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Atmospheric inverse modelling is a method for reconstructing historicalfluxes of green-house gas between land and atmosphere, using observedatmospheric concentrations and an atmospheric tracer transport model. The smallnumber of observed atmospheric concentrations in relation to the number ofunknown flux components makes the inverse problem ill-conditioned, andassumptions on the fluxes are needed to constrain the solution. A commonpractise is to model the fluxes using latent Gaussian fields with a meanstructure based on estimated fluxes from combinations of process modelling(natural fluxes) and statistical bookkeeping (anthropogenic emissions). Here,we reconstruct global \CO flux fields by modelling fluxes using Gaussian MarkovRandom Fields (GMRF), resulting in a flexible and computational beneficialmodel with a Mat\'ern-like spatial covariance, and a temporal covariancedefined through an auto-regressive model with seasonal dependence.  In contrast to previous inversions, the flux is defined on a spatiallycontinuous domain, and the traditionally discrete flux representation isreplaced by integrated fluxes at the resolution specified by the transportmodel. This formulation removes aggregation errors in the flux covariance, dueto the traditional representation of area integrals by fluxes at discretepoints, and provides a model closer resembling real-life space-time continuousfluxes.

Spatio-Temporal Reconstructions of Global CO2-Fluxes using Gaussian Markov Random Fields