Projection onto convex sets (POCS)

The principal problem of seismic data reconstruction is to solve the under-determined inversion problem:

$$\mathbf{Fm}=\mathbf{d},$$ (1)

where $\mathbf{m}$ is the well-sampled seismic data, $\mathbf{F}$ denotes the sampling operator, and $\mathbf{d}$ denotes the observed data Chen et al. (2014a). Because of the missing traces, the operator $\mathbf{F}$ is highly singular and thus make equation 1 highly under-determined.

There have existed many different algorithms for solving equation 1 by adding different constraints. The POCS algorithm Abma and Kabir (2006) is one of the most widely used methods to interpolate seismic data with irregularly missing traces, and has the following iterative expression:

$$\mathbf{m}_{n+1}=\mathbf{d}+(\mathbf{I}-\mathbf{F})\mathbf{A}^{-1}\mathbf{T}\mathbf{A}[\mathbf{m}_n],$$ (2)

where $\mathbf{m}_n$ denotes the reconstructed data after $n$th iteration, $\mathbf{A}$ and $\mathbf{A}^{-1}$ are the forward and inverse sparsity-promoting transforms, $\mathbf{I}$ is an identity matrix and $\mathbf{T}$ is a thresholding operator. However, for regularly missing traces, POCS can not obtain satisfying results because of the strong aliasing noise in the sparse domain