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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.
2 Modeling
Accepted 2017 September 1. Received 2017 August 30; in original form 2017 May 25
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.
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;
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
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
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.
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 ),
Mathematical morphological filtering
Figure 5. Time–frequency spectra of (a)–(e) denoised results using strong, moderate and weak high-pass filters, EMD method and MMF method, respectively.
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 ),
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
W. Huang et al.
by d • b) are defined respectively as:
d ◦ b = (d b) ⊕ b,
d • b = (d ⊕ b) b.
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,
d b = (d • b) ◦ b.
The average of open-closing and close-opening forms the MMF
Figure 6. Computing time costs with different methods.
Fb (d) = (d b + d b)/2,
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
Mathematical morphological filtering
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.
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
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
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),
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
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
+ Fbn−1 (d) − Fbn (d)
nth scale-band
+ ... + Fb1 (d) − Fb2 (d) + d − Fb1 (d) .
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
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.
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:
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.
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
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
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).
the local minima is zero (Huang et al. 1998; Chen & Ma 2014). The
mathematical principle of EMD can be briefly written as:
[en ],
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:
d =
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
[en ],
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:
[fn ],
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:
df =
[fn ],
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,
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,
[mn ],
dm =
[mn ],
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
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 ,
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
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.
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.
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
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
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.
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.
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
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
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.
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
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
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.
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-
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
Figure 23. Comparison of the eighth 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.
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Dilation operation:
d(t − τ ) + b(τ )
input Trace_in, SE
output Trace_out
2. for i = 1,2,3, . . . ,length(Trace_in)
3. find
5. end
6. end
7. return Trace_out
W. Huang et al.
Erosion operation:
d(t + τ ) − b(τ )
input Trace_in, SE
output Trace_out
2. for i = 1,2,3, . . . ,length(Trace_in)
3. find
5. end
6. end
7. return Trace_out
Opening–closing operation:
d b = (d ◦ b) • b
Opening operation:
d ◦ b = (d b) ⊕ b
input Trace_in, SE
output Trace_out
2. Temp=DILATION(Trace_in, SE)
3. Trace_out=EROSION(Temp,SE)
4. return Trace_out
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:
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
input Trace_in, SE
output Trace_out
2. Temp=CLOSING(Trace_in, SE)
3. Trace_out=OPENING(Temp,SE)
4. return Trace_out
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