ESURFEarth Surface DynamicsESURFEarth Surf. Dynam.2196-632XCopernicus GmbHGöttingen, Germany10.5194/esurf-3-291-2015Bedload transport controls bedrock erosion under sediment-starved conditionsBeerA. R.alexander.beer@wsl.chhttps://orcid.org/0000-0001-7538-6727TurowskiJ. M.https://orcid.org/0000-0003-1558-0565WSL Swiss Federal Institute for Forest, Snow and Landscape Research, 8903 Birmensdorf, SwitzerlandDepartment of Environmental System Sciences, ETH Zurich, 8092 Zurich, SwitzerlandGFZ German Research Centre for Geosciences, Telegrafenberg, 14473 Potsdam, GermanyA. R. Beer (alexander.beer@wsl.ch)13July2015332913092December20146January201526May20158June2015This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://esurf.copernicus.org/articles/3/291/2015/esurf-3-291-2015.htmlThe full text article is available as a PDF file from https://esurf.copernicus.org/articles/3/291/2015/esurf-3-291-2015.pdf
Fluvial bedrock incision constrains the pace of mountainous landscape
evolution. Bedrock erosion processes have been described with incision models
that are widely applied in river-reach and catchment-scale studies. However,
so far no linked field data set at the process scale had been published that
permits the assessment of model plausibility and accuracy. Here, we evaluate
the predictive power of various incision models using independent data on
hydraulics, bedload transport and erosion recorded on an artificial bedrock
slab installed in a steep bedrock stream section for a single bedload
transport event. The influence of transported bedload on the erosion rate
(the “tools effect”) is shown to be dominant, while other sediment effects
are of minor importance. Hence, a simple temporally distributed incision
model, in which erosion rate is proportional to bedload transport rate, is
proposed for transient local studies under detachment-limited conditions.
This model can be site-calibrated with temporally lumped bedload and erosion
data and its applicability can be assessed by visual inspection of the study
site. For the event at hand, basic discharge-based models, such as
derivatives of the stream power model family, are adequate to reproduce the
overall trend of the observed erosion rate. This may be relevant for
long-term studies of landscape evolution without specific interest in
transient local behavior. However, it remains to be seen whether the same
model calibration can reliably predict erosion in future events.
Introduction
Quantitative landscape evolution analysis is a fundamental domain in today's
geomorphological research. The hydrological system plays an important role in
landscape response to tectonics through the formation of drainage networks,
the adjustment of river channel shape and slope, and the routing of sediments
(e.g., Howard et al., 1994; Whipple and Tucker, 1999; Whipple, 2004). Thus,
a mechanistic understanding of the processes that are active in rivers and
their mathematical description is crucial in order to capture landscape
evolution as a whole (e.g., Lague, 2014). Bedrock rivers are particularly
frequent in mountainous regions and there have been many attempts to model
their erosional work (overviews by Sklar and Dietrich, 2006; Turowski, 2012;
Whipple et al., 2013). A number of physical fluvial erosion processes acting
on bedrock surfaces have been described, and abrasion by bedload and plucking
of blocks are thought to be the most important of these (cf. Whipple et al.,
2013). Both processes are driven by the impact of bedload particles on the
bedrock surface.
Bedrock incision is generally thought to depend on flow hydraulics and in
situ substrate properties. This notion forms the basis of the most commonly
used erosion models of the stream power incision model family (Howard and
Kerby, 1983; Seidl et al., 1994; Turowski, 2012; Lague, 2014), in which
erosion rate is a power function of stream power or bed shear stress (Whipple
and Tucker, 1999). However, mechanistically, it is known that fluvial bedrock
incision is driven by the impact of sediment particles (Sklar and Dietrich,
2001; Hartshorn et al., 2002; Turowski, 2012; Cook et al., 2013; Turowski
et al., 2015). Several effects of the transported sediment need to be
accounted for. These are the “thresholds of motion and suspension” that are
related to a characteristic grain size (Lague et al., 2003; Sklar and
Dietrich, 2004; Attal et al., 2011), the shielding of bedrock by sediments,
known as the “cover effect” (e.g., Gilbert, 1877; Turowski et al., 2008;
Johnson et al., 2009), and erosive pebble impacts on the bedrock that depend
on the amount of mobile sediment, known as the “tools effect” (e.g., Foley,
1980; Turowski and Rickenmann, 2009; Cook et al., 2013). Taking into account
these four effects, erosion rate E can be written as (cf. Sklar and
Dietrich, 2006)
E=KHyaSebFecQsd.
Here, the scaling of bedrock erodibility, sediment erosivity and,
consequently, the dominant erosional process, are lumped together in
a model-specific prefactor K (e.g., Howard, 1994; Sklar and Dietrich,
2006). Hy is a placeholder for an effective hydraulic parameter (e.g.,
discharge, stream power, bed shear stress), in which the term efficient means
it can incorporate a threshold of grain motion. Thus, Hy represents the
sediment motion effects in these cases. The suspension effect term
Se is the fraction of particles in suspension, Fe is
the fraction of bed exposure, which is related to the cover effect, and
Qs is the sediment transport rate, describing the availability of
erosive tools (see Appendix A for more details). The exponents a, b, c,
and d modulate the dependence of erosion rate on these four effects,
respectively.
Available fluvial erosion models were originally developed at the process
scale, and their application to whole stream sections or even catchments is
problematic (e.g., Lague et al., 2005). Spatial upscaling, from process to
reach scale, and from reach to catchment scale, is not completely understood.
Several factors should be taken into account explicitly, such as time
(Gardner et al., 1987; Mills, 2000; Finnegan et al., 2014), space (Hancock
et al., 1998; Wohl, 1998; Goode and Wohl, 2010), changing morphology (Inoue
et al., 2014; Johnson, 2014; Zhang et al., 2015), and variability in forcing
conditions, such as discharge, climate or sediment process interactions
(Hancock et al., 1998; Snyder et al., 2003; Lague et al., 2005; Whipple
et al., 2013). In addition, many models predict similar steady-state
morphology (e.g., Whipple and Tucker, 2002; Lague, 2014), but the transient
evolution of entire channels is difficult to reconstruct in the field, and
hence model validation is challenging.
The theoretical consideration of incision model sensitivity (Sklar and
Dietrich, 2006) and model assessment by means of field data have, to date,
largely focused on the steady-state geometry of entire channels (Lague,
2014). van der Beek and Bishop (2003) remodeled the long-profile evolution of
incising rivers in the Upper Lachlan catchment (southeastern Australia) based
on known paleo-profiles. They found that all of the tested models gave
reasonable predictions for the current long profiles with the application of
suitable parameter sets. In contrast, Tomkin et al. (2003) determined that
none of the tested models could satisfactorily explain the data from the
well-studied Clearwater River (northwestern Washington State, USA), which is
thought to exhibit steady-state incision. Tomkin et al. (2003), however,
attributed this failure more to the quality of their data, rather than to the
inadequacy of the applied incision models. A different approach was taken by
Turowski et al. (2013, 2015), who compared field measurements of energy
delivery to the streambed to predictions using the saltation–abrasion model
(Sklar and Dietrich, 2004). This approach, however, can only be applied to
specific model types because only one of the model's elements is compared to
field data.
The problem of model adequacy and potential study-site sensitivity can be
simplified if models and their behavior are examined at the process scale
(Whipple and Tucker, 1999; Tucker and Whipple, 2002). However, such
evaluations have not been possible to date due to the lack of data at the
appropriate resolution (Turowski, 2012). Hence, the transient validity of
fluvial erosion models at the process scale has neither been assessed in the
laboratory nor in the field. Here, we use field data of unprecedented detail
and quality (Beer et al., 2015) to directly evaluate available fluvial
incision models at the process scale, using a transient bedrock erosional
signal throughout a single sediment transport event. Thus, we obtain
constraints for the modeling of fluvial bedrock erosion at a scale that has
not been studied previously.
Observation site and data
The Erlenbach is a small pre-Alpine mountain stream in Switzerland that hosts
a well-instrumented bedload transport observatory. In 2011, the site was
supplemented with a novel setup for measuring bedrock erosion, which was
named “erosion scales” (Beer et al., 2015). The infrastructure, measurement
methods, and accuracy have been described in detail elsewhere (Rickenmann
et al., 2012; Turowski et al., 2013; Beer et al., 2015), and are only briefly
summarized here. Discharge is gauged with 15 % uncertainty, bedload mass
transport (henceforth referred to as bedload transport) can be determined to
an accuracy of 1kg±30% using the Swiss plate geophone
bedload sensor system (here called the geophone sensor), and at-a-point
erosion sensors in the streambed have a resolution of better than
0.1 mm with 5 % uncertainty. The measurements of these three
quantities are completely independent and all data are recorded at a time
resolution in minutes (Beer et al., 2015).
Overview of the erosive discharge event showing the three
independently measured data sets. The colored bars on top indicate the three
study periods for model sensitivity analysis.
Event statistics. Values are given for the event period, unless otherwise indicated.
DataValueSite data Local bed slope16 %Bed roughness (standard deviation of bed elevations)0.04 mevent data Length of event period900 minLength of bedload period649 minLength of erosion period363 minTotal discharge33′095 m3Maximum discharge1.1 m3 s-1Range of water depth0.01–0.17 mRange of stream width0.53–2.43 mEvent bedload grain size d30 (30 vol% grains are smaller)0.015 mEvent bedload grain size d50 (50 vol% grains are smaller)0.020 mEvent bedload grain size d90 (90 vol% grains are smaller)0.075 mTotal bedload transporteda8.8 tTotal at-a-point erosion of erosion sensor c30.85 mmObserved at-a-point erosion steps13Bedload motion (Hy) Calculated critical shear stress (Shields stress)0.006Critical discharge0.008 m3 s-1Critical shear stress1.9 N m-2Critical unit stream power5.9 W m-2Mean Froude number6.7 ± 3.3Range of Reynolds numbers105–106Range of Rouse numbers3.0–13.1Mean transport stage ts (shear stress/critical shear stress)75 ± 38Mean relative bedload supply (actual bedload/potential bedload)b0.07 ± 0.07Maximum relative bedload supply0.56Mean bedload concentration (bedload/discharge)b0.006 ± 0.006Maximum bedload concentration0.05Bedload suspension (Se) Mean suspension term Se0.64 ± 0.18Minimum suspension effect Se0.34Bedrock exposure (Fe) Mean fraction of exposure Fe0.95 ± 0.07Minimum fraction of exposure Fe0.44Bedload transport (Qs) Maximum bedload transport Qs103 kg min-1Mean bedload transportbQs14 kg min-1
a Over a bed area of 0.18 m2 surrounding the erosion sensor (the concrete slab).b Values for the bedload period.
The applied incision models. The second horizontal line subdivides
models that include the tools effect from models that do not. The notation of
variables in the explicit models is given in Appendix A.
Model class1Model name1Original referenceConsidered sediment effects2General model shape3Explicit model4MotionSusp.CoverToolsH0Constant erosionE∼1/(length of period)E∼1/(length of period)IUnit stream power (USP)Howard et al. (1994)––––E∼HyaE∼ωaIIExcess unit stream power (EUSP)Sklar and Dietrich (2006)√–––E∼HyexaE∼ωexaIIILinear decline (LD)Whipple and Tucker (2002)––√–E∼HyaFecE∼ωa(1-Qs/Qsc)cIVAlluvial bedload (AB)Sklar and Dietrich (2006)√–√–E∼HyexaFecE∼tsexa(1-Qs/Qsc)cVaTools only (TO)5This work–––√E∼QsdE∼QsdVTools (T)Foley (1980)√––√E∼HyexaQsdE∼tsexaQsdVIParabolic stream power (SPP)Whipple and Tucker (2002)––√√E∼HyaFecQsdE∼ωa(1-Qs/Qsc)cQsdVIISaltation–abrasion without suspension (SAws)Sklar and Dietrich (2006)√–√√E∼HyexaFecQsdE∼tsexa(1-Qs/Qsc)cQsdVIIISaltation–abrasion (SA)Sklar and Dietrich (2004)√√√√E∼HyexaSebFecQsdE∼tsexa(1-u*/wf)b(1-Qs/Qsc)cQsd
1 Based on the classification by Sklar and Dietrich (2006); model choice and description is given in Appendix A.2 Grain motion and suspension thresholds, cover and tools
effects.3 Based on Eq. (1).4 Based on Eq. (A1).5 Concept of this work.
For the following analysis, we study a rainfall-induced discharge event at
the Erlenbach stream, featuring a peak flow of 1.1 m3s-1 with
strong supercritical and turbulent flow conditions (the mean Froude number is
6.7 and the Reynolds number exceeds 105; Fig. 1, Table 1). The entire
investigation period (termed “event period” in Fig. 1) starts at the onset
of the stream's response to rainfall and ends some hours after the rainfall,
broadly defining a time frame around the actual bedload transport event. We
focus on the surface erosion measured on a dry-packed concrete slab (a test
“bedrock”) placed flush against the streambed of an artificially fixed
smooth chute channel with a slope of 16 %. This chute channel is attached
to a natural alluvial streambed section upstream that is shallower and is
characterized by a high macro-roughness in the form of step-pool units and
isolated boulders (Yager et al., 2012a, b). Thus, the broader setting
represents a convex knickpoint transition from an alluvial to a bedrock
streambed with a length of around 30 m, situated upstream from the
measurement installation. The concrete bedrock slab was overpassed by
8.8 t of bedload in nearly 11 h, as detected by the geophone sensor
located immediately downstream (Rickenmann et al., 2012; Beer et al., 2015).
The slab hosts three vertical at-a-point erosion sensors that continuously
record surface elevation. In addition, the slab was surveyed with close-range
photogrammetry (Rieke-Zapp et al., 2012) before and after the event to
confirm the measured cumulative erosion rates at the erosion sensor
positions.
Here, we restrict our analysis to erosion sensor c3, which was located in
the middle of the flow path and featured the highest erosion rate (cf. Beer
et al., 2015). The sensor recorded 13 erosion steps during the event, with
a total erosion of 0.85 mm. The temporal evolution of erosion before
the first recorded step is unknown. Hence, only subsequent data are used for
analysis (termed “erosion period” in Fig. 1). To account for the temporal
uncertainty of the occurrence of the individual erosion steps and to obtain
a transient curve, we use linearly interpolated data, hereafter referred to
as c3i. The course of this curve is a robust representation of the
erosional evolution of the sensor and its cumulative value is consistent with
the erosive pattern of the surrounding slab surface (cf. Beer et al., 2015).
The chute channel streambed approaching the erosion sensor c3 shows
a 8 m wide trapezoidal cross section of jointed rip-rap with low bed
roughness (the standard deviation of streambed elevations is 0.04 m).
In any case, using an at-a-point sensor, the spatial scale of the erosion
measurement is small. Hence, bed morphology or roughness does not have
a strong influence on the erosion signal and the flow properties are
homogeneous on the site of the sensor. These facts, together with its steeper
slope compared to the natural channel upstream, cause sediment-starved or
detachment-limited conditions in the chute channel. Overall, the situation at
the measurement site is close to the conditions that are typically assumed in
process-scale erosion models, and specifically to those for the derivation of
the saltation–abrasion model (Sklar and Dietrich, 2004).
Methods
Our purpose is to assess the ability of erosion models to predict observed
transient bedrock erosion c3i (linearly interpolated data from the erosion
sensor c3). Sklar and Dietrich (2006) classified the spectrum of existing
incision models according to their incorporation of the four types of
sediment effects (Eq. 1). We selected a representative model from each of
their classes (Table 2). These models are (I) unit stream power USP,
(II) excess unit stream power EUSP, (III) linear decline LD, (IV) alluvial
bedload AB, (V) tools T, (VI) parabolic stream power SPP, (VII) the
saltation–abrasion model neglecting the suspension effect SAws, and
(VIII) the saltation–abrasion model including the suspension effect SA. In
addition, we included a variant of T, a tools-only dependent model TO (Va),
in which erosion rate is proportional to cumulated bedload transport rate
(see Appendix A for model details). To evaluate the predictive quality of
these models against a standard, we further used the simplest possible model
as a null hypothesis that assumes a constant erosion rate during the entire
period of consideration: (H0) constant erosion. Note that a model based on
stream power can be converted into a model based on shear stress and vice
versa using simple assumptions of hydraulic geometry and flow velocity (cf.
Whipple and Tucker, 1999). Thus, models USP and EUSP can be seen as
representative of other members of the stream power model family.
Exponents of the four sediment effects (cf. Eq. 1) for the common
(com) and optimized (opt) model versions, respectively, given for the three
simulation time periods under consideration and the resulting model
performance relative to c3i. Implausible parameters are indicated in bold
and model groups with and without the tools effect are separated by a
horizontal line.
a Model denotations are given in Table 2.b Respective parameters for commonly used (_com) and optimized (_opt) values applied for the four sediment effects.c This exponent is used for entire stream power neglecting the
grain motion threshold effect.
Recently published models dealing with the interplay between bed roughness
and sediment cover regarding erosion rates (Johnson, 2014; Inoue et al.,
2014; Zhang et al., 2015) were not considered here. This is because the
channel at the experimental site features a steep and smooth bed downstream
from a rougher and shallower bed. This resulted in low relative sediment
supply (maximum 0.56) and bedload concentrations with high transport stages
(given the low mean grain size d50), and therefore high fractions of
exposure (Table 1). Consequently, sediment–roughness interactions and bed
cover are unlikely. We furthermore did not explicitly apply the elaborate
total-load model by Lamb et al. (2008a), due to its need for high shear
stress ratios combined with high relative sediment supply to deviate from the
SAws model (Lamb et al., 2008a; Scheingross et al., 2014), which is not the
case at the Erlenbach stream. Models focusing on plucking as the dominant
erosion process (Chatanantavet and Parker, 2009; Dubinski and Wohl, 2013)
were also not considered since abrasion can be assumed to be the dominant
process in our experimental setting (cf. Beer et al., 2015). For further
details on model choice and parametrization, see Appendix A.
The threshold of bedload motion on site was calculated based on Rickenmann
et al.'s equation for the critical discharge in steep torrent channels (see
Eq. 18 in Rickenmann et al., 2006), and resulted in a very low value of
0.008 m3s-1. This threshold was far exceeded during the
entire study period (cf. Fig. 1), and corresponds to a critical Shields
stress of 0.006, i.e., a nondimensionalized version of the bed shear stress
used for the initiation of particle motion (cf. Shields, 1936; Table 1). The
calculated value is below the commonly applied lower limit of 0.03 (e.g.,
Buffington and Montgomery, 1997; Sklar and Dietrich, 2004) and an order of
magnitude lower than that calculated with an empirical equation for steep
alluvial streams (0.09; Lamb et al., 2008b, Fig. 1 therein). However, the
standard deviation of bed elevations in the vicinity of the erosion sensor
c3 is 0.04 m (Table 1), implying a smooth bedrock surface, for
which a critical Shields stress of 0.006 is plausible (cf. Hodge et al.,
2011; Chatanantavet et al., 2013; Auel, 2014). This value is also consistent
with no visual observations of sediment deposits (neither sand nor cobbles)
in the chute channel section and on top of the geophones over many years.
Thus, the prevailing transport stage, defined as the ratio of acting
nondimensional bed shear stress over critical Shields stress, was generally
high.
As is apparent from Fig. 1, actual bedload transport started after the
exceedance of the calculated threshold of motion, at least for the particle
sizes that are detectable with the geophone sensors (> 0.01 m; cf.
Rickenmann et al., 2012). This is due to the fact that the detachment of the
bedload passing on site actually occurred in the upstream natural stream
section under completely different hydraulic conditions (cf. Turowski et al.,
2011, 2013; Yager et al., 2012a, b). There, the relative importance of the
four sediment effects presumably is different from the measurement site, and
the threshold of bedload motion is higher (cf. Schneider et al., 2014). Since
we focus on the chute channel, it would not be plausible to use a threshold
of motion from the alluvial section. However, to assess its downstream
influence on bedload transport and for the assessment of the motion effect,
we defined an additional virtual threshold of bedload motion (VTBM) at the
observed exceedance of a bedload transport rate of 1 kgmin-1
(which is the beginning of the bedload period; cf. Fig. 1), corresponding to
a critical discharge of 0.36 ms-1, a critical unit stream power
of 407 Wm-2 and a critical Shields stress of 0.26.
We calculated erosion rates with each model for the flood event under
consideration using independently observed hydraulic parameters and sediment
transport rates. The relationships between discharge, flow height, and stream
width that are required for the calculation of hydraulic parameters such as
local unit stream power at the position of the observed bedrock slab were
determined using the methods described by Beer et al. (2015). The mean grain
size of the transported sediment during the event considered here was
estimated at 0.02 m (using data by Rickenmann et al., 2012, Fig. 9
therein) at a mean discharge of 0.77 m3s-1 and a mean bedload
transport rate of 0.45 kgm-1s-1 for the period of observed
bedload transport. During the event, bedrock abrasion apparently was the
dominant erosional process, since neither direct observations nor surveying
results gave any indication of solution, plucking or cavitation (Beer et al.,
2015).
Evolution of the modeled erosion signals over the flood event
compared to discharge, bedload mass transport and interpolated erosion rate
(see Table 2 and Appendix A for model descriptions) given for the event period. Positions of all three
simulation time periods are shown on top like in Fig. 1. (a) Scaled
predictions of constant erosion and models neglecting the tools effect
(models USP to AB), (b) scaled predictions for models incorporating
the tools effect (models TO to SA) and (c) the transient evolution
of the four sediment effects (factors in Eq. 1; data resolution is in
minutes). Note that the threshold of motion term and the tools term are
binary, but the suspension term and the fraction of exposure term can
continuously vary between 0 and 1; see text for further explanations.
The interpolated erosion line c3i and the individual model outcomes were
scaled to unity to focus on transient behavior, ignoring the prefactors K
with their multivariate sensitivities to lithology, climate, and sediment
(Whipple and Tucker, 1999). Since we only have reliable erosion data some
time after the onset of bedload transport (cf. Fig. 1), we calculated model
performance sensitivity on bedload transport using three separate simulation
time periods. For the “event period”, the start
and end of the simulation were set at arbitrary times broadly including the
studied bedload transport event. For the “bedload period”, the start and
end of the simulation period coincide with the observed bedload transport
period as measured with the geophone sensor, while the “erosion period”
only covers the time span where c3i data exist.
For the evaluation of the transient individual model performance, overall
deviations between model predictions and c3i were quantified by means of
the root mean square error (RMSE; cf. van der Beek and Bishop, 2003; Valla
et al., 2010) of the cumulative values, and minute-by-minute differences were
considered in order to assess model feasibility and highlight dominant
processes. The proportion of explained variance PEV (a measure commonly known
as the Nash–Sutcliffe model efficiency coefficient in hydrology; e.g.,
Gyalistras, 2003) was used as a second measure to evaluate model predictive
quality, utilizing instantaneous rather than cumulative erosion values. In
addition, we optimized individual model performance by adjusting the
exponents a, b, c, and d (see Eq. 1), as long as they differed from
0, to minimize the RMSE based on the methods of Brent (1973) and Nelder and
Mead (1965). We are aware of the fact that the exponents of the
saltation–abrasion model SA (Sklar and Dietrich, 2004) and some of the
exponents in other models come from physics-based analyses. Similarly,
bedrock erosion can be expected to be linearly dependent on sediment supply
due to the tools effect (cf. Sklar and Dietrich, 2001, 2004; Johnson and
Whipple, 2010; Whipple et al., 2013; Auel, 2014; Jacobs and Hagmann, 2015).
However, we decided to consider all models as implementations of a generic
model equation (Eq. 1 or the more detailed Eq. A1; see Appendix A) and
therefore analyzed their performance with optimized exponents despite
potential mechanistic parametrizations.
Results
All incision models performed at least slightly better than the null
hypothesis of constant erosion (Fig. 2, Table 3). The tested models can be
roughly separated into two groups based on their transient behavior,
corresponding to those models that do and those that do not include the tools
effect Qs (Fig. 2a and b). Here, we focus on the event period
to describe the main observations. A detailed comparison of each model in
each simulation time period is given in Appendix B (Fig. B1).
The models of the first group (models USP to AB; Fig. 2a; Table 2) show
a smooth increase in cumulative erosion over the course of the event, while
those of the second group (models TO to SA; Fig. 2b) exhibit a wavy
pattern. With respect to the four sediment effects, we observed the
following.
Threshold of motion: due to the small value of this threshold,
it was exceeded during the whole study period and there is no effect visible
(e.g., no difference between USP and EUSP). All models in Fig. 2a predicted
erosion even when none was detected (cf. Sklar and Dietrich, 2006), which is
particularly obvious at the end of the observation period where c3i data
are available. However, applying the higher virtual threshold VTBM at the
actual onset of bedload transport (i.e., restriction of the scaled model
evolution to the limits of the bedload period with Hy=0 outside the
vertical lines in Fig. 2c) led to a smoother fit of models EUSP and AB to
c3i (not shown here). Hence, inclusion of a threshold of motion is of
distinct importance, especially if the tools effect is ignored.
Threshold of suspension: the status of complete suspension transport
(this corresponds to Se=0) was not reached during this event
for grain sizes equal to or greater than 0.02 m (cf. Fig. 2c). The
mean of the suspension term Se (for details on the calculation,
see Appendix A) was 0.64±0.18 throughout the event, with a minimum of
0.34 (Table 1), and thus substantial pebble saltation was predicted. The use
of model SA that includes the suspension effect term showed a larger
deviation from the data than the otherwise equivalent model SAws (Fig. 2b).
Cover effect: the fraction of exposure Fe average
was 0.95±0.07 with a minimum value of 0.44. Hence, there was no time when
erosion was completely prohibited (Fig. 2c) and consequently there was no
remarkable improvement in modeling performance when including the cover term,
e.g., when comparing models USP and LD (Fig. 2a).
Tools effect: the wavy pattern observed in the erosion record c3i,
as well as in the models that include the tools effect (TO to SA; Fig. 2b),
closely follows the evolution of cumulative bedload transport over the course
of the event with model SPP showing the largest deviation. Actual bedload
transport Qs initiated at VTBM here, but it receded before
falling below this threshold again at the end of the event period. This
recession is accompanied by the cessation of the continued increase in c3i.
Models including the tools effect (TO to SA, Fig. 2b) show smaller RMSE than
those that do not for all three simulation time periods, except model SPP
using the standard parameter set and applied during the erosion period
(Table 3). All model predictions except for model TO improved when optimized,
with the highest improvements for the erosion period (in this period, also
TO). However, for model SA, the optimized exponents of both threshold factors
(motion and suspension) show implausible values. For all other models,
exponents only adjusted moderately during optimization.
Analysis of the minute-by-minute differences of each model prediction from
c3i (Fig. 3a) revealed the same pattern of a noticeable improvement for
models that include the tools effect (cf. RMSE values from Table 3 in
Fig. 3b), since models neglecting it (USP–AB) did not perform better than
the null hypothesis of constant erosion. Remarkably, this pattern is also
visible in the PEV values (Fig. 3c), which are based on the instantaneous
erosion values that show greater variability. A PEV value < 0 means poor
model performance, since PEV = 0 modeling is comparable to the assumption
of constant erosion and > 0 indicates a reduced error variance compared to
variance in the original values of c3i (cf. Gyalistras, 2003).
Comparison of individual model performance for the event period:
(a) prediction differences from the course of bedrock erosion c3i,
(b) root mean square errors RMSE and (c) proportion of
explained variances PEV (calculated for the instantaneous erosion values).
For model denotations, see Table 2.
Generally, for both the event period and the bedload period we obtained
similar results, while for the erosion period models showed comparably worse
predictions (cf. Fig. B2). Models neglecting the tools effect underpredicted
observed erosion by 7 and 4 % (median) for the first two simulation time
periods (both for the common and optimized parameter sets), whereas models
including the tools effect showed medians of differences of nearly 0 %. For the
erosion period the median values were partially better, but the interquartile
ranges were larger by far. The interquartile ranges of the models SAws and SA
showed the smallest values when using the optimized parameter in the event
period, but their overall performance is comparable to the TO model. Model
SPP, which neglects the threshold of motion, had the worst performance of all
approaches that consider bedload tools, when standard parameters were used.
Model parameter optimization achieved the most improvements for the erosion
period, where the performance of models that include the tools effect could
be improved to a quality comparable to that achieved in the two other
simulation time periods using standard parameter sets (Fig. B2, Table 3).
DiscussionModel sensitivity to simulation time period
Incision model behavior was comparable for the event period and for the
bedload. For the erosion period, the performance of all models was notably
worse than in the other periods as visible from the largely increased
interquartile ranges of the model differences per minute (Table 3 and Fig. B2).
However, the worsened performance is an artifact of scaling all
erosion series to 1 for the model evaluation. In contrast to both of the
other periods, in which cumulated erosion rate c3i actually started at the
first certain erosion step (i.e., >0) within the time period under
consideration, analysis for the erosion period began with c3i set to 0 to
ensure a common initiation of all variables. Thus, any erosion that occurred
before the beginning of the erosion period was disregarded. With R2=0.96, the strength of the correlation between c3i and the cumulative
bedload during the erosion period is slightly smaller than for the event
period and the bedload period (R2=0.98 for both), and this smaller
correlation translates directly to the predictive power of the tools effect
for measured erosion. The decreased correlation strength may have various
causes. (i) If the bedload path in the channel bed systematically changes as
discharge increases, the correlation between bedload transport rates and
erosion rates may decrease, since small discharges were omitted in the
erosion period. (ii) Bedload transport rates are measured over the entire
slab surface (0.18 m2), but the erosion sensor records at-a-point.
(iii) The erosion sensor does not measure continuously, but in steps.
Therefore, temporal variability in pebble impacts can cause mismatches
between bedload transport rates and erosion rates. (iv) Due to the shorter
period of interest, and to the scaling of the total erosion rate to 1, the
sharp increase in c3i around 05:40 (Fig. 1) resulted in higher relative
deviations of the incision models.
Nevertheless, the pattern of model improvement by including the tools effect
was the same for all simulation time periods. In addition, the RMSE values
are reasonably similar for all three time periods, both for the common and
the optimized model versions. Hence, at least for the timescales investigated
here, there is no significant temporal sensitivity to model application
regarding actual bedload transport.
Relevance of the four sediment effects
In the following, we evaluate the predictive power of the four sediment effects in the same order as given in the section on results.
The inclusion of the negligible threshold of bedload motion (cf. Table 1)
did not have any effect on model performance. However, application of the
virtual motion threshold VTBM led to a smoother match between the models EUSP
and AB with the observations compared to models USP and LD. Even though the
details in the transient pattern of erosion could not be reproduced with
these models, the inclusion of the VTBM threshold enabled the prediction of
the general observed trend, at least for this event. Thus, as has been
previously suggested (cf. Lague et al., 2003; Chatanantavet et al., 2013;
Lague, 2014), the inclusion of a threshold of motion is necessary to obtain
a plausible temporal pattern of erosion. Therefore, if no direct information
on bedload transport is available, the threshold of motion in Hy might be
the most relevant parameter for erosion modeling (cf. Sklar and Dietrich,
2006; Attal et al., 2011). Furthermore, as has already been argued in
sediment transport studies (e.g., Rickenmann and Koschni, 2010), while the
local threshold of motion might not be relevant on site, the threshold of the
channel section upstream that is actually supplying the sediment may be. This
threshold can be used as a virtual discharge threshold to determine the
timing and the amount of bedload transport at the site of interest.
Within the data set analyzed here, the inclusion of the suspension effect
term decreased the predictive quality of the SA model compared to the
otherwise equivalent model SAws that does not include it. Increasing the
suspension threshold in the suspension effect term Se (cf.
Eq. A1) was not reasonable due to the already high values of the fraction of
exposure Fe. In contrast, threshold values below 0.65 (i.e., the
maximum of the squared term in the Se term) prohibited erosion
predictions at high shear velocities when actual erosion in c3i occurred.
Together with high Rouse numbers (cf. Table 1), this indicates that the
bedload transport mode for the mean grain size d50 was likely dominant
during this event, even with the high transport stages prevailing
(75±38). The scaling here is different: while bed shear stress scales
linearly with flow depth, shear velocity scales with the square of flow depth
and discharge velocity was nearly constant during the event period at
4–5.6 ms-1. The suspension term proposed by Sklar and Dietrich
(2004) could not be evaluated here in more detail, but its lack of
explanatory power is consistent with the assumption of Lamb et al. (2008a),
who extended the SA model to account for erosion by suspended load due to
turbulence-driven impacts. However, as discussed above, the total erosion
model of Lamb et al. (2008a) would not substantially deviate from the SA
model here, since relative bedload supply was very low (cf. Table 1).
Explicit consideration of the cover effect is recommended in the
literature (e.g., Lamb et al., 2008a; Nelson and Seminara, 2011; Whipple
et al., 2013). However, the influence of bed cover (in the form of the
Fe term) appears to be insignificant here. Given the site
characteristics, the absence of the cover effect is plausible. Due to the
increased transport capacity of the chute channel compared to the natural
streambed upstream, relative sediment supply was low and the mean transported
sediment size d50 was lower than the situation on site would allow.
Hence, detachment-limited conditions prevailed (cf. Turowski et al., 2013;
Beer et al., 2015) and sediment deposition did not occur. This means that
static cover did not occur and that dynamic cover (cf. Turowski et al., 2007)
was unlikely.
Erosion rate c3i smoothly followed the accumulated bedload
transport during the event period (cf. Fig. 2b). This indicates that erosion
is driven by particle impacts and that the dominant sediment effect was the
tools effect. Therefore, a simple empirical model in which the erosion rate
is proportional to bedload transport rate Qs (the TO model)
explains the data similarly to and as well as other models that incorporate
the tools effect (Fig. 3, Table 3), including highly developed mechanistic
process models such as the full saltation–abrasion model (SA). The
importance of the tools effect, and its linear dependency on bedload volume,
is in line with previous field and laboratory observations (Sklar and
Dietrich, 2001; Turowski and Rickenmann, 2009; Cook et al., 2013; Wilson
et al., 2013; Auel, 2014; Jacobs and Hagmann, 2015), and our data provide the
first direct field evidence for the tools effect at the process scale at high
temporal resolution (cf. Whipple et al., 2013; Beer et al., 2015).
Optimized model parameters
For some models, the optimization procedure resulted in substantial
improvements in their predictive power. Because of its considerable practical
importance, we discuss the behavior of the stream power incision model family
(USP) in detail, and make some general remarks on other models, especially
those where we found large predictive differences between the common and
optimized parameters.
For the USP model, the optimized exponent a on unit stream power ω
was 1.5 for the event period and 1.1 for the bedload period (Table 3).
However, the choice of this exponent did not significantly affect the
predictive power of the models (Fig. 4), at least when it remained within the
range of values reported in the literature (between 0 and 2; see Lague, 2014,
for a review). For the modeling with the higher threshold of motion VTBM, the
optimized parameter a of the EUSP model was 0.6, close to the common value
of 0.5. Both of these numbers support the common usage. However, our
observations at the process scale are not directly comparable to previously
published values, which are typically derived from measurements at the reach
or catchment scale. A proper upscaling and a comparison with reach-scale
measurements would be necessary to enable a complete interpretation of these
results. For the USP model, we obtained an optimized value of a close to 0
for the erosion period, but with no noticeable improvement over the common
parameter of 0.5 (Table 3). To summarize, at least for the specific case of
the chute channel and the studied event at the Erlenbach stream, the
inclusion of a motion threshold in the USP equation (the EUSP model) makes
the common parameter value of 0.5 acceptable for the modeling of the general
trend in the transient evolution of bedrock erosion.
Performance of the USP model expressed as RMSE deviation from
measured erosion c3i for the common range of the unit stream power exponent
a (increments of 0.1) for the three time periods. The dotted vertical line
indicates the commonly used exponent of 0.5; the diamonds show optimized
parameters and the dashed line is the RMSE value of constant erosion (model
HO) for the event period.
Model optimization for all three simulation time periods led to negative
values for the exponents of the fraction of exposure Fe for
models LD and AB. This resulted in a comparably good performance for the two
models, since the first “hump” in the c3i curve could be predicted (cf.
Fig. 2 at around 05:30). However, the cover effect was negligible in the
setting at hand and a negative exponent value for Fe contradicts
the physical assumptions of the cover effect (see Appendix A) since it
increased the value of this factor and, therefore, resembled the tools
effect. The observation thus underlines the importance of erosive tools.
Parameter optimization for models TO to SA in part led to a strong nonlinear
dependency of the erosion rate on the tools effect (cf. Table 3), together
with negative exponent values a for Hy and partly increased exponent
values c for Fe. As mentioned above, this can be related to an
adjustment of the model curve to predict the first “hump” in the c3i
curve (cf. Fig. B1). This short period of higher erosion rates can be traced
back to two subsequent erosion steps measured with the erosion sensor c3
(cf. Fig. 1; Beer et al., 2015, Fig. 5 therein). Since overall bedload
transport at this time was rather below average and comparable bedload
transport rates otherwise caused lower spatial surface erosion, the steps may
likely be related to a high-energy strike of a single bedload grain at the
precise position of the at-a-point erosion sensor. Hence, the nonlinearity
here was rather a confirmation than a falsification of the process physics,
i.e., that erosion is proportional to the sediment impact energy (cf. Sklar
and Dietrich, 2004). Furthermore, the tendency to optimize the parameter d
towards values exceeding 1 implies the absence of a dynamic cover effect, for
which d<1 would be expected, at least in the TO model. Finally, the
implausibly high values of the optimized exponents for model SA in comparison
to the otherwise equivalent model SAws may be related to the fact that the
definition of the suspension term might be incorrect (see discussion above).
The common exponents of the reduced model SAws are physics based, and
therefore optimization was not very effective. This might indicate that the
basic physics of bedrock erosion is fairly well captured by the
saltation–abrasion model. Similarly, Turowski et al. (2013, 2015) found that
the hydraulic forcing of the energy delivery to the bed, which is thought to
be proportional to the erosion rate, is well captured by the SA model.
Overall, optimizing of models that include the tools effect mainly resulted
in reductions of the interquartile ranges of model deviations. Values for the
event period and the bedload period were equally adjusted. Notable
improvement, leading to a similar performance in comparison to the other
models, was achieved in the SPP model, where the tools effect compensated for
the missing threshold of bedload motion. However, no model could clearly beat
the performance of the simple TO model (cf. Fig. 3), which is the most
effective indicator that the tools effect is the dominant driver of bedrock
erosion in our setup.
Generality of the results
Our results thus far are only based on a single erosive event with regard to
a mean sediment grain size. For analysis, the start and end of the event were
more or less fixed and the erosion rate was scaled to 100 %. Thus, model
performance was generally good. However, differences in individual
performances were visible and could be used to evaluate governing effects.
The specific situation of a steep smooth bedrock section downstream from an
alluvial channel resulting in detachment-limited conditions makes the
Erlenbach stream site ideally suited for the study of the tools effect, which
is the dominant erosion effect here. We confirmed the linear dependence of
bedrock abrasion on bedload flux (as expected by Whipple et al., 2013) using
independent transient field data at a temporal resolution in minutes. This
supports the assumption that erosion is driven by particle impacts (e.g.,
Sklar and Dietrich, 2001, 2004; Turowski et al., 2013). Deviations from
linear scaling are plausible over the given sediment size distribution, with
larger grains showing higher impact energy efficiencies and smaller grains
showing lower ones, which can be related to differing transport modes
(Turowski et al., 2015). However, focusing on the mean grain size d50
seemed to average out these relationships, and the simple TO model was shown
to be sufficient to predict bedrock erosion under the given conditions.
For comparable situations where mean values of bedload volumes and erosion
rate on bare bedrock sections are available, this model can be easily
calibrated by adjusting its prefactor K. Potential applications are steep
bedrock channels in detachment-starved catchments (e.g., Wohl, 1998, 1999),
channel knickpoint sections with exposed bedrock such as waterfalls (e.g.,
Miller, 1991; Wohl et al., 1994; Cook et al., 2013; Mackey et al., 2014;
DiBiase et al., 2015), high lateral bedrock sections above the channel or its
banks (e.g., Hartshorn et al., 2002; Turowski et al., 2008) or even
hydropower facilities that have to cope with natural sediment flux such as
sediment bypass tunnels (Jacobs and Hagmann, 2015).
Models of the commonly applied stream power model family USP were still found
to be feasible, which is important due to the rarity of measured bedload
transport rates. The excess unit stream power incision model EUSP using the
common exponent of 0.5 was shown to be adequate to reproduce the
tools-dominated incision, if the details of erosional evolution within the
events are not of interest. However, to define the effective threshold of
motion and the characteristic grain size, it is crucial to recognize the
streambed situation upstream, which restricts the timing and magnitude of
erosion on site.
Inherent channel morphology (e.g., Wohl, 1998; Johnson and Whipple, 2010)
crucially steers the dominance of the tools effect for whole-stream evolution
and linked landscape evolution in terms of bed roughness (Inoue et al., 2014;
Johnson, 2014), relative sediment supply (Turowski et al, 2008; Lague, 2010),
transport mode (Lamb et al., 2008a; Scheingross et al., 2014; Turowski
et al., 2015), and lithology (Sklar and Dietrich, 2001; Whipple et al.,
2013). Thus, the cover effect gains importance (Turowski et al., 2007) and a
more complex model such as SAws should be suitable. For the USP model family,
at least some aspects of temporal upscaling are understood (e.g., Lague
et al., 2005). However, our understanding of spatial upscaling in general,
and of temporal upscaling of sediment-flux-dependent incision models
specifically, is unclear to date (cf. Whipple and Tucker, 1999; Lague, 2010,
2014).
The model-specific prefactors K (cf. Eq. 1) could be calibrated by scaling
the absolute cumulative model predictions with the observed cumulative
erosion rate c3i. These factors are partly empirical (e.g., for the USP
model) or are based on known material properties (e.g., in the SA model; for
an overview, see Sklar and Dietrich, 2006). Analysis of these values,
however, is not the focus here, and would need to take more events into
consideration for a robust calibration on site.
Conclusions
Fluvial bedrock erosion is driven by the impacts of sediment particles. Out
of several sediment effects, the tools effect dominantly determines erosion
rates at the Erlenbach stream erosion observatory, which exemplifies a steep
bedrock channel downstream of an alluvial streambed. The pattern of transient
erosion during the course of a single flood event can be described by
a simple model in which erosion rate is proportional to bedload transport.
Moreover, this simple model performs similarly well or better than more
complex models from the literature, including the mechanistically based
saltation–abrasion model, and several models from the stream-power incision
model family. The model can be site calibrated with temporally lumped data,
and is applicable in, for example, detachment-limited steep and smooth
bedrock rivers, in bedrock knickpoint reaches as well as in bedload-exposed
hydropower infrastructure.
On the scale of the individual event, models from the stream-power incision
model family can adequately describe the generally observed erosion trend. In
our tests, the application of an excess shear stress model with an exponent
of 0.5 does not capture the detailed evolution of erosion throughout the
event, but is adequate to represent the overall form of the erosion curve, if
it is parameteterized with an adjusted threshold of bedload motion. Analysis
of more events is needed to verify whether this result can be generalized, or
whether it is specific to the event and field site considered here.
Additional data acquisition and analysis of transient erosion rates in other
settings are required (Tucker and Whipple, 2002) to, for example, study
interactions between different erosion processes that are not considered in
modeling to date (Whipple et al., 2013), to examine the model-specific
prefactors K, and to potentially provide guidance for site-specific model
choice based on locally active morphological processes.
Model selection
According to Sklar and Dietrich (2006), all fluvial bedrock incision models
published to date can be represented by a generic equation (a detailed
version of Eq. 1):
E=KHy-Hyca1-u*wf2b1-QsQsccQsd.
Here, Hy represents transport stage ts (the fraction of nondimensional
bed shear stress and nondimensional critical Shields stress) or unit stream
power ω. Hyc is a potentially associated threshold term
accounting for grain motion, which results in excess transport stage
tsex and excess unit stream power ωex (given
as Hyex in Table 2). Qsc is sediment transport
capacity, u* is flow shear velocity and wf is particle fall
velocity in still water for the mean grain size d50. The terms in
brackets from the left to the right represent the bedload motion effect, the
bedload suspension effect, and the fraction of bedrock exposure (referring to
the cover effect) and the tools effect, respectively (cf. Eq. 1).
There are 24=16 combinations of the four sediment effects controlled by
the exponents a, b, c and d in an incision model, that in turn can be
adjusted for specific dominant erosion processes such as abrasion, plucking
and macroabrasion (Whipple et al., 2000; Sklar and Dietrich, 2004; Lamb
et al., 2008a; Chatanantavet and Parker, 2009; Dubinski and Wohl, 2013). We
restricted our analysis to the eight model classes (i.e., combinations of
sediment effect parameters) identified by Sklar and Dietrich (2006) that were
proposed, analyzed and applied in several studies to date, but added a null
hypothesis model (class H0) and a simple bedload-dependent model (class Va).
We analyzed one representative of each bedrock incision model class whose
selection (Table 2) and parameterization (Table 3) were based on the
following reasons (cf. Sklar and Dietrich, 2006).
Class HO, constant erosion: this model served as a null hypothesis (H0)
and simply assumes a constant erosion rate over the time period considered.
The fixed instantaneous erosion rate equaled the cumulated erosion rate at
the end of the period (i.e., 1, since it was scaled) divided by the length of
the period.
Class I, unit stream power model (USP): this model (Seidl and Dietrich, 1992;
Howard, 1994; Howard et al., 1994) is most widely used in landscape evolution
modeling studies (Lague, 2014) and for the interpretation of channel long
profiles (e.g., Braun and Willett, 2013), although there is evidence
contradicting its predictions (Gasparini et al., 2007; Lague, 2014). It is
straightforward since it only incorporates discharge data, neglects any
sediment effects, and assumes Hyc=0. The single exponent a to
scale unit stream power ω (which is proportional to discharge) is
mainly set to 0.5 in modeling studies as done in the present study, but may
vary between 0 and 2 for field data (Croissant and Braun, 2014; Lague, 2014),
and most field cases suggest a=1 (e.g., Stock and Montgomery, 1999;
Snyder et al., 2000; see Lague, 2014, for a review). The USP model is
equivalent to the shear stress model (Howard and Kerby, 1983; Turowski, 2012)
and their model exponents are related by a factor of 2/3 (Whipple and
Tucker, 1999).
Class II, excess unit stream power model (EUSP): an extended version of
the USP model with non-zero Hyc (Sklar and Dietrich, 2006), thus
incorporating a threshold of unit stream power to permit grain motion. Here,
we applied the excess unit stream power ωex with the
calculated unit stream power value at the onset of bedload motion (cf.
Table 1) using the same model exponent a=0.5 as in the USP model (e.g.,
Tucker and Slingerland, 1997; Whipple et al., 2000).
Class III, linear decline model (LD): this model set was formulated by
Whipple and Tucker (2002) and is functionally equivalent to the undercapacity
model by Beaumont et al. (1992). However, the latter does not draw on the
cover effect. Instead, it draws on the consumption of discharge energy for
sediment transport that would otherwise be used for erosion. Erosion rate is
limited by the fraction of actual bedload Qs to bedload transport
capacity Qsc, i.e., the cover effect. If this fraction approaches
1, erosion decreases to 0 (Sklar and Dietrich, 2004). The exponential
dependency of the cover term proposed by Turowski et al. (2007) was not
applied due to the prevailing tools domain. We applied the bedload transport
equation of Rickenmann (2001, Eq. 3 therein) to calculate Qsc
with a prefactor calibrated for the Erlenbach stream, using a grain size
fraction d90/d30 based on data by Rickenmann et al. (2012, Fig. 9
therein; cf. Table 1). We restricted our analysis to the model version of
Whipple and Tucker (2002) with an exponent a=2.
Class IV, alluvial bedload (AB): Sklar and Dietrich (2006) proposed
this version of the linear decline model LD, based on excess transport stage
tsex and accounting for the threshold of motion Hyc=1.
Class Va, tools-only model (TO): this model simply relates the erosion rate to
the observed cumulative bedload transport rate. We introduce it here with the
tools exponent d=1 (based on, e.g., Sklar and Dietrich, 2001; Jacobs and
Hagmann, 2015) and rank it into the classification of Sklar and Dietrich
(2006), based on the top-down introduction system of the sediment effects
there.
Class V, tools (T): the model of Foley (1980) was applied following
the approximation given by Sklar and Dietrich (2006) using a=-0.5.
Class VI, parabolic stream power (SPP): in their attempt to include
both the tools and the cover effect, Whipple and Tucker (2002) developed this
model based upon considerations of Sklar and Dietrich (1998) using unit
stream power ω. We chose the proposed version with a=1.
Class VII, saltation–abrasion model without the suspension effect (SAws):
the same model as SPP, but using excess transport stage tsex
instead of unit stream power ω and additionally incorporating the
sediment motion threshold Hyc.
Class VIII, full saltation–abrasion model (SA): the complete
saltation–abrasion model (Sklar and Dietrich, 2004) additionally accounts
for the grain suspension effect. We adopted the threshold of ceasing erosion
(1 in the suspension term; Eq. A1) from Sklar and Dietrich (2004); however,
this value is controversial (cf. the review by Cheng and Chiew, 1999), and
indeed the whole conception has been questioned (Lamb et al., 2008a; Scheingross et al., 2014). The parameter b responsible for the
suspension term was set to 1.5 here, since this is consistent with the
original model (cf. Sklar and Dietrich, 2004).
Detailed model results
Separate comparison of each of the model-based erosion predictions
to c3i for (a) the event period, (b) the bedload period
and (c) the erosion period using the individual common and optimized
parameter sets, respectively.
Separate comparison of the differences between model predictions and
c3i given as boxplots (without outliers). Each model performance is shown
for the three different periods of consideration (the same colors as the time
periods indicated in Figs. 1 and 2) with runs using both common and optimized
parameter sets (wide and narrow boxes) for each model.
The individual parameter sets of all incision models (USP to SA) were
optimized for the three simulation time periods, respectively (Table 3). In
Fig. B1, a separate comparison of transient model behavior is shown for the
particular model predictions compared to the observed erosion course c3i.
For further visual assessment of the individual model performance, the
transient model differences from c3i for all three periods are provided as
boxplots without whiskers in Fig. B2.
Acknowledgements
The authors are grateful to Florian Heimann, James Kirchner, Joel
Scheingross, Colin Stark and Carlos Wyss for fruitful discussions and help
with data analysis. Comments by Alexandre Badoux and Johannes Schneider on an
earlier version greatly improved the text. The authors are further thankful
to an anonymous referee, Phairot Chatanantavet and Leonard Sklar for their
thorough reviews, helpful comments and hints. We thank Curtis Gautschi for
valuable suggestions that helped to improve the language of an earlier
version of the manuscript. This study was supported by SNF grant
200021_132163/1. Edited by: E. Lajeunesse
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