In the manuscript "Uncertainty quantification for overshoots of tipping thresholds", the authors investigate how uncertainty on the parameters of a dynamical system yielding a saddle-node bifurcation (namely the tipping threshold and restoring force) leads to uncertainty in the outcome of overshoot scenarios, in which the forcing is higher than the tipping threshold for a finite time.



Notably, the results draw upon the previously established inverse-square law. These are sound and certainly publishable, fitting well the scope of Earth System Dynamics. I however have a series of comments (labelled c#) that I would like to have considered before recommending publication of the manuscript (along with some more taste-dependant suggestions labelled s# at the end)

c1: The way that probabilities are computed from the inverse square law is not announced early enough in the manuscript. If I understand, the method is explained by the prior to last paragraph in Appendix A3. I may have missed it, but the reader is never directed to appendix A3 when mentioning "theoretical probabilistic boundaries" or "boundary levels derived from the theory". This especially brings confusion in the end of section 2.3 from line 317 when reading "we can make use of the theory to derive the probability of tipping", this time via a different method. I would suggest to have this part of appendix A3 in the main text, for example the first time that the method is used. 

c2: As much as possible, I would find it clearer if "prior distribution" and "posterior distribution" were systematically used, rather than "prior" and "posterior" alone.

c3: L19 - I find the first sentence weird as an introductory statement for this manuscript. Also, I see that climate tipping points are indeed mentioned in the  "17 sustainable development goals" but, as written now, it reads as if tipping points themselves are a development goal.

c4: L28 - "Assumed" is incorrect as it is no assumption but some approximation. I believe "thought" would fit better.

c5: L73-74 - As far as I can understand, the focus on bifurcation-induced tipping in this study is total. How is the previous paragraph on different tipping mechanisms related to the study? If it is because safe overshoot phenomena are inherently a rate-induced effect, it should be clearer. I believe that this paragraph could be reformulated to motivate that rate-induced effects can also impact tipping behaviours, with safe-overshoot being an example.

c6: L96 - Is it true that this parameter is always the inverse timescale? For the simple systems used I can see it, as 1/a0 multiplies the left hand side. However, for more complicated systems it is less clear, depending on the definition of the timescale of a system. Maybe "related to the inverse timescale" would be more correct?

c7: L111 - Here and in general, it would be nice to refer to the appendix when appendix content is involved (and also, not only to the equations of the appendix) with sentences like "more about that in appendix #". Also true for example for the forcing (A1) used, to which I see no mention in the main text.

c8: Bayesian inference should be appendix A6.

c9: Fig 1.c - It would be nice to mention somewhere in the text that overshoot occurs in the whole plane, and indeed every time such a figure is presented. It is somehow trivial because of the y-axis name, but it can be missed in a first read.

c10: L176 - I would not describe these systems as identical. They have a different bifurcation diagram, but a saddle-node occurring at the same forcing, and the same stable state at zero forcing. It implies that rate-induced effects are more important for one than the other, and the way it impacts the inverse square law (and especially the fact that it would work better for one than another, as one is further away from steady state) is not trivial to me.

c11: L224(and after) - the dynamical variable in the Cessi model is not the change in salinity flux, but the meridional salinity gradient. It is very confusing as the forcing applied to the model is actually the meridional gradient of freshwater flux (for example in L259).

c12: L305 - I do not see the asymmetry of the orange band in Fig 8.a, I must have missed something?

s1: L5 - I would replace "and avoid tipping" by "without tipping" (it can sound like the fact of overshooting in itself makes a tipping less probable).

s2: L7 - I would add "However, in the real world ...".

s3: L10 - I would write "separating safe from unsafe overshoot", for the same reason as s1.

s4: L16 - "in this conceptual model, ..." would be more straightforward than "In our study, ...".

s5: L123 - I would find it more natural to have "We now illustrate ..." at the beginning of the next paragraph.

s6: L179 - I would remove "quite".

The manuscript “Uncertainty quantification for overshoots of tipping thresholds” by Lux-Gottschalk & Ritchie looks at how uncertainties affect the probability of tipping. Motivated by the uncertainty of real-world models, they first conduct probabilistic assessments of how uncertainty within parameters can lead to uncertainty in the tipping response to overshoot trajectories. Their general framework is then highlighted in a box model of the Atlantic Meridional Overturning Circulation which has a similar underlying structure. Overall, this is good and interesting work, and it fits well into the scope of ESD. Additionally, this work has the potential to be the basis for more complex scenarios. Below I list some comments that should be considered before publication.

Specific comments:

As I was reading the manuscript, initially I had a hard time following what was happening as I was flipping back and forth to the appendix from the main text. While I understand why the authors put together the appendix, I think it would help the strength of the paper to present or at least describe in more detail many of the methods and derivation of the tipping probability from the inverse square law, the models, etc. It may be worth considering just putting some of the appendix within the main text rather than it being separate as it will help with readability. Another alternative is to better reference the appendix when needed.

I think there needs to be a clearer description on how uncertainty/overshoot trajectories/tipping probabilities all relate, and the motivation needs to be more clearly depicted in the introduction.

L19: There is a misspaced apostrophe. Also, this is the start of your introduction, and it reads awkwardly/a bit unclear. I suggest combining the first two sentences and making it more clear how tipping points relate to the sustainable development goals/climate action.

L20-21: Tipping events are not only in environmental areas or climate subsystems. While I recognize it is environmental for the context of this work, it is misleading to say tipping only occurs for environmental conditions.

L28: Replace the word assumed with proposed/suggested, etc.

L55-73: While I appreciate the details on the three main tipping mechanisms and a nod/summary to the work that has been conducted, I am not sure if an entire paragraph is needed. How does it relate to your work? You mention that you consider various profiles that have different impacts on the overshoot of the critical bifurcation threshold value. If there is a clearer connection regarding each tipping mechanism to overshooting, then make those connections. Otherwise, you do not need all of this.

L109: Bayesian statistics is referenced incorrectly. There doesn’t seem to be a Materials and Methods section? Then in the appendices themselves, you have two A1’s.

L119: Arguably, the biggest uncertainty to consider is the location of the tipping threshold. Why? You seem to just drop this at the start of 2.1. It would be helpful to say where this is coming from or foreshadow it.

L137-138: If we have some initial estimate of the uncertainty for the location of the tipping threshold, then a tipping probability based on this distribution can be assigned for any forcing profile. Make it more clear how the tipping probability is assigned as currently it is unclear the how.

L222: It would be helpful in the beginning of this section to make sure to highlight what is going to be relevant for overshooting scenarios/uncertainty.

L222: I would remind readers that the results/limitations of this section are based on low-dimensional box model, or you can put this into the conclusions. 

L234: Do results substantially change if the advective timescale is not fixed at 70 years? How was 70 exactly chosen?

General comment: Legends in figures for colors/lines would make the figures easier to decipher without needing to read the entire figure caption.

General comment: Are the equations supposed to be centered or is this just an artifact of the preprint (L94, L320)?

Technical comments:

L3: The sentence beginning with “One common mechanism for tipping to occur is via forcing a nonlinear system…” is awkward. The start of the sentence should be changed to make this sentence flow better.

L29-30: There is unnecessary commas.

L49: Specify the “those” in: … in addition to those, …

L81-82: Say datasets back-to-back, I would suggest rewording.

L120: I suggest losing the parentheses about calling it threshold later and just work that into this sentence. If you are changing notation/how you will refer to things, don’t make it a small note.

L124: This first half of the first sentence reads a little bit awkwardly. Eliminate at least one of the commas if possible.

L136: We have seen that for a given overshoot forcing profile… refer to where we just saw this profile.

L387-390: This paragraph feels out of place. It should perhaps be earlier in the conclusion.