The paper by Liang et al. compares regional inverse estimates of methane (CH₄) surface fluxes in East-Asia for the year 2019. The driver data are GOSAT and TROPOMI satellite observations of the column-average mole fractions XCH₄. Liang et al. describe the methodology based on the GEOS-CHEM transport model and a regularized inversion. They compare inversions of GOSAT and TROPOMI XCH₄  and find good agreement for some, substantial discrepancies for other regions. Comparisons to independent data sets serve as guidance to explain the regional differences.

Scope: I am a bit puzzled of what the overall goal of the study is. Is it a budget report of East Asian methane emissions or is it an evaluation of GOSAT and TROPOMI biases? The former would require more complete error analyses, more than a year of data, and more extensive discussions of previous work. For the latter, I would argue that the manuscript lacks completeness in terms of discusssing error sources (see comment below). I recommend making the overall goal of the study clearer and revising the paper in the view of that goal.

Proxy-CH4: Generally, the main (and, I believe, conceptually limiting) error source of the proxy method (GOSAT) must be discussed more thoroughly. It is the errors of the CO₂ fields that are used to construct XCH₄ from the raw CH₄/CO₂ ratio. Any (e.g. regionally correlated) errors in the prescribed CO₂ fields (typically taken from models) will map into respective errors in XCH₄. In fact, others [Schepers et al., JGR, 2012,  https://doi.org/10.1029/2012JD017549] have compared proxy and full physics methods in the early days of the GOSAT mission. They found that, in a case study for India, erroneous CarbonTracker CO₂ fields caused biases in proxy XCH₄ data [Fig. 9 and 10 and related discussion in Schepers et al.]. The paper must examine and discuss this source of error to balance the discussion of scattering induced errors of the full-physics method (TROPOMI). To the best of my knowledge, the current version of the UoL proxy algorithm uses a model ensemble for CO₂-rescaling. One could try to estimate the error by looking at the spread of these (and potentially other) models in the investigated regions.

Setup of the inverse problem: I wonder about the setup of the inverse problem. If I get it right, the parameter vector contains 600 spatial elements which represent spatially distributed annual surface fluxes. I find this a mismatch of spatial and temporal scales. While the inversion is free to optimize a lot of spatial detail, any sub-annual temporal variability of fluxes is imposed. Given further, that the measurement vector contains daily XCH4 data, I would argue that the temporal resolution of the inversion is at odds. The authors should discuss this aspect and provide sensitivity studies showing that their choice does not induce biases (e.g. by imposed seasonality).

Further, the authors have chosen to represent the prior covariance in relative terms (50%) with respect to the prior. This choice imposes that the spatial structure of posterior fluxes will be very similar to the one of the prior fluxes (simply because changing a small flux by 50% (or likewise) remains a small flux). This is clearly visible when comparing Fig. 2 and 4 (even though the log-scale in Fig. 2 needs some defiant eyeballing). The authors should clearly state the consequences of this assumption.

Inverse method: Equation 1 is the cost function of the inverse method. It is the classic regularization setup with a prior mismatch and a least squares measurement term where one term is scaled by a regularization parameter which the authors determine according to Figure S3. If I understand correctly, the condition on the selected regularization parameter is that the scaled least-squares term and the prior term impose equal cost. Why would one set such a condition when aiming at evaluating the information content of different data sets? In my understanding, this particular condition implies that whatever your measurement data are (be it dense or sparse, accurate or not), you force the inversion to deliver roughly the same degrees of freedom (for a given prior constraint). Figure 7 appears to confirm this conclusion: while GOSAT and TROPOMI have vastly different data density, the information content of the inversion is roughly the same. In consequence, the presented findings on degrees of freedom would not in any way represent the “natural” information content of the data but they are driven by design of the inverse method. Generally, I would think that an L-curve method should work better for getting a regularization parameter that actually represents the information content of the data [see the cover (or chapter 4.6) of the book by Per Christian Hansen cited in the manuscript].

Discussion: The posterior error bars of the satellite inversions (e.g. line 226f) are very small. I assume that they only represent the propagated measurement errors according to equation (4) (and line 185f) and that model transport errors, representativeness errors, more systematic measurement errors are neglected. When comparing the satellite-derived emissions to other studies (line 230ff), the reported error bars should be representative of the full error budget.

Inverse methane emissions over East Asia are estimated and compared in this research for the year 2019 using data from two satellite observations, GOSAT and TROPOMI. Based on the GEOS-Chem transport model and analytical Bayesian inversion, Liang et al. developed the regional inversion framework. Comparisons at the regional level reveal consistency overall, but significant variation in some areas. The authors analyze the observations and independent measurements in further detail and argue that the variations can be explained by the data coverage and various retrieval techniques. Some arguments, however, need more analysis or are not convincing enough. Only after the authors address the following concerns can I recommend the paper to be published.

General comments

1, The overall goal of this paper is not very clear. Several key points are mixed up in this manuscript, but the authors do not break it down into individual points for this study. If the authors intend to assess the impact of the various satellite retrievals, I would suggest using one satellite but two retrieval products (for GOSAT: “proxy” v.s. “full-physics”; TROPOMI: official dataset v.s. WMFD). If the authors want to show the robustness of the inversion system they built, I suggest discussing error sources in detail and showing intermitted results.

2, In both the abstract and discussion, the authors mention the large discrepancy over certain areas in East Asia is caused by retrievals. The cost imposed by the least-squares term and the prior term, however, appears to be equal. Thus, the changes in posterior emissions are mainly driven by the inverse system but not observations.

3, The comparisons between simulations (with a priori and a posterior) and other independent measurements also indicate that increasing the emission intensity is ineffective to improve the result (see specific comment 6) in background areas. Does it imply that the model is unable to adequately capture the variations in these areas or that certain sources are missing from the same grid cell? The common problem in inverse modeling is the missing sources in a priori emission inventory. Again, if the authors aim at evaluating the emissions in China, please add more discussion on this aspect.

Specific comments

1, Line 50-55: Please add more information about other methods to derive methane emissions as well as other satellites that are currently in service for methane monitoring (Sentinel-2, GHG-sat, etc.). The introduction here can be more comprehensive.

2, Line 95-100: As far as I know, the new version of TROPOMI has already been reprocessed. And they provide the data over the ocean (glint-mode). If the authors downloaded the official operational product, I strongly suggest using the reprocessing data. The operational product comes in a variety of versions, each of which contains various errors and biases that might cause inconsistency in error analysis. Please check/specify if the data in 2019 comes from the same version.

Additionally, retrieving data over the ocean (typically retrieved under sun-glint conditions) differs from retrieving data over land. It might cause discontinuity from land to ocean. Do authors check if there are any corresponding biases in GOSAT data?

3, The diverging colormap of Figure 2 causes confusion. It is better to use a monotonically increasing colormap.

4, Line 160. The anthropogenic emission from EDGAR v4.3.2 is relatively out of date, and which has also been found that the emissions have been overestimated in many areas. Why do authors not use the later version (latest: EDGAR v6)? At least, the authors should mention/estimate known biases in EDGAR v4.3.2.

5, About Figure 4, either in GOSAT or TROPOMI inversion, the spatial differences show a strong spatial correlation between a priori and a posterior (a v.s. c and b v.s. d). Is it caused by the assumption of a priori covariance?

6, About Table 1, there are no improvements in the values of R². The low R² may imply the model lack repetitiveness in some places (considering they are background stations). Additionally, after being constrained by satellite measurements, the negative biases with the a priori inventory simply turn to positive biases, demonstrating that adjusting the emission intensity does not improve the outcomes of simulations.

7, Line 310, section 4.3.1. Figure 3(a) show higher emission corrections than (b) while Figure 6(a) displays a small variation in concentration in IND. However, the variation of XCH₄ in EC demonstrates the consistency of Figures 3 and 6. Any explanations for this?

8, section 4.3.1. How do the sampling biases in different seasons and regions affect the comparisons between GOSAT and TROPOMI?

9, Line 390, Section 4.3.3. The authors argue that the lack of observations over the ocean leads to unrealistic enhancement of XCH₄. However, there are no sources over the ocean, the strong enhancement at the southeast corner is more likely caused by the model's erroneous processes of the transport. For example, the boundary condition of the regional model is updated by the output of the global model, which may contain bugs/errors.