How to average logarithmic retrievals?

Calculation of mean trace gas contributions from profiles obtained by retrievals of the logarithm of the abundance rather than retrievals of the abundance itself are prone to biases. By means of a system simulator, biases of linear versus logarithmic averaging were evaluated for both maximum likelihood and maximum a priori retrievals, for various signal to noise ratios and atmospheric variabilities. These biases can easily reach ten percent or more. As a rule of thumb we found for maximum likelihood retrievals that linear averaging better represents the true mean value in cases of large local natural variability and high signal to noise ratios, while for small local natural variability logarithmic averaging often is superior. In the case of maximum a posteriori retrievals, the mean is dominated by the a priori information used in the retrievals and the method of averaging is of minor concern. For larger natural variabilities, the appropriateness of the one or the other method of averaging depends on the particular case because the various biasing mechanisms partly compensate in an unpredictable manner. This complication arises mainly because of the fact that in logarithmic retrievals the weight of the prior information depends on abundance of the gas itself. ... mehrNo simple rule was found on which kind of averaging is superior, and instead of suggesting simple recipes we cannot do much more than to create awareness of the traps related with averaging of mixing ratios obtained from logarithmic retrievals.