IEEE Signal Processing - March 2018 - 43

the problem in a noisy system, while the regularization term is
a common solution for ill-posed problems. For instance, like
MED, [1] assumes that the reflectivity is sparse. Thus, they
propose in [1] a sparse MBD (SMBD) that employs a sparsitypromoting regularization function for SIMO deconvolution.

SIMO seismic deconvolution
In this section, we present the fundamentals of SIMO deconvolution. We begin by presenting the noiseless SIMO system model.
To that end, consider again the CSG shown in Figure 1. In this
case, the recorded data at the pth receiver, x p (k), can be modeled as the convolution of the reflectivity series seen by the signal
that reaches this receiver, h p (k), with the seismic wavelet s (k),
i.e., x p (k) = s (k) ) h p (k). In terms of z-transforms, we can write
X p ^ z h = H p ^ z h S ^ z h.

(1)

In the jargon of channel equalization used in [5], each trace
corresponds to a channel output, and each reflectivity series
corresponds to the impulse response of a channel.
To develop SIMO deconvolution, consider a pair of traces x p (k) and x q (k), with X p (z) = H p (z) S (z) and X q (z) =
H q (z) S (z). Then, by isolating the seismic wavelet, we have
S (z) =

Xq ^ z h
Xp ^zh
=
.
H p ^ z h Hq ^ z h

and h = [h T1 , h T2 ] T . If the null space of X has dimension one;
all of the nontrivial solutions to this equation are proportional
to each other. Thus, if the estimate ht is a nontrivial solution to
Xh = 0, then ht is proportional to h. That is, we have determined the reflectivity up to a scalar factor, which is the best
we can achieve in this case [5]. Due to this scalar ambiguity,
it makes no sense to measure the performance of the methods
2
using h - h since this value may be large purely due to a
scalar factor. Instead, as in [1] and [21], we will use the Pearson correlation coefficient (PCC) between h and ht given by
(h T ht / || h || || ht ||). The closer the PCC is to one, the better the
estimate. As shown in [13] and [14], this happens if H 1 (z) and
H 2 (z) are coprime, i.e., if they do not share common zeros.
If there are J traces, there are JK unknown samples of
the J reflectivity functions. But with J traces, we can now
form [J (J - 1) /2] equations like (4), each with M + K - 1
rows, by combining all possible pairs of traces, i.e., all possible combinations of p and q. This leads to a system of
[J (J - 1) /2] (M + K - 1) equations and JK unknowns:

As a consequence, we must also have
(3)

Note that, in the time domain, both products correspond to
the convolution of a trace with a reflectivity function. Thus, in
matrix notation, (3) can be expressed as
X p h q - X q h p = 0, 6p, q,

(4)

where X p is the convolution matrix of the pth trace,
Rx p ^ 0 h 0
S
Sx p ^ 1 h x p ^ 0 h
X p = SS h
h
0
S 0
S 0
0
T

V
g
0
0
W
W
g
0
0
W,
j
h
h
W
g x p ^ M - 1 h x p ^ M - 2 hW
g
0
x p ^ M - 1 hW
X

(5)

and h q = [h q (0), f, h q (K - 1)] T is the vector formed with
the samples of the qth reflectivity function. The size of X p is
(M + K - 1) # K, where M = (K + L - 1) is the length of the
trace, K is the length of the reflectivity function, and L is the
length of the wavelet.
Assume first that there are only two traces. In this case, (4)
can be written as
Xh = 0,
(6)
where
X = 6X 2 - X 1@

(7)

(8)

R
V
SX 2 - X 1
W
SX 3
W
- X1
S
W
- X1
SX 4
W
Sh
W
j
X=S
W
X
X
3
2
S
W
X4
- X2
S
W
S
W
X5
- X2
SS
W
h
jW
T
X

(9)

where

(2)

X p ^ z h H q ^ z h - X q ^ z h H p ^ z h = 0, 6p, q.

Xh = 0,

and h = [h T1 , f, h TJ ] T . As in the case of having only two traces, like in (6) and (7), the channels can be identified uniquely,
up to a scalar factor, if the null space of the data matrix X has
dimension one, which happens when X is of rank JK - 1 or,
equivalently, if the polynomials H p (z) do not share common
zeros [13], [14].
In the presence of noise, X will most likely have full column rank, so that Xh = 0 has only the trivial solution h = 0.
Since we cannot hope to find a reflectivity that makes Xh = 0,
we may seek instead a reflectivity that makes Xh as small as
possible [13]. Thus, in the noisy case, multichannel deconvolution becomes an optimization problem: the reflectivity is the
2
value of h that minimizes Xh . To avoid the trivial solution
and deal with the problem of the scale factor, one can impose
2
a constraint such as h = 1 [1].
Clearly, the great benefit of the procedure proposed by [12]
is that it results in equations that do not depend on the wavelet
s (k) . However, the requirement that the channels are coprime
is hard to verify in practice. In fact, as can be seen in Figure 1, the seismic wavelet in neighboring traces goes through
very similar paths, so that there is great similarity between
their reflectivity functions. In the next sections, we see how

IEEE Signal Processing Magazine

|

March 2018

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43



Table of Contents for the Digital Edition of IEEE Signal Processing - March 2018

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