Generalized Covariance Functions for Mesoscale Wind Analysis and Data Assimilation

The streamfunction and velocity potential have been widely used as control variables in many operational variational data assimilation systems. For limited-area high-resolution data assimilation, the use of these two control variables was often found to degrade the analyses and forecasts when compared with the direct use the two components of horizontal velocity as control variables with no multivariate correlation. As this finding was essentially experimental, reasons for the degradations were largely unexplored. Theoretically, if the random vector field of horizontal velocity error is homogenous and isotropic as often assumed or estimated, then its associated streamfunction and velocity potential are uncorrelated but the two velocity components are not uncorrelated. This theoretical consideration favors the use of streamfunction and velocity potential as control variables over the velocity components. However, when the auto- and cross-covariance functions for the two velocity components are derived from two uncorrelated Gaussian-type auto-covariance functions (mimicked by the operationally used recursive filter) for the streamfunction and velocity potential, the derived auto- and cross-covariance functions contain negative sidelobes. These negative sidelobes can cause unrealistic features in analyzed winds outside the data covered areas (as often observed in mesoscale and storm-scale wind analyses performed with high-resolution radar wind data). The negative sidelobes can provide a theoretical explanation for aforementioned degradations. It is thus necessary to eliminate the negative sidelobes in order to improve the use of streamfunction and velocity potential as control variables, but this requires the covariance functions to be formulated in terms of generalized functions (or distributions). The detailed formulations will be presented at the conference and their applications to radar wind data analysis will be discussed with experimental results.