Why Gaussian Approximations to Posteriors are More Appropriate than to Priors

We present mathematical arguments and numerical examples that suggest that Gaussian approximations of posterior distributions can be appropriate even if forecast distributions, generated by nonlinear models, are not at all Gaussian. In such a regime of “intermediate” non-Gaussianity, smoothing and variational methods tend to outperform ensemble Kalman filters (EnKF), while particle filters (PF) require larger ensembles. In short, the reasons are that variational methods make use of nearly Gaussian posterior distributions while PFs make no assumption about the underlying problem structure and EnKF implicitly uses Gaussian approximations of forecast distributions, which are less appropriate than Gaussian approximations of posterior distributions. Our examples suggest that nonlinearity of the numerical model and/or observation operators must be extreme for a fully non-Gaussian approach to be worthwhile. Extremely nonlinear problems, however, require larger ensembles (hundreds to thousands rather than tens of ensemble members) as well as a reformulation of the overall DA framework and assessment of algorithm performance.

High-dimensional problems require localization, i.e., that the DA algorithms reflect the fact that observations have local, not global effects. Localization is typically achieved in hybrid variational methods, EnKF or EnKS by setting small off-diagonal elements in forecast covariances equal to zero. In PF, localization is less straightforward and is a current research topic. Generally, localization for PF is “sub-optimal” in the sense that the various localization techniques proposed over the past years introduce errors and biases which, at this point in time, are not fully understood. We provide numerical examples where additional errors caused by localization are more severe than errors due to Gaussian approximations of posterior distributions. Along the way, we emphasize that inflation seems to be as important as localization for successful application of PFs.