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Temporal variability and memory in sediment transport in an experimental step-pool channel Saletti, Matteo; Molnar, Peter; Zimmermann, André; Hassan, Marwan A.; Church, Michael Nov 28, 2015

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RESEARCH ARTICLE10.1002/2015WR016929Temporal variability and memory in sediment transport in anexperimental step-pool channelMatteo Saletti1, Peter Molnar1, Andre Zimmermann2, Marwan A. Hassan3, and Michael Church31Institute of Environmental Engineering, ETH Zurich, Zurich, Switzerland, 2Northwest Hydraulic Consultants, NorthVancouver, British Columbia, Canada, 3Department of Geography, University of British Columbia, Vancouver, BritishColumbia, CanadaAbstract Temporal dynamics of sediment transport in steep channels using two experiments performedin a steep flume (8%) with natural sediment composed of 12 grain sizes are studied. High-resolution (1 s)time series of sediment transport were measured for individual grain-size classes at the outlet of the flumefor different combinations of sediment input rates and flow discharges. Our aim in this paper is to quantify(a) the relation of discharge and sediment transport and (b) the nature and strength of memory in grain-size-dependent transport. None of the simple statistical descriptors of sediment transport (mean, extremevalues, and quantiles) display a clear relation with water discharge, in fact a large variability between dis-charge and sediment transport is observed. Instantaneous transport rates have probability density functionswith heavy tails. Bed load bursts have a coarser grain-size distribution than that of the entire experiment.We quantify the strength and nature of memory in sediment transport rates by estimating the Hurst expo-nent and the autocorrelation coefficient of the time series for different grain sizes. Our results show thepresence of the Hurst phenomenon in transport rates, indicating long-term memory which is grain-sizedependent. The short-term memory in coarse grain transport increases with temporal aggregation and thisreveals the importance of the sampling duration of bed load transport rates in natural streams, especiallyfor large fractions.1. IntroductionSediment transport is a fundamental process in fluvial systems, which has been the focus of extensiveresearch for a wide range of purposes. Understanding and properly describing fluvial sediment transportdynamics has important implications in geomorphology and engineering; e.g., sediment erosion and depo-sition are the main drivers of landscape evolution [e.g., Dietrich et al., 2003; Tucker and Hancock, 2010], andthey can impact civil infrastructures and cause natural hazards [e.g., Totschnig et al., 2011; Badoux et al.,2014]. Sediment transport strongly influences river ecosystems and fish habitat [e.g., Lisle, 1989; Konrad,2009] and determines the morphology of channels [e.g., Hassan et al., 2005; Church, 2006]. Modeling sedi-ment transport is a challenging task, especially in channels steeper than 4–5%, where the methods devel-oped for lowland streams do not perform well [Rickenmann, 2001, 2012] because of complex boundaryconditions in channel morphology (due, for example, to step-pool structures and to the wide range of parti-cle sizes and shapes), turbulence of the flow (enhanced by roughness elements like boulders and log-jams),and the impossibility of defining uniform and steady conditions.Step-pool morphology is very often found in such steep channels, when large boulders create channel-spanning structures called steps, with downstream pools maintained by the tumbling water flow [e.g.,Montgomery and Buffington, 1997; Chin and Wohl, 2005; Church and Zimmermann, 2007]. Sediment transportin step-pool streams is particularly influenced by morphological changes within the channel [e.g., Grantet al., 1990; Lenzi, 2001; Hassan et al., 2005; Comiti and Mao, 2012], by the spatial and temporal variability ofsediment supply [e.g., Recking et al., 2012; Recking, 2012], by the geomorphic coupling with adjacent hill-slopes [Molnar et al., 2010], and extreme hydrologic events [Lenzi et al., 2004; Turowski et al., 2009].Alluvial sediment in step-pool streams is poorly sorted and boulders constitute a distinctive feature of thebed morphology. They enhance channel stability, acting as keystones in the step formation process [Juddand Peterson, 1969; Hayward, 1980; Whittaker, 1987; Church and Zimmermann, 2007; Curran, 2007]. WhenKey Points: Large variability between dischargeand sediment transport Assessment of nature and strength ofmemory in fractional transport rates Memory depends on sediment yield,grain size, and aggregation intervalSupporting Information: Supporting Information S1Correspondence to:M. Saletti,saletti@ifu.baug.ethz.chCitation:Saletti, M., P. Molnar, A. Zimmermann,M. A. Hassan, and M. Church (2015),Temporal variability and memory insediment transport in an experimentalstep-pool channel, Water Resour. Res.,51, 9325–9337, doi:10.1002/2015WR016929.Received 14 JAN 2015Accepted 12 NOV 2015Accepted article online 18 NOV 2015Published online 28 NOV 2015VC 2015. American Geophysical Union.All Rights Reserved.SALETTI ET AL. TEMPORAL VARIABILITY AND MEMORY IN SEDIMENT TRANSPORT 9325Water Resources ResearchPUBLICATIONSthese large grains are dislodged, they mobilize significant quantities of sediment which propagate down-stream. Step-pool channels have been studied extensively in recent decades, focusing mainly on channeland step geometry [Curran and Wilcock, 2005; Milzow et al., 2006; Chartrand et al., 2011], hydraulics [Comitiet al., 2009; Zimmermann, 2010; Wilcox et al., 2011], bed morphology [Zimmermann and Church, 2001; Wei-chert et al., 2008], step formation, and channel stability [Zimmermann et al., 2010; Waters and Curran, 2012].Einstein’s pioneering work [Einstein, 1937, 1950] pointed out almost 80 years ago that bed load transportshould be treated as a stochastic process. Since then, many studies have adopted a probabilistic view,focusing on the microstructural description of bed load and its implication for macroscopic observations[e.g., Ancey et al., 2006; Ancey and Heymann, 2014] or on a probabilistic description of the bulk sedimentflux [e.g., Furbish et al., 2012; Roseberry et al., 2012]. Effort has also been devoted to describe bed load withdifferent statistical formalisms [e.g., Ganti et al., 2009; Singh et al., 2009; Turowski, 2010], as well as to charac-terize transport fluctuations at different temporal scales [e.g., Heyman et al., 2013; Ma et al., 2014].Statistical analyses of transport rates, measured directly in the field or in laboratory experiments, could helpto improve our understanding of sediment transport and its prediction. For example, by computing theHurst exponent or similar parameters, as done extensively in hydrological studies [e.g., Hurst, 1951; Mandel-brot and Wallis, 1969; Koutsoyiannis and Montanari, 2007], the degree of memory (or persistence) in bedload transport could be assessed at different temporal scales as well as the clustering and burstiness intransport rates. Although the Hurst exponent has previously been calculated for suspended sediment con-centration of the Yellow River by Shang and Kamae [2005], to our knowledge, the memory effect for bedload data collected for different grain sizes has not been investigated.Our work provides new insight into the dynamics of bed load transport in step-pool streams, focusing ongrain-size-dependent effects. Flume experiments allow observations of sediment transport at high resolu-tion in channels where step formation is facilitated. We use part of a data set of experiments performedin a steep flume at The University of British Columbia [Zimmermann, 2009]. The novelty of these experi-ments lies in the combination of a realistic channel morphology (step-pool sequences with 12 differentgrain sizes) and long and high-frequency (1 Hz) sediment transport measurements for different grainsizes.In this paper, we use the experimental data to address three objectives. (1) We explore temporal variabilityin sediment transport rates in relation to sediment supply, flow conditions, and channel adjustment in astep-pool morphology. (2) We estimate the probability distribution of instantaneous transport rates, focus-ing on the effect of intense bursts in sediment transport. (3) We detect and describe memory in sedimenttransport: first by estimating the Hurst exponent and its dependence on grain size, and second by an auto-correlation analysis to establish the short-term memory signals in sediment transport as a grain-size-dependent phenomenon that is affected by the aggregation interval.2. Methods2.1. Description of the ExperimentsWe use part of a data set of 32 experiments performed at The University of British Columbia in a 5 m longand 0.83 m wide flume, with slopes ranging between 3% and 23% (all details about the experiments can befound in Zimmermann et al. [2008], Zimmermann [2009], Zimmermann et al. [2010], and Zimmermann[2010]). The length of the flume was sufficient to observe the formation of at least five step-pool units inevery run. In order to simulate natural gravel-bedded streams, gravel-sized sediment was employed andwalls made rough by attaching plywood to the flume sides to enhance the sidewall roughness. The bed sur-face was scanned every hour with a red laser profiler after stopping the water flow in order to obtain a DEMof the entire channel bed; repeated measurements of the same bed revealed an average standard error of61.9 mm (very good considering that the largest stones in the channel have a 128 mm diameter). Averageflow velocity was measured every minute with an automated salt conductance method by injecting approx-imately 250 mL of dilute salt solution into the upstream end of the flume and monitoring the downstreamconcentration with two conductivity probes located 1 and 4 m from the top of the flume. Water dischargewas measured with a Thermo Polysonics DCT 1088 Transit Time Flow Meter (the instrument error is 5%) at10 Hz frequency and then averaged to obtain a 1 Hz times series.Water Resources Research 10.1002/2015WR016929SALETTI ET AL. TEMPORAL VARIABILITY AND MEMORY IN SEDIMENT TRANSPORT 9326A sediment mixture was fed into the flume at a con-stant rate ranging from 43 to 301 g/s/m for a dura-tion of about 60 min, in four different pulses. Thisprocedure was found to be most effective in creatingan armored bed with a developed step-pool mor-phology. The sediment mixture used in the experi-ments was scaled (1:20) to match a prototype step-pool stream, Shatford Creek in British Columbia(Canada) [Zimmermann and Church, 2001]. The d25,d50, and d75 were 4, 9.8, and 20 mm, respectively.The grain-size distribution was divided into w/2 sizeclasses (where w52/5log2d) ranging from 2 to128 mm. Transport rates of the 12 sediment sizeclasses were acquired at 1 Hz frequency by photo-graphing sediment passage over a light table at theoutlet of the flume [Zimmermann et al., 2008]. In order to study the influence of different grain sizes, in thispaper, we integrate the sediment classes into four grain sizes for which transport rates (TR) were analyzed:TR1 (2–5.6 mm), TR2 (5.6–16 mm), TR3 (16–45 mm), and TR4 (>45 mm).In this study, the results of experiments 5 and 6 of the data set, characterized by the same channel slopeand same experimental setup, are analyzed. A summary of the two experiments is provided in Table 1 andFigure 1. In these two experiments, a feed phase was followed by an adjustment phase. In the feed phase, asequence of four periods with sediment input and slowly increasing flow rates lasting 1 h each (F1–F4) wasapplied with the purpose of developing a stable armored bed and a step-pool morphology. After everysediment feed pulse, a recovery phase (R1–R4) followed, during which the feed was shut off. In the adjust-ment phase, sediment input was turned off and water discharge was increased every hour (see Table 1)until either the sediment at the base of the flume at the channel outlet was completely scoured or the max-imum permitted flow rate (85 L/s) was reached. When one of these two conditions was reached, the experi-ment was terminated. In the adjustment phase, we analyzed separately periods with different dischargevalues (DQ1–DQ7), in order to observe the relation between water discharge and sediment transport in asupply-limited condition.2.2. Analysis of Instantaneous Transport RatesWe computed basic statistics for instantaneous (1 s) transport rates of fractional and total bed load trans-port, the former being the time series of the four grain-size classes chosen (TR1–TR4) and the latter beingF1 R1 F2 R2 F3 R3 F4 R4 DQ1DQ2DQ3DQ4DQ5DQ6Average Transport Rate (g/s)     Water Discharge (l/s)  0102030405060EXP 5TR1TR2TR3TR4QF1 R1 F2 R2 F3 R3 F4 R4 DQ1DQ2DQ3DQ4DQ5DQ6DQ7Average Transport Rate (g/s)     Water Discharge (l/s)  0102030405060EXP 6TR1TR2TR3TR4QADJUSTMENTADJUSTMENTFEED FEEDFigure 1. Phases and periods of experiments (left) 5 and (right) 6. Different grain-size classes (TR1–TR4) are shown with different colors, and water discharge (Q) is represented by a lightblue line. Different columns show average transport rate for different periods lasting 1 h each. During the feed phase, feed periods (F1–F4) alternate with recovery periods (R1–R4). Dur-ing the adjustment phase, hourly periods (DQ1–DQ7) represent an increase in water discharge (see Table 1).Table 1. Summary of Experiments 5 and 6aExp. 5 Exp. 6Duration (h) 20.1 21.4Flume slope (%) 8 8Mean channel width (m) 0.359 0.359Feed pulses (number) 4 4Total sediment fed (kg) 76 80Adjustment phase: Q valuesDQ1 (L/s) 19 19DQ2 (L/s) 23 23DQ3 (L/s) 28 28DQ4 (L/s) 34 34DQ5 (L/s) 41 41DQ6 (L/s) 50 50DQ7 (L/s) 60aMore details can be found in Zimmermann [2009].Water Resources Research 10.1002/2015WR016929SALETTI ET AL. TEMPORAL VARIABILITY AND MEMORY IN SEDIMENT TRANSPORT 9327their sum. An example of four timeseries of fractional transport rates isprovided in Figure 2. It can be seenthat fine grains (TR1 and TR2) movealmost continuously with pulse-likefeatures. In contrast, coarse grains (TR3and especially TR4) move intermit-tently, i.e., bed load bursts are followedby long periods of no movement.We analyzed the relation betweentotal instantaneous transport rates andwater discharge during the adjustmentphase. We fitted different probabilitydistributions to the total instantaneoustransport rates, finding that the best fitis given by a three-parameter General-ized Extreme Value (GEV) distribution.The GEV model is a versatile distribu-tion which is able to capture the heavytail characterizing phases of intensesediment transport. Special attentionwas devoted to bursts in transport rates (i.e., instantaneous values of intense transport rate) as these areresponsible for most of the bulk transport, despite being infrequent. Moreover, we determined the grain-size distribution of extreme transport rates and compare it with that of the entire experiment.2.3. Scaling Behavior, Memory, and AutocorrelationWe perform a scaling analysis of the time series of total and fractional transport rates, in order to estimatethe Hurst exponent H. The Hurst exponent is a parameter which quantifies the degree of memory (or per-sistence) of a process, by analyzing the properties of its time series over different temporal aggregationscales.We estimate H by calculating r, the standard deviation of the time series aggregated at different temporalscales T [Koutsoyiannis, 2006; Koutsoyiannis and Montanari, 2007] ranging from 1 to 90 s. H is then definedby:r5r0T12H(1)where r0 is the standard deviation of the original time series (with 1 s aggregation scale). In a log-log space,the slope of the straight line interpolating rr05rr0ðTÞ is equal to H2 1.Two extreme situations may occur. A process with H5 0.5 shows short-term persistence (or short-termdependence), typical of Markovian processes, having memory only of t2 1. On the other hand, a processwith H5 1 is said to have long-term persistence (long-term dependence or long memory) because the pastcontinuously influences the present. In such a process, there is a tendency toward clustering in time of simi-lar events [Koutsoyiannis and Montanari, 2007]. In terms of sediment transport rates, this means a tendencyof intense (or weak) instantaneous transport events to cluster into long periods of high (or low) transport[Koutsoyiannis, 2002; Shang and Kamae, 2005]. More details about the Hurst exponent and its relation tointermittency can be found in supporting information.The tendency toward short-term or long-term persistence in sediment transport can be observed in ourdata (Figure 2). A process having H ! 1, such as TR1 and TR2, shows increasing and decreasing tendenciesover a range of scales, while a process with H ! 0.5, such as TR3 and TR4, is more intermittent with veryshort memory of the past.We compute the Hurst exponent for the 13 periods of the adjustment phase (DQ1–DQ6 in experiment 5and DQ1–DQ7 in experiment 6), both for total and fractional transport rates in order to assess the propertiesof memory in sediment transport as a function of mean transport rate and grain size.0 100 200 300 400 500 600TR1 (g/s)050100150H=0.950 100 200 300 400 500 600TR2 (g/s)0100200300H=0.870 100 200 300 400 500 600TR3 (g/s)010002000H=0.67Time (sec)0 100 200 300 400 500 600TR4 (g/s)0100020003000H=0.56Figure 2. A 10 min time series for the four instantaneous fractional transport ratesduring the adjustment phase (experiment 5, DQ6). Finer fractions (TR1 and TR2)move continuously, with increasing and decreasing trends and Hurst exponentsclose to 1. Coarser fractions (TR3 and especially TR4) show an intermittentbehavior with bed load bursts followed by periods of no movement, and also asmaller H.Water Resources Research 10.1002/2015WR016929SALETTI ET AL. TEMPORAL VARIABILITY AND MEMORY IN SEDIMENT TRANSPORT 9328The strength of short-term memory can be determined from the temporal autocorrelation of the data. Weestimate the autocorrelation coefficient qs as a function of the time lag s:qs5csr2051NXN2st51ðYt2Y ÞðYt1s2Y Þ1NXNt51ðYt2Y Þ2(2)where cs is the autocovariance with lag s and r20 is the variance, Y is the mean of Y, Yt, and Yt1s are, respec-tively, the values of the time series at time t and t1 s, and N is the length of the time series Y. We calculatedq for every time lag s for every period DQ of the two experiments. Finally, we repeated the autocorrelationanalysis for the time series aggregated at different temporal scales, in order to show the effect of samplingduration on autocorrelation for different grain sizes.3. Results3.1. Effect of Discharge on Bulk Sediment TransportThe evolution of the mean sediment transport rates in different phases and periods indicates that bulk sedi-ment flux is poorly related to water discharge (Figure 1). When sediment is fed into the flume, the feed ratedictates the output: intense transport rates are recorded during feed pulses (F1–F4) while in the recoveryperiods (R1–R4) mean transport rates drop. When the feed is shut off in the adjustment phase (Figure 1)and discharge is gradually incremented, average sediment transport does not consistently increase withflow rate. There are periods of high transport rates during lower discharges (e.g., DQ1 in experiment 5 andDQ2 in experiment 6), related to morphological changes in the bed (e.g., the collapse of one or more steps)with release of sediment stored in steps and pools.The lack of a relation between instantaneous sediment transport rates and discharge is shown by box plotsin Figure 3. In both experiments, there is large variability in transport rates for the same value of flow dis-charge. Total instantaneous sediment transport rates range from 0 to more than 2000 g/s. Changes in thepercentiles of transport rates are also not consistently associated with changes in discharge. Bursts in trans-port rates, such as values larger than 1000 g/s, happen both with small (e.g., DQ2 in experiment 6) and largedischarges (e.g., DQ6 in experiment 5). Furthermore, there is a substantial difference between the sedimenttransport rates in the adjustment phases in experiments 5 and 6, despite identical discharge values. ThisDQ1 DQ2 DQ3 DQ4 DQ5 DQ610−210−1100101102103104Transport rate (g/s)Exp 5DQ1 DQ2 DQ3 DQ4 DQ5 DQ6 DQ710−210−1100101102103104Transport rate (g/s)Exp 6Figure 3. Box plots of instantaneous bulk sediment transport rates in the periods of the adjustment phase for experiments 5 and 6 as afunction of increasing water discharge (DQ1–DQ7). The box includes the values between the first and the third quartile; the second quar-tile (i.e., the median) is represented by a horizontal red line; the whiskers delimit minimum and maximum values; when the minimumvalue is zero, the line is not shown. Transport rates are plotted on a log scale to show the large variability for the same value of discharge.Water Resources Research 10.1002/2015WR016929SALETTI ET AL. TEMPORAL VARIABILITY AND MEMORY IN SEDIMENT TRANSPORT 9329shows the importance of random variations in sediment transport, conditioned by hydraulic fluctuations,incipient sediment motion conditions, sediment availability, and the formation of bed morphology.3.2. Bed Load Bursts: Probability Distributions and Fractional CompositionWe fit different probability distributions to the total instantaneous transport rates during the adjustmentphase, finding a good fit, especially for the tails (i.e., large transport rates), given by a three-parameter Gen-eralized Extreme Value distribution (Figure 4). We show the GEV model fits to data with a kernel smoothingfunction estimator applied in logarithmically spaced bins. In Figure 4 (top), we illustrate this comparison forthe entire adjustment phase of both experiments.It is very important to model accurately the tails of the transport rate distributions because they play animportant role in the total mass of the bulk sediment transport. For example, in the adjustment phase ofexperiment 5 (lasting 6 h), there are only 216 single measurements with an instantaneous transport rategreater than 195 g/s (which corresponds to an exceedence probability of 1%) but they are responsible for33% of the total sediment yield. Throughout 6 h, almost one third of the sediment was delivered in only216 s. This analysis was conducted during the adjustment phase, when extreme transport rates are particu-larly interesting because they are likely to be associated with changes in bed morphology and sedimentstorage.Figure 4. (top) Probability density function of total instantaneous transport rates and (bottom) grain-size distribution of sediment yield fordifferent exceedence probability values. The probability density functions of instantaneous transport rates are estimated with a kernelsmoothing function and fitted with a generalized extreme value distribution for the entire adjustment phase of both experiments. The per-centage of the sediment yield belonging to the four grain-size classes as a function of the exceedence probability shows a consistentcoarsening in bed load bursts of low exceedence probability.Water Resources Research 10.1002/2015WR016929SALETTI ET AL. TEMPORAL VARIABILITY AND MEMORY IN SEDIMENT TRANSPORT 9330The fractional composition of bed load bursts during the adjustment phase indicates size-selective transportlimited by the mobility of the coarsest grains (Figure 4, bottom). If only instantaneous transport rates with alow exceedence probability are considered, the grain-size distribution of the sediment yield is coarser thanthat of the whole experiment and the coarsening is inversely proportional to the selected exceedence prob-ability. Visual observations during the experiments confirm that large transport rates during phases of nofeed are directly linked to major channel adjustments. Movements of coarse grains and sometimes step col-lapses were unequivocally observed during periods with high average transport rates.3.3. Memory and Autocorrelation in Sediment TransportTo detect memory and long-range dependence in sediment transport rates in the adjustment phase, weestimate the Hurst exponent H following equation (1). In Figure 5, we plot r=r0 for the total transport rates.The values of H are listed in Table 2. The Hurst exponent is related to the total transport rate (see Table 2):periods characterized by a relatively intense transport (such as DQ1 and DQ6 in experiment 5 and DQ2 andDQ4 in experiment 6) display milder slopes in Figure 5, which means a larger Hurst exponent and thuslong-term persistence.We also find a grain-size dependence of the Hurst exponent (Table 2 and Figure 6). First, finer grain sizeshave a tendency toward long memory with H ! 1, while coarser grains tend toward the short-term persist-ence regime with H ! 0.5 (Figure 6a). Second, the memory in total sediment transport is limited bythe mobility of coarse grains. During periods in which coarse grains are showing short-range dependenceFigure 5. Relation between dimensionless standard deviation of total transport rates r=r0 and aggregation scale T for different periods ofexperiments 5 and 6. Standard deviation displays a power law decrease with aggregation interval in log-log scale, with a slope equal toH2 1, where H is the Hurst exponent.Table 2. Values of the Hurst Exponent for All the Periods in the Adjustment Phase and for the Four Grain-Size ClassesExperiment Period TR ðg=sÞ HTRtot HTR1 HTR2 HTR3 HTR45 DQ1 14.7 0.95 0.96 0.95 0.92 0.735 DQ2 5.3 0.78 0.95 0.94 0.66 0.575 DQ3 3.2 0.88 0.96 0.91 0.75 0.625 DQ4 2.7 0.77 0.93 0.87 0.61 0.505 DQ5 5.4 0.68 0.85 0.83 0.61 0.545 DQ6 31.2 0.95 0.97 0.96 0.91 0.756 DQ1 3.2 0.83 0.95 0.95 0.78 0.606 DQ2 20.3 0.91 0.96 0.96 0.90 0.696 DQ3 2.8 0.75 0.92 0.90 0.63 0.506 DQ4 20.4 0.93 0.96 0.95 0.84 0.666 DQ5 5.8 0.63 0.92 0.90 0.80 0.506 DQ6 2.8 0.67 0.91 0.77 0.56 0.526 DQ7 6.7 0.67 0.90 0.82 0.62 0.50Water Resources Research 10.1002/2015WR016929SALETTI ET AL. TEMPORAL VARIABILITY AND MEMORY IN SEDIMENT TRANSPORT 9331(H ! 0.5), regardless of the H of finer grains, the Hurst exponent for total transport is low (Figure 6b). Thiscan be seen also in Table 2, where H is compared to the average transport rate TR. In periods with low TR,memory is determined by H of coarse grains, while for periods with high TR, memory associated with themobility of finer grains starts to play a major role.The strength in memory of grain-size-dependent transport rates can be assessed by the autocorrelationcoefficient qs in equation (2). For the instantaneous (T5 1 s) transport rates, qs is shown in Figure 7 for theentire adjustment phase for the four grain-size classes in experiments 5 and 6. Finer fractions (TR1 and TR2)have a stronger autocorrelation, while for coarser grains (TR3) autocorrelation strength and decorrelationtime (i.e., time at which the autocorrelation coefficient drops close to zero) become smaller and vanish forthe largest grains (TR4). Here the intermittency of the signal, due to the infrequent mobility of these grains,is responsible for the absence of correlation.The final question is how the strength of memory relates to the temporal aggregation scales in sedimenttransport data. To address this, we analyze qs from equation (2) for s5 1 lag as a function of the aggre-gation interval T for the studied grain-size classes (we plot an example for period DQ6 in experiment 5 inFigure 8). For this analysis, we selected periods with large sediment transport so that movements of coarsegrains were present. The analysis shows that q1 increases with aggregation scale T for the largest fractions(TR3 and TR4), while it slightly decreases for finer fractions. This has implications for bed load transport sam-pling, because it shows that the intermittent nature of size-dependent grain transport can be observed onlyat high sampling resolutions. Already at a sampling resolution of T5 40 s, we practically lose any meaning-ful grain-size-dependent difference in autocorrelation. Long sampling may be important for total transportrates, but it dilutes meaningful grain-size effects on memory.4. Discussion4.1. Discharge and Sediment TransportThe first outcome of our analysis is the absence of a consistent relation between water discharge and sedi-ment transport in the experimental data set. The same Q can produce a large spectrum of possibleresponses in terms of bed load transport and sediment yield, as shown in Figure 3. Transport rates in theabsence of sediment feed show variability at three different levels: (1) for the same value of Q, instantane-ous values of transport rates span 3–5 orders of magnitude; (2) within the same experiment, progressiveincrements in discharge do not necessarily lead to larger values in sediment transport; (3) phases of equal Qin distinct experiments yield consistently different transport rates. These observations may have severalcauses.TR tot TR1 TR2 TR3 TR4Hurst Exponent0.50.60.70.80.91H Total0.5 0.6 0.7 0.8 0.9 1H Fractional0.50.60.70.80.91TR1TR2TR3TR4(b)(a)Figure 6. Dependence of the Hurst exponent on grain size. (a) Box plots of H for total and fractional transport rates for the 13 DQ periodsof the adjustment phase. (b) H for the four grain-size classes plotted as a function of H of the total transport for the 13 DQ periods.Water Resources Research 10.1002/2015WR016929SALETTI ET AL. TEMPORAL VARIABILITY AND MEMORY IN SEDIMENT TRANSPORT 9332As already pointed out in previous studies [Dietrich et al., 1989; Hassan et al., 2008; Recking et al., 2012; Reck-ing, 2012], sediment supply may dominate sediment transport: periods of intense transport are those duringwhich sediment is fed into the flume, even if the values of water discharge are smaller compared to theadjustment phase (Figure 1). When the feed is turned off, other mechanisms must be responsible for epi-sodic displacements of large volumes of sediment. The use of a grain-size mixture instead of uniform sedi-ment can partially explain the presence of bed load transport fluctuations [Recking et al., 2009]. Moreover,sediment storage strongly influences sediment transport [Hassan et al., 2008; Pryor et al., 2011]: step-poolsare natural sediment storage units and they can act to inhibit the effect of flow conditions in mobilizinggrains from the river bed. As directly observed during our experiments, some of the bursts in transport ratesare due to the displacement of large grains (e.g., a step collapse) and as a consequence to the increasedavailability of sediment. A similar mechanism has been observed by Ghilardi et al. [2014a, 2014b] in anexperimental channel with a cascademorphology, where a geotechnical fail-ure caused the breaking of the armorlayer, therefore causing bed loadbursts. Especially, in channels havingwell-structured and stable beds, flow-dependent parameters alone cannotexplain patterns of sediment transportwithout considering channel stabilityand sediment storage dynamics [Has-san and Zimmermann, 2012].Finally, in our experiments, channelshaving the same slope, same grainsize, and same discharge respondedquite differently because they experi-enced different bed-forming condi-tions in the past. The observedvariability in sediment transport canTime Lag (sec)100 101 102 103Autocorrelation Coefficient10-310-210-1100EXP 5TR1TR2TR3TR4Time Lag (sec)100 101 102 103Autocorrelation Coefficient10-310-210-1100EXP 6TR1TR2TR3TR4Figure 7. Autocorrelation coefficient qs as a function of time lag s for different grain sizes in the adjustment phase of experiments (left) 5 and (right) 6. Finer grain sizes (TR1 and TR2)have larger values of qT, while coarse grains show weaker autocorrelation (TR3) or even no autocorrelation at all (TR4). The horizontal dotted lines represent the 5% confidence boundon serially uncorrelated data.Aggregation interval T (sec)0 10 20 30 40 50 60Autocorrelation coefficient00.20.40.60.81Exp 5 - DQ6TR1TR2TR3TR4Figure 8. Autocorrelation coefficient for the first lag qs51 as a function of theaggregation scale T for different grain-size fractions. The largest fraction (TR3 andTR4) shows a general increase in autocorrelation when increasing the aggregationinterval. Finer fractions (TR1 and TR2) exhibit a small decrease with increasingaggregation intervals.Water Resources Research 10.1002/2015WR016929SALETTI ET AL. TEMPORAL VARIABILITY AND MEMORY IN SEDIMENT TRANSPORT 9333be explained as a consequence of the stochastic nature of bed stability [Zimmermann et al., 2010], meaningthat sudden channel instabilities, such as a step collapse, may cause the measured bursts in transport rates.It has been previously emphasized [Buffington and Montgomery, 1999; Hassan and Zimmermann, 2012] thatin most streams flows characterized by the same duration and magnitude can produce largely differentchannel morphologies, and we have shown here how this applies also to sediment transport.4.2. Stochastic Sediment TransportThe observed variability in transport rates highlights the stochastic nature of sediment transport [Einstein,1937] and strengthens the need for its statistical description [e.g., Furbish et al., 2012; Ancey and Heymann,2014]. That instantaneous transport rates are well fitted by a heavy-tailed distribution is in agreement withprevious work [e.g., Gomez et al., 1989; Singh et al., 2009] and identifies a process in which high-magnituderates play an important role in bulk sediment transport. This might be also the case when looking at singleparticle dynamics (e.g., in tracer studies), where both particles travel distances and resting times can showheavy-tailed probability density functions, identifying a so-called anomalous diffusion behavior [e.g.,Schumer et al., 2009; Bradley et al., 2010; Ganti et al., 2010; Martin et al., 2012].Incipient motion itself can be treated as a stochastic phenomenon [e.g., Papanicolaou et al., 2002], for exam-ple, in the context of an impulse formulation criterion [see Valyrakis et al., 2011, and references therein].Given the power law relation between magnitude and frequency, Valyrakis et al. [2011] used a GEV distribu-tion to model impulses leading to particle entrainment. Our results are consistent with this outcome: thecondition of incipient motion is directly linked to particle transport, so it is not surprising that impulsesrelated to entrainment follow the same family of statistical distributions as instantaneous transport rates.4.3. Hurst Phenomenon and Memory in Sediment TransportIn this work, we described the nature and strength of memory in bed load transport. As summarized inTable 2 and Figures 5 and 6, the tendency toward long memory in transport rates increases with sedimenttransport intensity and decreases with grain size.We propose that, in the absence of sediment feed, stages with increased transport rates are due to changesin bed storage, when grains that were stored in steps and pools and other formerly stable structures arereleased and transported downstream, and this large sediment delivery is responsible for long-rangedependence (H! 1). On the contrary, during stages of low transport (H! 0.5), we see mostly random fluc-tuations due to the stochastic nature of bed load transport.The values of the Hurst exponent and the autocorrelation analysis reveal that memory is grain-size depend-ent and this highlights the importance of fractional transport data for an accurate description of bed loaddynamics. Most notably, it is important to understand the contribution of coarse grains that are transportedwith high intermittency, especially because in natural steep streams their displacement can have major con-sequences as a natural hazard.The Hurst exponent and the autocorrelation structure found in our data are similar to those found in a pre-vious study of Shang and Kamae [2005]. By analyzing daily records of suspended sediment in the YellowRiver, the authors proved the presence of long-range dependence both in the slow decay of the autocorre-lation function and in the value of H. Despite using the same framework, our work provides a complemen-tary analysis, proving the existence of long memory in bed load transport and its strong dependence ongrain size.4.4. Temporal Scales and Sampling DurationOur results stress the importance of sampling duration, especially for an intrinsically intermittent processsuch as bed load transport. As pointed out by Bunte and Abt [2005], sampling time impacts the mean meas-ured transport rates. We showed in Figure 8 that by aggregating the signal at different temporal scales theeffect on grain-size-dependent correlation is significant. Coarse grain transport is uncorrelated at short sam-pling durations due to intermittency (see for example, the bottom plot for TR4 in Figure 2), but becomescorrelated at larger aggregation scales. Only by increasing the sampling time it is possible to capture thetime scales of sediment pulses connected to the release of large grains. This has important practical conse-quences: by studying the autocorrelation structure of sediment transport signals at different scales, it isWater Resources Research 10.1002/2015WR016929SALETTI ET AL. TEMPORAL VARIABILITY AND MEMORY IN SEDIMENT TRANSPORT 9334possible to determine the time scale characterizing the movements of different fractions and define a mini-mum sampling duration necessary to properly estimate them.4.5. Statistical Framework for Sediment TransportThe statistical methods applied in our study are similar to those used by Singh et al. [2009] in analyzinginstantaneous transport rates in laboratory experiments. Some differences in procedure and outcomesshould be noted. First, we restricted our analysis to the monoscaling case, because our records are not longenough to perform a multiscaling analysis. For the same reason, we could not study the effect of the chang-ing scale on the shape of the pdf function, even though this can be done with longer records, as shown bySingh et al. [2009]. Second, even if both studies deal with sediment transport rates, they cannot be directlycompared: in our case, grain size, the absence of feed, and bed morphology play a fundamental role in dic-tating the results. We were able to show the importance of the coarsest fractions in bed load bursts and thestrength of memory in different grain sizes captured by the Hurst exponent.Moreover, we provided a complementary evidence for the observed trend of increasing mean transportrate with increasing sampling duration for large water discharge. We showed in Figure 8 that, in periods ofintense transport, coarse fractions (which are responsible for bed load bursts) are uncorrelated at low sam-pling duration, i.e., the sampling duration is too short to capture the time scales characterizing their move-ment. It is only at sufficiently long sampling durations that they become correlated: at this point, as showedby Bunte and Abt [2005] and Singh et al. [2009], bed load bursts start to be measured and contribute to themean flux.Finally, we did not observe a decrease in intermittency with increasing discharge as in the experiments ofSingh et al. [2009]. Again, this can be explained by the strong connections between bed load transport andchanges in bed morphology in the experiments. As previously discussed, in well-armored channels insupply-limited conditions, the main factor dictating sediment transport response is the stability of the bed,thus masking the effect of the changing flow conditions.5. ConclusionsBy means of experiments performed in a steep flume with natural sediment, we have analyzed time seriesof fractional sediment transport rates measured at high frequency (1 Hz) in a step-pool channel. None ofthe simple statistical descriptors of sediment transport display a clear relation with water discharge. On thecontrary, we observed a large variability between discharge and sediment transport, due to the stochasticnature of bed load transport and the influence of bed morphology, sediment supply and storage, and indi-vidual channel histories. Further research is needed to explain this variability and its causes; this will help toimprove both sediment transport predictions and bed load modeling in general.We analyzed instantaneous transport rates, whose pdfs are heavy tailed. The fractional composition of sedi-ment yield reveals that the coarse fractions, which constitute a small part of the sediment yield of the entireexperiment, become the dominant part during most intense sediment transport. This highlights the impor-tance of large-grain dynamics, both in terms of channel stability and sediment budget estimates.We performed scaling and autocorrelation analyses on transport rates to assess the nature and strength ofmemory in the data. We showed the presence of long-term memory in sediment transport by the Hurstexponent H. Factors affecting the tendency toward a long-range dependence regime are mainly (1) themagnitude of sediment transport, (2) the grain size of the transported material, and (3) the aggregationinterval. We have shown that H is a function of grain size and that the long-term memory regime (Happroaching 1) is limited by the displacement of the coarsest fractions. This effect can only be detected atsufficiently high sampling frequency and tends to disappear by aggregating the data. To our knowledge,this is the first time that the Hurst phenomenon has been quantified in bed load transport data togetherwith other factors affecting the memory. Our work shows that fractional transport rates represent a key ele-ment for a better understanding of sediment transport and related changes in channel morphology, espe-cially in step-pool streams, where sediment transport is supply limited and strongly modulated by thecollapse of step structures and subsequent bed adjustment.Water Resources Research 10.1002/2015WR016929SALETTI ET AL. TEMPORAL VARIABILITY AND MEMORY IN SEDIMENT TRANSPORT 9335ReferencesAncey, C., and J. Heymann (2014), A microstructural approach to bed load transport: Mean behaviour and fluctuations of particle transportrates, J. Fluid Mech., 744, 129–168, doi:10.1017/jfm.2014.74.Ancey, C., T. B€ohm, M. Jodeau, and P. Frey (2006), Statistical description of sediment transport experiments, Phys. Rev. E, 74, 011302, doi:10.1103/PhysRevE.74.011302.Badoux, A., N. Andres, and J. M. Turowski (2014), Damage costs due to bedload transport processes in Switzerland, Nat. 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