Bayesian wavefield separation by transform-domain sparsity promotion

The separation algorithm

For appropriately chosen $\lambda_1,\lambda_2, \eta$ , and reasonably accurate SRME-predicted multiples, the minimization of the objective function $f(\mathbf{x}_1,\mathbf{x}_2)$ leads to a separation of the primaries and multiples. To minimize $f(\mathbf{x}_1,\mathbf{x}_2)$ in Equation 6, we devise an iterative thresholding algorithm in the spirit of the work by Daubechies et al. (2003). Starting from arbitrary initial estimates $\mathbf{x}_1^0$ and $\mathbf{x}_2^0$ of $\mathbf{x}_1$ and $\mathbf{x}_2$ , the $n^{\text{th}}$ iteration of the algorithm proceeds as follows

$$\begin{array}[l]{lll} \mathbf{x}_1^{n+1} &=&\tensor{T_{\frac{... ...tensor{A}^T\tensor{A}\mathbf{x}_1^{n}\big) \right], \end{array}$$ (7)

where $\tensor{T}_{\mathbf{u}}: \mathbb{R}^{\vert{\mathcal{M}}\vert} \mapsto \mathbb{R}^{\vert{\mathcal{M}}\vert}$ is the elementwise soft-thresholding operator--i.e., for each $\mu\in{\mathcal{M}}$ , ${T}_{u_\mu}(v_\mu):= \mathrm{sgn}(v_\mu) \cdot \max(0,\vert v_\mu\vert-\vert u_\mu\vert)$ . The proposed algorithm provably converges to the minimizer of $f(\mathbf{x}_1,\mathbf{x}_2)$ , provided all weights--i.e., all components of the vectors $\mathbf{w}_1$ and $\mathbf{w}_2$ , are strictly positive (Daubechies et al., 2003).