Technical Note: Variance-covariance matrix and averaging kernels for the Levenberg-Marquardt solution of the retrieval of atmospheric vertical profiles

The variance-covariance matrix (VCM) and the
averaging kernel matrix (AKM) are widely used tools to
characterize atmospheric vertical profiles retrieved from remote
sensing measurements. Accurate estimation of these
quantities is essential for both the evaluation of the quality
of the retrieved profiles and for the correct use of the
profiles themselves in subsequent applications such as data
comparison, data assimilation and data fusion. We propose a
new method to estimate the VCM and AKM of vertical profiles
retrieved using the Levenberg-Marquardt iterative technique.
We apply the new method to the inversion of simulated
limb emission measurements. Then we compare the
obtained VCM and AKM with those resulting from other
methods already published in the literature and with accurate
estimates derived using statistical and numerical estimators.
The proposed method accounts for all the iterations done
in the inversion and provides the most accurate VCM and
AKM. Furthermore, it correctly estimates the VCM and the
AKM also if the retrieval iterations are stopped when a physically
meaningful convergence criterion is fulfilled, i.e. before
achievement of the numerical convergence at machine
precision. The method can be easily implemented in any
Levenberg-Marquardt iterative retrieval scheme, either constrained
or unconstrained, without significant computational
overhead.