Bayesian wavefield separation by transform-domain sparsity promotion

Synthetic data example:

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

$$=20\log_{10} \frac{\Vert\frac{\mathbf{s}_1}{\Vert\mathbf{s}_1\Ver... ...\mathbf{s}}_1\Vert _2}-\frac{\mathbf{s}_1}{\Vert\mathbf{s}_1\Vert _2}\Vert _2},$$ (8)

where the energies of the `multiple-free' data, $\mathbf{s}_1$ , and the estimated primaries, ${\widetilde{\mathbf{s}}}_1$ , are both normalized to one. First, Bayesian separation with optimal parameters yields an SNR of $\old{12.133}\new{12.13}\,\mathrm{dB}$ compared to $\old{10.268}\new{10.27}\,\mathrm{dB}$ for single thresholding. These SNRs are computed with respect to the ``multiple-free'' data. Second, the result for $\lambda ^\ast _1=0.7$ , $\lambda ^\ast _2=2.0$ and $\eta ^\ast =0.5$ in Figure 1(e) shows less remnant multiple energy compared to the standard SRME result with an SNR of $\old{9.815}\new{9.82}\,\mathrm{dB}$ . These parameter choices were found empirically and the smaller value for $\lambda^\ast_1$ is consistent with the predicted multiples and hence the weights $\mathbf{w}_1$ 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 ($> 2^{31}$ 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 $\{\lambda_1^\ast,\, \lambda_2^\ast,\,\eta^\ast\}$ compared to other choices) and that control over the estimated multiples ( $\{\lambda_1^\ast,\, \lambda_2^\ast,\,\eta^\ast\}$ versus reduced control for $100\cdot\{\lambda_1^\ast,\, \lambda_2^\ast,\,\eta^\ast\}$ ) 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 ( $\lambda _1/\lambda _2$ ), and different levels of fidelity in the predicted multiples $\eta ^{-1}$ . Here, $(\lambda_1^\ast,\,\lambda_2^\ast,\,\eta^\ast)=(0.7,\,2.0,\,0.5)$ . 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 $12.13\,\mathrm{dB}$ , adds $\old{1.486}\new{1.48} \,\mathrm{dB}$ to the estimated primaries without this control, yielding an SNR of $10.65\,\mathrm{dB}$ .

SNR ( dB)	$\{\lambda_1^\ast,\, \lambda_2^\ast\}$	$\{2\cdot \lambda_1^\ast,\, \lambda_2^\ast\}$	$\{\lambda_1^\ast,\, 2\cdot\lambda_2^\ast\}$	$100\cdot\{\lambda_1^\ast,\, \lambda_2^\ast\}$
$\eta^\ast$	12.13	11.21	11.46	-
$\frac{1}{2}\cdot\eta^\ast$	11.36	9.43	11.46	-
$2\cdot\eta^\ast$	11.44	12.13	9.92	-
$100\cdot\eta^\ast$	-	-	-	10.65

Bayesian wavefield separation by transform-domain sparsity promotion