In recent years more and more organizations around the world are in the process of developing and implementing four-dimensional data assimilation systems for research and operational purposes. One of the important issues in four-dimensional data assimilation is the specification of the initial background error covariance. There are different ways to estimate the initial background error covariance, such as using a simplified Kalman filter or a cycling representer. The simplified Kalman filter is currently explored at ECMWF, while the cycling representer algorithm is currently studied at NRL. The cycling representer data assimilation algorithm proposed by Xu and Daley (2000) is a weak constraint four-dimensional variational data assimilation algorithm. One of the unique aspects of this algorithm is its ability to give an internally consistent estimation of the initial background error covariance for the subsequent cycles. In Xu and Daley (2000) the algorithm was successfully applied to a simple transportation problem. The algorithm, however, has not previously been applied to a multivariate, multi-dimension system with dynamic instability. In this paper a barotropically unstable shallow water system was used as the test bed to further examine the cycling representer algorithm. The evolution of singular vectors associated with the shallow water system during a 96-hour period served as the "truth". Several experiments were conducted to examine the impact of single cycle, cycling only, and cycling with covariance updating on the assimilation results, respectively. Results will be presented at the meeting.