The authors introduce a machine-learning approach for atmospheric inversion in which Lagrangian “footprints” are emulated on the fly, thereby avoiding tedious precomputations and storage.

The paper seems to be based on a report, He et al (2024b), quoted 12 times. The report has not been peer-reviewed, which can be easily understood when reading it. Its lack of scientific maturity is unfortunately shared by this submission as I will explain below. The paper remains interesting by reflecting the current rise in AI techniques, for instance in the field of operational meteorology, but I encourage the authors to revise their approach and their text thoroughly.

Major comments

• The framework is clearly Lagrangian and completely ignores the Eulerian framework that is usually implemented to address some of its limitations, for instance for the processing of large amounts of observations like satellite data. The two approaches, Lagrangian and Eulerian, are complementary and their respective limits should not be considered in isolation.
• The abstract does not reflect the approach fairly. For instance, the high spatial resolution of FootNet should not be reported without its near-absence of temporal resolution.
• The logic behind the use of Gaussian plume models in some studies (l. 68-70) has not been understood: it has nothing to do with “favourable atmospheric conditions” but rather with the possibility to adjust wind direction.
• The test dataset seems to have been extracted from the same time period as the training dataset. This may explain the very short list of input data (Table 1) but suggests a very limited generalization capability necessitating continuous retraining and therefore reducing the computational advantage.
• The absence of temporal resolution in the FootNet footprints is a major limitation that will appear to many as a show-stopper. Who would accept the authors’ empirical fix after having visualized what it means in practice in Figure 1?
• The authors do not discuss mass conservation in FootNet, but this is the core of atmospheric inversion. Who would accept footprints that do not conserve tracer mass?
• The explanation of the better performance of FootNet in one experiment blames the physical model, but could well be due to a compensation between FootNet errors, in particular in time, and errors in the assignment of prior error correlations.
• Sensitivity tests on some minor aspects of the initial non-peer-reviewed FootNet are more like filler.
• After 11 pages, the list of FootNet input data in Table 1 may come as a shock. The diurnal cycle of meteorology is only sampled four times. Wind fields in input are used at one level only (10 meters). Boundary layer turbulence is represented by planetary boundary layer height only. Convective exchanges are not represented at all. Surprisingly, the authors preferred to introduce some Gaussian plume variables without clearly understanding the need to make them explicit to FootNet.
• Joined with the absence of temporal resolution in output, we are closer to a toy model than to ensuring that climate goals are met (l. 309).

I am skipping minor comments.  

General comments

Authors further revised and improved a Footnet model that is used to efficiently calculate the surface flux footprints for the Bay Area urban CO2 network. The Footnet model footprints are used here in inversion and resultant fluxes and fit to observed CO2 are compared to the inverse model based on footprints calculated in a traditional way with a STILT LPDM. It is shown that the emulated footprints perform on par with LPDM-simulated when applied in inversion and may even have an advantage on some metrics. The work is a valuable contribution extending a still limited line of studies using neural networks learning from LPDM outputs to simulate flux footprints at much reduced CPU time. The technology is important for advancement of emission studies with high resolution (1-10 km) wide swath satellite observations of GHGs which involves processing of large volume of data, thus requiring enormous amount of CPU time and storage. The work relies on high resolution HRRR meteorology available over CONUS and well-established STILT transport, so it can be handily extended to many other regional applications. The paper is well written and can be accepted with minor revision, clarifying several points suggested in the detailed comments.

Detailed comments

Line 9-11 (abstract) Authors mention that, surprisingly, “the updated FootNet model out-performs the full-physics model when used in a flux inversion”. However, they don’t have a rational explanation for this and speculate as: “This improved performance is likely because atmospheric transport simulated with a full-physics transport model is not necessarily more accurate. The more simplistic representation of transport in the machine learning model helps to mitigate transport errors”. Suggest to drop this speculative discussion, as it may happen that in the next version or a case study the ML and full-physics footprints will have other biases and advantage of the ML will be lost.

Line 21-22 Giving the list of references here, authors implicitly limit the type of models useful for inverse modeling at high resolution to Lagrangian, while there are successful examples of using Eulerian models for the same purpose (eg Steiner et al, 2024). On the other hand, the possible applications of ML-based footprint simulators do not have to be regional as there are global models using Lagrangian footprints (eg Nayagam et al, 2024, Janardanan et al 2024) facing same or bigger computational challenges as for the regional ones.

Lines 28-30 Can add satellite-based studies of point sources (eg Janardanan et al 2016).

Lines 57-58 Note that for large n, one may opt to using forward transport instead, either Lagrangian or Eulerian or plume-based like PMIF (Wang et al, 2020)

Line 224 Table 1. Looking from the experience of applying limited set of parameters for describing PBL mixing before 3-D dynamic models of turbulence (eg Hanna, 1984), the choice of driving variables does not look optimal. Why don’t include surface stress, Monin-Obukhov length, 100 m or mid-PBL winds, for example?

Line 315-320 There is an impression that the ad hoc replacement of time-resolving footprints with a decay-based model will not be universally applicable, and it should be mentioned as a limitation of the proposed method.

References

Hanna SR, Applications in air pollution modeling. In: Nieuwstadt FTM, van Dop H (Eds) Atmospheric turbulence and air pollution modelling. Atmos Sci Library, vol 1. Springer, Dordrecht. 10.1007/978-94-010-9112-1_7, 1984

Janardanan et al., Comparing GOSAT observations of localized CO2 enhancements by large emitters with inventory-based estimates, GRL, 43, 3486-3493, 10.1002/2016GL067843, 2016

Janardanan et al., Country-level methane emissions and their sectoral trends during 2009–2020 estimated by high-resolution inversion of GOSAT and surface observations, Environ. Res. Lett., 19, 10.1088/1748-9326/ad2436, 2024

Nayagam et al., A top-down estimation of subnational CO2 budget using a global high-resolution inverse model with data from regional surface networks, Environ. Res. Lett., 19, 10.1088/1748-9326/ad0f74, 2024.

Steiner et al., European CH4 inversions with ICON-ART coupled to the CarbonTracker Data Assimilation Shell, Atmos. Chem. Phys., 24, 2759–2782, 10.5194/acp-24-2759-2024, 2024.

Wang et al.: PMIF v1.0: assessing the potential of satellite observations to constrain CO₂ emissions from large cities and point sources over the globe using synthetic data, Geosci. Model Dev., 13, 5813–5831, 10.5194/gmd-13-5813-2020, 2020.

General Comment:

The manuscript “High-Resolution GHG Flux Inversions Using a Machine Learning Surrogate Model for Atmospheric Transport” presents an important study showcasing the potential of ML-based emulators for atmospheric transport in enabling fast computation of observation footprints used in flux inversions. The demonstrated gains in computational speed and storage compared to traditional methods relying on full-physics models are convincing and could pave the way for near-real-time GHG flux monitoring using dense observational systems, such as the new generation of satellite instruments. Provided the comments below are addressed, this paper would be suitable for publication in this journal.

One key claim of this work is that the proposed ML approach (FootNet v2) outperforms the full-physics model in the flux inversion. The authors attribute this to the smoother spatial structure of the FootNet v2 footprints, which they hypothesize helps mitigate transport error. Given the broad implications of this finding for the field, this important statement should be supported with more evidence than the statistical results obtained from a single case study. Two potential avenues for further substantiation are:

1.Extending the comparison to other cases reflecting different meteorological conditions.

2.Conducting an OSSE (Observing System Simulation Experiment).

I suggest the authors conduct at least an OSSE experiment, which would entail:

a) Generating synthetic observations from a reference “true” emission field.

b) Generating a prior ensemble of HRRR meteorology (or from another model) and fluxes.

c) Performing multiple inversions using both STILT and FootNet v2.

Each inversion would use a different meteorological realization to simulate transport error. The hypothesis presented in this study suggests that the ML surrogate, by smoothing the transport patterns, would produce better inversion results on average than STILT. This could be evaluated by comparing the standard deviations of inversion errors (relative to the known true fluxes) and the biases.

Although this approach is idealized, it would provide more robust statistical evidence to support the claim that the ML surrogate yields better performance than the full-physics model.

Minor Comments:

Introduction:

The authors should discuss the use of variational methods as an alternative approach to address high-dimensional inversion problems involving large flux and/or observation spaces. In this framework, transport Jacobians do not need to be explicitly constructed, and efficient minimization algorithms enable rapid computation of mean posterior fluxes.

Conclusion:

It would be useful to discuss the potential application of ML surrogates for atmospheric transport in performing MCMC inversions. This approach could provide full posterior probability density functions (PDFs) without constraining the prior PDFs to specific forms, such as Gaussian priors.