# An Attempt to Take Into Account Natural Variability in 1D Bedload Prediction

A. Recking, A. Johannot, K. Horita, M. Nasr, D. Vázquez-Tarrío, S. Zanker,

F. Fontaine, G. Melun

► To cite this version:

A. Recking, A. Johannot, K. Horita, M. Nasr, D. Vázquez-Tarrío, et al.. An Attempt to Take Into Account Natural Variability in 1D Bedload Prediction. Journal of Geophysical Research: Earth Surface, 2024, 129 (7), pp.e2023JF007601. 10.1029/2023JF007601. hal-04801666

HAL Id: hal-04801666

https://hal.science/hal-04801666v1

Submitted on 26 Nov 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>
# JGR Earth Surface

## RESEARCH ARTICLE

10.1029/2023JF007601

**Key Points:**

* A new field data set is used for computing bedload in a Monte Carlo approach

* Bedload modeling accounting for input variability reproduces bedload fluctuation well

* Because of non-linearity, volumes computed with input variability are 20% higher

**Supporting Information:**

Supporting Information may be found in the online version of this article.

**Correspondence to:**

A. Recking,
alain.recking@inrae.fr

**Citation:**

Recking, A., Johannot, A., Horita, K., Nasr, M., Vázquez-Tarrío, D., Zanker, S., et al. (2024). An attempt to take into account natural variability in 1D bedload prediction. *Journal of Geophysical Research: Earth Surface, 129*, e2023JF007601. https://doi.org/10.1029/2023JF007601

Received 16 DEC 2023

Accepted 25 JUN 2024

**Author Contributions:**

**Conceptualization:** A. Recking

**Data curation:** A. Johannot, K. Horita, M. Nasr, D. Vázquez-Tarrío, F. Fontaine

**Formal analysis:** A. Recking, A. Johannot, K. Horita, M. Nasr, D. Vázquez-Tarrío, S. Zanker

**Funding acquisition:** A. Recking, G. Melun

**Investigation:** A. Recking, A. Johannot, K. Horita, M. Nasr, S. Zanker, F. Fontaine

**Methodology:** A. Recking

**Project administration:** A. Recking

**Resources:** A. Johannot, M. Nasr, S. Zanker, F. Fontaine

**Software:** A. Recking, G. Melun

**Supervision:** A. Recking

**Validation:** A. Recking, D. Vázquez-Tarrío

© 2024. The Author(s).
This is an open access article under the
terms of the Creative Commons
Attribution License, which permits use,
distribution and reproduction in any
medium, provided the original work is
properly cited.

<figure><img src="image_2.png" /></figure>

# An Attempt to Take Into Account Natural Variability in 1D Bedload Prediction

A. Recking¹ ID, A. Johannot¹ ID, K. Horita² ID, M. Nasr¹,³ ID, D. Vázquez-Tarrío⁴ ID, S. Zanker⁵, F. Fontaine¹, and G. Melun⁶ ID

¹University Grenoble Alpes, INRAE, CNRS, IRD, Grenoble INP, IGE, Grenoble, France, ²BGC Engineering Inc., Vancouver, BC, Canada, ³GINGER BURGEAP, R&D, Grenoble, France, ⁴Department of Geological Risks and Climate Change, IGME, CSIC, Madrid, Spain, ⁵EDF Hydro, Division Technique Générale (DTG), Saint-Martin-le-Vinoux, France, ⁶Office Français de la Biodiversité, Vincennes, France

**Abstract** Bedload transport can fluctuate considerably over relatively short periods of time and for a given quasi-constant flow rate. What are the implications of replacing the fluctuating signal with a smoothed signal when calculating bedload transport using averaged values, as is common practice? This question was investigated with the BedloadR code, which allows 1D bedload calculation as well as Monte Carlo simulations using a new data set collected in the Severaisse River (French Ecrins massif). Four bedload equations (Camenen & Larson, 2005, https://doi.org/10.1016/j.ecss.2004.10.019; Meyer-Peter & Mueller, 1948; Parker, 1990, https://doi.org/10.1080/00221689009499058; Recking, 2013a, https://doi.org/10.1061/(asce)hy.1943-7900.0000653) were selected for their performance relative to the measured bedload (except for and Meyer-Peter and Mueller) and because each equation has a different mathematical form and degree of nonlinearity. They were used in a Monte Carlo approach, with input probability distributions fitted to the measured river width, slope, bed grain-size distribution, and to the associated (computed) Shields stress. The results show that accounting for natural variability in the calculation reproduces bedload fluctuations well. But overall, when calculating the bedload volume transported by a flow event, accounting for variability systematically leads to higher estimated volumes (of the order of 20%) than those obtained with a deterministic approach using average input parameters. This is a direct consequence of the nonlinearity of the equations.

**Plain Language Summary**
Transport of bedload in a river, that is, sediment in motion that stays close to the riverbed, fluctuates a lot over relatively short time periods for a given near-constant discharge. In this paper, we try to reproduce this natural process. To do this, we use a protocol that involves running a bedload equation several thousand times, slightly and randomly modifying the input data with each run. The results show that by taking into account the natural variability of sediment size or channel geometry, we obtain realistic modeling of bedload transport fluctuations.

## 1. Introduction

Sediment transport in rivers is a very powerful driver of the landscape evolution. It not only controls channel morphodynamics, but sediments are also the support for many living communities, including vegetation, fishes, invertebrates, and indirectly, most living species (Wohl et al., 2015). Humankind is not left out. Its destiny has always been closely associated with this natural process, whether for agriculture, the quality of habitat, water resources, or energy (Parrinello & Kondolf, 2021). Despite its importance and more than one century of research, our knowledge of the physical processes controlling sediment transport is still sparse compared to many other scientific disciplines. This frustrating situation has many explanations (Ancey & Recking, 2023). The main one is that measuring sediment transport in rivers during floods is a very complicated task, and overall, beyond the technical difficulties, the very high natural variability of the river system (granulometry, geometry, hydraulics) makes the logistics very complex. The working solution is to compute sediment transport, and many equations have been proposed for computing bedload transport, which is the transport of coarse material in contact with the bed. The objective of this paper is to investigate how well simple empirical equations can capture and evaluate the average response of non-linear transport processes within a highly variable environment.

To get rid of the above-mentioned natural complexity, most of the bedload equations have been built with a simplified 1D configuration (often in flume experiments with a near uniform geometry, sediment, and hydraulics). When they are used with data (grain size, topography) also averaged at the reach scale, as is usually done in
**Writing – original draft:** A. Recking,
A. Johannot, D. Vázquez-Tarrío

**Writing – review & editing:** K. Horita,
S. Zanker, G. Melun

practice, the natural variability can be considered definitively excluded from the calculation protocol. This “missing dimension” (Ferguson, 2003) is not without consequences: smoothing the river by averaging both the data and the models gives a biased representation of a natural process (the bedload response to shear stress), which is by nature strongly nonlinear (Chen & Stone, 2008; Ferguson, 2003; Recking, 2013b). Not only does it limit the validity of 1D flume equations to 1D field computation, but the local cross-section average parameters are morphology-dependent, which suggests that different computational settings could be required for each morphology (Adams & Eaton, 2022; Francalanci et al., 2012; Monsalve et al., 2020; Recking et al., 2016).

Conscious of these limitations, the scientific community has progressively reintroduced some of the natural complexity into the computation strategies. Computational methods have been developed that allow one to account for spatial variability in multidimensional numerical models (Cienciala & Hassan, 2016; Schuurman et al., 2013; Segura & Pitlick, 2015). However, this practice is limited to qualified practitioners, and because the standard of practice still remains on the use of 1D modeling with reach averaged data, the variability must be accounted for in the bedload equations themselves (Fernández & Garcia, 2017). First attempts toward a better consideration of the complexity of natural systems have focused on a better consideration of the grain-size distribution [GSD] through the development of fractional equations computing bedload not only for a single characteristic grain size but for all sizes representative of each class present at the bed surface (Parker, 1990; Parker & Klingeman, 1982; Wilcock & Crowe, 2003). One step forward has consisted in reintroducing complexity into bedload computation with probability functions describing the spatial variability in grain size (Chen & Stone, 2008) and bed shear stress (Monsalve et al., 2020). An alternative to these approaches has consisted in directly fitting cross-section average field measurements (for bedload and shear stress) (Parker & Klingeman, 1982; Parker, 1990), and in all kinds of morphologies (Barry et al., 2004; Recking, 2010, 2013a). In doing so, the dimension lost by computing with the average parameters is implicitly reintroduced in the bedload function through the data used for its derivation. This approach appears to give interesting results (Ancey, 2019b; Barry et al., 2004; Guilherme Kruger Bartels et al., 2021; Hinton et al., 2018; Hosseiny et al., 2023; Pitlick et al., 2021; Recking, 2010), but it is not fully satisfactory as we still consider a one-to-one relationship between water discharge Q and bedload Qs (Ancey, 2019a). The reality is much more complex and measurements have shown that bedload is highly fluctuating over relatively short time periods, for a given near-constant discharge (Brownlie, 1981; Clayton & Pitlick, 2007; Cudden & Hoey, 2003; Dinehart, 1992; Downs et al., 2016; Gomez, 1983; Garcia et al., 2000; Gomez & Soar Philip, 2022; Kreisler et al., 2017; Mao et al., 2014; Misset, Recking, Legout, Bakker, et al., 2020; Møen et al., 2010; Powell et al., 1999; Rickenmann, 2020; Vericat et al., 2006).

The stochastic nature of bedload transport has been largely described (Ancey & Heyman, 2014) and bedload fluctuations in a given section can result from local processes (for instance bed armoring (Gomez, 1983; Powell, 1998; Vázquez-Tarrío et al., 2020), flow turbulence (Nelson et al., 1995), bedload layer activity (Parker, 2008; Recking et al., 2023)) or from variation of sediment supply from upstream (for instance the passage of dunes (Dinehart, 1992), of bedload sheets (Recking et al., 2009; Whiting et al., 1988), bar migration (Dhont & Ancey, 2018) or bank collapse (Molnar et al., 2010)). Whatever the reason, the fluctuating signal is replaced by a smoothed signal when computing bedload transport, and this likely impacts calculations of sediment transport integrated over time, for example, when computing a sediment budget. This is what this paper aims to discuss with 1D simulations that account for the natural variability through Monte Carlo simulations.

Previous papers have analyzed the effect of replacing the local shear stress with a cross-section-averaged shear stress and concluded that, without taking variability into account, bed load equations tend to under-predict, particularly for low to moderate transport (Adams & Eaton, 2022; Bertoldi et al., 2009; Chen & Stone, 2008; Monsalve et al., 2020; Recking, 2013b). The approach proposed here is different in that, instead of comparing the effect of replacing local (i.e., on each vertical) values with an average across a section (cross-sectional average), we consider how the spatial and temporal variability of cross-sectional average values affects bedload prediction. Another novelty is that this is done on the basis of a large field investigation in an alpine river. This paper is organized as follows: Section 1 presents the study site, materials, and methods; Section 2 presents the data collected; Section 3 presents the Monte Carlo simulations; and Section 4 presents a discussion of the results. We conclude that accounting for natural variability reproduces the bedload fluctuations well. But overall, when calculating the volume transported by a flow event, accounting for variability systematically leads to higher
<figure><img src="image_3.png" /><figcaption>**Figure 1.** The study site in the Severaisse River (Google earth, satellite image from 5 September 2020). Bedload produced by morphological changes in the upstream zone is efficiently transferred in the downstream reach as “traveling bedload” (Piton & Recking, 2017; Misset, Recking, Legout, Bakker, et al., 2020).</figcaption></figure>

volume estimates (on the order of +20%) than those obtained deterministically using average parameters because of the nonlinear relationship between bedload transport and shear stress.

## 2. Study Site, Material and Methods

### 2.1. The Study Site

The study site (Figure 1) is the Severaisse River, located in the Ecrins Massif (SE French Alps). Its 130 km² catchment is characterized by a low human impact and actively eroding hillslopes that supply large quantities of sediment to the valley bottom during debris-flow or rock-fall events. The river has a hydrologic regime characterized by snowmelt from late spring to the beginning of summer, and by ice-melt and storm-events during the late summer and autumn.

Measurements were taken (in 2–6 June 2021) at a 12 m wide laterally constrained section under a bridge in Villar-Loubiere where bedload can easily be monitored. The French Electricity Company (EDF) has a hydrometric gauge at the bridge that collects continuous discharge measurements. Despite large bedload fluxes passing through, we have not observed any permanent bar formation in the last 10 years (at most, ephemeral uniform deposits of gravel can be observed just after a flood). Previous bedload measurements suggest that the sediment pulses passing through this “transfer zone” come from the upstream “production zone” (Bakker et al., 2020; Misset, Gimbert, et al., 2019; Misset, Recking, Legout, Bakker, et al., 2020; Misset, Recking, Legout, Val-sangkar, et al., 2020; Misset et al., 2021), which is a 600 m long by 80 m wide braided reach where a multi-year time-lapse survey indicates that the local morphodynamics are very active. In this reach, both the channel position and local slope vary continuously with time, as illustrated by the time-lapse images in Figure 2.

### 2.2. Accounting for Variability

Our working hypothesis is that the morphologically active braided reach in Figure 1 produces bedload pulses that propagate efficiently through the downstream paved reach (grain size distributions are presented in next paragraphs) without exchange with the bed, following the concept of traveling bedload (Piton & Recking, 2017; Yu et al., 2009).

In 1D modeling, the simplest approach consists of reducing each “uniform” reach to one characteristic cross-section representative of the average parameters of the reach (grain size, channel geometry and slope). Following this approach, we considered a cross-section representative of the braided reach (in terms of average parameters). However, unlike the standard approach, the selected cross-section must also be representative of the trigger conditions responsible for successive pulses. This can be achieved by accounting for a temporal variability in the bed arrangement (surface GSD, section geometry, slope) which is captured in the dimensionless shear stress $\tau^*$ (Shields, 1936) given by:

$$ \tau^* = \frac{\tau}{g(\rho_s - \rho)D} = \frac{dS}{(s - 1)D} \qquad (1) $$
<figure><img src="image_4.png" /><figcaption>**Figure 2.** Time-lapse images (flow direction is from top to bottom) and hydrograph of the Severaisse River in October 2021 (5-year return period flood). Looking at the upstream part, a new channel was formed after the flood on the left side. In the paper we distinguish this “new channel” with the “old channel” still present on the right side.</figcaption></figure>

where g [ms⁻²] is the acceleration of gravity, ρ [kg/m³] is the water density, ρs [kg/m³] is the sediment density, D [m] is the representative grain diameter, d [m] is the flow depth, S [m/m] is the slope, and s = ρs/ρ.

Characterizing the temporal variability of τ*(Q, S, W, D) in a given cross-section would require a lengthy and challenging field program. This is why our second working hypothesis is that the spatial distribution of τ* in the reach is a reasonable proxy for the temporal distribution of τ* at one location (ergodic hypothesis). Characterizing the spatial variability of τ*(Q, S, W, D) requires measuring and quantifying the variability of in channel properties and the grain size. An intensive field survey was completed in spring and autumn of 2021, during which we measured hydraulics and bedload at the bridge, and channel topography (61 cross sections and associated local slope) and grain-size distributions (measurements of bed GSD at 67 locations) within the braided reach.

## 2.3. Measurement Techniques and Protocols

The bed topography of the braided reach was measured using a theodolite and a Leica Zeno 20 GPS. We first measured four sections in June 2021 to characterize the active channel associated with the bedload measurements. A second field campaign was organized at low flow in October 2021, when 57 additional cross sections were measured. Each cross-section was associated with a local bed slope measured over a length equal to 10 times the river width. There was a 5-year recurrence interval flood in early October 2021 (Figure 2) that resulted in migration of the active channel (i.e., the channel concentrating the main flow). Therefore, the active channel measured in the October field campaign was different from that measured in June. In the following section, we designate the “old channel” as the active channel during the bedload measurements in June, and the “new channel” as the active channel since October.

The bed GSD was measured following the Wolman count protocol (Wolman, 1954). Three operators participated, but the same protocol was prescribed for the random sampling to limit any bias. The sample locations were selected randomly in different morphological units (Bunte & Abt, 2001) to assess the variability in GSD: the old channel, the new channel, and the central bars recently reworked by the flows. Elevated bars were not sampled as they have not been inundated by recent floods. Almost half of the measurements were within the old or new channels. All GSD samples consisted of 100 pebbles, except for 10 samples collected in the new channel in March 2022, which were limited to 50 pebbles because the water level was still high and the water temperature was very low, preventing prolonged access to the sampling sites. Any areas covered by clay and silt (suspension deposition) were excluded.

Bedload was measured continuously using an HTI 96 hydrophone plugged into the EA-SDA14 card (RTSYS company) and installed in the left bank 100 m downstream of the bridge (Figure 1). The hydrophones used have a flat frequency response between 2 Hz and 30 kHz. Recordings were done with a 156 kHz sampling frequency and a 30 s duration repeated every 10 min, and were stored in .wav files. The .wav files were then processed in a frequency band associated with bedload (>2 kHz) to extract the acoustic power, following procedures presented in
Geay et al. (2021). Note that the bank hydrophone signal is used here as a proxy for the bedload signal fluctuations but was not converted to a bedload flux as the protocol is still under development (Nasr et al., 2023a, 2023b).

Direct bedload sampling was performed from the bridge in June 2021 with a pressure difference sampler (Elwha sampler 20.7 × 12 cm) equipped with a mesh size of 0.5 mm. The sampler was deployed over 10 verticals (approximately every 1.2 m) to integrate the bedload flux over the cross-section using Equation 2:

$$q_s = \frac{1}{W_t} \sum_{i=1}^{N} \frac{Q_s i}{W_e} W_i \quad (2)$$

Where $q_s$ is the bedload rate per unit width [g.s⁻¹.m⁻¹], N is the number of verticals, $Q_{si}$ is the bedload rate for each measurement point [g.s⁻¹], $W_t$ is the cross-section width [m], $W_e$ is the sampler width [m] and $W_i$ is the width considered representative of the sampling point i [m]. The sampling time for a given vertical varied (from 15 to 60 s) depending on the transport conditions to avoid a decrease in trapping efficiency, which depends on the sampler filling speed. The new data complement existing data measured during the melting season in 2018 (Misset, Recking, Legout, Bakker, et al., 2020).

Each bedload measurement was systematically associated with a measurement of the flow height distribution (measured each meter with a graduated pole from the bridge) from which we could compute the wetted area A, and the mean discharge $Q = A*U$ (wetted area times mean flow velocity U). The mean flow velocity U was estimated in two ways. First, the surface flow velocity was measured using a hand held Surface Velocity Radar (SVR). The measurements were performed from the bridge, almost simultaneously with bedload sampling, and in the same verticals. The average surface velocity was then adjusted with a coefficient of 0.8 to obtain the depth-averaged velocity (Dramais et al., 2013; Welber et al., 2016). Second, U was computed with a friction law from the measured cross sections, slope, GSD, and flow depth. The Ferguson (2007) flow resistance equation (presented in Text S2 in Supporting Information S1) was used as it has been shown to be particularly efficient for mountain streams (Rickenmann & Recking, 2011).

## 2.4. Bedload 1D Computation and Monte Carlo Simulations

All bedload computations were done with the BedloadR program (written in R), available online at https://en.bedloadweb.com/. BedloadR was initially developed to catalog bedload data from the literature (more than 10,500 field values and 1,300 flume values) and make it accessible. BedloadR also computes 1D bedload transport under uniform conditions and offers an alternative for practitioners to simple hand calculations and complex numerical models. For each cross section, the program discretizes the geometry and computes A(d) and R(d) functions linking the wetted area A and the hydraulic radius R to the flow depth d. It then uses the Ferguson (2007) friction law to compute a velocity function U(d) from which a discharge rating curve $Q(d) = A(d)U(d)$ can be developed. Finally, a relationship R(Q) is established for the considered cross section, which allows the program to compute the shear stress $\tau(Q) = \rho gSR(Q)$ and the corresponding Shield stress $\tau*(Q)$ (Equation 1). When combined with the bedload function $Q_s(\tau^*)$, we are left with a $Q_s(Q)$ function. A module was specifically designed within BedloadR for studying the effect of natural variability on bedload with Monte Carlo simulations (see Wilcock et al. (2009) for an example of application of this method). The Monte Carlo module allows the selection of several distribution functions (uniform, normal law, log-normal law) to characterize the variability associated with each input variable. From these probability distributions, the program generates random sets of variables (grain diameter, slope, shear stress) used to compute bedload transport. For a given discharge Q, the simulation is repeated 10,000 times to produce a statistical distribution of the computed bedload.

When computing a flow event, at each time step t, a Shields stress $\tau^*$ is attributed randomly to $Q(t)$ from the distribution and a transport rate $Q_s(\tau^*, t)$ is computed. Integration over time gives a transported volume V (m³). The flow event is computed 10,000 times, resulting in a distribution of the transported volumes.

# 3. Results of the Field Campaign and Data Analysis

## 3.1. Measurement of Water and Sediment Discharge

Figure 4 presents the continuous signal of water discharge and hydrophone for the period that coincides with direct sampling of bedload (discrete measured values are superimposed to the continuous signal) from June 10 to
<figure><img src="image_5.png" /><figcaption>**Figure 3.** (a) Location of the GSD samples (orthophoto of June 2021). (b) Location of the cross-section samples (post-flood “October 2021” orthophoto). The “old” channel is on the right side of the river, and the “new” channel is on the left side (the ID of each data is available in Figure S1 in Supporting Information S1).</figcaption></figure>

June 18 (the full period as well as spectra are presented in supplementary material, Figure S3 in Supporting Information S1).

Both the discharge and the acoustic power show a daily response, which is typical of the snowmelt process. Frequency analysis of acoustic power fluctuations shows the existence of continuous periods of fluctuation ranging in length from 20 min (the sampling period) to one day. Another measurement performed in the same place with a seismograph (not presented here) confirms these fluctuations lasting from a few minutes to several hours.

Direct bedload sampling (with the Elwha sampler) targeted the daily melt flows of June 2021 (we did not measured the large floods of May and October). Because bedload activity has a “memory” (East et al., 2018), we may expect that the transport of June was influenced by bed remobilization of the May 2021 flood. However, the data plotted as a function of discharge in Figure 5b shows good consistency with the data collected during melting discharges of 2018 (Misset, Recking, Legout, Bakker, et al., 2020). The measured bedload values suggest that large bedload differences can exist for a given discharge, and can span values up to two orders of magnitude. This is consistent with the data shown in Figure 4, with samples obtained at the same location alternating between weak and intense transport over very short time intervals (same discharge).

Each measured bedload value $Q_s$ is associated with a discharge value $Q$. Figure 5a shows $Q$ calculated using three methods: (a) the rating curve provided by EDF; (b) $Q$ calculated using the velocity $U$ obtained with the SVR radar; and (c) $U$ calculated with the Ferguson (2007) equation (used with $D_{50}$ = 66 mm and $D_{84}$ = 125 mm measured below the bridge). There is a small discrepancy between the three methods, which can likely be attributed to rating curve error resulting from changes in the channel section due to a thin deposition layer of gravel a few centimeters thick temporarily trapped between large pebbles under the bridge following a 2- to 5-year flood that occurred in May 2021 (Figure S3 in Supporting Information S1). Given the greater consistency between the $Q$ values obtained using the Ferguson (2007) equation and the SVR, these values were used.

## 3.2. The Bed Grain-Size Distribution

We did 57 Wolman counts in the dry wadable parts of the braided reach (“production zone”) with 33 collected on the bars and 24 in the old channel. Ten additional Wolman counts were collected in water in the “new” active channel (Figure 3a). We could sample the full channel including the talweg, but because of high velocities in the center causing access difficulties, these latter measurements are not of the same quality as the measurements done on the dry bed and are considered separately. All data are made available online (Recking & Johannot, 2023). Table 1 presents the results for the sand fraction $F_s$, the median diameter $D_{50}$, and the $D_{84}$ measured in the dry bed of the braided reach.

The characteristic grain sizes were very consistent between measurements collected in different areas of the channel and at different times. Because of the existence of large bed patchiness, we also computed the mean GSD
<figure><img src="image_6.png" /><figcaption>**Figure 4.** Bedload monitoring with hydrophone for the direct sampling period (red crosses are discrete cross-section average values measured with the Elwha sampler in June 2021).</figcaption></figure>

by weighting each sample by the surface concerned (Bunte & Abt, 2001), with little change in the result ($D_{50}$ = 45, $D_{84}$ = 113).

The full data set (57 values) was then used to analyze the probability distributions of $F_s$, $D_{50}$, and $D_{84}/D_{50}$, by fitting the values to the normal and log normal distributions (see procedure and figures in supplementary material Figure S4 in Supporting Information S1). Table 2 lists the scale and fit parameters of $F_s$, $D_{50}$ and $D_{84}/D_{50}$ for the two distributions tested.

The large coefficient of variation can be explained by the variability in different locations throughout the channel, which can also be influenced by the channel morphology (Figure 6a). Very small $D_{50}$ values in the bars

<figure><img src="image_7.png" /><figcaption>**Figure 5.** (a) Discharge-water-level (stage) rating curve using multiple methods and (b) the observed bedload $Q_s$ as a function of discharge $Q$ (the melting season 2018 data (Misset et al., 2021) are also plotted for comparison).</figcaption></figure>
<caption>**Table 1**</caption>

Characteristics of Grain Size Measured in the Dry Bed of the Severaisse River

<table><thead><tr><th>Date</th><th>Location</th><th>Number sample</th><th>F<sub>s</sub> (%)</th><th>D<sub>50</sub> (mm)</th><th>D<sub>84</sub> (mm)</th></tr></thead><tbody><tr><td>May 2021</td><td>Bars</td><td>16</td><td>12</td><td>38</td><td>103</td></tr><tr><td>July 2021</td><td>Bars</td><td>17</td><td>13</td><td>42</td><td>113</td></tr><tr><td>Oct 2021</td><td>Old Channel</td><td>24</td><td>7</td><td>45</td><td>113</td></tr><tr><td>All data</td><td>Bars</td><td>33</td><td>13</td><td>40</td><td>108</td></tr><tr><td>All data</td><td>Bars + old channel</td><td>57</td><td>11</td><td>42</td><td>108</td></tr></tbody></table>

correspond to large patches of fines located in the print of secondary flow paths (larger patches of sand located at the outlet of secondary channels), which can be associated with deposition during the hydrograph recession. On the contrary, the concentration of large pebbles and cobbles is usually associated with local bar erosion and local steep slopes. Boulders were also present and located above the mean bar level, downstream of the channel bend, suggesting grain deposition that occurred during past very high energetic flows. The same processes are suspected in the old channel, where sediment starvation during the flood recession causes strong bed reworking (especially if, as we suppose, sediments follow the new channel), producing local very steep and armored zones, occasionally covered with patches of fine sediments transported by secondary flows.

Patches were completely absent in the new active channel where the grain-size distributions were more uniform along the reach (Figure 6b). The absence of patches could be observed through the clear water in September 2021, and was confirmed by direct measurements when the channel was wadable again at low flows in March 2022. The March 2022 sampling also indicated that sand and fine gravels were present in water but could not be captured by the Wolman count because they were located below the pebbles or in their interstices. This surface armoring resulted in a bias toward a coarser GSD (especially for D<sub>50</sub>) measured in the new channel (Table 3).

To conclude, both the [in-water] active channel and the [dry] exposed bed presented a bias linked respectively to deficit (i.e., fines hidden in matrix) and excess (i.e., large patches) sand representation. For bedload computation, the GSD must be representative of in-channel conditions, and because most available data are from the dry bed, we considered the data collected on the exposed dry bed after excluding 4 GSD associated with the large patches of sand. The variability of this new data set composed of 29 samples, shown in Table 4, is reduced when compared to the previous analysis of Table 2.

### 3.3. Bed Topography

It was not possible to measure the bed topography below the water surface in June, but the top width was between 14 m (when considering the water surface elevation) and 16 m (top of bank) for the 4 cross sections. Fifty-seven additional cross sections were measured in the old and new channels when the river was wadable again in March 2022. All data are made available online (Recking & Johannot, 2023). Because the bankfull

width was difficult to detect in many of the cross sections, we characterized the variability by fitting the bottom channel width to the normal and log-normal probability distributions. The results presented in Table 5 show negligible differences between the old and the new channels.

The mean bed slopes measured over the full length of a reach in June were 0.01 m/m for the downstream constrained reach ("transfer zone") and 0.016 m/m for the braided reach. The measured local bed slopes (57 measurements) were fit to a normal and log normal distribution (Table 6) with a separate fit completed for the old and new channels. Both the old and new channels had a mean local slope of μ<sub>s</sub> = 0.016 m/m, but large differences were observed for the standard deviation and coefficient of variation (σ<sub>s</sub>, CV = σ<sub>s</sub>/μ<sub>s</sub>). Differences in variability here may be related to the variability observed with the grain-size distributions in the old channel that experienced sediment starvation compared to the new channel (Figure 6).

<caption>**Table 2**</caption>

Fit of the Normal and Log Normal Distribution (Mean μ, Standard Deviation σ, Coefficient of Variation CV = σ/μ*100) for the Sand Content Fs (Given in %), Diameters D<sub>50</sub> and D<sub>84</sub>

<table><thead><tr><th></th><th></th><th>F<sub>s</sub> (%)</th><th>D<sub>50</sub></th><th>D<sub>84</sub>/D<sub>50</sub></th></tr></thead><tbody><tr><td rowspan="3">Normal law</td><td>μ</td><td>10.6</td><td>43.3</td><td>2.5</td></tr><tr><td>σ</td><td>6.17</td><td>13.47</td><td>0.61</td></tr><tr><td>CV = σ/μ</td><td>58%</td><td>31%</td><td>25%</td></tr><tr><td rowspan="3">Log normal law</td><td>μ</td><td>2.2</td><td>3.7</td><td>0.9</td></tr><tr><td>σ</td><td>0.74</td><td>0.35</td><td>0.23</td></tr><tr><td>CV = σ/μ</td><td>34%</td><td>9%</td><td>26%</td></tr></tbody></table>
<figure><img src="image_8.png" /><figcaption>**Figure 6.** (a) Photos of the dry bed (high variability) and (b) the active channel (low variability).</figcaption></figure>

### 3.4. Shields Stress

The Shields stress is the ratio between the slope-depth product and grain size (Equation 1). One simple solution for deriving a Shields stress distribution would consist of combining the distribution for slope, grain size and water depth, as illustrated by scenario one in Figure 7.

The problem is that these parameters are not independent: for a given discharge $Q$, the local flow depth $d$ is controlled by the local slope $S$ and the local grain size $D$. This raises the question of covariation between flow depth $d$, slope $S$, and grain size $D$. Random association of these parameters can potentially lead to extreme and unrealistic values at both ends of the Shields stress distribution. To account for covariation, we chose the scenario 2 (Figure 7), which consists of building a Shields stress data set using the local parameters associated with each section. We used 57 complete samples with cross section, slope, and grain size data (21 samples in the old channel and 36 in the new channel). For each cross section $i$, a hydraulic radius function $R_i(Q)$, and a shear stress function $\tau_i^*(Q)$ are computed with BedloadR. A shear stress is then calculated for each discharge $Q$ considered between 1 and 50 m$^3$/s. This resulted in 50 Shields stress data sets (i.e. one for each discharge $Q$) comprising 57 samples each (57 sections). Each data set was fitted to normal and lognormal probability distributions, with the old and new channel samples considered separately. Figure 8 presents how the mean $\mu$, standard deviation $\sigma$ and coefficient of variation CV vary with $Q$.

Data from the old and new channels have nearly the same variability. The slightly higher variability in the new channel can be explained by local coarse grains generating a few abnormal low Shields stress values. For the Monte Carlo simulations in Section 4, a normal law is considered with a coefficient of variation of 15% (the objective being not to definitely solve this question, but rather to use a realistic representation of this variability for the Severaisse River).

## 4. Modeling

### 4.1. Bedload Computation With Mean Input Data

Sixteen bedload equations were tested in BedloadR with a self-formed cross-section measured in the central part of the braided reach, the mean GSD ($F_s = 11\%$, $D_{50} = 47$ mm, $D_{84} = 118$ mm), and the mean slope ($S = 0.016$).

<table><caption>**Table 3**

Characteristics Grain Size Measured in Water (in the Active Channel)</caption><thead><tr><th>Date</th><th>Reach</th><th>Location</th><th>Number sample</th><th>Pebbles/sample</th><th>F<sub>s</sub></th><th>D<sub>50</sub></th><th>D<sub>84</sub></th></tr></thead><tbody><tr><td>March 2022</td><td>Braiding reach</td><td>New channel</td><td>10</td><td>50</td><td>-</td><td>64</td><td>121</td></tr></tbody></table>
<caption>**Table 4**</caption>

Fit of the Normal and Log Normal Distribution on 29 GSD Samples (After Excluding 4 GSD With Large Patches of Sand) for the Sand Content Fs (Given in %), Diameters D<sub>50</sub>, and D<sub>84</sub>

<table><thead><tr><th></th><th></th><th>F<sub>s</sub> (%)</th><th>D<sub>50</sub></th><th>D<sub>84</sub>/D<sub>50</sub></th></tr></thead><tbody><tr><td rowspan="3">Normal law</td><td>μ</td><td>10.4</td><td>46.5</td><td>2.5</td></tr><tr><td>σ</td><td>5.77</td><td>11</td><td>0.6</td></tr><tr><td>CV</td><td>56%</td><td>24%</td><td>25%</td></tr><tr><td rowspan="3">Log normal law</td><td>μ</td><td>2.2</td><td>3.8</td><td>0.9</td></tr><tr><td>σ</td><td>0.7</td><td>0.24</td><td>0.2</td></tr><tr><td>CV</td><td>31%</td><td>6%</td><td>26%</td></tr></tbody></table>

Most equations overpredict the measured bedload transport (see details on the methodology and results Figure S5 in Supporting Information S1). For the Monte Carlo simulations we choose to retain four equations (Figure S6 in Supporting Information S1): Camenen-Larson (2005), Parker (1990), Recking (2013a) and Meyer-Peter and Mueller (MPM) equation [1948]. They were selected for their performance relative to the measured bedload (except for MPM) and because each equation has a different mathematical form and degree of nonlinearity (Figure S7 in Supporting Information S1). A summary of the main properties of each equation is presented in Table 7. The volumes computed for the measurement period (Figure 4) are also presented, and round to approximately 2–3 thousand cubic meters, except for MPM.

## 4.2. Monte Carlo Simulations

The final step of the study is to evaluate how the natural variability in the grain size and Shields stress quantified for the Severaisse River in Section 3 impacts the bedload computation using a Monte Carlo approach. For a given discharge $Q$, the program does not compute one unique mean Shields stress value $\tau^*$ but instead computes 10,000 $\tau^*$ values randomly selected from the Shields stress normal distribution with $\sigma/\mu = 0.15$ (Figure 8). The diameter $D$ also appears explicitly in either $\Phi = q_{sv}/\sqrt{g(s-1)D^3}$ (for the MPM, the Camenen Larson and the Recking equations), or in the fractional computation for the Parker equation (in the $\phi$ parameter). Covariation may exist between $D$ and $\tau^*$ but there were not enough data for such analysis and it was not considered. We used the normal distribution for $D_{50}$ and $F_s$, and lognormal distribution for $D_{84}/D_{50}$ (with parameters of Table 4). The full GSD is needed for the fractional computation used in the Parker equation: for each computation in BedloadR, the $F_s$, $D_{50}$, $D_{84}/D_{50}$ values are randomly selected and an empirical model (Recking, 2013b) is used to construct a new GSD (Figure S8 in Supporting Information S1).

Figure 9a presents an example of the bedload distribution obtained with the Parker equation for $Q = 25 \text{ m}^3/\text{s}$ Figure 9b plots the Parker bedload model (computed with mean input values) with sample simulation values randomly selected in the distributions for different discharges and with the direct bedload measurements. Most samples are within plus or minus one order of magnitude around the model, which is consistent with the direct measurements (Figure 5) and with most field data sets presented in https://en-bedloadweb.com/.

Figure 10 presents a simulation of bedload transported in the measuring period (10–18 June 2021) computed with the Parker equation. Bedload computed deterministically with the mean values (as usually done in 1D computation) is shown in green. The dashed line is the result of one Monte Carlo simulation accounting for the natural variability of the input data (computed with a characteristic time of 20 min, consistent with the signal in Figure 4 and with the literature (Rickenmann, 2023)). The result obtained when accounting for variability is qualitatively consistent with the continuous bedload monitoring shown in Figure 4 (quantitatively, the result in Figure 9 can be seen as a validation of the fluctuation envelope).

The hydrograph of 10–18 June 2021 was computed 10,000 times, and Figure 11 presents the resulting distribution of computed volumes. The statistical Monte Carlo approach may allow volumes to be provided with a given confidence interval rather than a unique mean value.

<caption>**Table 5**</caption>

Fit of the Normal and Log Normal Distribution to the Channel Width (57 Cross Sections)

<table><thead><tr><th></th><th></th><th>Old channel</th><th>New channel</th></tr></thead><tbody><tr><td rowspan="3">Normal law</td><td>μ</td><td>8.3</td><td>8.6</td></tr><tr><td>σ</td><td>2.27</td><td>2.0</td></tr><tr><td>CV</td><td>27%</td><td>23%</td></tr><tr><td rowspan="3">Log normal law</td><td>μ</td><td>2.1</td><td>2.1</td></tr><tr><td>σ</td><td>0.28</td><td>0.23</td></tr><tr><td>CV</td><td>13%</td><td>11%</td></tr></tbody></table>

Table 8 presents the volumes computed using the 4 equations, with and without accounting for the natural variability, for the measurement period (June 2021) and for all of 2021. One can see that the addition of variability consistently leads to higher estimated volumes, and that the increase differs from one equation to another. This result is consistent with results obtained when considering variability at the scale of the cross-section (Adams & Eaton, 2022; Bertoldi et al., 2009; Chen & Stone, 2008; Monsalve et al., 2020; Recking, 2013b) and is directly linked to the nonlinearity of the equation. When a time series is considered, a random Shields stress $\tau^*(t)$ is selected at each time step that is not the mean value. Rather it could be a low or high $\tau^*$ and the lower $\tau^*$ cannot balance the higher $\tau^*$ values: nonlinearity implies that
<table><caption>**Table 6**

*Fit of Local Channel Slope With the Normal Distribution (57 Measurements)*</caption><thead><tr><th></th><th></th><th>Old channel</th><th>New channel</th></tr></thead><tbody><tr><td rowspan="3">Normal law</td><td>μ</td><td>0.016</td><td>0.016</td></tr><tr><td>σ</td><td>0.0058</td><td>0.0032</td></tr><tr><td>CV</td><td>37%</td><td>20%</td></tr></tbody></table>

small changes in $\tau^*$ lead to a very sharp increase in $q_s$ (as illustrated with the asymmetric distribution in Figure 9b and with the bedload fluctuations shown in Figure 10). The threshold MPM equation is weakly affected for June 2021 because for these moderate transport conditions, many $\tau^*$ values fall below $\tau_c^*$ (the threshold Shields stress below which there is no transport), which means that excess prediction of zero transport helps to counterbalance the excess transport computed at the other end of the $\tau^*$ distribution (the differences are greater for the whole year, as shear stresses are higher).

The volumes obtained for the year 2021 should be compared with the results of Pitlick et al. (2021) who estimated an average annual transport of 8,000 m³ using the Parker (1990) equation and a statistical analysis of the local hydrology. The percentage differences with and without accounting for variability are higher than when computed only for June 2021. Over a longer time period, there is a larger range of Shields stress values that are sampled and the nonlinear effects of the equations compound.

Simulations were done with a referent fluctuation period of 20 min, but results were not overly sensitive when considering different times (e.g., 2,100 m³ instead of 2,130 m³ computed with the Parker equation when considering 2 hr instead of 20 min). Because of the unresolved question of covariation between $D$ and $\tau^*$, additional simulations were also performed with no variability in $D$ to test the impact on the results. Again, the results were not overly sensitive (e.g., 2,090 m³ instead of 2,130 m³ computed with the Parker equation).

# 5. Discussion

## 5.1. Natural Variability of the River Parameters

Previous research has documented diverse patterns of grain size sorting in gravel-bed rivers at different spatial scales, such as downstream fining (Sambrook Smith & Ferguson, 1996; Singer, 2008), downbar variations (Lunt & Bridge, 2004; Rice & Church, 2010), or at the bedform and particle cluster scale (Powell, 1998; Richards & Clifford, 1991). However, little work has been devoted to collecting replicate grain size samples across the same river reach to explore how these different sources of textural heterogeneity in gravel rivers combine and translate into spatial variability in the different percentiles or metrics of the GSD. In this regard, Mosley and Tindale (1985) sampled surface grain size in 12 channel sections over a ~1 km reach of the Ashley River (New Zealand). They documented that the mean size ranged from 11.7 to 18.6 mm. Clayton and Pitlick (2007) collected several Wolman samples along 20 cross sections in a 137 m meander bed of the Colorado River and observed that the $D_{50}$ and the $D_{84}$ ranged from 8 to 57 mm and 11 and 56 mm. More recently, Guerit et al. (2014) collected 30

volumetric samples in the Urumqi River (China) and reported how the $D_{50}$ ranged between 17 and 32 mm. Although the variance in grain-size metrics observed in these studies is comparable to that reported for the Severaisse River in the present work, none of them investigated the best distribution model for the spatial variability in grain-size percentiles; therefore, it is difficult to draw conclusions that can be compared with the results from our analysis.

Regarding the variability in topographic parameters (active width, slope), some previous studies have focused on determining how they covary with discharge, such as Meirovich et al. (1998) for slope or Ashmore et al. (2011) for active width. However, to our knowledge, few studies have attempted to quantify the spatial variability of the local slope or active width in gravel-bed rivers. It is a priori conceivable that this spatial variability in local values must be largely controlled by upstream sediment supply, channel shape and morphology (Penning-Roswell & Townshend, 1978; Prestegaard, 1983; Wohl et al., 1993), since the mechanisms explaining alluvial channel maintenance seem to be related to a topographic self-organization oscillating between shallow, wide sections and deeper, narrow ones (Brown & Pasternack, 2017; Eaton et al., 2006). In this regard, some classical field research has already described lower local bed slopes in pools and higher local slopes in riffles (Emmett et al., 1980; Sear, 1996). In the particular case

<figure><img src="image_9.png" /><figcaption>**Figure 7.** Two possible scenarios for computing the Shields stress probability distribution.</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>**Figure 8.** Fit of Shields stress $\tau^*$ with the normal and log-normal distributions, for the old channel (where was located the main flow before October 2021) and the new channel (where was located the main flow after October 2021). Evolution of parameters $\mu$, $\sigma$ and CV with discharge $Q$.</figcaption></figure>

of braided rivers (such as the Severaisse), local variations in channel geometry seem to covary with morphological and flow patterns (e.g., diffluences, confluences) (Bridge, 1993). Again, there has been little systematic analysis of which statistical distribution model best explains the reach-scale spatial variability in channel topography. Two remarkable exceptions are Moody and Troutman (2002) and Allen et al. (2018) who document that the spatial variability in channel width is approximately distributed as a log-normal distribution.

With regard to the spatial variability of hydraulic parameters (e.g., shear stress, velocity, depth), most work has concerned the distribution of local values but not the distribution of the mean values along the reach, which interests this study. Gamma distribution has often been proposed for the local shear stress (Bertoldi et al., 2009; Monsalve et al., 2020; Paola, 1996; Recking, 2013b). Lamouroux et al. (1992, 1995) and Lamouroux (1998) measured the distributions of point velocity, depth and shear stress in very different German and French natural streams and observed that they showed a similar shift from an exponential frequency distribution at low discharges to a normal distribution at high discharges. Since then, studies in freshwater ecology have shown the transferability of the velocity and depth distribution models of Lamouroux et al. (1995) and Lamouroux (1998) to

<table><caption>**Table 7**
Presentation of Equations Selected for the Monte Carlo Analysis</caption><thead><tr><th rowspan="2">Equation</th><th rowspan="2">Type</th><th colspan="2">Function</th><th colspan="3">Performance</th></tr><tr><th>Exponent</th><th>Threshold equation</th><th>Shear stress correction<sup>a</sup></th><th>E2<sup>b</sup></th><th>Volume<sup>c</sup> (m<sup>3</sup>)</th></tr></thead><tbody><tr><td>MPM</td><td>Power</td><td>1.5</td><td>Yes</td><td>Yes</td><td>29%</td><td>11,745</td></tr><tr><td>Recking</td><td>Power</td><td>6.5 to 2.5 when τ* increases</td><td>No</td><td>No</td><td>67%</td><td>3,300</td></tr><tr><td>Camenen-Larson</td><td>Exponential</td><td>—</td><td>No</td><td>Yes</td><td>79%</td><td>2,031</td></tr><tr><td>Parker</td><td>Power + exponential</td><td>14.5 at low q<sub>s</sub></td><td>No</td><td>Yes</td><td>86%</td><td>2,130</td></tr></tbody></table>

<footnote><sup>a</sup>Correction for form resistance (Figure S7 in Supporting Information S1). <sup>b</sup>Number of $Q_{sCal}/Q_{sMeas}$ ratios in the range (0.5–2). <sup>c</sup>Volume computed for the melting period.</footnote>
<figure><img src="image_11.png" /><figcaption>**Figure 9.** Bedload computation with the Monte Carlo approach and using the Parker, 1990 equation (a) example of simulated bedload distribution for $Q = 25 \text{ m}^3\text{/s}$ (b) Relationship between sediment mass flux and volumetric flux (dashed lines are plus or minus one order of magnitude around the median). Here the Parker (1990) model is plotted with samples randomly selected in the distributions (i.e., we repeat Figure (a) for several discharges and pick-up randomly one blue bar in each figure and reported it as a blue cross in Figure (b)) and with the measurements (figure extracted from Bedloadweb.com).</figcaption></figure>

streams in other environments (e.g., Girard et al. (2013, 2014); Lamouroux et al. (2013); Saraeva and Hardy (2009)), and this type of statistical model has been widely used to describe reach-scale habitats. In the same vein, Cienciala and Hassan (2016) analyzed the sensitivity of bedload calculations (using Wong and Parker's version of the Meyer-Peter-Mueller formulas) to variability in depth, velocity and shear stress, and reported that the coefficient of variation in predicted transport rates ranged from 50% to 150%. Based on these results, they suggest that propagation of uncertainty using the distribution of parameter values should be explicitly considered when modeling channel processes, which is in good agreement with the overall spirit of the present manuscript.

In summary, the available studies analyzing the spatial distribution of the variables used in this work are scarce and scattered (the literature review supporting this section is summarized in Table S2 in Supporting Information S1). We believe that more systematic studies focusing on analyzing the spatial variability in these parameters are needed to push forward the sediment-transport modeling strategy proposed here. In this regard, the recent development of new high-resolution topographic methods has exponentially increased our capacity for spatially distributed characterizations of topography (Wheaton et al., 2015) and grain size in gravel rivers (Vazquez-Tarrio et al., 2017). These methods provide new clues for the analysis of reach-scale spatial variability in the topographic

<figure><img src="image_12.png" /><figcaption>**Figure 10.** Hydrograph and bedload volumetric flux computed with the Parker equation for the monitoring period (10–18 June 2021). The green line (QsRef) corresponds to the bedload computed using the mean input values. The fluctuating dashed line corresponds to bedload computed with the Monte Carlo approach (figure extracted from Bedloadweb.com).</figcaption></figure>
<figure><img src="image_13.png" /><figcaption>**Figure 11.** Distribution of the volumes transported computed with the Parker (1990) equation for June 2021; blue lines indicate the 50% confidence interval [2,450–2,550 m³] (figure extracted from Bedloadweb.com).</figcaption></figure>

and grain-size parameters needed for sediment-transport modeling (see, for example, Storz-Peretz and Laronne (2013, 2018)).

## 5.2. Ergodic Hypothesis

We used the ergodic hypothesis, which consists of considering that “an infinite long record at one point has the same statistical properties of a record taken over an infinite number of spatial ensembles at a particular point in time” (Paine, 1985). This hypothesis has been widely used in geomorphology despite being subject to numerous limitations such as scale distortion (Church & Mark, 1980; Lane & Richards, 1997; Paine, 1985). Among requirements, what concerns the present study is the need of a near equilibrium state that ensures that the same process is acting in the whole domain concerned, and that this process is stable in time. A temporal evolution of the study site cannot be excluded. For example, Misset et al. (2020a) observed a seasonal effect for transport dynamics, explained mainly by the evolution of morphology (sedimentary supply and average bed slope) from the beginning to the end of the melt season. However, these changes affect slightly the mean

values but do not fundamentally affect the properties of the study site: the feedback processes between transport and morphology remain the same, and the overall variability of grain size and local slopes remain unchanged through space and time. Because the braided reach is continuously reworked by the flows almost every year (with channel migration through the active band), it ensures a near homogeneous bed texture (clusters and patches) and morphology (bars) everywhere. The main limitation of the ergodic approach is that it can lead to an over-representation of extreme values in the case where their presence in the spatial distribution is associated with long-term evolution (Elmendorf et al., 2015). These zones could be observed in our study site (elevated bars in margins with very coarse material) and were excluded from sampling. Long-term evolution implies imbricated effects (Lane & Richards, 1997) and remains an open question in all fluvial morphodynamics studies (even without considering variability), which is out of the scope of this paper.

## 5.3. Bedload Prediction

This study demonstrates that even when best efforts are made to collect very well documented data, 1D bedload computation remains a very challenging task. An equation valid for the Severaisse River may not be valid for another river. Calibration may be necessary in most situations. Considering the natural variability allows one to compute not only a median $Q_s(Q)$ curve but also the associated probabilistic distribution, and as illustrated in Figure 9a, allows one to reproduce random samples closer to what is measured in reality. The consequence is that the computed volumes are higher than the referent volume computed without considering variability. The discrepancy (volume difference with and without variability) is directly linked to the degree of nonlinearity. For each bedload equation, it depends on the form of the mathematical function used in the different transport regime (the phenomena could easily be replicated in an Excel file using a simple power law with different exponents and a random distribution for $\tau^*$).

In terms of recommendations for future works, several options can be considered. The first one concerns the equation calibration, which should also be done ideally with cumulated volume when possible. This poses

<table><caption>**Table 8**

*Transported Volumes Computed Both Deterministically (Mean Inputs) and With a Monte Carlo Approach*</caption><thead><tr><th rowspan="2">Equation</th><th colspan="3">Computed volume for June 2021 (m³)</th><th colspan="3">Computed volume for year 2021 (m³)</th></tr><tr><th>Deterministic approach</th><th>Monte Carlo approach</th><th>Difference</th><th>Deterministic approach</th><th>Monte Carlo approach</th><th>Difference</th></tr></thead><tbody><tr><td>MPM</td><td>11,745</td><td>12,240</td><td>+4%</td><td>28,040</td><td>34,000</td><td>+21%</td></tr><tr><td>Recking</td><td>3,300</td><td>3,580</td><td>+8%</td><td>12,770</td><td>14,300</td><td>+12%</td></tr><tr><td>Camenen-Larson</td><td>2,760</td><td>3,130</td><td>+13%</td><td>8,130</td><td>9,950</td><td>+22%</td></tr><tr><td>Parker</td><td>2,130</td><td>2,510</td><td>+17%</td><td>6,500</td><td>8,300</td><td>+28%</td></tr></tbody></table>
additional issues such as how to account for the contribution of suspension in the deposited volume (which can reach up to 80% (Misset, Recking, et al., 2019)).

This work also advocates for improved development of the coupling between bedload prediction and measurements using new surrogate techniques (such as hydrophones, geophones) that give access to high frequency and continuous data (at least it can inform on the fluctuations).

Finally, despite the topic of the paper was the use of 1D models, a huge potential of improvement also exists when coupling the investigation with multidimensional numerical models (Francalanci et al., 2012), which can be used for building the appropriate spatial distribution of shear stress (Adams & Eaton, 2022) or for accounting for the complex feedbacks that exist between shear stress distribution, erosion, and morphology (Eaton et al., 2006).

## 6. Conclusion

The bedload signal obtained using a hydrophone and through direct sampling with an Elwha sampler show that bedload transport fluctuates a great deal in the Severaisse River, a French alpine river. These large fluctuations are similar to what has been observed elsewhere and highlight the limitations of computing bedload with mean bed and grain-size values, which inevitably leads to a “smoothed” bedload signal. To explore alternatives, we took advantage of an intensive field survey in the Severaisse River (French Ecrins Massif) during the 2021 snowmelt season to test different scenarios using a Monte Carlo approach.

Our working hypothesis is that the observed fluctuations can be attributed to the bed and flow variability in the upstream braided reach. A large part of the paper was thus dedicated to a description of this variability and the fitting of measured data to probability functions. One challenge of bedload prediction is that the data must be representative of the active (in water) channel, whereas the measurements are more accessible out of this channel in the wadable dry bed. Part of the paper was dedicated to a comparison between data collected in the dry part of the floodplain (old abandoned channel and dry bars) and in the channel under water. The results show (for GSD and slope) similar mean values, but there was much greater variability in the dry areas. This can be explained by the fact that the abandoned channels are heavily reworked by the flows (supply limitation), and that large patches of fine sediment are deposited by secondary flood flows during overbank flows. We conclude that data collected in the dry bed can be representative for computing bedload with average values, but caution should be exercised when defining the bed variability for a probabilistic bedload computation.

The second major finding of the article is that, when calculating the volume of bedload transported by a flow, taking variability into account systematically leads to higher estimated volumes (on the order of 20%) than those obtained with a deterministic approach using average input parameters. This is a direct consequence of the non-linearity of the equations, where a small change in the mean Shields stress results in a large change in the transport response. Consequently, when randomly selecting Shields stress values from a distribution, the transport generated by Shields values below the median does not balance the transport generated by Shields values above the median. This effect is enhanced by the non-linear properties of the bedload equation used, and could reach 28% in our simulations.

## Data Availability Statement

The data used in this paper have been deposited at https://data.mendeley.com/datasets/vhfzm2fsvz/1 (Recking & Johannot, 2023). The code used for Monte Carlo simulations is available online at https://en-bedloadweb.com/.

## References

Adams, D. L., & Eaton, B. C. (2022). A comparison of 1D and 2D bedload transport functions under high excess shear stress conditions in laterally-constrained gravel-bed rivers: A laboratory study. *EGUSphere, 1–21. https://doi.org/10.5194/egusphere-2022-53

Allen, G. H., Pavelsky, T. M., Barefoot, E. A. e. a., Lamb, M. P., Butman, D., Tashie, A., & Gleason, C. J. (2018). Similarity of stream width distributions across headwater systems. *Nature Communications, 9*(1), 610. https://doi.org/10.1038/s41467-018-02991-w

Ancey, C. (2019a). Bedload transport: A walk between randomness and determinism 1. The state of the art. *Journal of Hydraulic Research*, 1–24.

Ancey, C. (2019b). Bedload transport: A walk between randomness and determinism 2. Challenges and prospects. *Journal of Hydraulic Research*, 1–24.

Ancey, C., & Heyman, J. (2014). A microstructural approach to bedload transport: Mean behaviour and fluctuations of particle transport rates. *Journal of Fluid Mechanics*, 744, 129–168. https://doi.org/10.1017/jfm.2014.74

Ancey, C., & Recking, A. (2023). Scaling behavior of bedload transport: What if bagnold was right? *Earth-Science Reviews*, 246, 104571. https://doi.org/10.1016/j.earscirev.2023.104571

## Acknowledgments

This research was funded by INRAE (National Institut for Research on Agriculture and Environment) and The French Office for Biodiversity (OFB), and the French CARNOT Institut for Water and Environment. The work by one of the authors (D. Vázquez-Tarrío) was supported by the 2023–2026 Grant signed between the Spanish Directorate General for Water (DGA-MITERD; Government of Spain) and the Spanish National Research Council (CSIC-Ministry of Science, Innovation and Universities), which includes action “Sedimentary Morphodynamics” (20233TE012: IGME-CSIC; Tarquín 2 Project). The authors would like to thank Pr John Pitlick who did an early review of the manuscript. We are grateful to Dr. Amy East (AE), Dr. Roberto Fernandez, and anonymous reviewers whose constructive comments greatly improved the manuscript.
Ashmore, P., Bertoldi, W., & Gardner, J. T. (2011). Active width of gravel-bed braided rivers. *Earth Surface Processes and Landforms*, *36*(11), 1510–1521. https://doi.org/10.1002/esp.2182

Bakker, M., Gimbert, F., Geay, T., Misset, C., Zanker, S., & Recking, A. (2020). Field application and validation of a seismic bedload transport model. *Journal of Geophysical Research: Earth Surface*, *125*(5). https://doi.org/10.1029/2019JF005416

Barry, J. J., Buffington, J. M., & King, J. G. (2004). A general power equation for predicting bed load transport rates in gravel bed rivers. *Water Resources Research*, *40*(W10401), 1–22. https://doi.org/10.1029/2004wr003190

Bertoldi, W., Ashmore, P., & Tubino, M. (2009). A method for estimating the mean bed load flux in braided rivers. *Geomorphology*, *103*(3), 330–340. https://doi.org/10.1016/j.geomorph.2008.06.014

Bridge, J. S. (1993). The interaction between channel geometry, water flow, sediment transport and deposition in braided rivers. *Braided rivers. J. Best and C. S. Bristow. Geological Society*, 75(1), 13–71. https://doi.org/10.1144/gsl.sp.1993.075.01.02

Brown, R. A., & Pasternack, G. B. (2017). Bed and width oscillations form coherent patterns in a partially confined, regulated gravel–cobble–bedded river adjusting to anthropogenic disturbances. *Earth Surface Dynamics*, 5, 1–20. https://doi.org/10.5194/esurf-5-1-2017

Brownlie, W. R. (1981). *Prediction of flow depth and sediment discharge in open channels*. Keck Laboratory of Hydraulics and Water Resources, California Institute of Technology.

Bunte, K., & Abt, S. R. (2001). *Sampling surface and subsurface particle-size distributions in Wadable and cobble bed streams for analyses in sediment transport, hydraulics and streambed monitoring* (p. 450). USDA.

Camenen, B., & Larson, M. (2005). A bedload sediment transport formula for the nearshore. *Estuarine, Coastal and Shelf Science*, *63*(1–2), 249–260. https://doi.org/10.1016/j.ecss.2004.10.019

Chen, L., & Stone, M. C. (2008). Influence of bed material size heterogeneity on bedload transport uncertainty. *Water Resources Research*, *44*(1), W01405. https://doi.org/10.1029/2006wr005483

Church, M., & Mark, D. M. (1980). On size and scale in geomorphology. *Progress in Physical Geography*, *4*(3), 342–390. https://doi.org/10.1177/03091338000400302

Cienciala, P., & Hassan, M. A. (2016). Sampling variability in estimates of flow characteristics in coarse-bed channels: Effects of sample size. *Water Resources Research*, *52*(3), 1899–1922. https://doi.org/10.1002/2015WR017259

Clayton, J. A., & Pitlick, J. (2007). Spatial and temporal variations in bed load transport intensity in a gravel bed river bend. *Water Resources Research*, *43*(W02426), 1–13. https://doi.org/10.1029/2006WR005253

Cudden, J. R., & Hoey, T. B. (2003). The causes of bedload pulses in a gravel channel: The implications of bedload grain-size distribution. *Earth Surface Processes and Landforms*, *28*(13), 1411–1428. https://doi.org/10.1002/esp.521

Dhont, B., & Ancey, C. (2018). Are bedload transport pulses in gravel bed rivers created by bar migration or sediment waves? *Geophysical Research Letters*, *45*(11), 5501–5508. https://doi.org/10.1029/2018GL077792

Dinehart, R. L. (1992). Evolution of coarse gravel bed forms: Field measurements at flood stage. *Water Resources Research*, *28*(10), 2667–2689. https://doi.org/10.1029/92wr01357

Downs, P. W., Soar, P. J., & Taylor, A. (2016). The anatomy of effective discharge: The dynamics of coarse sediment transport revealed using continuous bedload monitoring in a gravel-bed river during a very wet year. *Earth Surface Processes and Landforms*, *41*(2), 147–161. https://doi.org/10.1002/esp.3785

Dramais, G., Le Coz, J., Le Boursicaud, R., Hauet, A., & Lagouy, M. (2013). *Jaugeage par radar mobile: Protocole et résultats. SHF Hydro-metrie 2013*. SHF.

East, A. E., Logan, J. B., Mastin, M. C., Ritchie, A. C., Bountry, J. A., Magirl, C. S., & Sankey, J. B. (2018). Geomorphic evolution of a gravel-bed river under sediment-starved versus sediment-rich conditions: River response to the World's largest dam removal. *Journal of Geophysical Research: Earth Surface*, *123*(12), 3338–3369. https://doi.org/10.1029/2018JF004703

Eaton, B. C., Church, M., & Davies, T. R. H. (2006). A conceptual model for meander initiation in bedload-dominated streams. *Earth Surface Processes and Landforms*, *31*(7), 875–891. https://doi.org/10.1002/esp.1297

Elmendorf, S., Henry, G., Hollister, R., Fosaa, A., Gould, W., Hermanutz, L., et al. (2015). Experiment, monitoring, and gradient methods used to infer climate change effects on plant communities yield consistent patterns. *PNAS*, *112*(2), 448–452. https://doi.org/10.1073/pnas.1410088112

Emmett, W. W., Myrick, R. M., & Meade, R. H. (1980). *Field data describing the movement and storage of sediment in the East Fork river, Wyoming. Part 1: River hydraulics and sediment transport*, 1979. USGS.

Ferguson, R. (2007). Flow resistance equations for gravel and boulder bed streams. *Water Resources Research*, *43*(W05427), 1–12. https://doi.org/10.1029/2006wr005422

Ferguson, R. I. (2003). The missing dimension: Effects of lateral variation on 1-D calculations of fluvial bedload transport. *Geomorphology*, *56*(1–2), 1–14. https://doi.org/10.1016/s0169-555x(03)00042-4

Fernández, R., & Garcia, M. (2017). Input-variable sensitivity assessment for sediment transport relations. *Water Resources Research*, *53*(9), 8105–8119. https://doi.org/10.1002/2016WR020249

Francalanci, S., Solari, L., Toffolon, M., & Parker, G. (2012). Do alternate bars affect sediment transport and flow resistance in gravel bed rivers? *Earth Surface Processes and Landforms*, *37*(8), 866–875. https://doi.org/10.1002/esp.3217

Garcia, C., Laronne, J. B., & Sala, M. (2000). Continuous monitoring of bedload flux in a mountain gravel-bed river. *Geomorphology*, *34*(1–2), 23–31. https://doi.org/10.1016/s0169-555x(99)00128-2

Geay, T., Zanker, S., Misset, C., & Recking, A. (2021). Passive acoustic measurement of bedload transport: Towards a global calibration curve? *Journal of Geophysical Research*.

Girard, V., Lamouroux, N., & Mons, R. (2014). Modeling point velocity and depth statistical distributions in steep tropical and alpine stream reaches. *Water Resources Research*, *50*(1), 427–439. https://doi.org/10.1002/2013WR01389

Girard, V., Le Goulven, P., Calvez, R., & Lamouroux, N. (2013). Velocity and depth distributions in stream reaches: Testing European models inEcuador. *Journal of Hydraulic Engineering*, *139*(7), 794–798. https://doi.org/10.1061/(asce)hy.1943-7900.0000726

Gomez, B. (1983). Temporal variations in bedload transport rates: The effects of progressive bed armouring. *Earth Surface Processes and Landforms*, *8*(1), 41–54. https://doi.org/10.1002/esp.3290080105

Gomez, B., & Soar Philip, J. (2022). Bedload transport: Beyond intractability. *Royal Society Open Science*, *9*(3), 211932. https://doi.org/10.1098/rsos.211932

Guerit, L., Barrier, L., Narteau, C., Metivier, F., Liu, Y., Lajeunesse, E., et al. (2014). The grain-size patchiness of braided gravel-bed streams – Example of the Urumqi River (northeast tian Shan, China). *Advances in Geosciences*, *37*, 27–39. https://doi.org/10.5194/ageo-37-27-2014

Guilherme Kruger Bartels, G., dos Reis Castro, N. M., Loguercio Collares, G., & Mainardi Fan, F. (2021). Performance of bedload transport equations in a mixed bedrock–alluvial channel environment. *Catena*, *199*, 105108. https://doi.org/10.1016/j.catena.2020.105108

Hinton, D., Hotchkiss, R. H., & Cope, M. (2018). Comparison of calibrated empirical and semi-empirical methods for bedload transport rate prediction in gravel bed streams. *Journal of Hydraulic Engineering*, *144*(7), 17. https://doi.org/10.1061/(ASCE)HY.1943-7900.0001474
Hosseiny, H., Masteller, C. C., Dale, J. E., & Phillips, C. B. (2023). Development of a machine learning model for river bed load. *Earth Surface Dynamics*, 11(4), 681–693. https://doi.org/10.5194/esurf-11-681-2023

Kreisler, A., Moser, M., Aigner, J., Rindler, R., Tritthart, M., & Habersack, H. (2017). Analysis and classification of bedload transport events with variable process characteristics. *Geomorphology*, 291, 57–68. https://doi.org/10.1016/j geomorph.2016.06.033

Lamouroux, N. (1998). Depth probability distributions in stream reaches. *Journal of Hydraulic Engineering*, 124(2), 224–227. https://doi.org/10.1061/(asce)0733-9429(1998)124:2(224)

Lamouroux, N., Merigoux, S., Doledec, S., & Snelder, T. H. (2013). Transferability of hydraulic preference models of aquatic macroinvertebrates. *River Research and Applications*, 29(7), 933–937. https://doi.org/10.1002/rra.2578

Lamouroux, N., Souchon, Y., & Herouin, E. (1995). Predicting velocity frequency distributions in stream reaches. *Water Resources Research*, 31(9), 2367–2375. https://doi.org/10.1029/95WR01485

Lamouroux, N., Statzner, B., Fuchs, U., Kohmann, F., & Schmedtje, U. (1992). An unconventional approach to modeling spatial and temporal variability of local shear stress in stream segments. *Water Resources Research*, 28(12), 3251–3258. https://doi.org/10.1029/92WR01761

Lane, S., & Richards, K. (1997). Linking river channel form and process: Time, space and causality revisited. *Earth Surface Processes and Landforms*, 22(3), 249–260. https://doi.org/10.1002/(sici)1096-9837(199703)22:3<249::aid-esp752>3.0.co;2-7

Lunt, I. A., & Bridge, J. S. (2004). Evolution and deposits of a gravelly braid bar, Sagavanirktok River, Alaska. *Sedimentology*, 51(3), 415–432. https://doi.org/10.1111/j.1365-3091.2004.00628.x

Mao, L., Dell'Agnese, A., Huincache, C., Penna, D., Engel, M., Niedrist, G., & Comiti, F. (2014). Bedload hysteresis in a glacier-fed mountain river. *Earth Surface Processes and Landforms*, 39(7), 964–976. https://doi.org/10.1002/esp.3563

Meirovich, L., Laronne, J. B., & Reid, I. (1998). The variation of water-surface slope and its significance for bedload transport during floods in gravel-bed streams. *Journal of Hydraulic Research*, 36(2), 147–157. https://doi.org/10.1080/00221689809498630

Meyer-Peter, E., & Mueller, R. (1948). Formulas for bed-load transport. In *Proceedings 2nd meeting*. IAHR.

Misset, C., Gimbert, F., Geay, T., Recking, A., Legout, C., Bakker, M., et al. (2019). *Using turbidity and seismic measurements to unravel sources and transport interactions of fine and coarse sediments in a mountain river*. EGU.

Misset, C., Recking, A., Legout, C., Bakker, M., Bodereau, N., Borgniet, L., et al. (2020). Combining multi-physical measurements to quantify bedload transport and morphodynamic interactions in an Alpine braiding river reach. *Geomorphology*, 351, 106877. https://doi.org/10.1016/j.geomorph.2019.106877

Misset, C., Recking, A., Legout, C., Bakker, M., Gimbert, F., Geay, T., & Zanker, S. (2021). Using continuous turbidity and seismic measurements to unravel sediment provenance and interaction between suspended and bedload transport in an alpine catchment. *Geophysical Research Letters*, 48(4), e2020GL090696. https://doi.org/10.1029/2020GL090696

Misset, C., Recking, A., Legout, C., Valsangkar, N., Bodereau, N., Zanker, S., et al. (2020). The dynamics of suspended sediment in a typical alpine river reach: Insight from a seasonal survey. *Water Resources Research*, 55(12), 10918–10934. https://doi.org/10.1029/2019WR025222

Misset, C., Recking, A., Navratil, O., Legout, C., Poirel, A., Cazihlac, M., et al. (2019). Quantifying bed-related suspended load in gravel bed rivers through an analysis of the bedload-suspended load relationship. *Earth Surface Processes and Landforms*, 44(9), 1722–1733. https://doi.org/10.1002/esp.4606

Møen, K., Bogen, J., Zuta, J. F., Ade, P. K., & Esbensen, K. (2010). Bedload measurement in rivers using passive acoustic sensors. *S. I. R.* 2010-5091. U.S. Geological Survey.

Molnar, P., Densmore, A. L., McDardell, B. W., Turowski, J. M., & Burlando, P. (2010). Analysis of changes in the step-pool morphology and channel profile of a steep mountain stream following a large flood. *Geomorphology*, 124(1–2), 85–94. https://doi.org/10.1016/j.geomorph.2010.08.014

Monsalve, A., Segura, C., Hucke, N., & Katz, S. (2020). A bed load transport equation based on the spatial distribution of shear stress—Oak Creek revisited. *Earth Surface Dynamics*, 8(3), 825–839. https://doi.org/10.5194/esurf-8-825-2020

Moody, J. A., & Troutman, B. M. (2002). Characterization of the spatial variability of channel morphology. *Earth Surface Processes and Landforms*, 27(12), 1251–1266. https://doi.org/10.1002/esp.403

Mosley, M. P., & Tindale, D. S. (1985). Sediment variability and bed material sampling in gravel-bed rivers. *Earth Surface Processes and Landforms*, 10(5), 465–482. https://doi.org/10.1002/esp.3290100506

Nasr, M., Johannot, A., Geay, T., Zanker, S., Le Guern, J., & Recking, A. (2023a). Optimization of passive acoustic bedload monitoring in rivers by signal inversion. *Earth Surf. Dynam. Discuss*, 1–27. https://doi.org/10.5194/esurf-2022-68

Nasr, M., Johannot, A., Geay, T., Zanker, S., Le Guern, J., & Recking, A. (2023b). Passive acoustic monitoring of bedload with drifted hydrometer. *Journal of Hydraulic Engineering*, 149(7), 06023003. https://doi.org/10.1061/JHEND8.HYENG-13438

Nelson, J. M., Shreve, R. L., McLean, S. R., & Drake, T. G. (1995). Role of near-bed turbulence structure in bed load transport and bed form mechanics. *Water Resources Research*, 31(8), 2071–2086. https://doi.org/10.1029/95wr00976

Paine, A. D. M. (1985). Ergodic reasoning in geomorphology: time for a review of the term? *Progress in Physical Geography: Earth and Environment*, 9(1), 1–15. https://doi.org/10.1177/030913338500900101

Paola, C. (1996). In P. J. Ashworth, S. J. Bennet, J. L. Best, & S. J. McLelland (Eds.), *Incoherent structures: Turbulence as a metaphor for stream braiding*. John Wiley and sons.

Parker, G. (1990). Surface-based bedload transport relation for gravel rivers. *Journal of Hydraulic Research*, 28(4), 417–428. https://doi.org/10.1080/00221689009499058

Parker, G. (2008). Transport of gravel and sediment mixtures. In ASCE manual 54 "sedimentation engineering: Processes, measurements, modeling, and practice" (p. 1132). American Society of Civil Engineers.

Parker, G., & Klingeman, P. C. (1982). On why gravel bed streams are paved. *Water Resources Research*, 18(5), 1409–1423. https://doi.org/10.1029/wr018i005p01409

Parrinello, G., & Kondolf, G. M. (2021). The social life of sediment. *Water Hist*, 13, 1–12. https://doi.org/10.1007/s12685-021-00280-w

Penning-Roswell, E., & Townshend, R. G. (1978). The influence of scale on factors affecting stream channel slope. *Transactions of the Institute of British Geographers*, 3(4), 395–415. https://doi.org/10.2307/622120

Pitlick, J., Recking, A., Liebault, F., Misset, C., Piton, G., & Vazquez-Tarrio, D. (2021). Sediment production in French Alpine rivers. *Water Resources Research*, 57(12), e2021WR030470. https://doi.org/10.1029/2021WR030470

Piton, G., & Recking, A. (2017). The concept of “travelling bedload” and its consequences for bedload computation in mountain streams. *Geomorphology*, 42(10), 1505–1519. https://doi.org/10.1002/esp.4105

Powell, D. M. (1998). Patterns and processes of sediment sorting in gravel-bed rivers. *Progress in Physical Geography*, 22, 1–32. https://doi.org/10.1191/030913398666402127

Powell, D. M., Reid, I., & Laronne, J. B. (1999). Hydraulic interpretation of cross-stream variations in bed-load transport. *Journal of Hydraulic Engineering*, 125(12), 1243–1252. https://doi.org/10.1061/(asce)0733-9429(1999)125:12(1243)
Prestegaard, K. (1983). Variables influencing water-surface slopes in gravel-bed streams at bankfull stage. *GSA Bulletin*, *94*(5), 673-678. https://doi.org/10.1130/0016-7606(1983)94<673:viwsig>2.0.co;2

Recking, A. (2010). A comparison between flume and field bedload transport data and consequences for surface based bedload transport prediction. *Water Resources Research*, *46*(3), 1-16. https://doi.org/10.1029/2009wr008007

Recking, A. (2013a). A simple method for calculating reach-averaged bedload transport. *Journal of Hydraulic Engineering*, *139*(1), 70-75. https://doi.org/10.1061/(asce)hy.1943-7900.0000653

Recking, A. (2013b). An analysis of non-linearity effects on bedload transport prediction. *Journal of Geophysical Research - Earth Surface*, *118*(3), 1-18. https://doi.org/10.1002/jgrf.20090

Recking, A., Frey, P., Paquier, A., & Belleudy, P. (2009). An experimental investigation of mechanisms responsible for bedload sheet production and migration. *Journal of Geophysical Research*, *114*(F3), F03010. https://doi.org/10.1029/2008jf000990

Recking, A., & Johannot, A. (2023). Severaisse River data 2021 [Dataset]. *Mendeley Data VI*. https://doi.org/10.17632/vhfzm2fsvz.1

Recking, A., Piton, G., Vazquez-Tarrio, D., & Parker, G. (2016). Quantifying the morphological print of bedload transport. *Earth Surface Processes and Landforms*, *41*(6), 809-822. https://doi.org/10.1002/esp.3869

Recking, A., Vázquez Tarrio, D., & Piton, G. (2023). The contribution of grain sorting to the dynamics of the bedload active layer. *Earth Surface Processes and Landforms*, *48*(5), 1-18. https://doi.org/10.1002/esp.5530

Rice, S. P., & Church, M. (2010). Grain-size sorting within river bars in relation to downstream fining along a wandering channel. *Sedimentology*, *57*(1), 232-251. https://doi.org/10.1111/j.1365-3091.2009.01108.x

Richards, K., & Clifford, N. (1991). Fluvial geomorphology: Structured beds in gravelly rivers. *Progress in Physical Geography*, *15*(4), 407-422. https://doi.org/10.1177/030913339101500404

Rickenmann, D. (2020). Effect of sediment supply on cyclic fluctuations of the disequilibrium ratio and threshold transport discharge, inferred from bedload transport measurements over 27 Years at the Swiss Erlenbach stream. *Water Resources Research*, *56*(11), e2020WR027741. https://doi.org/10.1029/2020WR027741

Rickenmann, D. (2023). Bedload transport fluctuations, flow conditions and disequilibrium ratio at the Swiss Erlenbach stream: Results from 27 years of high resolution temporal measurements. *Earth Surface Dynamics*. https://doi.org/10.5194/egusphere-2023-964

Rickenmann, D., & Recking, A. (2011). Evaluation of flow resistance in gravel-bed rivers through a large field dataset. *Water Resources Research*, *47*(7), 1-22. https://doi.org/10.1029/2010wr009793

Sambrook Smith, G. H., & Ferguson, R. I. (1996). The gravel-sand transition: Flume study of channel response to reduced slope. *Geomorphology*, *16*(2), 147-159. https://doi.org/10.1016/0169-555x(95)00140-z

Saraeva, K., & Hardy, T. B. (2009). Prediction of fisheries physical habitat values based on hydraulic geometry and frequency distributions of depth and velocity. *International Journal of River Basin Management*, *7*(5), 31-41. https://doi.org/10.1080/15715124.2009.9635368

Schuurman, F., Marra, W. A., & Kleinhans, M. G. (2013). Physics-based modeling of large braided sand-bed rivers: Bar pattern formation, dynamics, and sensitivity. *Journal of Geophysical Research: Earth Surface*, *118*(4), 2509-2527. https://doi.org/10.1002/2013JF002896

Sear, D. A. (1996). Sediment transport processes in pool-riffle sequences. *Earth Surface Processes and Landforms*, *21*(3), 241-262. https://doi.org/10.1002/(sici)1096-9837(199603)21:3<241::aid-esp623>3.0.co;2-1

Segura, C., & Pitlick, J. (2015). Coupling fluvial-hydraulic models to predict gravel transport in spatially variable flows. *Journal of Geophysical Research: Earth Surface*, *120*(5), 834-855. https://doi.org/10.1002/2014JF003302

Shields, A. (1936). *Application of similarity principles and turbulence research to bed load movement*. *Hydrodynamics laboratory publ. J. C. V. U. W.P.Ott, trans.*, US Department of Agriculture, soil conservation service cooperative laboratory (p. 167). California Institute of Technology, Pasadena.

Singer, M. B. (2008). Downstream patterns of bed material grain size in a large, lowland alluvial river subject to low sediment supply. *Water Resources Research*, *44*(12), W12202. https://doi.org/10.1029/2008wr007183

Storz-Peretz, Y., & Laronne, J. (2018). The morpho-textural signature of large bedforms in ephemeral gravel-bed channels of various planforms. *Hydrological Processes*, *32*(5), 617-635. https://doi.org/10.1002/hyp.11437

Storz-Peretz, Y., & Laronne, J. B. (2013). Morphotextural characterization of dryland braided channels. *GSA Bulletin*, *125*(9-10), 1599-1617. https://doi.org/10.1130/b30773.1

Vazquez-Tarrio, D., Borgnier, L., Liebault, F., & Recking, A. (2017). Using UAS optical imagery and SfM photogrammetry to characterize the surface grain size of gravel bars in a braided river (Vénéon River, French Alps). *Geomorphology*, *17*-13. https://doi.org/10.1016/j geomorph. 2017.01.039

Vázquez-Tarrio, D., Piégay, H., & Menéndez-Duarte, R. (2020). Textural signatures of sediment supply in gravel-bed rivers: Revisiting the armour ratio. *Earth-Science Reviews*, *207*, 103211. https://doi.org/10.1016/j.earscirev.2020.103211

Vericat, D., Batalla, R. J., & Garcia, C. (2006). Breakup and reestablishment of the armour layer in a large gravel-bed river below dams: The lower Ebro. *Geomorphology*, *76*(1-2), 122-136. https://doi.org/10.1016/j geomorph.2005.10.005

Welber, M., Le Coz, J., Laronne, J., Zolezzi, G., Zamler, D., Dramais, G., et al. (2016). Field assessment of noncontact stream gauging using portable surface velocity radars (SVR). *Water Resources Research*, *52*(2), 1108-1126. https://doi.org/10.1002/2015WR017906, hal-01501577

Wheaton, J., Fryirs, K., Brierly, G., Bangen, S., Bouwes, N., & O'Brien, G. (2015). Geomorphic mapping and taxonomy of fluvial landforms. *Geomorphology*, *248*, 273-295. https://doi.org/10.1016/j geomorph.2015.07.010

Whiting, P., Dietrich, W. E., Leopold, L. B., Drake, T. G., & Sherve, R. L. (1988). Bedload sheets in heterogenous sediments. *Geology*, *16*(2), 105-109. https://doi.org/10.1130/0091-7613(1988)016<0105:bsihs>2.3.co;2

Wilcock, P., Pitlick, J., & Cui, Y. (2009). *Sediment transport primer, Estimating bed-material transport in gravel-bed rivers, Gen Tech Rep RMRS-GTR-226* (p. 78). U.S. Department of Agriculture, Forest service, Rocky Mountain Research Station.

Wilcock, P. R., & Crowe, J. C. (2003). Surface-based transport model for mixed-size sediment. *Journal of Hydraulic Engineering*, *129*(2), 120-128. https://doi.org/10.1061/(asce)0733-9429(2003)129:2(120)

Wohl, E. E., Bledsoe, B. P., Jacobson, R. B., LeRoy Poff, N., Rathburn, S. L., Walters, D. M. W., & Wilcox, A. C. (2015). The natural sediment regime in rivers: Broadening the foundation for ecosystem management. *BioScience*, *65*(4), 358-371. https://doi.org/10.1093/biosci/biv002

Wohl, E. E., Vincent, K. R., & Merritts, D. J. (1993). Pool and riffle characteristics in relation to channel gradient. *Geomorphology*, *6*(2), 99-110. https://doi.org/10.1016/0169-555x(93)90041-y

Wolman, M. G. (1954). Method of sampling coarse river bed material. *Transactions—American Geophysical Union*, *35*(6), 951-956.

Yu, G.-A., Wang, Z.-Y., Zhang, K., Chang, T.-C., & Liu, H. (2009). Effect of incoming sediment on the transport rate of bed load in mountains streams. *International Journal of Sediment Research*, *24*(3), 260-273. https://doi.org/10.1016/s1001-6279(10)60002-9
References From the Supporting Information

Garcia Lugo, G. A., Bertoldi, W., Henshaw, A. J., & Gurnell, A. M. (2015). The effect of lateral confinement on gravel bed river morphology. *Water Resources Research*, 51(9), 7145–7158. https://doi.org/10.1002/2015WR017081

Leduc, P. (2013). *Etude expérimentale de la dynamique des macroformes sédimentaires en territoires de montagne: cas des rivières en tresses*. phD thesis (p. 255). UJF.

Milhous, R. T. (1973). *Sediment transport in a gravel-bottomed stream*. phD thesis (p. 232). Oregon State University.

Strickler, K. (1923). *Beiträge zur Frage der Geschwindigkeitsformel und der Rauhigkeitszahlen für Ström, Kanäle und geschlossene Leitungen*. Eidgenössisches Amt für Wasserwirtschaft, N°16.