Jones et al. present an inverse modeling study on methane emissions from the city of Indianapolis. While a number of studies have been published to address the same problem, the study is unique and novel by using data from five portable solar-tracking Fourier transform infrared (FTIR) spectrometers and an inversion method devised to comprehensively account for uncertainties, especially those uncertainties that are not often adequately addressed in the literature, namely, uncertainties of the background methane concentration, and uncertainties in the spatial pattern of the prior inventory. As such, this study is positioned to complement previous efforts in painting a fuller picture on the issue and to help point to future directions of research. The paper is well written, including key information regarding the inversion method and a concise presentation of the key results in figures and tables. I recommend publication of the paper, and provide the following questions and comments for the authors to consider. I realize addressing the comments below might require some additional model experiments. I encourage the authors to do what they deem as appropriate with available resources and time, as I believe a better understanding and discussion of these issues would further strengthen the paper.

I think the section on the results (Section 3) should be expanded. In its current form, with text of less than 30 lines, this section does not go much further from summarizing the numbers from the figures and tables and comparison with previous estimates. I suggest the authors to include more discussions, highlighting the unique findings from the study with previous studies as the context, and offer some experiences and suggestions for future work based on the results. Here are some comments and questions for the authors to consider:

The authors offer an explicit approach to calculating the background concentration with STILT (Eq. 5), which to me is necessary and a preferred approach in future studies. I suggest the authors further explore and demonstrate the benefit of using this approach, e.g., by doing another inversion that does not treat the background with a background influence matrix. How would the emission estimates change?

One intriguing finding from the study is that two prior spatial patterns have led to drastically different results. This finding should be further explored and discussed. The study uses a small state vector with just a few terms as scaling factors for the emissions from different sectors; this approach implies considerable confidence/weight on the prior spatial distribution in fitting to the observations. An alternative approach that has been used previously would be to solve for the spatial pattern with a bigger state vector that leaves more degrees of freedom on top of the prior spatial distributions, i.e., scaling factors of emissions from different parts of the domain, for example, by dividing the domain into a number of squares a few kilometers in dimension. This approach would mean the data can have more freedom in informing the spatial pattern of the posterior. What are the considerations being such a choice of state vector? Is it because of the lack of data points from FTIR, or is it because the results make more sense this way? Have you tried, and if not, is it feasible to try another inversion with a different configuration of the state vector?

The results presented in the figures and tables need some discussions and explanations. For instance, Figure 15 shows large day-to-day variations of diffuse methane emissions of about a factor of 5 (if taking the minimum of 50 mol s^-1 on 5/13 and maximum of 250 mol s^-1 on 5/22). Can you explain the reasons behind this change? Figure 15 apparently shows that using roads as the spatial pattern for the prior, an inversion using all data gives a higher emission estimate than estimates on all individual days. Why is this?

The data used by the study, i.e., FTIR column data, is somewhat unique by its own right. It would be very beneficial for future studies if the authors can, based on their experiences, summarize and discuss a few key considerations future studies should bear in mind when using this type of data. For example, what are the favorable situations to use these data, what supplemental data are needed, and what would be the minimum number of sensors needed. 

Technical errors:

Line 54: “We focus of” should be “we focus on”

Line 55: “have proven” should be “have been proven”

Summary

The paper presents quantification of methane emissions using a network of five portable solar-tracking Fourier transform infrared (FTIR) spectrometers during a field campaign at the city of Indianapolis, USA in May of 2016. Methane emissions are estimated using a combination of Lagrangian transport model with a Bayesian inversion framework. The approach estimated both, surface emissions and background methane concentrations flowing into the city. Diffuse emissions, presumably leaks from the natural gas infrastructure, were found to be 73 ± 22 mol s^-1, 68% higher than estimated from bottom-up methods.   

General Comments

The paper performs a valuable experiment by trying to use only a several days of total column methane measurements from EM27/SUN solar-tracking total column Fourier transform infrared (FTIR) spectrometers to estimate methane emissions at Indianapolis. In comparison to the complexities of aircraft flights or of setting up long-term tower measurements, EM27/SUN instruments are relatively easy to move around and in principle could be used at multiple urban centers over the span of several months, which could significantly help with the goal of quick greenhouse gas emission estimation required by policymakers to efficiently address the issue of climate change. A novel Bayesian inversion framework is advocated in synergy with EM27/SUN instruments to perform the mentioned methane emission estimation. In this regard, the paper is appropriate for the journal and helps to advance the field of carbon cycle. However, despite these positive attributes, there are a number of critical omissions in the description of the work’s methodology that need to be fixed before this paper can be published. Additionally, the article lacks interpretation and proper discussion of the results.           

Specific Comments

Lines 174-175: If there are no particles released above 2500 m, is there an assumption that layers above are homogenous or somehow not important to the total column methane measurements? What about potential transport of significant methane plumes in the mid-troposphere?

Line 186: Variable s and indices i and j are not clearly defined. Please define.

Lines 205-215: the units of vector b and matrix B are not properly defined. Text indicates that units of vector b are ppb, while units of B are hr^-1 (according to Figure 5). Given such units, equation 5 does not work (since all the sums must maintain units of ppb). It is possible that authors treat B as unitless, a fraction of particles affected by the defined edges. But this needs to be clearly stated. Units and variables must always be defined in a proper mathematical notation.

Line 213: Please explain how the error term, ε_b is calculated?

Line 221: The definition implies that you are solving for matrix x, that is for an emission scaling factor a_s (unitless) and background time series b (ppb), is that correct?  

Line 224: Background error was not described previously (please see Line 213 comment).

Line 273: Part of a sentence is missing, please fix.

Equation 10: if a priori scaling factors are all 1 and a prior background mixing ratio is 1.84 ppm, what is the point of the following conditions: 1 ≤ i ≤ n_sec and otherwise? Define i.  

Equation 11: the conditions (such as i, j ≤ n_sec and i = j) are not explained.

Line 284: It is not clear what is meant by observations i and j. Are these the same as in section 2.4?

Lines 301-302: On these lines there is a following statement, “…the diagonals of which describe the model framework’s ability to reduce the uncertainties of the priors…” Please explain what do you mean by an ability to reduce the uncertainties of the priors? How an uncertainty could be reduced given that it is determined subjectively (as described in the methodology section)?

Lines 315-316: Although the diffuse emission value makes sense with all the days combined, individually the emissions look unrealistic as they are shown in Table 4. How can emissions jump from 40.4 mol/s to 146.3 mol/s in 2 days? And what is the meaning of negative emissions? It is not clear why the uncertainty is only 22 mol/s given a sample of only 5 days (each with significantly different values). Physically this is highly improbable. As an experimental product this result may be of an interest, but it should not be interpreted as an actual value of diffuse methane emissions for Indianapolis. As a cross check, it may be a good idea to take data from INFLUX towers and perform an inversion for the same days to see if the emission results will be comparable. Also, aircraft can be used to perform mass balance when available. Now, it is understandable that these comparisons may be outside of the scope of this work, but then the presented results must note that they are experimental in nature and are not to be interpreted as actual natural gas emissions from Indianapolis. Please address this in the results/discussion section.

Figure 11: What part of edge do dashed optimized background values represent? Is that combined optimization of background from all the instruments?   

Figure 13: This figure seems cluttered, is it possible to present these data in the time series format? Also, what is the significance of the selected bins?

Figure 14: It is not exactly clear how the model decides whether to adjust diffuse emissions or background mixing ratios (the choice seems subjective as it should depend on the prescribed a priori error). It would be great to have a sensitivity analysis where it could be clearly shown what happens when the errors of both, diffuse emissions and background mixing ratios, are allowed to vary to see if there is any stability in the shown solution.

Lines 326-331: Varying the prior of the spatial distribution of emissions significantly changes the result of the total diffuse emission estimation. That in itself shows that each posterior result should be approached with caution.   

Figure 16: Please define on one of the maps what is precisely meant by “Marion County”.