A mixed variational-smoothing data assimilation method is derived under the perfect-model hypothesis. An incremental 4D-Var analysis scheme is supplemented with a low-rank approximation of its equivalent Kalman smoother, under the perfect-model assumption. A consistent method results, where the analysis and forecast error covariances provided by the smoother part describe the errors performed respectively during the incremental 4D-Var analysis and the model prediction phases. This is because the whole method is built around a low-rank approximation of the forecast error covariance matrix, and different hypotheses for the computation of the analysis and that of its error covariances are avoided. The method provides both flow-depen- dent analysis and forecast error covariances. In addition, most current developments for pre-operational or operational variational data assimilation systems are either already embedded within the method or may be straightforwar- dly included to it.