A kernel-density based ensemble filter applicable to high-dimensional systems

A drawback of ensemble Kalman filters (EnKFs) that has been noted previously by several researchers is that some implementations of the EnKF can produce unrealistic background error-covariance estimates. In particular, under the presence of strong nonlinearity in the forecast, one ensemble member can become "separated from the pack." This is due to the linear, Gaussian assumptions in the EnKF update, where every member's deviation from the mean is reduced in the same proportion during the assimilation of a particular observation. In its most extreme form, after many cycles the resulting ensemble can consist of one isolated member, with the rest of the members being very tightly packed together. Such an ensemble does not provide a realistic model of the error covariances.

Several solutions have been proposed, such as rotating the ensemble (e.g., Sakov 2008 MWR). Anderson (2010 MWR, in press) recently proposed a "rank histogram filter" that relaxes the assumption of Gaussianity of the prior. However, this implementation of the filter did not perform well at small ensemble sizes.

The authors propose here an extension of Anderson's concept. Instead of modeling the prior distribution at the observation location using rank-histogram concepts, instead the prior is modeled with a nonparametric kernel density estimate. This permits updates to the observations that are non-Gaussian but which performs much better with small ensemble sizes.

The "Kernel Density Filter" will be demonstrated and compared to existing filters in a hierarchy of standard models, from Lorenz '96 to primitive-equation models. We will explain why this implementation may be preferable to the rank-histogram approach.