Bayesian wavefield separation by transform-domain sparsity promotion

Discussion

As shown in our derivation, the Bayesian formulation in Equation 5 leads to an optimization problem--i.e., the minimization of $f(\mathbf{x}_1,\mathbf{x}_2)$ , which involves a combined minimization of the weighted $\ell^1$ -norms of the coefficient vectors, the $\ell^2$ -misfit between the SRME-predicted and estimated multiples, and the $\ell^2$ -misfit between the sum of the estimated primaries and multiples and the observed total data. From this interpretation, it is clear that our Bayesian formulation is an extension of earlier work (Herrmann et al., 2007a) since our objective function includes an additional term. This new term acts as a safeguard by making sure that the estimated multiples remain sufficiently close to the SRME-predicted multiples.The lower our confidence is in the SRME-predicted multiples, the more emphasis we place on the total data misfit. The case $\eta\to \infty$ is analogous to an absolute lack of confidence on the SRME-predicted multiples, and thus includes a misfit concerning the total data only. This limiting unrealistic assumption underlaid our earlier formulation (Herrmann et al., 2007a). Away from this limit, Equation 6 leads to solutions that are not only sparse, but also produce estimated curvelet coefficients for the multiples that are required to fit the SRME-predicted multiples. The relative degrees of sparsity for the two signal components are controlled by $\lambda_1$ and $\lambda_2$ .

The performance of the presented separation algorithm depends on: (i) the sparsity of the coherent signal components in the transform domain: the sparser the two signal components, the smaller the chance that the supports of the two curvelet coefficient vectors overlap; (ii) the validity of the independence assumption, which is empirically established in Herrmann et al. (2007a); (iii) the accuracy of the SRME-predictions. Even though it was shown that curvelet-domain separation is relatively insensitive to errors in the SRME-predictions, significant amplitude errors (significant timing errors are assumed absent) lead to a deterioration of the separation. However, for smoothly varying amplitude errors, a remedy for this situation has recently been proposed that is based on introducing a secondary curvelet-domain matched filter (Herrmann et al., 2007b).

Bayesian wavefield separation by transform-domain sparsity promotion