Wind field reconstruction with adaptive random Fourier features

We investigate the use of spatial interpolation methods for reconstructing the horizontal near-surface wind field given a sparse set of measurements. In particular, random Fourier features is compared with a set of benchmark methods including kriging and inverse distance weighting. Random Fourier features is a linear model β(x)=∑Kk=1βk eiωkx approximating the velocity field, with randomly sampled frequencies ωk and amplitudes βk trained to minimize a loss function. We include a physically motivated divergence penalty |∇⋅β(x)|2, as well as a penalty on the Sobolev norm of β. We derive a bound on the generalization error and a sampling density that minimizes the bound. We then devise an adaptive Metropolis–Hastings algorithm for sampling the frequencies of the optimal distribution. In our experiments, our random Fourier features model outperforms the benchmark models.