In dual-Doppler analyses, accurate retrieval of the vertical velocities of the wind field is an ongoing problem in radar meteorology. Reliable synthesis of the vertical velocity field is desirable not only in studies of tornadogenesis and in initialization for numerical weather prediction models but also in many situations for which dual-Doppler analyses are commonly applied, such as in studies of the microphysical, kinematic, dynamic, and thermodynamics structures of thunderstorms. This study presents the vertical velocity field as the solution of a minimization problem which is based on variational formulations of the vertical vorticity equation and the mass continuity equation. The theoretical part discusses the well-posedness of the problem, the existence and uniqueness of the solution in a Hilbert space, and the stability of the solution' s approximation. Since the approximation is computed by means of the classical finite element method, the numerical part presents a method that reduces the computational time and studies the accuracy of the numerical solution and its sensitivity to the errors in the radar data.

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