# Methane Flux Inversion

Spatial prediction targets the conditional distribution of a latent spatial field given observations, which is rarely available in closed form and is hard to sample. Guided diffusion separates the two: an unconditional diffusion model is trained on the process model alone, and a differentiable likelihood gradient is added during reverse diffusion at prediction time, so one trained process model serves many data models. Across Gaussian processes, a Ginzburg–Landau amplitude field and a methane-flux inversion over the UK and Ireland, the predictive distributions match exact or MCMC results and scale linearly with the latent dimension at higher resolutions.

Spatial prediction targets the conditional distribution of a latent spatial field given observations, which is rarely available in closed form and is hard to sample. Guided diffusion separates the two: an unconditional diffusion model is trained on the process model alone, and a differentiable likelihood gradient is added during reverse diffusion at prediction time, so one trained process model serves many data models. Across Gaussian processes, a Ginzburg–Landau amplitude field and a methane-flux inversion over the UK and Ireland, the predictive distributions match exact or MCMC results and scale linearly with the latent dimension at higher resolutions.