A Diffusive Ensemble Kalman Filter

A new Ensemble Kalman Filter (EnKF), called a diffusive EnKF, is proposed in this study. The diffusive EnKF assumes that the perturbations orthogonal to the ensemble are uncorrelated with respect to the ensemble and have infinite covariance matrix. The assumption of infinite covariance matrix is equivalent to assuming that there is no forecast information in the space orthogonal to the ensemble. Generally for atmospheric applications, the ensemble size is less than the model dimension, so that the ensemble does not span the full model space. The full model space can be split into two subspaces: the space spanned by the ensemble, called the ensemble space, and the remainder, called the null space. In the ensemble space, the diffusive and traditional EnKFs are similar in that they optimally combine observations and forecasts. In the null space that projects on observations, however, the traditional EnKF approaches the forecast while the diffusive filter approaches the observations.

Data assimilation experiments with the Lorenz-96 model have been conducted to compare the diffusive EnKF and the traditional EnKF. For the ensemble sizes larger than the model dimension, the Diffusive EnKF is identical to the traditional one. For small ensemble sizes, e.g., 5-10 for Lorenz-96 model, the traditional EnKF diverges while the diffusive EnKF does not. It should be noted for the small ensemble size, the traditional EnKF diverges even with covariance inflation while the diffusive filter does not. The numerical experiments suggest that the diffusive EnKF can be an attractive alternative filter for the atmospheric data assimilation for small ensemble sizes.