Because bed load equations are nonlinear and because parameters describing the flow and the bed can have large variance, different results are expected when integrating bed load over a cross section with respect to spatially variable local data (2D), or when computing bed load from cross-section-averaged data, which reduces the problem to uniform conditions (1D). Evidence of these effects is shown by comparing 1D (flume-derived) equations with 2D field measurements, and by comparing a 2D (field-derived) equation with 1D flume measurements, leading to the conclusion that different equations should be used depending on whether local or averaged data are used. However, whereas nonlinearity effects are considerable for low-transport stages, they tend to disappear for higher-flow conditions. Probability distribution functions describing the variance in flow and bed grain size distribution (GSD) are proposed, and the width-integrated bed load data (implicitly containing the natural variance in bed and flow parameters) are used to calibrate these functions. The method consists of using a Monte Carlo approach to match the measured 2D bed load transport rates with 1D computations, artificially reproducing the natural variance associated with the mean input parameters. The Wilcock and Crowe equation was used for the 1D computation because it was considered representative of 1D transport. The results suggest that nonlinearity effects are mostly sensitive to the variance in shear stress, modeled here with a gamma function, whose shape coefficient α was shown to increase linearly with the transport stage. This variance in shear stress suggests that even for very low flow conditions, shear stress can locally exceed the critical shear stress for the bed armor, generating local armor breakup. This could explain why the bed load GSD is usually very similar to subsurface GSD, even in the presence of complete armor.
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 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 (D50, D84), 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).
 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].
Ferguson  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=ρs/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 qs(<τ*>), and its decomposition in local values τi* and qs(τi*); whereas the local shear stress τi* is twice the average value <τ* > in the figure, the corresponding computed bed load transport is plotted such that qs(τi*) > > 2qs(<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 qs∝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 qs∝(τ*−τc*)p, these effects would be maximum near the critical Shields stress τc*).
 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.
 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?
 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.
2 Data Set Presentation
 In this part, the data set used in the analyses is presented; field data are considered 2D data because they are width integrated, and they are distinguished from flume data that are considered 1D (near uniform) and therefore an analogue for local transport in natural rivers. Note that the term 1D should rigorously correspond to strictly uniform flows (constant depth and bed roughness); however, such flows are almost never fully observed even in the flume, where small bed forms can exist and sediment patches can develop [Meyer-Peter and Mueller, 1948; Dietrich et al., 1989]. Consequently, 1D flow is used here to designate quasi-uniform flows when compared with 2D flows in the field.
 All parameters considered width-averaged are noted between the symbols < >; other standard notations correspond to local values. The flow conditions were discriminated with the transport stage defined by the τ*/τc* ratio [Church and Zimmerman, 2007], where τc* is the critical Shields stress estimated with the following formula fitted to a compilation of field data from the literature [Recking, 2009]:
 Large uncertainties exist on τc* [Buffington and Montgomery, 1997], and a constant and arbitrary value τc* = 0.03 [Parker et al., 2003] or 0.047 [Meyer-Peter and Mueller, 1948] could have been used, but using a dependency with slope is more likely to represent reality [Mueller et al., 2005; Lamb et al., 2008; Recking, 2009; Ferguson, 2012]. The transport stage can be defined for different diameters Di. D50 was used because this diameter is often considered representative. However, because the exponent in equation (2) is near 1, the results would have been slightly changed with other diameters such as D84 (used for plotting the data in Recking et al. ). Nonetheless, this is not important because this choice impacts only the distribution of the results on the figures (the x-axes) and not the results themselves (all equations being used exactly as recommended by their authors).
2.1 1D Data
 Investigating local transport assumes knowledge of the local values. Whereas a few field data sets measure local transport and the associated flow velocity, the bed surface GSD is always highly uncertain and always averaged at the reach scale. Such local data sets would be very difficult to construct because measuring the local bed GSD associated with transport is almost impossible during flooding. A practical way to obtain such information is flume experiments, where the flow can be stopped and the bed can be sampled after each bed load measurement, as done by Wilcock et al. . They produced bed load data in a 0.6 m wide and an 8 m long tilting flume, with recirculation of poorly sorted sediment mixtures and flow conditions allowing partial transport. Five runs were produced with different sand contents, and the data are summarized in Table 1 and are available in Wilcock et al. . These data cover a large slope range (0.06%–2%) and are (to the best of the author's knowledge) the only published flume data that fully document partial transport [Wilcock and McArdell, 1993]. These flows are not strictly speaking 1D, as uniformity implies a perfectly constant water depth and bed roughness. However, as mentioned by the authors, “the sediment bed was essentially planar,” which allows for the hypothesis of near-uniform flows when compared with flows in a natural river reach.
Table 1. Experimental Runs From Wilcock et al. 
 An existing field data set comprising 6319 values (available in Recking ) was expanded with new data from the literature comprising 2614 measurements collected on 24 river reaches. The main characteristics of this new data set are given in Appendix A, and the data (including bed surface GSD) are available online as supplementary material. The complete data set comprises 8940 values collected at 109 river sites and is summarized in Table 2. Most sampling results (when specified) provided width-averaged data for the bed GSD, the flow characteristics (discharge, velocity), and the bed load transport. The following figures provide general descriptions of the data set and were used in the subsequent calculations of 2D bed load transport.
Table 2. Main Characteristics of Field Data
Diameter D50 (mm)
Diameter D84 (mm)
Bankfull depth (m)
Bankfull width (m)
 Figure 2 presents the cumulative distribution of diameter < D84>, slope < S>, and width < W > for the 109 reaches composing the data set. Only 5% are sand bed rivers, and 10% have a D84 smaller than 1 cm (not many published bed load data collected on sand bed rivers were found in the literature); 70% are gravel bed rivers, and 20% are cobble and boulder bed rivers. Slopes span a broad range of values from 0.01 to 8%. Most widths are in the range 0–15 m, but reaches as large as 500 m were also considered. The different ranges obtained for < D84>, slope < S>, and width < W > are consistent, with width and grain size evolving with slope, from large lowland sandy rivers to steep narrow boulder streams [Montgomery and Buffington, 1997; Church and Zimmerman, 2007]. Figure 3 plots slope < S > as a function of < D84 > for the data set considered; the trend roughly follows a power function.
 A selection of 121 GSDs is plotted in Figure 4. Figure 4a plots a selection of 78 GSDs collected in gravel bed rivers (from the Idaho data set; King et al. ) and the corresponding averaged GSD. It shows a similar shape between all curves. Figure 4b plots 42 additional GSDs, including sand bed rivers. Figure 4 indicates that the sand fraction at the bed surface 〈Fs〉 can vary greatly, and these data suggest 0 < 〈Fs〉 < 0.2 for most gravel bed rivers. This is confirmed in Figure 5, where 〈Fs〉 is plotted as a function of 〈D84〉: It is close to 1 for sand bed rivers and rapidly decreases to approximately 0.1–0.15 for gravel bed and cobble bed rivers.
 Figure 6 (that also includes additional data from Pitlick et al. ) indicates that the 〈D84〉/〈D50〉 ratio is approximately equal to 2, which was also found to be representative of gravel bed rivers by Rickenmann and Recking  with another data set. However, the ratio can also differ significantly from 2, and 〈D84〉/〈D50〉 = 2 is associated with a variance following approximately a log-normal distribution with a standard deviation σDR =0.3.
 Figure 7 presents relations between the transport stage and the bed surface < D84 > and the slope. It indicates a decrease of <τ*>/<τc* > with < D84 > following roughly a power law (R2 = 0.45); a similar relation does not seem to exist with the slope. Low <τ*>/<τc* > values for high < D84 > values is not surprising, as the Shields number <τ* > is known to barely exceed 120% of the critical value <τc* > in gravel bed rivers [Parker, 1978; Andrews, 1983; Ryan et al., 2002; Mueller et al., 2005; Parker et al., 2007]. In fact, only for sand and fine gravels can the transport stage be very high [Buffington, 2012].
 Figure 8a plots the unit transport rates < qs > (g/s/m), and Figure 8b plots the averaged dimensionless transport <Φ>, with Φ defined by [Einstein, 1950]:
where ρs is the sediment density. Transport rates cover a wide range; the scatter can be explained by nonlinearity effects (Figure 1), but also by grain sorting [Dietrich et al., 1989; Recking et al., 2009b] and bed form propagation [Gomez et al., 1989; Recking et al., 2009a]. It is interesting to note that the threshold value <τ*>/<τc* > =1 (computed for D50) corresponds to somewhat considerable transport and <τ*>/<τc* > =2 approximately delimits two groups with a change in trend, a result which is consistent with previous analyses [Parker et al., 1982; Buffington, 2000].
 To complete this description of the field data, the bed morphology should be presented. Unfortunately, except for a few cases [Hassan and Church, 2001; Church and Hassan, 2002], an exact description of the reach used for measurements was usually not included in the publications. Mostly, only general descriptions of river morphology were provided, but available information suggests that straight reaches were frequently chosen for bed load measurements. Nevertheless, the surrounding reach morphology of some sites may exhibit substantial topographic variation (e.g., pool-riffle or step-pool channels).
3 Looking for Evidence of Nonlinearity Effects
 In this part, the flume and the field data are used with bed load equations to look for evidence of nonlinearity effects. Bed load equations are used exactly as recommended by their authors.
3.1 From the Flume to the Field
 This part aims to analyze whether averaging the data produces under-prediction when 1D equations are used in the field.
 Sixteen flume-derived 1D bed load transport equations were compared with a large field data set in Recking et al. . The conclusion of this study was that for the lowest transport stages (corresponding to partial transport, with τ*/τc* < 1 for the coarser fractions), these threshold equations predicted zero transport whereas the measured transport was nonzero. When the nonzero transport computations were considered, the results showed overestimation, for all equations, often by several orders of magnitude, which is not in accordance with the expected under-estimation induced by nonlinearity. This resulted in a generally poor performance of most of the equations tested, except for high-flow conditions (when τ*/τc* > 2, which corresponds to sand and fine gravels in Figure 7a). One reason potentially responsible for overprediction is that these equations do not account for the changing critical shear stress with slope.
 To overcome this threshold problem (zero prediction and slope effects), a 1D bed load equation expressed as a function of τ*/τc* was fitted in Recking  on a large flume data set obtained with near-uniform sediments (compilation comprising most of the data that have served to build the equations reported in the literature, available as supplementary material in Recking ). This equation accounts for variation of the critical Shields stress with slope and is presented in section B1. When this 1D equation (used with the median diameter) is compared with the 1D flume data from Wilcock et al. , the results plotted in Figure 9 can be deemed satisfying, considering that there was no calibration; most particularly, the trend is very good, and there is no clear evidence for over- or under-prediction.
 On the other hand, when this equation is compared with the field data, the results plotted in Figure 10 clearly confirm underprediction when the transport stage is τ*/τc* < 0.7, as expected by nonlinearity effects. However, like the other equations tested in Recking et al. , this equation overestimates transport rates for higher-flow conditions and gives acceptable results only when τ*/τc* > 2 approximately. The possible reason for overestimation in the range 0.7 < τ*/τc* < 2 is that these equations do not correctly account for substantial form losses for low relative flow depths [Rickenmann, 2001]. The results of Figure 10 are improved in the range 1 < τ*/τc* < 2 when the Meyer-Peter and Mueller  shear stress correction is used; however, it also considerably increases under-prediction in the τ*/τc* < 1 range. Another reason for overestimation is that all these equations were derived to match as well as possible the transport capacity of nearly uniform fine materials in flume experiments [see Recking et al., 2008 for a review of these data]. In these conditions, each increment in shear stress was systematically balanced by an increment in transport. In gravel and cobble bed rivers, sediments are not always available for transport because of supply limitations and hiding effects, and small increments in shear stress may not be compensated by an immediate increment in bed load, which can explain overestimation [Bathurst, 2007; Recking, 2012].
 To overcome the above mentioned problem, fractional equations were developed for computing the transport of what is actually present on the bed surface [Parker and Klingeman, 1982; Parker, 1990; Wilcock and Crowe, 2003]. Among these equations, the Wilcock and Crowe  equation is considered a truly surface-based relation [Parker, 2008] because it was derived from the flume experiments of Wilcock et al. , with the exact surface GSD being measured immediately after each flow event. This equation (presented in section B2) was used as suggested by the authors, with the GSDs sampled at 1-ψ intervals, and by computing the grain shear stress with τ = 17(SD65)1/4U3/2 [Wilcock et al., 2009]. A comparison with the field data set (restricted to 82 reaches and 6239 values because the full GSD was not available for all data) showed no significant overprediction and a tendency for under-prediction at low shear stresses, in agreement with expected nonlinearity effects (Figure 11). Similar results were obtained by Gaeuman et al.  who compared the Wilcok and Crowe equation to bed load measurements of the Trinity river.
 To conclude, the above analysis suggests that the variance in flow and bed parameters produces under-prediction at low-transport stages when 1D equations are used in the field.
3.2 From the Field to the Flume
 A second way to verify the effects of nonlinearity is to use a 2D field-derived equation for computing local transport. By fitting the width-integrated transport measurements, these equations have a built-in allowance for the effects of spatial variability, which should improve the computation of field bed load transport with width-averaged input data, when compared with standard 1D equations. On the other hand, if they implicitly compensate for spatial variability, these equations should logically overpredict transport when compared with 1D measurements because of nonlinearity effects. This is what was tested here by comparison with the flume data from Wilcock et al.  (assuming that these data are representative of 1D transport in the field).
 Several 2D field-derived equations have been proposed in the literature [Parker et al., 1982; Barry et al., 2008; Recking, 2010]. Here, an equation (presented in section B3) proposed in a previous paper by the author [Recking, 2013] was used because it requires only a few parameters and permits a direct comparison with flume data. This equation was compared with the field data set (restricted to 5735 values not used in its construction, collected at 73 sites, for blind testing), and the results plotted in Figure 12 indicates only a slight under-prediction at very low Shields stress ratios. The comparison with a selection of river reaches did not indicate any particular dependency on the bed morphology (Figure 13). On the other hand, the comparison with 1D flume data confirmed overprediction for the low-transport stages, as expected by nonlinearity effects (Figure 14).
 To conclude this part, only low-transport stages are affected by nonlinearity effects. Assuming that the flume data are representative of local flows, this analysis suggests that the variance in flow and bed parameters produces under-prediction when 1D equations are used in the field. Field-derived equations compensate for these effects, but by contrast, they overpredict bed load transport when they are used for computing local transport with local flow values (e.g., in a numerical model). The following section aims to fill the gap between 1D and 2D transport by investigating the variance associated with each parameter.
4 Investigating the Missing Dimension
 The difference between 1D bed load computation and 2D bed load measurements results from variance in shear stress and bed properties. Each bed load value of the data set is by construction the average of several bed load values measured locally and produced by the variance in flow and bed GSD (as illustrated in Figure 1a); consequently, bed load data implicitly include the variance in flow and bed parameters, which evolve in both space and time. On the other hand, this variance is absent from the associated flow and bed data (discharge measured not simultaneously and often at a gauging station, in another location, and with a single mean GSD). This variance, which is present in bed load data and absent in the flow and bed data used in the equations, is what Ferguson  called the “missing dimension.”
 The only way to make the use of a 1D equation consistent for comparison with measured 2D field data would be to “artificially” reproduce the natural flow and bed variability. This is done in this part of the paper, using the averaged input data in the 1D Wilcock and Crowe equation, with their variance described by probability distribution through a Monte Carlo approach.
4.1 Probability Distribution Functions Describing the Variance in Flow and Bed Parameters
 The depth 〈d〉 can vary considerably in an irregular cross section, ranging from near zero close to the banks to a maximum at the thalweg, and is linearly related to shear stress for a given slope 〈τ〉 ∝ 〈S〉〈d〉. The variance associated with longitudinal slope across a given cross section is difficult to evaluate; however, as pointed out by Ferguson , available field studies have actually shown a positive correlation between d and τ (local deviations from τ ∝ d tending to cancel each other out), leading to a similar frequency distribution for d and τ. This is why in the following the variance was considered directly for the shear stress 〈τ〉 instead of separate variance for 〈d〉 and 〈S〉. This variance was described with an asymmetric gamma probability function, as already proposed in other works [Paola, 1996; Nicholas, 2000; Bertoldi et al., 2009]:
where , 〈τ〉 is the width-averaged bed shear stress, and α is a parameter describing the width of the distribution. The lower its value, the larger the variance in τ. A value α=1 was found to be a limiting value for highly irregular cross sections [Paola, 1996; Nicholas, 2000], for instance, in braided streams. It also predicts a shear stress in the range 0 < τ/〈τ〉 < 5 (Figure 15), which is approximately what Ferguson  obtained with his probability function when the shear stress is below its mean value of over 80% of the cross section. Values greater than 5 would be representative for single-thread irregular channels, and α tends toward infinity for a rectangular cross section.
 Fractional equations require a complete definition of the GSD. Instead of discussing the variance associated with each size class, GSD was reduced to three parameters: the sand fraction on the bed surface Fs, the median diameter D50, and the D84/D50 ratio. This model, presented in Table 3, was obtained by considering the proportionality ratios between each size class measured in the compilation of 78 GSDs, plotted in Figure 4a. Figure 16a shows three examples of measured GSDs and their approximation with the model, and Figures 16b and 16c indicate that the difference between the measured and computed percentage for each fraction of the whole data set is very weak. There is no intention here to reduce all natural GSDs to these three parameters, but it allows very realistic GSDs to be built.
aInput data are D50, D84, Fs (the sand fraction), and Dm if Fs = 0 (minimum grain size). Cn is a coefficient, and Di is the upper limit of the size class. For constructing a GSD, Fs and Cn are used for computing the limits of each size class (column 3) and the % in each class (column 2).
Dm = 2 mm but if Fs = 0 Dm = the minimum diameter of the GSD
3.3 replaced by 8 if Dm > 2 mm
2.5 replaced by 1.5 if Dm > 2 mm
 Only a few studies have investigated the variability of GSDs in a given reach [Church and Kellerhalls, 1978; Crowder and Diplas, 1997], and defining the variance for Fs, D50, and D84/D50 for the purpose of this study is not trivial. Considering the central limit theorem, the distribution of the means of a series of samples should be normally distributed regardless of the underlying distribution of the sample; therefore, it is hypothesized that for a given mean value 〈D50〉, diameters D50 vary randomly following a normal distribution (Figure 17). Several values were considered for standard deviation σD. The value 〈D84〉/〈D50〉 = 2 and the associated normal distribution (σDR = 0.3) observed between sites (Figure 6) was hypothesized as always valid, including for different locations in a given site. Fs = 0.1 was considered on average for gravel bed rivers (Figure 5); however, sand can be either uniformly distributed over the surface or concentrated in local patches [Buffington and Montgomery, 1999a; Dietrich et al., 2006], which can be described with an asymmetric beta function B(x1,x2), with an average set to x1/x2 = 0.1 (where x1 and x2 are the shape parameters of the function). Figure 18 plots B(x1, x2) for different values of x1 and x2. Low coefficients (x1 = 0.001; x2 = 0.01) simulate near-zero sand everywhere and high local sand concentration (patches); on the other hand, high coefficients (x1 = 100 and x2 = 1000) simulate a more uniform distribution of the sand fraction over the surface.
 Limited evidence of a clear correlation between grain size and depth led Ferguson  to consider two limiting scenarios: purely random patchiness, and perfect fining upward as in classic models of meander point bars. Here, only the case of near-straight reaches was considered, with no or little correlation between grain size and depth. Flows over typical morphological units such as riffle, pools, and bars are considered in the discussion.
4.2 A Monte Carlo Approach for Calibrating the Probability Functions
 In this section, the width-integrated bed load data (implicitly containing the natural variance in bed and flow parameters) were used to calibrate the probability functions describing the variance in flow and bed parameters. This approach is in a sense similar to that of Chiari and Rickenmann , who used bed load measurements for back-calculating the macroroughness of several alpine rivers; however, whereas they used a numerical model, a Monte Carlo approach was used here.
 For a given set of width-averaged input data (say, 〈Fs〉, 〈D〉, 〈τ〉), the Monte Carlo approach consists in performing a large number of random draws from the probability distribution of each input parameter to construct numerous sets of local values (Fs, D, τ) about the means. In the second step, each set of local values is used in the 1D bed load equation to compute the associated local bed load transport qs. Assuming the probability distribution functions are correct, averaging these local bed load values should retrieve (in a statistical sense) the measured (and width-averaged) bed load transport qs meas.
 For each flow range (considered through the ratio ) and given values of 〈S〉, 〈Fs〉, 〈D50〉, and 〈D84〉/〈D50〉, the different steps are:
 computation of 〈τ*〉 for each transport stage (considered for values 0.3, 0.5, 0.7, 1, 1.3, 1.5, 2, 5, and >5) knowing (given by equation (2)), from which we deduce 〈R/D84〉 = 1.65 < τ* >/< S > (Eq. (1));
 calculation of the flow velocity 〈U〉 for 〈R/D84〉 with the Ferguson  flow resistance equation (this equation was chosen because it has been shown to give satisfying values when compared with a large data set in Rickenmann and Recking );
 calculation of the appropriate mean grain shear stress 〈τ〉 = 17(〈S〉〈D65〉)1/4〈U〉3/2 [Wilcock et al., 2009]; <D65 > is given in millimeters and deduced from the model of Table 3;
 construction of several sets of local values for τ and GSD (with the model of Table 3) by random draws in the probability distributions of 〈τ〉 and 〈Fs〉, 〈D50〉, 〈D84/D50〉; computation of the associated bed load qs(τ,GSD) with the Wilcock and Crowe equation. Large data sets (N = 5000) were constructed to ensure a stable solution for the computed bed load probability distribution;
 calculation of the average value ;
 comparison of 〈qs(τ,GSD)〉 with qs meas.
4.3 Results of Simulations
 The simulations were tested with different values for α (for τ), σD (for D50), and x1 (with x2 = 10x1 for < Fs > = 0.1). Figure 19 compares the qs(〈τ*〉)/qsmeas ratios (where qs meas is the measured bed load transport and qs(〈τ*〉) is bed load computed with the Wilcock and Crowe equation used with average input data, as shown in the right panel of Figure 11) with qs(〈τ*〉)/< qs(τ*) >, where < qs(τ*) > is the average bed load deduced from the Monte Carlo computation. Adequate probability distributions should make it possible to fit < qs(τ*) > with qs meas (in a statistical sense).
 No stable solution could be obtained with σD > 0.3 (leading to near-zero D50), and simulations were considered with 0.1 < σD < 0.3. This range is consistent with Segura et al. , who measured σD in the range 0.1–0.36 on the basis of multipebble counts [Wolman, 1954] involving 2500–4700 particles on three reaches of the Williams Fork River (Colorado, USA). It is also consistent with measurements made on a large gravel bar on the Gunnison River [Barkett, 1998], a tributary to the Colorado River, consisting of 28 separate Wolman counts of 100 particles each (total of 2800 particles), and for which σD was 0.28 (John Pitlick, personal communication).
 Figure 19a shows that it was not possible to match the measurements for all flow conditions when setting a constant α-value (α = 1 actually seems to work fairly well as a lower limit to the data). As a consequence, α was empirically varied with the transport stage in Figures 19b and 19c. All combinations (α, σD, x1) were tested with within the range [0.3–100], for constant slopes < S > and < D50 > (Figure 19b) and also with the transport stage linked to the slope and GSD (Figure 19c) through relations plotted in Figures 3 and 7a. Whatever the hypothesis, the best fit between computed (with the Monte Carlo approach) and measured bed load was obtained by varying α linearly with a function given in the form (Figures 19b and 19c):
 The value of ξ in equation (5) is slightly sensitive to the variance in GSD. A value of 5 best fits the measured bed load transport with σD =0.1 (Figure 19c), and a value of approximately 8–10 (not shown in the figure) would be more appropriate for a higher variance in D50 (σD =0.25). Results plotted in Figure 19c indicate a weak dependency on whether sand is considered uniformly displayed over the surface (x1 = 100, x2 = 1000) or is concentrated into patches (x1 = 0.001, x2 = 0.01).
 The conclusion from these simulations is that predictions are weakly sensitive to the variance in GSD (Figures 19b and 19c) but very sensitive to the variance in shear stress (Figure 19a). Such an increase in the parameter α with flow strength was an expected result [Nicholas, 2000], and values within the range 2 < α < 10 are consistent with results for single-thread channels found by Tunnicliffe et al. .
5.1 Consequences for Field Application
 The above results suggest that all models are not adapted to all situations. 1D equations used with width-average input data are likely to under-estimate width average bed load transport. The Monte Carlo approach (used with appropriate probability distributions) can be used for compensating the “missing dimension.” It is illustrated hereafter with the example of the Lochsa River (using hydraulics and bed load data presented in King et al. ).
 For the given bed characteristics (Fs = 0.03, D50 = 0.126 m, D84 = 0.28 m, S = 0.0023), the procedure has consisted in defining, for each of the 71 runs characterized by a measured velocity U and depth d, the transport stage <τ>/<τc>, the GSD, the probability distribution for all parameters (no calibration was used here, and α=5, σD = 0.1, σDR = 0.3, x1 = 100, and x2 = 1000 following the above results), and in building bed load data with the Wilcock and Crowe equation (3000 values were computed for each run), which were averaged and compared with the measured bed load.
 The results presented in Figure 20 indicate that taking into account the variance in bed and flow parameters can greatly improve prediction when compared with a direct computation of the width-averaged data. The inconvenience of this approach is that thousands of calculations are needed for each width-averaged input data considered.
5.2 Local vs. Averaged Shear Stress
 Testing the equations (Figure 11) and the Monte Carlo results (Figure 19) showed that the under-prediction is greater for low-transport stages and decreases with increasing shear stress, as Ferguson  also concluded. This can be explained by a reduced variance in the shear stress when the flow increases, because depth variations with the local bed topography may become relatively negligible with regard to the mean flow depth (this is particularly true for moveable beds becoming flatter with increasing transport stage).
 Equations (4) and (5) were used to compute the distribution of the local transport stage τ/<τc>, for different values of the mean transport stage <τ>/<τc>. The results, plotted in Figure 21, indicate that for <τ>/<τc > ratios as low as 0.3, the shear stress may locally be higher than the critical shear stress for mobility of the bed surface (τ/τc > 1). This means that local armor breakup can always exist to some degree, exposing the subsurface material to the flow. This finding is consistent with the observation that even in the presence of a coarse armor, with a zero sand fraction at the bed surface (Fs = 0), the bed load GSD is always much finer than the surface GSD and equivalent to the subsurface GSD. This was, for instance, the case for several of the Idaho streams [King et al., 2004], for which bed load was measured for very low transport stages. However, the bed load material may also include upstream sediment supply that is not accounted for in the current analysis. Consequently, the sand fraction measured at the bed surface at rest could be an incorrect indicator of sediment availability for bed load computation.
5.3 Several Questions to Be Investigated Further
 The first question is how much are the above results dependent on the Wilcock and Crowe equation, used here as representative of 1D transport? To answer this question, another 1D equation was considered. In work not shown here, an attempt was made to adapt the 2D Recking  equation (section B3) to 1D transport by introducing a correction coefficient ζ < 1 such that:
 According to Figure 14, the function ζ(τ*, Fs) must converge to 1 when the sand fraction Fs converges to 1 or when the transport value is high (when all grain classes are moving, which can be expressed, for instance, with reference to the threshold conditions τc* for the coarser fraction considered through diameter D84 in equation (2)). Calibration with the Wilcock et al. flume data gave:
where τ* and τc* were calculated with equations (1) and (2) for Di = D84. This function performed as well as the Wilcock and Crowe equation for the bulk bed load transport when compared with the flume data of Wilcock et al., it produced results that are very similar to Figure 11 when compared with the field data and gave a very similar result for α (i.e., linear variation with <τ*>/<τc*>) when used in the Monte Carlo approach. Consequently, it is not the Wilcock and Crowe equation that should be questioned here, but the flume data from which it was derived: Are they representative of local transport in the field? Very precise local field measurements would be needed to confirm these results, not only for the bed shear stress and bed load transport, but also for the associated bed surface GSD (such data do not yet exist to the best of the author's knowledge).
 A second question concerns the effects of the local bed morphology. The case of near-straight reaches was considered here, with no or little correlation between the depth and GSD. Developing probability functions that are typical of morphological units such as riffle, pools, and bars [Buffington and Montgomery, 1999b; Bunte and Abt, 2001] may necessitate a more sophisticated approach, linking these two parameters [Ferguson, 2003]. However, investigating the correlation between depth and grain size is challenged by uncertainties on the variance in GSDs, which may change with hydraulic roughness [Buffington and Montgomery, 1999a] or other external factors, such as the transport stage: The streambed textures are typically documented at low flow when the bed is at rest and likely reflect preferential transport by secondary flows, during hydrograph recession. For instance, there is no reason why the bed patchiness measured on the bed at rest would be representative of the bed during flooding.
 To conclude, the investigation proposed in this paper considered variations around the mean input values, which themselves were assumed to be exact. However, an erroneous estimation of the width-averaged input data is likely to strongly impact the results. This aspect should be considered in further analysis. It should be noted that the Monte Carlo analysis can also be a useful tool for estimating the error on bed load prediction, with consideration of uncertainties attached to the measured (or estimated) field data [Wilcock et al., 2009].
 Because they are nonlinear, bed load transport equations should logically under-predict observed bed load transport when they are used with averaged (flow and bed) input data [Gomez and Church, 1989; Paola and Seal, 1995; Ferguson, 2003; Bertoldi et al., 2009; Francalanci et al., 2012]. However, this expectation has not been clearly demonstrated in prior studies comparing predicted transport rates with observed values, which show overprediction, usually by several orders of magnitude [Rickenmann, 2001; Barry et al., 2004; Bathurst, 2007; Recking et al., 2012].
 In this paper, evidence of nonlinearity was shown by comparing 1D (flume-derived) equations and 2D field measurements (Figures 10 and 11) and by comparing a 2D (field-derived) equation with 1D flume measurements (Figure 14). Comparison with a 1D nonthreshold equation derived for uniform sediments [Recking, 2010] and with no shear stress correction produced under-prediction for low-transport stages only (when τ*/τc* < 1), overprediction in the range 1 < τ*/τc* < 2, and near adequate transport for higher transport stages (Figure 10). Overprediction in the range 1 < τ*/τc* < 2 is typical of other 1D threshold equations [Recking et al., 2012]. The nonthreshold fractional equation from Wilcock and Crowe used with shear stress correction also under-predicted bed load for the low-transport stages (when τ*/τc* < 1) as expected by nonlinearity effects and adequately reproduced bed load for higher transport stages (Figure 11). On the other hand, the field-derived 2D equation [Recking, 2013] that implicitly takes into account the variance in flow and bed parameters correctly predicted the field data (Figure 12) but overpredicted 1D bed load transport measured in the flume by Wilcock et al.  (Figure 14).
 Under-prediction exists because each bed load data is by construction the average of several local bed load values containing the variance in shear stress and bed GSD, whereas such variance is absent from the mean input values used in bed load equations. The GSD was reduced to three parameters (Fs, D50, D84/D50), and probability distribution functions were proposed to describe the variance associated with the flow and the bed. Then the Wilcock and Crowe equation was compared with the field bed load data in a Monte Carlo approach, in order to calibrate the parameters of these functions. The conclusion is that bed load prediction is weakly sensitive to the variance in bed GSDs but is highly sensitive to the variance in shear stress (Figure 19). The shear stress was modeled by a gamma function, whose shape coefficient was found to vary linearly with the transport stage <τ*>/<τc*>. The variance in shear stress suggests that local shear stress can exceed the critical shear stress for the bed armor even for very low flow conditions (Figure 21); this may explain local armor breakup and why the bed load GSD is frequently equivalent to the subsurface GSD (much finer than the surface GSD).
 Several aspects of this study remain to be investigated. In particular, little is known about the variance in GSDs for a given reach, and new data (collected in the flume or in the field) are needed for comparison with the Wilcock et al.  data used here as representative of 1D transport.
This table presents the main characteristics of rivers used in the field data base, in complement to other data already available in Recking 2010. All the data are available online as supplementary material.
Appendix A: Main Characteristics of the Additional Data
Sediment discharge measured by trapping sediment in a mesh-covered hopper and pumping it into a weighing tank. Continuous water stage record. Discharge calculated from velocity (and depth) determined by means of floats.
Gaweesh and Van Rijn , Abdel-Fattah et al. 
Nozzle (width = 0.096 m; height = 0.055 m; length = 0.085 m; rear width = 0.105 m; rear height = 0.06 m) connected to a nylon bag with a mesh size of 150 or 250 10–6 m. Velocity and depth measurements.
 Appendix B presents bedload equations used in the paper: B1 is a capacity equation that best fits flume data having near-uniform sediments; B2 is a flume derived surface based equation; B3 is a field derived surface based equation.
B1 1D Equation for Uniform Sediments [Recking, 2010]
Critical Shields stress corresponding to grain entrainment [-].
 This study was supported by Irstea and the ANR GESTRANS project granted to the author. It would not have been possible without the USDA and the USGS data sets, as well as all the other published data sets presented in Recking . The author is particularly grateful to Mary-Ann Madej from the USGS, Randy Klein from Redwood National and State Parks, Dieter Rickenmann from WSL, Mike Church from UBC, Ian Reid from Loughborough University, Sandra E. Ryan-Burkett from the US Forest Service, John Pitlick from Boulder University, and François Métivier from IPGP for having generously and spontaneously agreed to share their data. The author would also like to thank Rob Ferguson, Mike Church, and other anonymous reviewers who greatly contributed to this paper by providing helpful reviews of an earlier version of this manuscript. Thanks are extended to John Buffington (Associate Editor), who contributed to this paper by providing additional reviews. Special thanks are addressed to John Pitlick for discussions and very helpful comments.