Review: Simulating out-of-sample atmospheric transport to enable flux inversions by Dadheech and Turner

This is a well-written manuscript of a nice study, and very well-organized. I especially appreciate that the authors extended the comparison analysis through performing flux inversions in order to understand the impact of using the FootNet footprints when deriving fluxes. I have only skimmed the previous two papers by this same group on this topic, but it seems that in this one they have extended the model to work on column (satellite) GHG data, and that they have evaluated/validated the results for out-of-sample data, which is probably the main contribution here. The editors can determine if perhaps the manuscript is better suited for GMD, as it is very much a model development & validation study. I recommend publication after addressing the comments below. 

Overall comments:

In regional or city-scale inversions, particle trajectories from ATD models like STILT are also often used to sample and/or optimize a background in some way.  Can this be done with Footnet? 

In the out-of-sample FootNet simulations (i.e. when the footprint was generated using a model that did not use the Barnett  region for training), were the simulations also from a different year or month than what was used in training? I.e., I am wondering how well Footnet performs for data (receptors) from a completely different year than the training. I would think that transport is correlated across very large spatial scales and it could be that if trained on data from the same time period, could allow the model to perform better, even if the receptor is hundreds of km away from the training data receptors?  Now reading L 195, perhaps this was already done in previous work, perhaps that could be mentioned either way.

Optional question for the authors to perhaps comment on in the Conclusion or future work: How can FootNet can be used for larger continental-scale inversions.  Will v4 simulate longer time periods (at coarser scales) to perform inversions for CONUS?

Lastly, it seems the code is available. Would the authors recommend that others use the already-trained FootNet code to generate footprints for their own use?  What would be the caveats about the use of this model (where would it perform well or not)?

Abstract:

I would add “dispersion” to “atmospheric transport and dispersion models” here in the abstract at least, as dispersion is a large part of the modeling that Hysplit/Stilt is doing, along with transporting the tracer, and that is a common term in the literature (ATD models).

L47-48 adjust grammar in this sentence appropriately – what does “which” refer to (L48). Perhaps omit “the” prior to “flux inversion”?

L75+ Perhaps the authors could clarify what they mean by “pseudo-observation” - is this a simulated GHG concentration, or is it a footprint (i.e. gridded and varying in space and time)?  Readying on in L86, it seems the observations were simulated with footprints but – I would think the observations are footprints themselves, right?  The output after all is a footprint.

L88 - why was HRRR re-gridded to 1-km?  Using STILT does not require that even if the STILT grid is at 1-km… does FootNet perform better when this is done– in which case I would guess it depends how the interpolation is done?

L94- How far back were the particles traced in the training footprints?  From this time step list, it seems it would only be 24 hours.  How does this affect the simulated column footprints, especially the influence of the upper altitudes where the particles may not have any footprint influence in the first 24 hours at times?

L169 - Is there a reference for how much uncertainty there is in the STILT footprints?  Thinking about the comparison between FootNet and STILT in the context of the overall uncertainty in STILT may be a useful framing and can more quantitatively make this point. The papers looking at differences between different models probably only compare in certain places and times making extrapolation or generalization difficult, but one could cite some here as a comparison- is the uncertainty 10%, 20%, 50%?  (e.g., Karion et al., 2019, https://doi.org/10.5194/acp-19-2561-2019  is in the Barnett so could be useful for making this point?).

Fig 5 - regarding the smoother footprints generated by FootNet: Were the STILT footprints run with the optional far-field smoothing (Gaussian Kernel method) that is provided with the University of Utah STILT footprinting codebase? If so, how was it set?

SI Fig S2 and Fig S3, perhaps note in the legend that GP refers to Gaussian Plume or define in the table in Fig 1 next to “Gaussian Plume (GP)”, for example, for consistency.

Appendix A: It would be useful to include a similarly short description of the details of the SF inversion so the reader does not need to refer to the previous papers, if possible. 

Fig 6, units should be included - presumably these are summed over space and time for each receptor (so each data point in the color scale is a full footprint?).

Fig 6A does indicate a high bias in the FootNet footprints - can the authors comment on this?

Figs 7 & 8: Another figure or panel should indicate the difference between the posterior fluxes for each case (relative to the STILT or XSTILT case, plus comparing the in-sample vs. out-of-sample FootNet, either absolute units of percent perhaps).  As is, the second column really looks identical.  Especially in Fig 7, the footprints look quite different between panel A vs. D and G, so it would be useful to see the magnitude of the flux difference. 

L215: given the importance of the Gaussian plume proxy, can the authors include the basics of how this was calculated (equation?) – perhaps in the SI or appendix.  For example, how was the stability class determined for each case?  It is indeed interesting that this is the most important input, showing that giving the model some basic understanding of the relationship between the inputs helps it perform better, rather than giving the model only the inputs to the GP proxy, for example. Perhaps this points to the model not actually “learning” relationships, since the GP gives it the basic expected relationship between wind, PBL, etc.

General Comments

This paper introduces FootNet v3. FootNet is a deep learning emulator of atmospheric transport that computes the sensitivity of passive atmospheric trace gas concentrations to upwind emissions (the “footprint”). The footprint is a key component of inverse analysis for emissions quantification, but is a computational bottleneck that limits the feasibility of the analysis. FootNet promises to provide footprints with 650x less wall clock time per footprint vs Lagrangian Particle Dispersion models or Eulerian transport models, while maintaining or even improving model fidelity. FootNet v3 specifically promises to provide footprints for locations and times outside the training data, allowing the model to be deployed to new column and point concentration observations without re-training. This would represent a major step forward in the field of inverse analysis for emissions quantification by greatly decreasing cost and expanding access to more researchers.

FootNet v3 improves upon previous versions of FootNet by:

• Using the improved U-Net++ architecture in place of the U-Net architecture of previous FootNet versions.
• Increasing the quantity of training footprints to 500,000 and increasing the breadth to cover the entire Continental United States across seasons and meteorological conditions. These training footprints were computed using XSTILT driven by HRRR meteorology with a

The paper makes the following key claim about FootNet v3 that must be justified:

                  FootNet v3 can produce footprints at locations and times outside the training data, towards generalization to new observations.

To justify this claim, the authors train and run FootNet v3 in a “out-of-sample” configuration, where training data from large regions in California and Texas are withheld and the resulting model is used to compute flux inversions using real observations of 1) surface CO2 measurements from the BEACO2N network in San Francisco, and 2) TROPOMI XCH4 observations in the Barnett Shale of Texas. The authors find that FootNet v3 performs comparably and even slightly better than STILT alone in the out-of sample inversions, demonstrating that FootNet v3 could indeed be used for regions out of the training sample. These experiments appear to be well done.

Additional claims include that the quantity of samples and sample strategy were sufficient to constrain the model, which was demonstrated by a validation loss experiment, and that the model appropriately conserved mass, which was tuned with a parameter for surface footprints.

This paper is well written and technical comments about the writing are minor. I support publication of this paper with minor revisions.

One question I do have: The authors demonstrate here and in their previous papers that the added diffusivity of FootNet enhances some properties of the predicted concentrations, but how does if affect the distributions of retrieved emitters? Does it induce more diffuse posterior emissions patterns? What are the implications for modeling emissions from the “fat tail” distribution of methane emissions from point sources? This is one of the key questions that could be answered by high resolution satellite data, which would benefit most from having such a computationally efficient transport model.

Specific Comments

How much influence in the out-of-sample footprints lies in the in-sample domain? If this is significant it could taint the results. A stronger experiment would be to fully separate the out of sample domain in time and space, though I suspect the spatiotemporal domain was chosen to align with previous work to conserve limited resources, which is understandable.

Line 116 “but also suggest that moderately sized, region-specific training efforts may be sufficient to fine-tune for local applications” It is not obvious to me that this point follows from the data provided.

Line 147: “While STILT’s spatial structure is more physically realistic (as it directly solves the governing equations), it is not necessarily more accurate due to potential biases in the driving meteorological fields. Highly localized but biased footprints could introduce artifacts in GHG flux inversions. In this context, the smoother prediction from FootNet may actually be preferable.” STILT has a feature that can artificaially increase the dispersion and induces similar smoothness. See Lin and Gerbig (2005) Accounting for the effect of transport errors on tracer inverrsions, Geophys Res Lett 32 L01802 doi:10.1029/2004GL021127. Also, correlated errors along the duration of a STILT footprint are implemented in Jones et al. (2021) Assessing urban methane emissions using column-observing portable Fourier transform infrared (FTIR) spectrometers and a novel Bayesian inversion framework Atmos. Chem. Phys., 21, 13131–13147, https://doi.org/10.5194/acp-21-13131-2021.

Line 167 “This means that FootNet could likely be used in domains outside of where it was trained.” I thought that this was tested directly by excluding footprints in this domain from the training set. I think that the conclusions that can be drawn from the HRRR vs GFS comparison are analogous to those that would be drawn from alternate met products used in traditional observation system simulation experiments.

Technical Corrections

In He et al., 2025, the FootNet version is named FootNet v1.0. In Dadheech et al 2025 the version number is omitted. Should this not be FootNet v2.0? What is the minor version number signifying in He et al., 2025?

In this paper, Eulerian transport models and Lagrangian Particle Dispersion models are referred to as “full-physics”. These models rely heavily on parameterizations to simulate dispersion, which is a fundamental property of the output, and therefore full-physics is not an appropriate term.

The descriptions of the domains in the caption of figure 2 is confusing and I think an error– It says that the gray dots indicate individual pseudo observations, and that the Domains A and B are withheld from the out-of-sample training configuration, but there is a wide margin of missing gray dots around Domains A and B. Should the full set of pseudo-observations cover all of CONUS (as is given in the supplement) and the out-of-sample set be the dots drawn? This would better align with the description in the text. Also, this caption refers generically to FootNet when it should refer to FootNet v3.

Line 96 “Table in Figure 1” -> “The ‘Inputs’ table in Figure 1”?

Figure 4 caption: “the surface model” should be replaced with “Surface FootNet” so that the figure stands alone.

Line 169: “For instance, STILT and FLEXPART can produce larger disagreement than observed here when run under similar conditions. This flexibility allows FootNet to support flux inversions using multiple sources of meteorology” citation required.

Figure 6: What is the property being evaluated? Sum of footprint weight?