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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 posteriordoi:10.5194/acp-11-921-2011 fatcat:ntcvrdbhy5bhdj336o7hkkabsq