Random walk models of fluvial bed load transport use probability distributions to describe the distance a grain travels during an episode of transport and the time it rests after deposition. These models typically employ probability distributions with finite first and second moments, reflecting an underlying assumption that all the factors that influence sediment transport tend to combine in such a way that the length of a step or the duration of a rest can be characterized by a mean value surrounded by a specific amount of variability. The observation that many transport systems exhibit apparent scale-dependent behavior and non-Fickian dispersion suggests that this assumption is not always valid. We revisit a nearly 50 year old tracer experiment in which the tracer plume exhibits the hallmarks of dispersive transport described by a step length distribution with a divergent second moment and no characteristic dispersive size. The governing equation of this type of random walk contains fractional-order derivatives. We use the data from the experiment to test two versions of a fractional-order model of dispersive fluvial bed load transport. The first version uses a heavy-tailed particle step length distribution with a divergent second moment to reproduce the anomalously high fraction of tracer mass observed in the downstream tail of the spatial distribution. The second version adds a feature that partitions mass into a detectable mobile phase and an undetectable, immobile phase. This two-phase transport model predicts other features observed in the data: a decrease in the amount of detected tracer mass over the course of the experiment and enhanced particle retention near the source. The fractional-order models match the observed plume shape and growth rates better than prior attempts with classical models.
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 Sediment carried by a current of wind or water tends to disperse and spread out as it moves downstream. While there are a great many formulas designed to predict the bulk discharge of sediment at a particular point in a flow [e.g., Yalin, 1977], the rich problem of sediment speed and dispersion has received much less attention. Yet there are many applications for which it is important to know the speed and rate of spread of a body of sediment. Examples include the fate and transport of solid-phase contaminants in streams, the time lag between erosional exhumation of a grain and its delivery to a sedimentary basin, and the accumulation of cosmogenic radionuclides in sediment grains during transport. Applications like these require stochastic models of sediment transport, and stochastic transport models require data sets against which to test their predictions. Of particular importance are data sets that can reveal the extent to which the dispersion process obeys traditional Fickian behavior, as compared with the “fractional” or “anomalous” dispersion that has been documented in a diverse range of natural systems. In this paper, we use data from a classic field experiment on sediment dispersion experiment to test the predictions of a family of stochastic transport models, and in particular to compare the predictions of models based on thin- versus heavy-tailed grain velocity distributions.
Einstein [1937, 1950] was among the first to recognize the appeal of a stochastic model of fluvial bed load transport. He conceptualized particle motion as a series of random length steps separated by rests of random duration. This model and numerous other similar models that treat fluvial sediment transport as a random walk [e.g., Sayre and Hubbell, 1965; Paintal, 1971; Yang and Sayre, 1971; Niño et al., 2003; Papanicolaou et al., 2002; Marion et al., 2008] share an underlying assumption that all of the complexity, interaction, and variability in the factors that affect the erosion, transport, and deposition of sediment can be “encapsulated” in a probability distribution that predicts the likelihood that a grain moves farther than a certain distance or is immobile longer than a certain time. Specifically, these models all assume that the distributions of step length and resting time have well defined mean values surrounded by characteristic amounts of variability. What if this assumption is incorrect? What if geomorphic transport processes are so variable and complex that a distribution with a finite mean or variance is not a good representation of the underlying process?
 In recent decades, a wide variety of transport systems have been identified that exhibit scale-dependent behavior, where the fitted parameters in classical governing equations appear to change depending on the spatial or temporal scale over which it is measured (for thorough reviews see Metzler and Klafter [2000, 2004]). One familiar example from the geosciences is the observation that the apparent dispersivity of a heterogeneous aquifer tends to increase with the scale of the tracer test [see Neuman, 1990, and references therein]. The dispersivity represents a characteristic length scale of random motion. In subsurface spreading of conservative solutes under ideal conditions, the Fickian dispersion coefficient was found to be approximately linearly related to velocity magnitude, leading to a decomposition of the dispersion coefficient into a dispersivity tensor that reflected the porous medium properties and the velocity vector (a review is given by Bear ). In one dimension, the dispersion coefficient is the product of the dispersivity and fluid velocity. Benson et al.  showed that the scale-dependent behavior of a solute tracer injected into an aquifer in Mississippi during the macrodispersion experiments (MADE) could be replicated with a random walk model in which the probability distribution of step length has a divergent second moment. This type of distribution has a heavy, power law tail that specifies a much higher probability of extreme values than thin-tailed distributions like the Gaussian or the exponential. Depending on the amount of probability mass in the tail of the distribution, one or both of the integrals that define the first and second moments of the distribution (the mean and variance) mq = ∫xqp(x)dx, q = 1,2 may not converge on a finite value. In practice, a divergent moment means that the sample statistic will tend to increase with the number of samples drawn from the distribution because the larger sample is more likely to include an extreme value [e.g., Schumer et al., 2001]. It is this effect that gives rise to the apparent scale-dependent behavior. These distributions also violate the familiar form of the Central Limit Theorem (CLT), which connects random walk processes with finite moments to the advection-diffusion equation (ADE). In one dimension the ADE is
where C is the spatial concentration of the independent random walkers, t is time, x is the spatial coordinate, v is the average drift velocity (v = 0 for pure diffusion), and D is the diffusion (or dispersion) coefficient with dimensions of L2/T. Dispersive transport which follows the ADE is commonly described as Fickian, or normal dispersion. The Fickian diffusion coefficient is proportional to the variance of the underlying step length distribution and the variance of the dispersing plume scales as the product of diffusion coefficient and transport time. Motion governed by underlying probability distributions with divergent first or second moments obey a more generalized form of the CLT and the dispersion follows a more generalized form of the ADE, where the orders of the derivatives on the dispersive and time terms need not be integers [Benson, 1998]. The simplest form of this type of equation is one dimensional and fractional in space only:
In the fractional advection-dispersion equation (fADE), 0 < α ≤ 2 and the dispersion coefficient D has dimensions of Lα/T. The fADE describes transport in which the underlying probability density function of step length decays like a power law for large values: p(Δx) ∝ Δx−1–α. A plume following this equation has two features: the apparent centered second moment (the variance) grows proportionally to t2/α, and for α < 2 the leading edge concentration falls off according to a power law C(x,t) ∝ x−1–α. For α = 2, the ADE is recovered and the variance scales linearly with time. One of the advantages of the fADE in describing non-Fickian transport is that the dispersion coefficient is not scale-dependent, as it would have to be if it were directly proportional to the divergent second moment of the underlying step length distribution. Instead, the super-Fickian growth rate of the spreading plume is captured by the fractional derivative. Benson et al.  and Schumer et al.  provide discussions of the derivation of the fADE, the meaning of fractional derivatives, and the connection to random walks governed by heavy-tailed probability distributions. Other examples of fractionally dispersive transport systems range from charge transfer in semiconductors [Scher and Montroll, 1975] to fluid turbulence [Shlesinger et al., 1987] to bioturbation of soil and sediment [Meysman et al., 2008]. Recent reviews were given by Metzler and Klafter [2000, 2004].
 The dynamics of fractional dispersion can depart significantly from normal, Fickian dispersion. In this paper, we focus primarily on fractionally dispersive plumes with a heavy downstream tail that result from a distribution of particle step lengths with a divergent second moment. This results in earlier arrival of the tracers at downstream locations earlier than expected from a Fickian model and a mean tracer position that moves faster than the peak concentration.
 There are several reasons to suspect that geomorphic transport systems might exhibit fractional dispersion. A common observation in geomorphology is that estimates of geomorphic rates tend to change depending on the interval over which the measurement technique integrates. For example, Gardner et al.  plotted estimates of surface elevation change due to various processes against the duration of the interval over which the change was measured and found a slope of less than one, meaning that the rate of uplift or erosion tended to decrease with the length of the measurement interval. They suggested that a possible explanation for the slowdown is that longer observation intervals are more likely to capture a long interval of no surface elevation change, resulting in a lower average rate. Kirchner et al.  found a similar effect but in the opposite direction: average erosion rates in a catchment in Idaho increased with the length of the measurement time scale. They proposed that this was because longer intervals were more likely to include an extreme erosional event such as a large landslide. More recently, Singh et al.  demonstrated that the sediment flux in a large experimental flume was dependent on the time over which the measurement was made. These observations are similar to the apparent scale dependence of dispersivity in an aquifer and suggest that the underlying processes might be heavy tailed.
 In particular, there are several pieces of evidence that suggest that bed load transport by rivers may be fractionally dispersive in some cases. First, Nikora et al. [2001, 2002] found that the sample variance of plumes of real and simulated tracer grains can grow nonlinearly with time under some circumstances. In Fickian transport, the variance of the tracer plumes scales linearly with time t, σ2 = 2Dt, so the nonlinear variance scaling σ2 ∼ tγ, γ ≠ 1 observed by Nikora and colleagues indicates an apparent scale-dependent diffusion coefficient and fractional dispersion. Second, tracer studies in rivers have commonly observed strongly right-skewed distributions of travel distance [e.g., Schmidt and Ergenzinger, 1992; Habersack, 2001; McNamara and Borden, 2004]. These are usually fitted with distributions such as the exponential or gamma, but that does not exclude the possibility that the transport is actually heavy tailed, if relatively rare events on the tail of the distribution were not captured by the experiment. An exception is Pyrce and Ashmore , who found that a Cauchy distribution (a symmetrical distribution with heavy tails) centered on the spacing between the pools and bars described the step length distribution of tracers in a flume. Third, Stark et al.  and Tucker and Bradley  presented arguments for nonlocal behavior in certain forms of sediment transport, implying a potential for heavy-tailed particle travel distances and fractional dispersion. Also, Ganti et al.  presented an argument for how heavy-tailed particle travel distances might arise in gravel bed rivers. Finally, there are the data from the tracer experiment performed by W.W. Sayre and D.W. Hubbell of the U.S. Geological Survey in the 1960s [Sayre and Hubbell, 1965]. Motivated in part by a need to develop and test a probabilistic transport model like Einstein's, the Sayre and Hubbell tracer experiment provides an unusually vivid picture of the advection and dispersion of a pulse of radioactively tagged tracer sand in a natural river. The tracer concentration profiles exhibit heavy (power law) downstream leading edges, similar to those observed in solute tracer tests in heterogeneous aquifers, suggesting fractional dispersion.
 In this paper, we reanalyze the Sayre and Hubbell data. We begin by reviewing the tracer experiment and then discuss their original transport and dispersion model, highlighting three weaknesses. Next, we introduce a transport model that is conceptually similar to the Sayre and Hubbell model, but uses a distribution of particle step lengths that is heavy tailed with a divergent variance. This model is able to reproduce the non-Fickian behavior of the plume, but the values of model parameters are almost entirely empirically determined. Finally, we add a feature to the model that partitions mass into a detectable, mobile phase and an undetectable, immobile phase. This allows us to reproduce another feature observed in the Sayre and Hubbell data, the decrease in the amount of detected mass over the course of the experiment. It also provides additional constraints on some of the model parameters, allowing us to derive them directly from the data rather than choosing values by matching the model results to the observed concentration curves visually. The success of the models in describing the data provides strong evidence for fractional dispersion of sediment by a river.
2. Sayre and Hubbell's Transport and Dispersion of Labeled Bed Material
2.1. Tracer Dispersion Experiment
 On 3 November 1960, W.W. Sayre and D.W. Hubbell [Sayre and Hubbell, 1965] used a small boat, an electric can opener, and a long funnel to spread 40 pounds (18.2 kg) of radioactive sand across the bed of the North Loup River near Purdum, Nebraska. The goal of the experiment was to track the transport and dispersion of the tracer sand along the bed of the river and to test a theoretical transport model. The experiment was part of a series of investigations that the U.S. Atomic Energy Commission and the U.S. Geological Survey performed in the early 1960s to study radioactive waste in the environment. The study reach was an artificial meander cutoff about 50 feet (15.2 m) wide and relatively straight, with a gentle bend to the left. Sayre and Hubbell monitored the tracer sand for 13 days over 1800 feet (548.8 m) of channel. The site map and a drawing of the reach are reproduced from the original publication in Figure 1.
 The tracer sand used in the experiment was nearly uniform in size and shape, with a median size (0.305 mm) slightly coarser than the median diameter of the bed material. Figure 2 shows the grain size distributions of the tracer sand, the bed material, and the suspended sediment. The tracer size distribution was chosen so that tracers were representative of the bed material and would not travel in suspension for “any significant part of the time,” though there is some overlap between the large end of the suspended load grain size and the tracer size distribution. The sand was plated with Iridium-192, which emits gamma rays and has a half-life of 74 days. In 1960, such labeled sand was commercially available in 2 pound cans. Handling the cans with tongs (for radiation safety), they opened them with the can opener and poured the contents into a funnel resting on the riverbed. In all, Sayre and Hubbell emptied 20 cans of tracer sand onto the bed in piles at 2 foot intervals across the channel.
 To measure the radiation emitted by the tracer sand, Sayre and Hubbell towed a sled-mounted waterproof scintillometer along the bed behind their boat. The maximum detection range of the scintillometer was on the order of 1 foot (0.33 m). The tracks of the two primary downstream traverses were along the right and left sides of the channel, approximately one third of a channel width from the nearest bank. They also performed downstream traverses along the center of the channel and lateral traverses on several days.
 To convert the measured gamma ray count rate (in detections per minute) to a tracer concentration (in grams of tracer per unit volume of bed material), the team developed empirical calibration curves in the laboratory. They mixed known quantities of tracer material through varying volumes of sand and measured the radiation detected. Using these curves, they estimated the tracer concentration in the riverbed by averaging the gamma ray count rate over some distance (5–100 feet (1.5–30.5 m), based on the rate of change of the count rate) and assuming that the tracer was uniformly mixed into the bed to an average depth of 1.45 feet. This was the average depth to which tracer particles were mixed in core samples of the bed material taken over the course of the study.
 Sayre and Hubbell traversed the right side of the channel on 11 of the 13 days of the study and the left side on 10 days. Only two traverses along the channel centerline were made. Our digitization of their data from the right side of the channel is shown in Figure 3. Traverses along the left side of the channel indicated that a significant fraction of the tracer sand was buried early in the experiment and released about a week later. Some of this material may have washed over to the right side of the channel and may be visible in Figure 3 at 170.1 h as a bump in the concentration profile behind the peak at about 100 feet (30.5 m). Modeling dispersive transport with extended rest periods may require a differential equation that is fractional in time [e.g., Schumer et al., 2003], so we exclude data from the left side of the channel in order to focus on the simpler case of transport that is fractional in space only. Despite the extra material washing over from the left side, the data along the right side of the channel appear to be less affected by extended deep burial (relative to the length of the experiment), so we limit our analysis to these data. We omit one right side traverse performed toward the end of the study (196 h, labeled 9R in the original publication) from our analysis because only the first 400 feet of the reach were scanned. The longitudinal concentration profiles shown in Figure 3 are strongly skewed in the downstream direction. The peak concentration moved only about halfway down the reach over the course of the study (∼300 h), but the leading edge of the plume had nearly reached the end of the study reach by 96.2 h.
2.2. Sayre and Hubbell Transport Dispersion Model
Sayre and Hubbell  used their tracer data to test a theoretical model of tracer transport that predicts the probability that a tracer occupies a spot along the riverbed. The model is ultimately equivalent to Einstein's  but their derivation differs somewhat from Einstein's. The model treats particle motion as a series of alternating steps and rests. The step distance and the duration of the rest intervals are assumed to be independent random variables that obey the conditions of a Poisson process, and the corresponding probability density functions (pdfs) of travel distance, Δx, and resting time, Δt, are exponential:
The parameter k1 has units of 1/L and is the inverse of the mean travel distance. Similarly, k2 is the inverse of the mean resting time, with units of 1/T. Assuming that all particles start at rest and the total amount of time spent in motion is negligible compared to the total time at rest, equations (3) and (4) can be combined to yield the conditional probability density function of position at any time:
where x is distance from tracer release point, t is elapsed time, and I1 is a modified Bessel function of the first kind of order one. The function f(x,t) has units of inverse length and it predicts the probability of finding a tracer per unit length of channel. An alternative derivation of equation (5) is given in Appendix A. The Sayre and Hubbell model predicts Fickian dispersion because the first and second moments of both the step length and resting time distribution are finite.
 To calibrate the model, Sayre and Hubbell compared two model predictions to the data. First, the model predicts that the peak concentration will move downstream at a rate of k2/k1, once all particles have taken at least one step. This condition is satisfied when k2t ≈ 3. At earlier times, the model does not conserve mass and the peak concentration is poorly defined (see Appendix A). Second, the model predicts that the rate of decay of the peak concentration is a complicated function of the two model parameters. Given these two constraints, Sayre and Hubbell used plots of the position of the observed peak concentration versus time and the magnitude of the peak concentration versus time to solve for k1 and k2 graphically. Their calibrated model predicts that a grain travels an average of 36 feet (11 m) during an episode of motion followed by an average stationary interval of 12 h:
Equation (5) can be scaled to predict tracer concentration ϕ1(x,t) (in mass per unit volume of bed material) if certain conditions are met. First, the number of tracers must be sufficiently large as to approximate infinity. Second, the tracers must be seeded in a way that approximates a uniform plane source extending across the channel and into the bed to some mixing depth. Third, the average mixing depth must remain constant in the downstream direction and the tracers must be mixed uniformly through this depth. Finally, no more than a small fraction of the particles can travel in suspension. If these conditions are met, then the model should be scaled by the total mass of tracer material W divided by the product of channel width B and average mixing depth d:
The model scaling given by equation (6) is shown in Figures 3 and 4 for W = 1800 g, B = 50 ft, and d = 1.45 ft. From 44.2 h onward, this version of the model overpredicts the observed tracer concentration. This is because the amount of tracer mass detected decreased over the course of the experiment, falling below W/Bd after the second measurement time. To correct for this, Sayre and Hubbell chose to scale their model at each measurement time by the area under the concentration curve, or zeroth moment, μ0(t):
This version of the model is also shown in Figures 3 and 4. It should be noted that in the original publication, Sayre and Hubbell chose to scale their data to the units of model prediction (relative concentration, 1/L) by dividing the data by μ0(t) rather than multiplying the model by μ0(t) as shown in equation (7). We choose to present their model scaled to an absolute concentration (M/L3) for the sake of clarity.
 Qualitatively, the Sayre and Hubbell model given in equation (7) and shown in Figure 3 predicts the observed concentrations reasonably well. The most obvious weakness of the model is that it fails to capture the amount of mass in the leading edge of the tracer plume. Though this is difficult to see on a plot using linear axes, it is clear when the data and the model are plotted using logarithmic axes (Figure 5). The concentration of tracer particles several hundred feet downstream of the peak is as much as several orders of magnitude higher than the model (7) predicts. This is especially clear during the middle of the study, from 70.9 h to 170.1 h. The model appears to predict the concentration in the leading edge of the plume slightly better very early in the study because the tail is cut off at 400 ft (131.2 m), the maximum distance measured. Late in the study, the model appears to catch up with the data for the same reason: the measurements do not extend past 1800 feet (590.4 m).
 A second weakness of the model is that it requires an ad hoc parameter, a 110 foot downstream shift, to match the location of the data. The model curves in Figures 3 and 5 include this shift. The downstream shift is necessary presumably because the model assumes that all particles start at rest for an average duration of 12 h, while in reality the tracer particles probably started moving much sooner, as a consequence of the dosing procedure. Figure 6 shows the position of the peak observed concentration (the mode), along with the mode of the unshifted model, and the model shifted downstream by 110 feet. The model predictions start at 36 h because the mode of equation (5) is poorly defined for earlier times. The observed mode appears to move backward at 287.4 h because of tracer material that appears upstream of the peak starting at 170.1 h. This is most easily seen in Figure 3. Sayre and Hubbell attributed the influx of tracer mass to lateral transport from the left side of the channel. A large fraction of the tracer mass along the left side disappeared early in the study and reappeared about halfway through, with some washing over to the right side of the channel. They argued that the missing tracers must have been buried in a particularly deep dune trough for the first half of the study and exhumed about a week later.
 A final drawback of the model is that it fails to predict that the total amount of mass detected along a traverse changes with time. Figure 7 shows the area under the concentration curve, or zeroth moment, at each measurement time. Here, as elsewhere, we present only data from the right side of the channel. The zeroth moment is the total mass detected per unit cross-sectional area along the traverse. The model prediction of the zeroth moment, W/Bd is also included. At the first measurement time, the observed zeroth moment is ∼1.5 times higher than the model prediction. This is probably because the tracer sand was seeded in piles on the surface rather than being spread evenly across and into the channel bed. By 70.9 h, the observed moment has dropped to an approximately constant value that is about 15% lower than the model prediction. Because of this decrease in the amount of tracer mass detected, Sayre and Hubbell chose to use the model scaling given in equation (7), rather than that in equation (6).
3. Reanalysis of Tracer Plume Data
 In this section, we reanalyze the Sayre and Hubbell  tracer plume using an alternative model. First we develop a random walk model that allows a much higher probability of long steps than the Sayre and Hubbell model. This model addresses the first two criticisms of the Sayre and Hubbell model. Then we discuss a way to scale the model that accounts for the decay in detected mass shown in Figure 6.
3.1. Heavy-Tailed Transport Model
 Concentration profiles that exhibit linear leading edges when plotted using double logarithmic axes may indicate fractional dispersion that is due to an underlying step length distribution with divergent first or second moments. This type of dispersion is governed by the fADE, equation (2). The fADE can be solved analytically [Benson et al., 2001] or approximated numerically by random walks [Zhang et al., 2006; Zhang and Benson, 2008]. The random walk solution is relatively easy to implement and can be superior to the analytic solution and traditional numerical methods in terms of performance and the ability to incorporate complicated boundary conditions and spatially variable transport coefficients.
 Our random walk solution to the fADE is a particle-tracking model that is conceptually similar to the Sayre and Hubbell  model described above, but differs in several important ways. First, the probability that a particle takes a large step is greater in our model. This helps capture the amount of tracer mass in the leading edge of the plume. Second, we start all the particles with a step rather than a rest. We think this is more representative of the initial condition in the experiment and it helps to reproduce the high initial velocity of the particle plume that required Sayre and Hubbell to apply an ad hoc downstream shift to their model. Third, grains in our model move with a finite velocity, taking time to complete a step, rather than executing a step instantaneously. Finally, the model is parameterized differently. The implementation is a similar to those described by Valocchi and Quinodoz , Zhang et al. , and Benson and Meerschaert . Particle motion is governed by the average amount of time spent during a mobile epoch, τm, the average duration of a rest τim, the mean mobile velocity m, and the dispersion coefficient D. The model assigns each particle a random “flight time,” tf, chosen from an exponential distribution with mean τm. At each model time step Δt, the particle advances downstream a distance of Δx = mtf + δx′, where δx′ is a random, zero-mean fluctuation with a specified probability distribution. The fluctuations are meant to mimic a particle's random velocity fluctuations as it skitters along the bed during an episode of transport. The average distance covered during one mobile step is mtf. When a particle reaches the end of its time in motion, it is assigned an exponentially distributed random resting time with mean τim. The model computes the particle concentration at specified intervals by binning the one-dimensional “study reach” into 1 foot segments and counting the number of particles in each bin. We run the model with one million particles and a time step size of two minutes for 300 h, the approximate duration of the Sayre and Hubbell study.
 The velocity fluctuations in our model are drawn from an α-stable distribution. The α-stable distribution is useful for modeling highly variable data sets because it can be skewed positively or negatively and at least one of the tails of the distribution decays with a power law tail when the tail index parameter α is less than 2. For α = 2, the stable density is a Gaussian. We refer the reader to discussions by Zolotarev , Samorodnitsky and Taqqu , Benson et al. , and Nolan  for more information about the α-stable distribution. We use the parameterization of the α-stable distribution described by McCulloch  and referred to by Nolan  as S(α, β, γ, δ; 1), where α is the index of stability, β is the skewness, γ is the scale, and δ is the location. We choose the maximally positively skewed form of the distribution, β = 1, to model heavy-tailed jumps exclusively in the downstream direction. The location parameter δ is set to zero so that the mean of the fluctuations is zero. The scale parameter γ is related to the dispersion coefficient of the tracer plume by γ = (DΔt∣cosπα/2∣)1/α where Δt is the length of the time step over which the velocity fluctuation is applied [Benson et al., 2001]. The remaining parameter α controls the likelihood of large deviations from the location parameter and sets the slope of the leading edge of the tracer plume in log space. When α is in the range 1 < α < 2, the variance of the distribution diverges. When α is in the range 0 < α ≤ 1, the mean also diverges. The heavy-tailed distribution of velocity fluctuations over the course of a step results in an overall step length distribution that is heavy tailed with the same tail index α. The motion of the tracer plume is governed by the fADE (equation (2)) where the order of the fractional derivative is equal to the tail index parameter of the α-stable distribution.
 To calibrate the model, we must choose values for five parameters: m, D, τm, τim, and α. The choices are largely empirical. In principal, we could use the position of the center of mass of the model particle plume to constrain the relative values of three of these parameters. The three parameters m, τm, and τim set the relationship between the observed mean position of the particle plume, , and time, t, according to
In this application however, the model cannot be calibrated using equation (8) because the observations are cut off at the maximum distance measured, and the fastest moving particles do not contribute to the measured center of mass. If, as we assume, the underlying step length distribution is heavy tailed, then the loss of the fastest moving particles will cause the observed mean velocity to appear lower than the actual mean velocity. Instead, we chose the value of m so that the positions of the model peak concentration match the data.
Figure 8 shows the model concentration profiles for m = 10.5 ft/h (3.2 m/h), τm = τim = 12 h, a = 1.1, and D = 6.0 fta/h (1.6 ma/h for a = 1.1). We chose α and D to match the shape of the tracer plume downstream of the peak. The time parameters are arbitrarily set equal to the resting time parameter Sayre and Hubbell used, τm = τim = 12 h. This version of the model is more sensitive to the ratio of the time parameters than to their actual values, so other choices could work as well. Our model predicts the fraction of tracer particles per unit length of stream, which can be interpreted as the probability of finding a tracer in the interval x + Δx if the number of tracers is sufficiently large. The units of our model (1/L) are the same as the Sayre and Hubbell model given in equation (5), so it must be scaled to predict absolute tracer concentrations (M/L3). In Figure 8, we used Sayre and Hubbell's theoretical initial zeroth moment, W/Bd, to scale the model profiles. Figure 9 shows the location of the peak concentration of our model compared to the data and Sayre and Hubbell model. In general, our model does a much better job of predicting the amount of mass in the leading edge of the tracer plume and it is at least as good as the Sayre and Hubbell model at predicting the position of the peak concentration, without the benefit of the ad hoc 110 foot downstream shift. The initial condition of the model is responsible for the high initial velocity. It is also responsible for the discontinuity in the model results at about 25 h. Starting all particles in motion creates a narrow spike in the tracer plume (see Figure 8, 3.2 h) that drops off sharply upstream of the peak. As tracers end their first step and begin their first rest, the center part of the plume broadens. The discontinuity appears when the initial spike in the tracer concentration finally disappears.
 The model does not address the third problem with the Sayre and Hubbell model, the decrease in the observed tracer mass with time. Because of this, the model slightly overpredicts the peak concentration when scaled by W/Bd (see Figure 10). Over most of the study, the Sayre and Hubbell model scaled by the zeroth moment does a better job of predicting the peak concentration. While our model does lose some mass (about 8%) as tracers move past the downstream measurement boundary, the shape of the decay curve, the rate of decay, and the magnitude of the change are all inconsistent with the data (Figure 11).
Harvey and Gorelick  described a single-rate mobile/immobile (MIM) model that predicts that the amount of detectable mass will decay exponentially toward an equilibrium in which the mass entering storage is matched by the mass exiting storage. The fraction of mass in the mobile phase, as given by the zeroth moment, μ0(t), depends on the initial mobile zeroth moment, μ0(t = 0), the average amount of time spent in the mobile phase, τm, and the average amount of time in the immobile phase, τim:
This model is very similar to the two-layer vertical exchange model of Schick et al.  except that the parameters are the time in each phase rather than the exchange rates between layers. Our particle-tracking implementation of the mobile/immobile model differs from the model considered in section 3.1 only in that it assumes that immobile particles are buried beyond the range of the gamma ray detector. Sayre and Hubbell estimated that the zone of particle movement in their study reach was between 1.4 and 1.5 feet (∼0.45 m) thick. They based this estimate on the average mixing depth from core samples, the depth of dune troughs (using the method of Hubbell ) and from theoretical considerations. This mixing depth is similar to the detection range of their equipment. They report that the signal strength of a point source of 192Ir buried under ∼10 inches (25 cm) of sand would be approximately 0.5% as strong as a source buried at a depth of ∼2.5 inches (6.4 cm). For an 192Ir source mixed uniformly through an infinite volume of sand, 95% of the detected gamma rays would be emitted from a distance less than 9.2 inches (23.4 cm) from the detector. While it is not exactly clear how the thickness of the transport layer relates to the maximum depth at which a particle could be detected, the length scales are similar enough that roughly equating mobile layer thickness and tracer detection depth seems reasonable.
 The mobile/immobile model may explain why Sayre and Hubbell's decision to scale their model as shown in equation (7) achieved the goal of matching the magnitudes of observed and predicted concentration. Because the zeroth moment is approaching equilibrium by 44.2 h, μ0(t) is approximately equal to the equilibrium mobile fraction given by equation (9), μ0(t = 0)τm/(τm + τim) thereafter. Therefore, Sayre and Hubbell's model scaled by μ0(t) closely approximates the prediction of equation (9) and the model is implicitly transformed to a prediction of the mobile phase concentration.
 If the MIM model is an appropriate choice, then equation (9) provides a constraint on the time parameters τm and τim as well as a way to scale the model prediction to an absolute concentration that accounts for the decrease in detected mass. We estimate the three parameters in equation (9) by fitting it to the observations, simultaneously minimizing the sum of squared errors in both the untransformed and log-transformed data. The solid black line in Figure 12 shows the best fit result of τm = 42.2 h and τim = 38 h, μ0(t = 0) = 400 g/ft2 (4303 g/m2). Using these parameters in the model with α = 1.3, D = 7.2 ftα/h (1.5 mα/h for α = 1.3), and m = 6 ft/h (1.82 m/h) results in the concentration profiles shown in Figure 13. The analytic solution to the fractional mobile/immobile model (for which there is no simple, closed form, see Benson et al.  and Benson and Meerschaert ) is included as a dashed black line. As in the previous version, the values we choose for D and m are largely empirical. The dispersion coefficient in this case was chosen to match the width of the concentration curves toward the end of the study. The position of the peak concentration and the magnitude of the peak are shown in Figures 14 and 15. Only the analytic solution is shown: the particle model results are omitted for clarity. In this version of the model, we estimated the value of α from the time decay of the peak concentration, which should (for later time) decrease as t−1/α. In Figure 15, the slope of the data after the third measurement time is about −1/1.3, providing an additional constraint on the model parameters. Overall, this version of the model is an improvement over the first version in that equation (9) provides an additional constraint on the parameter choices and the model reproduces the observed loss of tracer mass.
4.1. Comparison of the Model Versions
 Both versions of our model discussed above do a reasonably good job of predicting the overall behavior of the tracer plume, despite different values for some of the parameters. The first version of the model does not differentiate between detectable (mobile) and undetectable (immobile) particles and demonstrates that a fractional-order dispersive model describing a random walk with heavy-tailed step lengths can reproduce the heavy leading edge of the tracer plume. The second version improves on the first by assuming that immobile particles are undetectable, adding a constraint on the time parameters in the model and reproducing the observed decrease in tracer mass. In both versions, the overall shape of the leading edge of the particle plume is controlled by the dispersion coefficient D and the tail index α. Ultimately, it is α that dictates the slope of the leading edge of the plume when plotted on a log-log plot, but the location of the power law leading edge is partly a function of D. A lower value of D will cause the tracer concentration to drop off more rapidly downstream of the peak before the power law section begins, while a higher value moves the beginning of the power law tail upstream. The second model uses a slightly higher dispersion coefficient combined with a less heavy-tailed step length distribution to accomplish results similar to the first. One consequence of the lower value of α used in the first version is that a higher mean mobile velocity (10.5 ft/h (3.2 m/h) versus 6 ft/h (1.82 m/h)) is required to match the position of the data. This is because as α approaches 1 from above, more and more of the probability shifts to the left of zero in order to balance the increasing likelihood of long steps and keep the mean at zero [Zhang et al., 2006]. Thus, the majority of the fluctuations actually slow the particle rather than speed it up.
4.2. Physical Realism
 It is reasonable to ask how the parameters of either model relate to the hydrologic conditions in the study reach, and in particular, how they relate to dune migration. Sayre and Hubbell  reported that the bed of their study reach was dominated by dunes with a wavelength of 10–15 feet (3–4.6 m) and an average height of about 1.5 feet (0.45 m), but they do not report a dune migration velocity. However, they did estimate the total bed load discharge Qb of about 170 tons per day. An estimate of the dune migration velocity can be obtained with the geometric argument that the average dune velocity for idealized triangular dunes is related to the specific bed load discharge by
where qb is the mass flux per unit channel width, c is the dune migration velocity, h is the dune height, γs is the specific weight of the sediment, and λ is the porosity. For qb = Qb/B, B = 50 ft (15.24 m), γs (1 − λ) = 100 lbs/ft3 (1604 kg/m3), and h = 1.45 ft (0.44 m), the estimated dune migration velocity c is a little more than 4 ft/h (1.22 m/h). This is in reasonably good agreement with the overall mean velocity for the two versions of the model presented above (∼4 ft/h (1.2 m/h) and 3.2 ft/h (0.97 m/h)). In the MIM version, the time parameters suggest that a tracer buried deeply enough to escape detection remains buried for the passage of about 10–15 dunes before being reexhumed and traversing a similar number of dunes before being buried again. While this seems reasonable, there is no way to know if it is actually true.
4.3. Comparison With Nikora et al.'s Conceptualization of Diffusive Regimes
Nikora et al. [2001, 2002] defined three different time scales for bed load transport. The fundamental component of grain motion is the local trajectory, defined as the motion between two collisions with the bed. The range of times over which local trajectories occur is called the local range. This is the shortest time scale of grain motion. The mode of dispersion in this realm is ballistic, meaning that the variance scales as σ2 ∼ t2. The intermediate range is defined as the time between two significant collisions with the bed. A significant collision is one that traps the grain well enough that the time at rest is large relative to the time scale of the motion. An intermediate trajectory integrates tens to hundreds of local trajectories and dispersion in the intermediate range may be subdiffusive, normal, or superdiffusive, characterized by the variance scaling σ2 ∼ tγ, 0 < γ < 2. Subdiffusion (γ < 1) is slower than normal, Fickan diffusion (γ = 1) and superdiffusion (1 < γ < 2) is faster. The third range that Nikora et al.  defined is the global range. A global trajectory is composed of many intermediate trajectories and they define the boundary between the intermediate range and the global range as a nondimensional time tu*/d where t is time, u* is the shear velocity, and d is the diameter of the moving grain. They suggest that the global range is likely to be subdiffusive, because of the possibility that a grain can be trapped for long periods of time (in the extreme, becoming a fluvial sedimentary rock), resulting in a divergent first and second moment of the overall resting time distribution. While it is true that heavy-tailed resting times can result in subdiffusion, they can also result in superdiffusion if the random walk is strongly biased in one direction [Weeks and Swinney, 1998], as seems likely in a river.
Nikora et al.  argued that Einstein's transport model describes the intermediate range, in part because steps are separated by intervals that are long relative to the time necessary to complete an intermediate trajectory. The Sayre and Hubbell  model also addresses the intermediate range. Nikora et al.  proposed that the boundary between the intermediate range and the global range is where the nondimensional time tu*/d ≈ 15. For nondimensional time scales longer than this, grain motion is in the global range. They based this number on the position of a break in slope between variance plots (versus nondimensional time) from two different tracer experiments. In the first experiment, Nikora et al. videotaped gravel moving across a 20 cm × 23 cm patch of the bed of Balmoral Canal in New Zealand and extracted 159 complete particle trajectories. They defined a complete trajectory as one that began and ended in the measurement window. The maximum time for one of these trajectories was on the order of one second. To simulate a spreading particle cloud, they translated the particle trajectories to a common origin and then measured the variance scaling of this simulated particle plume. They found that variance of their simulated particle plume grew faster than linearly in both the downstream and cross-stream directions, indicating superdiffusion. They plotted this data set with the variance of tracer data from an experiment by Drake et al.  in which they filmed the trajectories of 125 painted particles released onto the bed of Duck Creek over a period of 240 s. The variance of the Duck Creek data appear to scale sublinearly, indicating subdiffusion. Nikora et al. extended lines fit to the variance of the two data sets until they intersected at a nondimensional time tu*/d ≈ 15. They proposed that this is the approximate boundary between the intermediate and global scales.
 The time scale for the N. Loup River tracer experiment appears to place it firmly in the global range. Sayre and Hubbell reported that the average water surface slope of the study reach over the course of the experiment was about S = 8 × 10−4 and the average water depth was around h = 2 ft (60 cm). Assuming that the average bed slope over the reach is approximately equal to the water surface slope and writing shear velocity as u* = , where g is gravity, the shear velocity for the Sayre and Hubbell experiment was approximately u* = 6.9 cm/s. Using the midpoint of the experiment (∼150 h) for the time and a grain size of 0.3 mm, we compute the nondimensional time scale of the experiment, tu*/d = 1.2 × 108. Even if the estimates of the hydraulic conditions used here are correct only to an order of magnitude, the conditions in the study reach appear to fall within the global range using the definition of Nikora et al. In dimensional terms, the time scale separating the intermediate and global ranges for the study reach is about 6.5 × 10−2 s.
 By the definitions of Nikora et al. , our model probably represents the global range. The mean velocity of a model grain is on the order of 1–10 ft/h (0.3 to 3 m/h), while the flow velocity in the river was about 7000 ft/h (∼2 km/h). This huge difference between the flow velocity and the average grain velocity means that a single episode of motion in our model represents many smaller steps with significant periods of rest between them. Also, the average duration of an episode of motion, tens of hours, is many orders of magnitude larger than the proposed dimensional time marking the boundary between the ranges. However, the nature of the dispersion in our model is not consistent with the predictions of Nikora et al. about the global range. Our model is superdiffusive, contrary to the prediction that the global range is likely to be subdiffusive. Instead, our model is more consistent with the intermediate range behavior that Nikora et al. observed in the Balmoral Canal tracer data. The variance scaling of a fractionally dispersive plume is related to the parameter α by σx2 ∼ t2/α [Benson et al., 2000a, 2000b]. Nikora et al.  found variance scaling of σx2 ∼ t1.54 and σx2 ∼ t1.7 (the higher value was for the tracer grains that were larger than the D50 of the bed, while the lower value was for grains smaller than the D50) corresponding to α in the range of 1.1 to 1.3, very consistent with our results. The reason for the inconsistency between the scale of our model (the global range) the predicted mode of dispersion is not clear. One possibility is that their model, developed for gravel bed rivers, is not directly applicable to a sand bed river dominated by migrating dunes.
4.4. Variance Scaling
 In theory, the variance scaling of a tracer plume can be used to estimate the parameter α, but the variance of the North Loup River data is not a good indicator of the mode of dispersion. This is illustrated in Figure 16. Because the data are censored at the maximum measurement distance, the fastest tracers are not included in the variance calculation. This effect is evident in the variance of the model tracer plume shown in Figure 16 (only the α = 1.1 model is shown). Instead of indicating fractional dispersion, the censored variance, measured over 3600 feet, twice actual the length of the study reach, is similar to the data and the Fickian prediction of the Sayre and Hubbell  model. Only when all the model tracers are included in the variance calculation is the slope of the variance consistent with the theoretical prediction, σ2 ∼ t2/α.
5. Summary and Conclusions
 The model Sayre and Hubbell  developed to describe the data from their tracer experiment has three weaknesses. First, it does not reproduce the amount of mass in the downstream end of the tracer plume. This is due to exponentially distributed particle step lengths. Second, the initial condition of the model does not seem to accurately represent the actual initial conditions that tracer grains experienced and consequently the model requires an ad hoc downstream shift of 110 ft (33.5 m) to compensate for the high initial velocity of the tracer grains. Finally, the model does not predict the observed decay in the amount of detected tracer mass over the course of the experiment.
 We present two versions of an alternate model. The first version of the model is able to reproduce the heavy-tailed concentration profiles by using a heavy-tailed distribution of particle step lengths. By starting all particles with a step, rather than a rest, it is also able to account for the high initial velocity of the particle plume. In order to accommodate the observed decrease in detectable mass, the second model assumes that only tracers in motion are detectable and that immobile tracers are buried beyond the range of the detector. This version of the model is able to address all three weaknesses of the Sayre and Hubbell model, though the correspondence between the mobile and immobile phases of the model and dune migration, presumably the dominant mode of sediment transport in the study reach, cannot not be rigorously defined. In spite of this, the key result is the ability of the model to reproduce most of the salient features in the data, providing evidence for fractional-order dispersion of sediment by a natural river.
Appendix A:: Alternate Derivation Sayre and Hubbell Transport Model
 The compound Poisson process with rate k2 (along with exponentially distributed jumps with mean 1/k1) is also known as a CTRW. The jumps and intervening waiting times are both nonnegative so we may write the Laplace-Laplace (t↦s; x↦p) transform of the particle position density as
Inverting the transform in s gives
and recalling that the inverse Laplace transform of e1/p = δ(x) + I1(2)/, where δ(x) is the Dirac delta function and I1(x) is a first-order modified Bessel function of the first kind, and using the scaling and shifting rules from Laplace transform pairs ebx/cf(x/c) ⇔F(cp − b) gives us
Sayre and Hubbell  ignore the exponentially decaying probability of a particle remaining at the origin (the Dirac portion) and hence their model conserves mass for large t only.
 This work was supported by an Army Research Office grant, proposal 47033-EV, and a Graduate Assistance in Areas of National Need (GAANN) fellowship to Nate Bradley. Thanks also to NCED and the University of Illinois Hydrologic Synthesis Activity for sponsoring the Stochastic Transport and Emergent Scaling in Earth-Surface Processes Workshop at Lake Tahoe, Nevada in November 2007. Many of the ideas in this paper originated at that meeting. D.B. acknowledges support from NSF grants DMS–0539176 and EAR–0749035 and USDOE Basic Energy Sciences grant DE-FG02-07ER15841. Any opinions, findings, conclusions, or recommendations do not necessary reflect the views of the NSF, DOE, or ARO. Thorough reviews from four anonymous reviewers were extremely helpful in sharpening the focus of this paper.