Geophysical Journal International Geophys. J. Int. (2017) 211, 1296–1318 Advance Access publication 2017 September 5 GJI Marine geosciences and applied geophysics doi: 10.1093/gji/ggx371 Low-frequency noise attenuation in seismic and microseismic data using mathematical morphological filtering Weilin Huang,1,2 Runqiu Wang,1 Shaohuan Zu1 and Yangkang Chen3 1 State Key Laboratory of Petroleum Resources and Prospecting, China University of Petroleum-Beijing, Beijing 102249, China and Imaging Laboratory, Earth and Planetary Sciences, University of California, Santa Cruz, CA 95064, USA 3 National Center for Computational Sciences, Oak Ridge National Laboratory, One Bethel Valley Road, Oak Ridge, TN 37831-6008, USA. E-mail: chenyk2016@gmail.com 2 Modeling Accepted 2017 September 1. Received 2017 August 30; in original form 2017 May 25 SUMMARY Low-frequency noise is one of the most common types of noise in seismic and microseismic data. Conventional approaches, such as the high-pass filtering method, utilize the lowfrequency nature and distinguish between signal and noise based on their different frequency contents. However, conventional approaches are limited or even invalid when the signal and noise shares the same frequency band. Moreover, high-pass filtering method will suppress not only low-frequency noise but also low-frequency signal when they overlap in a same frequency band, which is extremely important in the inversion process for building the subsurface velocity model. To overcome the drawbacks of conventional high-pass filtering approach, we developed a novel method based on the mathematical morphology theorem to separate signal and noise using their differences in morphological scale. We extracted empirical relation between morphological scale and frequency band so that the mathematical morphology based approach can be conveniently used in low-frequency noise attenuation. The proposed method is termed as the mathematical morphological filtering (MMF) method. We compare the MMF approach with high-pass filtering and empirical mode decomposition (EMD) approaches using synthetic, reflection seismic and microseismic examples. The various examples demonstrate that the proposed MMF method can preserve more low-frequency signal than the high-pass filtering approach, and is more efficient and causes fewer artefacts than the EMD approach. Key words: Image processing; Controlled source seismology. I N T RO D U C T I O N In seismic exploration, a high signal-to-noise ratio (S/N) is important for some of the processing tasks, such as amplitude variation with offset (AVO) analysis, seismic attribute analysis and microseismic monitoring, especially critical in detection of small microseismic events and consequent seismic attribute measurements. (Liu et al. 2012; Liu & Chen 2013; Chen 2015a,b; Chang et al. 2016; Gan et al. 2016b,c,d; Huang et al. 2016; Liu et al. 2016b; Mousavi & Langston 2016a; Naghadeh & Morley 2016; Ren & Tian 2016; Zhang et al. 2016b; Mortazavi et al. 2017; Siahsar et al. 2017b,c). Because of these problems, developing noise attenuation methods has a great importance. Researchers have put a lot of effort into developing advanced and efficient denoising techniques for seismic data (Jones & Levy 1987; Duncan & Beresford 1995; Deighan & Watts 1997; Sabbione & Velis 2010; Naghizadeh & Sacchi 2011, 2012; Chen et al. 2014, 2016b,d; Han & van der Baan 2015; Velis et al. 2015; Wang et al. 2015; Zhang et al. 2015a,b, 2016a; Zhuang et al. 2015; Chen 2016; Kong et al. 2016; Sun & Wang 2016; Liu et al. 2016c; Mousavi & Langston 2016c,b, 2017; 1296 C Mousavi et al. 2016; Wu et al. 2016; Huang et al. 2017a,b,c) through different approaches. Low-frequency noise is a common type of noise in seismic data. This type of noise, for example, ground rolls, low-frequency ambient noise, swell noise and cable strum noise, masks useful signals and negatively impacts on the post-processing. Existing approaches for suppressing low-frequency noise include single-channel and multichannel approaches. The single-channel approaches, such as 1-D high-pass filtering, empirical mode decomposition (EMD; Huang et al. 1998; Battista et al. 2007; Chen et al. 2012, 2015, 2016a, 2017b,d; Gan et al. 2016a), and variational mode decomposition (VMD; Dragomiretskiy & Zosso 2014; Liu et al. 2015, 2016a, 2017), utilize the temporal differences between signal and noise for the signal/noise separation. For the suppression of low-frequency noise, high-pass filtering is based on the low-frequency nature of noise, but suffers from the disadvantage that it also eliminates low-frequency signals when they overlap. The low-frequency signals play an important role in inversion (Pratt et al. 1998; Liu 1999; Prieux et al. 2013a,b; Qu et al. 2016; Xue et al. 2016c; Zhang et al. 2016c). EMD-based approach first decomposes a signal into a series of its sub-signals (or components), The Authors 2017. Published by Oxford University Press on behalf of The Royal Astronomical Society. Mathematical morphological filtering 1297 Figure 1. Five scale-band components of a full-band signal and corresponding amplitude spectra. (a) The first trace is the initial full-band signal; the second– sixth traces are five scale-band components of the initial signal. (b) The black, red, yellow, green, blue, and cyan curves are the amplitude spectra of the first–sixth traces, respectively. Figure 2. Experimental data for fitting eq. (21). namely intrinsic mode functions (IMFs), and then abandon those IMFs which correspond to the low-frequency during signal reconstruction (Chen & Fomel 2015; Chen et al. 2017a,c). This approach can preserve some low-frequency signals, but its implementation is computationally expensive. Moreover, this approach exhibits difficulty in separating weak signals from strong noise. This mostly occurs when a recorded trace contains strong, low-frequency noise with weak, high-frequency signals mixed (Battista et al. 2007). Multichannel approaches such as f–k filtering utilize the difference between signal and noise not only the in time direction but also in spatial trace direction. Because of the more information taken into consideration, f–k filtering method sometimes performs better than single-channel approaches. However, f–k filtering method tends to cause serious distortion of the signal, producing seismic sections with ‘wormy’ appearance, when the amplitude of noise is much stronger than the signal (Qu et al. 2015; Xue et al. 2016a; Zu et al. 2016). Besides, the design of optimal 2-D filters is nontrivial (Oppenheim 2009). Furthermore, f–k filtering method can only be used to remove regular noise like ground rolls, but is not suitable Figure 3. Comparison of MMF, EMD and high-pass filtering methods. From left to right: the first trace: signal, the second trace: low-frequency noise, the third trace: signal+noise (input data), the fourth–sixth traces: three denoised results using strong, moderate and weak high-pass filters, the seventh trace: denoised result using EMD method, the eighth trace: denoised result using MMF, and the 9th–13th traces: five corresponding errors. for those noises which have negligible coherency in adjacent traces, such as swell noise and cable strum noise. The method we have developed can suppress low-frequency seismic noise using mathematical morphological filtering (MMF). The mathematical morphology, since initially proposed by Matheron (1975) and further introduced by Serra (1982), has been developed into a formal method. Koskinen et al. (1991) introduced the soft mathematical operations, which can maintain most of the properties of standard morphological operations, and can obtain improved performance under certain conditions. Sinha & Dougherty (1992) developed a generalization of binary mathematical morphology based on fuzzy set theory, in which images are modelled as fuzzy subsets of the Euclidean plane or Cartesian grid, and the morphological operations are defined in terms of a fuzzy index function. The MMF is also widely used in the field of image processing such as image 1298 W. Huang et al. Figure 4. Time–frequency spectra of (a) signal, (b) low-frequency noise and (c) signal+noise (input data). The signal and noise share the same frequency band. classification (Wang et al. 1994; Li et al. 2016b), image segmentation (Pal & Pal 1993; Zhou et al. 2016), and noise suppression (Peters 1995; Li et al. 2016a; Yuan et al. 2016; Zhou et al. 2017). The mathematical morphology has attracted much attention in the signal and image processing communities, but it is still new in the seismological community. In this paper, we propose to use MMF as a single-channel technique for suppressing low-frequency noise in seismic and microseismic data. Attenuating low-frequency noise from seismic data without damaging useful information of underlying signal at lowfrequency bands, is a long-standing problem (Xue et al. 2016b, 2017). However, the low-frequency signals are extremely important in regularizing the seismic inversion problem and avoiding the cycle-skipping issue (Wu et al. 2014). If we simply apply a highpass filter to the frequency spectrum of the data, we can lose some useful information of reflected signals as well. The basic idea behind the proposed MMF method is that we utilize the structure element (SE) with a large morphological scale and slide it on each trace to reject the low-frequency noise. We systematically study the theory of the MMF method and discuss the connection and difference between scale and frequency that are used in the MMF method and high-pass filtering methods, respectively. As the first time, we investigate the optimal choice of SE and its influence on the final filtered results. To make the MMF method compatible for quantita- tive application in complicated situations, we provide an empirical relation between the scale and frequency by fitting both synthetic and field seismic data sets. This empirical relation let us conveniently use the MMF method to remove low-frequency noise of an arbitrary frequency band without damaging the useful signals. We compare the MMF method with the high-pass filtering and EMD approaches. Different synthetic, field seismic and microseismic data examples demonstrate that the MMF method can preserve more low-frequency signal than high-pass filtering and is more efficient and causing fewer artefacts noise than the EMD approach. B A S I S O F M AT H E M AT I C A L M O R P H O L O G I C A L F I LT E R I N G ( M M F ) Unlike the traditional signal processing methods used in the seismic community, MMF is based on the stochastic integral geometry theory (Schneider & Weil 2008) and the logic topology theory (Vickers 1996). Let set d = [d(t)] represent a seismic time-series. On the one hand, the morphological dilation d ⊕ b is the morphological operation that transforms d with respect to a given set, namely the SE b = [b(t)]: d(t − τ ) + b(τ ), (1) d⊕b= τ Mathematical morphological filtering 1299 Figure 5. Time–frequency spectra of (a)–(e) denoised results using strong, moderate and weak high-pass filters, EMD method and MMF method, respectively. where denotes supremum. Both t and τ are samples. On the other hand, the morphological erosion d b is the morphological operation that transforms d with the SE b: d(t + τ ) − b(τ ), (2) db= τ where denotes infimum. So far all morphological systems are based on parallel or serial interconnections of morphological di- lations or erosions (Serra 1982; Huang et al. 2017d). It can be seen that the morphological dilation is an operation that ‘grows’ or ‘thickens’ the object of interest (i.e. set d), while the morphological erosion is an operation that ‘shrinks’ or ‘thins’ the object of interest (i.e. set d). Combination of dilation and erosion operations will derive the compounded morphological operations: opening and closing. The opening of d by b (denoted by d ◦ b) and closing of d by b (denoted 1300 W. Huang et al. by d • b) are defined respectively as: d ◦ b = (d b) ⊕ b, (3) d • b = (d ⊕ b) b. (4) In addition, combination of opening and closing operations can produce the second-compounded morphological operations: openclosing and close-opening. The open-closing of d by b (denoted by d b) and closing-opening of d by b (denoted by d b) are defined respectively as: d b = (d ◦ b) • b, (5) d b = (d • b) ◦ b. (6) The average of open-closing and close-opening forms the MMF as: Figure 6. Computing time costs with different methods. Fb (d) = (d b + d b)/2, (7) where Fb denotes the MMF with SE b. For helping readers in implementing the MMF, appendix A provides pseudo-codes of the six basic morphological operations as expressed from eqs (1) to (6). Figure 7. Clean and noisy data. (a) and (b) Clean data and its amplitude spectrum. (c) and (d) Noisy data containing low-frequency noise and its amplitude spectrum. Mathematical morphological filtering 1301 Figure 8. Denoising comparison. (a), (c) and (e) Denoised results using high-pass filtering, EMD and MMF methods, respectively. (b), (d) and (f) The corresponding amplitude spectra. SCALE AND FREQUENCY MMF can give various scale structures of a signal using various SE. It acts as a morphological propagator propagating the initial signal into the scale space. Large-scale space (slowly varying energies) corresponds to the contoured information such as the tendency of energy propagating, small-scale space (fast varying energies) corresponds to the detailed information such as the subtle vibration. From one scale to the next, details vanish, but the contours of the remaining objects are preserved sharply and perfectly localized 1302 W. Huang et al. Figure 9. Errors of (a) high-pass filtering method, (b) EMD method and (c) MMF method. (Meyer & Maragos 2000). The MMF can separates the input signal into two sub-signals whose scales are greater or less than that of the SE, If we use a series of MMFs represented by Fb1 , Fb2 , . . . , Fbn , in which Scale(b1 ) < Scale(b2 ) < . . . < Scale(bn ), the various scale information of the signal can be obtain simultaneously: d = Fb (d) + d − Fb (d), d= >Scale(b) (8) <Scale(b) where d denotes the input signal, Scale(b) represents the scale of SE b. This filter is implicitly similar to the ‘low-pass’ or ‘highpass’ filter, but it is with scale, not frequency. If we use two MMFs represented by Fb1 and Fb2 in which Scale(b1 ) < Scale(b2 ), the seismic trace d can be written as: d = Fb2 (d) + Fb1 (d) − Fb2 (d) + d − Fb1 (d) . 3rd scale-band 2nd scale-band 1st scale-band (9) This filter is implicitly similar to ‘bandpass’ filter, but it is with scale-band, not frequency band. Apparently, in eq. (9), 1st scale-band < Scale(b1 ) < 2nd scale-band < Scale(b2 ) < 3rd scale-band. Fbn (d) (n+1)th scale-band (10) + Fbn−1 (d) − Fbn (d) nth scale-band + ... + Fb1 (d) − Fb2 (d) + d − Fb1 (d) . (11) ... 2nd scale-band 1st scale-band Eq. (11) refers to multiscale morphology analysis (Meyer & Maragos 2000; Li et al. 2016b). The morphological scale is similar to frequency (or wavelength). Generally speaking, large-scale signal corresponds to lowfrequency signal (or long-wavelength signal). On the contrary small-scale signal corresponds to high-frequency signal (or shortwavelength signal). However, different scale-bands may have overlapped frequency contents. MMF differs from frequency filtering in that filters of the filter bank do not correspond to frequency band filtering but instead to scale-band. In order to better show readers the relationship between scale and frequency, Fig. 1 gives an example of five decomposition levels of a full-band signal and Mathematical morphological filtering 1303 Figure 10. Clean and noisy data with varying frequency components and decreasing amplitude from shallow to deep. (a) and (b) Clean data and its amplitude spectrum. (c) and (d) Noisy data containing low-frequency noise and its amplitude spectrum. corresponding amplitude spectra. The detailed parameters to produce the decomposition will be given in the next section. The five decomposition levels of the full-band signal are shown in Fig. 1(a). The 1st trace is the initial signal with 1 maximum amplitude. The 2nd–6th traces are the five scale-band subsignals, whose scale increases correspondingly. Fig. 1(b) gives the corresponding amplitude spectra of initial signal and five scale-band components. The black, red, yellow, green, blue, cyan curves are the amplitude spectra of the initial signal and five scale-band components, respectively. It can be seen from Fig. 1 that the decomposed subsignals tend to capture longer period features of the signal at higher scales while they contain narrower frequency bands. CHOICE OF SE The SE is the only input parameter in the MMF method. In this section, we will discuss how to select SE appropriately. Essentially, SE is a small portion of signal whose length is generally much shorter than that of the signal to be processed. Theoretically, SE can be arbitrary forms of functions. For attenuation of low-frequency noise, we provide a type of SE, which is defined as: (12) b(A,L) (t) = A 1 − (t/L)2 , where t ∈ [−L, L]. b(A, L) (t) denotes SE, which is dependent on parameters A > 0 and L > 0. The reasons of choosing eq. (12) as the form of SE are (1) it is smooth and symmetrical, (2) it is simple and easy to construct and (3) it is effective working with MMF. Accordingly, MMF can be represented as: F(A,L) (d) = (d b(A,L) + d b(A,L) )/2. (13) Thus we can adjust the strength of MMF by setting different combinations of A and L. The MMF becomes rigorous (i.e. it passes fewer signals in a larger scale band) as A decreases, or L increases. On the contrary, the MMF becomes mild (i.e. it passes more signal in a smaller scale band) as A increases, or L decreases. An example is shown in Fig. 1 for better understanding the performance of MMF using difference parameters. Fig. 1(a) is the input signal (1st trace) and five scale-band components (2nd–6th). Eq. (11) is used to produce the decomposition, with four MMFs 1304 W. Huang et al. Figure 11. Denoising comparison. (a), (c) and (e) Denoised results using high-pass filtering, EMD and MMF methods, respectively. (b), (d) and (f) The corresponding amplitude spectra. Mathematical morphological filtering 1305 Figure 12. Errors of (a) high-pass filtering method, (b) EMD method and (c) MMF method. which are Fb1 = F(1.6,10ms) , Fb2 = F(1.6,30ms) , Fb3 = F(1.6,90ms) , and Fb4 = F(1.6,190ms) . If we use d to represent the 1st trace, the 2nd trace is d − F(1.6,10ms) (d), the 3rd trace is F(1.6,10ms) (d) − F(1.6,30ms) (d), the 4th trace is F(1.6,30ms) (d) − F(1.6,90ms) (d), the 5th trace is F(1.6,90ms) (d) − F(1.6,190ms) (d), and the 6th trace is F(1.6,190ms) (d). As we can see, increase of L makes MMF more rigorous, which passes longer period components (MMF removes the components whose scale is less than SE’s scale). C O M PA R I S O N O F M M F, E M D A N D F R E Q U E N C Y F I LT E R I N G the local minima is zero (Huang et al. 1998; Chen & Ma 2014). The mathematical principle of EMD can be briefly written as: [en ], (14) n=1 where d is the input data. en , n = 1, 2, . . . , Ne − 1 is the number of IMFs. The last component e N e is the residual. For low-frequency noise attenuation, one can reconstruct the decomposed signal by ignoring last several IMFs which mostly capture long period components: e d = e The MMF is similar to the well-known EMD algorithm and frequency filtering, all of which can decompose a signal into several components based in its characteristics. EMD is to empirically decompose a signal into a finite set of its subsignals (i.e. IMF), which satisfy two conditions: (1) in the whole data set, the number of extrema and the number of zero crossings must be either equal or differ at most by one; and (2) at any point, the mean value of the envelope defined by the local maxima and the envelope defined by N e d= N0 [en ], (15) n=1 where de represents the denoised data using EMD method, N0e < N e . Fourier transform can transform a signal into a sum of harmonic signals as: N f d= n=1 [fn ], (16) 1306 W. Huang et al. Figure 13. Denoising comparison of the first field data set. (a) Original data. (b) Denoised result using high-pass filtering method. (c) Denoised result using EMD method. (d) Denoised result using MMF method. where fn , n = 1, 2, . . . , Nf is the harmonic signals. Similarly, for lowfrequency noise attenuation, one can reconstruct the decomposed signal by ignoring last several IMFs which mostly capture long period components: N f df = [fn ], (17) ⎧ n = 1; ⎪ ⎨ d − Fb1 , 1 < n < Nm; mn = Fbn−1 − Fbn , ⎪ ⎩ Fb( N m −1) (...Fb2 (Fb1 (d))), n = N m . where mn , n = 1, 2, . . . , Nm is the multiscale components. The attenuation of low-frequency noise using morphology hold as: where df represent the denoised data using frequency filtering, f N0 < N f (in practise we generally use soft threshold which is gradual to 0, rather than ‘truncation’ threshold because of Gibbs phenomenon). Mathematical morphology can decompose a signal into several morphological scale components, N m n=1 [mn ], N m f n=N0 d= (19) (18) dm = [mn ], (20) n=N0m where dm represent the denoised data using frequency filtering, N0m < N m . It is worth mentioning that, in practise, we do not need to decompose data into multiscale components using several MMFs, instead, we only need to find the ‘threshold’ to separate useful signals and noise using one MMF. For the F(A,L) given in eq. (13), the ‘threshold’ depends on the two parameters A and L. In our Mathematical morphological filtering 1307 Figure 14. Removed noise sections using (a) high-pass filtering, (b) EMD approach, and (c) MMF. experience, it is more convenient that we fix A and change L to adjust MMF for seismic low frequency noise attenuation. If the amplitude of input data is normalized in advance, parameter A is usually chosen between 1 and 5. For the value of parameter L, based on our numerical experiments, the relationship between objective frequency band 0 − f and the choice of L approximately obeys the power function, and can be represented as: L ≈ 4.25 f −1.6 , (21) where the unit of L is second (s) and unit of f is Hertz (Hz). The experimental data for obtaining the empirical eq. (21) are shown in Fig. 2. Eq. (21) was obtained by fitting both field and synthetic data sets. For a better comparison of the performances of the three methods on the suppression of low-frequency noise, we show an experiment of a synthetic trace in Fig. 3. The synthetic signal is a Ricker wavelet with 100 Hz dominant frequency and π /2 initial phase, as the 1st trace shows. In Fig. 3, the 2nd trace is low-frequency noise, which shares a frequency band with the signal. This noise (the 2nd trace) is added to the synthetic signal (the 1st trace) to form the input data (the 3rd trace) for the test. The S/N of this input data is estimated to be −6.0746 dB using (Chen et al. 2016c; Zu et al. 2017a,b; Chen 2017; Siahsar et al. 2017a; Zhang et al. 2017): S/N = 10 log s 2F , n 2F (22) where s is the synthetic signal (trace 1) and n denotes the added noise (trace 2). · F denotes the Frobenius norm of an input matrix. The 4th–6th and 9th–11th traces show the three filtered data using strong (>100 Hz), moderate (>75 Hz), and weak (>50 Hz) high-pass filters, and the corresponding errors (the difference between the signal and denoised results), respectively. The corner frequencies (100, 75 and 50 Hz) are selected because the noise and signal overlap in the frequency band of 50−100 Hz. The 7th and 12th traces show the denoised signal using EMD approach and corresponding error respectively. The 8th and 13th traces show the denoised signal using MMF approach and corresponding error respectively. It is clear that MMF achieves the best performance of removing low-frequency noise, which removes most noise and preserves the signal well. 1308 W. Huang et al. Figure 15. Amplitude spectra of (a) original data and denoised results using (b) high-pass filtering method, (c) EMD method and (d) MMF method. The S/Ns of the denoised results using strong, moderate and weak high-pass filterings, EMD method and MMF are 3.3744, −0.7813, −1.7350, −3.1371 and 18.6402 dB, respectively. The calculation of S/N follows eq. (22), except that n denotes the error. Fig. 4 demonstrates the time–frequency spectra of clean signal, low-frequency noise and the contaminated signal (signal + noise), from which the overlapped frequency band can be clearly observed. Fig. 5 demonstrates the time-frequency spectra of the five denoised results. It can be observed that the time-frequency spectrum of denoised a result by MMF is closest to that of the synthetic signal, which indicates the superior performance of MMF in preserving spectral structure of underlying signal. Fig. 6 shows outcome of an experiment comparing the computing time costs of high-pass filtering, EMD, and MMF based methods. The input data is randomly created with size gradually increasing from 1000 to 10 000 samples. It can be observed that computing time costs of all the three methods increase approximately linearly, but the increasing rate of EMD method (black curve) is obviously greater than others. Due to the fast Fourier transform (FFT) algorithm, the frequency filtering is very efficient (red curve). The morphological operations consist of additions and subtractions, which need relatively small calculation effort. Thus the computation speed of MMF is satisfactory (blue curve). Besides, note that morphological operations are local operations in the sense that the outcomes of MMF at t0 only depend on the local properties of the initial signal d at t0 + / − τ and SE b at τ . This property indicates that MMF can be easily implemented in independent data bins, which raises a potential research topic about the parallel implementation of MMF to save computing time. SYNTHETIC EXAMPLE We first test the MMF technique on a synthetic example. The clean data is composed of three primary events, which are created by convolving a Rick wavelet with 60 Hz dominant frequency and 0 initial phase with three series of impulses, as shown in Fig. 7(a). We add low-frequency noise (band-limited Gaussian noise) to the clean data as shown in Fig. 7(c). The amplitude spectra of the clean and noisy data are shown in igs 7(b) and (d). As we can see, the clean data and noise share a frequency band of 0–60 Hz. The S/N of the noisy data is −5.4680 dB. The calculation of S/N follows eq. (22). We first use high-pass filtering to remove the low-frequency noise. The Mathematical morphological filtering 1309 Figure 16. Denoising comparison of the second field data set. (a) Original data. (b) Denoised result using high-pass filtering method. (c) Denoised result using EMD method. (d) Denoised result using MMF method. denoised result and its amplitude spectrum are shown in Figs 8(a) and (b), respectively. It is clear that high-pass filtering cuts off the low-frequency components directly, but removes the useful lowfrequency signal at the same time. We then apply EMD technique to the noisy data. The result and corresponding amplitude spectrum are shown in Figs 8(c) and (d), respectively. EMD can attenuate some noise and preserve some low-frequency signal. But it also leaves a lot of noise energy and causes heavy damage to the signal. The result of using MMF (A = 2, L = 80 ms) is shown in Fig. 8(e). Fig. 8(f) presents its amplitude spectrum. The strongest energy of the low-frequency noise is removed and the signal is preserved well. We can clearly see from Fig. 8(f) that the low-frequency components of the signal are kept very well. The denoising errors are presented in Fig. 9, which shows the difference between the clean data and denoised results. The S/Ns of the denoised results using highpass filtering, EMD and MMF approaches are 4.4224, −3.7284 and 12.3910 dB, respectively. Apparently, MMF obtains the small- est error except for a slight amplitude damage, and the highest S/N. In order to make the synthetic data closer to a real case, we adjust the three primary events: their the dominant frequencies and amplitudes decrease gradually from shallow to deep, as Fig. 10(a) shows. Similarly, the low-frequency noise (band-limited Gaussian noise) with shared frequency band (0−80 Hz) is added to the clean data, as Fig. 10(c) shows. The low-frequency noise is a band-limited Gaussian noise. The S/N of the input data is −5.8044 dB. Figs 10(b) and (d) show their amplitude spectra. The denoised results and corresponding amplitude spectra using high-pass filtering, EMD and MMF (A = 1, L = 50 ms) approaches are demonstrated in Fig. 11. The denoising errors are presented in Fig. 12. The S/Ns of the denoised results using the three approaches are 0.0807, −3.7108 and 8.9576 dB, respectively. We can see that the frequency filtering method fails when signal and noise share the same frequency band, because it cannot remove the noise from the mixed components. 1310 W. Huang et al. Figure 17. Removed noise sections using (a) high-pass filtering method, (b) EMD method and (c) MMF method. EMD method also shows the limited performance in low-frequency noise attenuation, because it is not elaborate enough to distinguish between signal and low-frequency noise, as we can observe from this example. The MMF method obtains the best performance in terms of suppressing most of the noise and well preserving the low-frequency signal. REFLECTION SEISMIC EXAMPLE To demonstrate how MMF works in practice, we apply MMF on two real reflection seismic data sets. The first real data set is shown in Fig. 13(a). There is a significant amount of noise corrupting the reflection signal because of the cable strum. The energy of the cable strum noise is much stronger than that of the seismic reflections from the seafloor and below. The high-pass filtering method helps uncovering the desired data as we can see from Fig. 13(b). The low-frequency noise is attenuated effectively. Fig. 13(c) presents the denoised result by the EMD approach. EMD also removes most energy of noise but it obviously introduces some artificial trends. This phenomenon arises from the fact that the EMD method is very sensitive to both the physical and digital characteristics of the signal being analysed. The sensitivity is high enough to pick up quantization and fidelity errors (Battista et al. 2007). The denoised data using the MMF method (A = 3, L = 21 ms) is shown in Fig. 13(d). MMF also successfully removes the low-frequency noise. Fig. 14 presents the corresponding removed noise sections by high-pass filtering, EMD and MMF approaches, respectively. Fig. 15 presents amplitude spectra corresponding to the frame boxes in original data (Fig. 13a) and denoised results by high-pass filtering (Fig. 13b), EMD (Fig. 13c) and MMF approaches (Fig. 13d), respectively. It seems that there is no significant difference between the performance of high-pass filtering and MMF methods from the profiles in the time domain (Figs 13b and d). However, from the amplitude spectrum of the denoised result by high-pass filtering (Fig. 15b), we Mathematical morphological filtering 1311 Figure 18. Amplitude spectra of (a) original data and denoised results using (b) high-pass filtering method, (c) EMD method and (d) MMF method. Figure 19. Field 3-C microseismic data. The 1st–12th traces correspond to the H1 component, the 13th–24th traces correspond to the H2 component, and the 25th–36th traces correspond to the V component. find that the high-pass filtered data obviously lose the low-frequency components as highlighted by the red arrow (Fig. 15b), which are extremely valuable for subsequent seismic inversion. Because of the introduced artificial trends by the EMD technique, there is a lot of false energy existing in the low-frequency band, as we can see from Fig. 15(c). The second example explores the performance of MMF on the ground roll suppression. The original data set contaminated by ground roll is shown in Fig. 16(a). The high-amplitude and lowfrequency ground roll masks the primaries. Ground roll is one of the most common types of low-frequency noise. It is composed of surface wave whose vertical components are mainly dispersive Rayleigh waves, whose frequency components travel at different velocities leading to long complex wave trains that change as the length of the path travelled increases (Deighan & Watts 1997). The denoised results using high-pass filtering, EMD and MMF (A = 3, L = 60 ms) approaches are presented in Figs 16(b)–(d), respectively. The corresponding removed noise sections are shown in Figs 17(a)–(c), respectively. All the three approaches successfully remove the most energy of ground roll. But the EMD approach faces the same problem of introducing artificial trends, as we can see from Figs 16(c) and 17(b). Fig. 18 shows the amplitude spectra corresponding to the frame boxes in the original data (Fig. 16a) and denoised results by high-pass filtering (Fig. 16b), EMD (Fig. 16c) and MMF approaches (Fig. 16d), respectively. It can be observed that high-pass filtering eliminates all low-frequency signal, as highlighted by the red arrow in Fig. 18(b). MMF attenuates the ground roll and preserves the low-frequency signal at the same time. 1312 W. Huang et al. Figure 20. Denoising comparison of the H1 component data. (a) Original data. (b–d) Denoising results using high-pass filtering, EMD and MMF methods, respectively. The red frame boxes highlight the weak P-wave of microseismic event. M I C RO S E I S M I C E X A M P L E In this section, we test the MMF approach with a real microseismic data set. In microseismic monitoring, the low-frequency noise can seriously impact the performance of the follow-up processes such as the signal detection and arrival picking. It is worth pointing out that in some cases the strong low-frequency features can be long-period-long-duration signals (Caffagni et al. 2014; Zecevic et al. 2016). Fig. 19 shows a piece (1s) of the initial data from a real microseismic monitoring project in the west of China. The data are obtained by 12 downhole 3-components (3-C) geophones sampling at 0.5 ms. The 1st–12th and 13th–24th traces are the two horizontal components data (H1 and H2), and the 25th–36th traces are the vertical component data (V). We can see this data set is contaminated with strong low-frequency noise and the signals in H1 and V components are severely masked by the noise. We use the H1 component data as an example to compare the performance of MMF and other competing alternative methods. Figs 20 demonstrates the initial data and denoised results using the high-pass filtering, EMD and MMF (A = 1, L = 10 ms) methods, respectively. Fig. 21 shows the removed low-frequency noise by the three methods. All the three methods perform well from the perspective of low-frequency noise suppression, as we can observe in Fig. 21 that lots of low-frequency noise is removed and almost no coherent energies can be observed in the noise sections. The extremely weak event (highlighted by the red frame boxes) is more visually detectable after using all the three approaches, especially in the 3rd trace, because of the removal of large low-frequency noise. To compare and evaluate the denoising performance by different methods in detail, the initial and denoised data are used to pick the arrival time of the event. Arrival time picking is an important step in Mathematical morphological filtering 1313 Figure 21. Removed low-frequency noise using (a) high-pass filtering method, (b) EMD method and (c) MMF method. the processing of microseismic data, which can provide the real-time information for the location of microseismic events and inversion of the source mechanism. We chose the commonly used short-term average to long-term average (Allen 1978; Baer & Kradolfer 1987; Vaezi & Van der Baan 2015) (STA/LTA) algorithm to pick the event. A signal with high S/N will have a strong peak in the STA/LAT ratio curve, which corresponds a good denoising performance. Figs 22 and 23 show the STA/LAT ratio comparison of the 3rd and 8th traces. The black curve corresponds to the data and the red curve denotes the STA/LTA ratio. The red arrow highlights the arrival time of the event. Fig. 22(a) shows the initial 3rd traces and the corresponding STA/LAT ratio curves. We can see that the low-frequency noise masks the signal, and there is no observable peak in the location marked by the arrow. After applying the denoising approaches, the event is much clearer and we can observe notable peaks emerging in the red arrow locations from Figs 22(b)–(d). The event after using MMF (Fig. 22d) is more detectable than those after using high-pass filtering (Fig. 22b) and EMD approaches (Fig. 22c) since we can see that the STA/LTA peak in Fig. 22(d) is apparently higher than others. Because MMF can preserve those low-frequency components of the event, it is most suitable for signal detection in practical applications. We can also note that the misleading fake STA/LTA peak caused by the EMD approach in Fig. 22(c) as highlighted by the blue arrow. Fig. 23 presents a comparison of the 8th trace. We can see that the event cannot be detected well from the initial data (Fig. 23a) as well as the denoised data by high-pass filtering (Fig. 23b) and 1314 W. Huang et al. Figure 22. Comparison of the third trace. (a) Original data. (b–d) Denoising results using high-pass filtering, EMD and MMF methods, respectively. The red curves are the STA/LTA outputs. The black curves are the data. EMD method (Fig. 23c), because not only the event peak (marked by the red arrow) is hidden in the STA/LTA curve but also the misleading fake peak (marked by the blue arrow) exists. By contrast, MMF (Fig. 23d) obtains a strong event peak indicating a superior denoising performance. sequent full waveform inversion for building the subsurface velocity model by preventing the cycle-skipping issue. Because of the nature of morphological operations, MMF has cheap computational cost and can be easily implemented in parallel. From the synthetic and the field reflection seismic and microseismic data examples, it is obvious that the proposed MMF achieves an outstanding performance on low-frequency noise attenuation. C O N C LU S I O N S The traditional high-pass filtering method will cause the removal of some useful information of signal at low-frequencies when used for removing low-frequency noise. The EMD method is an adaptive signal decomposition method and can be used to remove lowfrequency components but at the expense of high computational cost and introducing significant artefacts. The MMF method provides an efficient way for filtering seismic data in the morphological scale domain. It can effectively suppress low-frequency noise and preserve the low-frequency components of the signal at the same time. The preservation of low-frequency signal is significant in sub- AC K N OW L E D G E M E N T S This work is supported by the National Basic Research Program of China (973 Program), grant NO: 2013CB228602. The authors appreciate Shuwei Gan, Huijian Li, Dong Zhang and the anonymous reviewer for constructive suggestions and inspiring discussions. WH would like to thank the China Scholarship Council for the financial support, and Modeling and Imaging Laboratory at the University of California, Santa Cruz for the help. Mathematical morphological filtering 1315 Figure 23. 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Zu, S., Zhou, H., Li, Q., Chen, H., Zhang, Q., Mao, W. & Chen, Y., 2017a. Shot-domain deblending using least-squares inversion, Geophysics, 82(4), V241–V256. Zu, S., Zhou, H., Mao, W., Zhang, D., Li, C., Pan, X. & Chen, Y., 2017b. Iterative deblending of simultaneous-source data using a coherency-pass shaping operator, Geophys. J. Int., 211(1), 541–557. APPENDIX A: PSEUDO-CODE OF MAIN M O R P H O L O G I C A L O P E R AT I O N S Dilation operation: d(t − τ ) + b(τ ) d⊕b= (A1) τ 1. DILATION input Trace_in, SE output Trace_out 2. for i = 1,2,3, . . . ,length(Trace_in) 3. find 4. Trace_out(i)=max(Trace_in(i-length(SE)+1:-1:i)+SE) 5. end 6. end 7. return Trace_out 1318 W. Huang et al. Erosion operation: db= d(t + τ ) − b(τ ) (A2) τ 1. EROSION input Trace_in, SE output Trace_out 2. for i = 1,2,3, . . . ,length(Trace_in) 3. find 4. Trace_out(i)=min(Trace_in(i:i+length(SE)-1)-SE) 5. end 6. end 7. return Trace_out Opening–closing operation: d b = (d ◦ b) • b Opening operation: d ◦ b = (d b) ⊕ b 1. CLOSING input Trace_in, SE output Trace_out 2. Temp=DILATION(Trace_in, SE) 3. Trace_out=EROSION(Temp,SE) 4. return Trace_out (A3) (A5) 1. OPENING–CLOSING input Trace_in, SE output Trace_out 2. Temp=OPENING(Trace_in, SE) 3. Trace_out=CLOSING(Temp, SE) 4. return Trace_out Closing–opening operation: 1. OPENING input Trace_in, SE output Trace_out 2. Temp=EROSION(Trace_in, SE) 3. Trace_out=DILATION(Temp, SE) 4. return Trace_out d b = (d • b) ◦ b Closing operation: d • b = (d ⊕ b) b (A4) 1. CLOSING–OPENING input Trace_in, SE output Trace_out 2. Temp=CLOSING(Trace_in, SE) 3. Trace_out=OPENING(Temp,SE) 4. return Trace_out (A6)

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