A Stochastic-Dynamic Model for the Spatial Structure of Forecast Error Statistics

Abstract

A simple model that yields the spatial correlation structure of global atmospheric mass-field forecast errors is derived. The model states that the relative potential vorticity of the forecast error is forced by spatially multi-dimensional white noise. The forecast error equation contains a nondimensional parameter c0, which depends on the Rossby radius of deformation. From this stochastic-dynamic equation, a deterministic equation for the spatial covariance function of the 500 mb geopotential error field is obtained. Three methods of solution are examined: 1) an analytic method based on spherical harmonics, 2) a numerical method based on stratified sampling of Monte-Carlo realizations of the stochastic-dynamic equation, and 3) a combined analytic-numerical method based on two successive applications of a fast Poisson solver to the deterministic covariance equation. The three methods are compared for accuracy and efficiency, and the third (combined) method is found to be clearly superior. The model's covariance function is compared with global correlation data of forecast-minus-observed geopoteniial fields for the DST-6 period February?March 1976. The data are based on the GLAS forecast-assimilation system in use at that lime (Ghil et al., 1979). The model correlations agree well with the latitude dependence of the data correlations. The fit between model and data confirms that the forecast error between 24 and 36 h is largely random, rather than systematic; the value of the parameter c0 which gives the best fit suggests that much of this error can be attributed to baroclinic, rather than barotropic effects. Deterministic influences not included in the model appear at 12 and 48 h. They suggest possibilities of improving the forecast system by a better objective analysis and initialization procedure, and a better treatment of planetary-wave propagation, respectively. An analytic formula is obtained which locally approximates well the model's global correlations. This formula is convenient to use in the calculation of weighting coefficients for analysis and assimilation schemes. It shows that Gaussian functions are a poor approximation for the forecast error correlations of the mass field, and their derivatives an even poorer approximation to wind field correlations.