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

Abstract

Propagating correlations between on-line and off-line state variables in the problem of sequential inversion of surface fluxes for atmospheric CH<sub4</sub> is investigated with the fixed-lag Kalman smoother, the fixed-lag ensemble square root Kalman smoother and the fixed-lag square root sigma-point Kalman smoother. The formulae for the fixed-lag Kalman smoother in Bruhwiler et al. (2005) are further developed with a more concise representation of the posterior covariance. Extension to the fixed-lag ensemble square root Kalman smoother and fixed-lag square root sigma-point Kalman smoother allows us to treat nonlinear observation operators easily. To constrain the posterior fluxes within their feasible ranges, the constrained fixed-lag Kalman smoother is also presented, and the variable transform technique is implemented for the other two smoothers. Results with synthetic inversion of CH<sub4</sub> fluxes 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 of including the correlation between on-line and off-line state variables is case dependent

Technical Note: Methods for interval constrained atmospheric inversion of methane

“…Extension to including correlations between on-line (variables are still being updated) and off-line (variables that are no longer updated) state variables is straightforward (see Bruhwiler et al, 2005;Tang and Zhuang, 2010). However, for the specific problem in our study, the gain from accounting for such correlation is rather small, as we showed in Tang and Zhuang (2010).…”

“…Eq. (8) is solved iteratively with the active set method (Murty, 1988), just as documented in Tang and Zhuang (2010). When a set of active constraints are identified, the constraints are set to equality, such that c(s…”

“…The sampling is done with the rejection sampling algorithm (von Neumann, 1963), using a uniform distibution as the instrumental distribution. The remaining components of s j (i ) with i >n, are determined by A similar set up to that in Tang and Zhuang (2010) is used for the comparison experiment, so only necessary information is briefed here. Specifically, the sensitivity matrix is derived by running a group of tagged-CH 4 simulations using the GEOS-Chem model (Bey et al, 2001;Wang et al, 2004), driven by the GEOS-5 meteorology data at a resolution of 4…”

“…Sampling errors are simulated by using the relative residual error method (Palmer et al, 2003) derived by an optimized reference run against the globalview data. The prior fluxes are obtained by adding random perturbations to a set of known fluxes (11 seasonal fluxes and 7 yearly constant fluxes, see Tang and Zhuang, 2010), and are then used to obtain prior CH 4 concentrations at those sampling locations. The inversion is assessed by comparing the inverted fluxes and CH 4 concentrations to the known fluxes and CH 4 concentrations from the reference run.…”

“…To apply the interval constraints imposed over the state variables, the variable transform method can be applied efficiently, as was done in Tang and Zhuang (2010). However, the variable transform involved is non-linear, and is thus difficult to deal with using the KS, which is developed based on linear dynamics.…”