Adaptive subtraction of multiples using the L1-norm

A strategy for multiple removal consists of estimating a model of the multiples and then adaptively subtracting this model from the data by estimating shaping filters. A possible and efficient way of computing these filters is by minimizing the difference or misfit between the input data and the filtered multiples in a least-squares sense. Therefore, the signal is assumed to have minimum energy and to be orthogonal to the noise. Some problems arise when these conditions are not met. For instance, for strong primaries with weak multiples, we might fit the multiple model to the signal (primaries) and not to the noise (multiples). Consequently, when the signal does not exhibit minimum energy, we propose using the L-1-norm, as opposed to the L-2-norm, for the filter estimation step. This choice comes from the well-known fact that the L-1-norm is robust to 'large' amplitude differences when measuring data misfit. The L-1-norm is approximated by a hybrid L-1/L-2-norm minimized with an iteratively reweighted least-squares (IRLS) method. The hybrid norm is obtained by applying a simple weight to the data residual. This technique is an excellent approximation to the L-1-norm. We illustrate our method with synthetic and field data where internal multiples are attenuated. We show that the L-1-norm leads to much improved attenuation of the multiples when the minimum energy assumption is violated. In particular, the multiple model is fitted to the multiples in the data only, while preserving the primaries.