A simple model for faceted topographies at normal faults based on an extended stream-power law

Hergarten, Stefan

doi:https://doi.org/10.5194/egusphere-2024-336

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https://doi.org/10.5194/egusphere-2024-336

Preprints

26 Feb 2024

| 26 Feb 2024

A simple model for faceted topographies at normal faults based on an extended stream-power law

Stefan Hergarten

Abstract. Faceted topographies at normal faults have been studied for more than a century. Since the dip angle of the facets is typically much lower than the dip angle of the fault, it is clear that the facets are not just the exhumed footwall, but have been eroded considerably. It has also been shown that a constant erosion rate in combination with a constant rate of displacement can explain the occurrence of planar facets. Quantitatively, however, the formation of faceted topographies is still not fully understood. In this study, the shared stream-power model for ﬂuvial erosion and sediment transport is used in combination with a recently published extension for hillslopes. As a major theoretical result, it is found that the ratio of the tangent of the facet angle and the dip angle of the fault as well as the ratio of baseline length and horizontal width of perfect triangular facets mainly depends on the ratio of the horizontal rate of displacement and the hillslope erodibility. Numerical simulations reveal that horizontal displacement is crucial for the formation of triangular facets. For vertical faults, facets are rather polygonal and much longer than wide. While the sizes of individual facets vary strongly, the average size is controlled by the ratio of hillslope erodibility and ﬂuvial erodibility.

Received: 03 Feb 2024 – Discussion started: 26 Feb 2024

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Stefan Hergarten

Status: closed

RC1:
'Comment on egusphere-2024-336', Anonymous Referee #1, 10 Jun 2024

This paper describes the application of an extended, simple stream-power and hillslope erosion model to investigate the evolution and controls on triangular facets at range fronts. The results highlight specific relationships between vertical and horizontal normal fault displacement rates, and the dip angle and dimensions of facets.
The paper contains some useful insights and prompted some ideas for future application, so is worthy of publication. However I suggest (extensive) minor but important revisions, because aspects of the presentation make it harder to follow than it needs to be; several assumptions are simply asserted or referred to a recent paper by the same author, without restating the key justification for the assumption in the current paper; and a few existing relationships between ridge relief and drainage incision and drainage spacing would be helpful comparisons. Detailed suggestions and questions below by line number.

1. (Abstract) why is it important that they’ve been studied for a century? This first line would be better as a brief description of what they are and our most general understanding of them. This would help the reader understand the second sentence, which presumes a detailed understanding of the setting and geometry of these features, which many readers will not have.
11. what does “rather” polygonal mean? Approximately?
13. impressing -> impressive ?
13-15. Same issue as abstract: needs to define what facets are, how they relate to terrain and tectonics, and why we care about them, before jumping into details of geometry and research questions. Again, the length of time they’ve been studied isn’t really relevant as motivation for this study.
** To this end, I also suggest making Figure 4 the first figure, and adding to it 2-3 photos of different-looking range fronts with facets, so the reader can see clearly what geometry you’re referring to. **
34. Re-cite Tucker here at the end of this paragraph.
39. Why is the 15 years relevant?
42-43. This omits the very relevant work of Densmore et al from the late 1990s who numerically modeled these in the Basin and Range (US) and explicitly discussed the planarity and other geometric aspects of the facets.
- Ellis, Densmore, and Anderson, 1999, Development of mountainous topography in the Basin Ranges, USA: Basin Research, v. 11, p. 2141, doi:10.1046/j.1365-2117.1999.00087.x.
- Densmore, A.L., Ellis, M.A., and Anderson, R.S., 1998, Landsliding and the evolution of normal‐fault‐bounded mountains: Journal of Geophysical Research: Solid Earth (1978–2012), v. 103, p. 15203–15219, doi:10.1029/98jb00510.

54-58. Regarding nonlinear hillslope transport, I don’t think the slope needs to be as steep as the threshold for rock stability; it includes shallow slips in regolith, nonlocal transport e.g. rocks going all the way down the hill, etc., and so planarity of slopes is approached well before the ultimate threshold slope is reached.
57. delete “then” at the end of the sentence
59-62. This doesn’t follow the discussion earlier in the paragraph, ad could be relocated down to the model description ~ line 105.
98. Justification for m=0?
115. “Some kind of” -> “a”; or perhaps the “only” preferred state
108-116. Is there any empirical support for this model?
131. “sheared” -> “shared”
157. “Must be the same” must it? Fault movement is episodic and generates substantial and persistent transients (e.g. Yanites, B.J., Tucker, G.E., Mueller, K.J. and Chen, Y.G., 2010. How rivers react to large earthquakes: Evidence from central Taiwan. Geology, 38(7), pp.639-642.)
Section 4 - I really appreciate the inclusion of empirical constraint on n for this application, rather than simply asserting n=1 as many studies do.
195. The data set used here is limited, but surely there are slip rate constraints on many ranges in (for example) the US basin and range, and geometries can be measured from DEMs, if more constraint is helpful.
235. One mesh width? 60 km? Or one grid cell width, delta-x?
236. Plain -> plane
237. Evaluating in the middle of the displacement steps: did you do a sensitivity test on this? Why not at the end of the step when more time for adjustment has occurred since the last step?
Fig. 7 caption needs to explain that the lines are the vertical map projections of the facets.
Figure 8 caption or legend needs to indicate that the shaded areas are now the facets corresponding to the same colors/times as the drainage lines.
Section 5.2. I guess this is useful to compare to earlier models, but do vertically-slipping vertical-dipping faults exist in nature?
254. I’d be careful with the word “impossible” unless the scope is clearly defined. (i.e. under these model rules)
271-272. This is hard to follow. The fault to the profile - what profile? Where are the knickpoints arriving for both rivers and hillslopes?
283-286. Is it the original trace of the fault that is important, or the total vertical uplift so far? For incision = uplift, only after the total uplift is equal to the eventual steady height of the facets/ridgelines is the form fully adjusted to the new state. (e.g. for SPIM, Whipple and Tucker, 1999; but also more generally)
294-298. How does the density of drainages at the range front under this model compare with e.g. Perron 2008 and its nonlinear-diffusion equivalent?
Fig 11, others - the 4My drainage networks are very disorganized and densified relative to typical network evolution simulations, is this an artifact of the channel-head identifying routine, or something dynamical related to normal faults and horizontal displacement?
310-315, as before, if you have the same vertical uplift rate in the horst block between simulations, the time to equilibration of the form will also be similar.
335. What is the horizontal width, rather than the baseline? The facet length measured perpendicular to the range? This will be controlled by the height and angle, and therefore more related to the vertical motion, see previous comment.
342. I -> It
343. “Some kind of” -> “a”
Finally, I was left wondering: What would nonlinear-flux hillslope transport predict for the steady relief of the ridge lines between drainages (e.g. Roering 1999/2001) and how does this relate to facet height?

Citation: https://doi.org/10.5194/egusphere-2024-336-RC1
- RC2:
  'Reply on RC1', Anonymous Referee #2, 11 Jun 2024
  
  Dear Editors
  Unfortunately, I have not yet received the revised file and thus am unable to proceed with my review and final judgment on the manuscript.
  Could you please resend the revised file at your earliest convenience? I apologize for any inconvenience this may cause and appreciate your assistance in this matter.
  Thank you for your understanding and support.
  Thanks
  DEK
  
  Citation: https://doi.org/10.5194/egusphere-2024-336-RC2
  - AC2: 'Reply on RC2', Stefan Hergarten, 08 Jul 2024
    
    Dear Reviewer,
    unfortunately, there must have been some wrong information. We are still in the first round and there is no revised manuscript so far.
    Best regards,
    
    Stefan Hergarten
    
    Citation: https://doi.org/10.5194/egusphere-2024-336-AC2
- AC1: 'Reply on RC1', Stefan Hergarten, 08 Jul 2024
  
  Dear Reviewer,
  thank you for your constructive comments. Before proceeding with a revised version, let me briefly respond to those points that go beyond the presentation of the theory and the results.
  54-58. Regarding nonlinear hillslope transport, I don't think the slope needs to be as steep as the threshold for rock stability; it includes shallow slips in regolith, nonlocal transport e.g. rocks going all the way down the hill, etc., and so planarity of slopes is approached well before the ultimate threshold slope is reached.
  
  Correct in reality, but the respective models rather enforce the threshold. The model of Densmore et al. introduces a minimum slope angle below which slopes always remain stable. In turn, the nonlinear diffusion model proposed by Roering at all introduces a maximum slope that can never be exceeded.
  
  157-159. "Must be the same" must it? Fault movement is episodic and generates substantial and persistent transients (e.g. Yanites, B.J., Tucker, G.E., Mueller , K.J. and Chen, Y.G., 2010. How rivers react to large earthquakes: Evidence from central Taiwan. Geology, 38(7), pp.639-642.
  
  Of course, there will be knickpoints moving upstream if the displacement is not continuous. However, we are discussing a model with continuous displacement here, and the effect of discontinuous displacement would be much weaker at steep hillslopes than in a river.
  
  195. The data set used here is limited, but surely there are slip rate constraints on many ranges in (for example) the US basin and range, and geometries can be measured from DEMs, if more constraint is helpful.
  
  The validation is even more challenging than I thought when writing the manuscript. All relations include only ratios of slip rates and erodibilities, and estimating erodibilities involves a big uncertainty. Therefore, slip rates are less helpful than it seems. Practically, we need the fault angle, the slope of the facet and that of the hillslopes in the transverse valleys. Surely possible to derive these properties from DEMs, but not in an afternoon.
  237. Evaluating in the middle of the displacement steps: did you do a sensitivity test on this? Why not at the end of the step when more time for adjustment has occurred since the last step?
  
  I indeed started with a version that considered the topography at the end of the step, so immediately before the next displacement was applied. Then I followed theoretical arguments about differential equations (e.g., about source terms) that the middle of the interval should be better. I found that the difference is very small, but stayed at this version.
  283-286. Is it the original trace of the fault that is important, or the total vertical uplift so far? For incision = uplift, only after the total uplift is equal to the eventual steady height of the facets/ridgelines is the form fully adjusted to the new state. (e.g. for SPIM, Whipple and Tucker, 1999; but also more generally)
  
  It is indeed the original fault trace here, which separates the region with an inherited drainage pattern from a pristine surface. It is responsible for the conversion of multiangular facets into triangular facets. In turn, the vertical equilibration is responsible for the planarity of the facets, which is achieved faster. This will become visible in Fig. 15, where planarity has already been achieved at t = 4 Myr, while several facets are still multiangular at that time in Fig. 10.
  294-298. How does the density of drainages at the range front under this model compare with e.g. Perron 2008 and its nonlinear-diffusion equivalent?
  
  Similar to the model used here, the linear diffusion model allows for adjusting the drainage density and the relief to any values, given that the scaling problems of the diffusion approach are not crucial. For the model used here, the horizontal length scale (that is inversely proportional to the drainage density) is kappa/K, while it is sqrt(D/K) for the linear diffusion model. For the nonlinear diffusion model, however, I am not sure about its scaling properties.
  Fig 11, others - the 4My drainage networks are very disorganized and densified relative to typical network evolution simulations, is this an artifact of the channel-head identifying routine, or something dynamical related to normal faults and horizontal displacement?
  
  It is neither an immediate effect of normal faulting nor a clear artifact of the channel-head identifying scheme. In the footwall, it is a temporary incision of small tributaries into the hillslopes when the hillslopes become steep. Here I am not sure whether it is realistic or not. In the hanging wall, it is an effect of the rapidly increasing sediment flux. This sediment flux results in frequent avulsions of the rivers. As a result, there are many "dead" channels with small catchment sizes, but they are still channels morphologically.
  310-315, as before, if you have the same vertical uplift rate in the horst block between simulations, the time to equilibration of the form will also be similar.
  
  This is basically true, but I did not explain it here because I was sure that reviewers would complain that it is not explained sufficiently well. In principle, the results of my 2021 paper in JGR Earth Surface on knickpoints in the shared stream-power model can be transferred to hillslopes, but it remains more complicated than for the SPIM. In a nutshell, equilibration will take longer than
  
  in the SPIM, but the effect does not depend on the horizontal rate of displacement. I think it is not too bad that readers who are familiar with the SPIM may find this result not surprising even without an explicit explanation.
  Finally, I was left wondering: What would nonlinear-flux hillslope transport predict for the steady relief of the ridge lines between drainages (e.g. Roering 1999/2001) and how does this relate to facet height?
  
  I am not sure whether I got the question fully, but I guess it refers to the nonlinear diffusion model with D to infinity if the slope approaches a limit slope S_c. Then the facets and the hillslopes approach the same slope S_f = S_h = S_c if the fault is steep enough and displacement fast enough.
  
  Best regards,
  
  Stefan Hergarten
  
  Citation: https://doi.org/10.5194/egusphere-2024-336-AC1

Status: closed

RC1:
'Comment on egusphere-2024-336', Anonymous Referee #1, 10 Jun 2024

This paper describes the application of an extended, simple stream-power and hillslope erosion model to investigate the evolution and controls on triangular facets at range fronts. The results highlight specific relationships between vertical and horizontal normal fault displacement rates, and the dip angle and dimensions of facets.
The paper contains some useful insights and prompted some ideas for future application, so is worthy of publication. However I suggest (extensive) minor but important revisions, because aspects of the presentation make it harder to follow than it needs to be; several assumptions are simply asserted or referred to a recent paper by the same author, without restating the key justification for the assumption in the current paper; and a few existing relationships between ridge relief and drainage incision and drainage spacing would be helpful comparisons. Detailed suggestions and questions below by line number.

1. (Abstract) why is it important that they’ve been studied for a century? This first line would be better as a brief description of what they are and our most general understanding of them. This would help the reader understand the second sentence, which presumes a detailed understanding of the setting and geometry of these features, which many readers will not have.
11. what does “rather” polygonal mean? Approximately?
13. impressing -> impressive ?
13-15. Same issue as abstract: needs to define what facets are, how they relate to terrain and tectonics, and why we care about them, before jumping into details of geometry and research questions. Again, the length of time they’ve been studied isn’t really relevant as motivation for this study.
** To this end, I also suggest making Figure 4 the first figure, and adding to it 2-3 photos of different-looking range fronts with facets, so the reader can see clearly what geometry you’re referring to. **
34. Re-cite Tucker here at the end of this paragraph.
39. Why is the 15 years relevant?
42-43. This omits the very relevant work of Densmore et al from the late 1990s who numerically modeled these in the Basin and Range (US) and explicitly discussed the planarity and other geometric aspects of the facets.
- Ellis, Densmore, and Anderson, 1999, Development of mountainous topography in the Basin Ranges, USA: Basin Research, v. 11, p. 2141, doi:10.1046/j.1365-2117.1999.00087.x.
- Densmore, A.L., Ellis, M.A., and Anderson, R.S., 1998, Landsliding and the evolution of normal‐fault‐bounded mountains: Journal of Geophysical Research: Solid Earth (1978–2012), v. 103, p. 15203–15219, doi:10.1029/98jb00510.

54-58. Regarding nonlinear hillslope transport, I don’t think the slope needs to be as steep as the threshold for rock stability; it includes shallow slips in regolith, nonlocal transport e.g. rocks going all the way down the hill, etc., and so planarity of slopes is approached well before the ultimate threshold slope is reached.
57. delete “then” at the end of the sentence
59-62. This doesn’t follow the discussion earlier in the paragraph, ad could be relocated down to the model description ~ line 105.
98. Justification for m=0?
115. “Some kind of” -> “a”; or perhaps the “only” preferred state
108-116. Is there any empirical support for this model?
131. “sheared” -> “shared”
157. “Must be the same” must it? Fault movement is episodic and generates substantial and persistent transients (e.g. Yanites, B.J., Tucker, G.E., Mueller, K.J. and Chen, Y.G., 2010. How rivers react to large earthquakes: Evidence from central Taiwan. Geology, 38(7), pp.639-642.)
Section 4 - I really appreciate the inclusion of empirical constraint on n for this application, rather than simply asserting n=1 as many studies do.
195. The data set used here is limited, but surely there are slip rate constraints on many ranges in (for example) the US basin and range, and geometries can be measured from DEMs, if more constraint is helpful.
235. One mesh width? 60 km? Or one grid cell width, delta-x?
236. Plain -> plane
237. Evaluating in the middle of the displacement steps: did you do a sensitivity test on this? Why not at the end of the step when more time for adjustment has occurred since the last step?
Fig. 7 caption needs to explain that the lines are the vertical map projections of the facets.
Figure 8 caption or legend needs to indicate that the shaded areas are now the facets corresponding to the same colors/times as the drainage lines.
Section 5.2. I guess this is useful to compare to earlier models, but do vertically-slipping vertical-dipping faults exist in nature?
254. I’d be careful with the word “impossible” unless the scope is clearly defined. (i.e. under these model rules)
271-272. This is hard to follow. The fault to the profile - what profile? Where are the knickpoints arriving for both rivers and hillslopes?
283-286. Is it the original trace of the fault that is important, or the total vertical uplift so far? For incision = uplift, only after the total uplift is equal to the eventual steady height of the facets/ridgelines is the form fully adjusted to the new state. (e.g. for SPIM, Whipple and Tucker, 1999; but also more generally)
294-298. How does the density of drainages at the range front under this model compare with e.g. Perron 2008 and its nonlinear-diffusion equivalent?
Fig 11, others - the 4My drainage networks are very disorganized and densified relative to typical network evolution simulations, is this an artifact of the channel-head identifying routine, or something dynamical related to normal faults and horizontal displacement?
310-315, as before, if you have the same vertical uplift rate in the horst block between simulations, the time to equilibration of the form will also be similar.
335. What is the horizontal width, rather than the baseline? The facet length measured perpendicular to the range? This will be controlled by the height and angle, and therefore more related to the vertical motion, see previous comment.
342. I -> It
343. “Some kind of” -> “a”
Finally, I was left wondering: What would nonlinear-flux hillslope transport predict for the steady relief of the ridge lines between drainages (e.g. Roering 1999/2001) and how does this relate to facet height?

Citation: https://doi.org/10.5194/egusphere-2024-336-RC1
- RC2:
  'Reply on RC1', Anonymous Referee #2, 11 Jun 2024
  
  Dear Editors
  Unfortunately, I have not yet received the revised file and thus am unable to proceed with my review and final judgment on the manuscript.
  Could you please resend the revised file at your earliest convenience? I apologize for any inconvenience this may cause and appreciate your assistance in this matter.
  Thank you for your understanding and support.
  Thanks
  DEK
  
  Citation: https://doi.org/10.5194/egusphere-2024-336-RC2
  - AC2: 'Reply on RC2', Stefan Hergarten, 08 Jul 2024
    
    Dear Reviewer,
    unfortunately, there must have been some wrong information. We are still in the first round and there is no revised manuscript so far.
    Best regards,
    
    Stefan Hergarten
    
    Citation: https://doi.org/10.5194/egusphere-2024-336-AC2
- AC1: 'Reply on RC1', Stefan Hergarten, 08 Jul 2024
  
  Dear Reviewer,
  thank you for your constructive comments. Before proceeding with a revised version, let me briefly respond to those points that go beyond the presentation of the theory and the results.
  54-58. Regarding nonlinear hillslope transport, I don't think the slope needs to be as steep as the threshold for rock stability; it includes shallow slips in regolith, nonlocal transport e.g. rocks going all the way down the hill, etc., and so planarity of slopes is approached well before the ultimate threshold slope is reached.
  
  Correct in reality, but the respective models rather enforce the threshold. The model of Densmore et al. introduces a minimum slope angle below which slopes always remain stable. In turn, the nonlinear diffusion model proposed by Roering at all introduces a maximum slope that can never be exceeded.
  
  157-159. "Must be the same" must it? Fault movement is episodic and generates substantial and persistent transients (e.g. Yanites, B.J., Tucker, G.E., Mueller , K.J. and Chen, Y.G., 2010. How rivers react to large earthquakes: Evidence from central Taiwan. Geology, 38(7), pp.639-642.
  
  Of course, there will be knickpoints moving upstream if the displacement is not continuous. However, we are discussing a model with continuous displacement here, and the effect of discontinuous displacement would be much weaker at steep hillslopes than in a river.
  
  195. The data set used here is limited, but surely there are slip rate constraints on many ranges in (for example) the US basin and range, and geometries can be measured from DEMs, if more constraint is helpful.
  
  The validation is even more challenging than I thought when writing the manuscript. All relations include only ratios of slip rates and erodibilities, and estimating erodibilities involves a big uncertainty. Therefore, slip rates are less helpful than it seems. Practically, we need the fault angle, the slope of the facet and that of the hillslopes in the transverse valleys. Surely possible to derive these properties from DEMs, but not in an afternoon.
  237. Evaluating in the middle of the displacement steps: did you do a sensitivity test on this? Why not at the end of the step when more time for adjustment has occurred since the last step?
  
  I indeed started with a version that considered the topography at the end of the step, so immediately before the next displacement was applied. Then I followed theoretical arguments about differential equations (e.g., about source terms) that the middle of the interval should be better. I found that the difference is very small, but stayed at this version.
  283-286. Is it the original trace of the fault that is important, or the total vertical uplift so far? For incision = uplift, only after the total uplift is equal to the eventual steady height of the facets/ridgelines is the form fully adjusted to the new state. (e.g. for SPIM, Whipple and Tucker, 1999; but also more generally)
  
  It is indeed the original fault trace here, which separates the region with an inherited drainage pattern from a pristine surface. It is responsible for the conversion of multiangular facets into triangular facets. In turn, the vertical equilibration is responsible for the planarity of the facets, which is achieved faster. This will become visible in Fig. 15, where planarity has already been achieved at t = 4 Myr, while several facets are still multiangular at that time in Fig. 10.
  294-298. How does the density of drainages at the range front under this model compare with e.g. Perron 2008 and its nonlinear-diffusion equivalent?
  
  Similar to the model used here, the linear diffusion model allows for adjusting the drainage density and the relief to any values, given that the scaling problems of the diffusion approach are not crucial. For the model used here, the horizontal length scale (that is inversely proportional to the drainage density) is kappa/K, while it is sqrt(D/K) for the linear diffusion model. For the nonlinear diffusion model, however, I am not sure about its scaling properties.
  Fig 11, others - the 4My drainage networks are very disorganized and densified relative to typical network evolution simulations, is this an artifact of the channel-head identifying routine, or something dynamical related to normal faults and horizontal displacement?
  
  It is neither an immediate effect of normal faulting nor a clear artifact of the channel-head identifying scheme. In the footwall, it is a temporary incision of small tributaries into the hillslopes when the hillslopes become steep. Here I am not sure whether it is realistic or not. In the hanging wall, it is an effect of the rapidly increasing sediment flux. This sediment flux results in frequent avulsions of the rivers. As a result, there are many "dead" channels with small catchment sizes, but they are still channels morphologically.
  310-315, as before, if you have the same vertical uplift rate in the horst block between simulations, the time to equilibration of the form will also be similar.
  
  This is basically true, but I did not explain it here because I was sure that reviewers would complain that it is not explained sufficiently well. In principle, the results of my 2021 paper in JGR Earth Surface on knickpoints in the shared stream-power model can be transferred to hillslopes, but it remains more complicated than for the SPIM. In a nutshell, equilibration will take longer than
  
  in the SPIM, but the effect does not depend on the horizontal rate of displacement. I think it is not too bad that readers who are familiar with the SPIM may find this result not surprising even without an explicit explanation.
  Finally, I was left wondering: What would nonlinear-flux hillslope transport predict for the steady relief of the ridge lines between drainages (e.g. Roering 1999/2001) and how does this relate to facet height?
  
  I am not sure whether I got the question fully, but I guess it refers to the nonlinear diffusion model with D to infinity if the slope approaches a limit slope S_c. Then the facets and the hillslopes approach the same slope S_f = S_h = S_c if the fault is steep enough and displacement fast enough.
  
  Best regards,
  
  Stefan Hergarten
  
  Citation: https://doi.org/10.5194/egusphere-2024-336-AC1

Stefan Hergarten

Model code and software

A simple model for faceted topographies at normal faults Stefan Hergarten https://doi.org/10.5281/zenodo.10473156

Video supplement

Formation of triangular facets at normal faults Stefan Hergarten http://hergarten.at/openlem/facets

Stefan Hergarten

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Short summary

Faceted topographies are impressing footprints of active tectonics in geomorphology. This paper investigates the evolution of faceted topographies at normal faults and its interaction with the river network theoretically and numerically. As a main result beyond several relations for the the geometry of facets, the horizontal displacement associated to normal faults is crucial for the dissection of initially polygonal facets into triangular facets bounded by almost parallel rivers.


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