A variational method is developed to retrieve winds in the first step and then thermodynamic fields in the second step from Doppler radar observations. In the first step, wind fields are retrieved at two time levels: the beginning and ending times of the data assimilation period, simultaneously from two successive volume scans by using the weak form constraints provided by the mass continuity and vorticity equations.As the retrieved wind fields are expressed by Legendre polynomial expansions at the beginning and ending times, the time tendency term in the vorticity equation can be conveniently formulated, and the retrieved winds can be compared with the radar observed radial winds in the cost function at the precise time and position of each radar beam. In the second step, the perturbation pressure and temperature fields at the middle time are then derived from the retrieved wind fields and the velocity time tendency by using the weak form constraints provided by the three momentum equations. The merits of the new method are demonstrated by numerical experiments with simulated radar observations and compared with the traditional least squares methods which consider neither the precise observation times and positions nor the velocity time tendency. The new method is also applied to real radar data for a heavy rainfall event during the 2001 Meiyu season in China.
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