An adaptive multiple subtraction step is necessary for almost all methods that predict seismic multiple reflected waves. We aim at giving a better understanding of matching filters based on lq-norms and on statistical independence. We found that the formulation of all of these techniques can be gathered in a mutual framework by introducing a space-time operator, called the primary enhancer, acting on the estimated primaries. The differences between the considered matching filters become more intuitive because this operator behaves as a simple amplitude compressor. In this perspective, all the methods tend to uncorrelate the predicted multiples and the enhanced estimated primaries. The study of these matching-filter methods can be narrowed to the study of the primary enhancer operator because it is the only difference. Moreover, we have emphasized the role of using adjacent traces or windowing approaches in terms of statistics, and we show that an adequate windowing strategy may overbear the choice of the objective function. Indeed, our analysis showed that setting a good windowing strategy may be more important than changing the classical least-squares adaptation criterion to other approaches based on lq-norm minimization or independent component analysis.