“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.

“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.

“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.