15 Mar 2021
15 Mar 2021
An analytical solution for the exhumation of an orogenic wedge and a comparison with thermochronology data
 ^{1}Department of structural geology and geodynamics, University of Göttingen, Goldschmidtstrasse 3, 37077, Göttingen, Germany
 ^{2}Geological Institute, RWTH Aachen University, Wüllnerstr. 2, 52062, Aachen, Germany
 ^{3}Mathematisches Institut, University of Göttingen, Bunsenstraße 3–5, 37073, Göttingen, Germany
 ^{4}Department of Geography and Geology, University of Salzburg, Hellbrunnerstraße 34/III, 5020, Salzburg, Austria
 ^{1}Department of structural geology and geodynamics, University of Göttingen, Goldschmidtstrasse 3, 37077, Göttingen, Germany
 ^{2}Geological Institute, RWTH Aachen University, Wüllnerstr. 2, 52062, Aachen, Germany
 ^{3}Mathematisches Institut, University of Göttingen, Bunsenstraße 3–5, 37073, Göttingen, Germany
 ^{4}Department of Geography and Geology, University of Salzburg, Hellbrunnerstraße 34/III, 5020, Salzburg, Austria
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: open (extended)

RC1: 'Comment on se202122', Anonymous Referee #1, 27 Apr 2021
reply
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?

RC2: 'Comment on se202122', Anonymous Referee #2, 30 Apr 2021
reply
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…”
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.
Viewed
HTML  XML  Total  BibTeX  EndNote  

390  84  8  482  1  1 
 HTML: 390
 PDF: 84
 XML: 8
 Total: 482
 BibTeX: 1
 EndNote: 1
Viewed (geographical distribution)
Country  #  Views  % 

Total:  0 
HTML:  0 
PDF:  0 
XML:  0 
 1