## 1. Introduction

[2] Measurements of bed load transport rates are fundamental to estimating material transport in a river, yet even defining a representative time period over which to sample is difficult due to the inherent variability and stochastic character of sediment transport. This variability is present over a wide range of scales, from the movement of individual grains [*Iseya and Ikeda*, 1987; *Drake et al.*, 1988; *Nikora et al.*, 2002; *Schmeeckle and Nelson*, 2003; *Sumer et al.*, 2003; *Ancey et al.*, 2008] up to the propagation of dunes and bars [*Kuhnle and Southard*, 1988; *Gomez et al.*, 1989; *Cudden and Hoey*, 2003; *Jerolmack and Mohrig*, 2005], even under steady flow conditions. Computed statistics of instantaneous bed load transport rates (flux) have shown that probability distributions are often skewed toward larger values [e.g., *Gomez et al.*, 1989], implying a high likelihood of extreme fluctuations, the prediction of which is essential for protecting hydraulic structures and assessing the stability of riverine habitat [*Yarnell et al.*, 2006]. It has also been observed that the mean sediment flux depends on the time interval (sampling time) over which the mean is computed, and previous work has suggested that this time dependence is the result of large, infrequent transport events [see *Bunte and Abt*, 2005, and references therein].

[3] An analogous time dependence that has been more thoroughly studied is that of the sedimentary record, where apparent deposition rate (measured from two dated surfaces) diminishes rapidly with measurement duration in virtually all depositional environments [*Sadler*, 1981, 1999]. Models show that this scale dependence is a direct result of the statistics of transport fluctuations [e.g., *Jerolmack and Sadler*, 2007]. In the case of geologic rates the data have been assumed to obey simple scaling over a wide range of time scales; that is, the statistical moments can be fitted as power law functions of scale, with the exponents linear in moment order. This power law relationship provides a value for the Hurst exponent, *H*, which may be used to compare rates at one scale to rates at a different scale via a simple statistical transformation (see also section 5). However, many geophysical processes exhibit multiscaling (or multifractal behavior), which implies that a range of exponents (and not a single exponent) is required to describe the changes in the probability density function (pdf) with scale. Examples include rainfall intensities [e.g., *Lovejoy and Schertzer*, 1985; *Venugopal et al.*, 2006b], cloud structures [e.g., *Lovejoy et al.*, 1993; *Arneodo et al.*, 1999a], river flows [e.g., *Gupta and Waymire*, 1996], river network branching topologies [e.g., *Rinaldo et al.*, 1993; *Marani et al.*, 1994; *Lashermes and Foufoula-Georgiou*, 2007], braided river systems [e.g., *Foufoula-Georgiou and Sapozhnikov*, 1998], and valley morphology [e.g., *Gangodagamage et al.*, 2007]. This rich multiscale statistical structure includes extreme but rare fluctuations (“bursts”) that occur inhomogeneously over time, giving rise to the so-called “intermittency” and leading to a nontrivial scaling of the statistical moments. A prime example of this is the velocity fluctuations in fully developed isotropic turbulence [e.g., *Parisi and Frisch*, 1985; *Frisch*, 1995; *Arneodo et al.*, 1999b].

[4] To the best of our knowledge, bed load sediment transport series have not been analyzed before from the perspective of quantifying how the statistical moments of the series change with scale. In an early study, *Gomez et al.* [1989] acknowledged that the probability distribution of sediment transport rates depends on sampling time (scale) and extended the Einstein and Hamamori distributions to a scale-dependent form, without, however, attempting any scale renormalization. Knowledge of the variability inherent in bed load transport rates at all scales is essential for quantifying material flux, for designing appropriate measurement programs, and for comparison among different data sets and model predictions at different temporal and spatial scales. Also, quantifying the statistical structure of these fluctuations across scales may yield insight into the fundamental physics of sediment transport and provide a set of diagnostics against which to rigorously test competing theories and bed load transport models [see also *Ancey et al.*, 2006, 2008].

[5] One would expect that the statistics of bed load sediment transport would relate in some way to the statistics of the fluctuations in bed elevation. Although river bed elevations have been analyzed much more than sediment fluxes and have been found to exhibit fluctuations across a wide range of scales, in both sandy [e.g., *Nikora et al.*, 1997; *Nikora and Hicks*, 1997; *Jerolmack and Mohrig*, 2005] and gravelly [*Dinehart*, 1992; *Nikora and Walsh*, 2004; *Aberle and Nikora*, 2006] systems, the link between bed topography and sediment flux remains largely unexplored due to the difficulty in simultaneous data acquisition. Establishing a relationship between the statistics of bed elevations and sediment transport rates is important for effective modeling of river bed morphodynamics and also for understanding the physics of sediment transport. More practically, since bed elevation data are far easier to collect than sediment flux measurements, an understanding of how the statistics of the one variable relate to those of the other, at least over a range of temporal scales, could greatly facilitate estimating sediment transport rate in the field.

[6] To address these issues we present here an analysis of data from a unique experimental laboratory setup capable of mimicking transport conditions in the field (see section 3). High-resolution, long-duration time series of sediment transport rates and bed elevation were simultaneously collected in a suite of experiments with a heterogeneous gravel bed. We use the multifractal formalism, originally developed for fluid turbulence [*Parisi and Frisch*, 1985; *Frisch*, 1995; *Muzy et al.*, 1994], to quantify the “roughness” (the average strength of local burstiness in the signal) and the “intermittency” (the temporal variability or heterogeneity of bursts of different strengths) and relate those geometrical quantities to the statistics of sediment flux and bed topography over a range of time scales. (Note that throughout the paper the term “roughness,” as defined mathematically via the strength of local singularities, refers to the signal roughness being that sediment transport rates or bed elevation fluctuations and it is not to be confused with other uses of the term roughness such as bed roughness or hydraulic roughness.) We substantiate the findings of *Bunte and Abt* [2005] that mean sediment transport rate diminishes with increasing sampling time at low bed stress (slightly above critical) but does the opposite for high-transport conditions, and we relate this reversal in trend to the influence of large-scale bed forms. Our analysis also allows characterization of the sampling time dependence of all of the statistical moments, allowing thus the prediction of extremes at small scales from the statistics at larger scales.