Wind field reconstruction with adaptive random Fourier features

Abstract

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)=∑k=1Kβk 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.