In “An efficient approach for inverting rock exhumation from thermochronologic age-elevation relationship”, Tian and co-authors present a least-squares inversion method for solving exhumation history from thermochronologic age-elevation relationships. Their approach seems sound identifying that the a priori erosion rate, time-interval, and geothermal gradient are all important controls on erosion rates, similar to those that have come before (e.g., Braun, 2003, van der Beek et al., 2010 or Fox et al., 2014). Certainly, it is suitable for ESurf. However, I have some queries regarding how the model handles certain parameters.

Dependence on time interval length: The authors spend some time discussing the suitability to time step interval. This makes sense. With larger steps, one can average into larger changes in erosion rates. I am wondering if selecting a suitable interval would not this be a function of the uncertainty of the underlying chronometer? E.g., if using largely systems with high uncertainty (e.g., AFT) then only a larger bin-size should be used. Whereas systems with higher precision (Ar-Ar, He) can resolve finer variations?

Testing examples: I am curious as to why the background erosion rate for the Dhauladar range could be considered so low for so long? Okay, 0.2-0.4 mm/yr is maybe not "low" in other orogenic systems. However, this is the Himalaya. Additionally, in Deeken et al., 2011, they are considering that the region has experienced high erosion rates for an extremely long time, facilitating channel flow. Did the authors consider a high a priori erosion rate in this system? Regardless, given the importance of the prior erosion rate to the predicted erosion rate, I believe the authors should discuss possible approaches.

The KTB borehole seems to be most poorly performing of the 3 tests. Not horribly, but something to notice. Specifically, the model seemly predicting ages that are systematically younger than their observed ages. Particularly, getting into the lower T chronometers (AHe and AFT). Could this be a function of performing this study in a borehole where temperatures do increase with depth? Further, what is the a priori erosion rate and variance?

Detailed comments:

Lines 20-22: “Modelling experiments demonstrate the significant and systematic influence of initial geothermal model, the a priori exhumation rate and the time interval length on the a posterior exhumation history”. Here the authors state this correlation in the abstract, yet in their demonstrated examples, there is little discussion on these topics.

Lines 41-42: “..ranging from mountain building (e.g., Zeitler et al., 2001; Whipp Jr. et al., 2007; Cao et al., 2022) and its decay (e.g., House et al., 2001..”. suggestion: "orogenic growth and decay".

Lines 51-52: “such as mica Ar-Ar, apatite, zircon and titanite fission-track and (U-Th)/He analyses..”. Given the wide variety of accessory phases that these decay systems are found in, I would simply omit all the possible mineral examples. As there is 3 mineral phases for FT, none for U-Th and one for Ar-Ar.

Lines 64-65: “patiotemporal changes in exhumation (Sutherland et al., 2009; Herman et al., 2013; Fox et al., 2014; Willett et al., 2020)”. Please include: “van der Beek, P. and Schildgen, T. F.: Short Communication: age2exhume – A Matlab script to calculate steady-state vertical exhumation rates from thermochronologic ages in regional datasets and application to the Himalaya, EGUsphere [preprint], https://doi.org/10.5194/egusphere-2022-888, 2022.”

Line 75: “Because both the underground geothermal field..”. suggestion: “As both the subsurface geothermal field..”. Also, the authors use “underground” a bit. I think “subsurface” might be the better descriptor.

Line 78-80: “reliable estimates of exhumation rates require solving exhumation itself, together with the evolution of other influencing factors”. This sentence needs some clarification it is difficult to see what the authors are driving at here.

Line 200: “Same as used in Fox et al. (2014)..”. Suggestion: “following protocol outlined in”.

Line 206: “..results reveal an abrupt triple-four-fold increase of exhumation rate..”. a “tripe-four-fold” seems a little confusing. Just pick either 3-fold or 4-fold.

Lines 251-255: “Worth noting is that 252 the models using relatively lower (16-20 C/km, Figs. 4a-4b) and higher (30-34 oC/km, Figs. 4e-4f) initial geothermal gradients yield relatively worse misfits (>1) than those using medium initial gradients (22-26 C/km) (Figs. 3 and 4c-4d), suggesting that the modern geothermal gradient can be used as a constraint for the initial geothermal model. In the above section the authors suggest that the present day geothermal gradient is ~40 C/km (line 203). Given that the authors find lower misfit in the "medium initial gradients", how does that fit with the claim "modern geothermal gradient can be used as a constraint for the initial geothermal gradient?

Line 284: “We proposed to run a set of models..”. Here and throughout, the authors use “proposed” however, it is unclear what they mean by ‘proposed’. In my view, if one is proposing to do something, they crucially, have not done it. However, the authors have, presumably, done some of the things they are proposing. Thus, I would just have a think if that is specifically the word they would like to use.

Lines 286-287: “As to the a priori variance of erosion rates, we propose to use a relative uncertainty of 30-70% of the mean value.” Here I am confused. Do the authors propose that users should set relative uncertainty to 30-70% or are they say that they set relative uncertainty to this value within their models? Given that the authors discuss variance. Is there a mean value that gives better fits? It is likely system dependent, but perhaps making a statement is useful here.

Line 293: “Using a large time length..”. missing “interval”.

Line 314: “these aren’t really “new modelling strategies”, as many of the previously mentioned modelling systems involve the same parameters. Thus, I would suggest making this more specific. modelling guidelines?

Line 315: “Putting together the lessons learned from the..”. Following the modelling protocol outlined above.

Line 316: “modeling strategy develops..”. is developed.

Line 361: “..Dharladar range in the central Himalayas..”. northwestern himalaya?

Line 366-367: “AER slope regression suggests an increase in apparent erosion rates from ~0.2 km/Myr to ~2.8 km/Myr at ~3.7-6.4 Ma (Deeken et al., 2011).” Looking at the Deeken et al., age-elevation plots, the inflection point seems a little ambiguous. Does your approach document significant change if you use the "no clear break in slope' approach?

Line 386: “The KTB apatite, zircon and titanite (U-Th)/He (AHe, ZHe and THe) and AFT age data vary largely between 85-50 Ma.” Looking at figure 9, it seems that the models respond better to the less precise systems (AFT) then compared to the more precise systems of AHe. Do the authors also observe this? If so, why would this be the case?

I thank the authors for their time.

Dear Authors,

The topic of the manuscript, estimating robust estimates of exhumation from thermochronological data, is an important method to study the spatial and temporal relations of tectonic, climate and erosion in various tectonic settings. Numerous papers have been published on this, ranging from simple geometric models, thermal history modelling to 1D to 3D thermal-kinematic models. The contribution of Yuntao Tian et al. is based on the 1D inversion of thermochronological data from age-elevation profiles sharing the same exhumation history. They do largely use the approach of Fox et al. (2014) with, as I understand, only very slight modifications of the data covariance matrix. It is important that this is stated clearly, and that the novelty of their contribution is easy to digest from the reader. It does largely read as a new method, but in most cases simply reprints equations/ideas formerly stated by Fox et al. (2014).

Please see my scientific comments and technical corrections for more details:

1. In the abstract, it is unclear what is the outcome of your model application. You state that ‘lessons learned from the experiments’, and afterwards, you propose a modelling strategy. One could get the impression that you have explored the difficulties but have not applied it.
2. Your approach is largely based on (a reprint of) that from Fox et al. (2014). Please make sure to clearly communicate that you are using his approach with only very slight modifications. It is often not directly recognizable if your equations are new or have been defined elsewhere. In fact, it would be easier to recognize your contribution if you put everything out of the manuscript (in the supplement) that is not new and just focus on your contribution and refer to Fox et al. (2014) for the method.
3. You applied your suggested modelling strategy to three examples and yielded ‘consistent’ results. This is good, but what is the benefit of running these models if you could just reproduce what others predicted before with ‘simpler’ methods? Please clearly state what is new and why we should use this modelling strategy and also what is different from what Fox et al. (2014) suggested?
4. It would be very nice to extend the applications to a synthetic data with changing exhumation rate and pronounced topography. In Braun et al. (2012) you will find two datasets (Fig. 3 and Fig. 9) that you could use to show the performance of your dataset in comparison to 1D thermal-kinematic modelling. Whatever the outcome is, you could discuss the limitations of some of your model parameterizations, such as the closure depth.

Technical corrections:

Line 24: It is not clear what you mean with ‘lower inversion results’, please be specific.

Line 40: Say something like ‘Quantifying rock exhumation…’.

Line 48: Give references to modelling tools such as Pecube (Braun 2003; Braun et al. 2012) and Glide (Fox et al. 2014).

Line 53-58: Mention that this approach is only applicable for constant cooling rates.

Line 59: Change to ‘Many analytical …cooling history from thermochronological data.’

Line 59-67: You may also want to mention the Fourier approach for correction AER from Glotzbach et al. (2015).

Line 78-80: Change to ‘…Ehlers and Farley, 2003). Estimating reliable exhumation rates requires to account for temporal variations of the thermal field caused by changes in the thermal and kinematic boundary conditions.’

Line 86-87: I would delete the last sentence.

Line 90-122: Do mention at the beginning that you are using the approach from Fox et al. (2014) with slight modifications, e.g. covariance.

Line 126-128: Replace with ‘The latter can be determined modelling the temperature of the crust using a 1D thermal-kinematic model, which accounts for heat conduction, advection and production…’

Line 149: Please give a reference for this equation, I guess Mancktelow and Grasemann (1997) and Fox et al. (2014).

Line 169: Replace fitness with ‘difference between observed and predicted ages weighted by the observed analytical uncertainty’.

Line 176: Replace ‘data fitness’ with ‘model result’.

Line 190: Delete ‘change’.

Line 227: Give a reference for this equation.

Line 251-255: Simplify and say that ‘…suing the present-day geothermal gradient (38.9 °C/km) in the misfit calculations does exclude higher and lower prior geothermal gradients of >30 and <20…’

Line 355: What do you mean with ‘latter’?

Line 375-376: Your estimate of the most recent exhumation is much lower compared to the raw interpretation, is that due to the overestimation of exhumation due to the topographic perturbation of isotherms?