75th EAGE Conference & Exhibition incorporating SPE EUROPEC
2013 London, UK, 10-13 June 2013
Introduction
Deconvolution is an important and recurrent topic in seismic
data processing. Many signalsand images can be represented via the
convolution of an unknown signal of interest and asource signature.
In general, the unknown signal or image can be estimated via
inverse meth-ods when the source signature is known. This process
is called deconvolution. When thewavelet is unknown the process
requires the simultaneous estimation of two signals. This pro-cess
is often called blind deconvolution (Shalvi and Weinstein, 1990).
Two early attempts onblind deconvolution are Homomorphic
Deconvolution based on the work by Oppenheim andSchafer (1968) and
implemented for the first time in exploration seismology by Ulrych
(1971).Another technique for blind deconvolution was proposed by
Wiggins (1985) who coined thename Minimum Entropy Deconvolution
(MED) algorithm. MED assumes that the reflectivity issparse and
operates by finding an inverse filter that maximizes a measure of
sparsity (Donoho,1981). Both Homomorphic Deconvolution and MED
suffer from a variety of shortcomings. Forinstance, homomorphic
deconvolution is inclined to instability due to phase unwrapping
and byits inherent inability to incorporate an additive noise term
in the data. MED deconvolution oftentends to annihilate small
reflection coefficients (Ooe and Ulrych, 1979; Walden, 1985).
The history of seismic deconvolution is plagued by interesting
statistical methods for blinddeconvolution. These methods, however,
only work under ideal signal conditions. For instance,important
excitement was generated by methods based on higher order
statistics because oftheir ability to estimate phase (Petropulu and
Nikias, 1992, 1993; Lazear, 1993; Hargreaves,1994; Sacchi et al.,
1996; Velis and Ulrych, 1996). However, the conditions for robust
waveletestimation required by these methods are not often satisfied
by seismic data (Stogioglou et al.,1996).
Euclid deconvolution is a member of the plethora of methods that
have been proposed for blinddeconvolution of seismic data. The
method was first discussed in the geophysical literature byRietsch
(1997a) and tested with real data examples in Rietsch (1997b). The
method has alsobeen investigated by Xu et al. (1995) and Liu and
Malvar (2001) in communication theory anddeconvolution of
reverberations. The idea can be summarized as finding common
factors ofthe z-transform of the source embedded in a group of
seismograms with different reflectivitysequences. The problem leads
to the estimation of the multichannel seismic reflectivity viathe
solution of homogeneous system of equations (Mazzucchell and
Spagnolini, 2001). In theideal case, the eigenvector associated to
the minimum non-zero eigenvalue of the homogenoussystem of
equations is an estimator of the multichannel reflectivity.
However, small amounts ofnoise impinge on the identification of the
eigenvector associated to the impulse response. Wepropose an
improvement to Euclid deconvolution where the homogeneous equation
is satisfiedby a sparse solution (sparse impulse responses). In
other words, we are assuming that themultichannel impulse response
of the earth is a multichannel sparse series.
Theory
The earth can be modeled as a linear time invariant (LTI)
system. The input-output relationshipfor this system, assuming a
stationary source wavelet and a noise free condition, can be
writtenas
d j[n] = ∑k
w[n− k]r j[k] , j = 1 . . .J (1)
where the multichannel seismic data can be represented by the d
j = (d j[0],d j[2], . . . ,d j[N−1])T ,similarly the impulse
responses for channel j can be written as r j = (r j[0],r j[2], . .
. ,r j[M −
75th EAGE Conference & Exhibition incorporating SPE EUROPEC
2013 London, UK, 10-13 June 2013
1])T , and finally we represent the seismic source function (the
wavelet) via the vector w =(w[0],w[2], . . . ,w[L− 1])T . We stress
that N = M +L− 1. We also remind the readers that con-volution can
be expressed using the z-transform as follows
D j(z) =W (z)R j(z) , j = 1 . . .J . (2)
By virtue of equation 2, it is easy to show that
Dp(z)Rq(z)−Dq(z)Rp(z) = 0 , ∀ p,q (3)
which can be rewritten in matrix-vector form as follows
Dp rq −Dq rp = 0 (4)
where Dp and Dq in equation 4 are the convolution matrices of
channels p and q, respectively.Combining all possible equations of
type 4 leads to the following homogeneous system ofequations
A x = 0 (5)
where
A =
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝
D2 −D1D3 −D1D4 −D1...
. . .D3 −D2D4 −D2...
. . .DJ −DJ−2
DJ −DJ−1
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
(6)
andx = [r1,r2,r3, . . . ,rJ]T . (7)
Euclid method estimates the reflectivity by estimating the
eigenvector associated to the mini-mum non-zero eigenvalue of AT A
Rietsch (1997a). A small amount of noise in the data makesthe
solution impractical for real data applications Rietsch (1997b).
The addition of a noise termin our signal model leads to
Ax = e . (8)
It can be shown that e is white and we propose to find a
solution x that minimizes e. In additionwe will constraint x to be
sparse. To avoid the trivial solution, we must equip our problem
withan extra constraint xT x = 1. We propose to find the solution
by minimizing the cost J(x)
x̂ = argminx
J(x), subject to xT x = 1 (9)
whereJ(x) =
12||Ax||22 +λ Hμ(x) . (10)
The symbol Hμ(.) is used to indicate the Huber norm with Huber
parameter μ. The Hubernorm forces solutions that are sparse. Using
the Huber norm enables us to use a simpleoptimization method based
on steepest descent techniques. It is worth mentioning that
topreserve the constraint and keep the solution on the unit sphere,
one should use an educatedstep that can be derived from Rodrigues’
rotation formula Murray et al. (1994).
75th EAGE Conference & Exhibition incorporating SPE EUROPEC
2013 London, UK, 10-13 June 2013
ExamplesTo examine the performance of proposed method we applied
SMBD to a near offset sectionfrom the Gulf of Mexico seismic data
(Figure 1a). Figures 1a and b show the data beforeand after
deconvolution via SMBD, respectively. We also estimated the wavelet
using thefrequency-domain least squares estimator. The wavelet is
portrayed in Figure 2b. For com-pleteness, the average sea floor
first break source wavelet is extracted and compared with
theestimated wavelet. There is a strong resemblance of the
estimated wavelet with the averagefirst break pulse that was
extracted by flattening and averaging the water bottom
reflection.
Conclusions
We have presented a modification to Euclid’s blind deconvolution
method. The proposedmethod alleviates some of the problems
encountered in blind deconvolution via Euclid method.For instance,
SMBD can tolerate moderate levels of noise and does not require to
know theprecise length of the source duration. The method permits
to estimate broad-band (sparse)impulse responses that can be used
for wavelet estimation.
AcknowledgementsThe authors are grateful to the sponsors of
Signal Analysis and Imaging Group (SAIG) at theUniversity of
Alberta.
References
Donoho, D. [1981] On minimum entropy deconvolution. Academic
Press, 109(6), 2831–9.Hargreaves, N. [1994] Wavelet estimation via
fourth-order cumulants, chap. 432. 1588–1590.Lazear, G. [1993]
Mixed-phase wavelet estimation using fourth-order cumulants.
Geophysics, 58(7),
1042–1051, doi:10.1190/1.1443480.Liu, J. and Malvar, H. [2001]
Blind deconvolution of reverberated speech signals via
regularization.
Acoustics, Speech, and Signal Processing, 2001. Proceedings.
(ICASSP ’01). 2001 IEEE Interna-tional Conference on, vol. 5, 3037
–3040.
Mazzucchell, P. and Spagnolini, U. [2001] Least square
multichannel deconvolution. EAGE 63rd Con-ference and Technical
Exhibition.
Murray, R.M., Li, Z. and Sastry, S.S. [1994] A Mathematical
Introduction to Robotic Manipulation. BocaRaton, FL: CRC Press.
Ooe, M. and Ulrych, T.J. [1979] Minimum entropy deconvolution
with an exponential transformation.Geophys. Prosp, 27, 458–473.
Oppenheim, A. and Schafer, R. [1968] Homomorphic analysis of
speech. Audio and Electroacoustics,IEEE Transactions on, 16(2), 221
– 226, ISSN 0018-9278, doi:10.1109/TAU.1968.1161965.
Petropulu, A. and Nikias, C. [1992] Signal reconstruction from
the phase of the bispectrum. SignalProcessing, IEEE Transactions
on, 40(3), 601 –610, ISSN 1053-587X, doi:10.1109/78.120803.
Petropulu, A. and Nikias, C. [1993] Blind convolution using
signal reconstruction from partial higher ordercepstral
information. Signal Processing, IEEE Transactions on, 41(6), 2088
–2095, ISSN 1053-587X,doi:10.1109/78.218138.
Rietsch, E. [1997a] Euclid and the art of wavelet estimation,
part i: Basic algorithm for noise-free data.Geophysics, 62(6),
1931–1938.
Rietsch, E. [1997b] Euclid and the art of wavelet estimation,
part ii: Robust algorithm and field-dataexamples. Geophysics,
62(6), 1939–1946.
Sacchi, M., Velis, D.R. and Ulrych, T.J. [1996] Wavelets via
Polycepstra. 1583–1586.Shalvi, O. and Weinstein, E. [1990] New
criteria for blind deconvolution of nonminimum phase sys-
tems (channels). Information Theory, IEEE Transactions on,
36(2), 312 –321, ISSN 0018-9448, doi:10.1109/18.52478.
Stogioglou, A., McLaughlin, S. and Ziolkowski, A. [1996]
Asymptotic performance analysis for fourth-order cumulant based
deconvolution. SEG Technical Program Expanded Abstracts, 422,
1587–1590.
Ulrych, T.J. [1971] Application of homomorphic deconvolution to
seismology. Geophysics, 36(6), 650–660.
Velis, D. and Ulrych, T. [1996] Simulated annealing wavelet
estimation via fourth-order cumulant match-ing. Geophysics, 61(6),
1939–1948.
Walden, A.T. [1985] Non-gaussian reflectivity, entropy, and
deconvolution. Geophysics, 50(12), 2862–2888.
Wiggins, R.A. [1985] Minimum entropy deconvolution.
Geoexploration, 16, 21–35.Xu, G., Liu, H., Tong, L. and Kailath, T.
[1995] A least-squares approach to blind channel iden-
tification. Signal Processing, IEEE Transactions on, 43(12),
2982 –2993, ISSN 1053-587X, doi:10.1109/78.476442.