Bayesian wavefield separation by transform-domain sparsity promotion

Empirical choice of the weights

In practice, the appropriate choice of the weights $\mathbf{w}_1$ and $\mathbf{w}_2$ proves important. Motivated by empirical findings (Herrmann et al., 2007a), we choose the weights using the SRME predictions for the two signal components: we set $\mathbf{w}_1=\max\{\vert\tensor{A}^T\mathbf{b}_2\vert,\epsilon\}$ and $\mathbf{w}_2=\max\{\vert\tensor{A}^T\mathbf{b}_1\vert,\epsilon\}$ with the operations taken elementwise. Here, $\tensor{A}^T$ is the forward curvelet transform (the discrete curvelet transform based on wrapping is a tight frame, so the transpose of this transform is its inverse) and $\epsilon$ a noise dependent constant. This choice of weights guarantees that the weights are strictly positive, thus the algorithm converges. Furthermore, such a choice makes it less likely that the optimization algorithm produces large curvelet coefficients for the primaries at entries in the curvelet vector that exhibit large coefficients for the predicted multiples, and vice versa.

Remark: Within the Bayesian context, this particular choice breaks the assumption that the curvelet coefficients of the primaries and multiples are independent. However, our estimator can still be interpreted as Bayesian with a posterior probability of the form given in Equation 5 with the weights defined as mentioned above.

Bayesian wavefield separation by transform-domain sparsity promotion