  Atmospheric trace-gas inversion refers to any technique used to predict
spatial and temporal fluxes using mole-fraction measurements and atmospheric
simulations obtained from computer models. Studies to date are most often of a
data-assimilation flavour, which implicitly consider univariate statistical
models with the flux as the variate of interest. This univariate approach
typically assumes that the flux field is either a spatially correlated Gaussian
process or a spatially uncorrelated non-Gaussian process with prior expectation
fixed using flux inventories (e.g., NAEI or EDGAR in Europe). Here, we extend
this approach in three ways. First, we develop a bivariate model for the
mole-fraction field and the flux field. The bivariate approach allows optimal
prediction of both the flux field and the mole-fraction field, and it leads to
significant computational savings over the univariate approach. Second, we
employ a lognormal spatial process for the flux field that captures both the
lognormal characteristics of the flux field (when appropriate) and its spatial
dependence. Third, we propose a new, geostatistical approach to incorporate the
flux inventories in our updates, such that the posterior spatial distribution
of the flux field is predominantly data-driven. The approach is illustrated on
a case study of methane (CH$_4$) emissions in the United Kingdom and Ireland.
