  Given sparse observations of buoy velocities, oceanographers are interested
in reconstructing ocean currents away from the buoys and identifying
divergences in a current vector field. As a first and modular step, we focus on
the time-stationary case - for instance, by restricting to short time periods.
Since we expect current velocity to be a continuous but highly non-linear
function of spatial location, Gaussian processes (GPs) offer an attractive
model. But we show that applying a GP with a standard stationary kernel
directly to buoy data can struggle at both current reconstruction and
divergence identification, due to some physically unrealistic prior
assumptions. To better reflect known physical properties of currents, we
propose to instead put a standard stationary kernel on the divergence and
curl-free components of a vector field obtained through a Helmholtz
decomposition. We show that, because this decomposition relates to the original
vector field just via mixed partial derivatives, we can still perform inference
given the original data with only a small constant multiple of additional
computational expense. We illustrate the benefits of our method with theory and
experiments on synthetic and real ocean data.
