As ocean models improve, assimilation of data with the help of models becomes increasingly important. The Kalman filter provides a method for assimilation of data that are arbitrarily distributed in time and space and have differing error characteristics. Its desirable features are optimality in the least squares sense for a broad class of systems, and recursiveness, i.e., the algorithm depends only upon statistical quantities that are updated with each successive observation. The observations themselves may then be discarded, and no actual history of the system under study need be retained. The full Kalman filter, however, presents considerable demands on computing resources. There are few examples with solutions in closed from, relatively little is known about the case in which the system under study is governed by partial rather than ordinary differential equation, and the effects of nonlinearity are still incompletely understood. In this study a first step is undertaken toward the formulation of a suitably simplified, computationally efficient form of the Kalman filter for estimation and prediction of ocean eddy fields. In this step, the full Kalman filter is applied to simplified systems designed to capture some of the properties of open ocean models, and computational results are analyzed and interpreted in terms of realistic models are datasets.