Applications of Estimation Theory to Numerical Weather Prediction
Numerical weather prediction (NWP) is an initial-value problem for a system of nonlinear partial differential equations (PDEs) in which the initial values are known only incompletely and inaccurately. Data at initial time can be supplemented, however, by observations of the system distributed over a time interval preceding it. Estimation theory has been successful in approaching such problems for models governed by systems of ordinary differential equations and of linear PDEs. We develop methods of sequential estimation for NWP.
A model exhibiting many features of large-scale atmospheric flow important in NWP is the one governed by the shallow-fluid equations. We study first the estimation problem for a linearized version of these equations. The vector of observations corresponds to the different atmospheric quantities measured and space-time patterns associated with conventional and satellite-borne meteorological observing systems. A discrete Kalman-Bucy (K-B) filter is applied to a finite-difference version of the equations, which simulates the numerical models used in NWP.
The specific character of the equations’ dynamics gives rise to the necessity of modifying the usual K-B filter. The modification consists in eliminating the high-frequency inertia-gravity waves which would otherwise be generated by the insertion of observational data. The modified filtering procedure developed here combines in an optimal way dynamic initialization (i.e., elimination of fast waves) and four-dimensional (space-time) assimilation of observational data, two procedures which traditionally have been carried out separately in NWP. Comparisons between the modified filter and the standard K-B filter have been made.
The matrix of weighting coefficients, or filter, applied to the observational corrections of state variables converges rapidly to an asymptotic, constant matrix. Using realistic values of observational noise and system noise, this convergence has been shown to occur in numerical experiments with the linear system studied; it has also been analyzed theoretically in a simplified, scalar case. The relatively rapid convergence of the filter in our simulations leads us to expect that the filter will be efficiently computable for operational NWP models and real observation patterns.
Our program calls for the study of the asymptotic filter’s dependence on observation patterns, noise levels, and the system’s dynamics. Furthermore, the covariance matrices of system noise and observational noise will be determined from the data themselves in the process of sequential estimation, rather than be assigned predetermined, heuristic values. Finally, the estimation procedure will be extended to the full, nonlinear shallow-fluid equations.
KeywordsData Assimilation Slow Wave Numerical Weather Prediction Initial Error Influence Function
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