In this thesis, we make use of the weak formulation of the vorticity equation and the mass continuity equation to calculate the vertical component, w, of the wind field V&ar; , given the two horizontal components u and v. Since w will be depending on u and v and its approximation, wN, will be depending on approximations of u and v and the number of points in some mesh, we will examine the effect of the latter on wN as well as the effect of small perturbations of u and v on w and wN. Theoretically, the sensitivity of w to u and v can be investigated by studying the existence of the Frechet differential of w with respect to the couple (u, v). We develop methods that significantly reduce the computational time when using the finite element method in both linear and cubic approximations. The mentioned methods are the origin of an idea, presented in Chapter VIII, that can be applied to any variational problem in order to reduce the cost of the computation of the stiffness matrix and the load vector. We set the problem such that we obtain an estimation of w from estimations of u and v based on radar data. The first and obvious finding is that the finer the mesh, the more accurate the approximation; the second states that more accurate results are obtained when using both the vorticity equation and the mass continuity equation than when working with only one of them. And the third one tells us that the coarser the mesh, the less sensitive the approximation; of course, the latter is a result of using a smaller number of points; because when we work with the same values of u and v and the same number of mesh points but in a larger or smaller domain, the sensitivity of wN remains the same.