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ultichannel matching pursuit for seismic trace decomposition
anghua Wang1
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ABSTRACT
The technique of matching pursuit can adaptively decom-pose a
seismic trace into a series of wavelets. However, thesolution is
not unique and is also strongly affected by datanoise. Multichannel
matching pursuit �MCMP�, exploitinglateral coherence as a
constraint, might improve the unique-ness of the solution. It
extracts a constituent wavelet that hasan optimal correlation
coefficient to neighboring traces, in-stead of to a single trace
only. According to linearity theory, awavelet shared by neighboring
traces is the best match to theaverage of multiple traces, and
therefore it might effectivelysuppress the data noise and stabilize
the performance. It isfound that the MCMP scheme greatly improves
spatial conti-nuity in decomposition and can generate a plausible
time-fre-quency spectrum with high resolution for reservoir
detection.
INTRODUCTION
Matching pursuit adaptively decomposes a seismic trace into a
se-ies of constituent wavelets �Mallat and Zhang, 1993; Wang,
2007�.ach of these wavelets, selected from a dictionary consisting
ofbundant wavelets, also called atoms, has an optimal correlation
co-fficient with the trace. The intention is to overcome
limitations inonversional time-frequency spectrum generation
methods such ashe Gabor transform and the wavelet transform. Figure
1a displays aeismic trace, and Figure 1b-d shows the time-frequency
spectraenerated from the Gabor transform, the wavelet transform,
andatching pursuit, respectively. In the Gabor transform, the size
of
he time window sliding along the trace is predefined and usually
is aonstant, tapered with a Gaussian function �Gabor, 1946�.
Thereforehe spectrum depends on the predefined window size �Figure
1b�. Inhe wavelet transform �Mallat, 2009�, the time duration of a
constitu-nt wavelet is predefined also and is set to be inversely
proportionalo its dominant frequency �Figure 1c�. On the contrary,
matchingursuit, with a flexible wavelet size, can adaptively match
the true
Manuscript received by the Editor 31 October 2009; revised
manuscript re1Imperial College London, Department of Earth Science
and Engineer
mperial.ac.uk.2010 Society of Exploration Geophysicists.All
rights reserved.
V61
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ignature in the seismic trace. The resultant time-frequency
spec-rum �Figure 1d� shows that each wavelet has distinct durations
inhe time and frequency axes.
When applying single-channel matching pursuit �SCMP� to
realeismic data �Figure 2a�, however, a problem exists:
nonuniquenessf the decomposition. Because of this nonuniqueness,
the data noiselso severely affects the decomposition.As SCMP
performs decom-osition on each trace independently, the extracted
wavelets, and theeconstructed profile �the left panel of Figure
2b�, are lacking in spa-ial continuation between seismic traces,
which results in some re-iduals �the right panel of Figure 2b�.
Consequently, the time-fre-uency spectrum will lack lateral
continuation along a profile.
In this study, matching pursuit is implemented in a
multichannelashion so as to explore lateral continuity of seismic
events andeanwhile to suppress the noise effect in decomposition
�Durka et
l., 2005; Studer et al., 2006�. In this way, multichannel
matchingursuit �MCMP� might partially overcome the problem of
nonu-iqueness. With the constraint of lateral coherency, extracted
wave-ets and the time-frequency spectrum will have optimal lateral
con-inuation along the profile. Such constrained decomposition also
sta-ilizes the convergence, and consequently, the reconstructed
profilean accurately resemble the original seismic section �Figure
2c�.
In the following sections, first I summarize the basics of
theCMP algorithm, the core of which is that a constituent
wavelet
hould be shared by neighboring traces. Then I present a robust
im-lementation of MCMP based on linearity theory. This linearity
alsoeads to discussion on the stability of performance. Finally, I
demon-trate the application of MCMP with two examples: The first is
to re-ove a strong coal-seam reflection by exploiting the spatial
continu-
ty, and the other is to generate plausible time-frequency
spectra foras reservoir detection.
MULTICHANNEL MATCHING PURSUIT
Given a seismic trace f�t�, single-channel matching pursuit is
im-lemented iteratively. After n�1 iterations, a total of n�1
wave-ets are extracted from the trace and the residual trace is
R�n�1�� f�t��,
8 January 2010; published online 13August 2010.ntre for
Reservoir Geophysics, London, U. K. E-mail: yanghua.wang@
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where R is a linear operator, called the residuumoperator, and
R�0�� f�t��� f�t�. At the nth itera-tion, the extracted wavelet is
wn�t�, and the resid-ual trace is
R�n��f�t���R�n�1��f�t���wn�t� . �1�
The wavelet wn�t� is presented as
wn�t��ang� n�t�, �2�
where g� n�t� is a basic wavelet, or an atom, ob-tained after
the nth iteration, which has the opti-mal correlation coefficient
with the residual traceR�n�1�� f�t��; � n is a group of parameters
repre-senting the atom; and an is the amplitude scalar.
For multichannel matching pursuit, assuminghere is a group of
neighboring traces � f1�t�,f2�t�, . . . ,fL�t�� around aingle trace
f��t�, the single trace f��t� is decomposed into a series ofavelets
based on this group of traces. As defined in equation 1, the
esidual trace at the beginning is R�0�� f��t��� f��t�, and at
the nth it-ration is
R�n��f��t���R�n�1��f��t���a�,ng� n�t�, �3�
here g� n�t� is an atom extracted from the nth iteration. Note
herehat, although a�,n is the amplitude of the wavelet w�,n for
this indi-idual trace �, g� n is an atom shared by all L traces
within the group.fter N iterations, we might decompose the single
trace f��t� into Navelets as
f��t�� �n�1
N
a�,ng� n�t��R�N��f��t��, �4�
here R�N�� f��t�� is the final residual trace.The Morlet wavelet
�Morlet et al., 1982a, 1982b� is used as the
tom in matching pursuit decomposition for seismic traces. A
basicorlet wavelet m�t� centered at the abscissa t�u is defined
as
m�t��exp��� ln 2�2
��m2 �t�u�2� 2
expi��m�t�u�����,�5�
here �m is the mean angular frequency, � is the phase, and � is
aonstant value controlling the wavelet width. The Morlet waveletas
a constant shape ratio, diameter/mean period�constant, wherehe
diameter or duration is measured at half of the maximum am-pli-ude
of the wavelet envelope, or �6 dB in logarithmic scale. There-ore,
there are four parameters � n� �un,� n,�n,�n� presenting antom g�
n�t�: the time abscissa un, the scale � n, the central frequencyn
��m,n, and the phase �n.The use of Morlet wavelets as constituent
atoms in a matching
ursuit is based on the following effectiveness and efficiency
con-iderations �Wang, 2007�. First, the Morlet wavelet can
represent thettenuation behavior of wave propagation �Morlet et
al., 1982a, b�.econd, using the Morlet wavelet as the atom, instead
of searchingithin a vast wavelet dictionary, saves
wavelet-searching time.hird, by using an analytic form, analytic
expressions can be derived
or the computation of wavelet decomposition and
time-frequencypectrum generation, as shown in the following
sections.
ency (Hz)60 80 100
by the Gaborhe spectrum
0.5
0.7
0.9
1.1
1.3
1.5
1.7
1.9
Tim
e(s
)
Frequency (Hz)0 20 40 60 80 100
Frequency (Hz)0 20 40 60 80 100
Frequ0 20 40
a) b) c) d)
igure 1. �a� A single seismic trace. �b� Time-frequency spectrum
obtainedransform. �c� The spectrum generated from the wavelet
transform. �d� T
a)
b)
c)
igure 2. �a� A sample seismic profile for matching pursuit
�redrame indicates the zoom-in area shown in b and c�. �b�
Reconstruct-d profile by single-channel matching pursuit and the
residuals. �c�econstructed profile using multichannel matching
pursuit and the
esiduals. This demonstrates the completeness of multichannel
de-omposition.
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Multichannel matching pursuit V63
IMPLEMENTATION AND LINEARITY
Within each iteration, the implementation can be divided
intohree steps. The first step is a single-channel matching pursuit
to gen-rate an initial estimate of the atom from an average
residual trace.he second step is to refine the atom in a
multichannel fashion. The
hird step is to estimate the amplitude of the wavelet
correspondingo each trace.
In the first step, averaging L residual traces within the group
gen-rates a single residual trace
R�n�1��y�t���1
L���1
L
R�n�1��f��t��, �6�
here R�0��y�t��� 1L��f��t�. Then, performing a Hilbert
transformn R�n�1��y�t��, we can find the instantaneous frequency �n
and thenstantaneous phase �n, corresponding to the maximum of the
in-tantaneous envelope. The corresponding time is the abscissa
un.
ith fixed un, �n, and �n values, we search for an optimal
parametern using the following equation:
g� n�t��arg maxg� n�D
R�n�1��y�t��,g� n�t��
�g� n�t��, �7�
here D� �g� n�t��� n�� is a comprehensive dictionary of the
constit-ent wavelets, R�n�1��y�t��,g� n�t�� denotes the inner
product of theeismic trace residual R�n�1��y�t�� with atom g� n�t�,
and �g� n�t���g� n�t�,g� n�t�� normalizes the atom g� n�t�. For the
Morlet wave-et, an analytic expression for �g� n�t�� is given by
Wang �2007�:
�g� n�2�
�
2� �
2 ln 2
� n�n
�1�exp�� �2� n22 ln 2
cos �� .�8�
onsidering � n as a variable is a powerful feature of the
matchingursuit process.
In the second step, we refine the parameters � n� �un,�
n,�n,�n�ver a group of preselected, uniformly distributed values by
maxi-izing the sum of the correlation coefficients in each residual
trace:
g� n�t��arg maxg� n�D
���1
L
�R�n�1��f��t��,g� n�t���
�g� n�t��. �9�
hus, the atom g� n�t� is the best fit to all traces within a
group.In the third step, we estimate the amplitude a�,n
corresponding to
ach individual trace � by
a�,n��R�n�1��f��t��,g� n�t���
�g� n�t��2 , �10�
nd finally the matched wavelet is found as w�,n�t��a�,ng�
n�t�.These three steps are performed iteratively for n�1,2, . . .
,N. The
rocedure terminates when the residual energy is less than a
presethreshold or the number of iterations reaches a preset maximum
val-e.
The robust multichannel implementation scheme described
abovexploits the linearity of the residuum operator R�n� �Durka et
al.,005�. Initially, because R�0� is an identity operator,
R�0��y�t���y�t�,e have
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1
L���1
L
R�0��f��t���R�0�� 1L ���1L f��t�� . �11�or the first iteration,
the residuum operator R�1� is a linear operatoro that
1
L���1
L
R�1��f��t���1
L���1
L
R�0��f��t���a�,0g� 0�t��
�R�0�� 1L ���1L f��t��� 1L ���1L a�,0g� 0�t��R�1�� 1L ���1L
f��t��, �12�
here 1L��a�,0g� 0�t� is an average wavelet. By induction, for
the nthteration, R�n� is a linear operator, so
1
L���1
L
R�n��f��t���1
L���1
L
R�n�1��f��t���a�,ng� n�t��
�R�n�1�� 1L ���1L f��t��� 1L ���1L a�,ng� n�t��R�n�� 1L ���1L
f��t�� . �13�
his means that the sum of residual traces, ��R�n�� f��t��, is
equal tohe residual of the trace summation, R�n����f��t��.
This linearity analysis also leads to
1
L���1
L
R�n�1��f��t��,g� n�t��
�� 1L ���1L R�n�1��f��t��,g� n�t����R�n�1�� 1L ���1L f��t��,g�
n�t��, �14�
here the first equality is derived from the linearity of the
productperator �·,·�, and the second equality is derived from the
linearity ofhe residuum operator. Equation 14 indicates that the
sum of prod-cts across all traces is equal to the product of the
sum and the atom.
In equation 9, the atom g� n�t� is shared by all traces within a
group.ccording to equation 14, it is the best fit to the average of
residual
races or the residual of trace averages.Averaging over multiple
trac-s might effectively suppress data noise and thus stabilize the
pro-ess.
APPLICATION EXAMPLES
xtracting a coal-seam reflection by exploiting
spatialontinuity
The first application example is to extract a strong coal-seam
re-ection so that the remaining target reflections will be visible
from
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V64 Wang
he seismic profile. As shown in Figure 3a, the strong reflection
isbout 1.5 s across the profile, and the target gas reservoir
formations immediately above it, between 1.4 and 1.5 s.
To track the coherent event, we use the instantaneous phase
infor-ation �Figure 3b� as the input to calculate the
crosscorrelation be-
ween traces. As the phase section shows a clear continuation
fol-owing the horizon at a time of about 1.5 s, performing
crosscorrela-ion between phase traces can automatically track the
horizon. Thisorrelation provides us the initial time abscissa u�,
where � is therace reference.
Figure 3c displays the wavelets extracted along the horizon by
us-ng the multichannel matching pursuit process. In this
application,he number of traces within a spatial window is set to
be either threer five. The results are similar.
Figure 3d is the same section as Figure 3a but after the removal
ofhe strong coal-seam reflection. This means that it is the
differenceetween Figure 3a and c. This section clearly shows weak
reflectionscross the profile at times between 1.4 and 1.5 s.
nalyzing time-frequency distribution for gas detection
The second application example is to generate a
time-frequencypectrum for gas reservoir detection. Low-frequency
shadows in theime-frequency spectrum are a direct indication for
gas reservoir de-ection �Chakraborty and Okaya, 1995; Castagna et
al., 2003; Korn-ev et al., 2004; Sinha et al., 2005�.
After decomposing a signal f��t� into a series of wavelets g�
n�t�,or n�1,2, . . .N, the Wigner distribution can be used to
present themplitude distribution in the time-frequency space as
1.0
1.2
1.4
1.6
1.8
2.0
0 2 4 6 8 10 12 14Distance (km)
Tra
velti
me
(s)
1.0
1.2
1.4
1.6
1.8
2.0
0 2 4 6Dista
Tra
velti
me
(s)
1.0
1.2
1.4
1.6
1.8
2.0
0 2 4 6 8 10 12 14Distance (km)
Tra
velti
me
(s)
1.0
1.2
1.4
1.6
1.8
2.0
0 2 4 6Dista
Tra
velti
me
(s)
) b)
) d)
igure 3. �a� A seismic section extracted from a 3D seismic cube.
�bhase section, on which the strong coal-seam reflection �at about
1.5he strong coal-seam reflection is extracted from the seismic
sectionatching pursuit. �d� The seismic section after removing the
coal-seeak target reflections between 1.4 and 1.5 s.
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A�f��t,���� �n�1
Na�,n
�g� n��W�g� n�t,���, �15�
here W�g� n�t,��� is the Wigner distribution of a selected
wavelet� n
�t� and is given by
W�g� n�t,����1
2����
��
g� n�t� �2�ḡ� n�t� �2� exp� i�� �d� , �16�
nd ḡ� n�t� is the complex conjugate of g� n�t�. For the Morlet
wavelet�t� defined in equation 5, an analytic expression for the
time-fre-uency spectrum can be derived as �Wang, 2007�
A�f��t,���� �n�1
Na�,n
�g� n��� �
2 ln 2
� n�n�1/2
exp��� �24 ln 2
�� n2�� ��n�2�n
2 exp��� ln 2
�2��n2�t�un�2
� n2 , �17�
here �g� n�t�� can be estimated efficiently using the analytic
expres-ion in equation 8.
Figure 4a shows a crooked line across six wells, and Figure
4b-displays three frequency profiles obtained by using the
multichannelatching pursuit method. We can see that, below the
target gas reser-
voirs �yellow circles�, the low-frequency shad-ows at 10 Hz are
gradually reduced, but mean-while the target gas reservoirs show
strong ampli-tudes in 20 and 25 Hz.
DISCUSSION ON “WAVELET”
We first look at the difference in “wavelet” be-tween the
wavelet transform and the matchingpursuit decomposition. Figure 5a
gives an exam-ple profile extracted from a 3D seismic cube,
andFigure 5b and c shows two constant-frequencyprofiles generated,
respectively, by these twomethods. In the wavelet transform for a
constantdominant frequency, a wavelet with fixed size iscorrelated
with a seismic trace at each samplepoint along the time axis, and
the resultant contin-uous correlation coefficient gives the
spectrumwith respect to time. In matching pursuit, howev-er, the
size of wavelets along the time axis is flexi-ble and is estimated
adaptively from the seismicdata. Therefore, the matching pursuit
methodshows much higher temporal resolution than thewavelet
transform does.
However, the constituent wavelet in bothmethods is conceptually
different from the wave-let used in the convolutional modeling of a
seis-mic trace. The latter assumes a wavelet to be con-stant along
the time axis, at least within a certain
0 12 14m)
0 12 14m)
nstantaneouse tracked. �c�multichannelection shows
8 1nce (k
8 1nce (k
� The is� can busing
am refl
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25 Hz.
a
b
c
FTtt
Multichannel matching pursuit V65
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time window, on which a nonstationary filter �ac-counting for
the earth attenuation effect� could beattached. In the wavelet
transform, the wavelet isfixed according to frequency and thus is
differentfrom the wavelet for convolution. This is also thecase in
the present version of matching pursuitdecomposition.
Although the wavelet in a matching pursuit isadaptive to the
data, it might be possible to exploitthe wavelet concept in
convolutional modelingand to generate another �soft� constraint in
thetime axis, in addition to the spatial axis, to im-prove further
the uniqueness of matching pursuitdecomposition. Ideally, if so, an
atom or a basicwavelet g� n�t� from the matching pursuit
couldprovide the wavelet for convolutional modeling,and the
amplitude scale a�,n would be relevant tothe reflectivity
magnitude.
CONCLUSIONS
The technique of matching pursuit, with flexi-ble wavelet size,
is a powerful method to decom-pose a seismic trace into a series of
wavelets.However, with single-channel matching pursuit,the solution
is not unique and is also strongly af-fected by data noise.
Multichannel implementa-tion can improve the performance of
matchingpursuit decomposition, in which the lateral coher-ence
between neighboring seismic traces is ex-ploited as a constraint to
overcome the nonu-
iqueness of the solution. An extracted wavelet should have an
opti-al correlation coefficient to a group of traces, instead of to
a single
race only.A robust implementation is based on linearity theory.
According
o linearity, an atom shared by neighboring traces is the best
match tohe average over multiple traces. Averaging might
effectively sup-ress data noise and thus stabilize the
procedure.
The MCMPscheme greatly improves spatial continuity in
decom-osition and temporal resolution in the resultant
time-frequencypectrum. This spatial continuity is exploited to
remove a strongoal-seam reflection from seismic data so as to allow
the weak targeteflections immediately on the top of the coal seam
to be character-zed. The MCMP scheme also can generate a plausible
time-fre-uency spectrum for detecting low-frequency shadows
underneathas reservoirs.
ACKNOWLEDGMENTS
I am grateful to the sponsors of the Centre for Reservoir
Geophys-cs, Imperial College London, for supporting this research.
I alsohank Charles Jones and Tim Sears for their thorough test on
my pre-iously published algorithm, which motivated me to pursue the
re-earch of this paper.
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