This paper addresses the problem of spatiotemporal wind velocity field estimation for air traffic management applications. Using data obtained from aircraft, the eastward and northward components of the wind velocity field inside a specific air space are calculated as functions of time. Both short-term wind velocity field forecasting and wind velocity field reconstruction are performed. Wind velocity data are indirectly obtained from the states of the aircraft flying in the relevant airspace, which are broadcast by the ADS-B and Mode-S aircraft surveillance systems. The wind velocity field is estimated by combining two data-driven techniques: the polynomial chaos expansion and the Gaussian process regression. The former approximates the global behavior of the wind velocity field, whereas the latter approximates the local behavior. The eastward and northward wind components of the wind velocity field must be estimated, which causes the problem to be a multiple-output problem. This method enables the estimation of the wind velocity field at any spatiotemporal location using wind velocity observations from any spatiotemporal location, eliminating the need for spatial and temporal grids. Moreover, since the method proposed in this article allows for the probability distributions of the estimates to be computed, it causes the computation of the confidence intervals to be possible. Furthermore, since the method presented in this paper allows for data assimilation, it can be used online to continuously update the wind velocity field estimation. The method is tested on different wind scenarios and different training-test data configurations, by means of which the consistency between the results of the wind velocity field forecasting and the wind velocity field reconstruction is checked. Finally, the ERA5 meteorological reanalysis data of the European Centre for Medium-Range Weather Forecasts are used to validate the proposed technique. The results show that the method is able to reliably estimate the wind velocity field from aircraft-derived data.