the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 08 May 2025
A Bayesian statistical method to estimate the climatology of extreme temperature under multiple scenarios: the ANKIALE package
Abstract. We describe an improved method and the associated package for estimating the statistics of temperature extremes in a Bayesian framework. Building on previous work, this method uses a range of climate model simulations to provide a prior of the real-world changes, and then considers observations to derive a posterior estimate of past and future changes. The new version described in this study makes it possible to process several scenarios simultaneously, while keeping one single counterfactual world (i.e., the world without human influence). We offer a free licensed, easy-to-use command-line tool called ANKIALE (ANalysis of Klimate with bayesian Inference: AppLication to extreme Events), which can be used to reproduce the analyses presented here, as well as to process user-defined events. ANKIALE is based on a python code, but is designed to be used from the command line interface. ANKIALE is natively parallel, enabling it to be used on a personal computer as well as on a supercomputer. The potential of this method and tool is illustrated via an application to maximum temperature over Europe until 2100, at a 0.25°- resolution, for a range of four emission scenarios, including a particular focus on the city of Paris (France).
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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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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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Supplement
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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-2025-1121', Astrid Kerkweg, 18 Jun 2025
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AC1: 'Reply on CEC1', Yoann Robin, 25 Jun 2025
Dear Astrid Kerkweg,
Thank you for your comment. I have sent to GMD a corrected manuscript, which will be available with the next file upload.
Best regards,
Yoann Robin
Citation: https://doi.org/10.5194/egusphere-2025-1121-AC1
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AC1: 'Reply on CEC1', Yoann Robin, 25 Jun 2025
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RC1: 'Comment on egusphere-2025-1121', Anonymous Referee #1, 26 Jun 2025
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AC5: 'Reply on RC1', Yoann Robin, 23 Oct 2025
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2025-1121/egusphere-2025-1121-AC5-supplement.pdf
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AC5: 'Reply on RC1', Yoann Robin, 23 Oct 2025
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RC2: 'Comment on egusphere-2025-1121', Anonymous Referee #2, 11 Aug 2025
General comments:
This manuscript by Robin et al. presents a significant methodological advance for the attribution of extreme temperature events, introducing a Bayesian framework and an open-source tool that enables the simultaneous treatment of multiple climate scenarios. The authors’ approach addresses a key gap in the literature by ensuring consistency across scenarios and by rigorously propagating uncertainties from both models and observations. The manuscript is technically ambitious and, in my view, represents an important contribution toward making attribution studies more stringent and transparent. In particular, the explicit handling of the full range of uncertainties, rather than relying on single-scenario or point estimates, sets a new standard for rigor in this field. I commend the authors for this achievement, and I believe their work will be highly valuable for the climate science and risk assessment communities.Specific comments:
Code testing:
Since GMD encourages reproducibility, I attempted to install and test the method myself. I appreciate the effort the authors have made to explain function calls and to provide well-structured code repositories with installation instructions. However, I was unable to install the prerequisite package SDFC, and did not pursue troubleshooting further. I recommend that the authors test the installation process in a clean environment, without assuming prior package installations (such as a default conda setup) or advanced Python knowledge on the part of users, to ensure accessibility for a broader audience.
Statistical nomenclature:
The manuscript introduces X as covariates from L104 onward, which is standard in statistical modelling. However, in line 118, the definition of the parameter vector θ includes these covariates alongside the scalar GEV parameters (μ0,μ1,…,ξ). This may cause confusion for readers familiar with statistical notation, as covariates are typically considered as observed or input variables, while parameters are the quantities to be estimated (often scalars).
I understand that in your Bayesian framework, the covariates themselves are uncertain and inferred from the data (due to model differences) and thus are treated as random variables. However, it would be helpful to add a clarifying sentence or two to explicitly distinguish between:
- Parameters (e.g., the GEV parameters μ0, μ1, σ0, σ1, ξ), which are scalars to be estimated, and
- Uncertain covariates (e.g., XR,0, XR,N, XR,A), which, although treated as part of the parameter vector in the Bayesian synthesis, conceptually represent covariate trajectories or functions rather than fixed parameters. (Uncertain covariates may not be the best description, maybe you find a better one).
A brief clarification in the text would help readers understand why covariates appear in the parameter vector and how their uncertainty is handled in your framework.
There are numerous parameters, covariates, and coefficients defined throughout the manuscript, often annotated with various sub- and superscripts. If possible, I recommend retaining only those notations that are essential for understanding the material, and ensuring that their use is consistent throughout the text. For example, in L189, the m in νm appears as a subscript, but it likely should be a superscript. Inconsistent or unclear notation makes it difficult for the reader to follow the argument, as it is not always apparent whether a symbol refers to a new concept or simply a different aspect of an existing one. Careful attention to notation and a streamlined set of symbols would greatly improve the manuscript’s readability.
Grammar and spelling mistakes:
While reviewing the manuscript, I noticed a considerable number of grammar and spelling mistakes throughout the text. Many of these issues could likely have been avoided with more thorough proof-reading or by using automated spelling and grammar checking tools. This has also led to a certain fatigue during the review of this manuscript, so please be prepared for additional comments in a second review round. I understand from my own experience that such errors can easily slip through, my own manuscripts have certainly not been immune to this. Nevertheless, I would encourage the authors to carefully revise the manuscript for language quality, as this will significantly improve readability and the overall impression of the work.
Technical corrections:
L16: Please correct “which consists in establishing” -> “which consist in establishing”.
L19: The phrase “has made a specialty of producing attribution studies within a short time (delay of the order of a week) following the occurrence of an event” is awkward. Consider revising to: “has specialized in producing attribution studies within a short time (typically within a week) following the occurrence of an event.”
L24: Do you mean "the statistical distribution"?
L29: "compbining"
L30: "within a Bayesian framwork"?
L34-37: Which are the two inconsistencies? Could you please expand on this part a bit more. Assuming that everyone has the RR20 paper in mind is probably a bit of a stretch. In which sense can inconsistencies arise across scenarios? May it be that parameters that should be estimated the same way independently of the scenario (shape parameter in the counterfactual for example) are not the same?
L38: THE probability ratio.
L49: An an…
Fig1: referring to the UNSD (2020) M49 norm -> Not quite sure why this is important here.
L57: I believe "Section 4" should not be abbreviated in the beginning of a sentence.
L57: Is it not rather "where estimates are derived independently for different scenarios"? (can 'scenarios' be estimated?)
L60: Which 'specific definition'?
L67: refer -> referred
L85: 1850 – 2014
L94: "as well as the future projections of the four SSP scenarios described above".
L101: As T_t is a set of maxima?
L105: "The covariate X_t^R,F is the sum of.."
L107: Please add X_t^G,F also to Eq. 1, as it's not a different model (right?), otherwise it's not quite clear what you mean. => L108: Indirectly or independently?
L111: "so we are.." -> this sounds too colloquially.
L113: the choice […], is entirely
L115-116: In Robin.. => Please rephrase this sentence.
L124: "in the case" -> "for the case"?
L126: I believe does not fit an estimation, rather use infer?
L128: Please (already here!) provide some more context on what you mean with a "multi-model synthesis. Is it a random vector estimated across various models?
L134: There is no theta^m in Eq. (2)
L138: "Let us start" -> too colloquial, covariateS
L139: ".. are derived using GAMs" (as X_t^R,F and X_t^G,F are fits of a GAM model to data, I don't think one should refer to the fits as models).
L143: twice Energy Balance Model. Can you please provide more context what this refers to, as all GCMs are in some sense energy balance models.
L151: twice theta^R
L154: I.e. per time step, is it the average over four values?
L156: I would strongly suggest to keep Fig S2 in the main text.
L161: Draw the vectors -> Not rather 'estimate'?
L166: derive the GEV parameters
L167: with the T_t^SSP series as target variable?
L169: Why do we get different results? Different starting values for the MLE optimisation?
L171: no "is"
L184: Can you please outline briefly what this hypothesis means? Does it refer to the assumption that the various climate simulations could all be potential realisations of actual climate, aside from a bias term nu^m?
L203: Which model has been excluded?
L238-239: Why not detrend the observations with a GAM before calculating the variance?
L254-270: This is a quite general description of Bayesian sampling techniques. I would suggest to drastically shorten it and put the extended version into the supplementary material.
L311: Where do we see the thing about SSP370 in Fig S5?
L313: Missing full stop.
L312: "does a good job" -> too colloquial
L325: “allowing to reproduce” -> “allowing reproduction of the results presented in this paper.”
L344: model data
L344: This is quite a specific format for a NetCDF. Do I understand correctly that most NetCDFs will require reformatting before they could be used here?
L357: Figures S1 is only referenced now.
L361: Make it clear that this is an example application.
L372: The sentence “No form of spatial dependency is taken into account, so the existence of an event at one place does not imply anything at another.” could be clearer as: “No spatial dependency is considered, so the occurrence of an event at one location does not imply anything about another location.”
L375: Maybe add maps of parameter estimates, too see how much they different from grid point to grid point.
L378: Change in intensity -> Does that refer to a change in intensity if the return period of the event is assumed the same under both factual and counterfactual conditions?
L391/391: I would replace the lower value by lower bound.
L421: What is the end of the sentence supposed to mean?
L450: Maybe not directly applicable in this scenario, but I believe generally very relevant for this community: https://ascmo.copernicus.org/articles/11/1/2025/
Fig S1: What are BEST observations?
Citation: https://doi.org/10.5194/egusphere-2025-1121-RC2 -
AC2: 'Reply on RC2', Yoann Robin, 23 Oct 2025
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2025-1121/egusphere-2025-1121-AC2-supplement.pdf
-
RC3: 'Potentially interesting, but very hard to read and possibly overcomplicated', Richard Chandler, 11 Aug 2025
-
AC3: 'Reply on RC3', Yoann Robin, 23 Oct 2025
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2025-1121/egusphere-2025-1121-AC3-supplement.pdf
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AC3: 'Reply on RC3', Yoann Robin, 23 Oct 2025
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RC4: 'Comment on egusphere-2025-1121', Anonymous Referee #4, 12 Aug 2025
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2025-1121/egusphere-2025-1121-RC4-supplement.pdf
-
AC4: 'Reply on RC4', Yoann Robin, 23 Oct 2025
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2025-1121/egusphere-2025-1121-AC4-supplement.pdf
-
AC4: 'Reply on RC4', Yoann Robin, 23 Oct 2025
Interactive discussion
Status: closed
-
CEC1: 'Comment on egusphere-2025-1121', Astrid Kerkweg, 18 Jun 2025
Dear Authors,
please note that the headlines of your subsections are missing.
I was already about to write that the Code and Data availability section is missing, but than found out, that the Title was just missing.
Please correct that as soon as possible!
Best regards, Astrid Kerkweg (GMD Executive Editor)
Citation: https://doi.org/10.5194/egusphere-2025-1121-CEC1 -
AC1: 'Reply on CEC1', Yoann Robin, 25 Jun 2025
Dear Astrid Kerkweg,
Thank you for your comment. I have sent to GMD a corrected manuscript, which will be available with the next file upload.
Best regards,
Yoann Robin
Citation: https://doi.org/10.5194/egusphere-2025-1121-AC1
-
AC1: 'Reply on CEC1', Yoann Robin, 25 Jun 2025
-
RC1: 'Comment on egusphere-2025-1121', Anonymous Referee #1, 26 Jun 2025
-
AC5: 'Reply on RC1', Yoann Robin, 23 Oct 2025
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2025-1121/egusphere-2025-1121-AC5-supplement.pdf
-
AC5: 'Reply on RC1', Yoann Robin, 23 Oct 2025
-
RC2: 'Comment on egusphere-2025-1121', Anonymous Referee #2, 11 Aug 2025
General comments:
This manuscript by Robin et al. presents a significant methodological advance for the attribution of extreme temperature events, introducing a Bayesian framework and an open-source tool that enables the simultaneous treatment of multiple climate scenarios. The authors’ approach addresses a key gap in the literature by ensuring consistency across scenarios and by rigorously propagating uncertainties from both models and observations. The manuscript is technically ambitious and, in my view, represents an important contribution toward making attribution studies more stringent and transparent. In particular, the explicit handling of the full range of uncertainties, rather than relying on single-scenario or point estimates, sets a new standard for rigor in this field. I commend the authors for this achievement, and I believe their work will be highly valuable for the climate science and risk assessment communities.Specific comments:
Code testing:
Since GMD encourages reproducibility, I attempted to install and test the method myself. I appreciate the effort the authors have made to explain function calls and to provide well-structured code repositories with installation instructions. However, I was unable to install the prerequisite package SDFC, and did not pursue troubleshooting further. I recommend that the authors test the installation process in a clean environment, without assuming prior package installations (such as a default conda setup) or advanced Python knowledge on the part of users, to ensure accessibility for a broader audience.
Statistical nomenclature:
The manuscript introduces X as covariates from L104 onward, which is standard in statistical modelling. However, in line 118, the definition of the parameter vector θ includes these covariates alongside the scalar GEV parameters (μ0,μ1,…,ξ). This may cause confusion for readers familiar with statistical notation, as covariates are typically considered as observed or input variables, while parameters are the quantities to be estimated (often scalars).
I understand that in your Bayesian framework, the covariates themselves are uncertain and inferred from the data (due to model differences) and thus are treated as random variables. However, it would be helpful to add a clarifying sentence or two to explicitly distinguish between:
- Parameters (e.g., the GEV parameters μ0, μ1, σ0, σ1, ξ), which are scalars to be estimated, and
- Uncertain covariates (e.g., XR,0, XR,N, XR,A), which, although treated as part of the parameter vector in the Bayesian synthesis, conceptually represent covariate trajectories or functions rather than fixed parameters. (Uncertain covariates may not be the best description, maybe you find a better one).
A brief clarification in the text would help readers understand why covariates appear in the parameter vector and how their uncertainty is handled in your framework.
There are numerous parameters, covariates, and coefficients defined throughout the manuscript, often annotated with various sub- and superscripts. If possible, I recommend retaining only those notations that are essential for understanding the material, and ensuring that their use is consistent throughout the text. For example, in L189, the m in νm appears as a subscript, but it likely should be a superscript. Inconsistent or unclear notation makes it difficult for the reader to follow the argument, as it is not always apparent whether a symbol refers to a new concept or simply a different aspect of an existing one. Careful attention to notation and a streamlined set of symbols would greatly improve the manuscript’s readability.
Grammar and spelling mistakes:
While reviewing the manuscript, I noticed a considerable number of grammar and spelling mistakes throughout the text. Many of these issues could likely have been avoided with more thorough proof-reading or by using automated spelling and grammar checking tools. This has also led to a certain fatigue during the review of this manuscript, so please be prepared for additional comments in a second review round. I understand from my own experience that such errors can easily slip through, my own manuscripts have certainly not been immune to this. Nevertheless, I would encourage the authors to carefully revise the manuscript for language quality, as this will significantly improve readability and the overall impression of the work.
Technical corrections:
L16: Please correct “which consists in establishing” -> “which consist in establishing”.
L19: The phrase “has made a specialty of producing attribution studies within a short time (delay of the order of a week) following the occurrence of an event” is awkward. Consider revising to: “has specialized in producing attribution studies within a short time (typically within a week) following the occurrence of an event.”
L24: Do you mean "the statistical distribution"?
L29: "compbining"
L30: "within a Bayesian framwork"?
L34-37: Which are the two inconsistencies? Could you please expand on this part a bit more. Assuming that everyone has the RR20 paper in mind is probably a bit of a stretch. In which sense can inconsistencies arise across scenarios? May it be that parameters that should be estimated the same way independently of the scenario (shape parameter in the counterfactual for example) are not the same?
L38: THE probability ratio.
L49: An an…
Fig1: referring to the UNSD (2020) M49 norm -> Not quite sure why this is important here.
L57: I believe "Section 4" should not be abbreviated in the beginning of a sentence.
L57: Is it not rather "where estimates are derived independently for different scenarios"? (can 'scenarios' be estimated?)
L60: Which 'specific definition'?
L67: refer -> referred
L85: 1850 – 2014
L94: "as well as the future projections of the four SSP scenarios described above".
L101: As T_t is a set of maxima?
L105: "The covariate X_t^R,F is the sum of.."
L107: Please add X_t^G,F also to Eq. 1, as it's not a different model (right?), otherwise it's not quite clear what you mean. => L108: Indirectly or independently?
L111: "so we are.." -> this sounds too colloquially.
L113: the choice […], is entirely
L115-116: In Robin.. => Please rephrase this sentence.
L124: "in the case" -> "for the case"?
L126: I believe does not fit an estimation, rather use infer?
L128: Please (already here!) provide some more context on what you mean with a "multi-model synthesis. Is it a random vector estimated across various models?
L134: There is no theta^m in Eq. (2)
L138: "Let us start" -> too colloquial, covariateS
L139: ".. are derived using GAMs" (as X_t^R,F and X_t^G,F are fits of a GAM model to data, I don't think one should refer to the fits as models).
L143: twice Energy Balance Model. Can you please provide more context what this refers to, as all GCMs are in some sense energy balance models.
L151: twice theta^R
L154: I.e. per time step, is it the average over four values?
L156: I would strongly suggest to keep Fig S2 in the main text.
L161: Draw the vectors -> Not rather 'estimate'?
L166: derive the GEV parameters
L167: with the T_t^SSP series as target variable?
L169: Why do we get different results? Different starting values for the MLE optimisation?
L171: no "is"
L184: Can you please outline briefly what this hypothesis means? Does it refer to the assumption that the various climate simulations could all be potential realisations of actual climate, aside from a bias term nu^m?
L203: Which model has been excluded?
L238-239: Why not detrend the observations with a GAM before calculating the variance?
L254-270: This is a quite general description of Bayesian sampling techniques. I would suggest to drastically shorten it and put the extended version into the supplementary material.
L311: Where do we see the thing about SSP370 in Fig S5?
L313: Missing full stop.
L312: "does a good job" -> too colloquial
L325: “allowing to reproduce” -> “allowing reproduction of the results presented in this paper.”
L344: model data
L344: This is quite a specific format for a NetCDF. Do I understand correctly that most NetCDFs will require reformatting before they could be used here?
L357: Figures S1 is only referenced now.
L361: Make it clear that this is an example application.
L372: The sentence “No form of spatial dependency is taken into account, so the existence of an event at one place does not imply anything at another.” could be clearer as: “No spatial dependency is considered, so the occurrence of an event at one location does not imply anything about another location.”
L375: Maybe add maps of parameter estimates, too see how much they different from grid point to grid point.
L378: Change in intensity -> Does that refer to a change in intensity if the return period of the event is assumed the same under both factual and counterfactual conditions?
L391/391: I would replace the lower value by lower bound.
L421: What is the end of the sentence supposed to mean?
L450: Maybe not directly applicable in this scenario, but I believe generally very relevant for this community: https://ascmo.copernicus.org/articles/11/1/2025/
Fig S1: What are BEST observations?
Citation: https://doi.org/10.5194/egusphere-2025-1121-RC2 -
AC2: 'Reply on RC2', Yoann Robin, 23 Oct 2025
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2025-1121/egusphere-2025-1121-AC2-supplement.pdf
-
RC3: 'Potentially interesting, but very hard to read and possibly overcomplicated', Richard Chandler, 11 Aug 2025
-
AC3: 'Reply on RC3', Yoann Robin, 23 Oct 2025
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2025-1121/egusphere-2025-1121-AC3-supplement.pdf
-
AC3: 'Reply on RC3', Yoann Robin, 23 Oct 2025
-
RC4: 'Comment on egusphere-2025-1121', Anonymous Referee #4, 12 Aug 2025
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2025-1121/egusphere-2025-1121-RC4-supplement.pdf
-
AC4: 'Reply on RC4', Yoann Robin, 23 Oct 2025
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2025-1121/egusphere-2025-1121-AC4-supplement.pdf
-
AC4: 'Reply on RC4', Yoann Robin, 23 Oct 2025
Peer review completion
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Mathieu Vrac
Aurélien Ribes
Occitane Barbaux
Philippe Naveau
The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
- Preprint
(7616 KB) - Metadata XML
-
Supplement
(11473 KB) - BibTeX
- EndNote
- Final revised paper
Dear Authors,
please note that the headlines of your subsections are missing.
I was already about to write that the Code and Data availability section is missing, but than found out, that the Title was just missing.
Please correct that as soon as possible!
Best regards, Astrid Kerkweg (GMD Executive Editor)