Abstract
Oceanographers are interested in predicting ocean currents and identifying
divergences in a current vector field based on sparse observations of buoy
velocities. Since we expect current dynamics to be smooth but highly
non-linear, 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 prediction 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
on synthetic and real ocean data.