## 1 Introduction

[2] Quantification of sediment transport in streams has been dominated by the capacity approach, in which focus is on the overall ability of a stream to transport sediment. This approach was first developed for sand bed rivers, but may not be suitable for gravel bed rivers, where stream capacity can change with sediment supply, sediment composition, or state of bed armoring [e.g., *Hassan et al*., 2008a]. Characterization of individual particle displacement patterns in streams is an alternative approach for estimating sediment transport and channel stability. Since it is not currently possible to measure the deterministic forces driving a particle as it interacts with streamflow, the bed, and other particles, a stochastic approach to movement is used to understand bulk transport properties and their link to channel stability and morphology. Bedload transport in rivers consists of movements of individual particles. The motion of grains is not continuous, but consists of a series of steps and rest periods due to the irregular bed surface boundary and the turbulent nature of flows [e.g., *Einstein*, 1937; *Schick et al*., 1987; *Hassan et al*., 1991; *McEwan et al*., 2004; *Lajeunesse et al*., 2010], either under the condition of partial or full mobility [*Wilcock and McArdell*, 1993].

[3] Probabilistic models of transport generally begin with the concept that individual particle “steps” have random lengths and the sum of these step lengths represents total travel distance. Travel distance for these random walk models is a random variable itself and can be described by a probability density that incorporates our uncertainty in exactly how far a particle will travel through time. Stochastic theory shows that long term transport for particles undergoing random walks is governed by advection-diffusion equations. There are a number of reasons for which deviation from classical diffusive-type transport occurs. First, we consider sources of nondiffusive or “anomalous” transport while particles are in motion. These include particle steps with long-range correlation, particle steps with extreme deviation from “average” transport behavior, and deterministic step components such as harmonics. Even if mobile particle transport is diffusive, the long term bulk motion of, say, a group of tracers, may still be anomalous if the distribution of particle immobile periods is sufficiently wide (or heavy-tailed) such that the slowdown in particle virtual velocities cannot be described by an average value.

[4] A key distinction in applicability of governing partial differential equations for tracer dispersion is that of thin- versus heavy-tailed particle step lengths and rest periods. Specifically, the commonly used advection-diffusion equation (ADE), which describes diffusive change in particle concentration with time as a function of average velocity and spread around that average, is only valid for thin-tailed step and rest distributions [*Ganti et al*., 2010]. Classical diffusive spreading implies that the mean square displacement (MSD) of particle spread is a linear function of time. The standard form of the ADE applies only to this case. Probability density functions (PDFs) for random travel lengths or rest periods with heavy tails decay much more slowly with increasing travel length or rest time than thin tails. The dispersion associated with heavy tails can show an MSD of particle spread that deviates from linear, as characterized by either super-diffusive or subdiffusive anomalous behavior. If the MSD is a power-law function of time with an exponent greater than unity, the process is super-diffusive, in which case space-fractional advection-dispersion equations are well suited to describing the temporal evolution of relative tracer concentrations. If, on the other hand, the exponent is less than unity, the process is subdiffusive, in which case time-fractional advection-dispersion equations reproduce relevant advective-dispersive characteristics [*Ganti et al*., 2010]. The important point is that it is only the tail characteristics of the travel distance and waiting time PDFs, and not their overall shape, that determines the long-term scaling behavior of a sum of particle steps with “random” length and waiting time.

[5] Historically, it has been assumed that distributions of sediment step lengths and rest periods, which together affect the overall transport distribution, have well-defined mean values surrounded by a characteristic amount of variability, suggesting that an advection-dispersion (diffusion) equation should reproduce the relevant features of sediment transport. Recent work suggests that this may not be the case. For example, *Ganti et al*. [2010] argued that the evolution of a patch of nonuniform sediment may be super-diffusive as a result of mixtures of step length distributions that arise due to grain size variation. To explore the applicability of heavy-tailed distributed step lengths, *Bradley et al*. [2010] reanalyzed the classic data collected by *Sayre and Hubbell* [1965; see also *Hubbell and Sayre*, 1964] for sand-bed rivers. They developed a model similar to that proposed by *Sayre and Hubbell* [1965] but assumed a heavy-tailed distribution of particle step lengths. To improve the performance of their model, they partitioned the tracers into a detectable mobile phase and an undetectable immobile phase. They concluded that super-diffusive models (in the form of a space-fractional advection dispersion equation) match the observed plume shape and growth rates better than classical step length models. These results contrast with detailed observations in a gravel bed irrigation canal under uniform flow conditions, which suggest that anomalous super-diffusive transport exists only as a preasymptotic “local” range within which correlated particle motions dominate transport [*Nikora et al*., 2002]. This is followed by an intermediate period during which tracer dispersion becomes subdiffusive as the effects of particle immobilization dominate [*Bradley et al*., 2010, *Nikora et al*., 2012]. There is evidence suggesting that wide distributions of particle immobile periods slow down the virtual velocity of coarse sediment transport [*Ferguson et al*., 2002; *Nikora et al*., 2002; *Martin et al*., 2012]. Recent laboratory flume studies designed to study particle step and rest characteristics [*Martin et al*., 2012] have yielded super-diffusive step length distributions that persisted after sediment was well mixed across the flume. More detailed studies of particle dispersion suggest, on the other hand, that anomalous transport behavior may be a result of periodicities in particle motion rather than super-diffusive step lengths [*Furbish et al*., 2012], and that particle velocities, which incorporate effects of both step time and length, are exponential-like and do not have heavy tails [*Roseberry et al*., 2012].

[6] The central question addressed by this paper may be posed as follows. What happens to coarse particles subject to intermittent flow capacity in natural streams? Older, field-based research shows that the distribution of step lengths of individual particles follows the Einstein-Hubbell-Sayre compound Poisson model or a simple Gamma, or exponential distribution for small displacements [e.g., *Hassan et al*., 1991; *Ferguson and Wathen*, 1998; *Ferguson et al*., 2002; *Pyrce and Ashmore*, 2003, 2005]. These studies, however, did not consider the possibility of power-law distributed.

[7] Our objective is to identify the nature of downstream dispersion of tracer particles at long timescales. Many studies focus on transport characteristics while particles are moving. However, long-time transport properties of gravel in streams are a function of both mobile transport characteristics and the often long periods during which particles are immobilized at the surface or through burial. Although we are interested in the characteristics of gravel transport when gravel is actually moving, and also the long-term bulk transport characteristics that include effects of both motion and nonmotion, we can realistically only measure the latter in the field. Laboratory experiments of particle transport allow for real-time measurement of particle velocities, entrainment and rest times, but these measurements are not currently feasible in long-term field studies. Tracer location is typically surveyed at convenient intervals, and 100% recovery of tracer stones is rarely achieved during a monitoring event. Thus, the number of transport events for individual particles is known precisely only if it is zero. Particles that have moved since previous surveys may have taken one or more steps during one or more flow events. When particles are not observed during a monitoring event, they may be buried deeply within the surveyed reach or be located beyond the survey area. This can only be resolved if the tracer is identified in subsequent monitoring events.

[8] The ultimate purpose of characterizing the statistical nature of steps and rests in streams is the identification of, and elucidation of the underlying physics contained in the governing equation for coarse sediment transport. This requires distinguishing between step lengths or travel distances that follow distributions such that *P*(*X* > *x*) ~ *Cx*^{− α}, where *P*(⋅) denotes probability, *X* is the travel distance, *C* is a proportionality constant, and the tail parameter *α* is less than two for heavy-tailed steps and greater than or equal to two for thin-tailed steps.

[9] We now refine the question posed above. When coarse particles move as bedload in rivers with all the complexities associated with a natural setting, do their travel distances follow thin- or heavy-tailed distributions? To achieve our goal, we used magnetically and passive integrated transponders (PIT) tagged particle field data collected under a range of bed states and bed morphologies.