the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 30 Oct 2025
GEOS-Chem-hyd: enabling source-oriented sensitivity analysis with GEOS-Chem
Abstract. Effective environmental policymaking requires accurate quantification of the impacts on air quality of changes in emissions. Chemical transport models, such as GEOS-Chem, are widely used for such analysis. Traditional methods to compute the relative influence of a change in emissions, or the sensitivity of pollutant concentrations to a change in emissions, with these models often suffer from numerical inaccuracies, especially when evaluating nonlinear atmospheric processes. To overcome these limitations, we integrated a novel sensitivity analysis approach leveraging hyperdual numbers into GEOS-Chem, making GEOS-Chem-hyd. The hyperdual step method accurately calculates first- and second-order sensitivities simultaneously and avoids common numerical errors associated with traditional finite difference methods. The real concentrations as well as first- and second-order sensitivities with respect to emissions calculated with GEOS-Chem-hyd align with the values calculated with GEOS-Chem version 14.0.0 within expected error. Applying GEOS-Chem-hyd to assess how changes in emissions of oxides of nitrogen could be expected to alter ozone, particulate matter, ammonium, and biogenic organic aerosol concentrations demonstrated regional differences and nonlinear influences. As an example of regional variations, the emissions of oxides of nitrogen were shown to decrease biogenic organic aerosol in most areas, except in portions of the boreal forests in Siberia. The nonlinear influence of emissions of oxides of nitrogen on ozone, ammonium, and biogenic organic aerosol was evidenced by second-order sensitivities on the same order of magnitude as the first-order sensitivities. GEOS-Chem-hyd incurs approximately four times greater computational costs than GEOS-Chem, which is competitive with the three GEOS-Chem model executions required for less accurate second-order sensitivities using the central finite difference method. GEOS-Chem-hyd provides a framework for efficient assessment of the influence of new scientific modules, which are easily incorporated into the sensitivity analysis framework, and supports informed emission control policy development through accurate, source-oriented sensitivity analysis.
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Status: final response (author comments only)
- RC1: 'Comment on egusphere-2025-4543', Anonymous Referee #1, 19 Jan 2026
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RC2: 'Comment on egusphere-2025-4543', Daven Henze, 05 Mar 2026
The manuscript by Akinjole and Capps presents the implementation of hyperdaul sensitivities in GEOS-Chem. This is a significant and novel achievement that allows them to accurately compute second order sensitivities, and the approach also has benefits for computing first order sensitivities. The sensitive are evaluated, numerically, and the tool is briefly applied. Overall, the manuscript is very thoughtful and well written. My main suggestion at this point is to dig a bit deeper into the postulated reasons for the discrepancies between FD and hyperdual first-order sensitivities. Below I provide some additional comments, and I believe addressing these would make the work well suitable for publication in GMD.
Comments:
48: I know that DDM is effectively the same as tangent linear, but you might also call it out as an approach here, in particular as DDM methods have been expanded to 2nd order sensitivities (i.e., Amir Hakami’s work).
51: I think there’s actually quite a bit of work with tagging for reactive species. The MOZART model had a tagging scheme for O3 — see papers from Louisa Emmons. Tim Butler does tagging — with WRF-Chem I believe. And Mike Kleeman’s CTM includes tagging for aerosol species, all the way through the aerosol microphysics. That being said, I agree with the statement that tagging, as a method, can be somewhat set aside here given it is fundamentally different in objective and implementation than the sensitivity methods being discussed.
59: One of the biggest uses of FD sensitivities in the GC community is the analytic inversions of CH4, including the calculations within the IMI. These papers all compute the entire Jacobian through ensembles of 100s to 1000s of FD sensitivities. However, these are for linear tracers, so there would not be a benefit in moving to higher-order sensitivity methods.
126/Eq 3: Could you comment on this identity? It’s not clear to me why these terms can’t equal each other, or why they are non-zero. It also seems to me, naively, to conflict with later equations. When the perturbation Ha is introduced on line 131, the e12 term is zero (even though Eq 3 states that e12 is not zero). Also, as a1 and a2 are arbitrary and thus could be equal), wouldn’t that make e1 and e2 equal to other, from inspection of Eq 7? I don’t doubt that your final equations are correct, but there’s just something about this presentation that I’m not grasping, and it would be useful if the text could explain this aspect in a little more detail.
176: Does this mean that GEOS-Chem-hyd doesn’t support GCHP? I think that’s fine, just thought it would be good to state this clearly.
180: Is this correct? I’m familiar with the parlance of “active variables” in adjoint code development, and I would have assumed it would be the same set of variables needed to convert to hyper dual. This would not include variables like temperature, RH, etc., which are not impacted by the emissions or initial conditions. Is that not the case here? If not, what is an example of a variable that does not impact concentrations, for which hyper dual type wasn’t necessary?
Fig 1: Why was dry dep not modified? Because it is linear? This pertains to my comment regarding line 180, where I suspect the text is incorrect; otherwise, dry dep would need to be modified as terms like dry dep velocities influence the species concentrations.
208/Eq 17: When viewing the manuscript pdf on a mac, there are two “?” that appear in the numerator of the right hand side.
250: That is true. Or, there may be a mistake in the hyper-dual implementation, or some differences owing to the approximations made in omitting hyper dual calculations (lines 268-272) within the activity coefficients or kinetic rate constants. I’d expect the authors to do a bit more here to identify the sources of these differences. This could readily be done by several methods, and it would be worth exploring one or more of these. One could try changing the FD perturbation size and seeing how the FD sensitivities respond. If nonlinearity is the reason for the discrepancy, the agreement should improve for smaller perturbations. If discontinuities are the issue, then the agreement should improve for larger perturbations. One could examine one-sided FD sensitivities — hyper dual can be considered valid if they lie between these, even if they don’t match the central FD estimate. One could take a deep dive on a few of the extreme outlier points and examine the forward model response to verify if it really is responding nonlinearly, or if values are all so small that roundoff error is degrading the FD.
Ok, I see that this is somewhat investigated in Fig 4. However, this figure is used more as a demonstration of the weakness of FD rather than as a method to verify that the central FD estimates differ from hyper dual owing to nonlinearity. Also, Fig 4 and the manuscript text don’t state what sensitivity is being plotted. Is this again the sensitivity of O3 with respect to NOx emissions?
General: can the axis of the sensitivity plots be adjusted such that the 0,0 lines are visible? For example, I’m trying to understand Fig 5 (a) - (d), and it would be a lot easier I could clearly tell which clusters of sensitivities are around 0.
Sensitivity Equations: All of the sensitivity equations showing emissions in the denominator have subscripts i,j,l,t on the emissions. However, it seems that the applications only use emissions that are defined domain-wide, is that correct? Would it be useful / simpler to show the notation for that?
Figures: Following on my above comment, for any panel of e.g. Fig 2, 5, etc., are the different points the different i,j locations of the numerator, as the emissions perturbation is domain wide?
320-324: Plausible, however the amount of disagreement for the SO2 sensitivities in 2(c) seem comparable to those for NOx in 2(g), however the second order sensitivities for NOx seem fine. Is it true that in 5(c) the hybrid sensitivities are almost all negative? If so, can one think about the mechanism and chemistry, and make some arguments about whether or not the range of values spanned by the hyd sensitivities makes more physical sense (these seem to be clustered around 0) than the hybrid?
370-376: I appreciate the comparison of hyperdual sensitivities to adjoint sensitivities. However, it feels a bit out of place here, as section 3.4 should focus on results of the application, which does not include any adjoint modeling. Should this be move to / integrated into the introduction?
Minor:
6: suggest “quantification of how changes in emission impact air quality”
420: NO$_x$
Citation: https://doi.org/10.5194/egusphere-2025-4543-RC2
Data sets
GEOS-Chem-hyd v.0 Samuel Akinjole https://doi.org/10.5281/zenodo.16575028
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The manuscript titled “GEOS-Chem-hyd: enabling source-oriented sensitivity analysis with GEOS-Chem” presents the development of a new sensitivity calculation method using hyperdual numbers in the widely used GEOS-Chem chemistry transport model. The authors provide background on hyperdual numbers, use a 1-day simulation to evaluate the model’s sensitivities to finite difference sensitivity approaches, and apply it in 1-week simulation to demonstrate its capabilities.
As the authors advocate, source-oriented sensitivities of atmospheric composition to emissions (and other parameters) can be a very valuable tool for policy-making, and as a result this development is a very interesting one. The paper is structured well, and clearly written. I have the following suggestions that the authors may wish to consider:
General comments:
Specific comments: