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.

翻译：基于稀疏的浮标速度观测数据，海洋学家致力于重建浮标以外的洋流场，并识别流场中的散度。作为第一步模块化处理，我们聚焦于时间静态情形——例如通过限制在短时间周期内。由于预期洋流速度是空间位置的连续但高度非线性函数，高斯过程提供了具有吸引力的模型。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，会因先验假设在物理上不切实际而在洋流重建和散度识别方面表现不佳。为更好地反映洋流已知的物理特性，我们提出对通过亥姆霍兹分解得到的矢量场的散度分量和无旋分量施加标准平稳核。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，我们仍可基于原始数据进行推断，且额外计算成本仅为较小的常数倍数。我们通过理论和在合成与真实海洋数据上的实验验证了该方法优势。