In "Complementing XCO2 imagery with ground-based CO2 and 14CO2 measurements to monitor CO2 emissions from fossil fuels on a regional to local scale", Potier et al. investigate different approaches to estimating gridded fossil fuel emissions from different types of in situ and satellite observational data. These tests are often referred to as Observing System Simulation Experiments (OSSEs), although even the terminology is a subject of debate. The contents of the paper are clearly relevant to the 2015 Paris Climate Accord, the observational and modeling capabilities needed to support it, and the subject matter and scope of Atmospheric Measurement Techniques. OSSEs are often criticized for their lack of representativeness of real data. For example, this manuscript assumes all biases and spatial correlations are zero, while observational and transport biases (Schuh, Peiro, etc.) are arguably the greatest barrier to improvements in both surface flux estimates and the science they can enable. Nevertheless, the goals of the Paris agreement are rapidly approaching, and systems must develop even more rapidly to meet those needs. This paper is a necessary first step in that direction that could be strengthened in several ways.

Major issues:

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1) Woodbury matrix identity: Equation 1 and Lines 522-524. This is an unfortunate oversight that has persisted in the geosciences literature despite being a basic result from linear algebra. If you "push the inverse through", you get the more appropriate equation A = B + HRH^T, which not only is simpler, but you can compute! This is a fundamental result from data assimilation and is a little unnerving to see in a paper about data assimilation. It also makes the statement on Lines 522-524 false, which is a good thing, because you can in fact estimate uncertainties in much higher dimensions than considered in this paper.

2) Assumptions about uncertainties: Line 114, Section 2.4.2 and 4.1. The assumptions of zero bias and zero spatial correlation in the experiments are unrealistic, but perhaps necessary to do any meaningful analysis. Almost any assumptions made about uncertainties could be criticized, so it's hard to pinpoint exactly what matters and what doesn't. Maybe that's the whole point. I do think you could make an argument that retrieval and transport biases (Crowell, Peiro, Schuh refs below), combined with our lack of understanding of appropriate spatiotemporal correlations (my own speculation) are the major roadblocks for further development in the field, which leads me to have some serious reservations about the claims in Section 4.1 and to wonder if they're even supported by the paper's analysis. Fortunately, I think almost all of this argument is tangential to the goals and actual results of the paper. Section 4.2 alone is enough to motivate the work. All OSSEs are flawed, but they will be needed to develop our monitoring capabilities as done here.

3) Plumes? I was expecting to see a figure of the things you're actually observing here: XCO2, in situ surface CO2 and 14CO2, perhaps even showing a plume? It would be nice to see those from the model and data somewhere. Maybe right after Figure 1?

4) Data availability. This paper lacks a data availability section that I thought AMT required. Hopefully this was an oversight, but I cannot see how it would be appropriate to publish this paper without clear links to the surface flux priors, simulated XCO2 retrievals, and other input data used in this study.

5) Linearity assumption: Line 144. I do not understand this. Are you using a linear transport operator or are you just using the linearization to compute uncertainties? The latter is common, while I have a hard time understanding how the former would be justified, especially during the day in the Summer. Can you please explain/clarify?

Minor issues:

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Our COVID-19 paper (https://doi.org/10.1126/sciadv.abf9415) seems relevant to this work. It would be nice to address that in this paper, but not necessary.

Bulky notation: Coming from a math background I find multi-character sub and super scripts unnerving, but realize it is more common in the geosciences literature. Still, I think many of the equations could be clarified by removing the excessively verbose sub/super-scripting. For example: H_s, H_t, and H_d instead of H_{sample}, H_{transp}, H_{distr}; C, CO_2, or [CO_2] instead of C_{a,CO2}, F^{14}_N instead of F^{14}_{Nucl}, etc. This would be particularly helpful in Equation 9 that has something like 33 characters in subscripts and 7 on the baseline, making it particularly difficult to parse.

Line 138. Who is "they"?

Line 141. 300 hPa seems very low. The paper cites the results of Santaren et al. (2021) as saying that uncertainty in boundary conditions have a negligible impact. It might be helpful to still say where those boundary conditions came from (maybe I missed this) and why Santaren et al. concluded that their influence did not have a strong impact on the inferred fluxes.

Line 188. I'm assuming the daily partition coefficients are used just to apply a diurnal cycle, but I'm not sure. Could you be more explicit?

Line 199. "contains" I think this is maybe a typo.

Signs. Are the signs of the delta values and Equations 3 and 4 consistent? I think if you're using NPP instead of NPE a positive sign would imply a flux from the atmosphere to the land, but that would make the signs in Equation 3 incorrect. Can you please address this.

Section 4.1. I find much of this highly speculative and unrelated to the results in the paper.

Line 511: "...". Is this a typo?

Lines 540-541: "it hardly provides information on plants, cities and regions outside its FOV". Again, I find this speculative and not necessarily shown in the paper, or necessary. CO2 data can potentially have an impact on fluxes upwind of its observation. Please either support or remove.

Line 548: "precision". And accuracy too?

Lines 560-567: This seems to be the strongest part of the paper, but I'm not sure the abstract gives the same impression.

Line 569: "Tn". Definitely a typo.

References

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Crowell: https://doi.org/10.5194/acp-19-9797-2019

Schuh: https://doi.org/10.1029/2018GB006086

Peiro: https://doi.org/10.5194/acp-22-1097-2022

This paper quantifies the uncertainty reduction in fossil fuel emission estimates caused by different combinations of XCO2 satellite data, CO₂ surface observations and ¹⁴CO₂ observations. The study uses 1 day model simulations that map the CO₂ emissions from different regions and emission categories on the expected signal in the measurement network.

The paper has a clear focus, and presents the results in a clear and concise way. However, the paper is somewhat limited in scope: only one day in 2015, biases are not addressed, and no actual observations are simulated. There are a number of issues (listed below) that need to be addressed before the paper can be published in its final form.

I furthermore attach an annotated pdf in which minor (and some major) issues are addressed.

Model errors

I find the treatment of model errors particularly simplified (line 312: Here we assume that the uncertainty in the observation operator is dominated by that of the transport model and we ignore temporal and spatial auto-correlations in these uncertainties). Given the large role of errors in this paper, I would expect at least an analysis how the uncertainty reduction depends on the (sometimes) arbitrary choices of model error. 

Satellite track

A CO2M track is used to investigate the sensitivity of XCO2 data to CO₂ emissions. However, the track is from 2014 (including the cloudiness etc.), which differs from the simulated year 2015. This might potentially  introduce biases and the authors should at least  argue why they focus on 2015 while the  track is simulated for 2014.

Self-referencing

There is annoying self-referencing, while important work of other groups is not mentioned. For instance, the important paper by Basu et al. 2020 (PNAS) is missing, which is a severe oversight by the author team, and actually quite worrying. Instead, there is substantial self-referencing. I understand that this work builds on many existing activities in the group. However, it is a good tradition to give an overview of activities performed by other groups (e.g. in the introduction). Now, the introduction is used to already present a misplaced introduction of their own system (lines 60-65), which clearly belongs in the method section.