IEEE Signal Processing - March 2018 - 48
regularization parameter m, which controls the tradeoff
between the sparsity, measured by R e (h), and the data fitness,
measured by (1/2) Xh 2 . The optimization algorithm is the
same as the one used in [1], with the difference that the data
u . The number of
matrix X is replaced by the reduced matrix X
iterations was set to 1,000, and the parameter that controls the
smoothness of the hybrid , 1 /, 2-norm was set to e = 0.0005. As
seen in Figure 4, using fewer traces results in a sharper final
image, with broader frequency content.
The results are shown in Figure 5, where (a)-(c) presents
the data fitness (1/2) Xh 2, the sparsity measure R e (h), and
the PCC between the original and estimated reflectivity series,
respectively, for different values of the threshold t max and for
different values of m. To make the plots clearer and more meaningful, instead of showing t max, we directly show the percentage
of trace pairs that are used as a result of different values of t max .
Note that, for a fixed m, decreasing t max, which leads to more
traces being discarded, yields a worse misfit, i.e., an increased
value of (1/2) Xh 2 . It also decreased the regularization term,
i.e., increased the sparsity of the estimated reflectivity. We also
note that decreasing t max increased the correlation coefficient
between the original reflectivity series and the estimated reflectivity, indicating an improved solution.
To see that using fewer traces may indeed improve the
results, consider points A and B in Figure 5(a), points C and D
in (b), and points E and F in (c). Points A, C, and E correspond
to the misfit, sparsity, and correlation coefficient, respectively,
achieved by the SMBD method with m = 6. Points B, D, and
F correspond to the misfit, sparsity, and correlation coefficient, respectively, achieved using 60% of the trace pairs, with
m = 5. To achieve a misfit of 27, SMBD uses m = 6 (point A),
which yields a sparsity of 35 (point C), and a correlation level
of 0.55 (point E). On the other hand, by setting t max = 0.034,
we obtain a matrix X that uses only 60% of the pairs. Using
this matrix and m = 5 yields the same misfit as SMBD (point B),
70
0.75
C
0.7
R (h)
40
PCC
32
D
30
∋
1 || ||2
2 Xh
In this tutorial article, we reviewed multichannel seismic
deconvolution, a class of methods that work on several traces
simultaneously, under the assumption that these traces were
generated by the same seismic wavelet. We have seen that
these methods present some advantages when compared to
algorithms that operate in a single trace at a time. For instance,
under some conditions, multichannel methods can produce an
exact, algebraic estimation of the reflectivity. However, these
conditions are hard to meet in practice. The main problem is
that the reflectivity functions of neighboring traces are not sufficiently different and the seismic signal is usually corrupted
by noise. Through an example, we have illustrated in detail
these problems and some of their consequences. The main one
is that we may have "almost" ill-posed problems, in the sense
that we know that all of the estimates in a certain subspace are,
from the point of view of multichannel methods, almost equally good, but we cannot determine which solution is the best.
Then, we showed how sparsity is an interesting regularization
34
50
20
60
Summary
36
60
30
B
a more sparse estimated reflectivity (point D) and a higher correlation (point F).
We now present some results obtained with a stacked section
of the marine acquisition, as shown in Figure 6. The parameters
for the algorithm were m = 10, e = 0.0001, and the maximum
number of iterations for the optimization method was set to
200. We assumed that the wavelet has length L = 21 samples.
We have run the algorithm in windows of 25 traces with 20%
overlap [1]. The parameter t max was set so that, on average,
50% of the pairs were used in each window. The results were
compared to the ones obtained with the SMBD algorithm. The
seismograms of Figure 6 show the estimated reflectivities with
the reduced SMBD and with the SMBD, respectively. Note
that, as seen in Figure 6(b) and (c), using fewer rows produces
a sharper image with a broader spectrum.
28
26
24
A
20
60
λ=5
F
0.6
E
0.55
22
70
80
90
100
Total Combinations (%)
(a)
0.65
70
80
90
Total Combinations (%)
(b)
λ=6
100
λ=7
0.5
60
70
80
90
100
Total Combinations (%)
(c)
λ=8
Figure 5. Each plot shows the following figures of merit when we vary the regularization parameter m and the number of trace pairs used to form the
2
system of linear equations: (a) data fitness (1/2) Xh , (b) sparsity measure R e (h), and (c) the PCC between the original reflectivity series and the
estimated reflectivity series.
48
IEEE Signal Processing Magazine
|
March 2018
|
Table of Contents for the Digital Edition of IEEE Signal Processing - March 2018
Contents
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