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. Global high-resolution observations of atmospheric composition from satellites can greatly improve our understanding of surface emissions through inverse analyses. Variational inverse methods can optimizesurface emissions at any resolution but do not readily quantify the errorand information content of the posterior solution. The information contentof satellite data may be much lower than its coverage would suggest becauseof failed retrievals, instrument noise, and error correlations thatpropagate through the inversion. Analytical solution of the inverse problemprovides closed-form characterization of posterior error statistics andinformation content but requires the construction of the Jacobian matrixthat relates emissions to atmospheric concentrations. Building the Jacobianmatrix is computationally expensive at high resolution because it involvesperturbing each emission element, typically individual grid cells, in theatmospheric transport model used as the forward model for the inversion. Wepropose and analyze two methods, reduced dimension and reduced rank, toconstruct the Jacobian matrix at greatly decreased computational cost while retaining information content. Both methods are two-step iterativeprocedures that begin from an initial native-resolution estimate of theJacobian matrix constructed at no computational cost by assuming thatatmospheric concentrations are most sensitive to local emissions. Thereduced-dimension method uses this estimate to construct a Jacobian matrixon a multiscale grid that maintains a high resolution in areas with highinformation content and aggregates grid cells elsewhere. The reduced-rankmethod constructs the Jacobian matrix at native resolution by perturbing the leading patterns of information content given by the initial estimate. We demonstrate both methods in an analytical Bayesian inversion of Greenhouse Gases Observing Satellite (GOSAT) methane data with augmented information content over North America in July 2009. We show that both methods reproduce the results of the native-resolution inversion while achieving a factor of 4 improvement in computational performance. The reduced-dimension method produces an exact solution at a lower spatial resolution, while the reduced-rank method solves the inversion at native resolution in areas of high information content and defaults to the prior estimate elsewhere.

Reduced-cost construction of Jacobian matrices for high-resolution inversions of satellite observations of atmospheric composition