Wind velocity field approximation from sparse data

In this work, we do not want to compute a potential that could generate the vector field data. We only want to get a global approximation of the vector field dataset on a bounded domain, taking into account in the modeling that this approximation derives from a potential. Furthermore, contrary to interpolation methods, we prefer to fit the vector field dataset in the case of realistic data (when the number of vectors is large or when the data are corrupted by noise). To achieve this, we introduce a minimization problem defined as a regularized least-square problem formulated on a Sobolev space of potentials. Obviously, this problem has an infinite number of solutions, but we derive from it a problem expressed in terms of the gradient vectors. We prove that the associated problem in terms of vectors has a unique solution which is the corresponding approximation of the vector field dataset. Then, we give a convergence result when the number of vectors increases to infinity. We also give the discretization complemented by an approximation error estimate of the involved smoothing splines. We then focus on numerical examples (real data sets from METEO FRANCE).