From Surface to Depth: Diagnosing Ocean Vertical Velocity with 3D Frequency Operator
Ocean vertical velocity plays a crucial role in influencing heat exchange, water mass movement, and biogeochemical processes between the surface and deep waters. Due to the difficulty in directly measuring subsurface vertical velocity w, a promising approach is to diagnose w from high-resolution surface observations, like horizontal surface velocity and sea surface height anomaly, via remote sensing. However, both existing traditional dynamic methods and classic machine learning algorithms struggle to provide accurate estimates due to their inability to capture the multi-scale spatiotemporal structures and intense vertical fluctuations. To address these challenges, combined with theoretical ocean dynamics, we propose TriSEFormer, a novel approach that leverages frequency-embedded attention mechanism from a tri-dimensional (3D) frequency perspective. Specifically, TriSEFormer comprises cascaded TriSE blocks, each consisting of a neural dynamics cell following a refinement attention module. The 3D spectral transformation within the neural dynamics cell decomposes the attention-enhanced surface embedding into different spectral components, improving the understanding of multi-scale turbulent structures. Meanwhile, we propose depth-encoded vertical weights to selectively modulate both real and imaginary parts, enhancing the vertical representation compared to dynamic estimation. Extensive experiments on one ideal simulation and three regional ocean datasets demonstrate that TriSEFormer outperforms all baselines, achieving up to a 9.9% improvement, particularly enhancing deep-ocean w diagnosis from 300 m to 2100 m in the Indian Ocean. Code is available at https://github.com/JessiQi25/TriSEFormer.