A number of methods for using ensemble integrations of numerical models as integral parts of data assimilation have appeared in the atmospheric and oceanic literature. Many of these have been derived from the Kalman filter and have been called ensemble Kalman filters. A more general class of methods that includes these ensemble Kalman filters is derived from the nonlinear filtering problem. By working in a joint state / observation space, many features of ensemble filtering algorithms become easier to derive. The ensemble filter methods discussed make use of a (local) least squares assumption about the relation between the prior distribution of an observation variable and the model state variables. The update procedure applied when a new observation becomes available can be described in two parts. First, an update increment is computed for each prior ensemble estimate of the observation variable by applying a scalar ensemble filter. Second, a linear regression of the prior ensemble sample of each state variable on the observation variable is used to compute update increments for the state variables. The regression can be applied globally or locally using Gaussian kernel methods.

Several previously documented ensemble Kalman filter methods are developed in this context. Some new ensemble filters that are clearly beyond the Kalman filter context are also discussed. The two part method allows a computationally efficient implementation of ensemble filters and also provides a more straightforward comparison of these methods since they differ only in the solution of a scalar filtering problem.