A local least squares framework for ensemble filtering

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 performed to compute update increments for each state variable ensemble member from corresponding observation variable increments. The regression can be applied globally or locally using Gaussian kernel methods.

Several previously documented ensemble Kalman filter methods, the perturbed observation ensemble Kalman filter and ensemble adjustment Kalman filter, are developed in this context. Some new ensemble filters that extend beyond the Kalman filter context are also discussed. The two-part method can provide a computationally efficient implementation of ensemble filters and allows more straightforward comparison of methods since they differ only in the solution of a scalar filtering problem.