Data-Driven Quadrature for Longwave and Shortwave Absorption by Major Greenhouse Gases

Czarnecki, Paulina; Brath, Manfred

doi:10.5194/egusphere-2026-2096

Preprints

https://doi.org/10.5194/egusphere-2026-2096

Preprints

21 Jul 2026

| 21 Jul 2026

Data-Driven Quadrature for Longwave and Shortwave Absorption by Major Greenhouse Gases

Paulina Czarnecki and Manfred Brath

Abstract. Broadband radiation calculations are computationally expensive, and climate and weather models require fast parameterizations for computing the flow of energy through Earth's atmosphere. Data-driven quadrature is an alternative to traditional gas-optics parameterizations consisting of an optimal, sparse sample of representative spectral points (frequencies) and weights such that the weighted sum of monochromatic calculations approximates the broadband quantity, offering flexibility while maintaining the accuracy and efficiency of state-of-the-art schemes. Data-driven quadrature was originally developed in cloudless present-day conditions for longwave (thermal) radiation. In this work, we update the optimization algorithm to support shortwave (sunlight) calculations, which must be robust to variations in solar zenith angle and surface reflectivity. We additionally expand both the longwave and shortwave schemes to capture variability in major greenhouse gas concentrations, with potential application to different climate scenarios. The schemes are validated using ERA5 data with clear and cloudy skies and compared to a state-of-the-art radiation parameterization, showing comparable accuracy at lower computational cost. Furthermore, implementation in a single-column radiative-convective equilibrium model with interactive ozone chemistry demonstrates the versatility of the scheme and potential for online operationalization in dynamical models. Here we release and describe these optimized sets of quadrature points and associated weights, along with a tutorial to guide the optimization of new point and weight configurations for other applications.

Received: 13 Apr 2026 – Discussion started: 21 Jul 2026

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Paulina Czarnecki and Manfred Brath

Status: final response (author comments only)

CEC1:
'Comment on egusphere-2026-2096 - No compliance with the policy of the journal', Juan Antonio Añel, 07 Aug 2026

Dear authors,
Unfortunately, after checking your manuscript, it has come to our attention that it does not comply with our "Code and Data Policy".
https://www.geoscientific-model-development.net/policies/code_and_data_policy.html
The information that you provide in the Code and Data Availability section of your manuscript is incomplete, and does not allow to track to acceptable repositories all the code and data necessary to replicate the work presented in your manuscript. For example, you cite Hogan and Matricardi to get access to part of the assets, but the mentioned manuscript links an ECMWF site and refers to a FTP server that can only be accessed by contacting the corresponding author (admittedly, a paper with such failures should have never been published in the journal). Also, to get access to the ARTS code you provide a link in one of the Zenodo repositories that you cite, but it corresponds to a GitHub site. However, GitHub is not a suitable repository for scientific publication. GitHub itself instructs authors to use other long-term archival and publishing alternatives, such as Zenodo. You have used the datadrivenquadrature package, but you do not provide a version number for it in the text, and a repository for it is missing too.
Additionally, the repositories do not seem to contain all the data necessary to replicate your work. For example, you mention that you have used ERA5 data, but you do not provide a repository for the used data, but mention the Copernicus Data Store, which is not a suitable repository for scientific publication.
The GMD review and publication process depends on reviewers and community commentators being able to access, during the discussion phase, the code and data on which a manuscript depends, and on ensuring the provenance of replicability of the published papers for years after their publication. Please, therefore, publish your code and data in one of the appropriate repositories and reply to this comment with the relevant information (link and a permanent identifier for it (e.g. DOI)) as soon as possible. We cannot have manuscripts under discussion that do not comply with our policy.
Later, if the Topical Editor decides to continue with the review or publication process of your manuscript and you are requested to upload a new version of it, then The 'Code and Data Availability’ section of your manuscript must also be modified to cite the new repository locations, and corresponding references added to the bibliography.
I must note that if you do not fix this problem, we cannot continue with the peer-review process or accept your manuscript for publication in GMD.
Juan A. Añel

Geosci. Model Dev. Executive Editor

Citation: https://doi.org/10.5194/egusphere-2026-2096-CEC1
- AC1:
  'Reply on CEC1', Paulina Czarnecki, 10 Aug 2026
  
  Dear Dr. Añel,
  Thank you for your comment. We have made the following changes to our archiving of data and software and the Code and Data Availability section:
  For example, you cite Hogan and Matricardi to get access to part of the assets, but the mentioned manuscript links an ECMWF site and refers to a FTP server that can only be accessed by contacting the corresponding author.
  The Hogan and Matricardi, 2020 data is freely available on the ECMWF Auxiliary Data Store, with no need to contact the corresponding authors (https://aux.ecmwf.int/ecpds/home/ckdmip/). We have updated our Code and Data availability section to reflect this.
  Also, to get access to the ARTS code you provide a link in one of the Zenodo repositories that you cite, but it corresponds to a GitHub site.
  The ARTS code is archived on Zenodo (https://zenodo.org/records/15854825). We have updated the README on the manuscript's Zenodo repository as well as the Code and Data availability statement to reflect this.
  You have used the datadrivenquadrature package, but you do not provide a version number for it in the text, and a repository for it is missing too.
  The datadrivenquadrature package used was version 1.0.1. We have created a Zenodo archive of this version, and the corresponding DOI and citation are now referenced in the Code and Data availability section as well as throughout the text.
  For example, you mention that you have used ERA5 data, but you do not provide a repository for the used data, but mention the Copernicus Data Store, which is not a suitable repository for scientific publication.
  The ERA5 data used is available in the manuscript's repository under 'atmdata'. We have updated the README and the Code and Data availability statement to clarify this.
  Below is the updated Code and Data Availability section:
  Code and data availability. Code used to generate the figures and training datasets, cost functions used for the optimization algorithm, quadrature points and associated weights, and the ERA5 data used are available at https://doi.org/10.5281/zenodo.18958691 (Czarnecki and Brath, 2026). The optimization may be performed using the Python package datadrivenquadrature version 1.0.1, publicly available under an MIT license and additionally archived at https://doi.org/10.5281/zenodo.21874563 (Ma and Czarnecki, 2026). Atmospheric conditions used for training and testing are freely available from the CKDMIP project (https://aux.ecmwf.int/ecpds/home/ckdmip/ Hogan and Matricardi, 2020), and the line-by-line code ARTS (version 2.6.16) is available at https://doi.org/10.5281/zenodo.18958691 (Buehler et al., 2025a).
  
  Citation: https://doi.org/10.5194/egusphere-2026-2096-AC1
  - CEC2: 'Reply on AC1', Juan Antonio Añel, 11 Aug 2026
    
    Dear authors,
    Many thanks for your quick reply. Unfortunately, an issue remains outstanding: we can not accept the ecmwf.int site as a repository, as it does not comply with the requirements of the policy of the journal. Therefore, please, store the data taken from the ECMWF in a repository acceptable according to the policy, and reply to this comment with a new version of the Code and Data policy where this is fixed.
    Juan A. Añel
    Geosci. Model Dev. Executive Editor
    
    Citation: https://doi.org/10.5194/egusphere-2026-2096-CEC2
    
    CEC3: 'Reply on CEC2', Juan Antonio Añel, 07 Sep 2026
    
    After checking a README file in the ECMWF server containing the datasets here used and for which a permanent repository was required, we have reached the conclusion that the authors have not the legal capacity to restore the data in this case. Therefore, an exception to the requirements for data publication and storage is granted in this case, based on the policy of the journal, and the current version of the manuscript is considered in compliance with the Code and Data policy of the journal.
    juan A. Añel
    Geosci. Model Dev. Executive Editor
    
    Citation: https://doi.org/10.5194/egusphere-2026-2096-CEC3
RC1:
'Comment on egusphere-2026-2096', Anonymous Referee #1, 20 Aug 2026
Overall evaluation
This is a well-written and well-organized paper revisiting and expanding Data-Driven Quadrature (DDQ), a new method for calculating how radiation flows through the Earth’s atmosphere (and how it interacts with the surface), specifically the absorption of radiant energy by gases. DDQ offers a faster alternative to traditional k-distribution algorithms by selecting a small, optimized set of spectral points and weights to estimate gaseous absorption. This paper expands on Czarnecki et al. (2023), which was published in the Journal of Advances in Modeling Earth Systems (JAMES). Here, the authors expand the DDQ framework to handle both solar and thermal IR radiation. Expansion to solar requires consideration in the training of wide ranges of solar zenith angles and surface albedos. While the 2023 JAMES paper proved that data-driven optimization could replicate longwave radiative transfer in a clear-sky column, this paper turns DDQ into a fully-fledged, dual-band gas-optics package that may soon be adaptable for incorporation into an Earth System model.
Validation against detailed reference calculations shows that DDQ achieves comparable accuracy at much lower computational cost, making it highly promising as a faster alternative to RRTMG for climate modeling. The validation rigor is particularly noteworthy, moving beyond idealized profiles to test the scheme against realistic ERA5 all-sky snapshots containing complex clouds and hydrometeors. Implementation in a radiative convective model further demonstrates that the scheme remains stable and reliable even when interacting with complex ozone chemistry. To support broader scientific use, the authors released their specific datasets alongside a tutorial for customizing the tool to different atmospheric scenarios.
In summary, this paper is an important contribution to atmospheric radiative transfer science and a valuable and necessary extension of the previous DDQ paper.
Minor points
The authors should explicitly state that the each of the panels of Figs. 1 and 3 is a component of the cost function of eq. (3).

Speaking of the cost function, it seems to me that there is some subjectivity on how to define it and what weights to use. Is the version shown in this paper appropriate for an ESM? Should CRE be added as an additional term? Is the first term mainly included for the boundary (TOA and surface) fluxes, and should it actually be limited to the boundary fluxes since the heating rates come from the net flux profile (so the first and second term of the cost function are not completely independent)?

Speaking of CRE (and clouds), should all-sky (cloudy) profiles be included in the training dataset?

Regarding use of scheme for RT in exoplanet environments, it’s not just a matter of inserting exotic gaseous concentrations. Such an extension would require training where the LBL reference calculations use specialized high-temperature/pressure and/or non-Earth spectroscopic line lists to generate a representative training baseline.

There is a systematic issue throughout the manuscript where the space between a numeric value and its unit (either hPa or Kd^-1) is missing. In some sentences, the space is present (e.g., "0.1 Kd⁻¹"), while in many others it is missing. Also the authors should choose one of Wm^-2 and W/m².
Citation: https://doi.org/10.5194/egusphere-2026-2096-RC1
RC2:
'Comment on egusphere-2026-2096', Anonymous Referee #2, 01 Oct 2026
Overall comments
This manuscript is a follow-up to the Czarnecki et al. (2023) paper about the authors’ DDQ model. The current manuscript provides a re-introduction to the method used to develop the model’s coefficients, but then focuses mainly on the shortwave part of the model. The bulk of this new work provides extensive validation of the accuracy of its shortwave calculations and some new longwave validations. A brief section also provides a demonstration of the use of the DDQ in an RCE model for an ozone chemistry application.

The manuscript is well written and the results are clearly presented, effectively demonstrating the high quality of the model’s calculations. The authors also provide useful supporting codes and tutorials so the community could apply the DDQ method to applications other than the ones for which validations are shown. This manuscript will be a good resource for atmospheric modelers considering a radiative transfer code (although these days one has to wonder if this method/code will be shortly eclipsed by AI approaches trained directly on line-by-line calculations).

That being said, there isn’t much in this manuscript that is that novel or interesting. The goals of the authors to have the community become aware of their fine efforts might have been satisfied by a citable (e.g. with a doi) white paper rather than a peer-review publication. However, I do think this manuscript should be accepted for publication in EGUsphere (subject to resolving the main issue below). Model developers should be granted the benefit of the doubt since, without a publication documenting their model, they wouldn’t have the standard capstone for their efforts – a peer-reviewed publication.

Main comments
I am having trouble with Figure 1. The set of grey curves to the right in panel (a) seem clearly labeled as the results for RRTMGP. These curves provide clearly worse results than those for DDQ_LW, yet the text states that DDQ’s errors (ncluding flux) “are at worst similar to those of RRTMGP”. Unless I am misunderstanding something about this panel, this statement isn’t a correct description of this comparison and should be modified to more accurately describe it. Both this statement and comparing the grey curves in this figure to Figures 2 and 4 in Pincus et al. make me wonder if something is off either in my understanding of Figure 1(a) or in the figure itself. A typical value for net flux at the surface is 80-90 W/m2, so the RRTMGP RMMSE error seems to be about 0.07 x 85 = ~6 W/m2. This is quite a poor result and inconsistent with the typical errors for “LW flux down at surface” shown in figures 2 and 4 in Pincus et al., which are 0.5-1.0 W/m2. (A similar, but smaller, discrepancy is present for the results at TOA and the tropospheric heating rate results in Fig 1b.) A similar poor result is shown for RRTMGP in Figure 3a, clearly different then the SW result shown in Figure 4 of Pincus et al. It is difficult to believe that both the results in Figures 1, 3 of this manuscript and those shown in Pincus et al. are valid. It is quite possible that readers of this paper will view the discrepancy of the results in this paper and those in Pincus et al. as a cloud over the integrity of the authors of Pincus et al. I encourage the authors to double-check the RRTMGP results shown here to make sure no error has been made in running RRTMGP or presenting the output results. If they believe them to be valid, the author should contact the authors of Pincus et al. so they can weigh in on their validity (and as a courtesy). If these results end up being presented in this manuscript, the authors should understand why these results are so different and provide an explanation in this manuscript. It shouldn't be left unexplained that there might be something off about the results shown in Pincus et al.

A few other questions about Figure 1:
Is the black curve the average RRMSE for the 50 present-day atmospheres, and the colored curves the equivalent for each of the 6 perturbations x 6 gases = 36 variations. Are there also 36 grey curves shown, but only the magnitude of the perturbations is apparent (i.e. the gas involved in the perturbation isn’t denoted). That’s my best guess for what is shown in this plot – if it isn’t correct, the authors might want to clarify the description provided in the manuscript.

In panel b, the results in the lower troposphere (RRMSE of 0.003) for DDQ are extremely good, implying a typical error of < 0.01 K/d. A truly excellent result. Do the authors understand why these results are so much better than the heating rates closer to the surface or in the upper troposphere?

I’m confused about the “floor of 0.1”. If artificially high errors are replaced by 0.1, wouldn’t 0.1 be a ceiling and not a floor?

Technical comments / questions
Line 34 – In other places in the manuscript, the authors appropriately credit previous work for the idea of approximating a sum of monochromatic calculated radiances with a weighted sum of a far more sparse set. As the first instance in the manuscript where this method is referred to, line 34 would be the most prominent and effective location to cite their predecessors’ work. (In a similar vein, the use of the word “originally” on line 5 is meant by the authors to refer to their own previous work, but at that point readers may interpret “data-drive quadrature” as a technique (originated by Moncet et al.) rather than this model’s name. So possibly “originally” should be changed to “in a prior work” in “in our prior work”.)

60-62 – Are the elements of S selected first and then the weights, or are the frequencies and weights chosen optimally at the same time?

80 – Probably it’s better to say “by factors of 2 up to 8” instead of “by a factor of 8”.

96-97 – Since the fluxes and heating rates have units and the f_i values do not, it seems like some aspect of this equation is not explained. (Presumably C is unitless.) Are the F and H values normalized in some way?

108-115 – Is 64 chosen somewhat arbitrarily or because it is computationally efficient in a way that, say, 63 or 65 may not be? Perhaps adding a explanatory phrase here might be informative to the reader.

156 – Probably having the word “forcing” twice in the opening part of this sentence is unnecessary.

170 – Is there meant to be an apostrophe after “ARTS”?

170-173 – Shouldn’t similar information about gas optics be provided when ARTS is mentioned above in the context of its use in generating the DDQ coefficients?

191 – For clarity, “globally-averaged” should also appear in the table captions.

203 – I assume that means that DDQ also uses spectrally varying surface reflectivities and albedos? Section 3.3 is a little sparse when it comes to information about the DDQ calculations. Side question: Would it be too difficult to train DDQ using spectrally varying surface properties?

Figure 4 caption – “In panels b) and c) relative errors are only shown if the reference flux is higher than 1 Wm−2“. Isn’t that any different than the color scale being white for errors close to 0?

209 – Clear-sky heating rates were validated previously, but the clear-sky flux results shown in 3.3.1 are clearly worse than those shown above, which was stated to be due to using spectrally varying surface properties. Is there a reason to think that this also wouldn’t affect the clear-sky heating rate comparisons for the ERA5 profiles?

210 – Why refer to this here as “thermal” when “longwave” is used elsewhere in this manuscript?

Figure 5 and 6 captions – That these figures are for cloudy cases should be mentioned.

210-235 – A fair amount of this discussion is about the stratosphere, where clouds are presumably not or minimally present. Most likely these results have nothing or little to do with the fact that these profiles are cloudy. Wouldn’t these errors apply to the earlier clear-sky results and be better discussed there, leaving this section for discussion related to the presence and handling of clouds?

231-234 – It’s unclear why the results of RRTMGP are discussed here since they don’t seem to be relevant to this section or Figure 6.

247-248 – It’s not clear what the meaning of “and any discrepancies are within the bias of the default radiation scheme RRTMG (not shown)”. It would seem that RRTMG is not relevant to Figure 7. In what model is RRTMG the default radiation scheme?

273 – I am unclear on the difference between how the look-up tables are used in the method described here and the method it is contrasted with in lines 276-277.

290-295 – The authors may want to reference the works of Zender and collaborators about spectrally varying surface properties.
Citation: https://doi.org/10.5194/egusphere-2026-2096-RC2

Paulina Czarnecki and Manfred Brath

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Supplement to "Data-Driven Quadrature for Longwave and Shortwave Absorption by Major Greenhouse Gases" Paulina Czarnecki and Manfred Brath https://doi.org/10.5281/zenodo.18958691

Paulina Czarnecki and Manfred Brath

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Short summary

Quantifying the way radiation flows through the atmosphere is computationally expensive, and most applications require fast approximation. Here, we further develop and test an alternative scheme, data-driven quadrature, which calculates radiative flux as a weighted sum at a small number of spectral points. In addition to thorough validation, we make available the spectral points and weights described as well as a code that can be used to generate new points and weights.


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