Projection onto convex sets and its fast version

In exploration geophysics field, geophysicists prefer formulating the seismic data reconstruction problem as problem 3. The projection onto convex sets (POCS) algorithm is one of the most widely used methods for reconstructing missing seismic data, especially for those irregular sampled seismic data binned onto regular grids. The POCS (Abma and Kabir, 2006) aims to solve equation 3 by the following framework:

$$\mathbf{d}_{n+1} = \mathbf{d}_{obs} + (\mathbf{I}-\mathbf{S})\mathbf{A}^{-1}\mathbf{T}_{\tau}\left[\mathbf{A}(\mathbf{d}_n)\right],$$ (6)

where $\mathbf{d}_n$ denotes the estimated data after $n$th iteration.

Inspired from the FISTA, we propose the following faster version of POCS (FPOCS):

$$\begin{split} \mathbf{d}_{n}' &= \mathbf{d}_n + \frac{v_n-1}{... ...{T}_{\tau}\left[\mathbf{A}(\mathbf{d}_{n}')\right]. \end{split}$$ (7)

An obvious difference between IST and POCS (or between FPOCS and FISTA) is whether we make use of the known data. Briefly speaking, the IST (FISTA) treats all the data components as unknown while POCS (FPOCS) only treats the missing data components as unknown. The comparison between IST (FISTA) and POCS (FPOCS) in terms of the reconstruction performance and convergence rate can be done from two cases: irregularly sampled noisy data and irregularly sampled clean data. We will implement such two comparisons in the section of examples. The general conclusion can be given in advance: for irregularly sampled noisy dataset, the IST (FISTA) method can be superior because during the thresholding process, the extra random noise will be attenuated gradually; for irregularly sampled clean dataset, the POCS (FPOCS) method can be superior because the known sampled data help constrain the spatial coherency during the inversion. Here, the noisy level is relative. Those data acquired from the marine acquisition are much better than those data from land acquisition. In this paper, we only deal with the datasets from marine acquisition, thus we prefer the FPOCS instead of the FISTA.