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Mixed-phase wavelet estimation A case studySomanath Misra and
Satinder ChopraArcis Corporation, Calgary, Alberta, Canada
Introduction
An accurate estimation of wavelet is crucial in the
deconvo-lution of seismic data. As per the convolution model,
therecorded seismic trace is the result of convolution of theearths
unknown reflectivity series with the propagatingseismic source
wavelet along with the additive noise. Thedeconvolution of the
source wavelet from the recordedseismic traces provides useful
estimates of the earthsunknown reflectivity and comes in handy as
an aid to geolog-ical interpretation. This deconvolution process
usuallyinvolves estimation of a wavelet, before it is removed
bydigital filtering. Since the earths reflectivity and seismicnoise
are both unknown, the wavelet estimation process isnot easy. The
statistical methods estimate the wavelet usingthe statistical
properties of the seismic data and are based oncertain mathematical
assumptions. The most commonly usedmethod assumes that the wavelet
is minimum phase and thatthe amplitude spectrum and the
autocorrelation of thewavelet is the same as the amplitude spectrum
and the auto-correlation of the seismic traces, within a scale
factor, in thetime zone from where the wavelet is extracted. With
theassumption that the wavelet is minimum phase, an estima-tion of
the wavelet is done from the trace autocorrelation.This method
always estimates a minimum phase wavelet andso is suitable for
wavelet estimates from seismic dataacquired using explosive
sources, only if their source signa-ture is purely minimum phase
and that is retained throughduring processing. However, it is not
applicable for esti-mating wavelets from sources giving mixed phase
signature.For example, deconvolution of non-minimum phase
seismicdata with a minimum phase wavelet will leave behind
aspurious phase in the data. The reason for this is that
theautocorrelation function and the power spectrum mentionedabove
are second-order statistical measures. They contain nophase
information and so cannot identify non-minimumphase signals. Also,
these measures work well for Gaussianprobability distribution of
amplitudes, and so will not yieldaccurate results for non-Gaussian
or non-linear distributions.The non-Gaussianity in the seismic data
could arise from thenon-minimum phase source signature, noise in
the data likeswell noise, and a non-linear earth response.
Consequently,higher-order statistics have been used for dealing
with non-Gaussian distributions. These statistics, known as
cumulants,and their associated Fourier transforms known as
poly-spectra, not only reveal amplitude information, but also
thephase information (Mendel, 1991). In this paper, we addressthis
issue through the use of higher-order statistics such thatthe phase
component in the data are more accurately esti-mated and
removed.
Higher-order statistics for wavelet estimation
Cumulant, a higher-order statistical property, preserves
thephase information of the wavelet under the assumption that
the reflectivity series is a non-Gaussian, stationary and
statis-tically independent random process. The second-ordercumulant
of a zero-mean process is just the autocorrelation,which as stated
above, has no phase information. The third-order cumulant is a two
dimensional correlation function. Fora Gaussian process, all
cumulants above the second-order arezero, but are non-zero for
non-Gaussian processes. Thus thesetwo statistics are not suitable
for recovering a non-minimumphase from a convolution process such
as the seismic trace.The fourth-order cumulant is a
three-dimensional correlationfunction which contains information
about phase. Just as theFourier transform of the autocorrelation
function yields thepower spectrum, similarly, the trispectrum
relates to thefourth-order cumulant via the 3D Fourier
transformation.Lazear (1993) and Velis and Ulrych (1995) estimate
the phaseof a wavelet by fourth-order cumulant matching wherein
aninitial guess for the wavelet is iteratively updated until
itsfourth order statistics match those of the seismic data.
Misra and Sacchi (2006) suggest the parameterization of
theembedded mixed phase wavelet as a convolution of theminimum
phase wavelet with an all-pass operator. The all-pass operator can
further be parameterized as the ratio of amaximum phase time
sequence and corresponding minimumphase time sequence with the
necessary time delay required toenforce causality in the all-pass
operator (Porsani and Ursin,1998). The denominator term in the
paramerization of the all-pass operator is a minimum phase sequence
whose length andcoefficients are unknown. As discussed in a later
section, weoptimize for the unknown coefficients of the minimum
phasesequence keeping the length as a constant parameter.
Seismic data is represented as the convolution of the
reflec-tivity sequence with the unknown wavelet. The unknownwavelet
showing mixed phase characteristics is further repre-sented as the
convolution of minimum phase wavelet and theall-pass operator. Thus
the seismic data is further representedin terms of two
convolutions, namely convolution of reflec-tivity with the minimum
phase wavelet which in turn isconvolved with the all-pass operator.
Deconvolving the datawith the minimum phase wavelet increases the
bandwidth ofthe output data and we subsequently refer to it as
thewhitened data. The whitened data is thus represented as
theconvolution of the underlying reflectivity series and the
all-pass operator. Hence it is possible to make an estimation ofthe
underlying reflectivity series by estimating the all-passoperator
from the whitened data.
Development of the algorithm
The well known Barlett-Brillinger-Rosenblatt formula(Lazear
1993; Mendel 1991) links the fourth-order cumulantof the seismic
trace with the fourth-order moment of theembedded wavelet. For
non-Gaussian, statistically inde-pendent and identically
distributed reflectivity series, the
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fourth-order cumulant of the seismic trace is equal to, within
ascale factor, the fourth-order moment of the wavelet providedthat
the noise distribution is Gaussian. The optimization proce-dure
described in the following paragraph minimizes the costfunction
given by the L2-norm between the fourth-order normal-ized trace
cumulant and the fourth-order wavelet moment.
The optimization of the cost function thus involves computation
ofthe normalized 4th order trace cumulant of the whitened data
(dataobtained after deconvolution with a minimum phase wavelet)
andthe normalized 4th order moment of the all-pass operator.
The shape of the cost function is unknown and may containseveral
local minima. Local optimization methods based ongradient
computation always proceed to the minimum nearest tothe chosen
initial model. In the present optimization problemwhere the shape
of the cost function is not known, a global opti-mization algorithm
is a preferred choice. Simulated annealingalgorithm with a
Metropolis acceptance/ rejection criterion(Misra, 2008) is adopted
for the optimization of the cost function.The model parameters for
the simulated annealing optimizationare the coefficients of the
minimum phase sequence in the para-meterization of the all-pass
operator. The all-pass operator foreach of the generated model is
computed from equation bytaking the ratio of the corresponding
maximum phase sequenceand the minimum phase sequence.
Application to post stack seismic data
In order to test the stability and reliability of the algorithm,
themethod outlined above was applied to a seismic data volumefrom
North America. For this volume, the data are further subdi-vided
into smaller zones and for each of these zones a mixed-phase
wavelet is estimated. Thedata in each of these subdivisionsare
deconvolved using the esti-mated mixed-phase wavelet.Further, an
average estimatedmixed-phase wavelet is obtainedby taking the mean
of the indi-vidual mixed-phase wavelets. Themean mixed-phase
wavelet is thenused to deconvolve the data. Theresults are
correlated with the P-wave log curve.
Figure 1 shows the workflow forthe method outlined above.
Case study
A seismic data volume from aprovince in North America waspicked
up for testing the algo-rithm outlined above. The datavolume had
been processed usinga conventional processingsequence. Four
different locationson the post-stack volume wereselected for the
estimation of themixed-phase wavelets in the sametime interval
(500ms) for each.
Article ContdMixed-phase waveletContinued from Page 32
Figure 2. (a) A segment of a seismic section around location 1.
(b) The estimated minimum phase wavelet, (c) the estimatedall-pass
operator and (d) the estimated mixed-phase wavelet for location 1,
and (e) input seismic section in (a) after mixedphase wavelet
deconvolution, Notice, the phase-corrected section exhibits higher
frequency content than the input data asexpected and so exhibits
much higher resolution.
Figure 1. Workflow for the methodology developed for
deconvolution of mixed phasewavelets from seismic data.
Continued on Page 34
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34 CSEG RECORDER February 2011
Figure 2a shows a segment of aseismic section around location1.
Figures 2b, c and d show theestimated minimum-phasewavelet, the
estimated all-passoperator and the estimatedmixed-phase wavelet at
the loca-tion 1. The minimum-phasewavelets are estimated from
theaverage autocorrelation of theseismic traces by the
Wiener-Levinson algorithm. The esti-mated minimum-phase waveletis
deconvolved from the datawhich resulted in broadening ofthe
bandwidth. Figures 2e showsthe phase-corrected data for thelocation
1. Figures 3, 4 and 5show a similar set of images forlocations 2, 3
and 4. Notice thatfor each set of images, the mixed-phase wavelet
deconvolvedsections exhibit the highest levelof detail. Again, it
would beadvisable to correlate the decon-volved sections with the
P-wavelog curves to gain some confi-dence in ascertaining if
theresolved reflections correlatewell and if they are
authentic.Notice in Figure 6, the correlationof the section in
Figure 6b isbetter than the one shown inFigure 6c.
Conclusions
Deconvolution of seismic datawith a minimum phase
waveleteffectively removes the ampli-tude spectrum of the
waveletfrom the data. However, in situa-tions where the
minimum-phaseassumption about the wavelet isnot valid, the
deconvolutionleaves behind a spurious phasecomponent in the data.
Themethod adopted in estimatingand hence removing the spuriousphase
involves estimation of thecoefficients of an all-pass oper-ator
from the data that have beenwhitened by the deconvolution ofthe
minimum-phase wavelet. Thewhitened data are used to opti-mize the
cost function involvingthe 4th order normalized trace cumulant and
the 4th ordermoment of the all-pass operators. The optimization
procedureuses simulated annealing with the Metropolis
acceptance/rejec-tion criterion. The estimated all-pass operator is
subsequentlyconvolved with the earlier estimated minimum-phase
wavelet to
estimate the mixed-phase wavelet in the data. The
suggestedmethod is tested on a seismic data set belonging to a
province inNorth America. The data set is subsequently deconvolved
withthe estimated mixed-phase wavelets. Further, an average
mixed-phase wavelet is computed from the individual mixed-phase
wavelets. The data are then deconvolved with the average
Article Contd
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Mixed-phase waveletContinued from Page 33
Figure 3. (a) A segment of a seismic section around location 2.
(b) The estimated minimum phase wavelet, (c) the estimatedall-pass
operator and (d) the estimated mixed-phase wavelet for location 1,
and (e) input seismic section in (a) after mixedphase wavelet
deconvolution, Notice, the phase-corrected section exhibits higher
frequency content than the input data asexpected and so exhibits
much higher resolution.
Figure 4. (a) A segment of a seismic section around location 3.
(b) The estimated minimum phase wavelet, (c) the estimatedall-pass
operator and (d) the estimated mixed-phase wavelet for location 1,
and (e) input seismic section in (a) after mixedphase wavelet
deconvolution, Notice, the phase-corrected section exhibits higher
frequency content than the input data asexpected and so exhibits
much higher resolution.
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February 2011 CSEG RECORDER 35
mixed-phase wavelet are corre-lated with the P-wave log dataand
shows better correlation. R
Acknowledgements
We acknowledge Prof. MauricioSacchi for his help and
sugges-tions during the development ofthe algorithm. We also
acknowl-edge an anonymous provider ofthe data shown in and
ArcisCorporation, Calgary, Canada forsupport of this work.
ReferencesLazear, G. D., 1993, Mixed-phase waveletestimation
using fourth-order cumulants,Geophysics, 58, 1042-1051.
Mendel, J. M., 1991, Tutorial on higher-orderstatistics
(spectra) in signal processing andsystem theory: theoretical
results and someapplications, Proceedings of the IEEE,
79,278-305.
Misra, S. and Sacchi, M. D., 2006, Non-minimum phase wavelet
estimation by non-linear optimization of all-pass
operators,Geophysical Prospecting, 55, 223-234.
Misra, S., 2008, Global optimization withapplication to
Geophysics, PhD thesis,University of Alberta, Canada.
Porsani, M. J., and Ursin, B., 1998,Mixed phase deconvolution,
Geophysics, 63, 637-647.
Velis, D. R., and Ulrych, T. J., 1996, Simulated annealing
wavelet estimation via fourth-order cumulant matching, Geophysics,
61, 1939-1948.
Article ContdMixed-phase waveletContinued from Page 34
Figure 5. (a) A segment of a seismic section around location 4.
(b) The estimated minimum phase wavelet, (c) the estimatedall-pass
operator and (d) the estimated mixed-phase wavelet for location 1,
and (e) input seismic section in (a) after mixedphase wavelet
deconvolution, Notice, the phase-corrected section exhibits higher
frequency content than the input data asexpected and so exhibits
much higher resolution.
Figure 6. Segment of a seismic section from (a) input data, and
(b) data after phase correction with theaverage mixed phase
wavelet. The inserted red curve is the P-wave log. Notice the
higher level of correla-tion of the log curve with the section in
(b).
Somanath Misra wasborn and raised in thecoastal province
ofOrissa, India. Aftergraduating in Physicsmajor, Somanathjoined
the IndianSchool of Mines for his
Masters degree in Applied Geophysics,which he successfully
completed in 1991.Thereafter, Somanath preferred to jointhe
exploration department in Orissa,India and worked there for about
10years. In 2003, he joined the SignalAnalysis and Imaging Group at
theUniversity of Alberta and completed hisPh.D. in 2008. Since then
he is working asa Reservoir Geophysicist in the reservoirservices
group at Arcis Corporation,Calgary, Canada. He is a member ofCSEG
and SEG.
Satinder Chopra:See March 2011RECORDER issuepage 5.