Methods of Sequential Estimation for Determining Initial Data in Numerical Weather Prediction.

Numerical weather prediction (NWP) is an initial -value problem for a system of nonlinear partial differential equations, in which initial values are known incompletely and inaccurately. Observational data available at the initial time must therefore be supplemented by data available prior to the initial time, a problem known as meteorological data assimilation. A further complication in NWP is that solutions of the governing equations evolve on two different time scales, a fast one and a slow one, whereas fast scale motions in the atmosphere are not reliably observed. This leads to the so-called initialization problem: initial values must be constrained to result in a slowly evolving forecast. The theory of estimation of stochastic-dynamic systems provides a natural approach to such problems. For linear stochastic-dynamic models, the Kalman-Bucy (KB) sequential filter is the optimal data assimilation method. We show that, for linear models, the optimal data assimilation -initialization method is a modified version of the KB filter. This modified KB filter corresponds to combining the standard KB filter with a projection onto the slow solution subspace. The shallow-water equations are a simple system whose solutions exhibit many features of large-scale atmospheric flow important in NWP. We implement the standard and modified KB filters for a linearized version of these equations, given a simple observational pattern. The numerical results show that the modified filter produces a slowly evolving forecast, at the expense of forecast errors only slightly larger than those incurred by using the standard KB filter. A statistical data assimilation method widely used at NWP centers is known as optimal interpolation (OI). We implement OI with the shallow-water model, and we use the estimation-theoretic framework to compare the performance of OI with that of the standard and modified KB filters. Numerical results show that the simplifying assumptions involved in OI lead to relatively large errors near boundaries separating data-dense and data-sparse regions, and that proper initialization is a partial cure for this boundary effect. We show also how estimation theory can be used to tune the free parameters involved in OI, in such a way that the tuned scheme performs roughly as well as the modified KB filter.