# Probability distributions of bed load transport rates: A new derivation and comparison with field data

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2 Birnbau Saunders distribution [Birnbau and Saunders, 1968] arises as a general approxiation hen no explicit assuptions about the distribution of aiting ties are ade. By assuing exponentially or Poisson distributed aiting ties, one arrives at the Poisson and Gaa distributions, respectively. The distribution functions are copared to a large field data set fro the Pitzbach, Austria [Rickenann and McArdell,2008;Turoski and Rickenann, 2009]. 2. Distribution Functions of Bed Load Transport Rates 2.1. The Birnbau Saunders Distribution [5] In practice often the sedient flux at a channel cross section is of interest. Particles arrive at varying intervals, and e ant to kno the total sedient volue arriving ithin a certain tie period and its variability. For the derivation of the distribution function, I ake the folloing foral assuptions: [6] 1. The aiting tie beteen the arrivals of individual bed load particles (interarrival tie) at a cross section is a stochastic variable ith an unspecified distribution ith ell defined ean and variance s 2. [7] 2. Within each easureent interval, enough particles arrive such that the central liit theore and the la of large nubers are applicable. [8] Iplicit in these to assuptions is the notion that the particles are actually countable and that individual arrivals are statistically independent. The folloing derivation ay thus not be applicable to environents here transport by bed for otion is doinant. This constraint and the to assuptions are discussed in ore detail in section 4.1. [9] Consider the arrival of individual sedient particles at the easureent cross section under steady conditions. The interarrival tie is described by a rando variable t ith ean and variance s 2. The tie T N at the arrival of the N th particle is then T N ¼ T N 1 þ N : Assuing that there are a large nuber of particles, the central liit theore ay be applied. Hence, the probability density function (pdf) for the total tie T N (ith corresponding stochastic variable t N ) at the arrival of the N th particle is approxiately noral. pdfðt N ðnþþ ð1þ ( ) 1 ð pffiffiffiffiffiffiffiffiffi exp t N NÞ 2 2N 2N 2 : ð2þ Note that the noral distribution in equation (2) is not assued but arises as a general approxiation fro the central liit theore. So far, no assuptions have been ade on the underlying distribution of interarrival ties. The size of particles and thus their ass follo a certain site specific distribution. Since N is large, the la of large nubers applies and the total ass is N, here is the ean ass of a single particle. Hence, the cuulative distribution function cdf(t N )oft N is the cuulative noral distribution F((x )/s) of the rando variable x ith ean and standard deviation s, hich is given by cdfðtþ ¼PT ð N ðþ t N Þ 8 Z 1 t N 2 9 >< ¼ q ffiffiffiffiffiffiffiffiffiffiffi 1 2 exp 2 2 >: 0 t N 1 ¼ F rffiffiffiffi B A : >= >; dt The event {M(t N ) } is equivalent to the event {T N () t N }, here M(t N ) is the rando variable representing the ass at tie t N. The cdf() ofm at given t N is then 0 1 cdf ðþ ¼PMðtÞ ð Þ ¼ 1 PTðÞ ð tþ ¼ F t Brffiffiffiffi A ; here t N has no been replaced by a constant t denoting the easureent interval. The probability density function is 8 pdfðjtþ ¼ dgðþ d ¼ þ t 2 9 >< r ffiffiffiffiffiffiffiffiffi 2 2 exp t >= >: 2 2 >; : ð5þ Reparaeterizing ith b = t/ and g 2 = s 2 /t to get the standard for of the Birnbau Saunders distribution yields ( ) pdfðjtþ ¼ p þ 2 ffiffiffiffiffiffiffiffiffiffiffiffi exp ð Þ : ð6þ The Birnbau Saunders distribution has previously been proposed to odel the ties at failure due to fatigue of aterial under cyclic stresses and belongs to a to paraeter exponential faily [Birnbau and Saunders, 1968]. The foralis of the derivation given above closely follos the one developed by Desond [1985]. [10] Equation (6) can easily be reritten in ters of fluxes (defined as Q s = /t) and the pdf of the bed load transport rate Q s in a given easureent interval t is given by ( ) pdfðq s jtþ ¼ Q s þ 0 ð pffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp Q s 0 Þ 2 2Q s 2 0 Q s 2Q s 0 2 ð3þ ð4þ : ð7þ Here b = / is a scale paraeter, and g 2 = s 2 /t is a shape paraeter as before. The expectation value E(Q s )of equation (7) is given by EQ ð s Þ ¼ 0 1 þ 2 2 ¼ 1 þ t 2 : ð8þ And the variance var(q s ) is given by varðq s Þ¼ þ 52 ¼ t þ t 2 2 : ð9þ 2of10

3 The standard deviation std(q s ) is equal to the square root of var(q s ). More inforation on the properties of the distribution can be found in the ork of Johnson et al. [1995]. As ould be expected, the expectation value E(Q s ) approaches the constant value / as t goes to infinity. This gives a constraint on the iniu length of the easureent interval t: it needs to be uch larger than s 2 /2 2 to obtain a reliable estiate of transport rates hich is independent of the length of the easureent interval and thus coparable for different streas or for the sae strea at different discharges [cf. Singh et al., 2009]. [11] The Birnbau Saunders distribution as derived ithout assuptions about the probability distribution of interarrival ties. I ill no outline to additional derivations, assuing interarrival ties, hich are distributed according to the exponential distribution and a continuous for of the Poisson distribution. It is not y ai here to argue that either of these distributions is the correct one, although at least the exponential distribution has been predicted fro certain physically based stochastic transport theories [e.g., Ancey et al., 2008]. Rather, I ant to sho (1) ho physically based results can be incorporated into the fraeork developed above and (2) ho such assuptions can yield various plausible probability distributions for bed load transport rates Exaple 1: Exponentially Distributed Interarrival Ties [12] Let the interarrival ties be exponentially distributed ith a pdf of the folloing for: pdfðþ ¼ 1 exp : ð10þ The tie at the N th arrival is then distributed according to the gaa distribution pdfðt N ðnþþ ¼ 1 N t N 1 N GðNÞ exp t N : ð11þ The gaa function G(x) is defined by GðxÞ ¼ Z 1 0 z x 1 e z dz: ð12þ Folloing the reaining steps of the derivation as given above one obtains the cuulative distribution function (cdf) of asses G ; t cdfðþ ¼ G : ð13þ Here the upper incoplete gaa function G(x, y) is defined by Gðx; yþ ¼ Z 1 y z x 1 e z dz: ð14þ Equation (13) is a continuous for of the Poisson distribution ith the folloing pdf: exp t t = : pdfðþ ¼ G ð15þ Here the usual factorial is replaced by the gaa function (equation (12)), hich is a continuous interpolation of the factorial function. This generalizes the Poisson distribution to apply for continuous variables. In the reainder of the article henever a Poisson distribution is entioned, I generally refer to a continuous version analogous to equation (15) Exaple 2: Interarrival Ties Distributed After a Continuous Version of the Poisson Distribution [13] Alternatively, let the interarrival ties be distributed after a continuous for of the Poisson distribution ith a pdf analogous to equation (15) pdfðþ ¼ 1 expf 1g 1 = : ð16þ G = Here the expected nuber of occurrences is equal to one, because the interarrival tie t as noralized by the ean. Note that this noralization of t ith is necessary to keep the equation diensionally consistent. Of course, the ter 1 = evaluates to one; I left it in to ake the connection to equations (15) and (17) ore explicit. The tie at the N th arrival is also based on a continuous Poisson distribution pdfðt N ðnþþ ¼ 1 expf Ng N t N= : ð17þ G tn Folloing through the reaining steps of the derivation, the asses are then distributed according to the cdf 1 G t ; cdfðþ ¼ : ð18þ G t = Equation (18) gives the cdf for the gaa distribution, and the corresponding pdf is pdfðþ ¼ 1 t t = 1 n exp o : ð19þ G t = The Birnbau Saunders distribution (equation (7)) is an approxiation for both the continuous Poisson distribution (equation (15)) and the gaa distribution (equation (19)), hich orks ell for large transport rates. 3. Testing the Distributions: The Pitzbach Sedient Transport Observations 3.1. Field Site: Pitzbach, Austria [14] The Pitzbach is a sall glacially fed strea in southestern Austria near the village of Ist on the southern side of the Inn valley (Tyrol). It is a gravel bed river ith a 3of10

5 Figure 3 5of10

6 (equation (19)), the distributions due to Haaori [1962] and Carey and Hubbell [1986] are tested, hich ere derived for bed for doinated transport. The Haaori distribution is valid in the range fro zero to four ties the ean transport rate Q and has a pdf of the for The cdf has the for pdfðq s Þ ¼ 1 ln 4Q : ð20þ 4Q Q s cdfðq s Þ ¼ Q s 4Q 1 þ ln 4Q Q s : ð21þ The distribution is top bounded, i.e., transport rates ith values larger than four ties Q are assigned a probability of zero. Carey and Hubbell [1986] generalized the odel and derived the pdf pdfðq s = n 1 Þ ¼ Q1 ax Qs = 1 = n 1 Q ð1 nþq 1 = n ax : ð22þ Here n is a constant, hich is generally saller than one [Goez et al., 1989], and the axiu possible transport rate Q ax is given by Q ax ¼ 2ðn þ 1ÞQ : ð23þ Figure 4. Variation of the best fit ean value and the standard deviation ith discharge for each bin ith at least 15 data points. The coefficient of variation is approxiately constant for all bins ith at least 80 data points. The fit values for the exponential y = A exp(b Q) and the poer la y = aq b arefortheeanvaluein(a)a = /s, B =0.40s/ 3, a = ( 3 /s) 1 b,andb = 3.58, and for the standard deviation in (b) A = /s, B =0.31s/ 3, a = ( 3 /s) 1 b, and b = directly related to discharge at a single location) for bed load transport rates at Goodin Creek. Although this relationship needs to be confired for other streas, the assuption std(q s ) / E(Q s ) ay be a good first approxiation for the standard deviation Coparison to Other Distribution Functions [18] Next, the Pitzbach data is copared to other distributions. In addition to a continuous for of the Poisson distribution (equation (15)) and the gaa distribution The cdf corresponding to the pdf in equation (22) is given by pdfðq s Þ ¼ Q s = Q ð1 n 1 = n n Qs = Q ÞQ ax ð1 nþq 1 = n ax : ð24þ Siilarly to the Haaori distribution, the Carey Hubbell distribution is top bounded at Q ax. In the Pitzbach, axiu transport rates exceed ean transport rates by a factor of four at ost discharges. Thus, the Haaori and Carey Hubbell distributions are clearly of liited value to describe the data. Hoever, in the range for hich they are valid, both odels give a reasonable fit to the data (Figure 5). With a value of n = 0.5, the Carey Hubbell distribution closely traces the data for lo transport rates (belo about the 50 percentile; Figure 5c), hile the Haaori distribution fits ell over the hole range of its validity. [19] Both the continuous Poisson and the gaa distribution give better fits to the data than the Birnbau Saunders distribution for lo transport rates. The gaa distribution gives a good fit for the hole data range, ith axiu deviations of 5%. All three distributions (Birnbau Saunders, continuous version of the Poisson distribution, Figure 3. Cuulative probability distribution and probability density functions (large figures labeled A1, etc.) of the observed loads (open circles, histogra) and the best fit (solid line) using equation (7) for discharges of (a) /s (292 data points), (b) /s (1223 data points), (c) /s (477 data points), and (d) /s (335 data points). The corresponding Shields nuber estiated for the artificial cross section at the easureent site is given on the plot. Sall figures sho (left, 2) percent percent plots and (right, 3) probability ratio plots for the sae discharges. Percent percent plots allo a good optical evaluation of the fit in the loer percentiles, hile ratio plots allo the evaluation of the fits in the right hand tail. 6of10

7 Figure 5. Coparison of the Haaori, Carey Hubbell, Birnbau Saunders, gaa, and Poisson distributions to the Pitzbach data at a discharge of /s (1223 data points). (a) Probability density functions on a seilogarithic plot. (b) Cuulative probability functions. (c) Percent percent plots. This visualization allos an assessent of the goodness of fit for lo and ediu transport rates. (d) Percentileratio plots. This visualization allos the assessent of the goodness of fit in the tail region. The gaa distribution gives the best fit ith axiu deviations of 5%. The Birnbau Saunders distribution provides a reasonable approxiation especially to the right hand tail. gaa) converge onto the right hand tail at high transport rates (Figure 5d). 4. Discussion 4.1. The Birnbau Saunders Distribution [20] The assuptions ade in the derivation of the Birnbau Saunders distribution arrant a discussion of the generality of the function. Since the arguent is purely statistical, the function is independent of the physics of sedient transport and should be idely applicable. Hoever, the applicability is restricted to systes here individual particle arrivals are countable and statistically independent, and the function ay not be applicable in environents here transport is doinated by bed for otion, as in any sand bed streas. In such environents, distribution functions specifically developed for dune otion, such as the Haaori or the Carey Hubbell distributions, ay yield better results (see for exaple the ork of Carey [1985] for field testing of the Haaori distribution in a sand bed river). It ay be possible to adapt the derivation of the Birnbau Saunders distribution by not considering the arrival of individual particles but the arrival of individual bed fors. The final distribution of transport rates ould then have the sae for. Hoever, the assuptions in the derivation set a constraint on the length of the easureent interval needed to ake the distribution applicable to a data set: it needs to be long enough such that the nuber of particles arriving ithin it is large enough such that the central liit theore and the la of large nubers apply. In natural channels discharge can fluctuate quickly and it ay not be easy to find a suitable easureent interval that ensures that hydraulic conditions are approxiately constant hile a sufficient nuber of bed fors arrive. [21] The constraint on the easureent interval iplies that at sall transport rates, i.e., hen only fe particles arrive, the Birnbau Saunders distribution ill necessarily break don. This ay be one of the reasons for the unsatisfactory fit of the function to the Pitzbach data for sall transport rates (cf. Figure 3). The rate of convergence to the noral distribution in the central liit theore can be 7of10

10 Einstein, H. A. (1937), Der Geschiebetrieb als Wahrscheinlichkeitsproble, Mitt. Versuchsanst. Wasserbau Eidg. Tech. Hochsch. Zürich, Rascher, Zürich, Sitzerland. Esséen, C. G. (1942), On the Liapounoff Liit of Error in the theory of probability, Ark. Mat. Astr. Och Fys., 28A(9), Frey, P., C. Ducottet, and J. Jay (2003), Fluctuations of bed load solid discharge and grain size distribution on steep slopes ith iage analysis, Exp. Fluids, 35(6), , doi: /s Goez, B., and M. Church (1989), An assessent of bed load transport forulae for gravel bed rivers, Water Resour. Res., 25(6), Goez, B., R. L. Naff, and D. W. Hubbell (1989), Teporal variations in bed load transport rates associated ith the igration of bed fors, Earth Surf. Processes Landfors, 14, Haaori, A. (1962), A theoretical investigation on the fluctuations of bed load transport, Rep. R4, Delft Hydraulics Laboratory, Delft, Netherlands. Hassan, M. A., and M. Church (2000), Experients on surface structure and partial sedient transport on a gravel bed, Water Resour. Res., 36, , doi: /2000wr Hoey, T. (1992), Teporal variations in bed load transport rates and sedient storage in gravel bed rivers, Prog. Phys. Geog., 16, Hofer, B. (1987), Der Feststofftransport von Hochgebirgsbächen a Beispiel des Pitzbachs, Österreichische Wasserirtschaft, 39, Johnson, N. L., S. Kotz, and N. Balakrishnan (1995), Continuous univariate distributions, 2nd ed., vol. 2, pp , John Wiley, Ne York. Kirchner, J. W., W. E. Dietrich, F. Iseya, and H. Ikeda (1990), The variability of critical shear Stress, friction angle, and grain protrusion in aterorked sedients, Sedientology, 37, Kuhnle, R. A., and J. B. Southard (1988), Bed load transport fluctuations in a gravel bed laboratory channel, Water Resour. Res., 24, Kuhnle, R. A., and J. C. Willis (1998), Statistics of sedient transport in Goodin Creek, J. Hydrol. Eng., 124, Lague, D., N. Hovius, and P. Davy (2005), Discharge, discharge variability, and the bedrock channel profile, J. Geophys. Res., 110, F04006, doi: /2004jf Lisle, I. G., C. W. Rose, W. L. Hogarth, P. B. Hairsine, G. C. Sander, and J. Parlange (1998), Stochastic sedient transport in soil erosion, J. Hydrol., 204, Lisle, T. E., Y. Cui, G. Parker, J. E. Pizzuto, and A. M. Dodd (2001), The doinance of dispersion in the evolution of bed aterial aves in gravel bed rivers, Earth Surf. Processes Landfors, 26, , doi: /esp.300. Papanicolaou, A. N., P. Diplas, N. Evaggelopoulos, and S. Fotopoulos (2002), Stochastic incipient otion criterion of spheres under various bed packing conditions, J. Hydrol. Eng., 128(4), , doi: / (ASCE) (2002)128:4(369). Recking, A., P. Frey, A. Paquier, and P. Belleudy (2009), An experiental investigation of echaniss involved in bed load sheet production and igration, J. Geophys. Res., 114, F03010, doi: /2008jf Rickenann, D., and B. W. McArdell (2008), Calibration of piezoelectric bed load ipact sensors in the Pitzbach ountain strea, Geodin. Acta, 21, 35 52, doi: /ga Singh, A., K. Fienberg, D. J. Jerolack, J. Marr, and E. Foufoula Georgiou (2009), Experiental evidence for statistical scaling and interittency in sedient transport rates, J. Geophys. Res., 114, F01025, doi: / 2007JF Stark, C. P., and N. Hovius (2001), The characterization of landslide size distributions, Geophys. Res. Lett., 28, Turoski, J. M. (2009), Stochastic odeling of the cover effect and bedrock erosion, Water Resour. Res., 45, W03422, doi: / 2008WR Turoski, J. M., and D. Rickenann (2009), Tools and cover effects in bed load transport observations in the Pitzbach, Austria, Earth Surf. Processes Landfors, 34, 26 37, doi: /esp Wilcock, P. R., and B. W. McArdell (1993), Surface based fractional transport rates: Mobilization thresholds and partial transport of a sand gravel sedient, Water Resour. Res., 29(4), Willis, J. C., and G. C. Bolton (1979), Statistical analysis of concentration records, J. Hydrol. Div., 105(1), Zierann, A., M. Church, and M. A. Hassan (2008), Video based gravel transport easureents ith a flue ounted light table, Earth Surf. Processes Landfors, 33, , doi: /esp J. M. Turoski, Eidgenössische Forschungsanstalt WSL Birensdorf, Zürcherstrasse 111, 8903 Birensdorf, Sitzerland. ( 10 of 10

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