Iterative deblending of simultaneous-source seismic data using seislet-domain shaping regularization

Shaping regularization

The unknown $\mathbf{m}$ in equation 3 can be recovered iteratively using shaping regularization:

$$\mathbf{m}_{n+1}=\mathbf{S}[\mathbf{m}_n+\mathbf{B}(\mathbf{\tilde{d}}-\mathbf{Fm}_n)],$$ (5)

where operator $\mathbf{S}$ shapes the estimated model into the space of admissible models at each iteration (Fomel, 2008,2007) and $\mathbf{B}$ is the backward operator that provides an inverse mapping from data space to model space. Daubechies et al. (2004) prove that, if $\mathbf{S}$ is a nonlinear thresholding operator (Donoho and Johnstone, 1994), $\mathbf{B}=\mathbf{F}^T$ where $\mathbf{F}^T$ is the adjoint operator of $\mathbf{F}$ , iteration 5 converges to the solution of equation 6 with $L_1$ regularization term:

$$\min_{\mathbf{m}} \parallel \mathbf{Fm-\tilde{d}} \parallel_2^2 +... ...mathbf{A}^{-1}\mathbf{m}\parallel_1,%\epsilon\parallel \mathbf{m} \parallel_p,$$ (6)

where $\mu$ is the threshold and $\mathbf{A}^{-1}$ denotes a sparsity-promoting transform. A better choice for $\mathbf{B}$ is the pseudoinverse of $\mathbf{F}$ : $\mathbf{B}=(\mathbf{F}^T\mathbf{F})^{-1}\mathbf{F}^T$ (Daubechies et al., 2008).

Iterative deblending of simultaneous-source seismic data using seislet-domain shaping regularization