## 1 Introduction

[2] As in many fields with stochastic approaches to particle transport, the conceptual model used to represent sediment transport through time in gravel bed rivers equates the distribution of overall travel distance at time *t* of a single particle as the sum of a random number of randomly sized step lengths. This random walk concept is chosen because, although the physics of particle motion is not random, we are currently only able to measure the bulk properties of moving particles rather than predict exact travel distances or times. These bulk properties take the form of probability densities, which give the likelihood of the set of possible step length distances in a given time step. We assume that the transport statistics (mean travel distance and the spread about the mean) of a single particle represent the actual distribution of a cloud of particles released in a channel at the same time. While it is possible to include a zero-travel distance due to particle deposition on or within the bed in the collection of likely travel distances, it seems that sediment residence (also waiting or rest) time in the bed plays such a significant role in overall transport that from a modeling perspective, particle motion characteristics and sediment residence time in the bed must be considered individually. Residence time accounts for the difference between instantaneous particle velocity and streamwise virtual velocity, which has been observed to decrease through time [*Ferguson et al*., 2002; *Ferguson and Hoey*, 2002; *Haschenburger*, 2011]. Consideration of sediment residence time in the bed of stream channels is useful on its own because it impacts aqueous chemical exchange and controls the storage and release of contaminants and nutrients in the river. Sediment residence time in channel substrate is a function of entrainment physics, bed morphology, and characteristics of vertical mixing in the bed, but our inability to directly measure the forces acting on individual grains leads to characterization of particle immobile periods by residence time probability distributions [*Hassan et al*., 2013].

[3] There is a large class of probability models that are used to predict transport properties given a jump length distribution and a residence time distribution. One general model, known as a continuous time random walk (CTRW), incorporates many of the stochastic models of sediment transport proposed since *Einstein* [1937] (details below) and can be used one of two ways. First, if the exact jump length and residence time distributions are known, then the analytical form of their densities can be plugged into a master equation that produces the probability density describing travel distance through space and time [*Scher and Lax*, 1973]. Second, if only the tail properties of the jump length and residence time distributions are known—that is, whether the tails of the distributions decay at exponential or power law rates—then asymptotic (long-time) travel distance can be computed. In either case, particle transport that is well described by discrete CTRW models is governed in the long-time limit by advection-dispersion equations (ADE). The difference between physical regimes that are modeled using preasymptotic or transient equations versus long-time asymptotic equations has been identified for sediment transport using the terms “local,” “intermediate,” and “global” diffusion regimes [*Nikora et al*., 2001; *Nikora et al*., 2002]. That work calls for experimental and theoretical evaluation of transition between the regimes [*Nikora et al*., 2002].

[4] Radio transmitter tagged stones [*Ergenzinger et al*., 1989; *Habersack*, 2001], magnetic stones, and passive integrated transponder tagged stones [*Lamarre et al*., 2005; *Bradley and Tucker*, 2012] have been used in Lagrangian analysis of resting periods, where consecutive rests of a single particle are tracked. Recent technology used to measure bed topography such as sonar transducers and light detection and ranging (lidar) allows collection of bed elevation time series at a point or in the horizontal plane. While data describing particle step length and velocity characteristics are readily produced in both flume and field gravel tracer tests, sediment residence times are more difficult to record. As a result, residence time is frequently backed out from overall tracer transport characteristics rather than directly measured. The first goal of this study is to use bed elevation data to measure sediment residence time distributions (and entrainment probabilities) explicitly through laboratory measurements and describe appropriate theoretical residence time distributions for use in analytical models of particle transport.

[5] Some notes on the significance of certain types of probability distributions in stochastic models of particle transport will put historical studies in context. It has long been known that a stochastic description of gravel dispersion must be a function of a relative measure of time, where total time is scaled by the mean discrete time at rest [*Stelczer*, 1981, p. 177]. However, in recent decades, the significance of exponential-type (thin tailed or finite mean) versus power law (heavy tailed or infinite mean) residence time distributions in determining existence of an average time at rest has become clear [*Schumer et al*., 2009]. The effect of exponentially distributed residence time, or of any residence time distribution with well-defined (i.e., finite) mean (such as a gamma, Gaussian, or any truncated distribution), is to retard virtual velocity of sediment relative to the instantaneous in-motion velocities. If a collection of tracers is released into a channel, their velocity distribution will reflect the fact that the particles begin in motion. As particles deposit and reside on and within the bed, incorporation of zero velocities will lead to an exponential decrease in overall travel velocity. Eventually, as the longest possible residence times are encountered and an average residence time emerges, the overall average transport velocity, or virtual velocity *V*_{v}, will be related to the average mobile velocity *v* scaled by a retardation coefficient *V*_{v} = *v*/*r*, where *r* is set by the average residence time in the bed. If a dispersion coefficient is used to describe the distribution of velocities around the average, it will also be scaled by the retardation coefficient. Certain heavy-tailed power law distributions on the other hand do not produce samples that converge to a mean value because the high probability of extreme values leads to subsequent values that continually throw off the sample mean. If residence times have an infinite-mean power law distribution, there is no convergence to a long-term average residence time and the virtual velocity will continue to decay as a power law. The latter model is unsatisfying as it implies that all particles will eventually be immobile—an unattractive idea for a nonaggrading bed. Instead, it is more realistic to consider that if residence times vary over many orders of magnitude as a power law, it would only be up to some finite value, likely related to the depth of the bed. This would result in power law decay of the virtual velocity until the incorporation of a maximum residence time finally leads to emergence of an average residence time. These types of residence time distributions can be represented with truncated or tempered power law distributions [*Zhang*, 2010; *Zhang et al*., 2012] which have power law character up to a cutoff after which the distribution decays much more rapidly. The limiting, continuum analytical equations that are associated with random walk-type models are different depending on the type of residence time distribution. Exponential-type residence times combined with diffusive transport lead to classical advection-dispersion equations, while infinite-mean power law residence times lead to fractional-in-time advection-dispersion equations [*Schumer et al*., 2003]. Finally, tempered fractional ADEs can reproduce the features of transport when power law residence times up to a finite maximum exist [*Zhang et al*., 2012]. The transient (or preasymptotic) period before an average residence time is achieved may produce decay in virtual velocity with exponential or power law type decay. If the long-term average sediment residence time is not reached within the time of observation or model period of interest, then the virtual velocity appears transient and will continue to decay. The choice of transport model depends on whether the timescale of interest is within the transient period, the asymptotic period, or both.

[6] Early studies on sediment residence time included development of stochastic models that fit laboratory flume experiments incorporating painted tracer stones [*Einstein*, 1937]. It was observed that travel time of a stone was small relative to its resting period between movements so that overall travel could be modeled using alternating random jump lengths and random resting periods. The residence time distributions used to describe the collection of resting periods were assumed to be exponential based on conceptual-theoretical grounds but not on resting period data. *Stelczer* [1981] argued that in practice, Einstein did not actually treat resting time as a random variable, but as a constant. In decades to follow, the Einstein formulation was the basis for models of flume and field tracer transport and the assumption that sediment residence times are exponential persisted [*Hubbell and Sayre*, 1964; *Yang and Sayre*, 1971; *Sun and Donahue*, 2000]. Use of a compound Poisson Process as a stochastic model for sediment motion [*Hubbell and Sayre*, 1964], for example, defined particle travel distance as the sum of jumps of random length and by definition included exponentially distributed resting periods between jumps so that the number of steps that occur during a discrete time interval can be described by a Poisson distribution [*Ross*, 1997]. In the scaling limit, particle motion described by a Poisson Process is governed by a classical ADE. Interestingly, *Hubbell and Sayre* [1964] stated that they are not sure if the exponential resting periods assumption is valid and leave open the possibility that this assumption will be invalidated by experiments. An example of exponentially distributed residence time data in the literature exists for a plastic particle in a sand-bed flume [*Yang and Sayre*, 1971]. Our own review of a set of residence time data from *Habersack* [2001] suggests that the author used a qualitative fit to an exponential function rather than a distributional goodness of fit compared with other classes of distributions. We find power law resting times over the *Habersack* [2001] experiment up to 30 min, with a crossover to exponential resting times between 50 and 90 min (Figure 1). Recent flume experiments have documented power law residence time distributions and associated them with long-term subdiffusive transport [*Martin et al*., 2012]. There is also evidence that power law residence times occur in natural channels. For example, *Nikora et al*. [2002] proposed a conceptual model describing variance behavior (*σ*^{2} ∝ *t*^{2γ}, where *γ* is a scaling exponent) for particle plume growth across three distinct scales, where variance represents dispersive spread around average growth behavior. Significantly, at long timescales, particle travel distance was expected to be subdiffusive (*γ* < 0.5) because of a wide distribution of sediment residence times. Gravel tracer data from field experiments supported this hypothesis at a variety of timescales [*Nikora et al*., 2002; *Drake et al*., 1988]. However, others [*Zhang et al*., 2012] argued that subdiffusive sediment transport in channels will only exist until the long-time average residence time is reached. They supported this argument by showing improved model fits to the classic *Sayre and Hubbell* [1965] data when a transport model incorporates a tempered Pareto residence time distribution in which power law decay of residence times becomes exponential after some characteristic averaging period is exceeded. While the first goal of this study (stated above) is to measure sediment residence time distributions, the second goal is to describe the origins of transient power law residence times in a bed and our expectations for the transition time to exponential residence times.

[7] The remainder of this manuscript is organized as follows: we first describe translation of bed elevation time series into resting periods of the many particles deposited and entrained at a single location in the bed through time and develop an empirical model for overall sediment residence time distributions (section 2). Next, we discuss the theory of how residence time is linked to bed elevation for two cases: a hypothetical case using semi-infinite bed elevation and a realistic case using finite bed elevation (section 3). We derive sediment residence time distributions from two examples of bed elevation series from flume experiments: a simple case with homogeneous sediment under low flow conditions on plane-bed morphology [*Wong et al*., 2007] and a more complicated case with heterogeneous sediment under high flow conditions on large-scale bedform morphology [*Singh et al*., 2009] (section 4). Both cases are short duration experiments of several minutes to several hours of runtime. Finally, we discuss implications for modeling sediment residence time from a bed elevation series.