An efficient approach for inverting rock exhumation from thermochronologic age-elevation relationship

Tian, Yuntao; Pan, Lili; Zhang, Guihong; Yao, Xinbo

doi:https://doi.org/10.5194/egusphere-2023-2119

Preprints

https://doi.org/10.5194/egusphere-2023-2119

Preprints

09 Oct 2023

| 09 Oct 2023

An efficient approach for inverting rock exhumation from thermochronologic age-elevation relationship

Yuntao Tian, Lili Pan, Guihong Zhang, and Xinbo Yao

Abstract. This study implements the least-squares inversion method for solving the exhumation history from thermochornologic age-elevation relationship (AER) based on the linear equation among exhumation rate, thermochronologic age and total exhumation from the closure depth to the Earth surface. 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. Lessons learned from the experiments include that (i) the modern geothermal gradient can be used for constraining the initial geothermal model, (ii) a relatively higher a priori exhumation rate would lead to systematically lower inversion results, and vice versa, (iii) the variance of the a priori exhumation rate controls the variation of the inverted exhumation history, (iv) the choice of time interval length should be optimized for resolving the potential temporal changes in exhumation. Putting together these findings, we propose a new stepwise inversion modeling strategy for optimizing the model parameters to mitigate the model dependencies on the initial parameters. Finally, we use three examples of different exhumation rates and histories for method demonstration. It is shown that our new modelling strategy produces geologically reasonable exhumation histories and geothermal gradients that are consistent with both the observed AER and modern geothermal data. The code and data used in this work is available in GitHub (https://github.com/yuntao-github/code4modelAER).

Received: 15 Sep 2023 – Discussion started: 09 Oct 2023

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Journal article(s) based on this preprint

26 Mar 2024

An efficient approach for inverting rock exhumation from thermochronologic age–elevation relationship

Yuntao Tian, Lili Pan, Guihong Zhang, and Xinbo Yao

Earth Surf. Dynam., 12, 477–492, https://doi.org/10.5194/esurf-12-477-2024,https://doi.org/10.5194/esurf-12-477-2024, 2024

Short summary

Yuntao Tian, Lili Pan, Guihong Zhang, and Xinbo Yao

Interactive discussion

Status: closed

RC1:
'Comment on egusphere-2023-2119', Gilby Jepson, 20 Nov 2023

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.

Citation: https://doi.org/10.5194/egusphere-2023-2119-RC1
- AC1: 'Reply on RC1', Yuntao Tian, 05 Dec 2023
  
  Dear Editor and Reviewers,
  Many thanks for your editorial work and the constructive and/or critical comments from two reviewers. We have carefully revised the manuscript based on the comments received, and also provided a detailed response to the comments and suggestions. Please find the line-by-line responses from the attachment of this reply.
  Best regards,
  Yuntao
  
  Citation: https://doi.org/10.5194/egusphere-2023-2119-AC1
RC2:
'Comment on egusphere-2023-2119', Christoph Glotzbach, 28 Nov 2023
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:

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.

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.

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?

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?
Citation: https://doi.org/10.5194/egusphere-2023-2119-RC2
- AC2: 'Reply on RC2', Yuntao Tian, 05 Dec 2023
  
  Dear Editor and Reviewers,
  Many thanks for your editorial work and the constructive and/or critical comments from two reviewers. We have carefully revised the manuscript based on the comments received, and also provided a detailed response to the comments and suggestions. Please find the line-by-line responses from the attachment of this reply.
  Best regards,
  Yuntao
  
  Citation: https://doi.org/10.5194/egusphere-2023-2119-AC2
EC1:
'Comment on egusphere-2023-2119', Richard Gloaguen, 13 Dec 2023

Dear Authors
In further iterations of your amendment it is very important that you highlight 1- the novelty of this approach and 2- the added value (what makes it necessary to have a new model and how you tackle this challenge). I would also recommend that you specify how your model differ from the work published by Fox et al., published in 2014.
Kind regards

Citation: https://doi.org/10.5194/egusphere-2023-2119-EC1
- AC3: 'Reply on EC1', Yuntao Tian, 15 Dec 2023
  
  Dear Editor,
  Thanks for your comments.
  As acknowledged in the manuscript, the equations are the same as Fox et al. (2014). However, the modeling strategies are different. The code by Fox et al (2014) uses a specific and assumed initial geothermal gradient and a prior exhumation rate for the inversion. However, given the dependence of modeled results on these initial parameters (see figures 4-7 of our manuscript), using specific initial geothermal and prior exhumation parameters may lead to biased modeling results. To address the issue, we treat the initial geothermal gradient and prior exhumation rate as unknowns, and run a batch of models to find the best that fits both the modern geothermal gradient and the observed thermochonometric data (see the misfit equation 17). See Figure 8 for our new modeling guideline.
  Following the comments from Christoph Glotzbach, we tested our modelling method using four synthetic datasets generated by Pecube. The synthetic models are modified from the figure 9 of Braun et al. (2012). It is found that the inverted exhumation histories from the synthetic datasets match the whole picture of the “truth”, although the temporal changes in the magnitude of the exhumation are slightly underestimated. For details, see section 8 of the revised manuscript. This is also new in terms of testing the method.
  Lastly, we provide a MATLAB-based GUI application that would be more user-friendly to geologists not familiar with raw scripts. See the step-by-step user manual at https://github.com/yuntao-github/A2E_app
  Let me know if further information is needed.
  Yuntao
  
  Citation: https://doi.org/10.5194/egusphere-2023-2119-AC3

Interactive discussion

Status: closed

RC1:
'Comment on egusphere-2023-2119', Gilby Jepson, 20 Nov 2023

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.

Citation: https://doi.org/10.5194/egusphere-2023-2119-RC1
- AC1: 'Reply on RC1', Yuntao Tian, 05 Dec 2023
  
  Dear Editor and Reviewers,
  Many thanks for your editorial work and the constructive and/or critical comments from two reviewers. We have carefully revised the manuscript based on the comments received, and also provided a detailed response to the comments and suggestions. Please find the line-by-line responses from the attachment of this reply.
  Best regards,
  Yuntao
  
  Citation: https://doi.org/10.5194/egusphere-2023-2119-AC1
RC2:
'Comment on egusphere-2023-2119', Christoph Glotzbach, 28 Nov 2023
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:

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.

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.

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?

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?
Citation: https://doi.org/10.5194/egusphere-2023-2119-RC2
- AC2: 'Reply on RC2', Yuntao Tian, 05 Dec 2023
  
  Dear Editor and Reviewers,
  Many thanks for your editorial work and the constructive and/or critical comments from two reviewers. We have carefully revised the manuscript based on the comments received, and also provided a detailed response to the comments and suggestions. Please find the line-by-line responses from the attachment of this reply.
  Best regards,
  Yuntao
  
  Citation: https://doi.org/10.5194/egusphere-2023-2119-AC2
EC1:
'Comment on egusphere-2023-2119', Richard Gloaguen, 13 Dec 2023

Dear Authors
In further iterations of your amendment it is very important that you highlight 1- the novelty of this approach and 2- the added value (what makes it necessary to have a new model and how you tackle this challenge). I would also recommend that you specify how your model differ from the work published by Fox et al., published in 2014.
Kind regards

Citation: https://doi.org/10.5194/egusphere-2023-2119-EC1
- AC3: 'Reply on EC1', Yuntao Tian, 15 Dec 2023
  
  Dear Editor,
  Thanks for your comments.
  As acknowledged in the manuscript, the equations are the same as Fox et al. (2014). However, the modeling strategies are different. The code by Fox et al (2014) uses a specific and assumed initial geothermal gradient and a prior exhumation rate for the inversion. However, given the dependence of modeled results on these initial parameters (see figures 4-7 of our manuscript), using specific initial geothermal and prior exhumation parameters may lead to biased modeling results. To address the issue, we treat the initial geothermal gradient and prior exhumation rate as unknowns, and run a batch of models to find the best that fits both the modern geothermal gradient and the observed thermochonometric data (see the misfit equation 17). See Figure 8 for our new modeling guideline.
  Following the comments from Christoph Glotzbach, we tested our modelling method using four synthetic datasets generated by Pecube. The synthetic models are modified from the figure 9 of Braun et al. (2012). It is found that the inverted exhumation histories from the synthetic datasets match the whole picture of the “truth”, although the temporal changes in the magnitude of the exhumation are slightly underestimated. For details, see section 8 of the revised manuscript. This is also new in terms of testing the method.
  Lastly, we provide a MATLAB-based GUI application that would be more user-friendly to geologists not familiar with raw scripts. See the step-by-step user manual at https://github.com/yuntao-github/A2E_app
  Let me know if further information is needed.
  Yuntao
  
  Citation: https://doi.org/10.5194/egusphere-2023-2119-AC3

Peer review completion

AR: Author's response | RR: Referee report | ED: Editor decision | EF: Editorial file upload

AR by Yuntao Tian on behalf of the Authors (09 Dec 2023) Author's response Author's tracked changes

EF by Polina Shvedko (12 Dec 2023) Manuscript

ED: Referee Nomination & Report Request started (13 Dec 2023) by Richard Gloaguen

RR by Christoph Glotzbach (28 Dec 2023)

Suggestions for revision or reasons for rejection

Dear Authors,
Thanks for taking the time and implement the suggested corrections and consider my comments in the revised manuscript. The manuscript did improve significantly, but would benefit from a few more changes as outlined below:
1. You state that your inversion is following the approach of Fox et al. (2014). It is, however, still difficult to judge if the equations 1 to 6 are new/similar to Fox et al. (2014). Please state for each equation if this is similar or not to Fox et al. (2014). I give you an example for equation 1: ‘Following Fox et al. (2014) a set of correlated…’ or equation 2: ‘Fox et al. (2014) constructed these two matrices as…’. Please do so in each equation or state if different to Fox et al. (2014).
2. I do like the synthetic examples and would like you to add a discussion section focusing on the ability to retrieve the exhumation history with the approach and potential conclusions for real studies. Your model is performing well, predicting the thermochron data, but clearly the exhumation history is non-unique – there are numerous exhumation histories that fit the data perfectly even outside your posterior prediction. This is very important to highlight and does mean that we should not overinterpret the inversion results and clearly the methodological approach is not able to reproduce instantaneous increases/decreases in exhumation.

Technical corrections:
Line 17-19: Change to ‘… investigates the usage of the least-squares inversion…’.
Line 49: Change to ‘…acquired by noble gas…’.
Line 50: Give examples or most common minerals used, e.g. apatite and zircon fission track.
Line 57-66: This is partly a repetition of line 45-47, merge the two sections and remove repetition.
Line 67: Replace ‘convincing’ with ‘reliable’.
Line 77: Change to ‘… estimation reliable…’.
Line 100: Change to ‘…assuming Gaussian…’.
Line 108-109: Change to ‘…Previous studies used the analytical age uncertainties (Fox et al., 2014; Fox et al., 2015) or a constant value (Jiao et al., 2017; Stalder et al., 2020) for the constructions of the data covariance.
Line 192: Change to ‘…(between 4.3-2.0 km) suggests a present-day closure depth of -3.3 (y-intercept of fitted slope, Fig. 2b).
Line 198: Change to ‘… the cumulative exhumation since …’.
Line 279: Please clarify what you mean with ‘becomes stable’.
Line 280-282: Please rewrite, sentence is not clear. Do you mean that the uncertainty of the posterior exhumation rate should be roughly similar to that of the input age data?
Line 285-288: This does need some more explanation. I guess the choice of variance should be selected based on the knowledge available. For instance, if you do have an AER with apparent exhumation ranging from 0 to 2 km/Myr, the variance in the prior exhumation rate should cover the full range with some extra variance to allow the model to explore the full range of possible scenarios. The goal cannot be to make the prior variance small to have a nicely constrained output since that could exclude complex models that may be correct.

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RR by Gilby Jepson (17 Jan 2024)

Suggestions for revision or reasons for rejection

This is my second review of “An efficient approach for inverting rock exhumation from thermochronologic age-elevation relationship”. I have found that my major concerns have been addressed by the authors. I leave a series of minor comments which the authors may find useful in preparing the manuscript for publication.

Lines 22-27: This is a long run-on sentence and could be broken into several shorter sentences.

Line 30: Suggestion: “Finally, we performed a method demonstration using…”/

Line 31: Suggestion: “We show..”.

Line 34: “Geologically reasonable” can include a very broad array of possible results including erroneous ones. Could this be rephrased to say that it gives good agreement with the published studies?

Lines 40-41: “..surface is important information for better understanding many geological problems..”. Suggestion: “..is important for understanding many geological promblems..”.

Line 67: Suggestion: “To produce a reasonable estimate of the exhumation history..”.

Line 89: “Exhumation” is capitalized.

Line 108: Suggestion: “It is worth noting that..”.

Line 135: should “Helium” and “Argon” be capitalized?

Line 136: Suggestion: “..in associated mineral phases..”. Also, does Dodson cover the annealing equation?

Line 139: suggestion: “..see review by..”.

Line 188: “ Suggestion: “..for a demonstration of the method..”.
Line 194: should be “annealing temperature” as opposed to “closure temperature”. Also, I don’t think the authors have defined the “AFT” acronym thus far.

Line 197-198: This sentence is a little confusing as written, can it be clarified?

Line 218: Suggestion: “interval” instead of “span”.

Line 218-219: “..age date points..”. Do the authors mean “age elevation”?

Line 219: Suggestion: “..are less well resolved..”.

Line 220: “..shown by the nearly equal resolution values for the two phases..”. Can this be clarified? Does “equal” mean poorly resolved?

Line 230-231: I think this sentence could be smithed a little better, it’s confusing as read with a bit of word repetition.

Line 264: Suggestion: “..has a systematic influence on the..”.

Line 270: Sometimes the authors have a pace between the “>” sign and the number and sometimes they do not.

Line 276: should “posterior” be in italics here?

Line 291-292: Suggestion: “..is an important task for inverting..”

Line 345: is that the right way to cite a figure from another paper?

Line 355: I think “c” and “d” should be in italics.

Line 387-389: Perhaps the authors mention this earlier, but why would the variations in exhumation be underestimated from the synthetic datasets?

Line 444: Suggestion: “..vary between..”.

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ED: Publish subject to minor revisions (review by editor) (18 Jan 2024) by Richard Gloaguen

AR by Yuntao Tian on behalf of the Authors (23 Jan 2024) Author's response Author's tracked changes Manuscript

ED: Publish as is (05 Feb 2024) by Richard Gloaguen

ED: Publish as is (07 Feb 2024) by Andreas Lang (Editor)

AR by Yuntao Tian on behalf of the Authors (13 Feb 2024) Manuscript

Journal article(s) based on this preprint

26 Mar 2024

An efficient approach for inverting rock exhumation from thermochronologic age–elevation relationship

Yuntao Tian, Lili Pan, Guihong Zhang, and Xinbo Yao

Earth Surf. Dynam., 12, 477–492, https://doi.org/10.5194/esurf-12-477-2024,https://doi.org/10.5194/esurf-12-477-2024, 2024

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Yuntao Tian, Lili Pan, Guihong Zhang, and Xinbo Yao

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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.

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Rock exhumation from the Earth interior to the surface is important information for better understanding many geological problems, ranging from mountain building and its decay, to resource and hydrocarbon evaluation and exploration. We propose a new stepwise inverse modeling strategy for optimizing the model parameters to mitigate the model dependences on the initial parameters that are required to simulate the rock exhumation processes.


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