IEEE Signal Processing - March 2018 - 45

In these cases, the smallest eigenvalue yielded an average correlation coefficient of around 0.3.
Note that we can only determine that the best solution is
the second-smallest eigenvector because we can compare it to
the reflectivity. In other words, in practice, we will never know
that we should not use the smallest eigenvector. Still, inspired
by MED, the quality of the method can be improved by saying
that the reflectivity is the sparsest vector in the space spanned
by the eigenvectors associated with the group of smallest eigenvectors. Recall that a vector is sparse if it has few nonzero components. In the next section, we discuss one such strategy in detail.

Sparse multichannel seismic deconvolution
In the presence of noise, multichannel seismic deconvolution
leads to an optimization problem that may be hard to solve due
to the similarity between the reflectivity functions. Mathematically, in this case there will be more than one small eigenvalue,
so that, for any linear combination of the associated eigenvectors,
2
the cost function Xh will have approximately the same value.
As is usual practice in these cases, regularization terms may be
added to the problem to help us choose one of these combinations
as the estimated reflectivity [1].
Currently, the most popular form of regularization in the
literature seeks a reflectivity function h that, besides making
2
Xh small, is also sparse. To that end, we introduce a sparsitypromoting regularization term to our problem, which yields all of
the well-known advantages of sparsity [22], [23]. This approach
is, in a way, related to MED, in that it exploits the fact that the
reflectivity function is a sparse signal since there are few rock
interfaces in the subsurface.
Of particular interest to this article is the work of Kazemi
and Sacchi [1], which proposed the SMBD method. The authors
employ the hybrid , 1 /, 2 -norm, also known as pseudo-Huber
function [24]-[26], as a sparsity-promoting regularization function, leading to the following estimator:
2
ht = argmin 1 Xh + mR e ^hh,
2
h
subject to h T h = 1,

(12)

where R e (h) = R i R k ` h 2i (k) + e 2 - ej is the hybrid , 1 /, 2norm and the constraint h T h = 1 is imposed to avoid the trivial
solution h = 0. The regularization parameter m controls the
tradeoff between the level of sparsity of the reflectivity series,
R e (h), and the data fitness, (1/2) Xh 2 . For instance, large values of m increase the weight of R e (h) in the cost function, so that
solutions in this case tend to have a smaller value of R e (h), i.e.,
will be more sparse at the expense of the fitness term Xh 2 .
Usually, the most common sparsity-promoting function is
the , 1 norm [27], [28]. The main advantage of the hybrid , 1 /, 2norm over the , 1 norm is that it is smooth and differentiable, so we
can use simple steepest descent methods to solve (12), as the one
detailed in [1], which was used to produce the results of this article. Furthermore, R e (h) also promotes sparsity [29], [30]. In fact
[30], if h i (k) & e, then h 2i (k) + e 2 - e . | h i (k) | - e, so that
this term behaves as the absolute value used in the , 1-norm. On the

other hand, if h i (k) % e, then h 2i (k) + e 2 - e . h 2i (k) / (2e),
which is a quadratic term that ensures differentiability.

Further improvements to SMBD
In this section, we describe some methods that exploit the
similarity between reflectivities to decrease the computational
complexity of SMBD. We will see that, as a by-product, the performance of the method also improves. We begin by noting that
the complexity of SMBD is to a great extent related to the size of
X, which may be quite large. Indeed, by considering all of the
J (J + 1) /2 possible combinations of p and q in X, the size of
linear equations to be solved grows quadratically with the number of channels J. In seismic deconvolution, where a CSG can
have hundreds or even thousands of traces, this may lead to an
unfeasible complexity.
Note, however, that the system in (8) has more equations than
necessary. Consider, for instance, a SIMO system with three traces that has 100 samples each, and a reflectivity with 90 samples,
so that X 1, X 2, and X 3 in (5) have dimensions 189 × 90. Thus,
the linear system matrix X in (9) has dimensions 567 × 270:
X2 - X1
=
- X 1H .
X >X 3
X3 - X2

(13)

However, all we need is that the null space of X has dimension one, which can be achieved if X has at least one fewer
rows than columns and all the rows are linearly independent.
In other words, our system has more rows than required. For
instance, consider the matrix
u = =X 2 - X 1
G,
X
X3 - X2

(14)

formed by eliminating the middle block row of X. It is not hard
u also has a null space of dimension one, as long
to verify that X
as the channels do not share common zeros.
u is easier than working with X, since
Clearly, working with X
the latter is a larger matrix. In fact, the cost of computing X T Xh,
which is necessary for the computation of the gradient vector of
the cost function in (12), is O ((JK ) 2 (M + K - 1) N c + (JK ) 2),
where N c is the number of block rows of X. Thus, eliminating
block rows of X, or equivalently ignoring some pairs of traces,
is of interest. At this point, one could argue that the final cost
does not actually depend on the number of rows in X, since we
can compute X T X and use this matrix for the remainder of the
method. Even though this is true, the structure of X, with its
many convolution matrices and zero entries, can be exploited so
that computing first Xh and then multiplying the result by X T is
less costly then multiplying h by X T X.
Note, however, that not all pairs can be ignored. For
instance, if we wish to determine the reflectivity, we need to
preserve at least one row involving each trace. This condition
u has the desired dimenguarantees that the range space of X
sion. This still leaves us a choice: Among all of the pairs, which
ones should we ignore and which ones should we keep?

IEEE Signal Processing Magazine

|

March 2018

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45



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