How to average logarithmic retrievals?

If the logarithm of a concentration is retrieved rather than the concentration itself, subsequent averaging of results will lead to biases. Tests with a system simulator have been carried out for retrievals of CO (left panel) and H2O (right panel). The bias depends on the natural variability and the signal/noise ratio of the measurement. The bias cannot be systematically avoided by averaging in the log-space (dashed lines) instead of the linear concentration space (solid lines).

Things get even more complicated if regularization is applied to logarithmic retrievals, because in this case the weight of the regularization term depends on the concentration itself. For details, see www.atmos-meas-tech.net/5/831/2012/amt-5-831-2012.html