A mean field approximation in data assimilation for nonlinear dynamics

This paper considers the problem of data assimilation into nonlinear stochastic dynamic equations, from the point of view that the optimal solution is provided by the probabilities conditioned upon observations. An implementation of Bayes formula is described to calculate such probabilities. In the context of a simple model with multimodal statistics, it is shown that the conditional statistics succeed in tracking mode transitions where some standard suboptimal estimators fail. However, in complex models the exact conditional probabilities cannot be practically calculated. Instead, approximations to the conditional statistics must be sought. In this paper, attention is focused on approximations to the analysis step arising from the conditioning on observational data. A suboptimal mean-field conditional analysis is obtained from a statistical mechanics of time-histories. It is shown to have a variational formulation, reducing the approximate calculation of the conditional statistics to the minimization of the “effective action”, a convex cost function. This mean-field analysis is compared with a standard linear analysis, based on a Kalman gain matrix. In the simple model problem, the mean-field conditional analysis is shown to approximate well the exact conditional statistics.