| 05 Jun 2026
Implementation and evaluation of the lognormal prior probability distribution in a variational atmospheric inversion framework
Abstract. In this study, we investigate the use of a lognormal prior probability distribution in atmospheric inverse modelling. We present the formal implementation in a variational inversion framework and analyze how the choice of statistical optimization parameter (mean, median, or mode) affects the inversion outcome. Using a case study of inverse modelling of sulfur hexafluoride (SF6) in Europe, we evaluate the performance of the lognormal implementation through both synthetic and real data experiments, and compare the results to inversions using a normal prior probability distribution. We estimate the posterior uncertainties using a Monte Carlo approach and examine their distribution.
We find that optimizing for the mean or the mode can produce improved emission estimates under the condition of a strong observational constraint, however, this can lead to unstable and strongly biased inversion results under a weak constraint. In contrast, optimizing for the median consistently improves emission estimates and leads to physically plausible results across all tested cases, providing the most reliable option.
We show that inversions using a lognormal prior distribution produce a similar posterior emission pattern as when using a normal prior distribution, however, avoid non-physical negative emission values and occasionally allow for stronger positive emission adjustments. Posterior uncertainties can be estimated using interpercentile ranges from an ensemble of inversions with prior emission errors following a lognormal distribution. Due to the strong asymmetry of posterior distributions with respect to the sign of the inversion increments, error reduction is better assessed in log space, where it provides a clearer measure of the constraints imposed by the observations.
Competing interests: At least one of the (co-)authors is a member of the editorial board of Geoscientific Model Development.
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