Data assimilation combines information from an imperfect model, sparse and noisy observations, and error statistics, to produce a best estimate of the state of a physical system. Different observational data points have different contributions to reducing the uncertainty with which the state is estimated. Quantifying the observation impact is important for analyzing the effectiveness of the assimilation system, for data pruning, and for designing future sensor networks. This paper is concerned with quantifying observation impact in the context of four dimensional variational data assimilation. The main computational challenge is posed by the solution of linear systems, where the system matrix is the Hessian of the variational cost function. This work discusses iterative strategies to efficiently solve this system and compute observation impacts.