The study presents an inversion analysis with a well-established modelling system using both in situ measurements from the ECCC network and GOSAT satellite measurements. The targeted region for analysis is Canada, and only fluxes in this domain are adjusted and only measurements over this domain are assimilated, which may be problematic (see discussion below). Various state vector setups are presented, where the anthropogenic and biogenic fluxes are scaled separately in time (annually or monthly, respectively) and based on anthropogenic sector or province. Due to the limited number of measurements, these latter approaches are rightly judged to be less robust. The study is interesting and relevant, although I have some concerns about the approach.

My major concern with this study has to do with the use of a global simulation where only the Canadian emissions are allowed to be optimized, and where only measurements over Canada are assimilated. It is stated that the “initial conditions from January 2009 are from a previous GOSAT inversion by Turner et al. (2015) which was shown to be unbiased globally when compared to surface and aircraft data”. Is this only for the initial 3D fields, or are the optimized fluxes also used for the extra-Canada domain? If not, I am concerned that the flux adjustments seen in Canada might actually be the results of errors elsewhere being adjusted the only way the state vector allows. All validation data are within the Canadian domain, but it is not clear that the fit upstream and downstream of these flux increments is equally consistent. This could be easily tested by comparing a forward run of the prior and posterior fluxes to e.g. soundings outside of the optimized domain. If the adjustments lead to a significantly poorer match to these non-assimilated measurements, there may be a problem with this approach. Methane is long-lived, so not only the measurements immediately downwind should be considered. This is the minimum analysis necessary to test if this approach is reasonable.

Figure 3, and the discussion of the “acceptable reproducibility of background methane”: Again, I am not convinced by this argument that these background fields are so accurate that one can reasonable optimize only Canadian fluxes because everything else agrees so well. Yes, the variability is well matched, but a bias of 5.3 ppb is certainly larger than the measurement uncertainty and, more importantly, looking at the bias alone seems to underestimate the difference. In this plot there seems to be an overestimation of the observations in the earlier part of the record, but by the last year the model seems to be underestimating the measured concentrations. If this boundary condition accepted as is, this difference in the trend will be mapped entirely onto the optimized Canadian fluxes, and not the upstream mismatch where the correction belongs. This trend in the model-observation mismatch appears in Figure 6 as well, even though it is only showing a subset of the simulation period (why?), but at least some of this is already apparent in the un-optimized western boundary. (Notably, the trend of the emissions correction that would be needed to correct this error is positive, just as the trend seen in the ECCC inversion for wetland emissions.)  

Furthermore, the argument that Maasakkers et al. (2020) showed “relatively minimal” adjustments to US emissions near the Canadian border does not mean that US fluxes from further afield do not affect concentrations measured in Canada. Yes, the winds are generally westerly, but air certainly crosses the border in both directions. Not to mention that the stations Egbert and Sable Island have a great deal of US signal when considering only westerly flow, as they are well south of the 49^th parallel.

The suggested increase in biogenic fluxes from 2010-2015 from the in-situ network is massive – this is on the order of 10% per year! This would be an extraordinary finding, if it can be substantiated. How might this be tested? Did you consider looking at isotope measurements, for example? Why might this not be seen in the GOSAT-only inversion? Why were the GOSAT and ECCC measurements not combined in this “standard” inversion setup as well (as they were in the “policy-themed” inversions presented in Section 3.4). It seems an obvious natural step to do so, to see if this trend is still apparent.

Once these concerns have been addressed the study would be appropriate for publication, but until the robustness of this “regional adjustment only” approach has been tested against independent measurements upwind and downwind of Canada in forward runs of both the prior and posterior fluxes, the scientific conclusions cannot be considered sufficiently robust.

Minor/typographic comments

L17: have been conflicting -> conflict

L26: slower -> a slower

L35: specify anthropogenically-influenced GHG: CO2 is less significant than H2O…

L54: because only a 3% source-sink imbalance, -> because a source-sink imbalance of only 3%,

L58: Please specify that the “Canadian greenhouse gas inventory” is not just an inventory of some prior integrated over Canada, but rather the government report of emissions submitted to the UNFCCC. This is a bit confusing. It’s mentioned in the abstract and fully capitalized, as if it were the proper name: “the Canadian Greenhouse Gas Inventory”. But then it’s also called the “National GHG Inventory” (also capitalized, also in the abstract), and then here just “the Canadian greenhouse gas inventory”. None of these match the title of the actual document, which should be explicitly introduced in the introduction.

L73: compromising interpretation -> compromising the interpretation

L83: wetlands fluxes -> wetland fluxes

L86-87: „an increase in” and “a decrease in” would be clearer than “upscaling” and “downscaling” in this context, which could be interpreted as spatial extrapolation/(dis)aggregation.

L93: insert comma after first use of “emissions”

L113: insert commas before and after “Estevan Point (ESP)”

L126: mol -> mole

L127: local time for when -> local time, when the

L129: western most -> westernmost

L136: I guess you mean the largest methane fluxes from wetlands in North America?

Section 2.2: I was surprised to see biomass burning not mentioned explicitly in the text, but only listed in the table. It can have quite a bit of interannual and regional variability. I was also surprised to see that the termite emissions were identical to those of biomass burning (in Canada??), and also geologic seeps. Is this just a coincidence, or were these three small sources just distributed evenly over the three (rather different) prior spatial distributions? Please clarify this, also in the text.

L191: A couple concerns here, one minor and one major. Here it is optimistically stated that the spatial pattern of emissions “may” show less agreement: this is almost certainly the case, just from a statistical perspective. The major concern has to do with the use of a global simulation where only the Canadian emissions are allowed to be optimized, and where only measurements over Canada are assimilated, but this is discussed elsewhere.

Figure 2: it seems a mistake that the contiguous US/Greenland is not screened out in panel D (but Alaska is), while it is for the other three panels.

L250: remove “done”

L259: needs a connecter after the comma (e.g. “such that the”, “wherein”, “and” …)

L281: insert “and” before “other”

L293: space missing?

L294: Did I understand correctly that the in situ data were averaged over the local afternoon each day, essentially giving just one data point per day per station (as described in line 127)? If so, a mean observational error of 65 ppb seems rather massive! Can this be attributed to a poor representation of the spatial distribution of the fluxes, which is not optimized explicitly? The only way the model can adjust the spatial distribution is by changing the weighting of the various categories.

L374, L376 (and elsewhere – find and replace): change “wetlands emissions” to “wetland emissions”

L396: and compares -> and compares them

L400: from region -> either “from regions” or “from the region”

L406: Is Egbert really sensitive to emissions from the Hudson Bay Lowlands? This surprises me. Fraserdale, sure, and maybe Chibougamau, but Egbert? Out of curiosity: for the simulations shown in Figure 6, were the different WetCHARTS scenarios also used for the US fluxes, or were these fixed? Also, please specify that the anthropogenic and “other natural” fluxes from Table 2 were used in the forward simulations shown in Figure 6 (which I assume to be the case).

L588-590: I don’t understand this sentence entirely, there seems to be words missing. Perhaps you mean: While there are about 5 times more GOSAT observations than ECCC observations for use in the analysis and the in-situ observations have larger observational error in Sa (due to model error), the surface measurements are much more sensitive to surface fluxes, which offsets the weight of the larger amount of GOSAT data. Or something like that?

L688: should “or” be “and”?

In Supplement:

P7 L167: out -> our

Figures S4 and S5: I wonder if these figures might not be easier for the reader to interpret if they were presented as matrices/surface plots? The amplitude of e.g. the singular vector decomposition in the bottom plots could still be indicated somehow, or even kept as line figures, which would help avoid confusion about the interpretation of dashed lines in the middle and bottom panels of Figure S5.

Overall suggestions:

I felt that the article introduction could flow better with better connection/continuity among topics. Some of the information in the introduction felt out of place (see specific suggestions below). I would think about what argument you want to make in each paragraph of the introduction and use informative topic sentences to guide the reader through each of these arguments.

It seems like background estimation was a difficult and challenging process in this study. I would consider adding a second approach to estimating the background -- either by optimizing global fluxes as part of the inverse model or by using a background constructed using atmospheric observations (instead of a model-based background). Section 3.1 of the manuscript includes a lengthy discussion of the merits of the model-based background and whether it is sufficiently accurate for the task at hand. Instead of this lengthy discussion, a second background estimate might be a better way to succinctly quantify the impacts of background uncertainties on the estimated methane fluxes.

Several sections of the manuscript are relatively long and wordy, especially section 3. In many cases, I think you could cut or condense some of the written material to yield a leaner, punchier, more concise manuscript.

I also have some concerns about the inverse modeling setup. I understand that redesigning the inverse modeling framework would require large numbers of new GEOS-Chem runs; hence, I would strongly urge the authors to revise their inverse modeling setup for future studies (even if not the current study). The inverse modeling simulations used in this study either (1) optimize the temporal distribution of fluxes assuming the spatial distribution of the prior is correct, or (2) optimize the spatial distribution of fluxes assuming that the temporal distribution of the prior is correct. In reality, I think both the spatial and temporal distribution of the prior flux estimate could be improved through inverse modeling, and it would be ideal to design an inverse model that does both. Otherwise, I worry that errors in one could interfere your inferences about the other. Also, I think you would see higher model-data correlations in Fig. 10 if your inverse models had more flexibility to adjust both the spatial and temporal distribution of emissions. In future studies, I would also consider using nested North America GEOS-Chem runs instead of using much coarser 2 x 2.5 resolution global simulations.

Specific suggestions:

The abstract is very long at about 400 words. I would consider making the abstract punchier and more concise.

Line 37 "however recent trends in": I think this phrase should be a separate sentence from the previous sentence.

Line 47 - 56: This paragraph feels out of place. It doesn't flow with the previous or subsequent paragraphs, and it is not clear how global atmospheric trends are relevant to the current study on Canada. I would consider cutting this paragraph.

Line 58: I wasn't clear how the first and second sentences of this paragraph relate to one another. I would try to find a topic sentence for this paragraph that summarizes the overall objective of this paragraph. You might want to have one paragraph about Canada's emissions inventory and another paragraph about existing studies instead of putting both topics in the same paragraph.

Line 75 "however studies have": I think this should be the beginning of a new sentence.

Line 75 "have been showing": replace with "show"

Lines 75 - 95: The information in this paragraph overlaps with the information in the previous paragraph. I would either come up with a unique topic sentence for this paragraph to differentiate this paragraph from the previous one, or I would combine the discussion of top-down studies in this paragraph with the discussion of top-down studies in the previous paragraph.

Line 105 "intercomparison": Why not use "comparison" instead?

Line 116 "mean along other GEOS-Chem prior emissions": It feels like there is a word missing here.

Line 147: Was the Chibougamau site decommissioned in 2011, or did it come back online into operation after 2015? This distinction isn't clear in the wording of line 147.

Figure 1: I believe that ECCC has several observation sites in Northwest Territories and Nunavut. Why not include those sites in the inverse model? See the list of ECCC sites shown in Fig. 2.6: https://www.nrcan.gc.ca/sites/www.nrcan.gc.ca/files/energy/Climate-change/pdf/CCCR_FULLREPORT-EN-FINAL.pdf.

Equations 1, 2, and 3: I think that vectors should be displayed in bold-italic font and matrices in bold font.

Line 235: Should the dimensions of K be m by n, given the definitions for m and n in the article?

Line 248: What are you optimizing for in the monthly inversion? Are you estimating methane fluxes from each individual model grid box in each month? If that were the case, I think the value of n here would be larger. Or are you optimizing something else? 

Line 252: I wouldn't refer to a monthly inversion as "high temporal resolution". I have seen existing studies estimate daily methane fluxes in an inverse model, and numerous inverse modeling studies of CO2 estimate 3-hourly fluxes.

Line 474 - 475: I disagree that there's a tradeoff between spatial resolution and temporal resolution in the inverse model. Alternative approaches would be to (1) use the GEOS-Chem adjoint in the inverse model, or (2) use a Lagrangian model like Flexpart or STILT in the inverse model. Those approaches would not necessitate a trade-off between the spatial and temporal resolution of the inverse model.

Line 582 "magnitude emissions in Canada": Is there a word missing here?