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We have demonstrated that simultaneous-source data sets can be separated using an inversion formulation. As shown in the numerical examples from realistic 2D models, our method can be applied in very complex geological environments, to data from multiple seismic sources and to simultaneous-source time-lapse seismic data sets. Incorporating dip information in the inversion procedure through non-stationary directional Laplacians (and in the time-lapse case, temporal constraints) helps to attenuate residual artifacts from the separation process.