the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
An analytical solution for the exhumation of an orogenic wedge and a comparison with thermochronology data
Elco Luijendijk
Leo Benard
Sarah Louis
Christoph von Hagke
Jonas Kley
Abstract. Thermochronology data is key for quantifying the exhumation history and dynamics of mountain belts. Here we present a new analytical solution for the steadystate exhumation of an orogenic wedge that undergoes transport along a basal detachment, uniform internal deformation, basal and frontal accretion. The solution predicts an increase in exhumation towards the interior of the wedge, with the rate of increase dependent on the degree of internal deformation. Application of the solution to a cross section in the Himalayas shows that in spite of its simplicity the solution provides a good fit to thermochronology data, with a coefficient of determination (R^{2}) of 0.75. This implies that, although the solution does not capture the effects of individual faults and folds, at a large scale deformation can be described by uniform compression and transport. The results also imply that this part of the Himalayas may be in steadystate. The equations presented here can be used to quantify exhumation, deformation and shortening rates in mature orogens that are in steadystate.
Elco Luijendijk et al.
Status: closed

RC1: 'Comment on se202122', Anonymous Referee #1, 27 Apr 2021
Dear Authors,
The authors do present an analytical approach to model the particle motion in an compressional orogen and estimate related thermochronological ages. The model does allow to efficiently change the boundary condition and therefore adjust to fit observed thermochronological data and make inferences about the importance of e.g. compression, detachment velocities and frontal and basal accretion.
Although I personally like the approach and acknowledge the effort the authors did to implement this model, I do not like the focus of the manuscript given the simplicity of the their model.
For further information see my scientific comments:
Scientific comments
 There is a quite similar approach as the one presented here for accretionary wedges (not including ‘compression’) (Batt et 2001), but this should be mentioned in the introduction and the similarities and differences should be discussed in detail. I would also like to see both approaches applied and visually compared, e.g. for an example without compression.
 I am not very happy how the authors simplify the thermal model, using a constant geothermal gradient (lateral and temporally constant), especially because they apply their model to the Himalayas , which is characterised by significant amount of horizontal and vertical rock motion. This does strongly perturb the thermal field, both laterally and also temporally. Although the authors show that they can fit the general age trend, that does not mean the model is correct.
 Similar to the thermal model, the calculation of thermochronological ages is oversimplified and should not be used for the purpose stated. There are numerous of recent studies demonstrating the strong impact of different cooling rates and radiation damage on thermochronological ages. In this regard, exhumation of the Himalayas does include movement above ramps and flats that should result in cooling rates to vary temporally and different amount of radiation damage to accumulate. I do not see any reason why not using available annealing and diffusion models and include them in this modelling approach (even if the thermal field is set constant, which I do not like).
 The oversimplified thermal model and calculation of thermochronological ages does not allow to use this model to be applied to a real dataset. Instead if might be used to study the general trend in thermochronological ages in active orogens, and study the different age trends related to the importance of compression, frontal and basal accretion. If this model, however, should be applied to real datasets, it is mandatory to include a more exact treatment of the thermal field and stateoftheart calculation of thermochronological ages.
 The authors state that they fit the thermochron data along the Kuru Chu cross section better than McQuarrie and Ehlers (2015). The general trend is fitted by the authors, but it seems that McQuarrie and Ehlers (2015) do a better job in fitting the details of data between 0 to 80 km better. I would also like to ask the authors to contact McQuarrie and Ehlers to get the raw data instead of taken the values from a figure. Anyway getting a better fit with a simple model that does neglect part of the complexity of the geological setting, thermal structure and analytical method does not necessarily mean their model is correct. Indeed it is usually easier to fit data with a simple model, since it can be adjusted easily to optimize fitting. This is often time consuming if all available geological constraints and sophisticated thermalkinematic models are used. However, even complex models do become computational more efficient and can be adjusted with efficient search algorithms and running models in parallel. Please add a little paragraph and explain the applicability of the different model setups and limitation associated with it.
 The authors state that because they can reproduce the general trend in ages in the studied profile with a steadystate assumption, the Himalaya might be in steadystate. Showing that one model is fitting the data does not mean that others model do not fit the data as well or even better. Since the presented model is only working in steadystate the authors cannot prove that steadystate is the best model to fit the data. What there model however can be used for, is to study the relative importance of e.g. frontal and basal accretion, detachment transport and compression, which the authors correctly discussed.
In summary, I would suggest the authors to change the focus of their manuscript and state the drawbacks and benefits of such an approach. I do see the real application of such an approach in testing largescale boundary conditions that can afterwards be used in more sophisticated model setups. I do not see that this approach can ‘compete’ or ‘replace’ approaches like that of McQuarrie and Ehlers (2015). I hope you find my comments and suggestions helpful and fing more details in the technical corrections.
Technical corrections:
Line 3: State what you mean with exhumation, amount or rate?
Line 20: Change to ‘fault blocks’.
Line 2324: That is not correct a similar approach for accretionary wedges (frontal and basal accretion) has been published already in 2001 by Batt et al. Please cite their work and discuss the similarities and differences with your model.
Line 91: I guess it should be v_{xc }= …
Line 99: State that you also use Eq. 7 and use the normalized compression velocity.
Line 101 and 103: It is not clear why you are defining the horizontal and vertical components in Eq. 14, but not all equations are used in Eq. 8 and 9.
Line 106107: Should be ‘…internal deformation and basal and frontal accretion…’, this is what the figure captions says. In this case, shouldn’t the velocity vectors decrease towards the interior of the model since the accretion is only occurring along the tip of the wedge and vectors should show divergence!? The integral of the vectors along x should be similar along the wedge, or not?
Line 137139: It is not clear why you do provide the ‘simplified solution’, please explain and if there are no good reasons, just do not show. Can you provide the difference between the analytical and the numerical solutions as figures, that would help the reader to check what you have wrote here (instead of showing the difference between the two provided analytical solutions). Maybe you only show the difference between the simplified analytical (if you want to keep showing it) against numerical since the other one are nearly identical.
Line 165: A uniform geothermal gradient is really not what can be used in such setting. Depending on the model setup, and especially the horizontal and vertical rates the thermal field will be strongly perturbed. For instance do have a look on the cited manuscript from McQuarrie and Ehlers (2015) Fig. 1a).
Line 168: This approach is also much too simplistic especially since you do have significant horizontal particle motion and cooling rates of samples might vary significantly throughout their exhumation. This and the knowledge of the importance of the amount of radiation damage on He diffusion kinetics in apatite and zircon (e.g. Flowers et al. 2009; Guenthner et al. 2013) has to be taken into account for the transformation of particle trajectories, cooling histories and final thermochron ages.
Line 208209: Please provide details and justification how you calculate the misfit, what is the MAE? Your model indeed fits the general trend of the data, however, the details are better fitted by McQuarrie and Ehlers (2015), with the exception of data >100 km away from the frontal tip. You may want to mention this.
Line 209210: Please ask the authors to provide the data, fortunately we do not need to digitize data from figures anymore!!!
Line 212: Please not that a model that fits data better is not automatically correct. The model of McQuarrie and Ehlers (2015) is based on numerous independent geological information that has been incorporated in a much more realistic model, but complex model. It is always easier to fit a simple model to data compared to complex model, but does that mean the simple model is more realistic, I would say not! There is a tradeoff between complexity and fit to the data, your model and the one from McQuarrie and Ehlers (2015) are endmembers in this relation. From both model setups we can learn and they do have their eligibility. Please add a bit of this in your discussion and not just say your model is better…
Line 225: That orogens are in steadystate has been described already by others (e.g. Willett and Brandon 2003, Bernet et al. 2001) and that was highly discussed and even often this has been disproved by additional data and more indepth interpretation (e.g. Michel et al. 2019). Since the thermal field of the crust is slow in responding to changes in the boundary conditions the resulting thermochronological data are often ‘smooth’ and the real/complex exhumation history is difficult to constrain. Fitting a steadystate pattern through data is therefore often easier compared to finding the real/complex exhumation history.
Line 239: What kind of steadystate do you mean, flux, exhumation, topography?
Figures:
Fig. 1: Can you draw a few more particle path and continue them to the surface. From the figure it is also not clear what is above the wedge, water, air? Also add ‘erosion’ below the precipitation on the surface.
Fig. 3: In caption change to ‘Panel shows…’.
Fig. 4: The heading of the panels are wrong, please correct.
Fig. 5: Use ‘5 Ma’ instead of ‘5 Ma’. Change panel d to Error between numerical and analytical solution.
Fig. 6: Change panel d to Error between numerical and analytical solution.
Fig. 7: Is the scale correct, I thought you speak somewhere from 200 km profile length. Looks shorter?
Fig. 8: What does MAE mean?
Citation: https://doi.org/10.5194/se202122RC1 
AC1: 'Reply to RC1', Elco Luijendijk, 12 Sep 2021
Note: Our replies to the comments by Reviewer 1 are shown below each comment in italics.
Dear Authors,
The authors do present an analytical approach to model the particle motion in an compressional orogen and estimate related thermochronological ages. The model does allow to efficiently change the boundary condition and therefore adjust to fit observed thermochronological data and make inferences about the importance of e.g. compression, detachment velocities and frontal and basal accretion.
Although I personally like the approach and acknowledge the effort the authors did to implement this model, I do not like the focus of the manuscript given the simplicity of the their model.
For further information see my scientific comments:
Scientific comments
 There is a quite similar approach as the one presented here for accretionary wedges (not including ‘compression’) (Batt et 2001), but this should be mentioned in the introduction and the similarities and differences should be discussed in detail. I would also like to see both approaches applied and visually compared, e.g. for an example without compression.
Reply: We thank the reviewer for pointing us to this paper. The approach referred to by the reviewer is equation 5 in Batt et al. (2001, https://doi.org/10.1029/2001JB000288), with a correction published later (https://doi.org/10.1029/2003JB002897). This equation provides vertical velocity inside a wedge as a function of a predefined erosion rate, accretion rate, and slope of the base of the wedge. This is quite different from our approach in that Batt et al. (2001) use erosion rate at the surface as an input, whereas our equation predicts erosion rates. The Batt et al. (2001) equation therefore cannot used to predict exhumation rates or thermochronometer ages without prior knowledge of the erosion rates. And of course erosion rates are themselves usually calculated using thermochronometer data. For this reason we cannot include a quantitative comparison between our model and the Batt et al. (2001) approach in the manuscript. However, we did add a brief discussion of this model to the introduction section of the revised version of our manuscript.
 I am not very happy how the authors simplify the thermal model, using a constant geothermal gradient (lateral and temporally constant), especially because they apply their model to the Himalayas , which is characterised by significant amount of horizontal and vertical rock motion. This does strongly perturb the thermal field, both laterally and also temporally. Although the authors show that they can fit the general age trend, that does not mean the model is correct.
Reply: We agree that the thermal model that we used is highly simplified. We have changed the thermal model and have updated the model code to include a numerical solution of the steadystate heat advection & conduction equation. The code uses the calculated velocity field as an input (eqs. 8, 9 in the manuscript), along with published thermal parameters and boundary conditions by Coutand et al. (2014) and McQuarrie and Ehlers (2015). The calculated steadystate thermal field is then used in combination with particle tracks calculated using our new equation to calculate the thermal history of particles. The new thermal model, in combination with the new thermochronology model / equations results in a similar fit to the AFT and ZFT data. However, the new thermochronology model results in higher resetting temperatures for the MAr thermochronometer. The steadystate model cannot fit both datasets well at the same time. Potential reasons for this are discussed in the results section of the revised manuscript.
 Similar to the thermal model, the calculation of thermochronological ages is oversimplified and should not be used for the purpose stated. There are numerous of recent studies demonstrating the strong impact of different cooling rates and radiation damage on thermochronological ages. In this regard, exhumation of the Himalayas does include movement above ramps and flats that should result in cooling rates to vary temporally and different amount of radiation damage to accumulate. I do not see any reason why not using available annealing and diffusion models and include them in this modelling approach (even if the thermal field is set constant, which I do not like).
Reply: We agree that use of a fixed closure temperatures was a bit overly simplistic and have upgraded the model code to calculate cooling ages using an approach by Fox et al. (2014, https://doi.org/10.5194/esurf2472014), which is based on the Dodson (1973, https://doi.org/10.1007/BF00373790) equation. We acknowledge that this approach, at least for the AFT method is still a simplification. However, the rapid cooling experienced by the samples makes linear cooling models such as used by Fox et al. (2014) relatively accurate in this case because the samples spend relatively little time in the partial annealing zone where differences in annealing rates are important. In addition, we would like to reserve the use of more sophisticated thermochronology models for follow up manuscript because this would require significantly more work on the model code and much more space in the manuscript for additional discussions.
 The oversimplified thermal model and calculation of thermochronological ages does not allow to use this model to be applied to a real dataset. Instead if might be used to study the general trend in thermochronological ages in active orogens, and study the different age trends related to the importance of compression, frontal and basal accretion. If this model, however, should be applied to real datasets, it is mandatory to include a more exact treatment of the thermal field and stateoftheart calculation of thermochronological ages.
Reply: This point was addressed by using a more realistic thermal model and thermochronometer model.
 The authors state that they fit the thermochron data along the Kuru Chu cross section better than McQuarrie and Ehlers (2015). The general trend is fitted by the authors, but it seems that McQuarrie and Ehlers (2015) do a better job in fitting the details of data between 0 to 80 km better. I would also like to ask the authors to contact McQuarrie and Ehlers to get the raw data instead of taken the values from a figure. Anyway getting a better fit with a simple model that does neglect part of the complexity of the geological setting, thermal structure and analytical method does not necessarily mean their model is correct. Indeed it is usually easier to fit data with a simple model, since it can be adjusted easily to optimize fitting. This is often time consuming if all available geological constraints and sophisticated thermalkinematic models are used. However, even complex models do become computational more efficient and can be adjusted with efficient search algorithms and running models in parallel. Please add a little paragraph and explain the applicability of the different model setups and limitation associated with it.
Reply: We did request the original data from McQuarrie. We did not receive a reply previously, but have repeated the request and McQuarrie has now kindly supplied the data.
Following Ockham's razor we feel that given two model codes the simplest code that fits the data should be preferred. We are not saying that it would not be possible to fit the data better with a more complicated model. However, at present our simple model performs better in this particular crosssection than the published models. This may or may not be the case in other parts of the Himalayas or other mountain belts, which we feel is something that would be worth exploring in the future.
 The authors state that because they can reproduce the general trend in ages in the studied profile with a steadystate assumption, the Himalaya might be in steadystate. Showing that one model is fitting the data does not mean that others model do not fit the data as well or even better. Since the presented model is only working in steadystate the authors cannot prove that steadystate is the best model to fit the data. What there model however can be used for, is to study the relative importance of e.g. frontal and basal accretion, detachment transport and compression, which the authors correctly discussed.
Reply: We agree that we cannot test transient models with our approach. However, previous attempts with transient models do not show a better fit than our steadystate approach. We would therefore like to still state that there is a strong possibility that this part of the Himalayas is in steadystate. We did change the wording of this claim in the manuscript to state that "this crosssection in the Himalayas may be in steadystate".
In summary, I would suggest the authors to change the focus of their manuscript and state the drawbacks and benefits of such an approach. I do see the real application of such an approach in testing largescale boundary conditions that can afterwards be used in more sophisticated model setups. I do not see that this approach can ‘compete’ or ‘replace’ approaches like that of McQuarrie and Ehlers (2015). I hope you find my comments and suggestions helpful and fing more details in the technical corrections.
Reply: We do not intend to state that this approach should replace numerical model approaches, and hope we have made this more clear in the revised version of the manuscript. We felt that a simple model like the one we presented was missing from the literature and could be a useful additional tool to study the dynamics of mountain belts. We did add a paragraph to the discussion section that discusses the benefits and drawbacks of the approach.
Technical corrections:
Line 3: State what you mean with exhumation, amount or rate?
Reply: We mean exhumation rate here, now explicitly stated in the manuscript.
Line 20: Change to ‘fault blocks’.
Reply: Ok
Line 2324: That is not correct a similar approach for accretionary wedges (frontal and basal accretion) has been published already in 2001 by Batt et al. Please cite their work and discuss the similarities and differences with your model.
Reply: We did now add a brief discussion of this study, see reply to point 1.
Line 91: I guess it should be vxc = …
Reply: correct, thanks for noticing this.
Line 99: State that you also use Eq. 7 and use the normalized compression velocity.
Reply: Ok
Line 101 and 103: It is not clear why you are defining the horizontal and vertical components in Eq. 14, but not all equations are used in Eq. 8 and 9.
Reply: This is because vyc was replaced with the expression for vyc from eq. 4. We have rewritten the equation to correct this.
Line 106107: Should be ‘…internal deformation and basal and frontal accretion…’, this is what the figure captions says. In this case, shouldn’t the velocity vectors decrease towards the interior of the model since the accretion is only occurring along the tip of the wedge and vectors should show divergence!? The integral of the vectors along x should be similar along the wedge, or not?
Reply: The horizontal and vertical compression vectors decrease towards the tip of the wedge and are directed to the left (Fig. 4a). The accretion velocity is constant along the wedge and directed to the right (fig 4b). The combination of the two shown in 4c results in horizontal velocities that cancel out at a distance of 75 km from the tip of the wedge and a total velocity that first decreases with distance to the tip and then increases again as it passes the point where the two components cancel out. Velocity integrated over over x varies in our approach.
Line 137139: It is not clear why you do provide the ‘simplified solution’, please explain and if there are no good reasons, just do not show. Can you provide the difference between the analytical and the numerical solutions as figures, that would help the reader to check what you have wrote here (instead of showing the difference between the two provided analytical solutions). Maybe you only show the difference between the simplified analytical (if you want to keep showing it) against numerical since the other one are nearly identical.
Reply: We have removed the simplified solution from the manuscript.
Line 165: A uniform geothermal gradient is really not what can be used in such setting. Depending on the model setup, and especially the horizontal and vertical rates the thermal field will be strongly perturbed. For instance do have a look on the cited manuscript from McQuarrie and Ehlers (2015) Fig. 1a).
Reply: We agree and have revised the thermal model. See previous replies.
Line 168: This approach is also much too simplistic especially since you do have significant horizontal particle motion and cooling rates of samples might vary significantly throughout their exhumation. This and the knowledge of the importance of the amount of radiation damage on He diffusion kinetics in apatite and zircon (e.g. Flowers et al. 2009; Guenthner et al. 2013) has to be taken into account for the transformation of particle trajectories, cooling histories and final thermochron ages.
Reply: We agree and have revised the thermochronometer models, see previous replies.
Line 208209: Please provide details and justification how you calculate the misfit, what is the MAE? Your model indeed fits the general trend of the data, however, the details are better fitted by McQuarrie and Ehlers (2015), with the exception of data >100 km away from the frontal tip. You may want to mention this.
Reply: We used two metrics to calculate the model fit, R2 and MAE. R2 is the coefficient of determination, which we now clarified better in the main text and the caption. MAE is the mean absolute error, which is defined in line 179, and is now also mentioned in the caption. We have added a more extensive discussion of the model fit of our and McQuarrie and Ehlers (2015) model to the revised manuscript.
Line 209210: Please ask the authors to provide the data, fortunately we do not need to digitize data from figures anymore!!!
Reply: We did request the data and had not received a reply before submitted the previous version of the manuscript. However, we have sought contact again and have now received the data.
Line 212: Please not that a model that fits data better is not automatically correct. The model of McQuarrie and Ehlers (2015) is based on numerous independent geological information that has been incorporated in a much more realistic model, but complex model. It is always easier to fit a simple model to data compared to complex model, but does that mean the simple model is more realistic, I would say not! There is a tradeoff between complexity and fit to the data, your model and the one from McQuarrie and Ehlers (2015) are endmembers in this relation. From both model setups we can learn and they do have their eligibility. Please add a bit of this in your discussion and not just say your model is better…
Reply: We do not claim that our model is more correct. We did not use the term correct in the manuscript, and feel that correct is not a good term to describe models. If one follows Ockhams razor then a simple model that fits the data better should be preferred. We do agree that the model that we use may be overly simplistic in assuming uniform compression and transport. However, until a model comes along with a better fit to the data, we feel that the simple model should be preferred. We do think it is important to present an alternative model that provides a better fit to the thermochronological data, and let the reader judge, which is preferable.
Line 225: That orogens are in steadystate has been described already by others (e.g. Willett and Brandon 2003, Bernet et al. 2001) and that was highly discussed and even often this has been disproved by additional data and more indepth interpretation (e.g. Michel et al. 2019). Since the thermal field of the crust is slow in responding to changes in the boundary conditions the resulting thermochronological data are often ‘smooth’ and the real/complex exhumation history is difficult to constrain. Fitting a steadystate pattern through data is therefore often easier compared to finding the real/complex exhumation history.
Reply: In our humble opinion, whether or not the real exhumation would be more complex should be judged by the data. And for the time being, if more complex models in this particular crosssection result in poorer fit to data then these models should not be preferred. We do however agree with the reviewer that thermochronological data alone may not be able to resolve all complexities within the system. In the case of the Himalayas steadystate has also been suggested based on a compilation of cosmogenic nuclides and sediment yield data. Our analysis corroborates these findings.
Line 239: What kind of steadystate do you mean, flux, exhumation, topography?
Reply: We mean steadystate exhumation, and have changed the text accordingly.
Figures:
Fig. 1: Can you draw a few more particle path and continue them to the surface. From the figure it is also not clear what is above the wedge, water, air? Also add ‘erosion’ below the precipitation on the surface.
Reply: Thanks for this suggestion, we have modified the figure accordingly
Fig. 3: In caption change to ‘Panel shows…’.
Reply: Corrected, thanks for spotting this.
Fig. 4: The heading of the panels are wrong, please correct.
Reply: We have corrected this.
Fig. 5: Use ‘5 Ma’ instead of ‘5 Ma’. Change panel d to Error between numerical and analytical solution.
Reply: Ok
Fig. 6: Change panel d to Error between numerical and analytical solution.
Reply: See previous comment.
Fig. 7: Is the scale correct, I thought you speak somewhere from 200 km profile length. Looks shorter?
Reply: That is correct. The modelled crosssection is 200 km to avoid boundary effects on the results. However, the balanced crosssection shown here only covers the first part of the crosssection. We have added an explanation of this to the revised manuscript.
Fig. 8: What does MAE mean?
Reply: We have added an explanation of the mean absolute error to the caption.
Citation: https://doi.org/10.5194/se202122AC1

RC2: 'Comment on se202122', Anonymous Referee #2, 30 Apr 2021
Dear authors,
This work presents a new analytical solution for the steadystate exhumation of an orogenic wedge. The authors used the prediction of the timedepth path and a simple thermal model to predict lowtemperature thermochronologic ages that they compare with a cross section in eastern Himalaya. This comparison shows good fit between observations and predictions. This solution does not capture the effects of individual faults and folds, but the authors claim from this comparison that 1) in this example, a simple model is sufficient to explain the largescale observations, and 2) that here, the Himalaya may be in steadysate.
General comments
In general, this is well written and easy to read. I like the idea to predict lowtemperature thermochronologic ages from a 2D model of a critical wedge evolution, but I have important remarks that cannot be eluded:
 You claim that this is the first attempt for such a study. I disagree, at least Batt et al., 2001 did a similar 2D critical wedge model to predict lowtemperature ages. This model is not presented in the present paper from Luijendijk et al., they do not explain in what their model is different, and also, there is no comparison of these two similar models. You should present the Batt et al., 2001 model, and discuss it in terms of both construction and results.
 My other major concern is about the thermal model used in the present model. This is not clear to me, if the thermal model resolve the in 2D the heat equation with vertical and lateral advection. This is important because in the model, particles are advected, and thus heat also! For a similar particle path model, taking in account or not heat advection will lead to timetemperature path that can be very different, and that can impact the lowT age prediction! So, you need to precise the thermal model (and thus give the values of the complete set of heat parameters). And if the thermal model does not take in account the heat advection, I would push to update the thermal model so that it takes it in account. Or show that there is no consequence in your time temperature path, and thus that it does not impact your age prediction.
 In the same type of comments, I am not very happy of the cooling models used to compute thermochronological ages. Here, you do not use published cooling models, but only a closure temperature. This is over simplistic, and it finally asks if your predictions can really be compared to observations! Maybe a way to prove that this works, this is to show that the cooling paths you compute (with a correct thermal model!) is similar to the cooling paths published in previous papers (i.e., Long et al., 2012?), or better, implement the published routines to compute correct thermochronologic ages from cooling histories.
 In the method section, I do not understand why you present a simplified solution that does not gives very accurate results. First, this is noise in the paper, and second, it decrease the confidence we could have in your other strong simplifications (temperature model, cooling model,…). I would remove this simplified solution, or explain why this is important.
 In the validation section, I do not understand how you set up your numerical model, i.e., what is the difference between the numerical model and the full analytical solution. You probably need to better explain (with an additional sketch?) your numerical model with the parameters used.
 And finally, in your application to a mountain belt, you need to explain with better arguments why you choose this section, because for instance the Central Nepal section also matches the arguments you present here. Also, the comparison between the restored crosssection published and your prism model differs importantly in some places. So you need to discuss the impact on predicted ages of your approximations in the geometry of the prism.
So, because of the strong simplifications you made, I am not sure that you really can compare predictions with a real dataset, and, more, that you can use this comparison to extract strong conclusions on how to interpret lowtemparature data and if the Himalayas are in steay state or not, specially wihtout moderating your sentences.
Specific comments
L5: precise where (approx. longitude for instance) the cross section is located in the Himalayas
L67: change “at a large scale deformation” to “deformation at a large scale” or “large scale deformation”?
Somewhere in the abstract, you need to specify taht your model is only in 2D and that it does not take in account any relief (that is important for comparison with real dataset).
L11 & L13 & L14: add “e.g.,” before the reference(s)
L24: and Batt et al., 2001 model? You need to say in what your approach differs from Batt et al., 2001!
L2526: Expend the idea present in this sentence as the comparison is not straight forward.
L91: vxc ?
L94: add a ref to the appendix where you explain how you defined these variables
L106107: This is not what there is in the caption of Fig. 4. Correct it.
L137138: Why showing the simplified solution if the full solution is a lot better? Does this solution give advantages when computing? If they are reasons to show this simplified solution, explain them, or remove the simplified solution from the paper.
L139: “The error increases with depth”. This part is a bit confusing: on the figs. 5 and 6, the colors representing the errors are inversed, they do not show the same between figs. 5 and 6. Please, to limit the confusion, put the same color bar.
Also, this sentence is true for the fig. 6d, but not for fig.5d where I see first a decrease of the error with depth, and then a rapid increase.
L140142: So, again, why showing the simplified solution if it does not permit to match well the full solution?
L148151: here, you give your arguments (good thermochronologic data coverage and high convergence rates) to justify that you choose this section. But if you take the Kathmandu/Trisuli or Sutlej region, it matches the same argument, so why didn’t you took these sections also? You need to give more arguments on the choice of your section.
L159: m m1, not m m1
L160: m m1 not m m1
L160: 200 km? from the fig 7, it looks closer to 100 kmlong than 200 kmlong. So what is wrong, the text or the legend of Fig. 7?
L160: add ref to fig. 7
L157160: You may describe and quantify the differences of the complex geometry of the wedge build from geological and geophysical data, and your triangular simplified wedge geometry. You then need to explain what are the consequences of these differences to your model results, i.e. where your results will be accurate, and where they will not.
I ask this question because the Long et al., 2012 crosssection show an important ramp of the MHT (representing the base of the wedge), also, your model does not take in account the important frontal ramp. Though, Van der Beek et al., 2006, Robert et al. 2009, 2011 have shown that this geometry of the base of the wedge has a huge impact on the thermochronological record. So, such a simplification risks to make inaccurate predictions. You really need to reinforce this section.
L162164: I think that the set of thermochronometers from the Kathmandu/Trisuli section is larger (See Herman et al., 2010 for instance). So why did you choose the Bhutan section and not the central Nepal section?
L165: 15°C km1 (also, generally, remove the space between numbers and the °C sign)
L165167: I do not really understand the thermal model.
 Do you update this thermal model through time while you are advecting your particles? Do you take in account lateral thermal advection? If yes, you need to describe this model precisely in the method section. If not, I do not understand how you will be able to compute accurate thermochronological ages as in your wedge model, you have a lateral component that is important.
 Do you have heat production (i.e., radioactivity) in your model?
 What are the thermal characteristics (i.e. thermal diffusivity) of the rocks you are advecting in the model? This is crucial for the thermal model, you need to give them.
L167172: this is a strange choice. If you want to predict thermochronological ages, this is better to use existing cooling models. This is not so hard because you can find most of the routines in the literature or on the Web. This approach is oversimplified!
L178180: I do not understand what you do here. What do you input in the downhill simplex algorithm? You need to explicitly explain what you are doing here in details.
Also, what is the MAE of the calculated thermochronological ages?
L181182: where are the results/comparison of these models?
L186: With such an ultrasimplified thermal model that does not take in account lateral heat advection (does it?), and an ultrasimplified cooling calculation, I am not sure this is relevant to compare your predictions to the data. Before to do that, you need to refine your thermal model and cooling models. Ask any quantitative thermochronologist about what is often a critical point when submitting papers for publication: there is almost always a strong debate around the thermal model/parameters used to interpret thermochronologic data/model. This is really a critical point.
L188: the exploration of the parameter space is very rough. In the introduction, you explained that you want to propose a model that does not need a lot of computing time and resources. So if this is the case (is it the case? You do not mention it until now), it would be easy and rapid to refine your parameter search with a smaller stepping.
L208: I would not say a better fit: you fit well the general trend, but not the details, and McQuarrie and Ehlers better explain the different local ages variations (because their model is based on numerous geological observation, and is thermally more complex), except for the farthest samples. You need to moderate your sentence, or better explain the comparison.
L209201: Ask the authors the source of their graph.
L2122013: You should change the sentence to “This suggest that, for this crosssection, at large scale, deformation of the wedge is represented by uniform deformation”, or for something like that. This is because your fit does not reproduce the details of the data because 1) your thermal/cooling model is oversimplified, and 2) you do not take in account relief.
L269, eq A5: There is a problem in this equation. A dx is missing (in place of the L?) in the left member.
Figures
Figure 2:
The font is to small, increase it (it is hard to read the name of the vectors); in the caption, change the last sentence to “The symbols are listed and described in Table 1”
Figure 3 & 4:
Does the length of the arrows mean something? If yes, it maybe needs a scalebar, or at least an explanation in the caption
Figure 5 & 6;
See my specific comment about the colorbar for panels d). Why using a ““ in the isochrones? It will simplify the fig if you remove this ““
Figure 7:
 a) the writings inside the fig. are not readable. Please increase the size of the font.
On this fig., you may mark the MHT (which is the base of your wedge) with a thicker line, so the reader will better see what approximation you are doing in your model.
On a), you also may add the thermochronologic data (on a mapview also?) you use in the Himalayan case.
You also may write in the caption what is the red triangle…
Figure 8:
Why the MAE is in a and ages in Ma? Cleaner if you gibe MAE in Ma.
Figure 10:
In caption, correct “calculated themrochronometer…” by “calculated theRmochronometer…”
Citation: https://doi.org/10.5194/se202122RC2 
AC2: 'Reply to RC2', Elco Luijendijk, 12 Sep 2021
Note: Our replies to the comments by Reviewer 2 are shown below each comment in italics
Dear authors,
This work presents a new analytical solution for the steadystate exhumation of an orogenic wedge. The authors used the prediction of the timedepth path and a simple thermal model to predict lowtemperature thermochronologic ages that they compare with a cross section in eastern Himalaya. This comparison shows good fit between observations and predictions. This solution does not capture the effects of individual faults and folds, but the authors claim from this comparison that 1) in this example, a simple model is sufficient to explain the largescale observations, and 2) that here, the Himalaya may be in steadysate.
General comments
In general, this is well written and easy to read. I like the idea to predict lowtemperature thermochronologic ages from a 2D model of a critical wedge evolution, but I have important remarks that cannot be eluded:
 You claim that this is the first attempt for such a study. I disagree, at least Batt et al., 2001 did a similar 2D critical wedge model to predict lowtemperature ages. This model is not presented in the present paper from Luijendijk et al., they do not explain in what their model is different, and also, there is no comparison of these two similar models. You should present the Batt et al., 2001 model, and discuss it in terms of both construction and results.
Reply: Thanks for pointing us to this paper. We have added it to the manuscript, see reply to the same comment by Reviewer 1.
 My other major concern is about the thermal model used in the present model. This is not clear to me, if the thermal model resolve the in 2D the heat equation with vertical and lateral advection. This is important because in the model, particles are advected, and thus heat also! For a similar particle path model, taking in account or not heat advection will lead to timetemperature path that can be very different, and that can impact the lowT age prediction! So, you need to precise the thermal model (and thus give the values of the complete set of heat parameters). And if the thermal model does not take in account the heat advection, I would push to update the thermal model so that it takes it in account. Or show that there is no consequence in your time temperature path, and thus that it does not impact your age prediction.
Reply: Agreed, we have improved the thermal model. See reply to the same comment by Reviewer 1.
 In the same type of comments, I am not very happy of the cooling models used to compute thermochronological ages. Here, you do not use published cooling models, but only a closure temperature. This is over simplistic, and it finally asks if your predictions can really be compared to observations! Maybe a way to prove that this works, this is to show that the cooling paths you compute (with a correct thermal model!) is similar to the cooling paths published in previous papers (i.e., Long et al., 2012?), or better, implement the published routines to compute correct thermochronologic ages from cooling histories.
Reply: Agreed. We have improved the thermochronology model. See reply to the same comment by Reviewer 1.
 In the method section, I do not understand why you present a simplified solution that does not gives very accurate results. First, this is noise in the paper, and second, it decrease the confidence we could have in your other strong simplifications (temperature model, cooling model,…). I would remove this simplified solution, or explain why this is important.
Reply: We have removed the simplified solution from the manuscript
 In the validation section, I do not understand how you set up your numerical model, i.e., what is the difference between the numerical model and the full analytical solution. You probably need to better explain (with an additional sketch?) your numerical model with the parameters used.
Reply: we have tried to better describe the numerical model in the revised manuscript. All it is is particle tracking based on velocity field calculated by eq. 8 & 9. We feel that devoting a sketch to this would probably be overkill.
 And finally, in your application to a mountain belt, you need to explain with better arguments why you choose this section, because for instance the Central Nepal section also matches the arguments you present here. Also, the comparison between the restored crosssection published and your prism model differs importantly in some places. So you need to discuss the impact on predicted ages of your approximations in the geometry of the prism.
Reply: We did or did not add a 2^{nd} xsection. It is true that other cross sections would also be suitable to test our model – even outside the Himalayas. We chose this section, as it has a high amount of highquality data, and is relatively wellstudied. Our point of this manuscript is not to criticize the existing model, but to present an alternative interpretation as a second endmember, if you wish. Generally it has been shown that cross sections are often only poorly constrained at depth. (In)famous examples exist e.g. from the Alpine foreland, or the Zagros mountains, where even based on highresolution data, different geometric solutions are possible. This is also true for the chosen cross section. The internal geometry of the wedge is not that well constrained. See for instance Coutand et al. (2014) who explore various different models for this that are all consistent with the relatively sparse geophysical data.
So, because of the strong simplifications you made, I am not sure that you really can compare predictions with a real dataset, and, more, that you can use this comparison to extract strong conclusions on how to interpret lowtemparature data and if the Himalayas are in steay state or not, specially wihtout moderating your sentences.
Reply: The revised thermal model and thermochronology model provide more realistic thermochronology ages that can be compared better with the data. We did moderate the sentences concerning steady state in the revised version of the manuscript.
Specific comments
L5: precise where (approx. longitude for instance) the cross section is located in the Himalayas
Reply: this is shown in the inset map in Fig. 7b
L67: change “at a large scale deformation” to “deformation at a large scale” or “large scale deformation”?
Reply: Ok
Somewhere in the abstract, you need to specify taht your model is only in 2D and that it does not take in account any relief (that is important for comparison with real dataset).
Reply: We have added a clarification to the abstract that the model is 2D. We have also added a brief discussion that the model does not take into account relief, which may affect the comparison with thermochronometers.
L11 & L13 & L14: add “e.g.,” before the reference(s)
Reply: Ok.
L24: and Batt et al., 2001 model? You need to say in what your approach differs from Batt et al., 2001!
Reply: See reply to the same comment by Reviewer 1.
L2526: Expend the idea present in this sentence as the comparison is not straight forward.
Reply: We explain that this reference solved the advective transport equation. Which is the same equation that one would need to solve for rock particle transport in an orogenic wedge. Although the boundary conditions are admittedly quite different.
L91: vxc ?
Reply: Ok. See reply to the same comment by Reviewer 1.
L94: add a ref to the appendix where you explain how you defined these variables
Reply: The variables are defined in the two following equations. Which we feel would be better here than in the appendix.
L106107: This is not what there is in the caption of Fig. 4. Correct it.
Reply: See reply to the same comment by Reviewer 1.
L137138: Why showing the simplified solution if the full solution is a lot better? Does this solution give advantages when computing? If they are reasons to show this simplified solution, explain them, or remove the simplified solution from the paper.
Reply: See reply to the same comment by Reviewer 1.
L139: “The error increases with depth”. This part is a bit confusing: on the figs. 5 and 6, the colors representing the errors are inversed, they do not show the same between figs. 5 and 6. Please, to limit the confusion, put the same color bar.
Reply: Agreed, we have used the same colorbar for the 2 figures.
Also, this sentence is true for the fig. 6d, but not for fig.5d where I see first a decrease of the error with depth, and then a rapid increase.
Reply: agreed. However, we have removed the simplified solution, so this panel is no longer in the revised manuscript.
L140142: So, again, why showing the simplified solution if it does not permit to match well the full solution?
See previous reply
L148151: here, you give your arguments (good thermochronologic data coverage and high convergence rates) to justify that you choose this section. But if you take the Kathmandu/Trisuli or Sutlej region, it matches the same argument, so why didn’t you took these sections also? You need to give more arguments on the choice of your section.
Reply: The main focus of the manuscript is the new analytical solution. We intended to keep the comparison to field data limited to one example. Testing it with other dataset is the scope of follow up work. We also do not wish to claim that this would be the best dataset to compare it to. We simply took a high quality published study with a wide range of thermochronometers and numerical crosssection balancing in the Himalayas that we were aware of.
L159: m m1, not m m1
Reply: Ok. Thanks for spotting this error.
L160: m m1 not m m1
Reply: Ok. Thanks for spotting this error.
L160: 200 km? from the fig 7, it looks closer to 100 kmlong than 200 kmlong. So what is wrong, the text or the legend of Fig. 7?
Reply: See reply to similar comment by reviewer 1.
L160: add ref to fig. 7
Reply: Ok
L157160: You may describe and quantify the differences of the complex geometry of the wedge build from geological and geophysical data, and your triangular simplified wedge geometry. You then need to explain what are the consequences of these differences to your model results, i.e. where your results will be accurate, and where they will not.
Reply: We would like to refrain from a detailed discussion of geological data and resulting expectations of where our model may be accurate or not, and would like to focus purely on a comparison of the model results with the data.
I ask this question because the Long et al., 2012 crosssection show an important ramp of the MHT (representing the base of the wedge), also, your model does not take in account the important frontal ramp. Though, Van der Beek et al., 2006, Robert et al. 2009, 2011 have shown that this geometry of the base of the wedge has a huge impact on the thermochronological record. So, such a simplification risks to make inaccurate predictions. You really need to reinforce this section.
Reply: We thank the reviewer for pointing us to the studies by Van der Beek et al. and Robert et al. It is true that in multiple occasions the presence or absence of a ramp impacts exhumation rates. An exciting result of our study is that the thermochronological data can be fitted without such a ramp, and at present the fit is better than in the existing model including complex geometry and localized strain. We would like to propose our model as a viable alternative.
L162164: I think that the set of thermochronometers from the Kathmandu/Trisuli section is larger (See Herman et al., 2010 for instance). So why did you choose the Bhutan section and not the central Nepal section?
Reply: There’s certainly a lot of data in the Herman et al. (2010) paper. However, we would like to refrain from testing additional crosssections in this manuscript. The choice for the Bhutan crosssection was based on the availability of several studies and balanced crosssections, which we felt made this an attractive target to compare with our more simple approach.
L165: 15°C km1 (also, generally, remove the space between numbers and the °C sign)
Reply: This was corrected in the revised version of the manuscript.
L165167: I do not really understand the thermal model.
 Do you update this thermal model through time while you are advecting your particles? Do you take in account lateral thermal advection? If yes, you need to describe this model precisely in the method section. If not, I do not understand how you will be able to compute accurate thermochronological ages as in your wedge model, you have a lateral component that is important.
 Do you have heat production (i.e., radioactivity) in your model?
 What are the thermal characteristics (i.e. thermal diffusivity) of the rocks you are advecting in the model? This is crucial for the thermal model, you need to give them.
Reply: we have completely revised the thermal model used in the manuscript. See also the replies to reviewer 1.
L167172: this is a strange choice. If you want to predict thermochronological ages, this is better to use existing cooling models. This is not so hard because you can find most of the routines in the literature or on the Web. This approach is oversimplified!
Reply: We have completely revised the thermochronology model used in the manuscript. See also the replies to reviewer 1.
L178180: I do not understand what you do here. What do you input in the downhill simplex algorithm? You need to explicitly explain what you are doing here in details.
Also, what is the MAE of the calculated thermochronological ages?
Reply: This is a widely used optimization / model calibration algorithm that simply minimizes the error between the model and the data by adjusting the model parameters. As this approach has been used so extensively in the past, we would like to refer to the cited references for more information.
The MAE is the mean absolute error as defined earlier in the manuscript. See also reply to the same question by reviewer 1.
L181182: where are the results/comparison of these models?
Reply: The model fit for these runs is shown in fig. 9
L186: With such an ultrasimplified thermal model that does not take in account lateral heat advection (does it?), and an ultrasimplified cooling calculation, I am not sure this is relevant to compare your predictions to the data. Before to do that, you need to refine your thermal model and cooling models. Ask any quantitative thermochronologist about what is often a critical point when submitting papers for publication: there is almost always a strong debate around the thermal model/parameters used to interpret thermochronologic data/model. This is really a critical point.
Reply: The thermochronology model was revised completely, see previous replies.
L188: the exploration of the parameter space is very rough. In the introduction, you explained that you want to propose a model that does not need a lot of computing time and resources. So if this is the case (is it the case? You do not mention it until now), it would be easy and rapid to refine your parameter search with a smaller stepping.
Reply: The exhumation model is obviously very fast since it only involves calculating two equations, which is done in a few milliseconds. However, the bottleneck in the new implementation is the time required for the numerical heatflow model, which adds significant computational expense. Nonetheless we have revised the parameter space search and have added more steps.
L208: I would not say a better fit: you fit well the general trend, but not the details, and McQuarrie and Ehlers better explain the different local ages variations (because their model is based on numerous geological observation, and is thermally more complex), except for the farthest samples. You need to moderate your sentence, or better explain the comparison.
Reply: The better fit refers to the better model fit statistic of the analytical solution. We would like to refrain from a more qualitative discussion of local errors or not and base the assessment of model performance on statistics. In the new version of the manuscript we state more explicitly that our solution is an alternative approach.
L209201: Ask the authors the source of their graph.
Reply: We have repeated our request and have now received the data and incorporated this in the manuscript.
L2122013: You should change the sentence to “This suggest that, for this crosssection, at large scale, deformation of the wedge is represented by uniform deformation”, or for something like that. This is because your fit does not reproduce the details of the data because 1) your thermal/cooling model is oversimplified, and 2) you do not take in account relief.
Reply: We agree that the model probably works best at large scale and have added the word at large scale to this sentence. However, in our opinion, in general the simplest model that fits the data the best should be the preferred model. Whether or not a model would come along in the future with more complexity that includes relief or the movement along single faults would fit the data better is speculative at this point. Note that the thermochronology model and thermal model have been improved for the revised manuscript.
L269, eq A5: There is a problem in this equation. A dx is missing (in place of the L?) in the left member.
Reply: True, that was a typo. Thanks for spotting this!
Figures
Figure 2:
The font is to small, increase it (it is hard to read the name of the vectors); in the caption, change the last sentence to “The symbols are listed and described in Table 1”
Reply: ok, we have implemented these suggestions
Figure 3 & 4:
Does the length of the arrows mean something? If yes, it maybe needs a scalebar, or at least an explanation in the caption
Reply: The length of the arrow scales with the velocity. The velocity is also visible in the colors, which is why we feel that adding a scale bar would be a bit double. However, we did add a statement to the caption that arrow size scales with velocity.
Figure 5 & 6;
See my specific comment about the colorbar for panels d). Why using a ““ in the isochrones? It will simplify the fig if you remove this ““
Reply: ok
Figure 7:
 a) the writings inside the fig. are not readable. Please increase the size of the font.
On this fig., you may mark the MHT (which is the base of your wedge) with a thicker line, so the reader will better see what approximation you are doing in your model.
On a), you also may add the thermochronologic data (on a mapview also?) you use in the Himalayan case.
You also may write in the caption what is the red triangle…
Reply: Ok. We prefer to keep the thermochronology data in the model result figures to not overcrowd this figure. We did add the sample locations to this figure.
Figure 8:
Why the MAE is in a and ages in Ma? Cleaner if you gibe MAE in Ma.
Reply: Agreed, we have converted this to Ma.
Figure 10:
In caption, correct “calculated themrochronometer…” by “calculated theRmochronometer…”
Reply: ok
Citation: https://doi.org/10.5194/se202122AC2

RC3: 'Comment on se202122', Anonymous Referee #3, 08 Jul 2021
Dear authors,
This article shows a new analytical solution to predict thermochronological dataset in an orogenic wedge. This simple model assumes a transport only accommodates by basal detachment. I appreciate all the details available on the different parameters implemented in this model.
The authors applied this model on a profile perpendicular to the Himalayan wedge. Results make in evidence a good correlation between predicted ages and observed ages using an uniform model without effects of individual tectonic structures. Authors concludes that the principal implication of these results is a good reproducibility of the thermochronological data with a simple model and a possible steady state evolution of the Himalayan orogenic wedge.
The manuscript is well written and we have all the details on the model and the analytical procedure, however major points have to be clarified and discussed, see my general comments.
General comments
1/ The main topic of the article have to be clarified l. 154155 " Note that the goal here is not to provide a detailed geological case study, but to demonstrate the use of the equation to calculate deformation and exhumation rates” but authors highlight in abstract and conclusion “The results also imply that this part of the Himalayas may be in steadystate.” (l.78) and “This indicates that the Himalayas may be in steadystate and that, at a large scale, the exhumation of mature mountain belts may be approximated by a relatively simple model of uniform and steadystate deformation, accretion and transport” (l. 239241). This conclusion may be true but it is not relevant for the Himalaya Mountain range using one cross section.
2/ Similar model have been developed by Batt et al. 2001 (JGR), a comparative study of the 2 models or at least a discussion on the main differences between the models must be developed. “However, to our knowledge no analytical solutions exist for the relation between deformation and exhumation of mountain belts.” (l.2324), this sentence is partially true and must be modified to show the specificity of this new model.
3/ Constant geothermal gradient is a big assumption in the models and it is not realistic according studies of Coutand et al., 2014 (JGR) and McQuarrie and Ehlers, 2015 (tectonics). This point and the impact on the model must be developed in the discussion. New kinematic models are not implemented (see Ault et al., 2019, tectonics for a synthesis) specify a closure temperature is possible and easier to implemented in the models but residence time in the Partial Annealing Zone (PAZ) and Partial Retention Zone (PRZ) need to be low to assume a closure temperature.
4/ Discussion of the results must be more detailed using differences between the predicted and observed ages and to discuss potential fault activity along the cross section. The comparison with study of McQuarrie and Ehlers, 2015 (tectonics) based on the better fit of the new model seems a bit complex. Differences between the 2 models are not specified in the text.
5/ This model applied on only one cross section in orogenic wedge of the Himalayas does not allow to provide strong conclusion on the Himalayas tectonic regime.
You can find detailed comments in the sections below:
Specific comments
L.5 Precise the location of the transect.
L.67 Remove this sentence, themochronological data not explained by the model can result of individual fault activity. See my general comment
L.78 This conclusion is not relevant if we consider ages do not fit with the model, see my previous comment.
L.17 I agree, one of the big advantages of the model presented is the low computational cost compare to other thermokinematic models.
L.2324 Please consider Batt et al. 2001 (JGR), see my general comment
L.91 Vxc ?
L.104 Fig.3 to be homogeneous in the text
L.106107 It is not the good caption for the figure
L.135139 Develop the simplified solution is necessary to the discussion ? If not you can remove this part in appendix and just comment the among of differences between the 2.
L.139: Considering the color bar on Fig.5d error decrease in the first kilometers.
L.143145 Is it possible to merge the 2 figures ? Also a common color bar for the 2 models seems more adequate.
L.154155 It seems important to clarify the goal of the paper. This sentence suggests that the main goal of the paper is to highlight numerical approach if it is the case other journals are more relevant and therefore you cannot conclude on geological implications.
L.159160 m.m1 ; the choice of each input parameters have to be justified by references
L.162164 A table in appendix with all the thermochronological data implemented in the model can be helpful for the reader. You have not selected MAr partially resetted (Long et al., 2012, Tectonics), you have to add a sentence on that.
L.165 Uniform geothermal gradient is a big assumption on this surface. Important thermal perturbation can be observed at this scale (McQuarrie and Ehlers, 2015, tectonics).
L.165166 add °C km1
L.167169 Use references for the different “closure temperature” used and not “resetting temperature”, numerous models showed that is a quite large range of temperature in particular if exhumation rates are low (see Ault et al. 2019, Tectonics).
L.169172 I agree but it can be an important bias on the model and you can have an idea calculating time residence of each sample in PAZ for AFT and PRZ for ZHe. It can be a sentence to justify that samples do not stay long time in PAZ and PRZ, in this case proxy on closure temperature can be used.
L.179 Define MAE.
L.186 It can be helpful for the reader to locate the different ages and samples on the cross section and having the possibility to link ages with the different tectonostratigraphic units. Scale of the Fig. 7a seems false, it is not 200 km long with this scale.
L.198 More discussion about the data and differences between observed and predicted ages can be useful. I am very curious to see if AHe and AFT which not fit come from a particular tectonostratigraphic unit and If it is the case it can be associated to a major fault activity ?
L.200 Add °C km1
L.208 Comparison with model from McQuarrie and Ehlers (2015) seems difficult. Your new model fit a global trend but previous model includes more parameters and explain better 2^{nd} order trend in the dataset.
L.209210: Digitizing from figure is not acceptable for publication. Ask the data.
L.223229 As suggested previously, it can be relevant to discuss the differences between observed and predicted ages to discuss the reason of the differences geothermal gradient, fault activity ?
L.239240 This sentence is for a profile in the southern flank of Himalayas and cannot be in the conclusion.
Figures:
Fig 2 Increase the size of the figure and the font size
Fig 3 and 4 Increase size of a) b) and c). It can be more lisible same comment for Figs. 3,4,5,6, 8, 9 and 10. The caption of Fig 4 do not correspond at the figure.
Fig 5 and 6: Merge the 2 figures, use the same color bar with the same range of value. Delete minus before isochrons ages.
Fig 7 It can be helpful for the reader to locate the different ages and samples on the cross section and having the possibility to link ages with the different tectonostratigraphic units. Scale of the Fig. 7a seems false, it is not 200 km long with this scale.
Fig 10 thermochronometer in the caption
Fig 11 R2= 0.43 not 0.43
Citation: https://doi.org/10.5194/se202122RC3 
AC3: 'Reply to RC3', Elco Luijendijk, 12 Sep 2021
Note: Our replies to the comments by Reviewer 3 are shown below each comment in italics
Dear authors,
This article shows a new analytical solution to predict thermochronological dataset in an orogenic wedge. This simple model assumes a transport only accommodates by basal detachment. I appreciate all the details available on the different parameters implemented in this model.
The authors applied this model on a profile perpendicular to the Himalayan wedge. Results make in evidence a good correlation between predicted ages and observed ages using an uniform model without effects of individual tectonic structures. Authors concludes that the principal implication of these results is a good reproducibility of the thermochronological data with a simple model and a possible steady state evolution of the Himalayan orogenic wedge.
The manuscript is well written and we have all the details on the model and the analytical procedure, however major points have to be clarified and discussed, see my general comments.
General comments
1/ The main topic of the article have to be clarified l. 154155 " Note that the goal here is not to provide a detailed geological case study, but to demonstrate the use of the equation to calculate deformation and exhumation rates” but authors highlight in abstract and conclusion “The results also imply that this part of the Himalayas may be in steadystate.” (l.78) and “This indicates that the Himalayas may be in steadystate and that, at a large scale, the exhumation of mature mountain belts may be approximated by a relatively simple model of uniform and steadystate deformation, accretion and transport” (l. 239241). This conclusion may be true but it is not relevant for the Himalaya Mountain range using one cross section.
Reply: This is a good point, and the reviewer is right that based on our study alone, we cannot make inferences for the entire Himalayas. Our results are however valid for the particular region. We have modified the statement accordingly.
2/ Similar model have been developed by Batt et al. 2001 (JGR), a comparative study of the 2 models or at least a discussion on the main differences between the models must be developed. “However, to our knowledge no analytical solutions exist for the relation between deformation and exhumation of mountain belts.” (l.2324), this sentence is partially true and must be modified to show the specificity of this new model.
Reply: This oversight has also been raised by reviewers 1 & 2. A sentence on this previous model has been included in the introduction in the new version of the manuscript.
3/ Constant geothermal gradient is a big assumption in the models and it is not realistic according studies of Coutand et al., 2014 (JGR) and McQuarrie and Ehlers, 2015 (tectonics). This point and the impact on the model must be developed in the discussion. New kinematic models are not implemented (see Ault et al., 2019, tectonics for a synthesis) specify a closure temperature is possible and easier to implemented in the models but residence time in the Partial Annealing Zone (PAZ) and Partial Retention Zone (PRZ) need to be low to assume a closure temperature.
Reply: Agreed, we have updated the thermal and thermochronology model in the revised manuscript. See also our replies to comments by reviewer 1 and 2.
4/ Discussion of the results must be more detailed using differences between the predicted and observed ages and to discuss potential fault activity along the cross section. The comparison with study of McQuarrie and Ehlers, 2015 (tectonics) based on the better fit of the new model seems a bit complex. Differences between the 2 models are not specified in the text.
Reply: Agreed, we have expanded the discussion of the comparison between our results and the previous work by McQuarrie and Ehlers (2015).
5/ This model applied on only one cross section in orogenic wedge of the Himalayas does not allow to provide strong conclusion on the Himalayas tectonic regime.
Reply: We agree and would like to refrain from statements about the entire Himalayas, see also the replies to the comments by reviewer 1 and 2.
You can find detailed comments in the sections below:
Specific comments
L.5 Precise the location of the transect.
Reply: Ok.
L.67 Remove this sentence, themochronological data not explained by the model can result of individual fault activity. See my general comment
Reply: The model fit is relatively good. We therefore feel that claiming that at a large scale the mode fits the trend of the data is valid here.
L.78 This conclusion is not relevant if we consider ages do not fit with the model, see my previous comment.
Reply: We are not sure what the reviewer means here. The model fit is shown and discussed. Why would the conclusion not be relevant?
L.17 I agree, one of the big advantages of the model presented is the low computational cost compare to other thermokinematic models.
L.2324 Please consider Batt et al. 2001 (JGR), see my general comment
Reply: Agreed, see previous reply
L.91 Vxc ?
Reply: Fixed.
L.104 Fig.3 to be homogeneous in the text
Reply: Thanks for noticing, we have corrected this.
L.106107 It is not the good caption for the figure
Reply: It is actually, we are not sure what is wrong with the caption here? Could you specify the issues with the caption?
L.135139 Develop the simplified solution is necessary to the discussion ? If not you can remove this part in appendix and just comment the among of differences between the 2.
Reply: We have removed the simplified solution from the manuscript.
L.139: Considering the color bar on Fig.5d error decrease in the first kilometers.
Reply: Agreed, we have expanded the discussion of the errors here
L.143145 Is it possible to merge the 2 figures ? Also a common color bar for the 2 models seems more adequate.
Reply: We feel that merging these 2 figures would produce a figure that would be too large.
L.154155 It seems important to clarify the goal of the paper. This sentence suggests that the main goal of the paper is to highlight numerical approach if it is the case other journals are more relevant and therefore you cannot conclude on geological implications.
Reply: I presume the reviewer means the analytical approach not the numerical? It is true that ithe manuscript presents an advance in methods and not a case study or a new knowledge of a geological process. However, we feel that the new equation and its potential applications are of sufficient interest to thermochronologist and geologists to warrant publication in Solid Earth.
L.159160 m.m1 ; the choice of each input parameters have to be justified by references
Reply: The references are mentioned in the first part of the sentence (The geometry of the wedge is based on published geological crosssections (Long et al., 2012; Coutand et al., 2014; McQuarrie and Ehlers, 2015)
L.162164 A table in appendix with all the thermochronological data implemented in the model can be helpful for the reader. You have not selected MAr partially resetted (Long et al., 2012, Tectonics), you have to add a sentence on that.
Reply: Thanks for noticing. We have clarified why the nonreset MAr samples were not used. The thermochronology data are a 100% copy of data used by McQuarrie and Ehlers (2015) and as such we feel that it would be superfluous to include a table in the manuscript. However, the data are included in the GitHub repository and the zenodo publication of the model code, which we now refer to in the main manuscript.
L.165 Uniform geothermal gradient is a big assumption on this surface. Important thermal perturbation can be observed at this scale (McQuarrie and Ehlers, 2015, tectonics).
Reply: The thermal model was updated to include steadystate conduction and advection
L.165166 add °C km1
Reply: Ok.
L.167169 Use references for the different “closure temperature” used and not “resetting temperature”, numerous models showed that is a quite large range of temperature in particular if exhumation rates are low (see Ault et al. 2019, Tectonics).
Reply: We have revised the thermochronology model and have removed this part of the manuscript.
L.169172 I agree but it can be an important bias on the model and you can have an idea calculating time residence of each sample in PAZ for AFT and PRZ for ZHe. It can be a sentence to justify that samples do not stay long time in PAZ and PRZ, in this case proxy on closure temperature can be used.
Reply: We have revised the thermochronology model and have removed this part of the manuscript.
L.179 Define MAE.
Reply: There is a definition in the same sentence (mean absolute error).
L.186 It can be helpful for the reader to locate the different ages and samples on the cross section and having the possibility to link ages with the different tectonostratigraphic units. Scale of the Fig. 7a seems false, it is not 200 km long with this scale.
Reply: The modeled crosssection was larger than the geological crosssection to avoid boundary effects in the model. The sample locations have been added to the crosssecttion
L.198 More discussion about the data and differences between observed and predicted ages can be useful. I am very curious to see if AHe and AFT which not fit come from a particular tectonostratigraphic unit and If it is the case it can be associated to a major fault activity ?
Reply: We feel that this is out of scope of the manuscript. The model that we propose does not resolve individual faults. Alhtough it could be adjusted to do so, that would warrant a separate publication.
L.200 Add °C km1
Reply: We have revised this section of the manuscript
L.208 Comparison with model from McQuarrie and Ehlers (2015) seems difficult. Your new model fit a global trend but previous model includes more parameters and explain better 2^{nd} order trend in the dataset.
Reply: The general consensus on assessment of model performance is that all else being equal models with less parameters are preferred over more complex models.
L.209210: Digitizing from figure is not acceptable for publication. Ask the data.
Reply: After a repeat request we have now gotten the data.
L.223229 As suggested previously, it can be relevant to discuss the differences between observed and predicted ages to discuss the reason of the differences geothermal gradient, fault activity ?
Reply: We feel that this is out of scope for this manuscript and would require a separate more detailed study.
L.239240 This sentence is for a profile in the southern flank of Himalayas and cannot be in the conclusion.
Reply: Agreed, we have revised this sentence to state that this section in the Himalayas may be in steadystate.
Figures:
Fig 2 Increase the size of the figure and the font size
Reply: agreed, this suggestion has been implemented.
Fig 3 and 4 Increase size of a) b) and c). It can be more lisible same comment for Figs. 3,4,5,6, 8, 9 and 10. The caption of Fig 4 do not correspond at the figure.
Reply: agreed. We are not sure what is wrong with the caption of Fig. 4.
Fig 5 and 6: Merge the 2 figures, use the same color bar with the same range of value. Delete minus before isochrons ages.
Reply: We disagree with the merging of these figures. It would results in a very large figure in which the subpanels would be difficult to read and interpret.
Fig 7 It can be helpful for the reader to locate the different ages and samples on the cross section and having the possibility to link ages with the different tectonostratigraphic units. Scale of the Fig. 7a seems false, it is not 200 km long with this scale.
Reply: We have added the sample positions. And its true that the crosssection is not 200 km, we now explain the reason for this in the main manuscript.
Fig 10 thermochronometer in the caption
Reply: ok
Fig 11 R2= 0.43 not 0.43
Reply: The R2 is actually negative here. Note that the coefficient of determination (R2) can be negative if the variance of the model error exceeds the variance of the data, which is the case here.
Citation: https://doi.org/10.5194/se202122AC3

AC3: 'Reply to RC3', Elco Luijendijk, 12 Sep 2021
Status: closed

RC1: 'Comment on se202122', Anonymous Referee #1, 27 Apr 2021
Dear Authors,
The authors do present an analytical approach to model the particle motion in an compressional orogen and estimate related thermochronological ages. The model does allow to efficiently change the boundary condition and therefore adjust to fit observed thermochronological data and make inferences about the importance of e.g. compression, detachment velocities and frontal and basal accretion.
Although I personally like the approach and acknowledge the effort the authors did to implement this model, I do not like the focus of the manuscript given the simplicity of the their model.
For further information see my scientific comments:
Scientific comments
 There is a quite similar approach as the one presented here for accretionary wedges (not including ‘compression’) (Batt et 2001), but this should be mentioned in the introduction and the similarities and differences should be discussed in detail. I would also like to see both approaches applied and visually compared, e.g. for an example without compression.
 I am not very happy how the authors simplify the thermal model, using a constant geothermal gradient (lateral and temporally constant), especially because they apply their model to the Himalayas , which is characterised by significant amount of horizontal and vertical rock motion. This does strongly perturb the thermal field, both laterally and also temporally. Although the authors show that they can fit the general age trend, that does not mean the model is correct.
 Similar to the thermal model, the calculation of thermochronological ages is oversimplified and should not be used for the purpose stated. There are numerous of recent studies demonstrating the strong impact of different cooling rates and radiation damage on thermochronological ages. In this regard, exhumation of the Himalayas does include movement above ramps and flats that should result in cooling rates to vary temporally and different amount of radiation damage to accumulate. I do not see any reason why not using available annealing and diffusion models and include them in this modelling approach (even if the thermal field is set constant, which I do not like).
 The oversimplified thermal model and calculation of thermochronological ages does not allow to use this model to be applied to a real dataset. Instead if might be used to study the general trend in thermochronological ages in active orogens, and study the different age trends related to the importance of compression, frontal and basal accretion. If this model, however, should be applied to real datasets, it is mandatory to include a more exact treatment of the thermal field and stateoftheart calculation of thermochronological ages.
 The authors state that they fit the thermochron data along the Kuru Chu cross section better than McQuarrie and Ehlers (2015). The general trend is fitted by the authors, but it seems that McQuarrie and Ehlers (2015) do a better job in fitting the details of data between 0 to 80 km better. I would also like to ask the authors to contact McQuarrie and Ehlers to get the raw data instead of taken the values from a figure. Anyway getting a better fit with a simple model that does neglect part of the complexity of the geological setting, thermal structure and analytical method does not necessarily mean their model is correct. Indeed it is usually easier to fit data with a simple model, since it can be adjusted easily to optimize fitting. This is often time consuming if all available geological constraints and sophisticated thermalkinematic models are used. However, even complex models do become computational more efficient and can be adjusted with efficient search algorithms and running models in parallel. Please add a little paragraph and explain the applicability of the different model setups and limitation associated with it.
 The authors state that because they can reproduce the general trend in ages in the studied profile with a steadystate assumption, the Himalaya might be in steadystate. Showing that one model is fitting the data does not mean that others model do not fit the data as well or even better. Since the presented model is only working in steadystate the authors cannot prove that steadystate is the best model to fit the data. What there model however can be used for, is to study the relative importance of e.g. frontal and basal accretion, detachment transport and compression, which the authors correctly discussed.
In summary, I would suggest the authors to change the focus of their manuscript and state the drawbacks and benefits of such an approach. I do see the real application of such an approach in testing largescale boundary conditions that can afterwards be used in more sophisticated model setups. I do not see that this approach can ‘compete’ or ‘replace’ approaches like that of McQuarrie and Ehlers (2015). I hope you find my comments and suggestions helpful and fing more details in the technical corrections.
Technical corrections:
Line 3: State what you mean with exhumation, amount or rate?
Line 20: Change to ‘fault blocks’.
Line 2324: That is not correct a similar approach for accretionary wedges (frontal and basal accretion) has been published already in 2001 by Batt et al. Please cite their work and discuss the similarities and differences with your model.
Line 91: I guess it should be v_{xc }= …
Line 99: State that you also use Eq. 7 and use the normalized compression velocity.
Line 101 and 103: It is not clear why you are defining the horizontal and vertical components in Eq. 14, but not all equations are used in Eq. 8 and 9.
Line 106107: Should be ‘…internal deformation and basal and frontal accretion…’, this is what the figure captions says. In this case, shouldn’t the velocity vectors decrease towards the interior of the model since the accretion is only occurring along the tip of the wedge and vectors should show divergence!? The integral of the vectors along x should be similar along the wedge, or not?
Line 137139: It is not clear why you do provide the ‘simplified solution’, please explain and if there are no good reasons, just do not show. Can you provide the difference between the analytical and the numerical solutions as figures, that would help the reader to check what you have wrote here (instead of showing the difference between the two provided analytical solutions). Maybe you only show the difference between the simplified analytical (if you want to keep showing it) against numerical since the other one are nearly identical.
Line 165: A uniform geothermal gradient is really not what can be used in such setting. Depending on the model setup, and especially the horizontal and vertical rates the thermal field will be strongly perturbed. For instance do have a look on the cited manuscript from McQuarrie and Ehlers (2015) Fig. 1a).
Line 168: This approach is also much too simplistic especially since you do have significant horizontal particle motion and cooling rates of samples might vary significantly throughout their exhumation. This and the knowledge of the importance of the amount of radiation damage on He diffusion kinetics in apatite and zircon (e.g. Flowers et al. 2009; Guenthner et al. 2013) has to be taken into account for the transformation of particle trajectories, cooling histories and final thermochron ages.
Line 208209: Please provide details and justification how you calculate the misfit, what is the MAE? Your model indeed fits the general trend of the data, however, the details are better fitted by McQuarrie and Ehlers (2015), with the exception of data >100 km away from the frontal tip. You may want to mention this.
Line 209210: Please ask the authors to provide the data, fortunately we do not need to digitize data from figures anymore!!!
Line 212: Please not that a model that fits data better is not automatically correct. The model of McQuarrie and Ehlers (2015) is based on numerous independent geological information that has been incorporated in a much more realistic model, but complex model. It is always easier to fit a simple model to data compared to complex model, but does that mean the simple model is more realistic, I would say not! There is a tradeoff between complexity and fit to the data, your model and the one from McQuarrie and Ehlers (2015) are endmembers in this relation. From both model setups we can learn and they do have their eligibility. Please add a bit of this in your discussion and not just say your model is better…
Line 225: That orogens are in steadystate has been described already by others (e.g. Willett and Brandon 2003, Bernet et al. 2001) and that was highly discussed and even often this has been disproved by additional data and more indepth interpretation (e.g. Michel et al. 2019). Since the thermal field of the crust is slow in responding to changes in the boundary conditions the resulting thermochronological data are often ‘smooth’ and the real/complex exhumation history is difficult to constrain. Fitting a steadystate pattern through data is therefore often easier compared to finding the real/complex exhumation history.
Line 239: What kind of steadystate do you mean, flux, exhumation, topography?
Figures:
Fig. 1: Can you draw a few more particle path and continue them to the surface. From the figure it is also not clear what is above the wedge, water, air? Also add ‘erosion’ below the precipitation on the surface.
Fig. 3: In caption change to ‘Panel shows…’.
Fig. 4: The heading of the panels are wrong, please correct.
Fig. 5: Use ‘5 Ma’ instead of ‘5 Ma’. Change panel d to Error between numerical and analytical solution.
Fig. 6: Change panel d to Error between numerical and analytical solution.
Fig. 7: Is the scale correct, I thought you speak somewhere from 200 km profile length. Looks shorter?
Fig. 8: What does MAE mean?
Citation: https://doi.org/10.5194/se202122RC1 
AC1: 'Reply to RC1', Elco Luijendijk, 12 Sep 2021
Note: Our replies to the comments by Reviewer 1 are shown below each comment in italics.
Dear Authors,
The authors do present an analytical approach to model the particle motion in an compressional orogen and estimate related thermochronological ages. The model does allow to efficiently change the boundary condition and therefore adjust to fit observed thermochronological data and make inferences about the importance of e.g. compression, detachment velocities and frontal and basal accretion.
Although I personally like the approach and acknowledge the effort the authors did to implement this model, I do not like the focus of the manuscript given the simplicity of the their model.
For further information see my scientific comments:
Scientific comments
 There is a quite similar approach as the one presented here for accretionary wedges (not including ‘compression’) (Batt et 2001), but this should be mentioned in the introduction and the similarities and differences should be discussed in detail. I would also like to see both approaches applied and visually compared, e.g. for an example without compression.
Reply: We thank the reviewer for pointing us to this paper. The approach referred to by the reviewer is equation 5 in Batt et al. (2001, https://doi.org/10.1029/2001JB000288), with a correction published later (https://doi.org/10.1029/2003JB002897). This equation provides vertical velocity inside a wedge as a function of a predefined erosion rate, accretion rate, and slope of the base of the wedge. This is quite different from our approach in that Batt et al. (2001) use erosion rate at the surface as an input, whereas our equation predicts erosion rates. The Batt et al. (2001) equation therefore cannot used to predict exhumation rates or thermochronometer ages without prior knowledge of the erosion rates. And of course erosion rates are themselves usually calculated using thermochronometer data. For this reason we cannot include a quantitative comparison between our model and the Batt et al. (2001) approach in the manuscript. However, we did add a brief discussion of this model to the introduction section of the revised version of our manuscript.
 I am not very happy how the authors simplify the thermal model, using a constant geothermal gradient (lateral and temporally constant), especially because they apply their model to the Himalayas , which is characterised by significant amount of horizontal and vertical rock motion. This does strongly perturb the thermal field, both laterally and also temporally. Although the authors show that they can fit the general age trend, that does not mean the model is correct.
Reply: We agree that the thermal model that we used is highly simplified. We have changed the thermal model and have updated the model code to include a numerical solution of the steadystate heat advection & conduction equation. The code uses the calculated velocity field as an input (eqs. 8, 9 in the manuscript), along with published thermal parameters and boundary conditions by Coutand et al. (2014) and McQuarrie and Ehlers (2015). The calculated steadystate thermal field is then used in combination with particle tracks calculated using our new equation to calculate the thermal history of particles. The new thermal model, in combination with the new thermochronology model / equations results in a similar fit to the AFT and ZFT data. However, the new thermochronology model results in higher resetting temperatures for the MAr thermochronometer. The steadystate model cannot fit both datasets well at the same time. Potential reasons for this are discussed in the results section of the revised manuscript.
 Similar to the thermal model, the calculation of thermochronological ages is oversimplified and should not be used for the purpose stated. There are numerous of recent studies demonstrating the strong impact of different cooling rates and radiation damage on thermochronological ages. In this regard, exhumation of the Himalayas does include movement above ramps and flats that should result in cooling rates to vary temporally and different amount of radiation damage to accumulate. I do not see any reason why not using available annealing and diffusion models and include them in this modelling approach (even if the thermal field is set constant, which I do not like).
Reply: We agree that use of a fixed closure temperatures was a bit overly simplistic and have upgraded the model code to calculate cooling ages using an approach by Fox et al. (2014, https://doi.org/10.5194/esurf2472014), which is based on the Dodson (1973, https://doi.org/10.1007/BF00373790) equation. We acknowledge that this approach, at least for the AFT method is still a simplification. However, the rapid cooling experienced by the samples makes linear cooling models such as used by Fox et al. (2014) relatively accurate in this case because the samples spend relatively little time in the partial annealing zone where differences in annealing rates are important. In addition, we would like to reserve the use of more sophisticated thermochronology models for follow up manuscript because this would require significantly more work on the model code and much more space in the manuscript for additional discussions.
 The oversimplified thermal model and calculation of thermochronological ages does not allow to use this model to be applied to a real dataset. Instead if might be used to study the general trend in thermochronological ages in active orogens, and study the different age trends related to the importance of compression, frontal and basal accretion. If this model, however, should be applied to real datasets, it is mandatory to include a more exact treatment of the thermal field and stateoftheart calculation of thermochronological ages.
Reply: This point was addressed by using a more realistic thermal model and thermochronometer model.
 The authors state that they fit the thermochron data along the Kuru Chu cross section better than McQuarrie and Ehlers (2015). The general trend is fitted by the authors, but it seems that McQuarrie and Ehlers (2015) do a better job in fitting the details of data between 0 to 80 km better. I would also like to ask the authors to contact McQuarrie and Ehlers to get the raw data instead of taken the values from a figure. Anyway getting a better fit with a simple model that does neglect part of the complexity of the geological setting, thermal structure and analytical method does not necessarily mean their model is correct. Indeed it is usually easier to fit data with a simple model, since it can be adjusted easily to optimize fitting. This is often time consuming if all available geological constraints and sophisticated thermalkinematic models are used. However, even complex models do become computational more efficient and can be adjusted with efficient search algorithms and running models in parallel. Please add a little paragraph and explain the applicability of the different model setups and limitation associated with it.
Reply: We did request the original data from McQuarrie. We did not receive a reply previously, but have repeated the request and McQuarrie has now kindly supplied the data.
Following Ockham's razor we feel that given two model codes the simplest code that fits the data should be preferred. We are not saying that it would not be possible to fit the data better with a more complicated model. However, at present our simple model performs better in this particular crosssection than the published models. This may or may not be the case in other parts of the Himalayas or other mountain belts, which we feel is something that would be worth exploring in the future.
 The authors state that because they can reproduce the general trend in ages in the studied profile with a steadystate assumption, the Himalaya might be in steadystate. Showing that one model is fitting the data does not mean that others model do not fit the data as well or even better. Since the presented model is only working in steadystate the authors cannot prove that steadystate is the best model to fit the data. What there model however can be used for, is to study the relative importance of e.g. frontal and basal accretion, detachment transport and compression, which the authors correctly discussed.
Reply: We agree that we cannot test transient models with our approach. However, previous attempts with transient models do not show a better fit than our steadystate approach. We would therefore like to still state that there is a strong possibility that this part of the Himalayas is in steadystate. We did change the wording of this claim in the manuscript to state that "this crosssection in the Himalayas may be in steadystate".
In summary, I would suggest the authors to change the focus of their manuscript and state the drawbacks and benefits of such an approach. I do see the real application of such an approach in testing largescale boundary conditions that can afterwards be used in more sophisticated model setups. I do not see that this approach can ‘compete’ or ‘replace’ approaches like that of McQuarrie and Ehlers (2015). I hope you find my comments and suggestions helpful and fing more details in the technical corrections.
Reply: We do not intend to state that this approach should replace numerical model approaches, and hope we have made this more clear in the revised version of the manuscript. We felt that a simple model like the one we presented was missing from the literature and could be a useful additional tool to study the dynamics of mountain belts. We did add a paragraph to the discussion section that discusses the benefits and drawbacks of the approach.
Technical corrections:
Line 3: State what you mean with exhumation, amount or rate?
Reply: We mean exhumation rate here, now explicitly stated in the manuscript.
Line 20: Change to ‘fault blocks’.
Reply: Ok
Line 2324: That is not correct a similar approach for accretionary wedges (frontal and basal accretion) has been published already in 2001 by Batt et al. Please cite their work and discuss the similarities and differences with your model.
Reply: We did now add a brief discussion of this study, see reply to point 1.
Line 91: I guess it should be vxc = …
Reply: correct, thanks for noticing this.
Line 99: State that you also use Eq. 7 and use the normalized compression velocity.
Reply: Ok
Line 101 and 103: It is not clear why you are defining the horizontal and vertical components in Eq. 14, but not all equations are used in Eq. 8 and 9.
Reply: This is because vyc was replaced with the expression for vyc from eq. 4. We have rewritten the equation to correct this.
Line 106107: Should be ‘…internal deformation and basal and frontal accretion…’, this is what the figure captions says. In this case, shouldn’t the velocity vectors decrease towards the interior of the model since the accretion is only occurring along the tip of the wedge and vectors should show divergence!? The integral of the vectors along x should be similar along the wedge, or not?
Reply: The horizontal and vertical compression vectors decrease towards the tip of the wedge and are directed to the left (Fig. 4a). The accretion velocity is constant along the wedge and directed to the right (fig 4b). The combination of the two shown in 4c results in horizontal velocities that cancel out at a distance of 75 km from the tip of the wedge and a total velocity that first decreases with distance to the tip and then increases again as it passes the point where the two components cancel out. Velocity integrated over over x varies in our approach.
Line 137139: It is not clear why you do provide the ‘simplified solution’, please explain and if there are no good reasons, just do not show. Can you provide the difference between the analytical and the numerical solutions as figures, that would help the reader to check what you have wrote here (instead of showing the difference between the two provided analytical solutions). Maybe you only show the difference between the simplified analytical (if you want to keep showing it) against numerical since the other one are nearly identical.
Reply: We have removed the simplified solution from the manuscript.
Line 165: A uniform geothermal gradient is really not what can be used in such setting. Depending on the model setup, and especially the horizontal and vertical rates the thermal field will be strongly perturbed. For instance do have a look on the cited manuscript from McQuarrie and Ehlers (2015) Fig. 1a).
Reply: We agree and have revised the thermal model. See previous replies.
Line 168: This approach is also much too simplistic especially since you do have significant horizontal particle motion and cooling rates of samples might vary significantly throughout their exhumation. This and the knowledge of the importance of the amount of radiation damage on He diffusion kinetics in apatite and zircon (e.g. Flowers et al. 2009; Guenthner et al. 2013) has to be taken into account for the transformation of particle trajectories, cooling histories and final thermochron ages.
Reply: We agree and have revised the thermochronometer models, see previous replies.
Line 208209: Please provide details and justification how you calculate the misfit, what is the MAE? Your model indeed fits the general trend of the data, however, the details are better fitted by McQuarrie and Ehlers (2015), with the exception of data >100 km away from the frontal tip. You may want to mention this.
Reply: We used two metrics to calculate the model fit, R2 and MAE. R2 is the coefficient of determination, which we now clarified better in the main text and the caption. MAE is the mean absolute error, which is defined in line 179, and is now also mentioned in the caption. We have added a more extensive discussion of the model fit of our and McQuarrie and Ehlers (2015) model to the revised manuscript.
Line 209210: Please ask the authors to provide the data, fortunately we do not need to digitize data from figures anymore!!!
Reply: We did request the data and had not received a reply before submitted the previous version of the manuscript. However, we have sought contact again and have now received the data.
Line 212: Please not that a model that fits data better is not automatically correct. The model of McQuarrie and Ehlers (2015) is based on numerous independent geological information that has been incorporated in a much more realistic model, but complex model. It is always easier to fit a simple model to data compared to complex model, but does that mean the simple model is more realistic, I would say not! There is a tradeoff between complexity and fit to the data, your model and the one from McQuarrie and Ehlers (2015) are endmembers in this relation. From both model setups we can learn and they do have their eligibility. Please add a bit of this in your discussion and not just say your model is better…
Reply: We do not claim that our model is more correct. We did not use the term correct in the manuscript, and feel that correct is not a good term to describe models. If one follows Ockhams razor then a simple model that fits the data better should be preferred. We do agree that the model that we use may be overly simplistic in assuming uniform compression and transport. However, until a model comes along with a better fit to the data, we feel that the simple model should be preferred. We do think it is important to present an alternative model that provides a better fit to the thermochronological data, and let the reader judge, which is preferable.
Line 225: That orogens are in steadystate has been described already by others (e.g. Willett and Brandon 2003, Bernet et al. 2001) and that was highly discussed and even often this has been disproved by additional data and more indepth interpretation (e.g. Michel et al. 2019). Since the thermal field of the crust is slow in responding to changes in the boundary conditions the resulting thermochronological data are often ‘smooth’ and the real/complex exhumation history is difficult to constrain. Fitting a steadystate pattern through data is therefore often easier compared to finding the real/complex exhumation history.
Reply: In our humble opinion, whether or not the real exhumation would be more complex should be judged by the data. And for the time being, if more complex models in this particular crosssection result in poorer fit to data then these models should not be preferred. We do however agree with the reviewer that thermochronological data alone may not be able to resolve all complexities within the system. In the case of the Himalayas steadystate has also been suggested based on a compilation of cosmogenic nuclides and sediment yield data. Our analysis corroborates these findings.
Line 239: What kind of steadystate do you mean, flux, exhumation, topography?
Reply: We mean steadystate exhumation, and have changed the text accordingly.
Figures:
Fig. 1: Can you draw a few more particle path and continue them to the surface. From the figure it is also not clear what is above the wedge, water, air? Also add ‘erosion’ below the precipitation on the surface.
Reply: Thanks for this suggestion, we have modified the figure accordingly
Fig. 3: In caption change to ‘Panel shows…’.
Reply: Corrected, thanks for spotting this.
Fig. 4: The heading of the panels are wrong, please correct.
Reply: We have corrected this.
Fig. 5: Use ‘5 Ma’ instead of ‘5 Ma’. Change panel d to Error between numerical and analytical solution.
Reply: Ok
Fig. 6: Change panel d to Error between numerical and analytical solution.
Reply: See previous comment.
Fig. 7: Is the scale correct, I thought you speak somewhere from 200 km profile length. Looks shorter?
Reply: That is correct. The modelled crosssection is 200 km to avoid boundary effects on the results. However, the balanced crosssection shown here only covers the first part of the crosssection. We have added an explanation of this to the revised manuscript.
Fig. 8: What does MAE mean?
Reply: We have added an explanation of the mean absolute error to the caption.
Citation: https://doi.org/10.5194/se202122AC1

RC2: 'Comment on se202122', Anonymous Referee #2, 30 Apr 2021
Dear authors,
This work presents a new analytical solution for the steadystate exhumation of an orogenic wedge. The authors used the prediction of the timedepth path and a simple thermal model to predict lowtemperature thermochronologic ages that they compare with a cross section in eastern Himalaya. This comparison shows good fit between observations and predictions. This solution does not capture the effects of individual faults and folds, but the authors claim from this comparison that 1) in this example, a simple model is sufficient to explain the largescale observations, and 2) that here, the Himalaya may be in steadysate.
General comments
In general, this is well written and easy to read. I like the idea to predict lowtemperature thermochronologic ages from a 2D model of a critical wedge evolution, but I have important remarks that cannot be eluded:
 You claim that this is the first attempt for such a study. I disagree, at least Batt et al., 2001 did a similar 2D critical wedge model to predict lowtemperature ages. This model is not presented in the present paper from Luijendijk et al., they do not explain in what their model is different, and also, there is no comparison of these two similar models. You should present the Batt et al., 2001 model, and discuss it in terms of both construction and results.
 My other major concern is about the thermal model used in the present model. This is not clear to me, if the thermal model resolve the in 2D the heat equation with vertical and lateral advection. This is important because in the model, particles are advected, and thus heat also! For a similar particle path model, taking in account or not heat advection will lead to timetemperature path that can be very different, and that can impact the lowT age prediction! So, you need to precise the thermal model (and thus give the values of the complete set of heat parameters). And if the thermal model does not take in account the heat advection, I would push to update the thermal model so that it takes it in account. Or show that there is no consequence in your time temperature path, and thus that it does not impact your age prediction.
 In the same type of comments, I am not very happy of the cooling models used to compute thermochronological ages. Here, you do not use published cooling models, but only a closure temperature. This is over simplistic, and it finally asks if your predictions can really be compared to observations! Maybe a way to prove that this works, this is to show that the cooling paths you compute (with a correct thermal model!) is similar to the cooling paths published in previous papers (i.e., Long et al., 2012?), or better, implement the published routines to compute correct thermochronologic ages from cooling histories.
 In the method section, I do not understand why you present a simplified solution that does not gives very accurate results. First, this is noise in the paper, and second, it decrease the confidence we could have in your other strong simplifications (temperature model, cooling model,…). I would remove this simplified solution, or explain why this is important.
 In the validation section, I do not understand how you set up your numerical model, i.e., what is the difference between the numerical model and the full analytical solution. You probably need to better explain (with an additional sketch?) your numerical model with the parameters used.
 And finally, in your application to a mountain belt, you need to explain with better arguments why you choose this section, because for instance the Central Nepal section also matches the arguments you present here. Also, the comparison between the restored crosssection published and your prism model differs importantly in some places. So you need to discuss the impact on predicted ages of your approximations in the geometry of the prism.
So, because of the strong simplifications you made, I am not sure that you really can compare predictions with a real dataset, and, more, that you can use this comparison to extract strong conclusions on how to interpret lowtemparature data and if the Himalayas are in steay state or not, specially wihtout moderating your sentences.
Specific comments
L5: precise where (approx. longitude for instance) the cross section is located in the Himalayas
L67: change “at a large scale deformation” to “deformation at a large scale” or “large scale deformation”?
Somewhere in the abstract, you need to specify taht your model is only in 2D and that it does not take in account any relief (that is important for comparison with real dataset).
L11 & L13 & L14: add “e.g.,” before the reference(s)
L24: and Batt et al., 2001 model? You need to say in what your approach differs from Batt et al., 2001!
L2526: Expend the idea present in this sentence as the comparison is not straight forward.
L91: vxc ?
L94: add a ref to the appendix where you explain how you defined these variables
L106107: This is not what there is in the caption of Fig. 4. Correct it.
L137138: Why showing the simplified solution if the full solution is a lot better? Does this solution give advantages when computing? If they are reasons to show this simplified solution, explain them, or remove the simplified solution from the paper.
L139: “The error increases with depth”. This part is a bit confusing: on the figs. 5 and 6, the colors representing the errors are inversed, they do not show the same between figs. 5 and 6. Please, to limit the confusion, put the same color bar.
Also, this sentence is true for the fig. 6d, but not for fig.5d where I see first a decrease of the error with depth, and then a rapid increase.
L140142: So, again, why showing the simplified solution if it does not permit to match well the full solution?
L148151: here, you give your arguments (good thermochronologic data coverage and high convergence rates) to justify that you choose this section. But if you take the Kathmandu/Trisuli or Sutlej region, it matches the same argument, so why didn’t you took these sections also? You need to give more arguments on the choice of your section.
L159: m m1, not m m1
L160: m m1 not m m1
L160: 200 km? from the fig 7, it looks closer to 100 kmlong than 200 kmlong. So what is wrong, the text or the legend of Fig. 7?
L160: add ref to fig. 7
L157160: You may describe and quantify the differences of the complex geometry of the wedge build from geological and geophysical data, and your triangular simplified wedge geometry. You then need to explain what are the consequences of these differences to your model results, i.e. where your results will be accurate, and where they will not.
I ask this question because the Long et al., 2012 crosssection show an important ramp of the MHT (representing the base of the wedge), also, your model does not take in account the important frontal ramp. Though, Van der Beek et al., 2006, Robert et al. 2009, 2011 have shown that this geometry of the base of the wedge has a huge impact on the thermochronological record. So, such a simplification risks to make inaccurate predictions. You really need to reinforce this section.
L162164: I think that the set of thermochronometers from the Kathmandu/Trisuli section is larger (See Herman et al., 2010 for instance). So why did you choose the Bhutan section and not the central Nepal section?
L165: 15°C km1 (also, generally, remove the space between numbers and the °C sign)
L165167: I do not really understand the thermal model.
 Do you update this thermal model through time while you are advecting your particles? Do you take in account lateral thermal advection? If yes, you need to describe this model precisely in the method section. If not, I do not understand how you will be able to compute accurate thermochronological ages as in your wedge model, you have a lateral component that is important.
 Do you have heat production (i.e., radioactivity) in your model?
 What are the thermal characteristics (i.e. thermal diffusivity) of the rocks you are advecting in the model? This is crucial for the thermal model, you need to give them.
L167172: this is a strange choice. If you want to predict thermochronological ages, this is better to use existing cooling models. This is not so hard because you can find most of the routines in the literature or on the Web. This approach is oversimplified!
L178180: I do not understand what you do here. What do you input in the downhill simplex algorithm? You need to explicitly explain what you are doing here in details.
Also, what is the MAE of the calculated thermochronological ages?
L181182: where are the results/comparison of these models?
L186: With such an ultrasimplified thermal model that does not take in account lateral heat advection (does it?), and an ultrasimplified cooling calculation, I am not sure this is relevant to compare your predictions to the data. Before to do that, you need to refine your thermal model and cooling models. Ask any quantitative thermochronologist about what is often a critical point when submitting papers for publication: there is almost always a strong debate around the thermal model/parameters used to interpret thermochronologic data/model. This is really a critical point.
L188: the exploration of the parameter space is very rough. In the introduction, you explained that you want to propose a model that does not need a lot of computing time and resources. So if this is the case (is it the case? You do not mention it until now), it would be easy and rapid to refine your parameter search with a smaller stepping.
L208: I would not say a better fit: you fit well the general trend, but not the details, and McQuarrie and Ehlers better explain the different local ages variations (because their model is based on numerous geological observation, and is thermally more complex), except for the farthest samples. You need to moderate your sentence, or better explain the comparison.
L209201: Ask the authors the source of their graph.
L2122013: You should change the sentence to “This suggest that, for this crosssection, at large scale, deformation of the wedge is represented by uniform deformation”, or for something like that. This is because your fit does not reproduce the details of the data because 1) your thermal/cooling model is oversimplified, and 2) you do not take in account relief.
L269, eq A5: There is a problem in this equation. A dx is missing (in place of the L?) in the left member.
Figures
Figure 2:
The font is to small, increase it (it is hard to read the name of the vectors); in the caption, change the last sentence to “The symbols are listed and described in Table 1”
Figure 3 & 4:
Does the length of the arrows mean something? If yes, it maybe needs a scalebar, or at least an explanation in the caption
Figure 5 & 6;
See my specific comment about the colorbar for panels d). Why using a ““ in the isochrones? It will simplify the fig if you remove this ““
Figure 7:
 a) the writings inside the fig. are not readable. Please increase the size of the font.
On this fig., you may mark the MHT (which is the base of your wedge) with a thicker line, so the reader will better see what approximation you are doing in your model.
On a), you also may add the thermochronologic data (on a mapview also?) you use in the Himalayan case.
You also may write in the caption what is the red triangle…
Figure 8:
Why the MAE is in a and ages in Ma? Cleaner if you gibe MAE in Ma.
Figure 10:
In caption, correct “calculated themrochronometer…” by “calculated theRmochronometer…”
Citation: https://doi.org/10.5194/se202122RC2 
AC2: 'Reply to RC2', Elco Luijendijk, 12 Sep 2021
Note: Our replies to the comments by Reviewer 2 are shown below each comment in italics
Dear authors,
This work presents a new analytical solution for the steadystate exhumation of an orogenic wedge. The authors used the prediction of the timedepth path and a simple thermal model to predict lowtemperature thermochronologic ages that they compare with a cross section in eastern Himalaya. This comparison shows good fit between observations and predictions. This solution does not capture the effects of individual faults and folds, but the authors claim from this comparison that 1) in this example, a simple model is sufficient to explain the largescale observations, and 2) that here, the Himalaya may be in steadysate.
General comments
In general, this is well written and easy to read. I like the idea to predict lowtemperature thermochronologic ages from a 2D model of a critical wedge evolution, but I have important remarks that cannot be eluded:
 You claim that this is the first attempt for such a study. I disagree, at least Batt et al., 2001 did a similar 2D critical wedge model to predict lowtemperature ages. This model is not presented in the present paper from Luijendijk et al., they do not explain in what their model is different, and also, there is no comparison of these two similar models. You should present the Batt et al., 2001 model, and discuss it in terms of both construction and results.
Reply: Thanks for pointing us to this paper. We have added it to the manuscript, see reply to the same comment by Reviewer 1.
 My other major concern is about the thermal model used in the present model. This is not clear to me, if the thermal model resolve the in 2D the heat equation with vertical and lateral advection. This is important because in the model, particles are advected, and thus heat also! For a similar particle path model, taking in account or not heat advection will lead to timetemperature path that can be very different, and that can impact the lowT age prediction! So, you need to precise the thermal model (and thus give the values of the complete set of heat parameters). And if the thermal model does not take in account the heat advection, I would push to update the thermal model so that it takes it in account. Or show that there is no consequence in your time temperature path, and thus that it does not impact your age prediction.
Reply: Agreed, we have improved the thermal model. See reply to the same comment by Reviewer 1.
 In the same type of comments, I am not very happy of the cooling models used to compute thermochronological ages. Here, you do not use published cooling models, but only a closure temperature. This is over simplistic, and it finally asks if your predictions can really be compared to observations! Maybe a way to prove that this works, this is to show that the cooling paths you compute (with a correct thermal model!) is similar to the cooling paths published in previous papers (i.e., Long et al., 2012?), or better, implement the published routines to compute correct thermochronologic ages from cooling histories.
Reply: Agreed. We have improved the thermochronology model. See reply to the same comment by Reviewer 1.
 In the method section, I do not understand why you present a simplified solution that does not gives very accurate results. First, this is noise in the paper, and second, it decrease the confidence we could have in your other strong simplifications (temperature model, cooling model,…). I would remove this simplified solution, or explain why this is important.
Reply: We have removed the simplified solution from the manuscript
 In the validation section, I do not understand how you set up your numerical model, i.e., what is the difference between the numerical model and the full analytical solution. You probably need to better explain (with an additional sketch?) your numerical model with the parameters used.
Reply: we have tried to better describe the numerical model in the revised manuscript. All it is is particle tracking based on velocity field calculated by eq. 8 & 9. We feel that devoting a sketch to this would probably be overkill.
 And finally, in your application to a mountain belt, you need to explain with better arguments why you choose this section, because for instance the Central Nepal section also matches the arguments you present here. Also, the comparison between the restored crosssection published and your prism model differs importantly in some places. So you need to discuss the impact on predicted ages of your approximations in the geometry of the prism.
Reply: We did or did not add a 2^{nd} xsection. It is true that other cross sections would also be suitable to test our model – even outside the Himalayas. We chose this section, as it has a high amount of highquality data, and is relatively wellstudied. Our point of this manuscript is not to criticize the existing model, but to present an alternative interpretation as a second endmember, if you wish. Generally it has been shown that cross sections are often only poorly constrained at depth. (In)famous examples exist e.g. from the Alpine foreland, or the Zagros mountains, where even based on highresolution data, different geometric solutions are possible. This is also true for the chosen cross section. The internal geometry of the wedge is not that well constrained. See for instance Coutand et al. (2014) who explore various different models for this that are all consistent with the relatively sparse geophysical data.
So, because of the strong simplifications you made, I am not sure that you really can compare predictions with a real dataset, and, more, that you can use this comparison to extract strong conclusions on how to interpret lowtemparature data and if the Himalayas are in steay state or not, specially wihtout moderating your sentences.
Reply: The revised thermal model and thermochronology model provide more realistic thermochronology ages that can be compared better with the data. We did moderate the sentences concerning steady state in the revised version of the manuscript.
Specific comments
L5: precise where (approx. longitude for instance) the cross section is located in the Himalayas
Reply: this is shown in the inset map in Fig. 7b
L67: change “at a large scale deformation” to “deformation at a large scale” or “large scale deformation”?
Reply: Ok
Somewhere in the abstract, you need to specify taht your model is only in 2D and that it does not take in account any relief (that is important for comparison with real dataset).
Reply: We have added a clarification to the abstract that the model is 2D. We have also added a brief discussion that the model does not take into account relief, which may affect the comparison with thermochronometers.
L11 & L13 & L14: add “e.g.,” before the reference(s)
Reply: Ok.
L24: and Batt et al., 2001 model? You need to say in what your approach differs from Batt et al., 2001!
Reply: See reply to the same comment by Reviewer 1.
L2526: Expend the idea present in this sentence as the comparison is not straight forward.
Reply: We explain that this reference solved the advective transport equation. Which is the same equation that one would need to solve for rock particle transport in an orogenic wedge. Although the boundary conditions are admittedly quite different.
L91: vxc ?
Reply: Ok. See reply to the same comment by Reviewer 1.
L94: add a ref to the appendix where you explain how you defined these variables
Reply: The variables are defined in the two following equations. Which we feel would be better here than in the appendix.
L106107: This is not what there is in the caption of Fig. 4. Correct it.
Reply: See reply to the same comment by Reviewer 1.
L137138: Why showing the simplified solution if the full solution is a lot better? Does this solution give advantages when computing? If they are reasons to show this simplified solution, explain them, or remove the simplified solution from the paper.
Reply: See reply to the same comment by Reviewer 1.
L139: “The error increases with depth”. This part is a bit confusing: on the figs. 5 and 6, the colors representing the errors are inversed, they do not show the same between figs. 5 and 6. Please, to limit the confusion, put the same color bar.
Reply: Agreed, we have used the same colorbar for the 2 figures.
Also, this sentence is true for the fig. 6d, but not for fig.5d where I see first a decrease of the error with depth, and then a rapid increase.
Reply: agreed. However, we have removed the simplified solution, so this panel is no longer in the revised manuscript.
L140142: So, again, why showing the simplified solution if it does not permit to match well the full solution?
See previous reply
L148151: here, you give your arguments (good thermochronologic data coverage and high convergence rates) to justify that you choose this section. But if you take the Kathmandu/Trisuli or Sutlej region, it matches the same argument, so why didn’t you took these sections also? You need to give more arguments on the choice of your section.
Reply: The main focus of the manuscript is the new analytical solution. We intended to keep the comparison to field data limited to one example. Testing it with other dataset is the scope of follow up work. We also do not wish to claim that this would be the best dataset to compare it to. We simply took a high quality published study with a wide range of thermochronometers and numerical crosssection balancing in the Himalayas that we were aware of.
L159: m m1, not m m1
Reply: Ok. Thanks for spotting this error.
L160: m m1 not m m1
Reply: Ok. Thanks for spotting this error.
L160: 200 km? from the fig 7, it looks closer to 100 kmlong than 200 kmlong. So what is wrong, the text or the legend of Fig. 7?
Reply: See reply to similar comment by reviewer 1.
L160: add ref to fig. 7
Reply: Ok
L157160: You may describe and quantify the differences of the complex geometry of the wedge build from geological and geophysical data, and your triangular simplified wedge geometry. You then need to explain what are the consequences of these differences to your model results, i.e. where your results will be accurate, and where they will not.
Reply: We would like to refrain from a detailed discussion of geological data and resulting expectations of where our model may be accurate or not, and would like to focus purely on a comparison of the model results with the data.
I ask this question because the Long et al., 2012 crosssection show an important ramp of the MHT (representing the base of the wedge), also, your model does not take in account the important frontal ramp. Though, Van der Beek et al., 2006, Robert et al. 2009, 2011 have shown that this geometry of the base of the wedge has a huge impact on the thermochronological record. So, such a simplification risks to make inaccurate predictions. You really need to reinforce this section.
Reply: We thank the reviewer for pointing us to the studies by Van der Beek et al. and Robert et al. It is true that in multiple occasions the presence or absence of a ramp impacts exhumation rates. An exciting result of our study is that the thermochronological data can be fitted without such a ramp, and at present the fit is better than in the existing model including complex geometry and localized strain. We would like to propose our model as a viable alternative.
L162164: I think that the set of thermochronometers from the Kathmandu/Trisuli section is larger (See Herman et al., 2010 for instance). So why did you choose the Bhutan section and not the central Nepal section?
Reply: There’s certainly a lot of data in the Herman et al. (2010) paper. However, we would like to refrain from testing additional crosssections in this manuscript. The choice for the Bhutan crosssection was based on the availability of several studies and balanced crosssections, which we felt made this an attractive target to compare with our more simple approach.
L165: 15°C km1 (also, generally, remove the space between numbers and the °C sign)
Reply: This was corrected in the revised version of the manuscript.
L165167: I do not really understand the thermal model.
 Do you update this thermal model through time while you are advecting your particles? Do you take in account lateral thermal advection? If yes, you need to describe this model precisely in the method section. If not, I do not understand how you will be able to compute accurate thermochronological ages as in your wedge model, you have a lateral component that is important.
 Do you have heat production (i.e., radioactivity) in your model?
 What are the thermal characteristics (i.e. thermal diffusivity) of the rocks you are advecting in the model? This is crucial for the thermal model, you need to give them.
Reply: we have completely revised the thermal model used in the manuscript. See also the replies to reviewer 1.
L167172: this is a strange choice. If you want to predict thermochronological ages, this is better to use existing cooling models. This is not so hard because you can find most of the routines in the literature or on the Web. This approach is oversimplified!
Reply: We have completely revised the thermochronology model used in the manuscript. See also the replies to reviewer 1.
L178180: I do not understand what you do here. What do you input in the downhill simplex algorithm? You need to explicitly explain what you are doing here in details.
Also, what is the MAE of the calculated thermochronological ages?
Reply: This is a widely used optimization / model calibration algorithm that simply minimizes the error between the model and the data by adjusting the model parameters. As this approach has been used so extensively in the past, we would like to refer to the cited references for more information.
The MAE is the mean absolute error as defined earlier in the manuscript. See also reply to the same question by reviewer 1.
L181182: where are the results/comparison of these models?
Reply: The model fit for these runs is shown in fig. 9
L186: With such an ultrasimplified thermal model that does not take in account lateral heat advection (does it?), and an ultrasimplified cooling calculation, I am not sure this is relevant to compare your predictions to the data. Before to do that, you need to refine your thermal model and cooling models. Ask any quantitative thermochronologist about what is often a critical point when submitting papers for publication: there is almost always a strong debate around the thermal model/parameters used to interpret thermochronologic data/model. This is really a critical point.
Reply: The thermochronology model was revised completely, see previous replies.
L188: the exploration of the parameter space is very rough. In the introduction, you explained that you want to propose a model that does not need a lot of computing time and resources. So if this is the case (is it the case? You do not mention it until now), it would be easy and rapid to refine your parameter search with a smaller stepping.
Reply: The exhumation model is obviously very fast since it only involves calculating two equations, which is done in a few milliseconds. However, the bottleneck in the new implementation is the time required for the numerical heatflow model, which adds significant computational expense. Nonetheless we have revised the parameter space search and have added more steps.
L208: I would not say a better fit: you fit well the general trend, but not the details, and McQuarrie and Ehlers better explain the different local ages variations (because their model is based on numerous geological observation, and is thermally more complex), except for the farthest samples. You need to moderate your sentence, or better explain the comparison.
Reply: The better fit refers to the better model fit statistic of the analytical solution. We would like to refrain from a more qualitative discussion of local errors or not and base the assessment of model performance on statistics. In the new version of the manuscript we state more explicitly that our solution is an alternative approach.
L209201: Ask the authors the source of their graph.
Reply: We have repeated our request and have now received the data and incorporated this in the manuscript.
L2122013: You should change the sentence to “This suggest that, for this crosssection, at large scale, deformation of the wedge is represented by uniform deformation”, or for something like that. This is because your fit does not reproduce the details of the data because 1) your thermal/cooling model is oversimplified, and 2) you do not take in account relief.
Reply: We agree that the model probably works best at large scale and have added the word at large scale to this sentence. However, in our opinion, in general the simplest model that fits the data the best should be the preferred model. Whether or not a model would come along in the future with more complexity that includes relief or the movement along single faults would fit the data better is speculative at this point. Note that the thermochronology model and thermal model have been improved for the revised manuscript.
L269, eq A5: There is a problem in this equation. A dx is missing (in place of the L?) in the left member.
Reply: True, that was a typo. Thanks for spotting this!
Figures
Figure 2:
The font is to small, increase it (it is hard to read the name of the vectors); in the caption, change the last sentence to “The symbols are listed and described in Table 1”
Reply: ok, we have implemented these suggestions
Figure 3 & 4:
Does the length of the arrows mean something? If yes, it maybe needs a scalebar, or at least an explanation in the caption
Reply: The length of the arrow scales with the velocity. The velocity is also visible in the colors, which is why we feel that adding a scale bar would be a bit double. However, we did add a statement to the caption that arrow size scales with velocity.
Figure 5 & 6;
See my specific comment about the colorbar for panels d). Why using a ““ in the isochrones? It will simplify the fig if you remove this ““
Reply: ok
Figure 7:
 a) the writings inside the fig. are not readable. Please increase the size of the font.
On this fig., you may mark the MHT (which is the base of your wedge) with a thicker line, so the reader will better see what approximation you are doing in your model.
On a), you also may add the thermochronologic data (on a mapview also?) you use in the Himalayan case.
You also may write in the caption what is the red triangle…
Reply: Ok. We prefer to keep the thermochronology data in the model result figures to not overcrowd this figure. We did add the sample locations to this figure.
Figure 8:
Why the MAE is in a and ages in Ma? Cleaner if you gibe MAE in Ma.
Reply: Agreed, we have converted this to Ma.
Figure 10:
In caption, correct “calculated themrochronometer…” by “calculated theRmochronometer…”
Reply: ok
Citation: https://doi.org/10.5194/se202122AC2

RC3: 'Comment on se202122', Anonymous Referee #3, 08 Jul 2021
Dear authors,
This article shows a new analytical solution to predict thermochronological dataset in an orogenic wedge. This simple model assumes a transport only accommodates by basal detachment. I appreciate all the details available on the different parameters implemented in this model.
The authors applied this model on a profile perpendicular to the Himalayan wedge. Results make in evidence a good correlation between predicted ages and observed ages using an uniform model without effects of individual tectonic structures. Authors concludes that the principal implication of these results is a good reproducibility of the thermochronological data with a simple model and a possible steady state evolution of the Himalayan orogenic wedge.
The manuscript is well written and we have all the details on the model and the analytical procedure, however major points have to be clarified and discussed, see my general comments.
General comments
1/ The main topic of the article have to be clarified l. 154155 " Note that the goal here is not to provide a detailed geological case study, but to demonstrate the use of the equation to calculate deformation and exhumation rates” but authors highlight in abstract and conclusion “The results also imply that this part of the Himalayas may be in steadystate.” (l.78) and “This indicates that the Himalayas may be in steadystate and that, at a large scale, the exhumation of mature mountain belts may be approximated by a relatively simple model of uniform and steadystate deformation, accretion and transport” (l. 239241). This conclusion may be true but it is not relevant for the Himalaya Mountain range using one cross section.
2/ Similar model have been developed by Batt et al. 2001 (JGR), a comparative study of the 2 models or at least a discussion on the main differences between the models must be developed. “However, to our knowledge no analytical solutions exist for the relation between deformation and exhumation of mountain belts.” (l.2324), this sentence is partially true and must be modified to show the specificity of this new model.
3/ Constant geothermal gradient is a big assumption in the models and it is not realistic according studies of Coutand et al., 2014 (JGR) and McQuarrie and Ehlers, 2015 (tectonics). This point and the impact on the model must be developed in the discussion. New kinematic models are not implemented (see Ault et al., 2019, tectonics for a synthesis) specify a closure temperature is possible and easier to implemented in the models but residence time in the Partial Annealing Zone (PAZ) and Partial Retention Zone (PRZ) need to be low to assume a closure temperature.
4/ Discussion of the results must be more detailed using differences between the predicted and observed ages and to discuss potential fault activity along the cross section. The comparison with study of McQuarrie and Ehlers, 2015 (tectonics) based on the better fit of the new model seems a bit complex. Differences between the 2 models are not specified in the text.
5/ This model applied on only one cross section in orogenic wedge of the Himalayas does not allow to provide strong conclusion on the Himalayas tectonic regime.
You can find detailed comments in the sections below:
Specific comments
L.5 Precise the location of the transect.
L.67 Remove this sentence, themochronological data not explained by the model can result of individual fault activity. See my general comment
L.78 This conclusion is not relevant if we consider ages do not fit with the model, see my previous comment.
L.17 I agree, one of the big advantages of the model presented is the low computational cost compare to other thermokinematic models.
L.2324 Please consider Batt et al. 2001 (JGR), see my general comment
L.91 Vxc ?
L.104 Fig.3 to be homogeneous in the text
L.106107 It is not the good caption for the figure
L.135139 Develop the simplified solution is necessary to the discussion ? If not you can remove this part in appendix and just comment the among of differences between the 2.
L.139: Considering the color bar on Fig.5d error decrease in the first kilometers.
L.143145 Is it possible to merge the 2 figures ? Also a common color bar for the 2 models seems more adequate.
L.154155 It seems important to clarify the goal of the paper. This sentence suggests that the main goal of the paper is to highlight numerical approach if it is the case other journals are more relevant and therefore you cannot conclude on geological implications.
L.159160 m.m1 ; the choice of each input parameters have to be justified by references
L.162164 A table in appendix with all the thermochronological data implemented in the model can be helpful for the reader. You have not selected MAr partially resetted (Long et al., 2012, Tectonics), you have to add a sentence on that.
L.165 Uniform geothermal gradient is a big assumption on this surface. Important thermal perturbation can be observed at this scale (McQuarrie and Ehlers, 2015, tectonics).
L.165166 add °C km1
L.167169 Use references for the different “closure temperature” used and not “resetting temperature”, numerous models showed that is a quite large range of temperature in particular if exhumation rates are low (see Ault et al. 2019, Tectonics).
L.169172 I agree but it can be an important bias on the model and you can have an idea calculating time residence of each sample in PAZ for AFT and PRZ for ZHe. It can be a sentence to justify that samples do not stay long time in PAZ and PRZ, in this case proxy on closure temperature can be used.
L.179 Define MAE.
L.186 It can be helpful for the reader to locate the different ages and samples on the cross section and having the possibility to link ages with the different tectonostratigraphic units. Scale of the Fig. 7a seems false, it is not 200 km long with this scale.
L.198 More discussion about the data and differences between observed and predicted ages can be useful. I am very curious to see if AHe and AFT which not fit come from a particular tectonostratigraphic unit and If it is the case it can be associated to a major fault activity ?
L.200 Add °C km1
L.208 Comparison with model from McQuarrie and Ehlers (2015) seems difficult. Your new model fit a global trend but previous model includes more parameters and explain better 2^{nd} order trend in the dataset.
L.209210: Digitizing from figure is not acceptable for publication. Ask the data.
L.223229 As suggested previously, it can be relevant to discuss the differences between observed and predicted ages to discuss the reason of the differences geothermal gradient, fault activity ?
L.239240 This sentence is for a profile in the southern flank of Himalayas and cannot be in the conclusion.
Figures:
Fig 2 Increase the size of the figure and the font size
Fig 3 and 4 Increase size of a) b) and c). It can be more lisible same comment for Figs. 3,4,5,6, 8, 9 and 10. The caption of Fig 4 do not correspond at the figure.
Fig 5 and 6: Merge the 2 figures, use the same color bar with the same range of value. Delete minus before isochrons ages.
Fig 7 It can be helpful for the reader to locate the different ages and samples on the cross section and having the possibility to link ages with the different tectonostratigraphic units. Scale of the Fig. 7a seems false, it is not 200 km long with this scale.
Fig 10 thermochronometer in the caption
Fig 11 R2= 0.43 not 0.43
Citation: https://doi.org/10.5194/se202122RC3 
AC3: 'Reply to RC3', Elco Luijendijk, 12 Sep 2021
Note: Our replies to the comments by Reviewer 3 are shown below each comment in italics
Dear authors,
This article shows a new analytical solution to predict thermochronological dataset in an orogenic wedge. This simple model assumes a transport only accommodates by basal detachment. I appreciate all the details available on the different parameters implemented in this model.
The authors applied this model on a profile perpendicular to the Himalayan wedge. Results make in evidence a good correlation between predicted ages and observed ages using an uniform model without effects of individual tectonic structures. Authors concludes that the principal implication of these results is a good reproducibility of the thermochronological data with a simple model and a possible steady state evolution of the Himalayan orogenic wedge.
The manuscript is well written and we have all the details on the model and the analytical procedure, however major points have to be clarified and discussed, see my general comments.
General comments
1/ The main topic of the article have to be clarified l. 154155 " Note that the goal here is not to provide a detailed geological case study, but to demonstrate the use of the equation to calculate deformation and exhumation rates” but authors highlight in abstract and conclusion “The results also imply that this part of the Himalayas may be in steadystate.” (l.78) and “This indicates that the Himalayas may be in steadystate and that, at a large scale, the exhumation of mature mountain belts may be approximated by a relatively simple model of uniform and steadystate deformation, accretion and transport” (l. 239241). This conclusion may be true but it is not relevant for the Himalaya Mountain range using one cross section.
Reply: This is a good point, and the reviewer is right that based on our study alone, we cannot make inferences for the entire Himalayas. Our results are however valid for the particular region. We have modified the statement accordingly.
2/ Similar model have been developed by Batt et al. 2001 (JGR), a comparative study of the 2 models or at least a discussion on the main differences between the models must be developed. “However, to our knowledge no analytical solutions exist for the relation between deformation and exhumation of mountain belts.” (l.2324), this sentence is partially true and must be modified to show the specificity of this new model.
Reply: This oversight has also been raised by reviewers 1 & 2. A sentence on this previous model has been included in the introduction in the new version of the manuscript.
3/ Constant geothermal gradient is a big assumption in the models and it is not realistic according studies of Coutand et al., 2014 (JGR) and McQuarrie and Ehlers, 2015 (tectonics). This point and the impact on the model must be developed in the discussion. New kinematic models are not implemented (see Ault et al., 2019, tectonics for a synthesis) specify a closure temperature is possible and easier to implemented in the models but residence time in the Partial Annealing Zone (PAZ) and Partial Retention Zone (PRZ) need to be low to assume a closure temperature.
Reply: Agreed, we have updated the thermal and thermochronology model in the revised manuscript. See also our replies to comments by reviewer 1 and 2.
4/ Discussion of the results must be more detailed using differences between the predicted and observed ages and to discuss potential fault activity along the cross section. The comparison with study of McQuarrie and Ehlers, 2015 (tectonics) based on the better fit of the new model seems a bit complex. Differences between the 2 models are not specified in the text.
Reply: Agreed, we have expanded the discussion of the comparison between our results and the previous work by McQuarrie and Ehlers (2015).
5/ This model applied on only one cross section in orogenic wedge of the Himalayas does not allow to provide strong conclusion on the Himalayas tectonic regime.
Reply: We agree and would like to refrain from statements about the entire Himalayas, see also the replies to the comments by reviewer 1 and 2.
You can find detailed comments in the sections below:
Specific comments
L.5 Precise the location of the transect.
Reply: Ok.
L.67 Remove this sentence, themochronological data not explained by the model can result of individual fault activity. See my general comment
Reply: The model fit is relatively good. We therefore feel that claiming that at a large scale the mode fits the trend of the data is valid here.
L.78 This conclusion is not relevant if we consider ages do not fit with the model, see my previous comment.
Reply: We are not sure what the reviewer means here. The model fit is shown and discussed. Why would the conclusion not be relevant?
L.17 I agree, one of the big advantages of the model presented is the low computational cost compare to other thermokinematic models.
L.2324 Please consider Batt et al. 2001 (JGR), see my general comment
Reply: Agreed, see previous reply
L.91 Vxc ?
Reply: Fixed.
L.104 Fig.3 to be homogeneous in the text
Reply: Thanks for noticing, we have corrected this.
L.106107 It is not the good caption for the figure
Reply: It is actually, we are not sure what is wrong with the caption here? Could you specify the issues with the caption?
L.135139 Develop the simplified solution is necessary to the discussion ? If not you can remove this part in appendix and just comment the among of differences between the 2.
Reply: We have removed the simplified solution from the manuscript.
L.139: Considering the color bar on Fig.5d error decrease in the first kilometers.
Reply: Agreed, we have expanded the discussion of the errors here
L.143145 Is it possible to merge the 2 figures ? Also a common color bar for the 2 models seems more adequate.
Reply: We feel that merging these 2 figures would produce a figure that would be too large.
L.154155 It seems important to clarify the goal of the paper. This sentence suggests that the main goal of the paper is to highlight numerical approach if it is the case other journals are more relevant and therefore you cannot conclude on geological implications.
Reply: I presume the reviewer means the analytical approach not the numerical? It is true that ithe manuscript presents an advance in methods and not a case study or a new knowledge of a geological process. However, we feel that the new equation and its potential applications are of sufficient interest to thermochronologist and geologists to warrant publication in Solid Earth.
L.159160 m.m1 ; the choice of each input parameters have to be justified by references
Reply: The references are mentioned in the first part of the sentence (The geometry of the wedge is based on published geological crosssections (Long et al., 2012; Coutand et al., 2014; McQuarrie and Ehlers, 2015)
L.162164 A table in appendix with all the thermochronological data implemented in the model can be helpful for the reader. You have not selected MAr partially resetted (Long et al., 2012, Tectonics), you have to add a sentence on that.
Reply: Thanks for noticing. We have clarified why the nonreset MAr samples were not used. The thermochronology data are a 100% copy of data used by McQuarrie and Ehlers (2015) and as such we feel that it would be superfluous to include a table in the manuscript. However, the data are included in the GitHub repository and the zenodo publication of the model code, which we now refer to in the main manuscript.
L.165 Uniform geothermal gradient is a big assumption on this surface. Important thermal perturbation can be observed at this scale (McQuarrie and Ehlers, 2015, tectonics).
Reply: The thermal model was updated to include steadystate conduction and advection
L.165166 add °C km1
Reply: Ok.
L.167169 Use references for the different “closure temperature” used and not “resetting temperature”, numerous models showed that is a quite large range of temperature in particular if exhumation rates are low (see Ault et al. 2019, Tectonics).
Reply: We have revised the thermochronology model and have removed this part of the manuscript.
L.169172 I agree but it can be an important bias on the model and you can have an idea calculating time residence of each sample in PAZ for AFT and PRZ for ZHe. It can be a sentence to justify that samples do not stay long time in PAZ and PRZ, in this case proxy on closure temperature can be used.
Reply: We have revised the thermochronology model and have removed this part of the manuscript.
L.179 Define MAE.
Reply: There is a definition in the same sentence (mean absolute error).
L.186 It can be helpful for the reader to locate the different ages and samples on the cross section and having the possibility to link ages with the different tectonostratigraphic units. Scale of the Fig. 7a seems false, it is not 200 km long with this scale.
Reply: The modeled crosssection was larger than the geological crosssection to avoid boundary effects in the model. The sample locations have been added to the crosssecttion
L.198 More discussion about the data and differences between observed and predicted ages can be useful. I am very curious to see if AHe and AFT which not fit come from a particular tectonostratigraphic unit and If it is the case it can be associated to a major fault activity ?
Reply: We feel that this is out of scope of the manuscript. The model that we propose does not resolve individual faults. Alhtough it could be adjusted to do so, that would warrant a separate publication.
L.200 Add °C km1
Reply: We have revised this section of the manuscript
L.208 Comparison with model from McQuarrie and Ehlers (2015) seems difficult. Your new model fit a global trend but previous model includes more parameters and explain better 2^{nd} order trend in the dataset.
Reply: The general consensus on assessment of model performance is that all else being equal models with less parameters are preferred over more complex models.
L.209210: Digitizing from figure is not acceptable for publication. Ask the data.
Reply: After a repeat request we have now gotten the data.
L.223229 As suggested previously, it can be relevant to discuss the differences between observed and predicted ages to discuss the reason of the differences geothermal gradient, fault activity ?
Reply: We feel that this is out of scope for this manuscript and would require a separate more detailed study.
L.239240 This sentence is for a profile in the southern flank of Himalayas and cannot be in the conclusion.
Reply: Agreed, we have revised this sentence to state that this section in the Himalayas may be in steadystate.
Figures:
Fig 2 Increase the size of the figure and the font size
Reply: agreed, this suggestion has been implemented.
Fig 3 and 4 Increase size of a) b) and c). It can be more lisible same comment for Figs. 3,4,5,6, 8, 9 and 10. The caption of Fig 4 do not correspond at the figure.
Reply: agreed. We are not sure what is wrong with the caption of Fig. 4.
Fig 5 and 6: Merge the 2 figures, use the same color bar with the same range of value. Delete minus before isochrons ages.
Reply: We disagree with the merging of these figures. It would results in a very large figure in which the subpanels would be difficult to read and interpret.
Fig 7 It can be helpful for the reader to locate the different ages and samples on the cross section and having the possibility to link ages with the different tectonostratigraphic units. Scale of the Fig. 7a seems false, it is not 200 km long with this scale.
Reply: We have added the sample positions. And its true that the crosssection is not 200 km, we now explain the reason for this in the main manuscript.
Fig 10 thermochronometer in the caption
Reply: ok
Fig 11 R2= 0.43 not 0.43
Reply: The R2 is actually negative here. Note that the coefficient of determination (R2) can be negative if the variance of the model error exceeds the variance of the data, which is the case here.
Citation: https://doi.org/10.5194/se202122AC3

AC3: 'Reply to RC3', Elco Luijendijk, 12 Sep 2021
Elco Luijendijk et al.
Model code and software
wedgex: calculate the exhumation of an orogenic wedge Elco Luijendijk https://doi.org/10.5281/zenodo.4571576
Elco Luijendijk et al.
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