The authors would like to thank the reviewer for their thoughtful comments. Below, original comments by the reviewer are in italics.

"First, the bidirectional flux scheme implemented in this work primarily represents ammonia exchange between the atmosphere and soil or ground surfaces, which is most relevant for agricultural soils and fertilizer-related emissions. In the inversion, however, satellite NH3 retrievals appear to be assimilated over all land areas without screening by land-use type. Satellite observations measure the total atmospheric NH3 column, which integrates signals from all nearby sources. If the inversion does not explicitly separate or mask different land-use categories, there is a risk that ammonia originating from non-soil sources, such as livestock facilities or urban emissions, may be incorrectly attributed to soil-related parameters in the bidirectional scheme. In such regions, the retrieved atmospheric ammonia may reflect a combination of emission processes that are not fully represented by the soil-based bidirectional scheme."

In the submitted draft of the manuscript, this issue was touched upon in lines 244-248, but I can now see that the point made in these lines was not sufficiently explained and needs further clarification (which will be added to the next draft version of the paper).

Lines 244-248 of the (first) draft of the manuscript contained the text:

“Γp describes the sources of ammonium to the ground and can have contributions from, for example, fertilizer application or waste from livestock. … In this work, Γp will represent agricultural sources (fertilizer/livestock), while separate non-agricultural ammonia emissions that account for ≲ 1% of total emissions will be accounted for using the unidirectional emissions model.”

So the bidirectional flux model is to represent emissions from all agricultural source, which in this case are from both fertilizers and livestock, that account for ~99% of the total ammonia emissions. However, it was not explained why we treat the livestock emissions with the same model as the fertilizer emissions.

The assumption being made here is that livestock ammonia emissions primarily originate from excreted nitrogen as manure. This assumption is made for ammonia livestock emissions in CMAQ and GEOS-Chem (see Zhu et al. (2015a) and Zhu et al. (2015b)). The review article by Lee et al. (2025) examines ammonia emissions from beef cattle and concludes that most of the ammonia emissions from beef cattle are from manure (mostly from urine). This assumption is made for all livestock types, and so the livestock emissions can be thought of as a type of fertilizer emissions.

The diurnal pattern of the livestock ammonia emissions is given by Eqs. (1) to (3) of  Zhu et al. (2015a) and by Eq. (1) in Zhu et al. (2015b). These equations are essentially Eqs. (2) and (5) of this manuscript, except written in the form of an average or total emissions term modulated by a time-dependent modulation term. In Zhu et al. (2015a) and Zhu et al. (2015b), there isn’t an explicit dependence of the manure pH, but in these works the livestock emissions are being modeled in the same manner as the fertilizer emissions.

In this manuscript, the ammonia source terms are label as either ‘ground’ or ‘soil’ but this was meant to be inclusive of manure from livestock. However, this was not made sufficiently clear in the first version of the paper and so the next version of the manuscript will have text added to clarify this point.

The paragraph in the manuscript that was quoted above has been changed to:

“Γp, which describes the sources of ammonium to the ground, is taken as a spatially-varying 2D field on the same horizontal grid as described in Section 2, allowing for a unique field for every month. In this work, Γp will represent agricultural sources (both fertilizer and livestock), while separate non-agricultural ammonia emissions that account for ≲ 1% of total emissions will be accounted for using the unidirectional emissions model. As ammonia emissions from cattle originate primarily from excreted nitrogen in manure (Lee et al., 2025), we make the same assumption as in Zhu et al. (2015b) in which all ammonia emissions from livestock are assumed to originate from manure. We can then treat both the livestock and fertilizer emissions using the same emissions modeling framework. As such, model parameters that reference the ground or soil should be understood to be inclusive of manure produced by livestock.”

Zhu, L., Henze, D., Bash, J., Jeong, G. R., Cady-Pereira, K., Shephard, M., Luo, M., Paulot, F. & Capps, S. (2015a). Global evaluation of ammonia bidirectional exchange and livestock diurnal variation schemes. Atmospheric Chemistry and Physics, 15(22), 12823-12843.

Zhu, L., Henze, D. K., Bash, J. O., Cady-Pereira, K. E., Shephard, M. W., Luo, M., & Capps, S. L. (2015b). Sources and impacts of atmospheric NH3: Current understanding and frontiers for modeling, measurements, and remote sensing in North America. Current Pollution Reports, 1(2), 95-116.

Lee, M., Auvermann, B. W., Tedeschi, L. O., Koziel, J. A., Brandani, C. B., Gouvêa, V. N., Smith, J. K. & Casey, K. D. (2025). Ammonia emissions from beef cattle feedyards: a review. Frontiers in Animal Science, 6, 1608387.

"This could also potentially explain why the model performance improvements appear strongest in cropland-dominated agricultural regions, where soil-related emissions are more consistent with the assumptions of the bidirectional model."

I tried to check if the bidirectional flux model’s performance is in fact better in cropland regions. I made a scatter plot with the data from Fig. 11 that plotted ‘RMSE^unidi – RMSE^bidi‘ on a map, but instead plotted ‘RMSE^unidi – RMSE^bidi‘ on the y-axis and the ratio of the fertilizer NH₃ emissions E_fert to total NH₃ emissions E_tot for each point on the x-axis, where E_fert and E_tot are taken from the emissions inventory (plot is in the attached PDF). The correlation coefficient for this scatterplot is in the upper right corner. The value of the correlation coefficient is low and I can’t see much of a relationship in the scatterplot. So I’m not sure if the model does perform better in these regions as compared to regions where the livestock emissions dominate.

"I therefore suggest that the authors discuss whether any land-use filtering or weighting was applied when assimilating satellite observations, how non-soil ammonia sources may influence the inversion results, and whether the optimized parameters could be biased by the presence of mixed emission sources."

Because the bidirectional flux model was used both fertilizer and livestock emissions and urban emissions only represents ≲ 1% of total emissions, no land-use filtering was done.

"Second, the temporal resolution of the simulations and inversions is not entirely clear from the manuscript. The paper mentions that a month’s worth of satellite retrievals is used in each inversion, resulting in monthly mean parameters. However, it would be helpful if the authors could clearly state the temporal resolution of the model simulations, for example whether transport and chemistry are simulated at hourly time steps, the temporal aggregation used in the inversion procedure, and whether the satellite observations are assimilated at their native temporal resolution or aggregated before assimilation. Given the strongly episodic nature of ammonia emissions, particularly fertilizer-induced emission pulses following application events, the temporal resolution could influence the ability of the inversion system to properly constrain bidirectional exchange parameters."

In Section 2 that describes the GEM-MACH model, the following line has been added:

“Meteorological/physics variables are integrated using a time step of 5 minutes, whereas the chemistry variables use a time step of 15 minutes, and model output is saved at every 15 minutes.”

At the end of Section 7.1 (‘Inversion Procedure’), this line has been added:

“Retrievals are compared to the GEM-MACH model output (available at 15 minute increments) closest to the retrieval time.”

At the end of Section 7.2 (‘Inversion Parametrization’), this line has been added:

“The ensemble GEM-MACH output was saved only at the top of the hour, in contrast to the unperturbed GEM-MACH runs that had output saved at every 15 minute time step, to reduce the storage requirements of the ensemble.”

There was no aggregation of the retrievals prior to input into the inversion system.

"Third, the model evaluation focuses primarily on surface ammonia concentration observations from the AMoN and NAPS monitoring networks. While these datasets are valuable for evaluating atmospheric ammonia concentrations, ammonia deposition observations are also widely used to constrain ammonia emissions and nitrogen budgets. Because bidirectional exchange influences both emissions and deposition processes, deposition measurements, such as ammonium wet deposition observations, could provide an additional and independent constraint on model performance. It would therefore be useful for the authors to discuss whether deposition observations could be used in future work to further evaluate the optimized bidirectional scheme."

This line has been added to the end of the conclusions section:

“Future evaluations could extend these validations by including comparisons to ammonium wet deposition observations to provide an additional validation data set.”

"Line 110: Please clarify the version and type of the “2011-based projected 2017 inventory for the United States.” The 2011 US NEI inventory has undergone several updates, and it would be helpful to specify whether the inventory used here is the emission factor–based inventory or one generated using a bidirectional scheme."

This has been changed to:

“This set of emissions were generated using a 2013 emissions inventory for Canada (https://www.canada.ca/en/environment-climate-change/services/pollutants/air-emissions-inventory-overview.html, last access: 14 March 2025), a 2011-based projected 2017 inventory (version 6.3) for the United States (https://www.epa.gov/air-emissions-modeling/2011-version-63-platform, last access: 24 February 2025) from the US Environmental Protection Agency (EPA), and a 2008 inventory (version 6.2) for Mexico (https://www.epa.gov/air-emissions-modeling/2011-version-62-platform, last access: 24115 February 2025) from the US EPA. The emissions inventories from the EPA were derived using a traditional emission factor–based method instead of the bidirectional flux-based inventories used in later versions of the National Emissions Inventory (NEI).”

"Line 175: “nonzero” should be written as “non-zero.”"

Changed.

"It would also be helpful if the authors included a discussion of uncertainties associated with the inversion results. A brief discussion of how uncertainties in satellite retrievals, prior emissions, and model parameter assumptions may influence the inversion results and the retrieved parameters would help readers better interpret the robustness of the conclusions."

A new section (Section 8.5) has been added to address this comment (as well as a comment made by Reviewer #2). The first paragraph of this section is:

“          We conclude with a brief examination of the relative sensitivity of the inversions to the inversion model parameters. While there is a noticeable reduction in the a posterior uncertainties for both Γp and pHg, the reduction in uncertainties for Γp was much larger than that for pHg in many regions (see Figure S5 of the Supplement). Furthermore, the number of degrees of freedom for the inversions (Rodgers, 2000) associated with Γp are 2 to 8 times larger than that associated with pHg. Overall, the inversions are better able to constrain Γp then pHg .”

This new text references a new figure added to the Supplement (Fig. S5, included in the PDF attached).

Rodgers, C. D.: Inverse methods for atmospheric sounding: theory and practice, vol. 2, World scientific, 2000.

The response to these comments contain a number of equations which are not easily displayed by text, so all response are contained in the PDF attached.