the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 21 Feb 2025
Accurate and fast prediction of radioactive pollution by Kriging coupled with Auto-Associative Models
Abstract. Uncertainty estimation is a key issue in nuclear crisis situations. Probabilistic methods for taking uncertainties into account in assessments are often costly in terms of the number of simulations and computation time. This is why emulation methods, which enable rapid estimation of numerical model outputs, represent a promising solution. The main limitation of emulation methods is that they can only predict scalar quantities. In a crisis context, decisions are often based on dose maps, which are mathematically represented by high-dimensional data. In this study, we use the Auto-Associative Model method to reduce the dimension of dose results, in order to then predict these reduced data by Kriging. We also compare this prediction method with others used by the French Nuclear Safety and Radiation Protection Authority (ASNR) to predict the consequences of a nuclear accident.
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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.
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Preprint
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The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
- Preprint
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- Final revised paper
Journal article(s) based on this preprint
Interactive discussion
Status: closed
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CEC1: 'Comment on egusphere-2024-3838', Juan Antonio Añel, 05 Mar 2025
Dear authors,
For the records, I note here that the code used in this manuscript is archived in https://zenodo.org/records/14856799.
Unfortunately, this information provided by you in a previous version of the manuscript was omitted in this Discussions version. Please, be sure it is included in any potentially reviewed version of your work.
Juan A. Añel
Geosci. Model Dev. Executive Editor
Citation: https://doi.org/10.5194/egusphere-2024-3838-CEC1 -
RC1: 'Comment on egusphere-2024-3838', Anonymous Referee #1, 07 Apr 2025
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2024-3838/egusphere-2024-3838-RC1-supplement.pdf
- RC2: 'Comment on egusphere-2024-3838', Anonymous Referee #2, 16 Apr 2025
-
AC1: 'Comment on egusphere-2024-3838', Raphaël Périllat, 15 May 2025
# Author Response to Reviewer #1
We thank Reviewer #1 for their constructive comments. Please find below our responses to each point raised.
General comments:
We thoroughly reviewed and corrected incorrect internal references (figures and sections), and improved the English language throughout the paper.
1. Section 1.1 – Broaden context beyond internal work
We added a citation to the IAEA report "Considerations in the Development of a Protection Strategy for a Nuclear or Radiological Emergency" (2021) to broaden the contextual background beyond our own institute's previous studies.
2. Line 35 – Missing reference for pX code
We added two references for the pX code in Section 1.1 to clarify its origin and documentation.
3. Line 55 – Typo: double "both"
Corrected.
4. Section 1.2 – Incorrect section references
We fixed all references to later sections and ensured that the reference to Section 5 is correct and relevant.
5. Table 1 – Clarify parameter ranges
We added explanations in the text and in the table caption for each parameter range. For example, we clarified that 0–100 m for release height is consistent with typical nuclear accident scenarios and excludes extreme or implausible configurations. We also provided references to justify these ranges.
6. Line 120 – Improve Kriging description
We revised Section 3.2 to improve the explanation of Kriging. While we chose not to enter into too much technical detail, we added references to Rasmussen and Williams (2006), Roustant et al. (2012), and Girard et al. (2016) for readers seeking more information.
7. Equation line 126 – Improve mathematical notation
We corrected the notation to $\forall i \in \{1, \dots, N\}$ and ensured consistency between left- and right-hand sides. The explanation now matches the equation.
8. Section 3.2 – Add BFGS reference and explanation
We ultimately decided not to mention BFGS to avoid unnecessary detail at that point in the manuscript.
9. Line 143 – Specify range values
We added a sentence in Section 3.3 to explain that dose values vary by up to a factor of $10^5$ depending on distance.
10. Line 145–148 – Justify AAM dimension choice
We added Figure\~\ref{fig\:error\_vs\_dimension} and an accompanying paragraph in Section 3.3 to explain how the number of AAM coordinates was selected.
11. Line 149 – Typo
Removed "Once these emulators built."
12. Line 157 – Missing closing parenthesis
Corrected.
13. Lines 160–162 – Figure reference and description
We clarified the reference to Figure 4 and expanded its caption.
14. Figure 7 caption – Inconsistencies with text
We corrected the figure (previously incorrect due to file mix-up) and aligned the caption and text.
15. Line 188–189 – Case 3 and 4 descriptions swapped
Corrected.
16. Line 221 – Improve Figure 9 and Table 3 explanations
We expanded the captions and added explanatory paragraphs at the end of Section 4.2 to help interpret both elements.
17. Line 240 – Typo
"allow" was replaced with "allows".
18. Section 5 – Too succinct
We significantly expanded Section 5 to better explain the objectives and results. We illustrated practical applications and cited the CONFIDENCE project to emphasize the relevance for operational use.
19. Line 243 – Typo
Corrected.
---# Author Response to Reviewer #2
We thank Reviewer #2 for the constructive and detailed feedback. Please find below our point-by-point responses.
## Major Comments
1. Regarding the use of AAM:
(a) More context on the method has been introduced in Section 1.2. However, we intentionally kept Section 3.1 concise to avoid going too deep into technical details. We provide relevant references for readers who wish to explore further.
(b) Section 1.2 was significantly expanded to clarify what we mean by "nonlinear structures". In particular, we now explicitly contrast the linear subspace assumption of PCA with the curved manifold approximation used in AAM.
(c) Section 1.2 also includes a reference to Girard et al. (2020), which compares AAM and PCA for reconstructing radiological dispersion maps — a case very similar to our own.
2. Regarding the construction of the emulator:
(a) We now include Figure~\ref{fig:error_vs_dimension}, which shows the evolution of reconstruction error as a function of AAM dimension. The discussion is expanded in Section 3.3.
(b) Yes, the emulators are fit independently for each coordinate. This is now clarified in Section 3.3.
3. Regarding the validation metrics:
(a) We clarified in the text and caption that Figure 7 shows FMS results computed on the test set. We also corrected the figure, which was previously showing the wrong version due to a file mix-up. The caption of Figure 4 has also been improved for consistency.
(b) We did not modify the manuscript on this point, as the FMS is a well-established metric in the radioactive dispersion modeling community. A reference to its use in the CONFIDENCE project has been added to support this.
(c) The SMSE of some scores is strongly degraded due to a single outlier that is poorly reconstructed and falls outside the axes limits of Figure 5. To mitigate this, we added the 95% quantile of the relative absolute error in Table 2, providing a more robust alternative metric that is less sensitive to extreme values.
(d) To clarify Figure 9, we expanded its caption and improved the caption of Table 3. Additional explanation was added at the end of Section 4.2 to help interpret both the figure and the table.
## Minor Comments1. We modified the abstract and Section 2 to clarify that scalar-only emulators are common in radioactive dispersion but not a limitation of emulators in general. The sentence around line 53 was removed for consistency.
2. We chose not to cite Cartwright et al. (2023) to keep the focus on directly relevant literature and avoid overloading the manuscript with peripheral references.
3. We chose not to cite the other suggested reference, as we have not read it in detail and preferred to reference sources that were directly used and understood in our work.
4. We expanded the caption of Figure 6 for clarity.
## Typographical Comments
1. Parenthetic citations have been reformatted according to GMD style.
2. The stray parenthesis at line 35 has been removed.
3. "sets of map" corrected to "sets of maps".
4. Table 1 units have been harmonized.---
# Response to Community Comments
Response to Juan Antonio Añel (CEC1):
Thank you for the reminder. The code DOI is now correctly included in the \codedataavailability{} section of the revised manuscript.---
We hope that the revised manuscript addresses all concerns raised. We thank the reviewers and the community for their constructive comments.
Citation: https://doi.org/10.5194/egusphere-2024-3838-AC1
Interactive discussion
Status: closed
-
CEC1: 'Comment on egusphere-2024-3838', Juan Antonio Añel, 05 Mar 2025
Dear authors,
For the records, I note here that the code used in this manuscript is archived in https://zenodo.org/records/14856799.
Unfortunately, this information provided by you in a previous version of the manuscript was omitted in this Discussions version. Please, be sure it is included in any potentially reviewed version of your work.
Juan A. Añel
Geosci. Model Dev. Executive Editor
Citation: https://doi.org/10.5194/egusphere-2024-3838-CEC1 -
RC1: 'Comment on egusphere-2024-3838', Anonymous Referee #1, 07 Apr 2025
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2024-3838/egusphere-2024-3838-RC1-supplement.pdf
- RC2: 'Comment on egusphere-2024-3838', Anonymous Referee #2, 16 Apr 2025
-
AC1: 'Comment on egusphere-2024-3838', Raphaël Périllat, 15 May 2025
# Author Response to Reviewer #1
We thank Reviewer #1 for their constructive comments. Please find below our responses to each point raised.
General comments:
We thoroughly reviewed and corrected incorrect internal references (figures and sections), and improved the English language throughout the paper.
1. Section 1.1 – Broaden context beyond internal work
We added a citation to the IAEA report "Considerations in the Development of a Protection Strategy for a Nuclear or Radiological Emergency" (2021) to broaden the contextual background beyond our own institute's previous studies.
2. Line 35 – Missing reference for pX code
We added two references for the pX code in Section 1.1 to clarify its origin and documentation.
3. Line 55 – Typo: double "both"
Corrected.
4. Section 1.2 – Incorrect section references
We fixed all references to later sections and ensured that the reference to Section 5 is correct and relevant.
5. Table 1 – Clarify parameter ranges
We added explanations in the text and in the table caption for each parameter range. For example, we clarified that 0–100 m for release height is consistent with typical nuclear accident scenarios and excludes extreme or implausible configurations. We also provided references to justify these ranges.
6. Line 120 – Improve Kriging description
We revised Section 3.2 to improve the explanation of Kriging. While we chose not to enter into too much technical detail, we added references to Rasmussen and Williams (2006), Roustant et al. (2012), and Girard et al. (2016) for readers seeking more information.
7. Equation line 126 – Improve mathematical notation
We corrected the notation to $\forall i \in \{1, \dots, N\}$ and ensured consistency between left- and right-hand sides. The explanation now matches the equation.
8. Section 3.2 – Add BFGS reference and explanation
We ultimately decided not to mention BFGS to avoid unnecessary detail at that point in the manuscript.
9. Line 143 – Specify range values
We added a sentence in Section 3.3 to explain that dose values vary by up to a factor of $10^5$ depending on distance.
10. Line 145–148 – Justify AAM dimension choice
We added Figure\~\ref{fig\:error\_vs\_dimension} and an accompanying paragraph in Section 3.3 to explain how the number of AAM coordinates was selected.
11. Line 149 – Typo
Removed "Once these emulators built."
12. Line 157 – Missing closing parenthesis
Corrected.
13. Lines 160–162 – Figure reference and description
We clarified the reference to Figure 4 and expanded its caption.
14. Figure 7 caption – Inconsistencies with text
We corrected the figure (previously incorrect due to file mix-up) and aligned the caption and text.
15. Line 188–189 – Case 3 and 4 descriptions swapped
Corrected.
16. Line 221 – Improve Figure 9 and Table 3 explanations
We expanded the captions and added explanatory paragraphs at the end of Section 4.2 to help interpret both elements.
17. Line 240 – Typo
"allow" was replaced with "allows".
18. Section 5 – Too succinct
We significantly expanded Section 5 to better explain the objectives and results. We illustrated practical applications and cited the CONFIDENCE project to emphasize the relevance for operational use.
19. Line 243 – Typo
Corrected.
---# Author Response to Reviewer #2
We thank Reviewer #2 for the constructive and detailed feedback. Please find below our point-by-point responses.
## Major Comments
1. Regarding the use of AAM:
(a) More context on the method has been introduced in Section 1.2. However, we intentionally kept Section 3.1 concise to avoid going too deep into technical details. We provide relevant references for readers who wish to explore further.
(b) Section 1.2 was significantly expanded to clarify what we mean by "nonlinear structures". In particular, we now explicitly contrast the linear subspace assumption of PCA with the curved manifold approximation used in AAM.
(c) Section 1.2 also includes a reference to Girard et al. (2020), which compares AAM and PCA for reconstructing radiological dispersion maps — a case very similar to our own.
2. Regarding the construction of the emulator:
(a) We now include Figure~\ref{fig:error_vs_dimension}, which shows the evolution of reconstruction error as a function of AAM dimension. The discussion is expanded in Section 3.3.
(b) Yes, the emulators are fit independently for each coordinate. This is now clarified in Section 3.3.
3. Regarding the validation metrics:
(a) We clarified in the text and caption that Figure 7 shows FMS results computed on the test set. We also corrected the figure, which was previously showing the wrong version due to a file mix-up. The caption of Figure 4 has also been improved for consistency.
(b) We did not modify the manuscript on this point, as the FMS is a well-established metric in the radioactive dispersion modeling community. A reference to its use in the CONFIDENCE project has been added to support this.
(c) The SMSE of some scores is strongly degraded due to a single outlier that is poorly reconstructed and falls outside the axes limits of Figure 5. To mitigate this, we added the 95% quantile of the relative absolute error in Table 2, providing a more robust alternative metric that is less sensitive to extreme values.
(d) To clarify Figure 9, we expanded its caption and improved the caption of Table 3. Additional explanation was added at the end of Section 4.2 to help interpret both the figure and the table.
## Minor Comments1. We modified the abstract and Section 2 to clarify that scalar-only emulators are common in radioactive dispersion but not a limitation of emulators in general. The sentence around line 53 was removed for consistency.
2. We chose not to cite Cartwright et al. (2023) to keep the focus on directly relevant literature and avoid overloading the manuscript with peripheral references.
3. We chose not to cite the other suggested reference, as we have not read it in detail and preferred to reference sources that were directly used and understood in our work.
4. We expanded the caption of Figure 6 for clarity.
## Typographical Comments
1. Parenthetic citations have been reformatted according to GMD style.
2. The stray parenthesis at line 35 has been removed.
3. "sets of map" corrected to "sets of maps".
4. Table 1 units have been harmonized.---
# Response to Community Comments
Response to Juan Antonio Añel (CEC1):
Thank you for the reminder. The code DOI is now correctly included in the \codedataavailability{} section of the revised manuscript.---
We hope that the revised manuscript addresses all concerns raised. We thank the reviewers and the community for their constructive comments.
Citation: https://doi.org/10.5194/egusphere-2024-3838-AC1
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Raphaël Périllat
Sylvain Girard
Irène Korsakissok
The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
- Preprint
(782 KB) - Metadata XML