We propose a contribution that combines model reduction with data assimilation. A dedicated Parametrized Background Data-Weak (PBDW) [1] approach has been introduced in the literature so as to combine numerical models with experimental measurements. We extend the approach to a time-dependent framework by means of a POD-greedy reduced basis construction. Since the construction of the basis is performed offline, the algorithm addresses the time dependence of the problem while the time stepping scheme remains unchanged. Moreover, we devise a new algorithm that exploits offline state estimates in order to diminish both the dimension of the online PBDW statement and the number of required sensors collecting data. The idea is to exploit in situ observations in order to update the best-knowledge model, thereby improving the approximation capacity of the background space.