THE BEHAVIOR OF FORECAST ERROR COVARIANCES FOR A KALMAN FILTER IN 2 DIMENSIONS

A Kalman filter algorithm is implemented for a linearized shallow-water model over the continental United States. It is used to assimilate simulated data from the existing radiosonde network, from the demonstration network of 31 Doppler wind profilers in the central United States, and from hypothetical radiometers located at five of the profiler sites. We provide some theoretical justification of Phillips' hypothesis, and we use the hypothesis, with some modification, to formulate the model error covariance matrix required by the Kalman filter. Our results show that assimilating the profiler wind data leads to a large reduction of forecast/analysis error in heights as well as in winds, over the profiler region and also downstream, when compared with the results of assimilating the radiosonde data alone. The forecast error covariance matrices that the Kalman filter calculates to obtain this error reduction, however, differ considerably from those prescribed by the optimal interpolation schemes that are employed for data assimilation at operational centers. Height-height forecast error correlation functions spread out broadly over the profiler region. Height-wind correlation functions for a base point near the boundary of the profiler region are not antisymmetric with respect to the line of zero correlation, nor does the zero-line pass through the base point. We explain why these effects on forecast error correlations are to be expected for wind profilers, which provide abundant wind information but no height information. Our explanation is supported by further experiments in which height observations assimilated from radiometers at just a few profiler sites reduce these effects.