Introduction

A classical approach for attenuating multiples consists of building a multiple model Verschuur et al. (1992) and adaptively subtracting this model from the multiple infested-data by estimating shaping filters Dragoset (1995); Liu et al. (2000); Rickett et al. (2001). The estimation of the shaping filters is usually done in a least-squares sense making these filters relatively easy to compute. In some cases, however, a least-squares criterion can lead to undesirable artifacts. This happens when, for example, the relatively strong primaries are surrounded by multiples, such that the filter tends to distort primary energy as well.

In this paper we show that the estimation of shaping filters with the $\ell^1$-norm gives better results than with the $\ell^2$-norm when multiples and primaries have noticeable amplitude differences. We first illustrate this with a simple 1D problem that highlights the limits of the least-squares approach. In a second synthetic example, we attenuate internal multiples and show that the $\ell^1$-norm gives far better results than does $\ell^2$.To finish, we utilize shaping filters on a multiple contaminated gather from a seismic survey showing that the $\ell^1$-norm leads to a substantial attenuation of the multiples.