Gaussian Approximations in Filters and Smoothers for Data Assimilation
Journal Article - Open Access
NAVAL RESEARCH LAB WASHINGTON DC WASHINGTON United States
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We present mathematical arguments and experimental evidence that suggest that Gaussian approximations of posterior distributions are appropriate even if the physical system under consideration is nonlinear. The reason for this is a regularizing effect of the observations that can turn multi-modal prior distributions into nearly Gaussian posterior distributions. This has important ramifications on data assimilation DA algorithms in numerical weather prediction because the various algorithms ensemble Kalman filterssmoothers, variational methods, particle filters PFsmoothers PS apply Gaussian approximations to different distributions, which leads to different approximate posterior distributions, and, subsequently, different degrees of error in their representation of the true posterior distribution. In particular, we explain that, in problems with medium nonlinearity, i smoothers and variational methods tend to outperform ensemble Kalman filters ii smoothers can be as accurate as PF, but may require fewer ensemble members iii localization of PFs can introduce errors that are more severe than errors due to Gaussian approximations. In problems with strong nonlinearity, posterior distributions are not amenable to Gaussian approximation. This happens, e.g. when posterior distributions are multi-modal. PFs can be used on these problems, but the required ensemble size is expected to be large hundreds to thousands, even if the PFs are localized. Moreover, the usual indicators of performance small root mean square error and comparable spread may not be useful in strongly nonlinear problems. We arrive at these conclusions using a combination of theoretical considerations and a suite of numerical DA experiments with low- and high-dimensional nonlinear models in which we can control the nonlinearity.
- Statistics and Probability