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),

The morphology of landscapes adjusts to conform to exogenic perturbations such as uplift and climate as well as spatial variations in lithology,
geologic structure, and biology. As such, numerous studies have taken
advantage of landscape morphology to estimate rates and timing of
perturbations to these landscape properties. In bedrock rivers, for
instance, geomorphic transport laws have been formulated to allow for
linkages between landscape form and process, including from measurements
such as channel steepness and

Similarly, hillslope geomorphic transport laws formulated for soil-mantled landscapes allow for estimation of uplift and erosion rates as well as prediction of the migration of hillcrests in response to landscape transience (Forte and Whipple, 2018; Mohren et al., 2020; Mudd, 2017; Mudd and Furbish, 2007, 2005; Roering, 2008; Roering et al., 2007, 2001, 1999). Over 100 years ago, it was proposed that hillslope form, specifically slope and curvature, may be an effective predictor of erosion rate, as hillslopes steepen and lengthen to accommodate increases in baselevel lowering (Gilbert, 1909, 1877). However, hillslopes do not continue to steepen as baselevel lowering progressively increases to faster and faster rates (e.g., Howard, 1994; Penck, 1953; Schumm, 1967; Strahler, 1950). Rather, hillslope gradients approach a threshold value as erosion rate increases, such that gradient becomes invariant and insensitive to further increases in baselevel lowering (Andrews and Bucknam, 1987; Burbank et al., 1996; DiBiase et al., 2012; Larsen and Montgomery, 2012; Montgomery, 2001; Roering et al., 1999). In such cases, sediment flux varies nonlinearly with slope due to threshold-dependent processes such as landsliding as well as granular creep (BenDror and Goren, 2018; Deshpande et al., 2021; DiBiase et al., 2012; Ferdowsi et al., 2018; Gabet, 2000; Larsen and Montgomery, 2012; Montgomery, 2001; Ouimet et al., 2009; Roering et al., 2001).

Despite the insensitivity of hillslope gradient in rapidly eroding
landscapes, soil-mantled hillslopes remain an effective record of landscape
transience and uplift. Specifically, hilltop curvature continues to respond
to baselevel lowering when uplift and erosion rates are high, even as slope
becomes insensitive to ever-increasing erosion rate (Hurst et al., 2012;
Mohren et al., 2020; Roering et al., 2007). For a one-dimensional hillslope
at steady state, erosion rate,

Past studies that couple geomorphic transport laws and hilltop curvature
have typically relied on curvature calculated from 2D polynomial functions
fit to the topographic surface (PFTs; i.e., polynomials fit to topography;
e.g., Roering et al., 1999). While a variety of polynomial forms and types of
curvature (i.e., tangential, planform, Laplacian, etc.) have been utilized
(e.g., Minár et al., 2020; Moore et al., 1991), Hurst et al. (2012) found
that six term functions were sufficient for measuring curvature to estimate
erosion rate. Specifically, Hurst et al. (2012) used least squares regression
to fit a surface,

While the application of PFTs has proven useful for calculating curvature to
estimate erosion rate and predict spatial and temporal variations in uplift
(e.g., Clubb et al., 2020; Godard et al., 2020; Hurst et al., 2019, 2013,
2012; Mohren et al., 2020; Roering et al., 2007), PFTs are computationally
cumbersome, hindering large-scale exploitation of high-resolution
topographic datasets that have become increasingly available. Here, we
demonstrate that 2D continuous wavelet transforms (CWTs) provide an
alternative and computationally efficient approach to calculating hilltop
curvature, operating at least

We selected the Oregon Coast Range (OCR) to compare CWTs and PFTs as hilltop
curvature measurement techniques, as it is a region that has been
extensively studied in the geomorphic literature, exhibits relatively
uniform topography over intra-catchment scales while exhibiting diversity in
hillslope form and erosion rate across the axis of the range, and has
negligible spatial variability in climate. The OCR is an unglaciated humid
landscape that parallels the Cascadia Subduction Zone and is characterized
by cool, wet winters when the majority of the annual 1–2 m of precipitation
falls, and warm, dry summers (PRISM Climate Group, 2016). The dominant tree
populations are composed of Douglas-fir (

We pinpointed catchments in the OCR that exhibit a range of hilltop
curvatures for analysis. Specifically, we focus on Hadsall Creek
(43.983

Oregon Coast Range study sites. Note the drainage divide (red)
between catchments that flow directly to the Pacific Ocean and those that
flow east into the Willamette River, which then flows northward to the
Columbia River.

Oregon Coast Range hillslope profiles. Example lidar hillshades of hillslopes from Hadsall Creek

The spatial proximity of Bear Creek, Hadsall Creek, and the NFSR makes them
well-suited to compare

We used PFTs to calculate curvature of the Hadsall and Bear Creeks and NFSR
lidar DEMs as enumerated in Eqs. (2) and (3). Each DEM has a grid spacing
of 0.9144 m (3 ft). The lidar for Bear Creek was collected in 2009 (average
point density: 8.14 pulses m

In contrast to PFTs, CWTs are computationally efficient and can provide a
variety of outputs depending on the analysis and type of wavelet used (e.g.,
Foufoula-Georgiou and Kumar, 1994, and references therein). Here, we applied
a 2D CWT using the Ricker wavelet (often known as the Mexican hat wavelet).
The Ricker wavelet has been used in geomorphology to map and estimate
landslide ages based on surface roughness (Booth et al., 2009; LaHusen et
al., 2020), identify dominant landforms at particular wavelengths (Struble
et al., 2021), and extract channel heads and drainage networks (Lashermes et
al., 2007; Passalacqua et al., 2010) and other topographic spectral
analyses including mapping faults and predicting lithospheric thickness
(e.g., Audet, 2014; Jordan and Schott, 2005; Malamud and Turcotte, 2001). In
applying the Ricker wavelet, we take advantage of a useful property of
convolutions that allows for simultaneous removal of topographic noise and
calculation of derivatives. Specifically,

The Ricker wavelet is the negative second derivative of a 2D Gaussian
function [L

Similar to the application of PFTs to estimate erosion rate, it is necessary
to select a measurement scale that effectively smooths over stochastic
sediment transport perturbations and noise that is inherent to topographic
datasets and DEMs and does not represent long-term morphology reflective of
baselevel lowering (Hurst et al., 2012; Roering et al., 2010). Thus, it is
important to utilize an appropriately scaled wavelet,

We applied the CWT and PFT for

We compared the efficiency of calculating curvature with a PFT to the CWT,
including both definitions of wavelet smoothing scale,

We also tested how DEM size affects the relative speed of the CWT and PFT
algorithms. We selected a DEM of size

We calculated curvature at every pixel of our DEMs, but

Curvature extracted from representative hilltop at Hadsall Creek, NFSR, and Bear Creek for a range of

We applied the CWT and PFT to the Hadsall Creek, NFSR, and Bear Creek lidar
DEMs and calculated curvature. We utilized the hilltop masks to extract
curvature at the hilltops (

To test the efficacy of

Erosion rate at OCR study sites calculated with Eq. (1), assuming

Recalculated CRN erosion rates. We used the CRONUS online calculator (Balco et al., 2008) to determine catchment-averaged erosion rates from

New CRN erosion rate at Bear Creek. We used the CRONUS online calculator (Balco et al., 2008) to determine catchment-averaged erosion rates from

We utilized synthetic hillslopes generated from a theoretical model to compare the accuracy of hilltop curvature calculated using the PFT and CWT
as well as test how well these approaches can predict erosion rate. We used
the functional form for a 1D hillslope experiencing nonlinear diffusion
given as

In addition, to account for natural topographic roughness that the CWT and
PFT smooth over to estimate

We find that the CWT is dramatically more efficient at calculating hilltop
curvature than the PFT. Curvature calculation time depends on smoothing
scale,

Speed of the CWT compared to the PFT for different smoothing scales,

In addition to the CWT outpacing the PFT at a large range of

We utilized 2D CWTs and PFTs to calculate

Comparison of

We additionally plot probability density functions (PDFs) of measured

Probability density functions of

We utilized

CRN erosion rate vs.

We calculated

Synthetic hillslopes constructed using Eq. (11). Upper row shows pink noise surfaces that are added to the original hillslope form (left column); yellow colors correspond with positive deviations from the hillslope (convex noise) and blue with negative deviations (concave noise). Each row of hillslopes corresponds with range of dimensionless erosion rates, from

Synthetic hillslopes constructed using Eq. (11). Same as Fig. 8, but with red noise added (see the Supplement for white noise example). Upper row shows red noise surfaces added to the original hillslope form (left column); yellow colors correspond with positive deviations from the hillslope (convex noise) and blue with negative deviations (concave noise). Each row of hillslopes corresponds with dimensionless erosion rates from

We observe that for

Ratio of

We observe that the uncertainty in

We find that when no noise is added to the synthetic hillslopes,

We observe that both the CWT and PFT produce biased

Application of CWTs and PFTs to measure

We utilized CWTs and PFTs to estimate erosion rate in a landscape that has
been thoroughly studied in past geomorphology studies. Encouragingly,

We have demonstrated that CWTs calculate

We find that both the CWT and PFT are unable to reproduce accurate

The grid resolution of digital topographic data has been recognized to
affect measurements of topographic curvature and hillslope sediment flux
(e.g., Ganti et al., 2012; Grieve et al., 2016b). However, the deviation
between known and measured

Importantly, we stress that neither CWTs nor PFTs are, at this time, capable
of accurately estimating hilltop curvature at moderate to high

The systematic underestimation of

We observe that erosion rate and

Current hilltop curvature measurement techniques do not have a well-defined
capability to filter topographic noise that is inherent to all landscapes
and topographic datasets while maintaining an unbiased value of

We utilized 2D continuous wavelet transforms to calculate hilltop curvature in three catchments in the Oregon Coast Range that exhibit a diversity of hillslopes. We found that the measured hilltop curvature values are comparable to those calculated from fitting 2D polynomial functions to topography to calculate curvature, a method that has been commonly applied elsewhere. Both techniques produce estimates of erosion rate that are consistent with those independently constrained from cosmogenic
radionuclides in stream sediments. Specifically, we find that erosion rate
calculated with the CWT is

We additionally test the accuracy of both the wavelet transform and
polynomial by constructing synthetic hillslopes following a nonlinear
diffusive hillslope geomorphic transport law. Synthetic hillslopes were
constructed with and without added surface noise of various types (white,
pink, red/Brownian) and exhibited various forms corresponding to a range of
dimensionless erosion rates. We find that both the wavelet transform and
polynomial are able to reproduce hilltop curvature for slow dimensionless
erosion rates (

We utilized TopoToolbox (

The supplement related to this article is available online at:

WTS and JJR conceived of and designed the study. WTS developed the study and completed the analysis. WTS prepared the manuscript with contributions from JJR.

The contact author has declared that neither they nor their co-author have any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Thank you to Brooke Hunter, Danica Roth, Fiona Clubb, and Odin Marc for
helpful discussions. Tyler Doane, an anonymous reviewer, and associate
editor Simon Mudd provided insightful reviews that improved the quality of
the manuscript. We are particularly grateful to Adam Booth, who provided
several helpful and enlightening conversations about wavelets.

This paper was edited by Simon Mudd and reviewed by Tyler Doane and one anonymous referee.