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 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 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：海洋学家关注的是基于稀疏浮标速度观测数据预测洋流并识别流速矢量场中的散度。由于我们预期流速是空间位置的连续但高度非线性的函数，高斯过程（GPs）提供了一种有吸引力的模型。但我们发现，将具有标准平稳核的高斯过程直接应用于浮标数据，在洋流预测和散度识别方面可能效果不佳——这是由于一些物理上不切实际的先验假设所致。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中的散度和无旋分量。研究表明，由于该分解仅通过混合偏导数与原始矢量场相关联，我们仍能基于原始数据进行推理，且额外计算量仅为较小的常数倍数。我们通过合成数据和真实海洋数据展示了该方法的优势。