Bayesian wavefield separation by transform-domain sparsity promotion

Field data example:

Figure 2(a) contains the common-offset section (at offset $200\,\mathrm{m}$ ) that we selected from a North Sea field dataset. SRME-predicted multiples and primaries are plotted in Figure 2(b) and 2(c). Comparison of the SRME-predicted primaries with the section obtained by our Bayesian separation method using the 3-D curvelet transform shows a clear improvement in the removal of shallow multiples and improved continuity for late primary arrivals. These results were obtained for $\lambda _1=1.0$ , $\lambda _2=2.0$ and $\eta =1.0$ . These parameter choices are consistent with less-than-ideal real data for which we can not expect significant changes in sparsity between primaries and multiples and for which we can not completely trust the predicted multiples. In this example, the solution is attained after only ten iterations. Again, refer to the ancillary material for a more detailed study of this example.

Figure1-a,Figure1-b,Figure1-c,Figure1-d,Figure1-e,Figure1-f
Figure 1. Primary-multiple separation on a synthetic data volume. (a) The total data, $\mathbf{b}$ . (b) Reference surface-related multiple-free data modeled with an absorbing boundary condition. (c) SRME-predicted primaries, $\mathbf{b}_1$ . (d) Estimate for the primaries, using 3-D curvelets and single thresholding. (e) The same but with Bayesian thresholding for $\lambda ^\ast _1=0.7$ , $\lambda ^\ast _2=2.0$ and $\eta ^\ast =0.5$ . (f) The same as (e) but now for $\{\lambda _1,\lambda _2,\eta \}=100\cdot \{\lambda ^\ast _1,\lambda ^\ast _2,\eta ^\ast \}$ . Notice the improvement in the estimated primaries by controlling the estimated multiples. By multiplying the $\eta$ and the other control parameters by a large factor, we diminished the control over the multiple prediction while keeping the sparsity penalties the same. Less control clearly adversely affects the estimated primaries, which is confirmed by the SNRs computed with respect to (b), i.e, $12.13\,\mathrm{dB}$ versus $10.65\,\mathrm{dB}$ .

Figure2-a,Figure2-b,Figure2-c,Figure2-d
Figure 2. Field data example of curvelet-domain primary-multiple separation. (a) Near-offset ( $200\,\mathrm{m}$ ) section for the total data plotted with automatic-gain control. (b) Estimate for the multiples, yielded by optimized multi-window SRME. (c) Corresponding estimate for the primaries using SRME. (d) Estimate for the primaries computed by Bayesian iterative thresholding with $\lambda _1=1.0$ , $\lambda _2=2.0$ and $\eta =1.0$ . Notice the improvement of the Bayesian curvelet-domain separation compared to SRME. Not only are the shallow multiples better removed but we also observe an improved continuity for the late primary arrivals.

Bayesian wavefield separation by transform-domain sparsity promotion