Can averaging kernels be averaged?
Often scientists do not use individual remotely sensed profiles of atmospheric state constituents but work with mean profiles instead. As with individual profiles, also comparison of mean profiles requires the application of the averaging kernels. On the face of it mean averaging kernels could be used, but unfortunately this is not quite correct if gas abundances and averaging kernels are correlated. The figure shows for several latitude bands the covariance term by which high-resolution profiles of the trace gas HCN after treatment with the mean averaging kernel have to be corrected. If climatological prior information has been used for the retrieval of the coarser resolved profile, then the covariance profile has to be added to the high-resolution profile after application of the averaging kernel.
As long as radiative transfer is sufficiently linear, there will usually be no correlation between trace gas abundance and averaging kernel. The situation is different for nonlinear radiative transfer, because here averaging kernels depend on the trace gas abundance. Since application of the averaging kernel matrix and calculation of mean profiles are non-commutative, a correction is needed. Formally, this correction is the covariance profile of the averaging kernel and the trace gas profile. This problem has been investigated by Thomas von Clarmann and Norbert Glatthor.
For more information see: https://doi.org/10.5194/amt-12-5155-2019