Climate projections should be able to reproduce long-term and low-frequency variability of the system. A method is proposed to estimate the low-frequency component of the ocean variability from observations using statistics from a prior long-term simulation. The paper presents a significant step forward in combining numerical ensembles together with satellite data to provide relevant reconstructions of a given ocean variable (in this case the low frequency variability of the sea surface height, SSH) in a way that reduces systematic errors of the prior pdf. It is likely that the application will lead to future improvement in the context of data assimilation. The paper includes significant theoretical advances but at the same time provides a concrete application for SSH estimation from observation and model results.