the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 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.
Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors. Views expressed in the text are those of the authors and do not necessarily reflect the views of the publisher.- Preprint
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Status: open (until 31 Jul 2026)
- RC1: 'Comment on egusphere-2026-2125', Anonymous Referee #1, 14 Jun 2026 reply
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RC2: 'Comment on egusphere-2026-2125', Anonymous Referee #2, 01 Jul 2026
reply
Review of egusphere-2026-2125:
Implementation and evaluation of the lognormal prior probability distribution in a variational atmospheric inversion framework
General comments
This paper presents a study on the use of lognormal probability distributions as part of a variational inversion system. The derivations for cost functions using lognormal prior PDFs and the mean, median or mode of the posterior solution are clearly presented and the impact of this choice on posterior emissions is tested using synthetic data tests with different prior and observational uncertainties. The impact of using lognormal and normal prior PDFs on posterior emissions and posterior emission uncertainties is tested using one year of SF6 observations to estimate SF6 emissions across Europe. An ensemble of inversions is used with Monte Carlo analysis to determine the posterior emission uncertainties.
As noted in the paper, the use of lognormal PDFs in atmospheric inversions is not a novel concept, but this investigation of the impact of a lognormal distribution on the inversion results could act as a useful reference for future inversion studies, and the method and experiments are explained clearly and in enough detail to enable other researchers to reproduce the method.
The impact of each posterior option (mean, mode, median) on the synthetic data inversions are presented clearly in combined figures, which allows for easy interpretation of the results.
Overall, despite the limited new science presented here, I think this manuscript meets the scope of GMD by presenting a summary of the technical aspects involved with choosing prior PDFs for inversions, which could act as a useful reference for future inversion studies. Therefore, I think this manuscript could be accepted for publication, after the minor comments below have been addressed.
Specific comments
Posterior emissions are analysed across the whole of Europe, despite the authors noting that emission sensitivity from the observations is poor for most of eastern and southern Europe. To address this, a more detailed comment should be included about the uncertainty of results beyond the region of good sensitivity. Or, results should only be analysed and presented over a smaller region that does have good sensitivity to emissions. Showing spatial emission results over a smaller region would also make all of the spatial emission figures easier to read.
Some topics that would be unfamiliar to non-experts (such as using atmospheric models to link surface emissions to atmospheric concentrations) are treated as assumed knowledge, which is a reasonable choice considering the scope and target audience of this paper. If the authors want to make this paper relevant to a wider audience, they would need to introduce some terms (such as atmospheric transport) in more detail, but I will leave this to the authors discretion.
Technical corrections and questions
Line 13: missing word after “however”? Should this read “…however, they avoid…”
Line 19: this line may be clearer if “based on its measurement in the atmosphere” is replaced with “based on measurements of its concentration in the atmosphere”.
Line 23: The repeat of “(or mole fractions)” could be removed, as this is already included on line 22.
Line 24: replace “determining” with “determine”.
Line 27: what does “variation, especially due to climate” mean here? Spatial or temporal variation or trends in emissions?
Line 37: “and prior estimate of x” should be added after “observations, y”.
Line 49: is there a reference for the line “The prior emission estimates for many atmospheric species have a lognormal distribution, strongly suggesting that the errors follow a lognormal distribution as well”?
Line 51: “cost function” has not been introduced yet. A line could be added earlier in this section, describing how a cost function is used to find a most likely estimate of x.
Line 79: add a line here, explaining why the PDF needs to be expressed in terms of the error, otherwise the reader has to wait for the presentation of the cost function for this explanation.
Line 81: add brackets to these equations (e.g. ln(ε) = ln(x) – ln(xb)) to maintain consistency with where this equation is used again later (e.g. line 101).
Line 108: I think the footnote here could be moved to after equation 13, as this is useful information and it would not break the flow of the text here. However, this is just a personal preference, so I’ll leave this to the authors’ discretion.
Line 115: some discussion of the choice of mean/median/mode in relation to atmospheric inversions could be included here. How does this choice differ compare to the data assimilation context in the quoted references?
Line 125: ‘transport function H(x)” is referenced here, when H has only been referred to as an “operator” before this point. To resolve this, more detail on H could be added at line 90, explaining how this could contain output from a Lagrangian or Eulerian transport model.
Line 127: reference or derivation for this equation? Other equations have been derived in detail, (e.g. with clear references to what substitutions and transformations are being used) but there is less detail on the derivation of Equation 15 from Equation 14.
Line 132: why does using this substitution improve the convergence rate?
Line 158: some more explanation of how this equation was derived would be helpful.
Line 161: you could add a real-world example of what this correlation means, presumably spatial correlation between emissions from different grid cells?
Line 165: add a reference for the statement “ land sinks are negligibly small”.
Lines 166 and 167: add the full names of the EDGAR and GAINS databases.
Line 169: for reproducibility, it might be useful to explain how you produced the normal and lognormal fit lines, discussed at this point in the text and shown in Figure 1.
Line 183: the footnote here could be included in the caption for Figure 2 instead.
Line 185: when reading this section, I wondered what observational uncertainty you were using, but this is covered later in sections 3.3 and 3.4. So it might be helpful to add a line here stating where observational error is defined.
Line 193: add a reference for the statement “SF6 is almost inert up to the middle stratosphere”.
Line 233: did you use any methods to test for convergence, in either the synthetic or real data tests?
Line 243: you could add some information on whether these prior and observation uncertainties are realistic, compared to the uncertainties ‘traditionally’ placed on these terms in inversions. Where does model error (e.g. uncertainties from the transport model) fit into these definitions of uncertainty?
Line 253: what inversion period are you using? Just one inversion over the whole of 2020, or multiple shorter inversion periods across 2020?
Line 254: EDGAR’s distribution seems to be closer to a lognormal, so why was GAINS chosen as the prior for these tests, instead of EDGAR?
Line 257: why was this observation error of 0.09 ppt chosen? And is it a reasonable assumption to apply this error value to all observations from all sites? I know this choice is currently under discussion in some inverse modelling groups, so I don’t know the ‘correct’ answer for this myself, but it might be useful to comment on why these observational uncertainties were chosen for this work in particular.
Line 273: “Synthetic data experiment setup” is confusing as a subheading under the “Results” heading, should “setup” be removed?
Line 313: missing space in “Fig.5j”.
Line 320: you note that the results are influenced by the specific random realization. Did you try rerunning these tests with different random perturbations, to see how the results varied?
Line 360: Does this claim that negative emission increments favours error reduction apply in all areas or just these 4 cases and in the two representative grid cells discussed in the next paragraph? Did you investigate this for all grid cells?
Figures 9 and 10: these figures may be easier to interpret if fewer histogram bins are used.
Line 476: should “hx” be “H(x)” here?
Citation: https://doi.org/10.5194/egusphere-2026-2125-RC2
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