Sea-Surface Wave Field Reconstruction from X-Band Marine Radar Imagery Using a Physics-Informed Three-Dimensional Fourier Neural Operator

X-band marine radar gives continuous, high-resolution sea-surface observation, but recovering phase-resolved elevation from backscatter is ill-posed, and its amplitude step classically needs an empirical, site-specific calibration. This study places wave-related constraints in the training objective of a three-dimensional Fourier Neural Operator, rather than in its architecture, and diagnoses where a standard operator fails. The operator maps radar sequences and a spatially constant wind-speed channel to elevation over two spatial dimensions and time, trained with a composite WaveLoss that combines a pointwise term, a dispersion-informed term and supervised gradient and wave-height terms. All data come from a simplified, noise-free radar simulator, so the operator learns to invert the simulator rather than the ocean, and validation with field data remains to be explored in future work. Normalization by wind condition makes the reconstruction wind-conditioned. On eight independent simulated realizations, changing the objective alone lowers the significant wave height error by about 10%, in every realization for the three-seed mean, with no detected change in elevation RMSE or pattern correlation (0.82). Single-term models trace this gain mainly to the wave-height term. The two-dimensional-plus-time mapping also yields wave direction and wavenumber–frequency dispersion consistency. A spectral-coverage analysis shows that the classical three-cycle window criterion did not predict failure in this configuration, while the failure at the lowest wind speed has several co-occurring causes that were not separated.