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.

翻译：海洋学家对基于稀疏浮标速度观测预测洋流并识别矢量场中的散度感兴趣。由于我们预期洋流动力学是光滑但高度非线性的，高斯过程提供了一个有吸引力的模型。但我们证明，直接将具有标准平稳核的高斯过程应用于浮标数据，在洋流预测和散度识别方面可能都会遇到困难——这是由于一些物理上不现实的先验假设。为了更好地反映已知的洋流物理特性，我们提出将标准平稳核应用于通过亥姆霍兹分解得到的矢量场的散度和无旋分量。我们证明，由于这种分解仅通过混合偏导数与原始矢量场相关联，我们仍然可以在仅增加少量恒定倍数计算开销的情况下，基于原始数据进行推断。我们在合成数据和真实海洋数据上展示了我们方法的优势。