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

Performance evaluation and convergence rate

Combining equations 3, 4 and 5, we get:

$$\left[\begin{array}{cc} \mathbf{d}_1 \mathbf{d}_2 \end{array}\r... ...gin{array}{cc} \mathbf{d}_1 \mathbf{d}_2 \end{array}\right]_n \right]\right],$$ (7)

where $\mathbf{S}$ is a block operator that takes the form $[\mathbf{S}_1\quad \mathbf{0};\mathbf{0}\quad\mathbf{S}_2]$ , and $\mathbf{S}_1$ and $\mathbf{S}_2$ correspond to the shaping operators for $\mathbf{d}_1$ and $\mathbf{d}_2$ , respectively. Using equation 7 and performing appropriate iterations, we aim to get an estimate of unblended seismic data.

To test the convergence performance and accuracy of iteration 7, we use the following measure (Hennenfent and Herrmann, 2006):

$$SNR_{i,n}=10\log_{10}\frac{\Arrowvert \mathbf{d}_i \Arrowvert_2^2}{\Arrowvert \mathbf{d}_i-\mathbf{d}_{i,n}\Arrowvert_2^2},$$ (8)

where $SNR_{i,n}$ stands for signal-to-noise ratio of $i$ th source after $n$ th iteration, $\mathbf{d}_{i,n}$ denotes the estimated model of $i$ th source after $n$ iterations, $\mathbf{d}_i$ denotes the true model for $i$ th source and $\parallel \cdot \parallel_2^2$ denotes the squared $L_2$ norm of a function.

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