Reduced Order Gaussian Smoothing for Nonlinear Data Assimilation

We investigate Gaussian filtering for data assimilation in numerical weather prediction (NWP). Data assimilation is the process of combining prior forecasts and observations to produce a system estimate. The prevailing data assimilation method in operational NWP centers is variational data assimilation. This method involves solving a cost function over a time window forming a maximum likelihood estimate. This method, however, requires the use of linearized models which in practice are difficult to produce and maintain. As an alternative we propose Gaussian smoothing for derivative-free, nonlinear data assimilation. Gaussian filters and their corresponding smoothers use numerical integration to evaluate the recursive Bayesian formulas for optimal filtering under Gaussian assumptions. This numerical integration typically requires many model evaluations making conventional Gaussian filtering/smoothing impractical for use in NWP. We will present a reduced order method for forming a Rauch-Tung-Striebel (RTS) type smoother. To do so we review the Bayesian filtering and smoothing equations and discuss an efficient numerical method for evaluating them. We will then discuss a numerical example using the Korteweg-de Vries equation to compare our technique to the standard variational approach.