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**Earth Surface Dynamics**
An interactive open-access journal of the European Geosciences Union

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- Editorial board
- Articles
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- About
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**Research article**
03 Jun 2020

**Research article** | 03 Jun 2020

Parameterization of river incision models requires accounting for environmental heterogeneity: insights from the tropical Andes

^{1}Helmholtz Centre Potsdam, GFZ German Research Centre for Geosciences, Potsdam, Germany^{2}CSDMS, Institute for Arctic and Alpine Research, University of Colorado at Boulder, Boulder, CO, USA^{3}Research Foundation Flanders (FWO), Egmontstraat 5, 1000 Brussels, Belgium^{4}Earth and Life Institute, Georges Lemaître Centre for Earth and Climate Research, University of Louvain, Place Louis Pasteur 3, 1348 Louvain-la-Neuve, Belgium^{5}Institute of Earth Surface Dynamics, University of Lausanne, 1015 Lausanne, Switzerland^{6}University of Liege, UR SPHERES, Department of Geography, Clos Mercator 3, 4000 Liège, Belgium^{7}Institute of Environmental Science and Geography, University of Potsdam, Germany^{8}Facultad de Ciencias Agropecuarias, Universidad de Cuenca, Campus Yanuncay, Cuenca, Ecuador^{9}Department of Earth and Environmental Sciences, KU Leuven, Celestijnenlaan 200E, 3001 Leuven, Belgium^{10}Department of Civil Engineering – Hydraulics Section, KU Leuven, Kasteelpark 40 box 2448, 3001 Leuven, Belgium

^{1}Helmholtz Centre Potsdam, GFZ German Research Centre for Geosciences, Potsdam, Germany^{2}CSDMS, Institute for Arctic and Alpine Research, University of Colorado at Boulder, Boulder, CO, USA^{3}Research Foundation Flanders (FWO), Egmontstraat 5, 1000 Brussels, Belgium^{4}Earth and Life Institute, Georges Lemaître Centre for Earth and Climate Research, University of Louvain, Place Louis Pasteur 3, 1348 Louvain-la-Neuve, Belgium^{5}Institute of Earth Surface Dynamics, University of Lausanne, 1015 Lausanne, Switzerland^{6}University of Liege, UR SPHERES, Department of Geography, Clos Mercator 3, 4000 Liège, Belgium^{7}Institute of Environmental Science and Geography, University of Potsdam, Germany^{8}Facultad de Ciencias Agropecuarias, Universidad de Cuenca, Campus Yanuncay, Cuenca, Ecuador^{9}Department of Earth and Environmental Sciences, KU Leuven, Celestijnenlaan 200E, 3001 Leuven, Belgium^{10}Department of Civil Engineering – Hydraulics Section, KU Leuven, Kasteelpark 40 box 2448, 3001 Leuven, Belgium

**Correspondence**: Benjamin Campforts (benjamin.campforts@gfz-potsdam.de)

**Correspondence**: Benjamin Campforts (benjamin.campforts@gfz-potsdam.de)

Abstract

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Landscape evolution models can be used to assess the impact of rainfall variability on bedrock river incision over millennial timescales. However, isolating the role of rainfall variability remains difficult in natural environments, in part because environmental controls on river incision such as lithological heterogeneity are poorly constrained. In this study, we explore spatial differences in the rate of bedrock river incision in the Ecuadorian Andes using three different stream power models. A pronounced rainfall gradient due to orographic precipitation and high lithological heterogeneity enable us to explore the relative roles of these controls. First, we use an area-based stream power model to scrutinize the role of lithological heterogeneity in river incision rates. We show that lithological heterogeneity is key to predicting the spatial patterns of incision rates. Accounting for lithological heterogeneity reveals a nonlinear relationship between river steepness, a proxy for river incision, and denudation rates derived from cosmogenic radionuclide (CRNs). Second, we explore this nonlinearity using runoff-based and stochastic-threshold stream power models, combined with a hydrological dataset, to calculate spatial and temporal runoff variability. Statistical modeling suggests that the nonlinear relationship between river steepness and denudation rates can be attributed to a spatial runoff gradient and incision thresholds. Our findings have two main implications for the overall interpretation of CRN-derived denudation rates and the use of river incision models: (i) applying sophisticated stream power models to explain denudation rates at the landscape scale is only relevant when accounting for the confounding role of environmental factors such as lithology, and (ii) spatial patterns in runoff due to orographic precipitation in combination with incision thresholds explain part of the nonlinearity between river steepness and CRN-derived denudation rates. Our methodology can be used as a framework to study the coupling between river incision, lithological heterogeneity and climate at regional to continental scales.

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How to cite.

Campforts, B., Vanacker, V., Herman, F., Vanmaercke, M., Schwanghart, W., Tenorio, G. E., Willems, P., and Govers, G.: Parameterization of river incision models requires accounting for environmental heterogeneity: insights from the tropical Andes, Earth Surf. Dynam., 8, 447–470, https://doi.org/10.5194/esurf-8-447-2020, 2020.

1 Introduction

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Research on how climate variability and tectonic forcing interact to make a landscape evolve over time has long been limited by the lack of techniques that measure denudation rates over sufficiently long time spans (Coulthard and Van de Wiel, 2013). Consequently, the relative role of climate variability and tectonic processes could only be deduced from sediment archives (e.g., Hay et al., 1988). However, whether sediment archives offer reliable proxies remains contested because sediment sources and transfer times to depositional sites are often obscured by stochastic processes that shred environmental signals (Bernhardt et al., 2017; Jerolmack and Paola, 2010; Romans et al., 2016; Sadler, 1981).

Nowadays, cosmogenic radionuclides (CRNs) contained in quartz minerals of
river sediments provide an alternative tool for determining catchment-wide
denudation rates on a routine basis
(Codilean et al., 2018; Harel et al., 2016; Portenga and Bierman, 2011). In
sufficiently large catchments, detrital CRN-derived denudation rates
(*E*_{CRN}) integrate over timescales that average out the episodic nature
of sediment supply (Kirchner et al.,
2001). Hence, benchmark or natural denudation rates can be calculated for
disturbed as well as pristine environments
(Reusser et al., 2015; Safran et al., 2005; Schaller et al., 2001; Vanacker et al.,
2007).

Catchment-wide denudation rates have been found to correlate with a range of topographic metrics including basin relief, average basin gradient and elevation (Abbühl et al., 2011; Kober et al., 2007; Riebe et al., 2001; Safran et al., 2005; Schaller et al., 2001). However, in tectonically active regimes, hillslopes tend to evolve towards a critical threshold gradient, which is controlled by mechanical rock properties (Anderson, 1994; Roering et al., 1999; Schmidt and Montgomery, 1995). Once slopes approach this critical gradient, mass wasting becomes the dominant process controlling hillslope response to changing base levels (Burbank et al., 1996). In such a configuration, hillslope gradients are no longer an indication of denudation rates (Binnie et al., 2007; Korup et al., 2007; Montgomery and Brandon, 2002), and hillslope metrics (Hurst et al., 2012) often require high-resolution topographic data that are not widely available.

Contrary to hillslope gradients, rivers and river longitudinal profiles are more sensitive to changes in erosion rates (Whipple et al., 1999). Bedrock rivers in mountainous regions mediate the interplay between uplift and erosion (Whipple and Tucker, 1999; Wobus et al., 2006). They incise into bedrock and efficiently convey sediments, thus setting the base level for hillslopes and controlling the evacuation of hillslope-derived sediment. Quantifying the spatial patterns of natural denudation rates in tectonically active regions therefore requires detailed knowledge of the processes driving fluvial incision (Armitage et al., 2018; Castelltort et al., 2012; Finnegan et al., 2008; Gasparini and Whipple, 2014; Goren, 2016; Scherler et al., 2017; Tucker and Bras, 2000).

River morphological indices, such as channel steepness (*k*_{sn})
(Wobus et al., 2006), have successfully been applied
as a predictor for catchment denudation and thus *E*_{CRN} by Safran et al. (2005) and many others, commonly identifying
a monotonically increasing relationship between channel steepness
(*k*_{sn}) (Wobus et al., 2006) and *E*_{CRN}
(Cyr
et al., 2010; DiBiase et al., 2010; Mandal et al., 2015; Ouimet et al.,
2009; Safran et al., 2005; Vanacker et al., 2015). Several authors
identified a nonlinear relationship between *k*_{sn} and *E*_{CRN} in both
regional (e.g.,
DiBiase et al., 2010; Ouimet et al., 2009; Scherler et al., 2014; Vanacker
et al., 2015) and global compilation studies
(Harel et al., 2016). Theory
suggests that this nonlinear relationship reflects the dependency of
long-term denudation on hydrological variability
(Deal
et al., 2018; Lague et al., 2005; Tucker and Bras, 2000). Hydrological
variability affects both temporal and spatial variations in river discharge,
and the effect of river discharge on denudation and river incision rates can
be approximated by theoretical model derivations. However, the impact of
hydrological variability on incision rates in natural environments has,
until now, only been successfully identified in a limited number of case
studies (DiBiase
and Whipple, 2011; Ferrier et al., 2013; Scherler et al., 2017).

We identify two limitations hampering the large-scale application of river incision models that include hydrological variability. First, the necessary high-resolution hydrological data are usually unavailable. Mountain regions are typically characterized by large temporal and spatial variation in runoff rates (e.g., Mora et al., 2014). Yet, most of the observational records on river discharge in mountain regions are fragmented and/or have limited geographic coverage. Second, large catchments are often underlain by variable lithologies. Studies exploring the role of river hydrology in controlling river incision have hitherto mainly focused on regions underlain by rather uniform lithology (DiBiase and Whipple, 2011; Ferrier et al., 2013) or they have considered lithological variations to be of minor importance (Scherler et al., 2017). However, tectonically active regions have usually experienced tectonic accretion, subduction, active thrusting, volcanism and denudation, resulting in a highly variable lithology over > 100 km distances (Horton, 2018). Rock strength is known to control river incision rates and is a function of its lithological composition and stratigraphic age (Brocard and van der Beek, 2006; Lavé and Avouac, 2001; Stock and Montgomery, 1999), as well as its rheology and fracturing (Molnar et al., 2007). If we want to use geomorphic models not only to emulate the response of landscapes to climatic and/or tectonic forces but also to predict denudation rates, then we need to account for variations in physical rock properties (Attal and Lavé, 2009; Nibourel et al., 2015; Stock and Montgomery, 1999). Even more importantly, these variations in rock erodibility can potentially obscure the relation between river incision and discharge (Deal et al., 2018). Therefore, the climatic effects on denudation rates can only be correctly assessed if the geomorphic model accounts for physical rock properties and vice versa. Based on current limitations, we formulate two main objectives: we want (i) to assess the impact of lithological heterogeneity on river incision and (ii) to unravel the role of allogenic (spatial and/or temporal runoff variability) versus autogenic (incision thresholds) controls on river incision. We develop and evaluate our approach in the southern Ecuadorian Andes, for which detailed lithological information is available as is a database of CRN-derived denudation rates (Vanacker et al., 2007, 2015).

Bedrock rivers are shaped by processes including weathering,
abrasion–saltation, plucking, cavitation and debris scouring
(Whipple et al., 2013). However, explicitly
accounting for these processes renders models too complex at the spatial and
temporal scales relevant to understanding landscape evolution of entire
mountain ranges. Therefore, a broad variety of models have been proposed to
simplify the complex nature of river incision dynamics (Armitage
et al., 2018; Lague et al., 2005; Shobe et al., 2017; Venditti et al.,
2019). Most river incision models assume a functional dependence of river
incision on the shear stress (*τ*; Pa) exerted by the river on its bed
(Sklar and Dietrich, 1998; Whipple and
Tucker, 1999). However, within the family of shear stress–stream power
models, several approaches exist. Most commonly used is the Area-Based
Stream Power Model (A-SPM), explicitly representing the universally observed
inverse power relation between channel slope and drainage area
(Howard, 1994; Whipple and
Tucker, 1999). Parametrization of the A-SPM is purely empirical and involves
the calibration of three incision parameters (an erosion efficiency parameter,
an area exponent and a slope exponent). Given the interdependency of these
parameters (e.g.,
Campforts and Govers, 2015; Croissant and Braun, 2014; Roberts and White,
2010), there is an ongoing effort to calibrate river incision models using a
process-oriented strategy whereby small-scale observations and physical
mechanisms are upscaled to the landscape scale
(Venditti et al., 2019). In particular and
not exclusively, ongoing efforts evaluate how the three incision parameters
are affected by the presence of incision thresholds
(e.g., DiBiase and Whipple, 2011; Lague, 2014), discharge variability
(DiBiase
and Whipple, 2011; Lague et al., 2005; Snyder et al., 2003; Tucker and Bras,
2000), and the spatial and temporal distribution of runoff
(Deal et al., 2018; Ferrier et al., 2013; Lague et al., 2005; Molnar et al.,
2006). In this paper, we evaluate how two of such derived models (the
Stochastic-Threshold and Runoff-Based Stream Power Model – ST-SPM and R-SPM, respectively) can be used to explain measured variations in denudation rates at the landscape scale.

The Area-Based Stream Power Model (A-SPM; Howard, 1994) is a first, lumped statistical approach to represent river incision:

$$\begin{array}{}\text{(1)}& E={K}^{\prime}{A}^{m}{S}^{n},\end{array}$$

in which *E* is the long-term river erosion (L T^{−1}), *K*^{′} (L^{1−2m} T^{−1}) is
the erosional efficiency as a function of rock erodibility and erosivity, *A* (L^{2}) is the upstream drainage area, *S* (L L^{−1}) is the channel slope, and *m* and
*n* are exponents whose values depend on lithology, rainfall variability and
sediment load. Equation (1) can be rewritten as a function of the steepness index, *k*_{s},

$$\begin{array}{}\text{(2)}& E={K}^{\prime}{k}_{\mathrm{s}}^{n},\end{array}$$

where *k*_{s} can be written as the upstream area-weighted channel gradient:

$$\begin{array}{}\text{(3)}& {k}_{\mathrm{s}}={\mathrm{SA}}^{\mathit{\theta}},\end{array}$$

in which $\mathit{\theta}=m/n$ is the concavity index (Snyder
et al., 2000; Whipple and Tucker, 1999). In order to compare steepness
indices from different locations, *θ* is commonly set to 0.45 and
referred to as the normalized steepness index, *k*_{sn} (Wobus et al., 2006). Variations in *k*_{sn} are
often used to infer uplift patterns by assuming a steady state between
uplift and erosion (Kirby and Whipple, 2012). In
transient settings, in which steady-state conditions are not necessarily met,
the *k*_{sn} values can be used to infer local river incision rates
(Harel et al.,
2016; Royden and Perron, 2013).

When using the A-SPM, the effect of autogenic (caused by intrinsic river
dynamics such as incision thresholds and changes in channel width) and
allogenic (originating from the transient response of river dynamics to
extrinsic changes such as climate variability) controls is assumed to be
accounted for in the model parameters (*K*^{′}, *m* and *n*). For example, it has been
shown that incision thresholds translate into a slope exponent *n* greater than
unity when applying the A-SPM (Lague, 2014). Notwithstanding empirical
evidence supporting the A-SPM, such as the scaling between drainage area and
channel slope in steady-state river profiles (Lague, 2014) or its capability
to simulate transient river incision pulses (Campforts and Govers, 2015),
the lumped modeling approach of the A-SPM cannot be used to evaluate the
role of autogenic or allogenic river response.

The Stochastic-Threshold Stream Power Model (ST-SPM; Crave and Davy, 2001; Deal et al., 2018; Lague et al., 2005; Snyder et al., 2003; Tucker and Bras, 2000) simulates the impact of hydrological variability and incision thresholds on river incision and thus enables us to evaluate the role of autogenic or allogenic river response.

The ST-SPM is calculated in two consecutive steps. First, instantaneous
river incision *I* (L t^{−1}) is calculated as

$$\begin{array}{}\text{(4a)}& {\displaystyle}I\left({Q}^{*}\right)=K{Q}^{*\mathit{\gamma}}{k}_{\mathrm{s}}^{n}-\mathit{\psi},\text{(4b)}& {\displaystyle}K={k}_{\mathrm{e}}{k}_{t}^{a}{k}_{\mathrm{w}}^{-a\mathit{\alpha}}{\stackrel{\mathrm{\u203e}}{R}}^{m};\phantom{\rule{0.125em}{0ex}}\mathit{\psi}={k}_{\mathrm{e}}{\mathit{\tau}}_{\mathrm{c}}^{a},\text{(4c)}& {\displaystyle}\mathit{\gamma}=a\mathit{\alpha}\left(\mathrm{1}-{\mathit{\omega}}_{s}\right);\phantom{\rule{0.125em}{0ex}}m=a\mathit{\alpha}\left(\mathrm{1}-{\mathit{\omega}}_{b}\right);\phantom{\rule{0.125em}{0ex}}n=a\mathit{\beta},\end{array}$$

in which *Q*^{*} represents the dimensionless normalized daily discharge calculated
by dividing daily discharge *Q* (L^{3} T^{−1}) by mean annual discharge
$\stackrel{\mathrm{\u203e}}{Q}$ (L^{3} T^{−1}), *k*_{e} (L^{2.5} T^{2} m^{−1.5})
is the erosional efficiency constant, $\stackrel{\mathrm{\u203e}}{R}$ (L T^{−1}) is the
mean annual runoff, *a* is the shear stress exponent reflecting the nature of
the incision process (Whipple et al.,
2000), *ψ* is the threshold term (L T^{−1}), and *k*_{t},
*k*_{w}, *α*, *β*, *ω*_{a} and *ω*_{b} are the channel hydraulic parameters described in Table 1.

In a second step, long-term river incision is calculated by multiplying
instantaneous river incision, *I*, calculated for a discharge of a given
magnitude (*Q*^{*}) with the probability for that discharge to occur (pdf(*Q*^{*})), and
subsequently integrating this product over the range of possible discharge
events specific to the studied timescale
(DiBiase
and Whipple, 2011; Lague et al., 2005; Scherler et al., 2017; Tucker and
Bras, 2000; Tucker and Hancock, 2010):

$$\begin{array}{}\text{(5)}& E={\int}_{{Q}_{\mathrm{c}}^{*}}^{{Q}_{\mathrm{m}}^{*}}I\left({Q}^{*}\right)\mathrm{pdf}\left({Q}^{*}\right)\mathrm{d}{Q}^{*},\end{array}$$

in which ${Q}_{\mathrm{c}}^{*}$ is the minimum normalized discharge
required to exceed the critical shear stress (*τ*_{c}), and
${Q}_{\mathrm{m}}^{*}$ is the maximum possible normalized discharge over the time
considered.

The Runoff-Based Stream Power Model (R-SPM) is a simplified version of the
Stochastic-Threshold Stream Power Model (ST-SPM). The R-SPM assumes that the
incision thresholds are negligible (*ψ*=0) and that discharge is
constant over time (${Q}^{*}=\mathrm{1})$, simplifying Eq. (5) to

$$\begin{array}{}\text{(6)}& E=K{k}_{s}^{n}.\end{array}$$

In the following sections, we first describe the study area, characterize the lithological configuration by developing a lithological erodibility index and compile a database to represent runoff variability. Second, we present the methods and assumptions used for calibrating and simulating river incision. In a third section, the modeling results are presented at the catchment scale: we start by evaluating the impact of lithological heterogeneity on river incision rates using an area-based river incision model (A-SPM). We then evaluate to what extent the variability in denudation rates can be explained by spatial and/or temporal runoff variability and the existence of incision thresholds using the R-SPM and ST-SPM. In a final section, we discuss our findings, highlight the implications of our work and discuss further perspectives.

2 Study area

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The Paute catchment is a 6530 km^{2} transverse drainage basin
(2.9^{∘} S, 79^{∘} W): the Paute River has its source in the eastern flank
of the Western Cordillera, traverses the Cuenca intramontane basin and cuts
through the Eastern Cordillera before joining the Santiago River, a
tributary of the Amazon (Fig. 1; Hungerbühler et al., 2002; Steinmann et al., 1999). Where the Paute
River cuts through the Eastern Cordillera, the topography is rough with
steep hillslopes (90th percentile of slope gradients: 0.40 m m^{−1}) and deeply incised river valleys (Guns
and Vanacker, 2013).

The oblique accretion of terranes to the Ecuadorian margin during the Cenozoic
resulted in a diachronous exhumation and cooling history along the
Ecuadorian cordillera system (Spikings et al.,
2010). South of 1.5^{∘} S, where the Paute basin is situated, three
distinct periods with a higher cooling rate have been reported during the
Paleogene at 73–55, 50–30 and 25–18 Ma, corresponding to a total
cooling from ca. 300 to ca. 60^{∘} C (Spikings et al.,
2010). In the Western Cordillera, no elevated cooling is observed during the
Paleogene and extensional subsidence of the Cuenca basin allowed
synsedimentary deposition of marine, lacustrine and terrestrial facies until
the Middle to Late Miocene (Hungerbühler
et al., 2002; Steinmann et al., 1999). The collision between the Carnegie
ridge and Ecuadorian trench at some time between the Middle to Late Miocene
(Spikings et al., 2001) resulted in uplift of the Western
Cordillera and caused a tectonic inversion of the Cuenca basin
(Hungerbühler
et al., 2002; Steinmann et al., 1999). Based on a compilation of mineral
cooling ages available for the Cuenca basin, Steinman et al. (1999) estimated a mean rock uplift
rate of ca. 0.7 mm yr^{−1} and a corresponding surface uplift of ca. 0.3 mm yr^{−1} from 9 Ma to present. Uplift patterns are assumed to be
reflected in the river steepness and not explicitly simulated in this paper.

The Paute basin is characterized by a tropical mountain climate (Muñoz et al., 2018). Despite the presence of mountain peaks up to ca. 4600 m (Fig. 1), the region is free of permanent snow and ice (Celleri et al., 2007). The region's precipitation is regulated by its proximity to the Pacific Ocean (ca. 60 km distance), the seasonal shifting of the Intertropical Convergence Zone (ITCZ) and the advection of continental air masses sourced in the Amazon basin, giving rise to an orographic precipitation gradient along the eastern flank of the Eastern Cordillera (Bendix et al., 2006). Total annual precipitation is highly variable within the Paute basin and ranges from ca. 800 mm in the center of the basin up to ca. 3000 mm in the eastern parts of the catchment (Celleri et al., 2007; Mora et al., 2014).

The erodibility map was developed using an empirical, hybrid classification
method: it combines information on the lithological composition
(Aalto et al., 2006) and the age of
non-igneous formations assuming higher degrees of diagenesis and increased
lithological strength for older formations (see Kober et al.,
2015). Adding age information to evaluate lithological strength has
advantages because lithostratigraphic units are typically composed of
different lithologies but mapped as a single entity because of their
stratigraphic age. The lithological erodibility (*L*_{E}) is calculated as

$$\begin{array}{}\text{(7a)}& \begin{array}{rl}& {L}_{\mathrm{E}}={\displaystyle \frac{\mathrm{2}}{\mathrm{7}}}L{}^{\prime}\\ & L{}^{\prime}=\left\{\phantom{\rule{0.125em}{0ex}}\begin{array}{l}\frac{\left({L}_{\mathrm{A}}+{L}_{\mathrm{L}}\right)}{\mathrm{3}},\phantom{\rule{0.25em}{0ex}}\text{non-igneousrocks}\\ \frac{{L}_{\mathrm{L}}}{\mathrm{2}},\phantom{\rule{0.25em}{0ex}}\text{igneousrocks}.\end{array}\right.\end{array}\end{array}$$

*L*_{A} is a dimensionless erodibility index based on stratigraphic age (Fig. 2a), and *L*_{L} is a dimensionless erodibility index based on
lithological strength (Table 3), similar to the erodibility indices
published by Aalto (2006). Note that *L*_{A} varies between 1
(Carboniferous) and 6 (Quaternary), whereas *L*_{L} ranges between 2 (e.g.,
granite) and 12 (e.g., unconsolidated colluvial deposits). The lithological
strength thus has a double weight, resulting in *L*^{′} values ranging
between 1 and 6. For igneous rocks, only *L*_{L} is considered, assuming that
the lithological strength of igneous rocks remains constant over time. For
river incision parameters to be comparable to other published ranges,
*L*_{E} is finally scaled around 1 by multiplying *L*^{′} with 2∕7. *L*_{E}
therefore ranges between 2∕7 and 12∕7. A description of the lithological
units, the age of the formations and their lithological strength (*L*_{A},
*L*_{L} and *L*_{E}) is provided in Supplement Table S3.

Using Eq. (7), we developed the erodibility map of Ecuador (Fig. S1) and
the Paute catchment (Fig. 2c) based on the 1M geological map of Ecuador
(Egüez et al., 2017). The lithological erodibility
values were compared with field measurements (*n*=9) of bedrock rheology
by Basabe (1998). An overview of measured lithological
strength values is provided in Table S4 (e.g., uniaxial compressive
strength). Figure 2b shows good agreement (*R*^{2}=0.77) between the
lithological erodibility index, *L*_{E}, and the measured uniaxial
compressive strength.

Catchment-wide denudation rates are derived from in situ produced ^{10}Be
concentrations in river sand. At the outlet of 30 sub-catchments (Fig. 1,
Table 2), fluvial sediments were collected. We refer to Vanacker et al. (2015) for details on sample processing and derivation of CRN denudation
rates taking into account altitude-dependent production, atmospheric scaling
and topographical shielding (Dunai,
2000; Norton and Vanacker, 2009; Schaller et al., 2002). CRN concentrations
are not corrected for snow or ice coverage because there is no evidence of
glacial activity during the integration time of CRN-derived denudation rates
(Vanacker et al., 2015). Three data
points were excluded from model optimization runs: two catchments with a basin
area smaller than 0.5 km^{2} (MA1 and SA) and one catchment
with an exceptionally low ^{10}Be concentration that can be attributed to
recent landslide activity (NG-SD; see
Vanacker et al., 2015).

Based on a gap-filled SRTM v3 digital elevation model (DEM) with 1 arcsec resolution (Farr et al., 2007), we calculate
river steepness for all channels with drainage areas > 0.5 km^{2} and average it over 500 m reaches. The optimized concavity *θ* for the Paute catchment (0.42; Text S1) is close to the frequently used
value of 0.45, so we fix concavity to the reference value of 0.45 and report
river steepness as normalized river steepness (*k*_{sn}) in the remainder of
this paper. The spatial pattern of *k*_{sn} values (Fig. 3) is a result of
the transient geomorphic response to river incision initiated at the Andes
Amazon transition zone (Vanacker et al.,
2015). To evaluate the extent to which transient river features influence
simulated denudation rates, chi plots (*χ*) for all studied sub-catchments are calculated following Royden and Perron (2013) and given in the Supplement (Text S1; Fig. S4; Royden and Perron, 2013).

To constrain the value of *k*_{w} used in the process-based incision models
(Eqs. 4 and 6), we calibrate the relationship between bankfull river width
(*W*_{b}) and discharge (Leopold and Maddock, 1953):

$$\begin{array}{}\text{(8)}& {W}_{\mathrm{b}}={k}_{\mathrm{w}}{\stackrel{\mathrm{\u203e}}{Q}}^{{\mathit{\omega}}_{b}},\end{array}$$

in which *k*_{w} (${L}^{\mathrm{1}-\mathrm{3}{\mathit{\omega}}_{b}}{t}^{{\mathit{\omega}}_{b}}$) and *ω*_{b} are scaling parameters regulating the interaction between mean annual
discharge $\stackrel{\mathrm{\u203e}}{Q}$ and incision rates (Eq. 4). We constrain *k*_{w} by
analyzing downstream variations in bankfull channel width for a fraction of
the river network (see Scherler et
al., 2017). River sections are selected based on the availability of
high-resolution optical imagery in Google Earth, and river width was derived
using the ChanGeom toolset (Fisher et al., 2013; Fig. S5).

The power-law fit between *Q* and *W* yields a value of 0.43 for the scaling
exponent, *ω*_{b}, with an *R*^{2} of 0.51 (Fig. 4). The value
of this exponent lies within the range of published values of 0.23–0.63
(Fisher et al., 2012;
Kirby and Ouimet, 2011). To maintain a dimensionally consistent stream power
model, *ω*_{b} was fixed to a value of 0.55. When doing so, the
fit remains good (*R*^{2}=0.5) and we obtained a *k*_{w} value of 3.7 m^{−0.65} s^{0.55} that is used in the remainder of the paper.

Evaluating the role of spatial and temporal runoff variability (Eqs. 5 and
6) requires estimates of catchment-specific runoff ($\stackrel{\mathrm{\u203e}}{R}$, spatial
variability) and discharge (temporal variability). Although measured runoff
data and discharge records are available for the Paute basin
(Molina
et al., 2007; e.g., Mora et al., 2014; Muñoz et al., 2018), the
monitoring network of existing hydrological stations does not capture the
spatial variability present in the different sub catchments of the 6530 km^{2} Paute basin (Fig. 1). To estimate runoff variability for all 30 sub-catchments, we use hydrological data derived in the framework of the
Earth2Observe Water Resource Reanalysis project
(WRR2; Schellekens et al.,
2017) available from 1979 to 2014. Specifically, we use the hydrological
data calculated with the global water model WaterGAP3 (Water – Global
Assessment and Prognosis: Alcamo et al., 2003; Döll et al., 2003) at a
spatial resolution of 0.25^{∘} and a daily temporal resolution
(http://www.earth2observe.eu, last access: 19 May 2020). Uncertainties associated with the WaterGAP3 data originate
from hydrological model assumptions and spatially distributed input data
(Beck et al., 2017). We revisit the
impact of uncertainties in the climatological data on our model runs in the
Discussion section of this paper. In the following paragraphs, we explain how we
derive (i) a high-resolution runoff map by spatially downscaling these coarse
data and (ii) catchment-specific magnitude frequency distributions of
discharge (pdf_Q^{*}) characterizing the temporal variability of runoff.

A global hydrological reanalysis dataset such as WaterGAP provides daily
runoff data over several decades and makes our methodology transferable to
other regions. However, a spatial resolution of 0.25^{∘} is
insufficient to represent highly variable regional trends in water cycle
dynamics over mountainous regions (Mora et al.,
2014) and in small catchments. Therefore, we downscale the Ecuadorian
WaterGAP3 data to a resolution of 2.5 km by amalgamating rain gauge data
with the reanalysis product. The procedure consisted of the following steps
and is presented in Figs. 5 and 6.

The relationship between precipitation (*P*) and runoff (*R*) is constrained from
the fit between monthly mean values for *P* and *R* available for all Ecuadorian
WaterGAP 0.25^{∘} pixels (Fig. 5).

A high-resolution mean annual precipitation map (*P*_{RIDW}) is calculated by
downscaling the WaterGAP precipitation data (*P*) using a series of rain gauge
observations (338 stations, 1990–2013) from the Ecuadorian national
meteorological service (INAMHI; available from
http://www.serviciometeorologico.gob.ec/biblioteca/, last access: 19 May 2020). A residual inverse
distance weighting (RIDW) method is applied to amalgamate mean annual gauge
data with the mean annual WaterGAP3 precipitation map. First, the
differences between the gauge and WaterGAP data are interpolated using an
IDW method (Fig. S6). Second, the resulting residual surface is added back
to the original *P* data. A similar approach is often applied to integrate gauge
data with satellite products, and we refer to the literature for further details
on its performance (e.g.,
Dinku et al., 2014; Manz et al., 2016). Figure 6a shows *P* for the Paute
region, and Fig. 6c shows its downscaled equivalent (*P*_{RIDW}).

Daily precipitation data (12 784 daily grids between 1979 and 2014) are
downscaled to 2.5 km using the ratio between *P*_{RIDW} and *P*, thereby
assuming that the mean annual correction for precipitation also holds for
daily precipitation patterns.

The relationship between *P* and *R* (Fig. 5) is used to derive daily runoff
values from the downscaled precipitation data for every day between 1979 and
2014.

The mean annual runoff map for the Paute basin is shown in Fig. 6b and
its downscaled equivalent in Fig. 6d. Mean annual values are further used
to calculate mean catchment runoff ($\stackrel{\mathrm{\u203e}}{R}$) and the discharge
variability (next paragraph) for every sub-catchment described in Table 2.
The mean catchment-specific runoff averaged for all catchments equals 0.82±0.35 m yr^{−1}.

Runoff variability is typically cast in terms of spatial runoff variability (Sect. 2.4.1). However, the temporal pattern of runoff might also influence river incision and is typically represented by discharge magnitude frequency distributions. Constraining the shape of these distributions is important because the number of large storm events determines the frequency with which thresholds for river incision to occur are exceeded (see Sect. 1.2.2 and references therein).

The probability distribution of discharge magnitudes consists of two components: at low discharges, the frequency of events increases exponentially with increasing discharge (Lague et al., 2005), whereas at high discharge, the frequency of events decreases with increasing discharge following a power-law distribution (Molnar et al., 2006). An inverse gamma distribution captures this hybrid behavior and can be written as (Crave and Davy, 2001; Lague et al., 2005)

$$\begin{array}{}\text{(9)}& \mathrm{pdf}\left({Q}^{*}\right)={\displaystyle \frac{{k}^{k+\mathrm{1}}}{\mathrm{\Gamma}\left(k+\mathrm{1}\right)}}{e}^{-{\scriptscriptstyle \frac{k}{{Q}^{*}}}}{Q}^{*-\left(\mathrm{2}+k\right)},\end{array}$$

in which Γ is the gamma function and *k* is a discharge variability
coefficient; *k* represents the scale factor of the inverse gamma distribution
and (*k*+1) the shape factor. Previous studies used a single average
*k* value to characterize regional discharge: DiBiase and Whipple (2011) use a constant *k* value for the San Gabriel
Mountains, whereas Scherler et al. (2017) use a constant *k* value for high and
low discharge but distinguish between eastern Tibet and the Himalaya.
However, given the strong variation in temporal precipitation regimes in the
Paute basin (Celleri et al., 2007;
Mora et al., 2014), we explicitly evaluated the role of temporal runoff
variability by calculating catchment-specific discharge distributions from
the WRR2 WaterGAP dataset.

Daily variations in discharge at the sub-catchment outlets (Fig. 1) were
calculated by weighing flow accumulation with runoff (*R*_{RIDW}; see Sect. 5.1.1). For every catchment, the complementary cumulative distribution
function (CCDF) of the daily discharge was fitted through the observed discharge
distribution as

$$\begin{array}{}\text{(10)}& \mathrm{CCDF}\left({Q}^{*}\right)=\mathrm{\Gamma}\left(k/{Q}^{*},k+\mathrm{1}\right),\end{array}$$

where Γ is the lower incomplete gamma function. Figure S7
illustrates the fit between the WaterGAP-derived discharge distribution and
the optimized CCDF for one of the catchments. Site-specific discharge
variability values (*k*) are calculated for all catchments and listed in Table 2. The obtained *k* values range between 0.8 and 1.2 with a mean of 1.01±0.12.

3 Methods

Back to toptop
The presented river incision models (A-SPM, R-SPM and ST-SPM in Sect. 1.2)
all depend on river steepness, *k*_{sn}, which is known to correlate well with
*E*_{CRN} (DiBiase
et al., 2010; Ouimet et al., 2009; Scherler et al., 2017; Vanacker et al.,
2015). Moreover, *E*_{CRN} integrates over time spans that average out
temporal fluctuations of denudation rates and over spatial extents that are
sufficient to average out the erratic nature of hillslope processes.
Therefore, *E*_{CRN} can be used to constrain models of river incision
provided a set of assumptions that we first describe below.

The use of CRN-derived denudation rates to calibrate river incision relies
on three main assumptions, summarized by Scherler et al. (2017). A first
assumption is that the catchment-wide denudation rates derived from CRN are
representative for long-term fluvial incision. Positive correlations between
river steepness, *k*_{sn}, and CRN-derived denudation rates support this
assumption (Vanacker et al., 2015), except for very small catchments where
CRN-derived denudation rates are sensitive to the occurrence of deep-seated
landslides during which material shielded at depth is supplied to the river
(Niemi et al., 2005; Yanites et
al., 2009). A second assumption when using CRN data to calibrate river
incision models is that the sediment cosmogenic nuclide budget is at steady
state at the catchment scale so that the input of CRN via in situ production
equals the export of CRN via sediment export and radioactive decay. Given
the size of the studied basins, this assumption seems to be reasonable. A
third assumption, in particular when using the process-based R-SPM and
ST-SPM, is that the runoff data used to calibrate the incision parameters
are uniform within the sampled sub-catchments and representative of the
time span over which CRN data integrate (1–100 kyr). This is a challenging
assumption given that available hydrological data only cover the recent
past. While spatial patterns of runoff, mainly controlled by orographic
precipitation, could be assumed to be broadly similar over the integration time of
CRN-derived denudation, this is not necessarily true for the temporal
variation in runoff. We will revisit the validity and implications of these
three assumptions in the Discussion section of this paper.

In a first set of model runs, we evaluate the performance of the area-based
SPM (A-SPM) in predicting *E*_{CRN} rates. To account for rock strength
variability Eq. (2) is rewritten as

$$\begin{array}{}\text{(11)}& E={k}_{a}\phantom{\rule{0.125em}{0ex}}{\stackrel{\mathrm{\u203e}}{L}}_{\mathrm{E}}{k}_{\mathrm{sn}}^{n},\end{array}$$

where *k*_{a} (L^{1−2m} T^{−1}) is the erosional efficiency parameter
and ${\stackrel{\mathrm{\u203e}}{L}}_{\mathrm{E}}$ is a dimensionless catchment mean lithological
erodibility value. Given its empirical nature, wherein the effect of allogenic
(e.g., runoff variability) and autogenic (e.g., incision thresholds and river
width dynamics) controls of fluvial processes is integrated within the
empirical scaling parameters (*K*, *m* and *n*), the A-SPM does not enable us to
identify the role of spatial or temporal runoff variability and incision
thresholds. Note that, at any point in the paper, lithological heterogeneity
within the Paute catchment is represented using the average values of
*L*_{E} for the individual sub-catchments indicated with
$\stackrel{\mathrm{\u203e}}{{L}_{\mathrm{E}}}$ and listed in Table 2. If lithological heterogeneity is
not considered, ${\stackrel{\mathrm{\u203e}}{L}}_{\mathrm{E}}$ is fixed to a value of 1.

In a second set of model runs, we evaluate to what extent more advanced SPMs can be used to understand the role of these allogenic and autogenic processes. We start by evaluating the performance of a runoff-based SPM (R-SPM). To account for rock strength variability Eq. (6) is rewritten as

$$\begin{array}{}\text{(12)}& E=K{\stackrel{\mathrm{\u203e}}{L}}_{\mathrm{E}}{k}_{\mathrm{sn}}^{n}.\end{array}$$

An overview of the parameter values required to solve the R-SPM is given in
Table 1. Only the value of *k*_{w} is based on a regional calibration of the
hydraulic geometry scaling (see Sect. 2.3). Other parameters are set to
theoretical values (reported by
Deal et al., 2018; DiBiase and Whipple, 2011; Scherler et al., 2017).
Actively incising bedrock channels are often covered by a layer of sediment
(Shobe et al., 2017). Therefore, we assume that river incision is scaled to
the bed shear stress as for bedload transport (Meyer-Peter
and Müller, 1948) and set *a* to 3∕2
(see DiBiase and
Whipple, 2011; Scherler et al., 2017). We use the Darcy–Weisbach resistance
relation and coefficients ($\mathit{\alpha}=\mathit{\beta}=\mathrm{2}/\mathrm{3}$) to calculate shear
stress exerted by the river flow on its bed and assume a friction factor of
0.08, resulting in a flow resistance factor *k*_{t} of 1000 kg m${}^{-\mathrm{7}/\mathrm{3}}$ s${}^{-\mathrm{4}/\mathrm{3}}$ (e.g., Tucker, 2004). The use of
Darcy–Weisbach friction coefficients in combination with $a=\mathrm{3}/\mathrm{2}$ results in
a value for the slope exponent equal to unity (*n*=1; see Eq. 4). Based on
these theoretical derivations, we fix *n* to unity when constraining the R-SPM.
Note that this contrasts with the first set of model runs (application of the
A-SPM) in which we allow *n* to vary. By fixing *n* to unity, we want to verify
whether spatial variations in runoff (incorporated in *K* from Eq. 12) can
explain variations in incision rates otherwise ascribed to nonlinear river
incision. The only parameter not fixed to a constant value is the erosivity
coefficient *k*_{e}, which is optimized as described in Sect. 3.3.

In a final set of model runs, we apply the Stochastic-Threshold SPM (ST-SPM) to evaluate the role of temporal precipitation variability and thresholds for incision (Eq. 4). Here, we adjust the ST-SPM to account for rock strength variability as

$$\begin{array}{}\text{(13)}& I=K{\stackrel{\mathrm{\u203e}}{L}}_{\mathrm{E}}{Q}^{*\mathit{\gamma}}{k}_{\mathrm{sn}}^{n}-\mathit{\psi}.\end{array}$$

To derive long-term erosion rates (*E*), Eq. (13) is integrated over the
probability density function of discharge magnitudes (Eq. 5), which requires
values for the lower (${Q}_{\mathrm{c}}^{*}$) and the upper (${Q}_{\mathrm{m}}^{*}$) limit
of the integration interval. Constraining ${Q}_{\mathrm{m}}^{*}$ is difficult based
on observational records alone as they might miss some of the most extreme
flooding events. However, when simulating incision rates over long time
spans and thus considering long return times of ${Q}_{\mathrm{m}}^{*}$
(> 1000 years), the solution of Eq. (5) is insensitive to the choice of
${Q}_{\mathrm{m}}^{*}$ (Lague et al., 2005). We
therefore set ${Q}_{\mathrm{m}}^{*}$ to infinity in all our model runs. The critical
discharge (${Q}_{\mathrm{c}}^{*}$) for erosion to occur can be derived from Eq. (13) by setting *I* equal to 0:

$$\begin{array}{}\text{(14)}& {Q}_{\mathrm{c}}^{*}={\left({\displaystyle \frac{\mathit{\psi}}{{K}_{\mathrm{st}}\stackrel{\mathrm{\u203e}}{{L}_{\mathrm{E}}}{k}_{\mathrm{sn}}^{n}}}\right)}^{{\scriptscriptstyle \frac{\mathrm{1}}{\mathit{\gamma}}}}.\end{array}$$

The impact of spatial variations in runoff and discharge variability is
evaluated by setting $\stackrel{\mathrm{\u203e}}{R}$ and *k* to the sub-catchment-specific values or the mean of these values (listed in Table 2; Eq. 4).
The parameters left free during optimization are the erosivity coefficient
*k*_{e} and the critical shear stress ${\mathit{\tau}}_{\mathrm{c}}^{*}$. Parameter values
of both variables are optimized as described in Sect. 3.3.

We propose three metrics to evaluate the performance of the river incision models. The first one is the commonly used model error (ME),

$$\begin{array}{}\text{(15)}& \mathrm{ME}={\sum}_{i=\mathrm{1}}^{i=\mathrm{nb}}\sqrt{{\left({\displaystyle \frac{({O}_{i}-{M}_{i})}{{\mathit{\sigma}}_{i}}}\right)}^{\mathrm{2}}},\end{array}$$

where nb is the number of *E*_{CRN} data points, *O*_{i} represents the catchment-specific measured *E*_{CRN} denudation rates, *M*_{i} represents the
catchment-specific modeled river incision and *σ*_{i} represents
the catchment-specific standard deviation of *E*_{CRN}. The advantage of the
ME is that it explicitly incorporates the error on the analytical data
(*E*_{CRN}) by weighing the model error with the analytical error. However,
errors on CRN data are heteroscedastic: they systematically increase with
increasing denudation rates. Although the ME thus provides a good metric to
evaluate overall model performance, the metric is not well suited to
optimize model parameters in an optimization procedure: too much weight will
be given on optimization of the model in the lower regime of the denudation
spectrum in which measured errors on *E*_{CRN} are low, whereas higher measured
*E*_{CRN} data will not be approximated well because of large associated
errors. To compensate for the effect of heteroscedasticity we rescale values
*O*_{i}, *M*_{i} and *E*_{i} using a logarithm with base 10 when calculating ME (Herman et al., 2015). In this paper, ME will be
used to evaluate model performance but not to optimize model parameters.

A second metric is the coefficient of determination, *R*^{2}:

$$\begin{array}{}\text{(16)}& {R}^{\mathrm{2}}=\mathrm{1}-{\displaystyle \frac{{\sum}_{i=\mathrm{1}}^{i=nb}({O}_{i}-{f}_{i}{)}^{\mathrm{2}}}{{\sum}_{i=\mathrm{1}}^{i=nb}({O}_{i}-\stackrel{\mathrm{\u203e}}{O}{)}^{\mathrm{2}}}},\end{array}$$

where *f*_{i} represents the fitted *E*_{CRN} denudation rates. Contrary to
ME, *R*^{2} evaluates the explained variance of the model by giving all
observations the same weight, regardless their analytical error. However,
when model parameters result in an offset between simulated and observed
data (i.e., the intercept of the fit), this can still result in a high
*R*^{2}.

We therefore use the Nash–Sutcliffe model efficiency to optimize model parameters (NS; Nash and Sutcliffe, 1970):

$$\begin{array}{}\text{(17)}& \mathrm{NS}=\mathrm{1}-{\displaystyle \frac{{\sum}_{i=\mathrm{1}}^{i=\mathrm{nb}}({O}_{i}-{M}_{i}{)}^{\mathrm{2}}}{({O}_{i}-\stackrel{\mathrm{\u203e}}{O}{)}^{\mathrm{2}}}}.\end{array}$$

The NS coefficient ranges between −∞ and 1, where 1 indicates optimal model performance explaining 100 % of the data variance. When NS =0, the model is as good a predictor of the mean of the observed data. When NS < =0, the model performance is unacceptably low. The NS coefficient was developed in the framework of hydrological modeling but has been applied in a wide range of geomorphologic studies (e.g., Jelinski et al., 2019; Nearing et al., 2011).

4 Comparing model results with CRN-derived denudation rates

Back to toptop
In the following sections, we compare simulated erosion rates, obtained with
the river incision models presented in Eqs. (11)–(13), with measured
CRN-derived denudation rates. We start with the use of the A-SPM (Eq. 11) to
evaluate the extent to which lithological variability controls denudation
rates. Once the impact of lithological heterogeneity on river incision is
clarified, we evaluate whether runoff variability and incision thresholds
can explain variations in *E*_{CRN}-derived denudation rates. To this end,
two river incision models are evaluated (the R-SPM and ST-SPM, presented in
Eqs. 12 and 13, respectively). The optimized parameters and model performance
of all model scenarios are listed in Table 4. Best-fit results of a selected
number of model runs are presented in Figs. 7 and 8. An overview of
model fits for all the scenarios listed in Table 4 is given in Figs. S8,
S9 and S10.

^{a} If ${\stackrel{\mathrm{\u203e}}{L}}_{\mathrm{E}}$ is variable, catchment-specific values for
*L*_{E} are used (Table 2).
^{b} If *R* is fixed, a uniform mean runoff value of 0.8 m yr^{−1} is
used for all catchments. If *R* is variable, catchment-specific values are used
(Table 2).
^{c} If *k* is fixed, a uniform mean discharge variability value of 1.01 is
used for all catchments. If *k* is variable, catchment-specific values are used (Table 2).
^{d} The slope exponent (*n*) is optimized as a free parameter in A-SPM 1–2. It is fixed to 1 in A-SPM 3–4 (see text).

In a first set of model runs we evaluate the use of an area-based stream
power model (A-SPM) to explain observed variations in CRN-derived denudation
rates (*E*_{CRN}). We optimize river incision parameters for four scenarios
(Table 4: A-SPM scenarios 1–4): in the first two scenarios, the slope
exponent, *n*, is left as a free parameter. In the second two scenarios, the
slope parameter is fixed to unity (*n*=1). Figure 7 illustrates both the
*k*_{sn}−*E*_{CRN} (Fig. 7a and b) and corresponding *E*_{Mod}−*E*_{CRN}
relationships, wherein *E*_{Mod} represents the simulated river incision
(Fig. 7c and d).

In A-SPM scenario 1 (Table 4, Fig. 7c), we assume a spatially uniform
erodibility (${\stackrel{\mathrm{\u203e}}{L}}_{\mathrm{E}}$ fixed to 1 in Eq. 11) and leave the erosion
efficiency coefficient (*K*^{′}) and the slope parameter *n* as free parameters during
model optimization. The optimized fit between simulated erosion (*E*; Eq. 2)
and *E*_{CRN} is shown in Fig. 7c. The optimized fit results in a
high degree of data scattering, resulting in an NS model efficiency of 0.5, an
*R*^{2} of 0.5, an ME of 3.25, and optimized values for *K*^{′} and *n* of respectively 0.73 m^{0.1} s^{−1} and 1.07. The fit between *k*_{sn} and *E*_{CRN} (Fig. 7a) or simulated river incision and measured denudation rates (Fig. 7c)
hints at the existence of a correlation between *E*_{CRN} and river incision
rates. The fit shown in Fig. 7c illustrates that modeled erosion
rates for catchments with a low mean erodibility index (high resistance
to erosion) are mostly overpredicted (plotting below the 1:1 line), whereas
modeled erosion rates of catchments with a high erodibility index are
mostly underpredicted (plotting above the 1:1 line).

In A-SPM scenario 2 (Table 4, Fig. 7d), we quantify the impact of varying
lithology by using sub-catchment-specific values for the lithological
erodibility (${\stackrel{\mathrm{\u203e}}{L}}_{\mathrm{E}}$ in Eq. 11) and leaving *k*_{a} and *n* as free
optimization parameters. The optimized fit between simulated river incision
(*E*, Eq. 11) and *E*_{CRN} is shown in Fig. 7d. Optimization results in an
NS model efficiency of 0.73, an *R*^{2} of 0.73, an ME of 2.23, and optimized values
for *k*_{a} and *n* of respectively 0.07 m^{0.1} s^{−1} and 1.63.
Considering lithological erodibility strongly reduces data scatter
surrounding the fit. The importance of lithological strength in controlling
the A-SPM and the *k*_{sn}−*E*_{CRN} relation (Fig. 7b) confirms that
strong metamorphic and plutonic rocks erode at slower rates than lithologies
that are less resistant to weathering such as volcaniclastic deposits. The
erodibility index appears to provide an appropriate scaling of relative rock
strength: analysis of residuals did not reveal any significant relation of
residuals with lithology. When using spatially variable, sub-catchment-specific lithological erodibility values (${\stackrel{\mathrm{\u203e}}{L}}_{\mathrm{E}}$) (Fig. 7d), the
*n* coefficient of the SPM is considerably larger than unity (*n*=1.63) and the
*k*_{sn}−*E*_{CRN} relationship becomes nonlinear (Fig. 7b),
corroborating earlier empirical findings
(DiBiase
et al., 2010; Harel et al., 2016; Lague, 2014; Whittaker and Boulton, 2012).
To evaluate the impact of a variable *n* exponent on the performance of the
empirical A-SPM, we executed two more model optimizations.

In A-SPM scenario 3 (Table 4, Fig. S8c), we assume a spatially uniform
lithology and erodibility (${\stackrel{\mathrm{\u203e}}{L}}_{\mathrm{E}}$ fixed to 1 in Eq. 11), fix *n* to 1 and
only leave *K*^{′} to be optimized as a free model parameter. With an NS model
efficiency of 0.5, an *R*^{2} of 0.5, an ME of 3.2 and an optimized value for
*K*^{′} of 1.00 m^{0.1} s^{−1}, the model fit and performance are similar to the
values obtained in scenario 1.

In A-SPM scenario 4 (shown in Table 4, Fig. S8d), lithological
variability is considered (using sub-catchment-specific values for
${\stackrel{\mathrm{\u203e}}{L}}_{\mathrm{E}}$ in Eq. 11), *n* is fixed to 1 and *K*^{′} is a free model parameter. With
an NS model efficiency of 0.51, an *R*^{2} of 0.56, an ME of 3.05 and an optimized
value for *K*^{′} of 1.4 m^{0.1} s^{−1}, the model performance is much lower
than when leaving the slope exponent *n* as a free parameter (A-SPM scenario
2).

The results from the four scenarios show that a nonlinear relationship
between river steepness (*k*_{sn}, representing river incision rates) and
*E*_{CRN} is unveiled when the lithological heterogeneity is explicitly
taken into account (Fig. 7b). Likewise, a nonlinear river incision model
(A-SPM scenario 2; Fig. 7d) in which lithological heterogeneity is
considered outperforms the other evaluated A-SPM scenarios (Table 4).

The previous analysis shows that the explanatory power of the A-SPM model,
and therefore the *k*_{sn}−*E*_{CRN} relationship, improves when considering
spatial variations in lithology. Moreover, when considering variations in
lithological erodibility, river incision is found to be nonlinearly
dependent on the channel slope (*S*), with *n*=1.63. In a next step we evaluate
whether this nonlinear relation can be explained by spatial and/or temporal
rainfall variability and/or the existence of thresholds for river incision
(Table 4: R-SPM scenarios 1–2 and ST-SPM scenarios 1–8; Fig. 8).

In a first set of model runs, we evaluate the performance of the
Runoff-Based Stream Power Model (R-SPM Eq. 12) to evaluate the role of
spatially variable runoff using catchment-specific values for mean runoff
(*R* derived from the WaterGAP data; reported in Table 2 and shown in Fig. 6).

In R-SPM scenario 1 (Table 4, Fig. S9a), lithological variability is not
considered ($\stackrel{\mathrm{\u203e}}{{L}_{\mathrm{E}}}$ fixed to 1 in Eq. 12). With an NS model efficiency of
0.49, an ME of 3.57 and an *R*^{2} of 0.51, model performance is comparable to
the regular A-SPM under uniform lithology with *n* fixed to 1 (NS = 0.5). This
illustrates that studying spatial runoff variability is not feasible when
ignoring the confounding role of lithological erodibility in controlling denudation
rates.

In R-SPM scenario 2 (Table 4, Fig. 8a), lithological variability is
considered (using sub-catchment-specific values for ${\stackrel{\mathrm{\u203e}}{L}}_{\mathrm{E}}$ in Eq. 12). With an NS model efficiency of 0.7, an ME of 2.61 and an *R*^{2} of 0.75,
model performance is close to that of the regular A-SPM under uniform
lithology with *n*≫1 (NS =0.72). This model
simulation therefore suggests that spatial variations in runoff can account
for the nonlinearity in the *k*_{sn}−*E*_{CRN} relationship: while slope
dependency in the R-SPM is fixed to unity (see derivation in Eq. 4a–c),
the model is capable of explaining the spatial pattern in denudation rates.
This implies that orographic rainfall and thus runoff gradient as shown in
Fig. 6 influence the efficiency of river incision. The offset between the
*R*^{2} (0.75) and NS (0.70) values can be attributed to the way in which these
metrics work: whereas *R*^{2} evaluates the goodness of the linear fit between
modeled and measured observations, NS evaluates the absolute differences
between modeled and observed denudation rates. Hence, for the NS model
efficiency to be high, observations must fit on the 1:1 line (Fig. 8a).
However, most of the simulated values for low denudation rates are
overestimated when using the optimized parameter values of the R-SPM and
plot below the 1:1 line (Fig. 8a). Therefore, we conclude that the R-SPM
performs well in predicting measured denudation rates but low denudation
rates are overestimated, resulting in an NS and ME value respectively
slightly lower and higher than those of the empirical A-SPM. In the
following section we evaluate whether introducing temporally variable runoff
coefficients and/or incision thresholds can further improve the performance
of a river incision model.

In a final series of model runs, we use the Stochastic-Threshold Stream
Power Model (ST-SPM, Eq. 13) to evaluate the role of spatially variable
runoff (catchment-specific *R*; reported in Table 2 and shown in Fig. 6) in
combination with catchment-specific runoff variability (*k*; reported in Table 2)
and the presence of incision thresholds (*τ*_{c} in *ψ* in Eqs. 4 and 10). Table 4 reports details on the different model scenarios in which
ST-SPM is optimized to the observed *E*_{CRN} data considering all possible
combinations (4) of uniform or spatially variable catchment mean runoff
(*R*) and uniform or spatially variable catchment mean runoff variability
(*k*). For reference, the four scenarios include both uniform and spatially
variable lithological erodibility, *L*_{E} (eight scenarios in total).

In ST-SPM scenarios 1–4 (Table 4, Fig. S10a–d), the ST-SPM is optimized
assuming a constant erodibility (*L*_{E} fixed to 1). Similar to what has
been found for the R-SPM, model performance is not any better compared to
the use of a simple A-SPM when not considering lithological variability.
This confirms that optimizing more complex river incision models (such as
the ST-SPM) has little added value when the heterogeneity in environmental
conditions (lithological heterogeneity) is not considered.

In ST-SPM scenarios 5 and 6 (Table 4, Fig. S10e–f), catchment mean
runoff ($\stackrel{\mathrm{\u203e}}{R})$ is fixed to the average value of all catchments (0.82 m yr^{−1}) in order to evaluate the role of (i) variations in observed
temporal runoff variability (*k*) and (ii) optimized values for the incision
threshold (*τ*_{c}). In scenario 5, *k* is fixed to the average value for
all catchments (*k*=1.01), whereas in scenario 6, *k* is set to the catchment-specific values as listed in Table 2. Both scenarios (5 and 6) perform well
with an NS value equalling 0.71, indicating that temporal runoff variability
(*k*) is not influencing model performance. Regardless of the lack of spatially
variable runoff (*R*), both scenarios perform as well as R-SPM scenario 2,
in which runoff variability was considered. The good performance of ST-SPM
scenarios 5 and 6 can be attributed to the presence of an incision threshold
(*ψ* *>* 0 in Eq. 13) at which *τ*_{c} is optimized to a
value of ca. 30 Pa (Table 4). The fact that the use of the ST-SPM with constant
runoff values yields a good model fit suggests that part of the nonlinear
relationship between river steepness, *k*_{sn}, and *E*_{CRN} can be attributed
to the presence of thresholds for river incision to occur
(Lague, 2014).

ST-SPM scenarios 7 and 8 (Table 4, Figs. S10e–f and 8b) are
similar to scenarios 5 and 6, with the difference that spatial runoff
variability is considered by using catchment-specific values for runoff
($\stackrel{\mathrm{\u203e}}{R}$; Table 2). Similarly to scenarios 5 and 6, using catchment-specific values for *k* does not improve model performance, resulting in a
similar model performance for scenarios 7 and 8. Overall, ST-SPM scenarios 6
and 7 result in the highest model performance of all tested scenarios, with
an NS model efficiency of 0.75, an ME of 2.22 and 2.21, and an *R*^{2} of 0.75. The
optimized model fit for ST-SPM scenario 7 is shown in Fig. 8b and
corresponds well with the 1:1 line between modeled and observed denudation
rates. Optimized values for *τ*_{c} are ca. 14–15 Pa, which is in the
range but at the lower spectrum of earlier documented values for critical
shear stress (e.g., Shobe et al.,
2018, report *τ*_{c} values between 10 and 1000 Pa). Contrary to the R-SPM
with which low denudation rates are overestimated (Fig. 8a), the ST-SPM does
predict low denudation rates better due to the consideration of an incision
threshold that mainly influences simulated river denudation rates at the
lower end of the spectrum.

ST-SPM scenarios 7 and 8 have a model error (ME of 2.22 and 2.21, respectively)
similar to the model error of A-SPM scenario 2 (ME =2.23). Hence, we
conclude that an ST-SPM considering spatial variations in runoff and
simulating a critical threshold for river incision performs as well as an
A-SPM with the effect of allogenic (runoff variability) and autogenic
(incision thresholds) response cast in the lumped empirical incision
parameters. While the R-SPM and ST-SPM do not necessarily predict spatial
patterns in observed *E*_{CRN} rates better than an A-SPM, they do enable
one to simulate the effect of runoff variability and incision thresholds,
therefore providing an operational tool to simulate past and future climate
changes. Note that differences in model performance between R-SPM scenario 2
and ST-SPM scenarios 5–8 are existent but not very pronounced. To evaluate
the significance of these differences, our analysis should be repeated on
larger datasets capturing a wider variability in denudation rates and
hydrology.

5 Discussion

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In theory, rates of hillslope denudation equal rates of river incision if landscapes are either in a steady state or if transient landscapes are characterized by rapid hillslope response (e.g., threshold hillslopes). Steady-state landscapes can only be achieved under stable climatic and tectonic settings that prevail over millions of years. Such stability is rarely met in tectonically active regions where landscapes continuously respond to environmental perturbations (Armitage et al., 2018; Bishop et al., 2005; Campforts and Govers, 2015).

The downstream reaches of the Paute catchment are a good example of a transient landscape where a major knickzone is propagating upstream, resulting in steep threshold topography downstream of the knickzone (Fig. S3 and Vanacker et al., 2015). Facing a sudden lowering of their base level after river rejuvenation, soil production and linear hillslope processes (Campforts et al., 2016) are no longer in equilibrium with rapidly incising rivers (Fig. 15 in Hurst et al., 2012). In steep topography, hillslopes may transiently evolve to their mechanically limited threshold slope whereby any further perturbation will result in increased sediment delivery through mass-wasting processes such as rockfall or landsliding (Bennett et al., 2016; Blöthe et al., 2015; Burbank et al., 1996; Larsen et al., 2010; Schwanghart et al., 2018). Given the erratic nature of landslides, not all threshold hillslopes will respond simultaneously to base-level lowering depending on local variations in rock strength, hydrology, land use and seismic activity (Broeckx et al., 2020; Guns and Vanacker, 2014). Therefore, catchments in transient landscapes might experience hillslope denudation with highly variable rates (Vanacker et al., 2020).

We argue that CRN-derived denudation rates in the Paute basin both overestimate and underestimate long-term incision rates in these catchments. Overestimation may result from the occurrence of recent, deep-seated landslide events, that deliver sediments with a low CRN concentration to rivers (Tofelde et al., 2018). Underestimation, in turn, may occur if long-term hillslope lowering is accomplished by rare and large landslides whose return periods exceed the integration time of CRN-derived denudation rates (Niemi et al., 2005; Yanites et al., 2009).

Longitudinal profiles of rivers draining to the knickzone in the Paute
catchment show marked knickpoints. This is particularly evident in
catchments 9–16 (Fig. 1) where *k*_{sn} values are high (Fig. 2) and
knickpoints appear in the longitudinal profiles (Figs. S3 and S4).
Simulated erosion rates for some of these catchments deviate from
CRN-derived denudation rates (Fig. 8b; IDs 13, 14 and 16), whereas for
others (e.g., IDs 9 and 11), predictions from the stochastic-threshold river
incision model show good agreement with *E*_{CRN} data. For
catchments with a sufficiently large drainage area, modeled incision rates
correspond well with *E*_{CRN} (IDs 9 and 11 being both ca. 700 km^{2}), most likely because the mechanisms that potentially
cause overestimation and underestimation cancel each other out at this
scale. For smaller catchments (IDs 8, 13, 14 and 16 all being < 12 km^{2}) there is a discrepancy between simulated river incision
rates and *E*_{CRN}.

Although river incision models can be used to simulate denudation patterns
in large transient catchments (> 10 km^{2}), there
is a need to develop alternative approaches including landslide
mechanisms in long-term landscape evolution models such as the TTLEM (Topo Toolbox Landscape Evolution Model; Campforts et al., 2017) or Landlab
(Hobley et al., 2017).

Our analysis reveals the potential role of temporal and spatial variations
of rainfall in long-term landscape evolution. Integration times of
CRN-derived denudation rates measured in the Paute basin are of the order of
1.5–175 kyr. In contrast, response times of longitudinal river profiles
generally range 0.25–2.5 Myr (Campforts
et al., 2017; Goren et al., 2014; Snyder et al., 2003; Whipple, 2001; Wobus
et al., 2006). During thousand-year to million-year timescales, it is unlikely that temporal
rainfall distributions remain stationary. Thus, there is little reason to
assume that the hydrometeorological data that we inferred from 35 years of
data fully capture rainfall variability over the response times of river
profiles and hillslopes. Contrary to temporal variations, the spatial
patterns in orographic precipitation are characteristic of the formation of
a mountain range at geological timescales
(Garcia-Castellanos and Jiménez-Munt,
2015). In the southern Ecuadorian Andes, moist air advection via the South
American low-level flow generates pronounced patterns of orographic
precipitation (Campetella and Vera, 2002). These
patterns might have persisted since at least the most recent phase of
Andean uplift in the Late Miocene (Spikings
et al., 2010; Spikings and Crowhurst, 2004). Present-day rainfall and runoff
spatial gradients (Fig. 6) are thus deemed to be informative for spatial
patterns of discharge at longer timescales (Sect. 3.1). The performance
of the stream power models underscores this interpretation. While accounting
for spatial patterns in runoff improves the performance of a
stochastic-threshold SPM (Table 4 and Sect. 4.2.2), incorporating proxies
for temporal discharge variability leads to no improvement of model
performance (the role of *k* in Sect. 4.2.2).

In all our simulations, model efficiency improves when incorporating rock
strength variability (Table 4), which is consistent with earlier studies (Lavé
and Avouac, 2001; Stock and Montgomery, 1999). In the absence of generally
accepted metrics of erodibility, we employ an empirically derived
lithological erodibility index (*L*_{E}; Eq. 7) based on the age and lithological
composition of stratigraphic units. Owing to its simplicity, this or a
similar index can potentially be applied at continental to global scales
at which information on rock physical properties is usually lacking the detail
available at smaller spatial scales (Attal and Lavé,
2009; Nibourel et al., 2015). Notwithstanding, river incision also depends
on other rock properties such as the density of bedrock fractures, joints
and other discontinuities (Whipple et
al., 2000). Fracture density has in turn been linked to spatial patterns of
seismic activity (Molnar et al., 2007). Given
the limited variability of seismic activity within the Paute basin
(Petersen et al., 2018; Fig. S2), seismicity was not
considered in our regional analysis but could be considered when applying
our approach to other regions characterized by more spatial seismic
variability.

Incorporating spatial patterns of rock strength not only reduces the scatter
surrounding the modeled river incision versus *E*_{CRN}-derived denudation
rates, but also controls the degree of nonlinearity between river steepness
(*k*_{sn}) and denudation rates, expressed by the slope exponent *n* in the A-SPM
(Fig. 7). Omitting rock strength variability results in a
*k*_{sn}−*E*_{CRN} relation that is close to linear in the Paute catchment
(with *n*=1.07). This contradicts other studies in which lithology was assumed
to be uniform and *n* has been reported to be larger than 1
(e.g.,
DiBiase et al., 2010; Lague, 2014; Whittaker and Boulton, 2012).

The A-SPM performs well in explaining *E*_{CRN} when lithology is
considered and *n* ≫1 (Fig. 9; high NS model
efficiency, low ME). For *n*=1, the performance of the A-SPM is low. The
result is consistent with earlier studies reporting *n*≫1 (e.g., DiBiase et al., 2010; Harel et al., 2016; Ouimet et al., 2009; Scherler et
al., 2014), which Lague (2014) attributes to discharge variability and
incision thresholds. We tested this hypothesis using the R-SPM and ST-SPM.
Our results underscore the fact that the nonlinear relationship between
*k*_{sn} and *E*_{CRN} can be attributed to the spatial variability of
mean annual runoff. Figure 9 shows that the R-SPM (in which *n* is fixed to the
theoretically obtained value of 1) performs better than an A-SPM when *n* is
fixed to 1. In tectonically active regions, steep river reaches often
spatially coincide with the edge of the mountain range over which mean annual
rainfall rates are highest. Accordingly, if variations in runoff are not
considered, the effects of orographic precipitation will be partly
accommodated by a nonlinear relationship between river steepness and
denudation rates. The R-SPM accounts for this effect but results in an
underestimation of low river incision rates (Fig. 8a). Moreover, the
model error (Fig. 9b) shows that the R-SPM does not perform as well as
the A-SPM. In a final set of model runs, we apply the ST-SPM with the
explicit simulation of a threshold, which improves model performance, especially
for low denudation rates, resulting in an overall model error equal
to the one obtained with the A-SPM with *n*≫1
(Fig. 9). This finding points to the potentially important role of
thresholds for river incision to occur.

The model performance of the ST-SPM equals the performance of an empirical A-SPM
with a slope exponent ≫1 (Fig. 9). Our
interpretation is that (i) spatial variations in runoff and (ii) the
incision thresholds are the causes of an observed nonlinear relation
between *k*_{sn} and *E*_{CRN}. With a seemingly equal model
performance, one could wonder what the benefit of the more complex ST-SPM
model is over a simple, nonlinear A-SPM. The aim of using an ST-SPM is,
however, beyond fitting observed denudation rates: we want to identify to
what extent the system is forced by internal allogenic dynamics such as the
presence of incision thresholds or external autogenic forces such as runoff
variability. The use of the ST-SPM illustrated that both processes can be
accounted for in a quantitative way so that future studies can explicitly
consider their role when reconstructing past landscape response to external
perturbations (e.g., climate change).

To further explore the interdependency between incision thresholds and
spatial runoff variability, our approach can be applied to CRN datasets
covering regions characterized by more pronounced rainfall gradients
(e.g., in
Chile: Carretier et al., 2018). Accounting for spatial variations in
temporal discharge distributions (with *k* characterizing the stochastic flood
occurrence) did not improve or deteriorate model performance (ST-SPM
scenario 8 in Table 4). This is likely due to data limitations: the
necessary data to characterize temporal variations in discharge within a
given catchment over a timescale that is relevant for CRN-derived denudation
rates are, at present, not available.

Our finding that spatial patterns in precipitation are related to river
incision patterns corroborate findings in Hawaii
(Ferrier et al., 2013), the Himalaya
(Scherler et al., 2017) and in the Andes (Sorensen and Yanites, 2019).
Sorensen and Yanites (2019) evaluated
the role of latitudinal rainfall variability in the Andes in erosional
efficiency using a set of numerical landscape evolution model runs. They
show that erosion efficiency in tropical climates at low latitudes, where
the Paute basin is located, is well captured by the spatial pattern of mean
annual precipitation and thus runoff. At higher latitudes (25–50^{∘}) where mean annual precipitation decreases but erosivity is still high due
to the intensity of storms (Sorensen and Yanites, 2019),
river erosivity is likely better captured by spatial patterns in storm
magnitude and frequency.

6 Conclusions

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Numerous studies report a nonlinear relationship between channel steepness and CRN-derived denudation rates. Based on the growing mechanistic understanding of river incision processes, this nonlinear relationship is often attributed to incision thresholds. Rainfall variability controls the frequency of river discharges that exceed incision thresholds. Although the dynamic interplay between stochastic runoff and incision thresholds theoretically results in a nonlinear relationship between channel steepness and denudation rates, coupling theory with field data has been challenging. We address this issue in the Paute basin where we scrutinize the relationship between CRN-derived denudation rates and river incision using three different stream power models. We show that lithological variability obscures the relationship between channel-steepness-based river incision and CRN-derived denudation rates.

In order to account for rock strength variability, which for the Paute basin is mainly ascribed to variations in lithological strength in the study area, we propose the use of an empirical lithological strength index that is based on the lithology and age of lithostratigraphic units. Including lithological variability in the models increases the correlation between river steepness and denudation rates and reveals a nonlinear relation, which we seek to explain using a stochastic-threshold SPM (ST-SPM). Using a downscaled version of a hydrological reanalysis dataset, we show that the combination of spatially varying runoff and incision thresholds explains the observed nonlinear relationship. We do not detect, however, an impact of temporal discharge patterns on river incision. We attribute this to the integration time of CRN data and response times of river longitudinal profiles, which extend beyond the timescales at which discharge distributions can be assumed to be stationary.

Our study shows the potential of an ST-SPM to infer regional and,
potentially, continental to global differences in rainfall variability.
However, we emphasize that its application needs to account for confounding
environmental variables such as rock strength. Simplified process
representation of stream-power-based incision models (e.g., lack of
sediment–bedrock interactions) might explain part of the remaining scatter
between predicted and measured denudation rates. However, residual analysis
shows that most of the remaining scatter occurs in small transient
catchments (up to 10 km^{2}) where sporadic mass-wasting
processes on hillslopes likely obscure the relation between measurements and
predictions. Elucidating this relation further could potentially be fostered by
dynamic numerical landscape evolutions models that explicitly simulate the
coupling between transient river adjustment and hillslope response.

Data availability

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Data availability.

All data used in this paper are freely available from the referenced agencies.
The hydrological data used in this paper were created in the framework of the Earth2ObserveWater Resource Reanalysis project (WRR2; Schellekens et al., 2017) and are available from 1979 to 2014. Specifically, we use the hydrological data calculated with the global water model WaterGAP3 (Water – Global Assessment and Prognosis: Alcamo et al., 2003; Döll et al., 2003) at a spatial resolution of 0.25 and a daily temporal resolution (http://www.earth2observe.eu, last access: 19 May 2020) (Schellekens et al., 2017). A high-resolution mean annual precipitation map (PRIDW) is calculated by downscaling the WaterGAP precipitation data (*P*) using a series of rain gauge observations (338 stations, 1990–2013) from the Ecuadorian national meteorological service (INAMHI; available from http://www.serviciometeorologico.gob.ec/biblioteca/, last access: 19 May 2020) (INAMHI, 2020). Topographic data are
available from NASA (Farr et al., 2007). Lithological data are
provided in the Supplement. Calculations were done in MATLAB
using the Topo Toolbox Software (Schwanghart and
Scherler, 2014).

Supplement

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Supplement.

The supplement related to this article is available online at: https://doi.org/10.5194/esurf-8-447-2020-supplement.

Author contributions

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Author contributions.

BC conceived the project in collaboration with VV, MVM and GG. BC performed the statistical analysis and took the lead in writing the paper. All authors contributed to shaping the research and analyses, as well as writing the paper.

Competing interests

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Competing interests.

The authors declare that they have no conflict of interest.

Financial support

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Financial support.

Benjamin Campforts received a postdoctoral grant from the Research Foundation
Flanders (FWO, reference number 12Z6518N). Gustavo E. Tenorio received a PhD grant
from ARES – the Académie de recherche et d'enseignement supérieur
de la Belgique – as part of the PRD research cooperation project
“Strengthening the scientific and technological capacities to implement
spatially integrated land and water management schemes adapted to local
socio-economic, cultural and physical settings”, as well as a grant from the
Secretaría de Educación Superior de Ciencia, Tecnología e
Innovación de la República del Ecuador. The article processing charges for this open-access

publication were covered by a Research Centre

of the Helmholtz Association.

Review statement

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Review statement.

This paper was edited by Simon Mudd and reviewed by two anonymous referees.

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Short summary

In this contribution, we explore the spatial determinants of bedrock river incision in the tropical Andes. The model results illustrate the problem of confounding between climatic and lithological variables, such as rock strength. Incorporating rock strength explicitly into river incision models strongly improves the explanatory power of all tested models and enables us to clarify the role of rainfall variability in controlling river incision rates.

In this contribution, we explore the spatial determinants of bedrock river incision in the...

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