IEEE Signal Processing - March 2018 - 44

the large similarity between neighboring traces may lead to
ill-posed problems and how regularization may improve the
numerical stability of this problem.

Impact of common zeros on multichannel
seismic deconvolution
One of the main assumptions of multichannel deconvolution
is that the channels do not share common zeros. To show how
violating this assumption may lead to ill-posed problems,
consider two polynomials H 1 (z) and H 2 (z), as illustrated in
Figure 2. If they share common zeros, they can be factored as
H 1 ^ z h = H 1l ^ z h H 1, 2 ^ z h,
H 2 ^ z h = H 2l ^ z h H 1, 2 ^ z h,

(10)

where H 1l (z) and H 2l (z) are coprime, i.e., they do not share
common zeros, and H 1, 2 (z) is the polynomial corresponding
to the common zeros of H 1 (z) and H 2 (z). In this case, we can
write (2) as
S ^ z h H 1, 2 (z) =

X2 ^ z h
X1 ^ z h
=
.
H 1l ^ z h H 2l ^ z h

(11)

This is the same equation that we would obtain in a normal SIMO
scenario, but now the wavelet S (z) is replaced by S (z) H 1, 2 (z).
Since H 1l (z) and H 2l (z) do not share common zeros, we can
proceed as before to determine these two polynomials.
However, when traces share common zeros, we are not able
to identify the actual channels, since the common zeros have now
been mixed with S (z). In other words, in this case the method
behaves as if the wavelet was S (z) H 1, 2 (z), as shown in Figure 2.
Interestingly, all of the aforementioned analysis may be
made in terms of linear algebra. As mentioned previously, we
want to find the reflectivity by solving Xh = 0, and, if the null
space of X has dimension one, we can determine the reflectivity up to a scalar factor. However, when the channels have

H1(z )

common zeros, the null space of X has a dimension larger than
one. All we can determine now is that h lies in this null space.
To extract more information, we have to move beyond multichannel methods in this case. For instance, inspired by MED,
in SMBD the reflectivity is chosen as the sparsest vector in the
null space of X [1].
The presence of similar zeros is also interesting. It will still
lead, in the noiseless case, to X having a null space of dimension one. However, X will also have some small singular values, which may lead to numerical difficulties. The presence of
noise exacerbates this problem. Recall that the solution in this
2
case is the vector h that minimizes Xh . In other words, h is
T
the eigenvector of X X associated with its smallest eigenvalue. With some abuse, we will call this the smallest eigenvector.
Now, if there are similar zeros, X T X actually has two or more
small eigenvalues, all close to each other. This may lead to cases
where, due to a particular noise realization, the smallest eigenvector produces a poor estimate of the reflectivity, whereas the
eigenvector associated to the second smallest eigenvalue produces a good estimate of the reflectivity. This is because, in this
case, both eigenvalues are similar and the choice of the smallest
one is determined more by the noise than by the reflectivities.
To see that the second-smallest eigenvalue may yield the
best solution in some cases, consider a nonrealistic example
with three reflectivities with two zeros each, at 0.5 and - 0.5,
0.5 and 0.2, and 0.51 and 0.2. All reflectivities have zeros close
to 0.5. Note that the reflectivities have much shorter duration than in practice but were chosen so as to guarantee that
they have similar zeros. Also, to ensure a long trace, our input
was 1,000 samples of a unit-variance white Gaussian signal,
instead of a seismic wavelet. We also added white Gaussian
noise with variance 0.1. In 96% of our simulations, the best
estimate presented a correlation coefficient larger than 0.9
with the actual reflectivities. However, in 48% of the cases this
good result was achieved by the second-smallest eigenvector.

x1(k )

s (k )

x1(k )

H2′(z )

x2(k )

H1, 2(z )

s (k )

H2(z )

H1′(z )

x2(k )

(a)

(b)

Figure 2. A block diagram of (a) a SIMO system with transfer functions H 1 (z) and H 2 (z) that share the common zeros represented by the squares. This

system can be decomposed as shown in (b): a SISO system with transfer function H 1, 2 (z), which contains the common zeros of H 1 (z) and H 2 (z), followed by a SIMO system with the remaining zeros of H 1 (z) and H 2 (z).

44

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