25 May 2021
25 May 2021
Hilltop curvature as a proxy for erosion rate: Wavelets enable rapid computation and reveal systematic underestimation
 ^{1}Department of Geosciences, University of Arizona, Tucson, Arizona 85721, USA
 ^{2}Department of Earth Sciences, University of Oregon, Eugene, Oregon, 97403, USA
 ^{1}Department of Geosciences, University of Arizona, Tucson, Arizona 85721, USA
 ^{2}Department of Earth Sciences, University of Oregon, Eugene, Oregon, 97403, USA
Abstract. Estimation of erosion rate is an important component of landscape evolution studies, particularly in settings where transience or spatial variability in uplift or erosion generates diverse landform morphologies. While bedrock rivers are often used to constrain the timing and magnitude of changes in baselevel lowering, hilltop curvature (or convexity), C_{HT}, provides an additional opportunity to map variations in erosion rate given that average slope angle becomes insensitive to erosion rate owing to threshold slope processes. C_{HT} measurement techniques applied in prior studies (e.g. polynomial functions), however, tend to be computationally expensive when they rely on high resolution topographic data such as lidar, limiting the spatial extent of hillslope geomorphic studies to small study regions. Alternative techniques such as spectral tools like continuous wavelet transforms present an opportunity to rapidly document trends in hilltop convexity across expansive areas. Here, we demonstrate how continuous wavelet transforms (CWTs) can be used to calculate the Laplacian of elevation, which we utilize to estimate erosion rate in three catchments of the Oregon Coast Range that exhibit varying slope angle, slope length, and hilltop convexity, implying differential erosion. We observe that C_{HT} values calculated with the CWT are similar to those obtained from 2D polynomial functions. Consistent with recent studies, we find that erosion rates estimated with C_{HT} from both CWTs and 2D polynomial functions are consistent with erosion rates constrained with cosmogenic radionuclides from stream sediments. Importantly, our CWT approach calculates curvature 10^{2}–10^{3} times more quickly than 2D polynomials. As such, continuous wavelet transforms provide a compelling approach to rapidly quantify regional variations in erosion rate as well as lithology, structure, and hillslope sediment transport processes, which are encoded in hillslope morphology. Finally, we test the accuracy of CWT and 2D polynomial techniques by constructing a series of synthetic hillslopes generated by a theoretical nonlinear transport model that exhibit a range of erosion rates and topographic noise characteristics. Notably, we find that neither CWTs nor 2D polynomials reproduce the theoretically prescribed C_{HT} value for hillslopes experiencing moderate to fast erosion rates, even when no topographic noise is added. Rather, C_{HT} is systematically underestimated, producing a power law relationship between erosion rate and C_{HT} that can be attributed to artifacts from the increasing prominence of planar hillslopes that narrow the zone of hilltop convexity as erosion rate increases. As such, we recommend careful consideration of measurement length scale when applying C_{HT} to estimate erosion rate in moderate to fasteroding landscapes, where curvature measurement techniques may be prone to systematic underestimation.
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William T. Struble and Joshua J. Roering
Status: closed

RC1: 'Comment on esurf202140', Tyler Doane, 14 Jun 2021
The comment was uploaded in the form of a supplement: https://esurf.copernicus.org/preprints/esurf202140/esurf202140RC1supplement.pdf

RC2: 'Comment on esurf202140', Anonymous Referee #2, 17 Jun 2021
This ms by Struble and Roering explores various approaches for the calculation of landscape curvature, in particular along ridglines. Quantitative analysis of hillslope morphology has been a major advance in geomorphology, and is based on the theoretical framework proposed by Roering et al. (2007) as well as on the increasing availability of high resolution topographic data.
Such analysis of hillslope properties have allowed to extract high density information on the relative distribution of erosion rates, and to discuss their significance in terms of landscape transience (e.g. Hurst et al. 2012, 2013). One key attribute of hillslope, required in such analysis is hilltop curvature (CHT), which is theoretically linearly related with erosion rate in the vicinity of hilltops when slopes are sufficiently shallow. Computing curvature across landscapes has often been a tricky business, which is usually done by fitting a second order polynomial surface over a defined neighborhood. The procedure requires careful consideration of the size of this neighborhood and can become very resource intensive for high resolution Digital Elevation Models when this size increases.
In this contribution Struble and Roering provide a systematic exploration of an alternative approach based on continuous wavelet transforms, which they demonstrate to be much more efficient from a computational point of view. They compare CHT estimates obtained with both methods for the hillslopes of the Oregon Coast Range. They also test the accuracy of CHT calculation using synthetic hillslope topographies with various degrees of prescribed noise, and highlight the systematic shortcomings of all types of calculation when dealing with sharp ridges. As a follow up they discuss the recent results of Gabet et al. (2021), showing a deviation of the relationship between CHT and erosion rate from theoretical prediction (linear), suggesting that the underlying systematic bias in the calculation of CHT they identified might be involved here.
This manuscript presents important results of high significance for the growing community of geomorphologist interested in extracting and interpreting hillslope properties from high resolution topographic data. It is clear and convincing, and will make a great contribution to Esurf once some relatively minor issues have been addressed.
A necessary addition is one or two figures presenting a closer view of the DEMs in the vicinity of hilltops (with corresponding topographic profiles), in order to give the readers a better sense on hillslope morphology, hilltop structure and the roughness of the surfaces. You could for example focus on the representative hilltops for each basins.
The method used for the identification of hilltops, prior to CHT computation, and the associated cutoff parameters should be presented explicitly.
At some point in the discussion you should develop how the CWT methods could (or not) be used for the calculation of other types of curvature, which relevant for quantitative geomorphology studies. For example planar curvature used in some approaches for the delineation of channel heads (Clubb et al. â€¦).
The quality of some figures could be substantially improved. Iâ€™ve noticed a lot of rasterization effects, see for example the nonexistent Xaxis line on figure 2i. Same remark for figures 5 and 9.
A great addition at the end of the paper would be a few guidelines and rules of thumb allowing to quickly identify situations where there is a substantial risk of bias in CHT calculation for a given field setting. This could be supported by a concluding figure allowing to visually make this assessment. For example what about generalizing Fig 10b with a measured/actual CHT surface as a function of estimated E (for which we can have some prior knowledge for a given setting) and Lambda (or D).
Specific comments
73Â : Â PFTÂ : acronym not clear
82Â : how important are the specifics of the fitting procedureÂ ?
 Circular vs rectangular window?
 z forced at central pixel elevation?
88Â : yes PFT calculation are computationally expensive, however in many situations curvature calculation only occurs at hilltops, which makes it manageable in most cases.
144: you could add a bit more background information on these LiDAR DEMs (date, data source, point density â€¦.).
164165 : unclear this sentence sounds like the Laplacian and CHT are different things, as you need some extraction procedure to get the latter.
166 : I find this formulation a bit ambiguous (estimating erosion rate). By itself CHT will provide relative variations in erosion rates, and you will need an estimation of the sediment transport efficiency to derive absolute values. It might lead to confusion for some readers, so you should make that explicit here.
192 : CPU characteristics?
194 : be more explicit : float32?
199 : recall explicitly the procedure used to extract hilltops. Would the divide order metrics introduced by Scherler and Schwanghart be of interest here?
205 : how do you measure this asymmetry?
210 : more information needed to characterize this single representative hilltop, for each catchment (nb of pixels, summary statistics , etc â€¦)
3.4.1 : See comment above on terminology (erosion rate estimation)
The title of this section is actually confusing as most of it deals with the scaling breaks
220 : in order to rule out any risk of circularity with what is presented below, recall the data used to infer this value of D, and explain to what extent they are independent from the newly acquired dataset.
265 : so what does the +/ 1 m mentioned above (262) refers to?
320 : is the slope of this relationship consistent with independently constrained transport coefficients?
406 Note that most well designed studies will limit the computation of CHT to the vicinity of identified hilltops and not the whole DEM, which implies a substantial reduction of computation times for all methods
412 high resolution topographic data are usually not pertinent and rarely used in such longwavelength analysis of river profiles.
421: for PFT this effect is easy to visualize and correspond to the inability of the quadratic surface to provide an appropriate fit to the narrow and sharp hilltop in terms of mathematical form. It would be interesting to have a similar explanation for CWT? Ok seen below, but maybe add a simple scheme to give a better visual representation of this bias.
436 : could we think of a dedicated filtering procedure that could be applied before calculating CHT? Could you just make a simple test of applying a lowpass filter to the topography before CHT calculation and see under what condition does it improve the comparison?
Figure 9 : this figure us quite difficult to read due to the changes in Yaxis
Overall quality of the figure could be improved : lots of rasterization effects on this one
Table 3 : even if it just one new sample there is a lot of missing information. Use standard approaches for presenting CRN data (isotopic ratios, standardization ...)

AC1: 'Comment on esurf202140  Response to Reviewers', William Struble, 07 Jul 2021
The comment was uploaded in the form of a supplement: https://esurf.copernicus.org/preprints/esurf202140/esurf202140AC1supplement.pdf
Status: closed

RC1: 'Comment on esurf202140', Tyler Doane, 14 Jun 2021
The comment was uploaded in the form of a supplement: https://esurf.copernicus.org/preprints/esurf202140/esurf202140RC1supplement.pdf

RC2: 'Comment on esurf202140', Anonymous Referee #2, 17 Jun 2021
This ms by Struble and Roering explores various approaches for the calculation of landscape curvature, in particular along ridglines. Quantitative analysis of hillslope morphology has been a major advance in geomorphology, and is based on the theoretical framework proposed by Roering et al. (2007) as well as on the increasing availability of high resolution topographic data.
Such analysis of hillslope properties have allowed to extract high density information on the relative distribution of erosion rates, and to discuss their significance in terms of landscape transience (e.g. Hurst et al. 2012, 2013). One key attribute of hillslope, required in such analysis is hilltop curvature (CHT), which is theoretically linearly related with erosion rate in the vicinity of hilltops when slopes are sufficiently shallow. Computing curvature across landscapes has often been a tricky business, which is usually done by fitting a second order polynomial surface over a defined neighborhood. The procedure requires careful consideration of the size of this neighborhood and can become very resource intensive for high resolution Digital Elevation Models when this size increases.
In this contribution Struble and Roering provide a systematic exploration of an alternative approach based on continuous wavelet transforms, which they demonstrate to be much more efficient from a computational point of view. They compare CHT estimates obtained with both methods for the hillslopes of the Oregon Coast Range. They also test the accuracy of CHT calculation using synthetic hillslope topographies with various degrees of prescribed noise, and highlight the systematic shortcomings of all types of calculation when dealing with sharp ridges. As a follow up they discuss the recent results of Gabet et al. (2021), showing a deviation of the relationship between CHT and erosion rate from theoretical prediction (linear), suggesting that the underlying systematic bias in the calculation of CHT they identified might be involved here.
This manuscript presents important results of high significance for the growing community of geomorphologist interested in extracting and interpreting hillslope properties from high resolution topographic data. It is clear and convincing, and will make a great contribution to Esurf once some relatively minor issues have been addressed.
A necessary addition is one or two figures presenting a closer view of the DEMs in the vicinity of hilltops (with corresponding topographic profiles), in order to give the readers a better sense on hillslope morphology, hilltop structure and the roughness of the surfaces. You could for example focus on the representative hilltops for each basins.
The method used for the identification of hilltops, prior to CHT computation, and the associated cutoff parameters should be presented explicitly.
At some point in the discussion you should develop how the CWT methods could (or not) be used for the calculation of other types of curvature, which relevant for quantitative geomorphology studies. For example planar curvature used in some approaches for the delineation of channel heads (Clubb et al. â€¦).
The quality of some figures could be substantially improved. Iâ€™ve noticed a lot of rasterization effects, see for example the nonexistent Xaxis line on figure 2i. Same remark for figures 5 and 9.
A great addition at the end of the paper would be a few guidelines and rules of thumb allowing to quickly identify situations where there is a substantial risk of bias in CHT calculation for a given field setting. This could be supported by a concluding figure allowing to visually make this assessment. For example what about generalizing Fig 10b with a measured/actual CHT surface as a function of estimated E (for which we can have some prior knowledge for a given setting) and Lambda (or D).
Specific comments
73Â : Â PFTÂ : acronym not clear
82Â : how important are the specifics of the fitting procedureÂ ?
 Circular vs rectangular window?
 z forced at central pixel elevation?
88Â : yes PFT calculation are computationally expensive, however in many situations curvature calculation only occurs at hilltops, which makes it manageable in most cases.
144: you could add a bit more background information on these LiDAR DEMs (date, data source, point density â€¦.).
164165 : unclear this sentence sounds like the Laplacian and CHT are different things, as you need some extraction procedure to get the latter.
166 : I find this formulation a bit ambiguous (estimating erosion rate). By itself CHT will provide relative variations in erosion rates, and you will need an estimation of the sediment transport efficiency to derive absolute values. It might lead to confusion for some readers, so you should make that explicit here.
192 : CPU characteristics?
194 : be more explicit : float32?
199 : recall explicitly the procedure used to extract hilltops. Would the divide order metrics introduced by Scherler and Schwanghart be of interest here?
205 : how do you measure this asymmetry?
210 : more information needed to characterize this single representative hilltop, for each catchment (nb of pixels, summary statistics , etc â€¦)
3.4.1 : See comment above on terminology (erosion rate estimation)
The title of this section is actually confusing as most of it deals with the scaling breaks
220 : in order to rule out any risk of circularity with what is presented below, recall the data used to infer this value of D, and explain to what extent they are independent from the newly acquired dataset.
265 : so what does the +/ 1 m mentioned above (262) refers to?
320 : is the slope of this relationship consistent with independently constrained transport coefficients?
406 Note that most well designed studies will limit the computation of CHT to the vicinity of identified hilltops and not the whole DEM, which implies a substantial reduction of computation times for all methods
412 high resolution topographic data are usually not pertinent and rarely used in such longwavelength analysis of river profiles.
421: for PFT this effect is easy to visualize and correspond to the inability of the quadratic surface to provide an appropriate fit to the narrow and sharp hilltop in terms of mathematical form. It would be interesting to have a similar explanation for CWT? Ok seen below, but maybe add a simple scheme to give a better visual representation of this bias.
436 : could we think of a dedicated filtering procedure that could be applied before calculating CHT? Could you just make a simple test of applying a lowpass filter to the topography before CHT calculation and see under what condition does it improve the comparison?
Figure 9 : this figure us quite difficult to read due to the changes in Yaxis
Overall quality of the figure could be improved : lots of rasterization effects on this one
Table 3 : even if it just one new sample there is a lot of missing information. Use standard approaches for presenting CRN data (isotopic ratios, standardization ...)

AC1: 'Comment on esurf202140  Response to Reviewers', William Struble, 07 Jul 2021
The comment was uploaded in the form of a supplement: https://esurf.copernicus.org/preprints/esurf202140/esurf202140AC1supplement.pdf
William T. Struble and Joshua J. Roering
William T. Struble and Joshua J. Roering
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