In
Figure 1,
we make comparisons between primaries derived from the total data in
Figure 1(a) using standard SRME
(Figure 1(c)), 3-D curvelets with single thresholding
(Figure 1(d)), and our Bayesian method with
(Figure 1(e)) and without control on the estimated
multiples (Figure 1(f)). For reference, we included
``multiple-free'' data in Figure 1(b). This data was
modeled with an absorbing boundary condition, removing the
surface-related multiples. The SRME-predicted primaries plotted in
Figure 1(c) were obtained with a multiple-window
matched-filtering procedure. Comparing these results with the
curvelet-domain separations shows significant improvements. This
behavior is confirmed by signal-to-noise ratios (SNRs) that measure
the performance of our algorithm for various parameter settings. To
accommodate lack of correct scaling for the different signal
components, we define the SNR as
SNR
(8)
where the energies of the `multiple-free' data,
, and
the estimated primaries,
, are both normalized to
one.
First, Bayesian separation with optimal parameters yields an SNR of
compared to
for single thresholding. These
SNRs are computed with respect to the ``multiple-free'' data. Second,
the result for
,
and
in Figure 1(e) shows less remnant
multiple energy compared to the
standard SRME result with an SNR of
. These parameter choices were
found empirically and the smaller value for
is
consistent with the predicted multiples and hence the weights
being less sparse than the predicted primaries. For
this example, the solution converged in only five iterations of
Equation 7, which is quite remarkable given the
extreme problem size (
unknowns). Table 1
summarizes the performance in SNRs compared to the ground truth shown
in Figure 1(b). These numbers lead us to conclude that
our separation scheme is relatively robust against changes in the
control parameters ( cf. the value for
compared to other choices) and that
control over the estimated multiples (
versus reduced control for
) leads to
better results. However, our method leads to minor loss of
primary energy especially in regions of high curvature. By changing
the parameters, this loss can be reduced at the expense of more
residual multiple energy. For a more detailed study of this example,
refer to the ancillary material which includes difference plots. A
more fundamental solution to this problem will be investigated in
future work.
Table 1:
Sensitivity analysis for the performance of Bayesian
separation for different sparsity ratios
(
), and different
levels of fidelity in the predicted multiples
. Here,
. The
SNRs are computed via Equation 8 and are relatively robust
against changes in parameters. Parameter combinations for which
the ratios of the sparsity over level
of fidelity are not preserved were omitted since
these lead to extremely low SNRs. Note that the inclusion of control
on the estimated multiples, yielding an SNR of
, adds
to the estimated primaries without this control, yielding an SNR
of
.
SNR ( dB)
12.13
11.21
11.46
-
11.36
9.43
11.46
-
11.44
12.13
9.92
-
-
-
-
10.65
Bayesian wavefield separation by transform-domain sparsity
promotion