## 1 Introduction

[2] Bed load transport prediction is important for many applications, including river engineering, hazard prediction, and environmental monitoring and management. Sophisticated equations have been proposed in recent decades [*Parker*, 1990; *Wilcock and Crowe*, 2003], and when the quality of the required input data is good and the flow hydraulics are calculated in sufficient detail to take into account shear stress variations, they have been shown to adequately predict transport rates, changes in bed topography, and downstream fining [*Ferguson and Church*, 2009; *Camenen et al*., 2011]. However, the requisite data (detailed grain size distribution [GSD], topography, discharge, or depth) are not always available, and in many practical situations bed load must be computed with limited information and width-averaged river characteristics: The GSD is reduced to a few surface diameters (*D*_{50}, *D*_{84}), often estimated by surface counting [*Wolman*, 1954], the bed topography is assumed to be trapezoidal or rectangular and reduced to a mean width *W* and slope *S*, and the flow is considered uniform at the reach scale (a single water depth *d* for a given discharge and the energy slope equal to the bed slope).

[3] Despite reflecting the reality of many practical situations, the proposed approach of computing bed load with simple models and width-averaged data has been widely criticized for two main reasons: First, equations using limited input data are assumed to be incapable of reproducing the full complexity of transport [*Habersack and Laronne*, 2002], and second, since bed load equations are nonlinear with exponents that may exceed values of 10, width-averaged bed load calculation has been suspected of under-estimating the true bed load flux if there is any local and/or spatial variation in either the bed material size distribution or in the flow hydraulics [*Gomez and Church*, 1989; *Paola and Seal*, 1995; *Ferguson*, 2003; *Bertoldi et al*., 2009; *Francalanci et al*., 2012].

[4] *Ferguson* [2003] demonstrated nonlinearity effects with an analytical model based on the Meyer-Peter and Mueller formulation (derived for local transport in a flume). Using a probability function describing the shear stress variation around its mean value, he showed that additional flux locally induced by high shear stress outweighs the lower flux induced by low shear stress and that, consequently, the total flux (the sum of all local fluxes) should be higher than the flux computed with the averaged shear stress. These effects are illustrated in Figure 1, where *τ ^{*}* is the Shields number, which for diameter

*D*is:

where *R* is the hydraulic radius, *S* is the slope, and *s*=*ρ _{s}*/

*ρ*is the ratio between the sediment and the water density. Figure 1a illustrates a river section, the averaged Shields stress <

*τ** > and computed bed load transport

*q*(<

_{s}*τ**>), and its decomposition in local values

*τ** and

_{i}*q*(

_{s}*τ**); whereas the local shear stress

_{i}*τ*

_{i}* is twice the average value <

*τ** > in the figure, the corresponding computed bed load transport is plotted such that

*q*(

_{s}*τ*

_{i}*) > > 2

*q*(<

_{s}*τ**>). These effects occur because bed load has been shown to be a power function of the shear stress and the value of the exponent is greater than 1. Considering

*q**

_{s}∝τ^{p}, Figure 1b shows that the higher the value of the exponent

*p*, the greater these effects (with a threshold equation of the form

*q**−

_{s}∝(τ*τ**)

_{c}^{p}, these effects would be maximum near the critical Shields stress

*τ**).

_{c}[5] In contrast to the above expectation, most studies comparing bed load equations to measured bed load transport rates report large overestimates instead of under-estimates when equations are used with width-averaged data, especially for gravel bed rivers [*Rickenmann*, 2001; *Barry et al*., 2004; *Bathurst*, 2007; *Recking et al*., 2012]. In addition, because equations derived on the basis of field data are supposed to have a built-in allowance for the effects of spatial variability, they should considerably improve the computation of bed load transport when compared with standard 1D equations; however, many equations based on field data are also site specific, and *Barry et al*. [2004, 2007] did not draw any conclusions about the superiority of one category of equation when compared with field data.

[6] Consequently, the questions this paper aims to answer are: How do the nonlinear effects influence predicted transport rates? Can a single equation, used with either the exact local shear stress or with width-averaged river characteristics, reproduce local transport and width-averaged transport, respectively, or should we consider two distinct families of equations, depending on whether bed load must be computed with local shear stress (as in numerical models) or with width-averaged data? Can we relate nonlinearity effects to the natural variance in flow and bed parameters?

[7] First, flume and field data are presented. Second, they are used with several bed load transport equations (1D capacity equation, 1D surface-based equation, and 2D field-derived equation) to look for evidence of nonlinearity effects. Third, the variance associated with each flow and bed parameter is described, and a Monte Carlo approach is used for statistically investigating (calibrating) the shape parameter of each probability distribution function. Finally, the results are used to discuss the use of equations in field applications.