Hilltop curvature as a proxy for erosion rate: wavelets enable rapid computation and reveal systematic underestimation

Abstract

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), CHT, 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. CHT 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 CHT 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 CHT 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 at least 103 times more quickly than 2D polynomials. This efficiency advantage of the CWT increases with domain size. 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 CHT value for hillslopes experiencing moderate to fast erosion rates, even when no topographic noise is added. Rather, CHT is systematically underestimated, producing a power law relationship between erosion rate and CHT that can be attributed to 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 CHT to estimate erosion rate in moderate to fast-eroding landscapes, where curvature measurement techniques may be prone to systematic underestimation.

Tan DEM‐X Preserves the Relationship Between Hilltop Curvature and Erosion Rate in the Qilian Shan

“…In this study, we compared C HT extracted from two terrain data sets with different resolutions for three typical sites to evaluate the performance of the different data sets at different scales. Our findings indicate that the C HT is influenced by the size of the moving window, which was also found by and Struble and Roering (2021). The magnitudes of the rises and falls are different, and there are inconsistent spatial changes between individual areas at the ridge scale (Figures 5g and 5h), which are caused by the inconsistent rises and falls.…”

Debris‐Flow Process Controls on Steepland Morphology in the San Gabriel Mountains, California

“…Similarly, hilltop curvature has been demonstrated as a predictor of erosion rate in soil‐mantled landscapes (e.g., Hurst et al., 2012; Roering et al., 2007). However, fluvial metrics lose their predictive power in steeplands, as channels reach a threshold steepness and debris flows dominate incisional processes at low drainage areas (Hilley et al., 2019; Stock & Dietrich, 2003), and hilltop curvature becomes an ineffective estimate of erosion rate in steep, rapidly eroding landscapes where soils become patchy and hilltops become conspicuously narrow and sharp (Gabet et al., 2021; Heimsath et al., 2012; Neely et al., 2019; Struble & Roering, 2021). As such, A df , S 0 , and a 1 present an exciting framework for estimating relative uplift and erosion rates between catchments in landscapes where fluvial and hillslope metrics become ineffective.…”

Optimising global landscape evolution models with 10Be

“…Spatial heterogeneity in erosion rate is often controlled by steep areas in the catchment, such as knickpoints, and higher values of n amplify the proportion of erosion derived from steep areas relative to the rest of the catchment (Fig.1).While linear diffusion (p = 1) is commonly applied in landscape evolution studies (e.g., Braun and Willett, 2013), our optimised p ~ 2 for the diffusion-only model is consistent withGabet al (2021) in which erosion rate correlates best with the square of hillslope convexity. In response toGabet et al (2021),Struble and Roering (2021) point to a systematic underestimation of curvature in natural landscapes that may be an artefact of the numerical methods used for estimating curvature from DEMs Gabet et al (2021). employ high-resolution (~1 m) LIDAR data, but the broader point made byStruble and Roering (2021) poses a serious limitation for largescale LEM analyses that are typically restricted to lower-resolution DEMs.…”

“…In response toGabet et al (2021),Struble and Roering (2021) point to a systematic underestimation of curvature in natural landscapes that may be an artefact of the numerical methods used for estimating curvature from DEMs Gabet et al (2021). employ high-resolution (~1 m) LIDAR data, but the broader point made byStruble and Roering (2021) poses a serious limitation for largescale LEM analyses that are typically restricted to lower-resolution DEMs. In such cases, the need for mass conservation and numerical stability are important considerations.…”

“…And yet, a diffusion equation with exponent p ≠ 1 is numerically unstable, physically unexplained, and does not accommodate deposition (the result of negative curvature). What does it say about the utility of running LEMs on natural landscapes if the optimised parameter value (p ~ 2) cannot be implemented?Struble and Roering (2021) suggest that p ~ 2 enhances the influence of steep, rapidly eroding areas on average curvature, which are commonly underestimated by many methods.…”