Subordinated Brownian motion model for sediment transport

Vamsi Ganti, Arvind Singh, Paola Passalacqua, and Efi Foufoula-GeorgiouDepartment of Civil Engineering, St. Anthony Falls Laboratory and National Center for Earth-Surface Dynamics,

University of Minnesota, Minnesota, USA�Received 29 November 2008; revised manuscript received 4 May 2009; published 9 July 2009�

Following the introduction of the Brownian motion model for sediment transport by Einstein, severalstochastic models have been explored in the literature motivated by the need to reproduce the observednon-Gaussian probability density functions �PDFs� of the sediment transport rates observed in laboratoryexperiments. Recent studies have presented evidence that PDFs of bed elevation and sediment transport ratesdepend on time scale �sampling time�, but this dependence is not accounted for in any previous stochasticmodels. Here we propose an extension of Brownian motion, called fractional Laplace motion, as a model forsediment transport which acknowledges the fact that the time over which the gravel particles are in motion isin itself a random variable. We show that this model reproduces the multiscale statistics of sediment transportrates as quantified via a large-scale laboratory experiment.

DOI: 10.1103/PhysRevE.80.011111 PACS number�s�: 05.40.�a, 92.40.Gc, 92.40.Cy

I. INTRODUCTION

Stochastic theories of sediment transport were initiatedwith the seminal work of Einstein �1�, who introduced aBrownian motion model for particle motion. Since then,these theories were advanced by the need to reproduce theobserved statistics of sediment transport rates or particlemovement. In �2�, a birth-death process was proposed forsediment transport, which was later shown, in �3�, to be in-adequate as it failed to predict the heavy tails found in theprobability density functions �PDFs� of the number of mov-ing particles in a given observation window. In �4�, the birth-death model was extended to a birth-death-immigration-emigration model to reproduce the experimentally observednegative binomial distributions for the number of movingsediment particles. The stochastic nature of sediment particleentrainment has been widely recognized and considerable ef-forts have been invested in modeling this behavior �5–8�.The underlying assumption of these models is that the shearstress, which is the initiator for sediment entrainment, fol-lows a Gaussian distribution. However, many experimentalstudies have shown that the shear stress fluctuations do notfollow a Gaussian distribution, and in particular it has beenshown that they follow a Gamma distribution �e.g., �9,10��.The role of near-bed turbulence in sediment transport hasalso been recognized to play an important role �11,12�. How-ever, turbulence is well known to exhibit variability over arange of scales, and it is reasonable to ask whether this mul-tiscale variability shows its effect on sediment transport se-ries and bed elevation fluctuations.

In a recent study �13�, the dependence of the statistics ofsediment transport on time scale �sampling time� akin to thescale-dependent statistics of fully developed turbulence �14�was documented. Specifically, it was shown that the PDF ofsediment transport rates at small sampling times exhibits aheavy-tailed distribution which however approaches aGaussian distribution as the sampling time increases. To thebest of our knowledge, no stochastic model of sedimenttransport exists which reproduces this observed multiscalestatistical structure of sediment transport series. It is thescope of this paper to present such a model and discuss itsmathematical properties and its physical relevance to model-ing sediment transport.

The paper is structured as follows. In the following sec-tion a brief review of multiscale statistics of sediment trans-port series observed in a large-scale laboratory experiment isgiven. In Sec. III the application of a stochastic model, calledthe fractional Laplace motion, is proposed to characterize thesediment transport series and is shown that it is able to re-produce the observed statistics. In Sec. IV the proposedmodel is validated against the sediment transport series ob-tained from a large-scale laboratory experiment. Finally, dis-cussion and conclusions are given in Secs. V and VI.

II. MULTISCALE STATISTICS OF SEDIMENTTRANSPORT SERIES

A large-scale laboratory experiment was recently con-ducted in the Main Channel facility at the St. Anthony FallsLaboratory, University of Minnesota, in order to study sedi-ment transport dynamics in gravel and sand-bed rivers. Thedetails of the experimental facility can be found in�13,15,16�. Here we briefly describe one of the experimentsfrom which data were used in this study. The flume is 2.74 mwide and 55 m long, with a maximum depth of 1.8 m �seeFig. 1�. Gravel with a median particle size �D50� of 11.3 mmwas placed in a 20-m-long mobile-bed section of the 55-m-long channel. A constant discharge of water at 4300 liters persecond was released into the flume. At the downstream end

FIG. 1. Experimental flume facility at the St. Anthony FallsLaboratory, University of Minnesota.

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of the test section was located a bedload trap, consisting offive weighing pans of equal size that spanned the width ofthe channel. Any bedload sediment transported to the end ofthe test section of the channel would fall into the pans, whichautomatically recorded the weight of the accumulated sedi-ment every 1.1 s. Data were collected over a period of 30 honce a state of statistical equilibrium was reached �see�13,16��. The original series of 1.1 s sampling interval wereconverted to 2 min sediment accumulations via moving av-eraging in order to remove mechanical �due to vibration�noise present in the raw data �see �13,16��. Let us denote byS�t� the 2 min sediment accumulation series which is shownin Fig. 2. In this section, we present the multiscale analysisperformed on this sediment transport series.

The goal of a multiscale analysis is to quantify the mannerin which the statistics of the local fluctuations, or variabilityin a series, changes with scale. In order to investigate themultiscale structure of S�t� over a range of scales, differences�or increments� were computed at different scales �lags� r,denoted by �S�t ,r�, as

�S�t,r� = S�t + r�t� − S�t� , �1�

i.e., �S�t ,r� is the incremental sediment accumulation withina time interval r�t, where �t=2 mins. In �13�, “generalizedfluctuations” were used defined via wavelet transforms �act-ing as a differencing filter�. Notice that while S�t� can onlybe positive, the fluctuation series �S�t ,r� will have zeromean and can be both positive and negative. The estimates ofthe qth-order statistical moments of the absolute values ofsediment transport increments at scale r, also called the par-tition functions or structure functions, M�q ,r� are defined as

M�q,r� =1

Nr�t=1

Nr

��S�t,r��q, �2�

where Nr is the number of data points of sediment transportincrements at a scale r. The statistical moments M�q ,r� forall q completely describe the shape of the PDFs as the scaler changes. Statistical scaling, or scale invariance, requiresthat M�q ,r� is a power-law function of the scale, that is,

M�q,r� � r��q�, �3�

where ��q� is the so-called scaling exponent function. For ascale-invariant series, it has been shown that the function��q� completely determines how the PDF of the variablechanges with scale �e.g., �17,18��. The simplest form of scal-ing, known as simple scaling or monoscaling, is when thescaling exponents are a linear function of the moment order,i.e., when ��q�=Hq. In this case, the shape of the PDF re-mains the same over scales apart from a rescaling by a de-terministic function which depends on the single parameterH. If ��q� is nonlinear, the shape of the PDF changes overscales and more than one parameter is required to describethis change �e.g., �17,18��. In this case, the series is called amultifractal. For most processes the nonlinear relationship of��q� with q can be parameterized as a polynomial, and thesimplest form is a quadratic approximation,

��q� = c1q −c2

2q2. �4�

The multiscale analysis in this framework provides a com-pact way, using two parameters c1 and c2, of parametrizingthe change of the PDF over a range of scales. In parallel tothe statistical interpretation of these parameters, there is alsoa geometrical interpretation. Specifically, the parameter c1 isa measure of the average “roughness” of the series and c2,the so-called intermittency coefficient, is a measure of thetemporal heterogeneity of the abrupt local fluctuations in theseries �in fact, it relates to the variance of the so-called localHölder exponent which measures the local degree of nondif-ferentiability of the series �e.g., �18���. It is noted that using ahigher than second degree polynomial approximation of��q�, say a third degree polynomial, introduces a third pa-rameter c3, which is a measure of the third moment of thelocal differentiability of the series and it might be hard toaccurately estimate from a limited sample size of data. Thus,in most practical applications the approximation of ��q�curve is restricted to a quadratic function which is parameter-ized by c1 and c2. Estimation of the multifractal parameters,c1 and c2, can be performed in various ways. For example,one can use a quadratic fit to the whole ��q� curve �estimatedfor several values of q from the slopes of M�q ,r� vs r inlog-log space� or use the first two scaling exponents only,��1� and ��2�, or use the cumulant analysis method �e.g., �18�and references therein�. In this study, we use the quadratic fitto the ��q� curve for the estimation of the parameters c1 andc2.

The multiscale analysis described above was performedon the sediment transport series shown in Fig. 2. Figure 3�a�shows the scaling of the moments of the sediment transportincrement series �S�t ,r� with scale r. It is to note that thestructure functions follow a power-law relation in r over arange of scales from r=4 to 64 �8 to 128 min�. The scalingexponents of the structure functions M�q ,r� are plotted as afunction of the order of moments q in Fig. 3�b� for q=0.5,1 ,1.5, . . . ,3. We observe that ��q� has a nonlinear de-pendence on q, which is an indication of the presence ofmultifractality and the fact that the shape of the PDF changeswith scale. Figure 3�c� displays the PDFs of sediment trans-port increments at two scales, r=10 and r=60 �i.e., 20 and

0 50 100 150 200 250 300 350 400 450 500

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Time (min)

S(t

)

FIG. 2. Sediment transport series S�t� �in kgs� at a samplinginterval of 2 min, i.e., series of 2 min sediment accumulation.

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120 min sediment accumulations, respectively�. It is notedthat at smaller scales the PDF of the sediment incrementsdeviates from a Gaussian distribution and is close to a doubleexponential. The PDF, eventually, becomes Gaussian atlarger scales. The PDFs reported in Fig. 3�c� are for scalesthat fall within the scaling regime of the sediment data series�see Fig. 3�a��. The dependence of the statistics of the sedi-ment transport rates on scale has also been documented infield observations �see �19� and a discussion in �13��. Asdiscussed above, we estimated the parameters of multifrac-

tality by approximating the ��q� curve in Fig. 3�b� as a qua-dratic function in q and the estimates obtained together withtheir 95% standard errors were c1=0.41�0.005 and c2=0.04�0.004. It is noted for comparison that the c2 estimateof velocity fluctuations in fully developed turbulence is ofthe order of 0.03 �14�. We emphasize that no existing sto-chastic model for sediment transport addresses the issue ofstatistical scale dependence documented in experimental andfield observations. In the following section, we propose astochastic model for sediment transport which exhibits the

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log2(r)

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mins

(a)

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0.2

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q

τ(q)

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10−1

100

δS(t,r)/<(δS(t,r))2>1/2

Pdf[δS(t,r)/<(δS(t,r))2>1/2]

(c)

FIG. 3. �a� Structure functionsof sediment transport series. Verti-cal lines delineate the scaling re-gime which is between 8 and 128min �see top horizontal axis�. �b�Estimated ��q� curve �solidpoints� from the slopes of struc-ture functions and a quadratic fit�solid line�. Deviation from thestraight line establishes the pres-ence of multifractality �see textfor parameter values�. �c� Changein PDF of sediment transport in-crements with scale. The soliddots correspond to PDF at incre-ments of r=10 �20 min� and +correspond to increments at r=60�120 min�. The solid line indicatesa Gaussian PDF.

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observed change in PDFs of sediment transport incrementsover scales, reproduces the multifractal behavior of the ex-perimental data series, and provides the potential for relatingthe observed macroscale statistics to the microscale dynam-ics of sediment particle movement.

III. PROPOSED MODEL: FRACTIONAL LAPLACEMOTION

A. Brownian motion

Brownian motion is widely recognized to be a specialcase of a continuous time random walk �CTRW�. In general,CTRWs specify the particle location xi at a time ti by theiterative discrete equations �e.g., �20,21��,

xi+1 = xi + �i, �5a�

ti+1 = ti + �i, �5b�

where ��i ,�i� is a set of random numbers drawn from a PDF��� ,��. One can recast the above equations in the followingform:

tn = �i=1

n

�i, �6a�

x�t� = �i=1

n

�i, �6b�

where t� �tn , tn+1�. The CTRW is said to be decoupled whenthe random variables �i and �i are mutually independent.Brownian motion is a special case of a decoupled CTRWwhere �i are independent identically distributed �i. i. d� ran-dom variables drawn from a Gaussian distribution and �i arei. i. d random variables sampled from an exponential distri-bution. It is to note that the increments of Brownian motionfollow a Gaussian distribution. However, the increments ofmost natural phenomena often show deviation from GaussianPDFs and this has prompted the introduction of other sto-chastic processes such as Lévy walks and continuous-timeLévy flights, where the random variables �i and/or �i aresampled from heavy-tailed PDFs. However, such processesdo not have all of their statistical moments convergent. Forexample, Lévy walks and Lévy flights do not have conver-gent second moments �22�. It is also noted that modeling realdata with such processes typically requires an exponentialtruncation of the algebraic decays �23� or sometimes evenmilder than algebraic decay �24�. Correlation and long-rangedependence in the observed data can be modeled by relaxingthe independence assumption in sampling �i and/or �i or byrelaxing the independence assumption of a decoupledCTRW. Fractional Brownian motion, denoted by BH�t�, is adecoupled CTRW starting at zero and has the following cor-relation function:

E�BH�t�BH�s�� =1

2��t�2H + �s�2H − �t − s�2H� , �7�

where E� . � denotes the expectation operator and H is a pa-rameter of fractional Brownian motion called the Hurst ex-

ponent. For H=0.5, the fractional Brownian motion reducesto the standard Brownian motion with independent incre-ments. For other values of 0�H�1, BH�t� is called the frac-tional Brownian motion and its increments are positively cor-related for H0.5 and negatively correlated for H�0.5.

An extension of Brownian motion, or fractional Brownianmotion, can be obtained via subordination. The notion ofsubordination was originated by Bochner �25�. One can ob-tain a subordinated stochastic process Y�t�=X�T�t�� by ran-domizing the clock time of a stochastic process X�t� using anew time t�=T�t�. The resulting process Y�t� is said to besubordinated to the so-called parent process X�t��, and t� iscommonly referred to as the operational time �26�. We pro-pose the application of subordination of fractional Brownianmotion �called fractional Laplace motion� as an extension tothe Brownian motion model proposed by Einstein for sedi-ment transport �2�. In the following subsection, we describethe properties of subordinated fractional Brownian motion.

B. Fractional Laplace motion

Fractional Laplace motion is a subordinated stochasticprocess, whose parent process is fractional Brownian motionand the operational time is a Gamma process �27�,

L�t� = BH�t� , �8�

where BH�t� is fractional Brownian motion with Hurst expo-nent 0�H�1 and t represents a Gamma process for anyt�0. The increments of the Gamma process �t+s−t� havea gamma distribution with shape parameter �=s and scaleparameter =1, i.e.,

f�x� =1

����x�−1e−x. �9�

For H=0.5 the subordinated process L�t�=BH�t� is calledthe Laplace motion.

Increments of the fractional Laplace motion defined byY�t ,r�=L�t+r�−L�t�, called the fractional Laplace noise,form a stationary process. Fractional Laplace noise has threeparameters, namely, the Hurst exponent of the parent processH, the variance of the parent process BH�t� at the smallestscale t=1, i.e., �2=Var�BH�1��, and the shape parameter ofthe Gamma process �t�, �. The variance of the fractionalLaplace noise can be expressed as a function of the scale rand its parameters as �27�

Var�Y�t,r�� = �2�2H + r/���r/��

. �10�

The covariance function of the fractional Laplace noise at agiven scale r, defined as ��n�=E�Y�t ,r�Y�t+n ,r��, can beexpressed in terms of its parameters for any n�1 as

��n� =�2

2��2H + �n + 1�r/��

��n + 1�r/��+

�2H + �n − 1�r/����n − 1�r/��

− 2�2H + nr/��

��nr/��� . �11�

Fractional Laplace noise is positively correlated for H0.5

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and is negatively correlated for H�0.5. In particular, frac-tional Laplace noise exhibits long-range dependence for H0.5.

The fundamental difference between fractional Laplacemotion and other similar stochastic processes such as frac-tional Brownian motion and Lévy motion is that in the lattertwo cases the PDFs of the increments remain Gaussian andLévy stable, respectively, at all scales. In fractional Laplacemotion, the PDFs of the increments are variable with scalewith Laplace PDFs at small scales and as the scale increasesthe PDFs approach Gaussian. In particular, fractionalLaplace motion deviates from the classical self-similarityand shows stochastic self-similarity �27�. The Laplace PDFemerges from a different and less well-known central limittheorem called the geometric central limit theorem, whichstates that the sum of a random number of independent iden-tically distributed variates with finite variance is asymptoti-cally Laplace if the random count is geometrically distrib-uted �28�. In fact, the Laplace PDF can be considered as aGaussian PDF with a random variance or spread �29�. Giventhe stochastic self-similarity extensively documented in sedi-ment transport series �in �13� and also in Sec. II of this pa-per�, the subordination of the fractional Brownian motionmodel proposed herein offers an attractive and simple exten-sion to Brownian motion for particle movement, as demon-strated in more detail in the next section.

IV. FRACTIONAL LAPLACE MOTION MODEL FORSEDIMENT TRANSPORT

The physical relevance of the fractional Laplace motion tomodel sediment transport is argued on the basis that the no-tion of operational time acknowledges the randomness in theentrainment time experienced by sediment particles whichare subject to a varied range of velocities in turbulent flows.It is known that turbulent velocity fluctuations themselvesexhibit intermittency and possess a multifractal behavior�e.g., �14��. Turbulent velocity “sweeps” and “bursts” areexpected to influence particle motion and introduce a multi-scale variability in the fluctuations of the resulting sedimenttransport series. In groundwater hydrology, the notion of op-erational time has been used to acknowledge the fact thattime passes faster for particles in higher velocity zones�30,31�. Along these lines, a subordinated Brownian motionmodel has been proposed to model hydraulic conductivity�28� and connections between turbulent velocities and het-erogeneous sediment properties have been proposed �32�.

In the following subsections we study the multiscale prop-erties of fractional Laplace motion and show that fractionalLaplace motion reproduces the intricate stochastic structureshown by the sediment transport series. Further, we elaborateon the model parameter fitting to the sediment transport se-ries.

A. Multifractal properties of fractional Laplace motion

In order to study the self-similar behavior of fractionalLaplace motion, we first study the analytical behavior of thestructure functions of fractional Laplace motion. The struc-

ture functions of fractional Laplace motion for �=1 can bewritten in terms of its parameters H and � as �27�

M�q,r� =2q

��1 + q

2��Hq + r/��

�r/��. �12�

Statistical scaling or self-similar behavior requires that thestructure functions follow a power-law relationship in scales.Figure 4�a� shows the structure-function dependence on

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log2(M

(q,r))

q = 0.5

q = 1.0

q = 1.5

q = 2.0

q = 3.0

q = 2.5

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τ(q)

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δL(t,r)/<(δL(t,r))2>1/2

Pdf[

δL(t,r)/<(

δL(t,r))2 >

1/2 ]

(b)

(a)

(c)

FIG. 4. �a� Structure functions of fractional Laplace motion fora set of chosen parameters H=0.4, �=3.0, and �=1 computed fromEq. �12�. The vertical lines correspond to the scaling regime of thesediment transport series which is from scales of r=4 to r=64. �b�Estimated ��q� curve �solid points� from the fitted slopes of thestructure functions. The solid line indicates a quadratic fit and thenonlinear dependence of ��q� on q establishes that fractionalLaplace motion shows a multifractal behavior in the scales underconsideration. �c� Change of PDF of increments of simulated frac-tional Laplace motion series. Solid dots correspond to PDF of in-crements at r=10 and + to r=60. Solid line indicates a GaussianPDF.

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scales in log-log space for an arbitrary choice of the param-eter values, H=0.4 and �=3.0. �These values of H and � areused for illustration of the model properties and the estima-tion of these parameters is discussed more thoroughly in thenext subsection�. It is to note that from Fig. 4�a� that al-though Eq. �12� does not analytically accept a power-lawexpression on r, for all practical purposes, fractional Laplacemotion can be approximated by a self-similar process, i.e.,the structure functions show a power-law relationship inscales at least for the range of scales which coincide with thescaling regime of sediment transport series �scales or lags ofr=4 to r=64�. Plotting the ��q� curve �estimated from theslopes of M�q ,r� vs r in log-log space within the abovescaling regime� one can see that the scaling exponents, ��q�,show a nonlinear dependence on the order of moments q �seeFig. 4�b��. It is to note that the scaling exponents ��q� areindependent of the variance of the parent process �2. Thechange in PDF of the increments of fractional Laplace mo-tion with scale is shown in Fig. 4�c�, where the PDF at smallscales �r=10 in Fig. 4�a�� shows a double-exponential behav-ior and it eventually tends to a Gaussian distribution for largescales �r=60 in Fig. 4�a��. The above results document thatfractional Laplace motion can be approximated by a stochas-tic self-similar process in an intermediate range of scales andwithin those scales it exhibits a multifractal behavior. At thelimit of very large time scales, i.e., as r→�, fractionalLaplace motion tends to a fractional Brownian motion with��q� a linear function of q �i.e., monofractal behavior�.

It is interesting to note from Eq. �12� that the second-order structure function of Laplace motion �H=0.5 and q=2� obeys a power-law relationship in scales and in particu-lar it shows a linear dependence on scales

M�2,r� = 2

���1.5��r , �13�

yielding an exponent of ��2�=1. This implies that Laplacemotion has self-similar second-order moments, i.e., it showsa log-log linear power spectrum �although higher order mo-ments are not exact power laws�. In the next subsection weelaborate on the parameter estimation of the fractionalLaplace motion from the sediment transport series.

B. Model fitting

As seen in the previous section, fractional Laplace motionhas three parameters H, � and �. The scale parameter of theoperational time PDF, , is 1 by the definition of fractionalLaplace motion �27�. Estimation of the parameters H and �from the sediment transport series is performed by minimiz-ing the mean squared error between the empirical and theo-retical ��q� curves. The mean squared error, denoted byMSE, is a function of H and � and is independent of �,

MSE�H,�� = �q

��m�q� − �̂�q��2, �14�

where �̂�q� are the estimated scaling exponents of the sedi-ment transport series �see Fig. 3�b�� and �m�q� are the scalingexponents of the fractional Laplace motion model which arecomputed from the slopes of the theoretical M�q ,r� versus r

within the scaling regime of the sediment transport series�4�r�64� in the log-log space �see Fig. 4�b��. Minimiza-tion of the mean squared error for the sediment transportseries yields a Hurst exponent of H=0.39 and a shape pa-rameter of �=6.8. It is to note that the multiscale structure offractional Laplace motion model is determined by the param-eters H and �. Further, we estimate the parameter � by mini-mizing the mean squared error between the variance of theincrements of sediment transport series and the fractionalLaplace noise for H=0.39 and �=6.8 over the scaling regime�4�r�64�,

� = Min �r=4

r=64

�Var��S�t,r�� − Var�Y�t,r���2, �15�

where Var��S�t ,r�� is the variance of the increments of sedi-ment transport series and Var�Y�t ,r�� is the variance of frac-tional Laplace noise at the scale r, given by Eq. �10�. Thevalue of � estimated using Eq. �15� was �=0.296. The mul-tifractal parameters of the fractional Laplace motion modelcomputed with the estimated parameters of H=0.39 and �=6.8 were c1=0.41 and c2=0.041, which compare very wellto the values estimated from the sediment transport data ofc1=0.41 and c2=0.04. �Note that c1 and c2 were not useddirectly in the model fitting which was done via Eq. �14� onthe whole ��q� curve�. As a result the model and the data-estimated ��q� curves are indistinguishable. Figure 5�a�shows the increments of sediment transport series at a scaleof r=20 or 40 min �note that this scale lies within the scalingregime of the sediment transport series�. For visual compari-son, the fractional Laplace noise simulated series with theestimated parameters H=0.39, �=6.8, and �=0.296 at thesame scale is shown in Fig. 5�b�.

As noted in the previous section, fractional Laplace noiseis negatively correlated for H�0.5. Figure 6�a� shows theautocorrelation function of the increments of sediment trans-port series at the scale r=20 �40 min�. The data show anegative correlation in the scaling regime of the sedimenttransport series for small lags. This is qualitatively consistentwith the fractional Laplace noise model which shows a nega-tive correlation for the estimated parameter values �see Fig.6�b��. The increments of fractional Laplace motion at smallscales follow a Laplace PDF which eventually becomesGaussian at larger scales. Figure 7�a� shows the PDF of sedi-ment transport increments at a scale of r=4 which is thebeginning of the scaling regime of the sediment transportseries. A Laplace PDF provides a good fit to the incrementsat that scale. As noted in Fig. 7�b�, the PDF of sedimenttransport increments at a scale of r=64 �128 min� tends to aGaussian PDF. Thus, one can see that the sediment transportseries are consistent with the properties of fractional Laplacemotion within the scaling regime.

V. DISCUSSION

In the previous section we established the fact that thefractional Laplace motion model is able to reproduce theintricate stochastic structure of the observed sediment trans-port series over a range of scales and also reproduce the

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change of the PDFs of increments of sediment transport se-ries in the scaling regime. In this section, we discuss thephysical significance of the notion of operational time insediment transport series. Near-bed turbulence is known toplay an important role in sediment transport �12�. Turbulentvelocity fluctuations pick up sediment particles and transportthem over long distances. However, since the turbulent ve-locities themselves are known to exhibit variability over alarge range of scales, the entrainment time experienced bythe sediment particles is also expected to carry some of thisvariability. This consideration leads to a randomization oftime over which a sediment particle is operated upon, assediment particles in different velocity zones experience timeto move faster or slower depending on whether they are in ahigh- or low-velocity zone, respectively. Thus, the notion ofoperational time can arise due to the stochastic nature ofsediment particle entrainment. It is interesting to note thatthe turbulent velocity fluctuations themselves exhibitLaplace and stretched Laplace distributions at small scalesand their PDFs become Gaussian at larger scales �33�. It isalso interesting to note that the rate of sediment particle en-trainments, which are proportional to the shear stress fluctua-tions at the bed, have been reported to follow a Gammadistribution �9�. Both these observations are qualitativelyconsistent with the fractional Laplace motion model for sedi-ment transport proposed in this paper.

The observed multiscaling and intermittency in sedimenttransport series �macroscale behavior� was shown to arise by

the introduction of the notion of operational time inBrownian-type particle movement �microscale behavior�.Thus, while the model parameters H and � relate to the �un-observed� particle movement statistics, they are estimatedfrom the �observed� sediment transport statistics, and specifi-cally from their multiscale behavior concisely parameterizedvia the parameters c1 and c2. It is of interest to study how theparameter space of �H ,�� relates to that of �c1 ,c2� in order togain insight on model sensitivity and the physical meaning ofthe parameter � which characterizes the variability of theparticle motion. We compute the multifractal parameters c1and c2 for different values of the model parameters H and �by evaluating M�q ,r� from Eq. �12�, estimating ��q� in therange 4�r�64, and approximating the ��q� curve as a qua-dratic function in q �Eq. �4��. Figure 8 shows the contourplots of c1 and c2 for different values of H and �. It is to notethat the average “roughness” of the sediment series, quanti-fied by the parameter c1, is strongly dependent on the Hurstexponent of the fractional Brownian motion H �see Fig. 8�a��and not as much on the parameter � of the operational time.On the other hand, from Fig. 8�b�, one can see that the inter-mittency coefficient c2 is strongly dependent on the shapeparameter � of the distribution of operational time for agiven value of H. In particular, for a given value of H, thevalue of c2 is higher for a higher value of �. One way tounderstand this is to note that for higher values of � theGamma distribution has a higher variance. Thus, for higher

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FIG. 5. �a� Comparison of the increments of the sediment trans-port series in kgs at scale r=20 �40 min� and �b� the same scaleincrements of simulated fractional Laplace motion series with H=0.39, �=6.8, and �=0.3. The values of H and � were obtained byminimizing the mean squared error defined in Eq. �14�. The value of� was obtained using Eq. �15�. The scale of r=20 was chosen forcomparison as it lies within the scaling regime of the sedimenttransport series.

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values of � the operational time is sampled from a distribu-tion with higher variance and this variability in the opera-tional time shows up as a higher intermittency coefficient inthe sediment transport series �larger degree of temporal het-erogeneity in bursts of sediment transport increments�. It isemphasized that estimation of the parameter values of thefractional Laplace motion, H and �, was performed throughthe scaling exponents of the structure functions of the sedi-ment transport series �Eq. �14��. Direct estimation of the pa-rameters H and �, or for that matter direct assessment of thewhole statistical structure of operational time from observa-tions, would require access to series of particle entrainmentwhich are difficult to make and are not available in the ex-perimental setting studied here. Rather, the present study at-tempted a physical insight via relating the macroscale statis-tics of the sediment series to the microscale dynamics ofparticle movement.

VI. CONCLUDING REMARKS

In this work we proposed the adaptation of fractionalLaplace motion as a stochastic model for sediment transport.Fractional Laplace motion arises from randomization of theclock time in fractional Brownian motion, and introduces thenotion of operational time. The physical significance of op-

erational time in the context of sediment transport was rea-soned on the basis that the stochastic nature of turbulentvelocity fluctuations near the bed induces stochasticity inparticle entrainment and, therefore, the time over which par-ticles are in motion. The proposed model was shown able toreproduce the multiscale statistics of sediment transport se-ries and was validated against a data set from a large-scalelaboratory experiment. The effect of the model parameters onthe multifractal parameters of sediment transport series wasalso discussed. Although direct estimation of the model pa-rameters would require particle-scale observations, it wasshown here that an indirect estimation based on the statisticsof sediment transport series is possible. We see this work asa step toward relating the microscale dynamics of particlemovement to the macroscale statistics of sediment transportvia minimum complexity stochastic models.

ACKNOWLEDGMENTS

This work has been supported by the National Center forEarth-surface Dynamics �NCED�, a Science and TechnologyCenter headquartered at the University of Minnesota andfunded by NSF under Agreement No. EAR-0120914NGAand also under Grants No. EAR-0824084 and No. EAR-0835789. We would like to thank Mark M. Meerschaert forhelpful discussions on analytical results of the fractionalLaplace motion. Computer resources were provided by theMinnesota Supercomputing Institute, Digital TechnologyCenter, at the University of Minnesota.

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FIG. 7. Change in PDF of sediment transport increments in thescaling regime. �a� Laplace PDF �solid line� provides a good fit tothe PDF of sediment transport increments at r=4 �8 min; beginningof the scaling regime� and �b� the PDF of sediment transport incre-ments becomes Gaussian �solid line� at r=64 �128 min; ending ofthe scaling regime�.

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