Pattern-based approach

The pattern-based approach separates noise and signal according to their multivariate spectra. In this Chapter, the patterns are approximated with 3-D prediction-error filters (PEFs). Calling ${\bf N}$ the PEFs for the multiples estimated from the multiple model with SRMP, ${\bf S}$ the PEFs for the signal estimated from the Spitz approximation Spitz (1999) and ${\bf s}$ the primaries, the objective function to minimize becomes  

$$g({\bf s})=\Vert{\bf MN(s - d)}\Vert^2+\epsilon^2\Vert{\bf MSs}\Vert^2,$$ (70)

where ${\bf M}$ is a masking operator that preserves the signal where no multiples are present. The signal is estimated iteratively with a conjugate gradient method, and not with the Huber norm of Chapter . In few words, the Spitz approximation consists in (1) convolving the noise PEFs N with the data d, and (2) estimating the signal PEFs S from the convolution result. As demonstrated by Abma (1995), the least-squares inverse of ${\bf s}$ in equation () is an optimal Wiener filter. Similar to Chapter , the dataset is divided into macro-patches of 50 consecutive shots with an overlap of 5 shots before the separation. Then the macro-patches are reassembled to form the final result.