Technical Note: Incorporating topographic deflection effects into thermal history modelling

Ketcham, Richard A.

doi:https://doi.org/10.5194/egusphere-2025-901

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https://doi.org/10.5194/egusphere-2025-901

Preprints

07 Mar 2025

| 07 Mar 2025

Technical Note: Incorporating topographic deflection effects into thermal history modelling

Richard A. Ketcham

Abstract. This contribution describes a set of equations and relations to calculate accurate cooling paths through the 2D temperature field of an exhuming region with periodic topography. A 1D model adequately captures the time-varying component of the system, making the computation efficient. A series of 2D finite element models demonstrate how temperatures below the periodic mid-slope, or mean topography, can be mapped to those below ridges and valleys, and how these transitions vary with topographic period and amplitude and the ratio of the near-surface geotherm to the atmospheric lapse rate. These new calculations are implemented into HeFTy to support multi-sample modelling of samples collected along topographic profiles, particularly for terranes with long-lived topography that exhumed through an inflected temperature field.

Received: 26 Feb 2025 – Discussion started: 07 Mar 2025

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Journal article(s) based on this preprint

18 Sep 2025

Technical note: Incorporating topographic deflection effects into thermal history modelling

Richard A. Ketcham

Geochronology, 7, 449–458, https://doi.org/10.5194/gchron-7-449-2025,https://doi.org/10.5194/gchron-7-449-2025, 2025

Short summary

Richard A. Ketcham

Interactive discussion

Status: closed

RC1:
'Comment on egusphere-2025-901', Jean Braun, 26 May 2025

\documentclass[11pt]{report}

\begin{document}

I have read the technical note entitled: "Incorporating topographic deflection effects into thermal history modelling" by Richard Ketcham with great interest. I believe this is a very useful contribution to the field of quantitative thermochronology, especially because it is incorporated in a widely used software for the interpretation of thermochron data (HeFTy).

The note proposes semi-analytical solutions, i.e., combining approximate analytical solutions to the 2D heat equation to empirical relationships derived, for the most, from numerical (finite element) solutions of the same equation. These solutions provide useful corrections for the cooling history of rocks being exhumed to the Earth's surface that are then used to compute thermochronological ages and other quantities to be compared with observational constraints.

I have a few points that I believe need to be addressed (all of a technical nature) and that would greatly help future users of the software in their understanding of the method and the advantages and limitations of its use in the interpretation of their data.

1. The corrections that are proposed here are meant to represent the effect of a finite amplitude, potentially time-dependent, topography on the geometry of the underlying isotherms, including the effect of vertical advection. As demonstrated by the authors these effects can be substantial. However, little mention is made of the effect of horizontal advection, which can be, in many cases, dominant over other effects. Indeed, motion of rock particles parallel to dipping faults can cause them to experience cooling paths that are drastically different from those obtained assuming pure vertical advection. Example of this can be seen in our interpretation of thermochron data in the Southern Alps of New Zealand (I refer here to the PhD works of G Batt and F Herman under my supervision many years ago). I do not suggest here that this perturbation be added to the current note/work, but that this fact should be made clear to the user.

2. The proposed correction(s) are all based on the assumption that the topography has a single wavelength. However, surface topography is often composed of more than a single wavelength. It would be very useful to better describe what the user should use as a topographic wavelength and amplitude in a given situation. For this, it would be useful to indicate that each terhmochronometric system (as defined by its closure temperature or temperature sensitivity) is sensitive to different topographic wavelengths because the depth (and thus the temperature) to which the topographic perturbation propagates is strongly (exponentiallY) dependent on the wavelength. This warrants a short paragraph in this report and, potentially in the user interface of the software, to help the user decide what wavelength and which amplitude is to be used in a real case application.

3. Alternatively, the software could be adapted to use a multi-wavelength topography but this would require that the corrections (for each wavelength and amplitude) be simply combined (added?). This would require additional work to implement and could be quite useful, but potentially beyond the scope of this short note. I leave it to the author to appreciate whether this should be done.

4.My last point, and maybe the most critical one, concerns the time-dependent solution. If I understand well the author proposes to use the steady-state solutions derived analytically and empirically to cases where the topography grows with time by simply using a time varying value for $H_0$. As shown by the author, this seems to provide relatively good results but my suspicion is that this is because the rate of topographic change remains relatively low compared to the rate of heat diffusion, i.e., the cases he explores must be at relatively low Peclet number. I suspect that a very rapid incision event (that would make the topographic relief grow rapidly with time) would produce a thermal response that cannot be adequately approximated by a series of steady-state solutions. Again, I do not suggest to the author to improve his solution but to warn the users of the limitation (in terms of how fast topography can change) of the method proposed here.

I also have a few minor comments:

a. I have tried to reproduce the predictions of this semi-analytical approach to compare them to solutions of the heat equation obtained by Pecube but failed to do so because the value of some constants, or the type or value of basal boundary conditions were not clearly given. It would be good if the author could include all the information necessary to reproduce the results shown in the note.

b. I do not think there is an analytical solution of the heat equation in Fox et al, 2014.

c. It is not clear what the "relative vertical rotation" mentioned at line 38 is; a few words of explanation would be useful (or a small sketch).

d. On line 78, there is a reference to "Finite element modelling"; this sounds like it requires a reference or it should be modified to "Our finite element modelling, shown in Figure X", if this is indeed the case.

e. On Line 106 the statement that "a constant basal gradient condition never converges to a steady-state with continuous erosion" is not correct, in my opinion. In 1D, the solution is:
$$T(z) = T_0+\frac{G\kappa}{u}e^{Lu/\kappa}(1-e^{zu/\kappa})$$
where $G$ is the assumed basal gradient at depth $L$.

Citation: https://doi.org/10.5194/egusphere-2025-901-RC1
- AC1:
  'Reply on RC1', Richard A. Ketcham, 19 Jun 2025
  
  See attached PDF for response.
  
  Citation: https://doi.org/10.5194/egusphere-2025-901-AC1
  - RC2: 'Comment on egusphere-2025-901', Jean Braun, 20 Jun 2025
    
    Many thanks for the very thorough discussion of my review by the author. Please find attached a short comment in response to the author's comment on the last point (e) I raised in my review.
    
    Citation: https://doi.org/10.5194/egusphere-2025-901-RC2
    
    AC2: 'Reply on RC2', Richard A. Ketcham, 21 Jun 2025
    
    I greatly thank the reviewer for his constructive and instructive response, which is a resource unto itself. The negative sign in the final exponential term had been omitted from the solution provided in the initial review, leading to my confusion.
    
    Citation: https://doi.org/10.5194/egusphere-2025-901-AC2
RC3:
'Comment on egusphere-2025-901', Christoph Glotzbach, 25 Jun 2025

Dear Authors,
I enjoyed reading your manuscript. It presents an important contribution to the community, and given that this functionality is implemented in HeFTy, I am confident it will be widely used in the future to enable more accurate interpretation of thermochronological data.
The technical note is well written and pitched at a level that will be understandable and useful to the intended audience from the thermochronological community.
One point I would encourage the authors to consider relates to the automatic assignment of samples to positions along the sine-shaped topography. While the approach is elegant, real topography is often more complex. I could foresee potential issues where samples collected from short-wavelength features (e.g., secondary peaks) might be incorrectly positioned, particularly in landscapes where the dominant wavelength is much larger. A brief discussion of this limitation or potential strategies to mitigate it would strengthen the manuscript. Please also see my technical comments:

Technical corrections:
Line 47-50: A quite similar dataset was compiled by Glotzbach et al. (2015) in combination with a Fourier approach to empirically estimate the perturbation of isotherms in complex 2D-3D topographic situations. You may want to cite this work.
Line 54-55: The choice of parameterisation would require some justification or showing a few alternative models, e.g. running until steady-state or with a different thermal conductivity. This would allow for the estimation of the uncertainty introduced by the choice of parameterisation.
Line 155: This is not very clear, maybe you can also give the relative deviation and have a figure showing the difference between numerical and empirical solution, taking into account the uncertainty in c2.
Line 211-212: Would be good to investigate under which boundary conditions the temperature difference is larger than 10°C and report this, to prevent users from stating that they can model their data without taking into account the topographic deflection effects.

Citation: https://doi.org/10.5194/egusphere-2025-901-RC3
- AC3: 'Reply on RC3', Richard A. Ketcham, 09 Jul 2025
  
  See attached PDF for response
  
  Citation: https://doi.org/10.5194/egusphere-2025-901-AC3

Interactive discussion

Status: closed

RC1:
'Comment on egusphere-2025-901', Jean Braun, 26 May 2025

\documentclass[11pt]{report}

\begin{document}

I have read the technical note entitled: "Incorporating topographic deflection effects into thermal history modelling" by Richard Ketcham with great interest. I believe this is a very useful contribution to the field of quantitative thermochronology, especially because it is incorporated in a widely used software for the interpretation of thermochron data (HeFTy).

The note proposes semi-analytical solutions, i.e., combining approximate analytical solutions to the 2D heat equation to empirical relationships derived, for the most, from numerical (finite element) solutions of the same equation. These solutions provide useful corrections for the cooling history of rocks being exhumed to the Earth's surface that are then used to compute thermochronological ages and other quantities to be compared with observational constraints.

I have a few points that I believe need to be addressed (all of a technical nature) and that would greatly help future users of the software in their understanding of the method and the advantages and limitations of its use in the interpretation of their data.

1. The corrections that are proposed here are meant to represent the effect of a finite amplitude, potentially time-dependent, topography on the geometry of the underlying isotherms, including the effect of vertical advection. As demonstrated by the authors these effects can be substantial. However, little mention is made of the effect of horizontal advection, which can be, in many cases, dominant over other effects. Indeed, motion of rock particles parallel to dipping faults can cause them to experience cooling paths that are drastically different from those obtained assuming pure vertical advection. Example of this can be seen in our interpretation of thermochron data in the Southern Alps of New Zealand (I refer here to the PhD works of G Batt and F Herman under my supervision many years ago). I do not suggest here that this perturbation be added to the current note/work, but that this fact should be made clear to the user.

2. The proposed correction(s) are all based on the assumption that the topography has a single wavelength. However, surface topography is often composed of more than a single wavelength. It would be very useful to better describe what the user should use as a topographic wavelength and amplitude in a given situation. For this, it would be useful to indicate that each terhmochronometric system (as defined by its closure temperature or temperature sensitivity) is sensitive to different topographic wavelengths because the depth (and thus the temperature) to which the topographic perturbation propagates is strongly (exponentiallY) dependent on the wavelength. This warrants a short paragraph in this report and, potentially in the user interface of the software, to help the user decide what wavelength and which amplitude is to be used in a real case application.

3. Alternatively, the software could be adapted to use a multi-wavelength topography but this would require that the corrections (for each wavelength and amplitude) be simply combined (added?). This would require additional work to implement and could be quite useful, but potentially beyond the scope of this short note. I leave it to the author to appreciate whether this should be done.

4.My last point, and maybe the most critical one, concerns the time-dependent solution. If I understand well the author proposes to use the steady-state solutions derived analytically and empirically to cases where the topography grows with time by simply using a time varying value for $H_0$. As shown by the author, this seems to provide relatively good results but my suspicion is that this is because the rate of topographic change remains relatively low compared to the rate of heat diffusion, i.e., the cases he explores must be at relatively low Peclet number. I suspect that a very rapid incision event (that would make the topographic relief grow rapidly with time) would produce a thermal response that cannot be adequately approximated by a series of steady-state solutions. Again, I do not suggest to the author to improve his solution but to warn the users of the limitation (in terms of how fast topography can change) of the method proposed here.

I also have a few minor comments:

a. I have tried to reproduce the predictions of this semi-analytical approach to compare them to solutions of the heat equation obtained by Pecube but failed to do so because the value of some constants, or the type or value of basal boundary conditions were not clearly given. It would be good if the author could include all the information necessary to reproduce the results shown in the note.

b. I do not think there is an analytical solution of the heat equation in Fox et al, 2014.

c. It is not clear what the "relative vertical rotation" mentioned at line 38 is; a few words of explanation would be useful (or a small sketch).

d. On line 78, there is a reference to "Finite element modelling"; this sounds like it requires a reference or it should be modified to "Our finite element modelling, shown in Figure X", if this is indeed the case.

e. On Line 106 the statement that "a constant basal gradient condition never converges to a steady-state with continuous erosion" is not correct, in my opinion. In 1D, the solution is:
$$T(z) = T_0+\frac{G\kappa}{u}e^{Lu/\kappa}(1-e^{zu/\kappa})$$
where $G$ is the assumed basal gradient at depth $L$.

Citation: https://doi.org/10.5194/egusphere-2025-901-RC1
- AC1:
  'Reply on RC1', Richard A. Ketcham, 19 Jun 2025
  
  See attached PDF for response.
  
  Citation: https://doi.org/10.5194/egusphere-2025-901-AC1
  - RC2: 'Comment on egusphere-2025-901', Jean Braun, 20 Jun 2025
    
    Many thanks for the very thorough discussion of my review by the author. Please find attached a short comment in response to the author's comment on the last point (e) I raised in my review.
    
    Citation: https://doi.org/10.5194/egusphere-2025-901-RC2
    
    AC2: 'Reply on RC2', Richard A. Ketcham, 21 Jun 2025
    
    I greatly thank the reviewer for his constructive and instructive response, which is a resource unto itself. The negative sign in the final exponential term had been omitted from the solution provided in the initial review, leading to my confusion.
    
    Citation: https://doi.org/10.5194/egusphere-2025-901-AC2
RC3:
'Comment on egusphere-2025-901', Christoph Glotzbach, 25 Jun 2025

Dear Authors,
I enjoyed reading your manuscript. It presents an important contribution to the community, and given that this functionality is implemented in HeFTy, I am confident it will be widely used in the future to enable more accurate interpretation of thermochronological data.
The technical note is well written and pitched at a level that will be understandable and useful to the intended audience from the thermochronological community.
One point I would encourage the authors to consider relates to the automatic assignment of samples to positions along the sine-shaped topography. While the approach is elegant, real topography is often more complex. I could foresee potential issues where samples collected from short-wavelength features (e.g., secondary peaks) might be incorrectly positioned, particularly in landscapes where the dominant wavelength is much larger. A brief discussion of this limitation or potential strategies to mitigate it would strengthen the manuscript. Please also see my technical comments:

Technical corrections:
Line 47-50: A quite similar dataset was compiled by Glotzbach et al. (2015) in combination with a Fourier approach to empirically estimate the perturbation of isotherms in complex 2D-3D topographic situations. You may want to cite this work.
Line 54-55: The choice of parameterisation would require some justification or showing a few alternative models, e.g. running until steady-state or with a different thermal conductivity. This would allow for the estimation of the uncertainty introduced by the choice of parameterisation.
Line 155: This is not very clear, maybe you can also give the relative deviation and have a figure showing the difference between numerical and empirical solution, taking into account the uncertainty in c2.
Line 211-212: Would be good to investigate under which boundary conditions the temperature difference is larger than 10°C and report this, to prevent users from stating that they can model their data without taking into account the topographic deflection effects.

Citation: https://doi.org/10.5194/egusphere-2025-901-RC3
- AC3: 'Reply on RC3', Richard A. Ketcham, 09 Jul 2025
  
  See attached PDF for response
  
  Citation: https://doi.org/10.5194/egusphere-2025-901-AC3

Peer review completion

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

ED: Publish subject to minor revisions (further review by editor) (20 Jul 2025) by Pieter Vermeesch

AR by Richard A. Ketcham on behalf of the Authors (28 Jul 2025) Author's response Author's tracked changes Manuscript

ED: Publish as is (29 Jul 2025) by Pieter Vermeesch

ED: Publish as is (01 Aug 2025) by Georgina King (Editor)

AR by Richard A. Ketcham on behalf of the Authors (01 Aug 2025)

Journal article(s) based on this preprint

18 Sep 2025

Technical note: Incorporating topographic deflection effects into thermal history modelling

Richard A. Ketcham

Geochronology, 7, 449–458, https://doi.org/10.5194/gchron-7-449-2025,https://doi.org/10.5194/gchron-7-449-2025, 2025

Short summary

Richard A. Ketcham

Model code and software

HeFTy v.2.2.0 installer Richard A. Ketcham https://www.dropbox.com/t/aMk8eN7fr65zJVMy

Richard A. Ketcham

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This technical note develops and demonstrates an improvement in how to calculate the temperatures experienced by rocks as they come to the Earth surface due to erosion in mountainous regions. The solution is fast and flexible, and works even in areas where erosion rates have varied through time. The new method has been added to software used to interpret geochronologic data to help discern the history of mountain ranges.


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