Technical Note: Propagating correlations in atmospheric inversions using different Kalman update smoothers

The scheme to propagate correlations between online and off-line state variables in atmospheric inversions using the fixed-lag Kalman smoother proposed in Bruhwiler et al. (2005) is explained as a process to impose a balanced constraint on the on-line state variables. It is then extended to the fixed-lag ensemble square root Kalman smoother and fixed-lag square root sigma-point Kalman smoother, allowing us to treat nonlinear observation operators easily. Further, to constrain the posterior fluxes within their feasible ranges, the constrained fixed-lag Kalman smoother is presented and the variable transform technique is proposed for the other two smoothers. Comparisons between various methods and observational data are conducted using a synthetic inversion of atmospheric CH4 fluxes. The results indicate that our developed methods are good alternatives to existing methods for conducting sequential inversion of atmospheric trace gases. It is also shown that the benefit to include the correlations between on-line and off-line state variables is case dependent.