Adaptive filtering with non-stationary helical filters

To handle the inherent non-stationarity of seismic data, we estimate a filter bank of non-stationary filters using helical boundary conditions Claerbout (1998). This approach has been successfully utilized by Rickett et al. (2001) to attenuate surface-related multiples. We use a NLCG solver with the $\ell^1$-norm and a standard conjugate gradient solver with the $\ell^2$-norm. The filter coefficients vary smoothly across the output space thanks to a preconditioning of the problem Rickett et al. (2001). In the following results, the non-stationary filters are 1D. We estimate the same number of coefficients per filter with the $\ell^2$- and $\ell^1$-norm.