the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 03 Nov 2025
EigenFlux: A Stable Multi-Stream Radiative Transfer Method for Strongly Backward-Scattering Media
Abstract. The radiative transfer (RT) of light and radiation through a medium such as an atmosphere, pigment, or water is of interest to many research communities, such as atmospheric physics and chemistry, metereology, climate research, astronomy, remote sensing, painting and coating material science, oceanography, hydrology, and graphics rendering. Despite its many uses, to the best of our knowledge there is not a non-commercial multi-stream algorithm capable of handling strongly backwards scattering systems with asymmetries in excess of -0.95. In this paper we present an derivation and implementation of the EigenFlux system which incorporates use of a Mesh Approximates multistream and eigenvalue decomposition with numerical stability achieved through the use of a natural reflectance condition. We conclude with numerical demonstrations of the range and precision of the method.
-
Notice on discussion status
The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
-
Preprint
(0 KB)
-
The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
Journal article(s) based on this preprint
Interactive discussion
Status: closed
-
RC1: 'Comment on egusphere-2025-4516', Anonymous Referee #1, 24 Mar 2026
-
AC1: 'Author Responses to Reviewer Comments', Matthew Johnson, 13 Jun 2026
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2025-4516/egusphere-2025-4516-AC1-supplement.pdf
-
AC1: 'Author Responses to Reviewer Comments', Matthew Johnson, 13 Jun 2026
-
RC2: 'Comment on egusphere-2025-4516', Anonymous Referee #2, 25 Mar 2026
Overall review
This manuscript presents a multi-stream radiative transfer solver capable of handling strong backward scattering surfaces. While there does appear to be some scientific merit to this paper, the authors do not clearly motivate their claims or justify their stated significance with results. They do not present any significant inter-comparison or applications with measurements that is required to meet the scope of AMT. They also do not clearly state their conclusions. There is no clear structure in their methods, results, and conclusions. There is no clearly defined conclusion or results sections. The methods section are broken into discrete section. These sections should be organized into clearly stated methods, results, conclusions. Additionally conclusions drawn should be explicitly demonstrated in the results section with tables, figures, and metrics. Currently their is no quantifiable justification for their abstract statement. I recommend major revision or submission to a different journal with a scope focused on the basic methodological presentation. Below are general comments and suggested changes. In addition to general comments and suggestions, I have also made specific comments and suggestions within the attached manuscript PDF file.General comments
1) There seems to be motivation for a non-commercial radiative transfer model (i.e., a novel method) that can handle strongly scattering media with absolute values of asymmetry that are high. While stated in the manuscript, there is no sources or comparisons to clearly identify this gap they are interested in addressing. Suggest being explicit in what is the problem is, how is it handled before your method, and how is your method faster, better, or simpler using error metrics and accounting for uncertainties.2) The method is novel and appears to have some advantages over the current standards, but there appears to be few results demonstrating the improvements. The tool being open source compared to a commercial option is a great idea but the commercial options are not compared nor is the current state of the open source tools. Suggest adding a results and conclusion section to demonstrate some of the improvements over current models or comparing against measured data for a specific application.
3) This seems substantial but it is not clear from their manuscript what aspects of their method are superior to models that can handle snow and ice. The authors do not motivate their work to fit into AMT's scope. There is no significant inter-comparison of methods in this work.
4) There method is explained in detail but can be hard to follow. Suggest rearranging and renaming most of the sections into: introduction, methods, results, and conclusions. Also, clearly state where the motivation is coming from using specific sources.
5) The authors do not clearly motivate their claims or justify their stated conclusions with results. \\\
6) The numerical experiments lack error analysis. Improvements should include analysis showing the sensitivity analysis, reports numerical error, and quantifies convergence time/reliability with mesh refinement. The authors should clearly show the limitation of their model and the improvements over other models.
7) The authors do appear to misrepresent some of the related work. For example, why is ``HydroLight, Radiative Transfer Software'' cited when it wasn't designed for this role of atmospheric radiative transfer with highly scattering media. Suggest comparing with models used for ice and snow that might be better comparisons.
8) The title does correctly represent the manuscript's contents. No suggestions.
9) The abstract claims are concise but not represented well in the manuscript's contents. Suggest clearly demonstrating the claimed results or altering the abstract to align with the manuscript.
10) The structure of this manuscript is a bit disorganized and much of the text is hard to understand as a result. Several paragraphs are 1 or 2 sentences. Suggest combining these short paragraphs to make the text more readable and understandable. The science question does not have a clear link to any research gaps.
11) The language is fluent, but typographical errors and unclear language are present throughout.
12) The formulae, symbols, abbreviations, and units appear to be correct, however no external comparisons are made to verify this. Certain terms like convergence are not defined. Suggest external comparisons and explicitly defining convergence.
13) Several figures have unclear captions and their results are unclear. Tables appear to be labeled as figures. Suggest improving the captions throughout to ensure these tables and figures are understandable on their own, adding more comparison figures, removing or combining figures that do not add to the scientific analysis, and ensuring tables are labeled and reference correctly.
14) The manuscript has some quality references. Suggest adding more references for comparisons to other multilayer radiative transfer models like those used for snow and ice.
15) The tool is available for download and is freely accessible. This is appropriate for this method focused paper. No suggestions
-
AC2: 'Author Responses to Reviewer Comments', Matthew Johnson, 13 Jun 2026
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2025-4516/egusphere-2025-4516-AC2-supplement.pdf
-
AC2: 'Author Responses to Reviewer Comments', Matthew Johnson, 13 Jun 2026
-
RC3: 'Comment on egusphere-2025-4516', Anonymous Referee #3, 30 Mar 2026
The paper introduces a mathematically rigorous numerical framework for solving the Radiative Transfer Equation (RTE) within plane-parallel, homogeneous, non-absorbing media. The paper has clear mathematical motivation. The EigenFlux method successfully models systems with extremely strong forward or backward scattering properties, handling asymmetries up to 0.95. The strength of the paper is that it effectively deals with the endpoint convergence limitations of traditional methods by using a Mesh Approximates approach with piecewise-linear subdivisions. The piecewise-linear subdivisions allow the use of the “a natural reflectance condition” which mathematically stabilizes the problem. The algorithm is computationally efficient and validated against a more commonly used algorithm DISORT.
While the math is rigorous and the method works reliably, the work currently lacks sufficient physical intuition. To ensure the paper is accessible to a broader scientific audience, a major revision is required. This should include a more a concise plain-language summary, and detailed definitions of all variables. Some suggestions are:
1. Readability of the text could be improved by defining and giving more explanation on commonly used terms like the single scatter albedo (ω), the asymmetry parameter (g), and the extinction path length (σ).
2. To align the paper with the journal’s scope, the author should look at more real-world uses than just pigments. It would be helpful to mention which specific research communities could use this algorithm. It will really help to specify which particles (like ice crystals or dust) cause the heavy forward and backward scattering mentioned, and what physical rules make that happen.
3.Clarify more on the model's limitations in absorptive conditions and define the specific scenarios (how does a model fail if the particles are absorptive) where the algorithm is most reliable.
4. The figure captions are extremely concise and should include more explanation to help readers understand the figures more easily.
5. Include error metrics, uncertainties and a conclusion section to effectively summarize the work.
6. Few mathematical typos are present in the text but otherwise its mathematically sound.
Citation: https://doi.org/10.5194/egusphere-2025-4516-RC3 -
AC3: 'Author Responses to Reviewer Comments', Matthew Johnson, 13 Jun 2026
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2025-4516/egusphere-2025-4516-AC3-supplement.pdf
-
AC3: 'Author Responses to Reviewer Comments', Matthew Johnson, 13 Jun 2026
Peer review completion
Interactive discussion
Status: closed
-
RC1: 'Comment on egusphere-2025-4516', Anonymous Referee #1, 24 Mar 2026
a { text-decoration: none; color: #464feb; } tr th, tr td { border: 1px solid #e6e6e6; } tr th { background-color: #f5f5f5; }
This manuscript presents EigenFlux, a multi-stream radiative transfer solver capable of handling media with extremely strong backward scattering (asymmetry < −0.95). The authors claim that no comparable non-commercial multi-stream algorithm exists for this scattering regime, and they demonstrate numerical stability using a natural reflectance boundary condition combined with eigenvalue decomposition and mesh approximations.
The work is technically deep, well structured, and fills a meaningful gap between traditional DO methods (e.g., DISORT) and the needs of strongly anisotropic scattering applications such as pigment modeling and atmospheric backscattering. The mathematical exposition is extensive and complements the numerical demonstrations.
The method introduces an abstract mesh-based approximation and eigenvalue decomposition. However, for readers in atmospheric physics or hydrology, the connection to physical interpretation (e.g., energy conservation, reciprocity, flux closure) needs more clarity.
Minor Suggested improvements:
- Show explicitly that the method conserves energy for typical phase functions
- Explain how the natural reflectance assumption relates to physical boundary conditions
- Provide discussion on limitations of the method for non-homogeneous or non-plane-parallel media
a { text-decoration: none; color: #464feb; } tr th, tr td { border: 1px solid #e6e6e6; } tr th { background-color: #f5f5f5; }
Numerical Experiments Are Extensive but Lack Error MetricsFigures show intensity distributions, eigenvalues, and transparency depths, but there is no table or section that:
- reports numerical error norms,
- quantifies convergence with mesh refinement,
- shows sensitivity to the choice of Chebyshev points or mesh density.
To strengthen the numerical section, consider adding:
- L2 or L∞error convergence plots
- Benchmark comparisons using known analytic solutions (e.g., Rayleigh scattering)
- Mesh resolution sensitivity analysis
Citation: https://doi.org/10.5194/egusphere-2025-4516-RC1 -
AC1: 'Author Responses to Reviewer Comments', Matthew Johnson, 13 Jun 2026
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2025-4516/egusphere-2025-4516-AC1-supplement.pdf
-
RC2: 'Comment on egusphere-2025-4516', Anonymous Referee #2, 25 Mar 2026
Overall review
This manuscript presents a multi-stream radiative transfer solver capable of handling strong backward scattering surfaces. While there does appear to be some scientific merit to this paper, the authors do not clearly motivate their claims or justify their stated significance with results. They do not present any significant inter-comparison or applications with measurements that is required to meet the scope of AMT. They also do not clearly state their conclusions. There is no clear structure in their methods, results, and conclusions. There is no clearly defined conclusion or results sections. The methods section are broken into discrete section. These sections should be organized into clearly stated methods, results, conclusions. Additionally conclusions drawn should be explicitly demonstrated in the results section with tables, figures, and metrics. Currently their is no quantifiable justification for their abstract statement. I recommend major revision or submission to a different journal with a scope focused on the basic methodological presentation. Below are general comments and suggested changes. In addition to general comments and suggestions, I have also made specific comments and suggestions within the attached manuscript PDF file.General comments
1) There seems to be motivation for a non-commercial radiative transfer model (i.e., a novel method) that can handle strongly scattering media with absolute values of asymmetry that are high. While stated in the manuscript, there is no sources or comparisons to clearly identify this gap they are interested in addressing. Suggest being explicit in what is the problem is, how is it handled before your method, and how is your method faster, better, or simpler using error metrics and accounting for uncertainties.2) The method is novel and appears to have some advantages over the current standards, but there appears to be few results demonstrating the improvements. The tool being open source compared to a commercial option is a great idea but the commercial options are not compared nor is the current state of the open source tools. Suggest adding a results and conclusion section to demonstrate some of the improvements over current models or comparing against measured data for a specific application.
3) This seems substantial but it is not clear from their manuscript what aspects of their method are superior to models that can handle snow and ice. The authors do not motivate their work to fit into AMT's scope. There is no significant inter-comparison of methods in this work.
4) There method is explained in detail but can be hard to follow. Suggest rearranging and renaming most of the sections into: introduction, methods, results, and conclusions. Also, clearly state where the motivation is coming from using specific sources.
5) The authors do not clearly motivate their claims or justify their stated conclusions with results. \\\
6) The numerical experiments lack error analysis. Improvements should include analysis showing the sensitivity analysis, reports numerical error, and quantifies convergence time/reliability with mesh refinement. The authors should clearly show the limitation of their model and the improvements over other models.
7) The authors do appear to misrepresent some of the related work. For example, why is ``HydroLight, Radiative Transfer Software'' cited when it wasn't designed for this role of atmospheric radiative transfer with highly scattering media. Suggest comparing with models used for ice and snow that might be better comparisons.
8) The title does correctly represent the manuscript's contents. No suggestions.
9) The abstract claims are concise but not represented well in the manuscript's contents. Suggest clearly demonstrating the claimed results or altering the abstract to align with the manuscript.
10) The structure of this manuscript is a bit disorganized and much of the text is hard to understand as a result. Several paragraphs are 1 or 2 sentences. Suggest combining these short paragraphs to make the text more readable and understandable. The science question does not have a clear link to any research gaps.
11) The language is fluent, but typographical errors and unclear language are present throughout.
12) The formulae, symbols, abbreviations, and units appear to be correct, however no external comparisons are made to verify this. Certain terms like convergence are not defined. Suggest external comparisons and explicitly defining convergence.
13) Several figures have unclear captions and their results are unclear. Tables appear to be labeled as figures. Suggest improving the captions throughout to ensure these tables and figures are understandable on their own, adding more comparison figures, removing or combining figures that do not add to the scientific analysis, and ensuring tables are labeled and reference correctly.
14) The manuscript has some quality references. Suggest adding more references for comparisons to other multilayer radiative transfer models like those used for snow and ice.
15) The tool is available for download and is freely accessible. This is appropriate for this method focused paper. No suggestions
-
AC2: 'Author Responses to Reviewer Comments', Matthew Johnson, 13 Jun 2026
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2025-4516/egusphere-2025-4516-AC2-supplement.pdf
-
AC2: 'Author Responses to Reviewer Comments', Matthew Johnson, 13 Jun 2026
-
RC3: 'Comment on egusphere-2025-4516', Anonymous Referee #3, 30 Mar 2026
The paper introduces a mathematically rigorous numerical framework for solving the Radiative Transfer Equation (RTE) within plane-parallel, homogeneous, non-absorbing media. The paper has clear mathematical motivation. The EigenFlux method successfully models systems with extremely strong forward or backward scattering properties, handling asymmetries up to 0.95. The strength of the paper is that it effectively deals with the endpoint convergence limitations of traditional methods by using a Mesh Approximates approach with piecewise-linear subdivisions. The piecewise-linear subdivisions allow the use of the “a natural reflectance condition” which mathematically stabilizes the problem. The algorithm is computationally efficient and validated against a more commonly used algorithm DISORT.
While the math is rigorous and the method works reliably, the work currently lacks sufficient physical intuition. To ensure the paper is accessible to a broader scientific audience, a major revision is required. This should include a more a concise plain-language summary, and detailed definitions of all variables. Some suggestions are:
1. Readability of the text could be improved by defining and giving more explanation on commonly used terms like the single scatter albedo (ω), the asymmetry parameter (g), and the extinction path length (σ).
2. To align the paper with the journal’s scope, the author should look at more real-world uses than just pigments. It would be helpful to mention which specific research communities could use this algorithm. It will really help to specify which particles (like ice crystals or dust) cause the heavy forward and backward scattering mentioned, and what physical rules make that happen.
3.Clarify more on the model's limitations in absorptive conditions and define the specific scenarios (how does a model fail if the particles are absorptive) where the algorithm is most reliable.
4. The figure captions are extremely concise and should include more explanation to help readers understand the figures more easily.
5. Include error metrics, uncertainties and a conclusion section to effectively summarize the work.
6. Few mathematical typos are present in the text but otherwise its mathematically sound.
Citation: https://doi.org/10.5194/egusphere-2025-4516-RC3 -
AC3: 'Author Responses to Reviewer Comments', Matthew Johnson, 13 Jun 2026
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2025-4516/egusphere-2025-4516-AC3-supplement.pdf
-
AC3: 'Author Responses to Reviewer Comments', Matthew Johnson, 13 Jun 2026
Peer review completion
Journal article(s) based on this preprint
Viewed
Since the preprint corresponding to this journal article was posted outside of Copernicus Publications, the preprint-related metrics are limited to HTML views.
| HTML | XML | Total | BibTeX | EndNote | |
|---|---|---|---|---|---|
| 1,944 | 0 | 13 | 1,957 | 0 | 0 |
- HTML: 1,944
- PDF: 0
- XML: 13
- Total: 1,957
- BibTeX: 0
- EndNote: 0
Viewed (geographical distribution)
Since the preprint corresponding to this journal article was posted outside of Copernicus Publications, the preprint-related metrics are limited to HTML views.
| Country | # | Views | % |
|---|
| Total: | 0 |
| HTML: | 0 |
| PDF: | 0 |
| XML: | 0 |
- 1
Daniel P. Johnson
The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
a { text-decoration: none; color: #464feb; } tr th, tr td { border: 1px solid #e6e6e6; } tr th { background-color: #f5f5f5; }
This manuscript presents EigenFlux, a multi-stream radiative transfer solver capable of handling media with extremely strong backward scattering (asymmetry < −0.95). The authors claim that no comparable non-commercial multi-stream algorithm exists for this scattering regime, and they demonstrate numerical stability using a natural reflectance boundary condition combined with eigenvalue decomposition and mesh approximations.
The work is technically deep, well structured, and fills a meaningful gap between traditional DO methods (e.g., DISORT) and the needs of strongly anisotropic scattering applications such as pigment modeling and atmospheric backscattering. The mathematical exposition is extensive and complements the numerical demonstrations.
The method introduces an abstract mesh-based approximation and eigenvalue decomposition. However, for readers in atmospheric physics or hydrology, the connection to physical interpretation (e.g., energy conservation, reciprocity, flux closure) needs more clarity.
Minor Suggested improvements:
a { text-decoration: none; color: #464feb; } tr th, tr td { border: 1px solid #e6e6e6; } tr th { background-color: #f5f5f5; }
Figures show intensity distributions, eigenvalues, and transparency depths, but there is no table or section that:
To strengthen the numerical section, consider adding: