Discrete Diffusion Priors for Rain-Field Reconstruction from Commercial Microwave Links

Reconstructing rain fields from commercial microwave links (CMLs) requires inferring spatially distributed intensities from sparse, path-integrated attenuation measurements. Rainfall intermittency and spatial dependence make the prior central to this inverse problem. We introduce discrete diffusion priors for rain fields: masked and uniform diffusion models trained on fields encoded with an explicit dry class, which preserves rainfall occurrence exactly, and a codebook learned on positive intensities. We formulate CML reconstruction under three observation models (midpoint and extended virtual rain gauges, and a path-integrated likelihood) and benchmark conditioning methods: masked in-filling, CML adaptations of G2D2 and GILC, and D-MGPS, a new discrete variant of midpoint guidance. On the OpenMRG dataset, masked in-filling reaches an RMSE comparable to that of continuous diffusion priors at one to two orders of magnitude lower runtime and attains the highest wet/dry accuracy. With midpoint gauges, its $95\%$ intervals are much narrower than kriging intervals and cover $0.94$ of all locations, but only $0.42$ of wet locations. Likelihood-guided discrete samplers, including D-MGPS, are currently less accurate than in-filling.