2010, Time-frequency seismic data de-noising: Geophysical Prospecting, 58, 441–453, http://dx.doi.org/10.1111/j.1365-2478.2009.00846.x.

Hoyer, P.O., 2002, Non-negative sparse coding: Presented at IEEEWorkshop on Neural Networks for Signal Processing, 557–565, http://dx.doi.org/10.1109/NNSP.2002.1030067..

Lee, D. D., and H. S. Seung, 1999, Learning the parts of objects by non-negative matrix factorization: Nature, 401, 788–791, http://dx.doi.org/10.1038/44565.

Mehmood, A., T. Damarla, and J. Sabatier, 2012, Separation of human and animal seismic signatures using non-negative matrix factorization: Pattern Recognition Letters, 33, 2085–2093, http://dx.doi.org/10.1016/j.patrec.2012.06.015.

Nagarajappa, N., 2012, Coherent noise estimation by adaptive Hankel matrix rank reduction: 82nd Annual International Meeting, SEG, Expanded Abstracts, 1–5, http://dx.doi.org/10.1190/segam2012-0063.1.

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Oropeza, V., and M. D. Sacchi, 2011, Simultaneous seismic data denoising and reconstruction via multichannel singular spectrum analysis: Geophysics, 76, no. 3, V25–V32, http://dx.doi.org/10.1190/1.3552706.

Sacchi, M. D., 2009, FX singular spectrum analysis: Presented at CSPG CSEG CWLS Convention. Ulrych, T., S. Freire, and P. Siston, 1988, Eigenimage processing of seismic sections: 58th Annual

International Meeting, SEG, Expanded Abstracts, 1261–1265, http://dx.doi.org/10.1190/1.1892508.

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