The use of surrogate modeling techniques for estimating the state of systems described by parameterized partial differential equations has been of great interest in recent years in the data assimilation community. The parameter associated with the observed state is also usually of great interest, e.g. in optimal control problems. Ideally, the introduction of a surrogate model should reduce the computational cost, particularly in comparison to the Full Order Model (FOM), while introducing the minimal possible approximation error. In particular, surrogate models based on standard estimation techniques such as the least-squares or Galerkin method introduce approximation errors on the predicted observations. Reducing such errors ensures that the results of the data assimilation will be closer to the one obtained using the FOM. In our work, we propose to build a reduced order model that minimises the error on the predicted observations. This method is based on a variation of the proper orthogonal decomposition method, where the inner product used in the computation of the snapshot correlation matrix implies the observation operator. The reduced space built is then able to reproduce exactly the quantities of interest in a projective sense. A challenge associated to the method we propose is to correctly exploit the newly produced reduced space, in order to preserve its properties, while using it in a Reduced Order Model (ROM) for estimation. We propose techniques to address this current limitation, enhancing estimations while keeping a negligible computational cost.