Bayesian stability analysis of the AMOC using nested time-dependent autoregressive model

Hallali, Luc; Myrvoll-Nilsen, Eirik; Franzke, Christian L. E.

doi:https://doi.org/10.5194/egusphere-2025-2461

Preprints

https://doi.org/10.5194/egusphere-2025-2461

Preprints

05 Jun 2025

| 05 Jun 2025

Bayesian stability analysis of the AMOC using nested time-dependent autoregressive model

Luc Hallali, Eirik Myrvoll-Nilsen, and Christian L. E. Franzke

Abstract. The Atlantic Meridional Overturning Circulation (AMOC) is a major climate element subject to possible ongoing loss of stability. Recent studies have found evidence of a gradual weakening in circulation, including early warning signals (EWS), such as increased fluctuations and correlation time of the system, which are both known to be indicators of a possible forthcoming tipping point. To assess these changes in statistical behavior we propose a robust and general statistical model based on a second-order autoregressive process with time-dependent parameters that allow for the statistical changes from increased external variability and destabilization to be accounted for separately. We estimate the time evolution of the correlation parameters using a hierarchical Bayesian modeling framework which also yields uncertainty quantification through the posterior distribution. To assess possible changes in AMOC stability we apply the model to an AMOC fingerprint proxy based on the Sub-Polar Gyre and the global mean temperature anomaly. We find statistically significant EWS which suggests that AMOC is indeed undergoing a loss of stability and is getting closer to a tipping point. The methodology developed in this study is made publicly available as an extension of the R-package INLA.ews.

Received: 26 May 2025 – Discussion started: 05 Jun 2025

Competing interests: At least one of the (co-)authors is a member of the editorial board of Nonlinear Processes in Geophysics. The peer-review process was guided by an independent editor, and the authors also have no other competing interests to declare.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors. Views expressed in the text are those of the authors and do not necessarily reflect the views of the publisher.

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Luc Hallali, Eirik Myrvoll-Nilsen, and Christian L. E. Franzke

Status: closed

RC1:
'Comment on egusphere-2025-2461', Anonymous Referee #1, 01 Jul 2025

In this paper, a recently developed Bayesian approach (Myrvoll-Nilsen et al., 2024) to

early warning signals (EWS) of tipping is applied to the time series of the Ceasar et al. (2018)

AMOC fingerprint. The main result is that also this method confirms that, based on this

fingerprint, the present-day AMOC is undergoing a loss of stability.
The paper is poorly written, its content is below standard for NPG and hence I

recommend to reject it. The main reasons are
1. The methodology in both sections 1 and 2 has a lot of overlap with the Myrvoll-Nilsen

et al., (2024) paper. Moreover, it is very poorly presented with many errors and typos (e.g.

errors in equations (7) and (11)) and symbols which are only defined later in the paper

(e.g. \kappa_f in (9), F below (1), etc.).
2. The context of the AMOC tipping problem is also poorly covered, with inappropriate

references, wrong terminology (e.g. in the title, this is no 'Bayesian stability analysis' of the

AMOC). There is no critical evaluation of the time series in Figure 4, e.g. it does not even have

units on the y-axis. The input of freshwater by the Greenland Ice Sheet has been so small

over this period that a response of the AMOC is questionable.
3. The results on the DO time series are already in Myrvoll-Nilsen et al., (2024) and cannot

be understood here without consulting that paper (which data, etc.?). The results for the

AMOC fingerprint are in terms of application the only new results. These are poorly described

and one would at least expect a comparison with other methods.
I would recommend to the authors to add the AMOC fingerprint example in the Myrvoll-Nilsen

et al., (2024) paper.

Citation: https://doi.org/10.5194/egusphere-2025-2461-RC1
- AC1: 'Reply on RC1', Luc Hallali, 15 Jul 2025
  
  “In this paper, a recently developed Bayesian approach (Myrvoll-Nilsen et al., 2024) to
  
  early warning signals (EWS) of tipping is applied to the time series of the Ceasar et al. (2018) AMOC fingerprint. The main result is that also this method confirms that, based on this fingerprint, the present-day AMOC is undergoing a loss of stability.
  The paper is poorly written, its content is below standard for NPG and hence I
  
  recommend to reject it. The main reasons are”
  Answer : We thank the referee for the comments, they will be addressed point-by-point below.
  “The methodology in both sections 1 and 2 has a lot of overlap with the Myrvoll-Nilsen
  
  et al., (2024) paper.”
  Answer : Indeed, we recognize that there is methodological overlap with Myrvoll-Nilsen et al., (2024), particularly in Sections 1 and 2. This is because our new model builds upon and extends the model presented in that paper. Moreover, we believe it is necessary to include sufficient details in order to explain the limitations of the first model, and to properly motivate how our new model addresses these limitations.
  Specifically, the old model was susceptible to false positive early warning signals caused by increased external variability, instead of loss of stability. By replacing the noise component with another time-dependent AR(1) process, the new model is able to separate these signals to avoid these issues.
  In a revised manuscript we will put more emphasis on how our model differs from the one from Myrvoll-Nilsen et al., (2024).
  “Moreover, it is very poorly presented with many errors and typos (e.g. errors in equations (7) and (11)) and symbols which are only defined later in the paper (e.g. \kappa_f in (9), F below (1), etc.).”
  Answer : It is not clear which errors in Equations (7) and (11) the reviewer is referring to.
  Equation (7) is the continuous-time expression of the dynamical system assumed in this paper. It is similar to the one introduced in Boettner and Boers (2022) and Morr and Boers (2024). This expression differs slightly from the more common variation of this model, where the Wiener process dW(t) is replaced by an OU process here denoted as U(t).
  For Equation (11) the reviewer might refer to our inclusion of the 0.5 shift in the exponent. This is used to improve the accuracy of the integral discretization.
  Definition of symbols used will be addressed.
  “The context of the AMOC tipping problem is also poorly covered, with inappropriate
  
  references, wrong terminology (e.g. in the title, this is no 'Bayesian stability analysis' of the AMOC).”
  Answer : We will add some context regarding current evidence and understanding surrounding a potential AMOC tipping point. We will also welcome any other suggestions the reviewer may have.
  The reference “(Boettner and Boers, 2022)” in line 15 of the introduction has been removed. If there are other inappropriate or missing references we would appreciate it if the reviewer could kindly point them out.
  Since early warning signals are related to stability, our time-dependent model provides an evolution of the stability of the system. The framework detailed in this paper thus performs Bayesian analysis of the AMOC stability. Of course, one limitation is that we assume the evolution of the autocorrelation parameter is linear. While our model is efficient at identifying whether or not there have been changes in stability, it is likely not an accurate representation of the actual evolution of the stability. In a revision we will make this limitation more clear and update the title to “Assessing AMOC stability using a Bayesian nested time-dependent autoregressive model ”.
  “There is no critical evaluation of the time series in Figure 4, e.g. it does not even have units on the y-axis.”
  Answer : The cumulative surface melt across years, based on the CWG melt stack from Trusel et al. (2018) is used here to reflect the total freshwater forcing from GrIS. CWG surface melt is directly linked to GrIS runoff which is known to be one major component in the possible destabilization of the AMOC.
  We will add details on this in the manuscript, and update the y-axis appropriately.
  “The input of freshwater by the Greenland Ice Sheet has been so small over this period that a response of the AMOC is questionable.”
  Answer : While the freshwater by the GrIS was small prior to 1990, it has increased non-linearly since then, as shown by Trusel et al., (2018) or Horhold et al., (2023). Moreover recent studies suggest that the acceleration of GrIS melting over the past decades may have already-observable impact on the AMOC strength (Devilliers et al.,2024), (Martin and Biastoch 2023), (Castro de la Guardia et al., 2015).
  “The results on the DO time series are already in Myrvoll-Nilsen et al., (2024) and cannot be understood here without consulting that paper (which data, etc.?).”
  Answer : The demonstration of our model on DO time series is meant to serve as a benchmark to compare with other approaches on a real data example, in addition to the other two tests assessing the accuracy of our model fitted onto simulated nested AR(1) processes and simulated tipping processes respectively.
  The results from the approach of Myrvoll-Nilsen et al.,(2024) are shown in Table 2, as are the results of Rypdal (2016) and Boers (2018). While there is no ground truth regarding which (if any) DO events are bifurcation-induced and should therefore exhibit EWS, they present a classical example of real tipping points and have been extensively studied in the literature. They therefore provide a natural and informative case study for comparing against other approaches. Our results appear to more or less corroborate previous results. In a revised manuscript we will provide more extensive descriptions of this experiment.
  “The results for the AMOC fingerprint are in terms of application the only new results. These are poorly described and one would at least expect a comparison with other methods.”
  Answer : In terms of application, this paper focuses indeed solely on the AMOC. However, the paper also introduces new methodology using a time-dependent nested AR(1) process that accounts for biases arising from structured external variability, addressing a core limitation of Myrvoll-Nilsen (2024). The methodology introduced is robust and general enough to be applied to other climate systems prone to tipping, therefore it constitutes another greatly significant result of this paper.
  We do provide a comparison in Table 3 with the model introduced in Myrvoll-Nilsen (2024) and show that using a forcing response as detrending our new model is able to detect EWS while the previous one is not. Furthermore we mention at the end of Section 3 that the results of this study are consistent with the ones of Boers (2021) and Ditlevsen and Ditlevsen (2023). We will add more discussion on the comparison in the revised manuscript.
  “I would recommend to the authors to add the AMOC fingerprint example in the Myrvoll-Nilsen et al., (2024) paper.”
  Answer : The study presented here indeed builds upon the hierarchical Bayesian framework developed in Myrvoll-Nilsen et al., (2024). However, the methodology differs significantly from that paper. As stated in the introduction, regular AR(1) processes (including the time-dependent version presented in Myrvoll-Nilsen et al., (2024)) do not account for structured external variability. On the other hand, this new nested AR(1) process can account for such biases. The revised version of the paper will make sure to emphasize the important differences between these two methodologies.
  
  Citation: https://doi.org/10.5194/egusphere-2025-2461-AC1
RC2:
'Comment on egusphere-2025-2461', Anonymous Referee #2, 01 Jul 2025

General comments:

The Atlantic Meridional Overturning Circulation (AMOC) is a key climate tipping element. The author claims that they proposed a robust and general statistical model based on a second-order autoregressive process featuring time-dependent parameters. These parameters separately account for the statistical changes arising from increased external variability and destabilization. By applying the model to an AMOC fingerprint proxy, the author detected statistically significant early warning signals (EWS) of declining AMOC stability and an approaching tipping point. This manuscript is well written, with clear descriptions, an appropriate length, and high-quality figures. The analysis is appropriate. It represents a valuable contribution to the field. I recommend accepting it after some minor revisions.
Minor comments:

1. Some sentences in the manuscript are too long in length; consider revising them to improve readability. (e.g. Line 5-7)
2. Line 30, add missing letter “s” in “called early-warning signal (EWSs)”
3. Line 148, are 500 independent simulations enough?
4. The subscripts in the captions of Fig. 2(a) and (c) appear to be incorrect, please check.
5. Line 191-192, the citation style for references needs to be modified. It is recommended to consolidate them into a single bracket.
6. There is an unexplained discrepancy: the abstract mentions “a second-order autoregressive process with time-dependent parameter”, but Section 3.1 exclusively discusses “time-dependent nested AR(1) model”. What’s the relationship between the AR(2) model and the time-dependent nested AR(1) model? This should be clarified.

Citation: https://doi.org/10.5194/egusphere-2025-2461-RC2
- AC2:
  'Reply on RC2', Luc Hallali, 15 Jul 2025
  
  “General comments:
  
  The Atlantic Meridional Overturning Circulation (AMOC) is a key climate tipping element. The author claims that they proposed a robust and general statistical model based on a second-order autoregressive process featuring time-dependent parameters. These parameters separately account for the statistical changes arising from increased external variability and destabilization. By applying the model to an AMOC fingerprint proxy, the author detected statistically significant early warning signals (EWS) of declining AMOC stability and an approaching tipping point. This manuscript is well written, with clear descriptions, an appropriate length, and high-quality figures. The analysis is appropriate. It represents a valuable contribution to the field. I recommend accepting it after some minor revisions.”
  Answer : We thank the referee for the constructive review. The comments will be addressed point-by-point below.
  "1. Some sentences in the manuscript are too long in length; consider revising them to improve readability. (e.g. Line 5-7)”
  Answer : We will revise the manuscript to make it more concise.
  “2. Line 30, add missing letter “s” in “called early-warning signal (EWSs)””
  Answer : Correction will be added.
  “3. Line 148, are 500 independent simulations enough?”
  Answer : For each test, Root Mean Squared error (RMSE) has been plotted against the number of independent simulations. Results show that the RMSE values tend to converge around 300 independent simulations. Therefore we believe that 500 independent simulations is enough.
  “4. The subscripts in the captions of Fig. 2(a) and (c) appear to be incorrect, please check.”
  Answer : Correction will be added.
  “5. Line 191-192, the citation style for references needs to be modified. It is recommended to consolidate them into a single bracket.”
  Answer : Correction will be added.
  “6. There is an unexplained discrepancy: the abstract mentions “a second-order autoregressive process with time-dependent parameter”, but Section 3.1 exclusively discusses “time-dependent nested AR(1) model”. What’s the relationship between the AR(2) model and the time-dependent nested AR(1) model? This should be clarified.”
  Answer : We mention in line 98 that Morr and Boers (2024) show that this nested AR(1) process is equivalent to an AR(2). To improve clarity, we will consistently use “nested time-dependent AR(1) model” in the revised manuscript.
  
  Citation: https://doi.org/10.5194/egusphere-2025-2461-AC2
  - RC4: 'Reply on AC2', Anonymous Referee #2, 31 Jul 2025
    
    So, has the revised manuscript been uploaded? I can't see any track changes.
    
    Citation: https://doi.org/10.5194/egusphere-2025-2461-RC4
RC3:
'Comment on egusphere-2025-2461', Anonymous Referee #3, 09 Jul 2025

This study proposed a nested time-dependent autoregressive model to investigate the early warning signals (EWS) of AMOC, following the work of Myrvoll-Nilsen et al. (2024). The proposed model was tested to find the EWS. The results showed that it can indeed better find warning signals. But some issues need to be addressed before considering for publication.
1)The math symbols are different in the formula and the text. For example, line 49 and formula (4). Line 65 and formula (9). Besides, what does the k_f denote in formula (9)?
2)Line 193. Please explain why 0.95 is used as a detection of EWS. In Table 2, which events are real occurrences. If this information is provided, it may be easier to understand the disadvantage of the proposed model.
3)Line 196. 2 should be Table 2? Line 214. 1971 should be 1871?
4)In this study, different detrending strategies were applied. Table 3, Figures 5 and 6 show that the results seem to depend on the detrending strategies. Then, do you think which strategy should be used in finding the EWS?

Citation: https://doi.org/10.5194/egusphere-2025-2461-RC3
- AC3: 'Reply on RC3', Luc Hallali, 15 Jul 2025
  
  “This study proposed a nested time-dependent autoregressive model to investigate the early warning signals (EWS) of AMOC, following the work of Myrvoll-Nilsen et al. (2024). The proposed model was tested to find the EWS. The results showed that it can indeed better find warning signals. But some issues need to be addressed before considering for publication.”
  Answer : We appreciate this helpful review. The comments will be addressed point-by-point below.
  “1)The math symbols are different in the formula and the text. For example, line 49 and formula (4). Line 65 and formula (9). Besides, what does the k_f denote in formula (9)?”
  Answer : The notation mistakes will be corrected.
  k_f is a scaling parameter used in computing the forcing response. Its definition will be added to the revised manuscript.
  “2)Line 193. Please explain why 0.95 is used as a detection of EWS. In Table 2, which events are real occurrences. If this information is provided, it may be easier to understand the disadvantage of the proposed model.”
  Answer : The reason why 0.95 was chosen is because that is the conventional threshold for statistical significance. It is ultimately arbitrary, and we will add a comment on this in the revised manuscript on why this threshold was chosen.
  Table 2 presents the results of testing our model on real data, from the Dansgaard-Oeschger (DO) events. It is debated which of these events (if any) are actually caused by bifurcations, hence there is no known truth to compare to. However, the model appears to corroborate previous results, lending credence to our model relative to existing approaches.
  “3)Line 196. 2 should be Table 2? Line 214. 1971 should be 1871?”
  Answer : This will be corrected.
  “4)In this study, different detrending strategies were applied. Table 3, Figures 5 and 6 show that the results seem to depend on the detrending strategies. Then, do you think which strategy should be used in finding the EWS?”
  Answer : While the results might differ between the detrending strategies, they all agree on the presence of early warning signals. Generally, one should select the detrending strategy that is most appropriate for the problem at hand. If forcing data is available, and physically meaningful, it would often be beneficial to incorporate that into the model, over the more basic detrending strategies.
  
  Citation: https://doi.org/10.5194/egusphere-2025-2461-AC3

Status: closed

RC1:
'Comment on egusphere-2025-2461', Anonymous Referee #1, 01 Jul 2025

In this paper, a recently developed Bayesian approach (Myrvoll-Nilsen et al., 2024) to

early warning signals (EWS) of tipping is applied to the time series of the Ceasar et al. (2018)

AMOC fingerprint. The main result is that also this method confirms that, based on this

fingerprint, the present-day AMOC is undergoing a loss of stability.
The paper is poorly written, its content is below standard for NPG and hence I

recommend to reject it. The main reasons are
1. The methodology in both sections 1 and 2 has a lot of overlap with the Myrvoll-Nilsen

et al., (2024) paper. Moreover, it is very poorly presented with many errors and typos (e.g.

errors in equations (7) and (11)) and symbols which are only defined later in the paper

(e.g. \kappa_f in (9), F below (1), etc.).
2. The context of the AMOC tipping problem is also poorly covered, with inappropriate

references, wrong terminology (e.g. in the title, this is no 'Bayesian stability analysis' of the

AMOC). There is no critical evaluation of the time series in Figure 4, e.g. it does not even have

units on the y-axis. The input of freshwater by the Greenland Ice Sheet has been so small

over this period that a response of the AMOC is questionable.
3. The results on the DO time series are already in Myrvoll-Nilsen et al., (2024) and cannot

be understood here without consulting that paper (which data, etc.?). The results for the

AMOC fingerprint are in terms of application the only new results. These are poorly described

and one would at least expect a comparison with other methods.
I would recommend to the authors to add the AMOC fingerprint example in the Myrvoll-Nilsen

et al., (2024) paper.

Citation: https://doi.org/10.5194/egusphere-2025-2461-RC1
- AC1: 'Reply on RC1', Luc Hallali, 15 Jul 2025
  
  “In this paper, a recently developed Bayesian approach (Myrvoll-Nilsen et al., 2024) to
  
  early warning signals (EWS) of tipping is applied to the time series of the Ceasar et al. (2018) AMOC fingerprint. The main result is that also this method confirms that, based on this fingerprint, the present-day AMOC is undergoing a loss of stability.
  The paper is poorly written, its content is below standard for NPG and hence I
  
  recommend to reject it. The main reasons are”
  Answer : We thank the referee for the comments, they will be addressed point-by-point below.
  “The methodology in both sections 1 and 2 has a lot of overlap with the Myrvoll-Nilsen
  
  et al., (2024) paper.”
  Answer : Indeed, we recognize that there is methodological overlap with Myrvoll-Nilsen et al., (2024), particularly in Sections 1 and 2. This is because our new model builds upon and extends the model presented in that paper. Moreover, we believe it is necessary to include sufficient details in order to explain the limitations of the first model, and to properly motivate how our new model addresses these limitations.
  Specifically, the old model was susceptible to false positive early warning signals caused by increased external variability, instead of loss of stability. By replacing the noise component with another time-dependent AR(1) process, the new model is able to separate these signals to avoid these issues.
  In a revised manuscript we will put more emphasis on how our model differs from the one from Myrvoll-Nilsen et al., (2024).
  “Moreover, it is very poorly presented with many errors and typos (e.g. errors in equations (7) and (11)) and symbols which are only defined later in the paper (e.g. \kappa_f in (9), F below (1), etc.).”
  Answer : It is not clear which errors in Equations (7) and (11) the reviewer is referring to.
  Equation (7) is the continuous-time expression of the dynamical system assumed in this paper. It is similar to the one introduced in Boettner and Boers (2022) and Morr and Boers (2024). This expression differs slightly from the more common variation of this model, where the Wiener process dW(t) is replaced by an OU process here denoted as U(t).
  For Equation (11) the reviewer might refer to our inclusion of the 0.5 shift in the exponent. This is used to improve the accuracy of the integral discretization.
  Definition of symbols used will be addressed.
  “The context of the AMOC tipping problem is also poorly covered, with inappropriate
  
  references, wrong terminology (e.g. in the title, this is no 'Bayesian stability analysis' of the AMOC).”
  Answer : We will add some context regarding current evidence and understanding surrounding a potential AMOC tipping point. We will also welcome any other suggestions the reviewer may have.
  The reference “(Boettner and Boers, 2022)” in line 15 of the introduction has been removed. If there are other inappropriate or missing references we would appreciate it if the reviewer could kindly point them out.
  Since early warning signals are related to stability, our time-dependent model provides an evolution of the stability of the system. The framework detailed in this paper thus performs Bayesian analysis of the AMOC stability. Of course, one limitation is that we assume the evolution of the autocorrelation parameter is linear. While our model is efficient at identifying whether or not there have been changes in stability, it is likely not an accurate representation of the actual evolution of the stability. In a revision we will make this limitation more clear and update the title to “Assessing AMOC stability using a Bayesian nested time-dependent autoregressive model ”.
  “There is no critical evaluation of the time series in Figure 4, e.g. it does not even have units on the y-axis.”
  Answer : The cumulative surface melt across years, based on the CWG melt stack from Trusel et al. (2018) is used here to reflect the total freshwater forcing from GrIS. CWG surface melt is directly linked to GrIS runoff which is known to be one major component in the possible destabilization of the AMOC.
  We will add details on this in the manuscript, and update the y-axis appropriately.
  “The input of freshwater by the Greenland Ice Sheet has been so small over this period that a response of the AMOC is questionable.”
  Answer : While the freshwater by the GrIS was small prior to 1990, it has increased non-linearly since then, as shown by Trusel et al., (2018) or Horhold et al., (2023). Moreover recent studies suggest that the acceleration of GrIS melting over the past decades may have already-observable impact on the AMOC strength (Devilliers et al.,2024), (Martin and Biastoch 2023), (Castro de la Guardia et al., 2015).
  “The results on the DO time series are already in Myrvoll-Nilsen et al., (2024) and cannot be understood here without consulting that paper (which data, etc.?).”
  Answer : The demonstration of our model on DO time series is meant to serve as a benchmark to compare with other approaches on a real data example, in addition to the other two tests assessing the accuracy of our model fitted onto simulated nested AR(1) processes and simulated tipping processes respectively.
  The results from the approach of Myrvoll-Nilsen et al.,(2024) are shown in Table 2, as are the results of Rypdal (2016) and Boers (2018). While there is no ground truth regarding which (if any) DO events are bifurcation-induced and should therefore exhibit EWS, they present a classical example of real tipping points and have been extensively studied in the literature. They therefore provide a natural and informative case study for comparing against other approaches. Our results appear to more or less corroborate previous results. In a revised manuscript we will provide more extensive descriptions of this experiment.
  “The results for the AMOC fingerprint are in terms of application the only new results. These are poorly described and one would at least expect a comparison with other methods.”
  Answer : In terms of application, this paper focuses indeed solely on the AMOC. However, the paper also introduces new methodology using a time-dependent nested AR(1) process that accounts for biases arising from structured external variability, addressing a core limitation of Myrvoll-Nilsen (2024). The methodology introduced is robust and general enough to be applied to other climate systems prone to tipping, therefore it constitutes another greatly significant result of this paper.
  We do provide a comparison in Table 3 with the model introduced in Myrvoll-Nilsen (2024) and show that using a forcing response as detrending our new model is able to detect EWS while the previous one is not. Furthermore we mention at the end of Section 3 that the results of this study are consistent with the ones of Boers (2021) and Ditlevsen and Ditlevsen (2023). We will add more discussion on the comparison in the revised manuscript.
  “I would recommend to the authors to add the AMOC fingerprint example in the Myrvoll-Nilsen et al., (2024) paper.”
  Answer : The study presented here indeed builds upon the hierarchical Bayesian framework developed in Myrvoll-Nilsen et al., (2024). However, the methodology differs significantly from that paper. As stated in the introduction, regular AR(1) processes (including the time-dependent version presented in Myrvoll-Nilsen et al., (2024)) do not account for structured external variability. On the other hand, this new nested AR(1) process can account for such biases. The revised version of the paper will make sure to emphasize the important differences between these two methodologies.
  
  Citation: https://doi.org/10.5194/egusphere-2025-2461-AC1
RC2:
'Comment on egusphere-2025-2461', Anonymous Referee #2, 01 Jul 2025

General comments:

The Atlantic Meridional Overturning Circulation (AMOC) is a key climate tipping element. The author claims that they proposed a robust and general statistical model based on a second-order autoregressive process featuring time-dependent parameters. These parameters separately account for the statistical changes arising from increased external variability and destabilization. By applying the model to an AMOC fingerprint proxy, the author detected statistically significant early warning signals (EWS) of declining AMOC stability and an approaching tipping point. This manuscript is well written, with clear descriptions, an appropriate length, and high-quality figures. The analysis is appropriate. It represents a valuable contribution to the field. I recommend accepting it after some minor revisions.
Minor comments:

1. Some sentences in the manuscript are too long in length; consider revising them to improve readability. (e.g. Line 5-7)
2. Line 30, add missing letter “s” in “called early-warning signal (EWSs)”
3. Line 148, are 500 independent simulations enough?
4. The subscripts in the captions of Fig. 2(a) and (c) appear to be incorrect, please check.
5. Line 191-192, the citation style for references needs to be modified. It is recommended to consolidate them into a single bracket.
6. There is an unexplained discrepancy: the abstract mentions “a second-order autoregressive process with time-dependent parameter”, but Section 3.1 exclusively discusses “time-dependent nested AR(1) model”. What’s the relationship between the AR(2) model and the time-dependent nested AR(1) model? This should be clarified.

Citation: https://doi.org/10.5194/egusphere-2025-2461-RC2
- AC2:
  'Reply on RC2', Luc Hallali, 15 Jul 2025
  
  “General comments:
  
  The Atlantic Meridional Overturning Circulation (AMOC) is a key climate tipping element. The author claims that they proposed a robust and general statistical model based on a second-order autoregressive process featuring time-dependent parameters. These parameters separately account for the statistical changes arising from increased external variability and destabilization. By applying the model to an AMOC fingerprint proxy, the author detected statistically significant early warning signals (EWS) of declining AMOC stability and an approaching tipping point. This manuscript is well written, with clear descriptions, an appropriate length, and high-quality figures. The analysis is appropriate. It represents a valuable contribution to the field. I recommend accepting it after some minor revisions.”
  Answer : We thank the referee for the constructive review. The comments will be addressed point-by-point below.
  "1. Some sentences in the manuscript are too long in length; consider revising them to improve readability. (e.g. Line 5-7)”
  Answer : We will revise the manuscript to make it more concise.
  “2. Line 30, add missing letter “s” in “called early-warning signal (EWSs)””
  Answer : Correction will be added.
  “3. Line 148, are 500 independent simulations enough?”
  Answer : For each test, Root Mean Squared error (RMSE) has been plotted against the number of independent simulations. Results show that the RMSE values tend to converge around 300 independent simulations. Therefore we believe that 500 independent simulations is enough.
  “4. The subscripts in the captions of Fig. 2(a) and (c) appear to be incorrect, please check.”
  Answer : Correction will be added.
  “5. Line 191-192, the citation style for references needs to be modified. It is recommended to consolidate them into a single bracket.”
  Answer : Correction will be added.
  “6. There is an unexplained discrepancy: the abstract mentions “a second-order autoregressive process with time-dependent parameter”, but Section 3.1 exclusively discusses “time-dependent nested AR(1) model”. What’s the relationship between the AR(2) model and the time-dependent nested AR(1) model? This should be clarified.”
  Answer : We mention in line 98 that Morr and Boers (2024) show that this nested AR(1) process is equivalent to an AR(2). To improve clarity, we will consistently use “nested time-dependent AR(1) model” in the revised manuscript.
  
  Citation: https://doi.org/10.5194/egusphere-2025-2461-AC2
  - RC4: 'Reply on AC2', Anonymous Referee #2, 31 Jul 2025
    
    So, has the revised manuscript been uploaded? I can't see any track changes.
    
    Citation: https://doi.org/10.5194/egusphere-2025-2461-RC4
RC3:
'Comment on egusphere-2025-2461', Anonymous Referee #3, 09 Jul 2025

This study proposed a nested time-dependent autoregressive model to investigate the early warning signals (EWS) of AMOC, following the work of Myrvoll-Nilsen et al. (2024). The proposed model was tested to find the EWS. The results showed that it can indeed better find warning signals. But some issues need to be addressed before considering for publication.
1)The math symbols are different in the formula and the text. For example, line 49 and formula (4). Line 65 and formula (9). Besides, what does the k_f denote in formula (9)?
2)Line 193. Please explain why 0.95 is used as a detection of EWS. In Table 2, which events are real occurrences. If this information is provided, it may be easier to understand the disadvantage of the proposed model.
3)Line 196. 2 should be Table 2? Line 214. 1971 should be 1871?
4)In this study, different detrending strategies were applied. Table 3, Figures 5 and 6 show that the results seem to depend on the detrending strategies. Then, do you think which strategy should be used in finding the EWS?

Citation: https://doi.org/10.5194/egusphere-2025-2461-RC3
- AC3: 'Reply on RC3', Luc Hallali, 15 Jul 2025
  
  “This study proposed a nested time-dependent autoregressive model to investigate the early warning signals (EWS) of AMOC, following the work of Myrvoll-Nilsen et al. (2024). The proposed model was tested to find the EWS. The results showed that it can indeed better find warning signals. But some issues need to be addressed before considering for publication.”
  Answer : We appreciate this helpful review. The comments will be addressed point-by-point below.
  “1)The math symbols are different in the formula and the text. For example, line 49 and formula (4). Line 65 and formula (9). Besides, what does the k_f denote in formula (9)?”
  Answer : The notation mistakes will be corrected.
  k_f is a scaling parameter used in computing the forcing response. Its definition will be added to the revised manuscript.
  “2)Line 193. Please explain why 0.95 is used as a detection of EWS. In Table 2, which events are real occurrences. If this information is provided, it may be easier to understand the disadvantage of the proposed model.”
  Answer : The reason why 0.95 was chosen is because that is the conventional threshold for statistical significance. It is ultimately arbitrary, and we will add a comment on this in the revised manuscript on why this threshold was chosen.
  Table 2 presents the results of testing our model on real data, from the Dansgaard-Oeschger (DO) events. It is debated which of these events (if any) are actually caused by bifurcations, hence there is no known truth to compare to. However, the model appears to corroborate previous results, lending credence to our model relative to existing approaches.
  “3)Line 196. 2 should be Table 2? Line 214. 1971 should be 1871?”
  Answer : This will be corrected.
  “4)In this study, different detrending strategies were applied. Table 3, Figures 5 and 6 show that the results seem to depend on the detrending strategies. Then, do you think which strategy should be used in finding the EWS?”
  Answer : While the results might differ between the detrending strategies, they all agree on the presence of early warning signals. Generally, one should select the detrending strategy that is most appropriate for the problem at hand. If forcing data is available, and physically meaningful, it would often be beneficial to incorporate that into the model, over the more basic detrending strategies.
  
  Citation: https://doi.org/10.5194/egusphere-2025-2461-AC3

Luc Hallali, Eirik Myrvoll-Nilsen, and Christian L. E. Franzke

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Short summary

We present an alternative statistical methodology to detect whether the Atlantic Ocean’s circulation system is approaching a tipping point. Our approach separates natural variability from real early warning signals of tipping, reducing false alarms. When applied to proxy of Atlantic Ocean’s circulation strength , we found significant signs that the system is ongoing destabilization . This suggests it may be approaching a tipping point, which could have major impacts on global climate patterns.


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