Determining topography from surface waves using a condition diffusion model

Inferring seabed topography from wave observations is fundamental to tsunami hazard assessment, coastal planning, and large-scale ocean circulation modelling. Classical inversion approaches (4DVar) typically rely on an optimization process. However, the strongly nonlinear coupling between free-surface dynamics and topography, together with perturbations and measurement noise, often induces instability and exacerbates ill-posedness. In contrast, data-driven methods can learn complex nonlinear mappings by exploiting the underlying data distribution. This study introduces a conditional diffusion model (DiffTopo) that reconstructs topography from surface wave fields governed by the shallow water equations. Using classifier free guidance, DiffTopo generates diverse candidate solutions, and a solver-based thresholding criterion subsequently admits only physically plausible realizations. We evaluate DiffTopo under fully observed wave fields and across three representative bathymetric configurations, demonstrating robust generalization performance. Moreover, DiffTopo complements variational data assimilation: warm-starting 4DVar with the best candidate reduces the relative error by approximately 60%, yielding faster convergence and higher quality reconstructions. Classical variational approaches incur substantial overhead from repeated gradient evaluations either through neural networks or gradient-based solvers. In comparison, direct generation provides fixed cost inference and often yields reconstructions with higher morphological fidelity and sharper structures. As a probabilistic generative initializer, DiffTopo is data-dependent and sensitive to training coverage, conditional guidance, and the physics-based screening threshold. Consequently, under distribution shift or severely sparse and noisy observations, solver-based refinement or fallback initialization may still be necessary. These results underscore the promise of diffusion-based generative modelling for addressing inverse problems in geophysical fluid dynamics.