  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 to a set of benchmark methods
including Kriging and Inverse distance weighting. Random Fourier features is a
linear model $\beta(\pmb x) = \sum_{k=1}^K \beta_k e^{i\omega_k \pmb x}$
approximating the velocity field, with frequencies $\omega_k$ randomly sampled
and amplitudes $\beta_k$ trained to minimize a loss function. We include a
physically motivated divergence penalty term $|\nabla \cdot \beta(\pmb x)|^2$,
as well as a penalty on the Sobolev norm. We derive a bound on the
generalization error and derive a sampling density that minimizes the bound.
Following (arXiv:2007.10683 [math.NA]), we 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.
