Optimally exploiting observing resources to obtain the best possible state estimate for the purpose of forecast initialization requires developing practical methods for implementing approximations to the Kalman filter. The impediment to direct application of the KF is that it requires the time dependent forecast error covariance the calculation of which presents a daunting computational challenge because of the large dimension of the forecast state. However, the dynamically relevant dimension of the forecast error system is far smaller than the state dimension and a number of reduced rank approximations to the KF have been developed. A method is described for implementing a new approximate KF using an optimal reduced rank representation of the forecast error system obtained by truncating the error system in Hankel operator form. Model examples are used to illustrate the method and a comparison with alternative suboptimal KF's is made.