Both random and structured perturbations affect seismic data. Their removal, to unveil meaningful geophysical information, requires additional priors. Seismic multiples are one form of structured perturbations related to wave-field bouncing. In this paper, we model these undesired signals through a time-varying filtering process accounting for inaccuracies in amplitude, time-shift and average frequency of available templates. We recast the problem of jointly estimating the filters and the signal of interest (primary) in a new convex variational formulation, allowing the incorporation of knowledge about the noise statistics. By making some physically plausible assumptions about the slow time variations of the filters, and by adopting a potential promoting the sparsity of the primary in a wavelet frame, we design a primal-dual algorithm which yields good performance in the provided simulation examples.