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


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1 WATER RESOURCES RESEARCH, VOL. 46,, doi: /2009wr008488, 2010 Probability distributions of bed load transport rates: A ne derivation and coparison ith field data Jens M. Turoski 1 Received 5 August 2009; revised 16 February 2010; accepted 24 February 2010; published 3 August [1] It has been knon for a long tie that sedient transport rates can vary strongly even if the abient hydraulic conditions reain steady. In this article, a ne approach is described to derive probability distributions of bed load transport rates, starting fro the aiting tie beteen the arrivals of individual sedient particles. The foralis should be valid hen transport is not doinated by bed for otion. Without any assuptions about the distribution of interarrival ties, the approach yields the Birnbau Saunders distribution, a to paraeter distribution previously used in lifetie odeling. Observed dependence of ean transport rates on the sapling tie and ultiscaling are predicted by the distribution. Assuing exponentially and Poisson distributed interarrival ties, the sae approach yields the Poisson and the gaa distributions. Using a high resolution bed load transport data set fro the Pitzbach, Austria, the distribution functions are tested on field data. The gaa distribution best describes the data, ith axiu deviations of 5%. Hoever, the Birnbau Saunders distribution ay be ore useful in certain applications, as it is a general approxiation in the proposed foralis and no debated assuptions are necessary for its derivation. Citation: Turoski, J. M. (2010), Probability distributions of bed load transport rates: A ne derivation and coparison ith field data, Water Resour. Res., 46,, doi: /2009wr Introduction [2] Bed load transport, the ater driven rolling, sliding, or hopping otion of coarse particles in a strea, is an iportant ode of sedient transfer in rivers and a key process in shaping the Earth s surface. Accurate calculation of transport rates is necessary both in engineering applications such as flood hazard itigation and in pure science. Bed load transport rates are knon to fluctuate strongly both in nature [e.g., Goez and Church, 1989; Hassan and Church, 2000] and in the laboratory [e.g., Kuhnle and Southard, 1988; Ancey et al., 2008] even under steady flo conditions. Several different causes for these fluctuations have been identified [Goez et al., 1989; Hoey, 1992], including variations in sedient supply [e.g., Benda and Dunne, 1997], spatially and teporally varying distribution of grain sizes, relative grain arrangeent, and grain sorting processes [e.g., Kirchner et al., 1990; Chen and Stone, 2008], and the passage of bed fors [e.g., Lisle et al., 2001; Recking et al., 2009]. [3] Several odels have been proposed to describe the probability distribution functions of bed load transport rates. Einstein [1937] considered bed load transport as a series of rest periods of rando length, interrupted by short periods of otion of rando distances. He assued that both step lengths and rest ties are exponentially distributed and 1 Eidgeno ssische Forschungsanstalt WSL Birensdorf, Birensdorf, Sitzerland. Copyright 2010 by the Aerican Geophysical Union /10/2009WR derived distribution functions for the aount of sedient transported over a cross section. Hoever, it is difficult to easure the distribution functions of rest periods and transport distances directly in the field or laboratory and any of the assuptions underlying Einstein s [1937] odel have not yet been validated. Guided by laboratory observations, Haaori [1962] developed a distribution function of bed load transport rates for cases hen bed for otion is doinant. In his odel, secondary dunes are responsible for the total transport. These secondary dunes entrain aterial and gro linearly ith distance hile oving up the stoss slope of priary dunes. Thus, the assuptions underlying his distribution are physically restrictive and apply only in liited circustances. More recent stochastic odels of bed load transport often describe the entrainent and deposition of particles in a control volue using Markov birth death odels [e.g., Lisle et al., 1998; Papanicolaou et al., 2002; Ancey et al., 2006, 2008; Turoski, 2009]. These odels typically feature a large nuber of paraeters that need to be calibrated on data but are hard to easure even under controlled laboratory conditions [cf. Ancey et al., 2008]. Thus, it is challenging to test and validate such odels directly. [4] Here I take a different approach. Instead of trying to devise an accurate description of the physics of bed load transport, I consider the distribution of aiting ties beteen particle arrivals (interarrival ties) at a cross section to derive probability distributions for bed load transport rates. Using odern equipent such as light tables [Frey et al., 2003; Zierann et al., 2008] or video caeras [Drake et al., 1988], it should be possible to directly easure this distribution in the laboratory and the field. The 1of10
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
4 Figure 1. Scheatic plan of the eir and ater intake at the Pitzbach. The ater flos through a Tyrolean eir ith a grid size of 15 c and is led into the sedient settling basin on the orographic left. There, several load cells are used to onitor the deposited volues every 15 in. If a threshold eight is exceeded, the basin is autoatically flushed and the sedient is reinserted into the channel donstrea of the eir. edian grain size of 3 4 c on the bed surface and an average channel bed slope of 9%. At an elevation of 1811 asl the Tyrolean Water Copany (TIWAG) aintains a ater intake in the strea for a hydro poer plant (Figure 1). Approxiately 60% of the total drainage area of 26.8 k 2 is covered by glaciers. As a result, ost bed load is transported throughout the suer onths, hen discharge varies ith the daily cyclicity typical for glaciated catchents. At the ater intake, the ater flos over a Tyrolean ear consisting of a etal grid ith a bar spacing of 15 c into a sedient settling basin. For the suers 1994 and 1995, the basin as equipped ith five load cells and a pressure sensor, hich ere used to easure the aount of accuulated sedient every 15 in. When a threshold eight as exceeded, the sedient in the basin as autoatically flushed out and reinserted into the channel donstrea of the eir. Since discharge is largely elt ater driven, it generally varies sloly over tie and the assuption that discharge is constant ithin a easureent interval can be ade. More inforation on the strea, the instruents and the easureent capaign can be found in the orks of Hofer [1987], Rickenann and McArdell [2008], and Turoski and Rickenann [2009]. For the survey period ore than 10,600 easureents ith nonzero bed load transport rates are available (Figure 2). The instruents reliably pick up a signal hen the deposited volue exceeds Saller volues are often issed and are thus underrepresented in the easureents. At a given discharge, easured transport rates scatter over up to four orders of agnitude around the ean. Nonlinear regression as often used to analyze bed load transport easureents neglects this diension and thus is inadequate to provide a full description of the data. described by the Birnbau Saunders distribution in all cases; hoever, for sall transport rates there are systeatic differences: at very sall and ediu transport rates equation (7) underpredicts the occurrence probability, for sall rates it overpredicts it. [16] At the Pitzbach, both the expected value E(Q s ) and the standard deviation std(q s ) are onotonically increasing functions of discharge, ell described by either an exponential or a poer function (Figures 2 and 4). Hoever, the good fit featured by the exponential function ay be isleading, as for lo transport rates, an iportant fraction of the data ay be cut off due the easureent threshold of the instruents, hich often iss sall deposited values (Figure 2). [17] For discharge classes ith a large nuber of data points (>80) the coefficient of variation cva(q s ) = std(q s )/ E(Q s ) is approxiately constant ith an average value of 1.2. Siilarly, Willis and Bolton [1979] observed cva(q s ) 1.6 for experiental sedient transport data of sand, and Kuhnle and Willis [1998] reported approxiately equal ean and standard deviation at a given shear stress (hich is 3.2. The Pitzbach Data and the Birnbau Saunders Distribution [15] The Pitzbach easureents ere classified into logarithically distributed bins in discharge, and equation (7) as fitted to the data ithin each bin using axiu likelihood estiation (Figure 3). Such a ethod is preferable to nonlinear regression as coonly used to analyze bed load transport data, since it explicitly acknoledges the spread of transport rates at a given discharge and allos the investigation of statistics other than the ean. For the Pitzbach data, the tail behavior at large transport rates is ell Figure 2. Bed load volue accuulated in 15 in periods as function of discharge. Volues saller than are often issed by the instruents and are thus underrepresented. Exponential and poer la fits ere done on binned eans (black circles), instead of on the hole set. Fit values are given in the caption of Figure 4. 4of10
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
8 quantified ith the Berry Esséen Theore [Berry, 1941; Esséen, 1942], hich states that the axiu difference beteen the real distribution and its noral approxiation scales ith N 1/2. Thus, the approxiation gets better as ore particles are trapped ithin the easureent interval. In general, the error depends on the ean interarrival tie and varies ith discharge. If the underlying distribution is exponential (section 2.2), the construction of a distribution function fro added rando nubers is reasonably approxiated by a noral distribution for N > 100 for the conditions at the Pitzbach. For coparison, assuing that the ean grain size of the aterial deposited in the retention basin at the Pitzbach is close to the 65 percentile of the size distribution of 6 [Rickenann and McArdell, 2008], around 2 illion particles are present in 1 3 of bed load, assuing 50% pore space and 50% fines. A volue of 1 3 is close to the average yield delivered ithin 15 in at a discharge of 8 3 /s (Figure 2). Thus, for ost practical conditions, the rate of convergence to the noral distribution should not be a proble. [22] The assuption of the la of large nubers leads to siilar probles. At the Pitzbach, the ean particle diaeter is around 6 for the aterial deposited in the retention basin. At sall transport rates often large grain sizes are underrepresented in the bed load [Wilcock and McArdell, 1993], and for the Pitzbach there is no inforation available on ho the grain size distribution (and the ean grain size) changes ith discharge and bed load transport rate. An accurate quantification of the rate of convergence is thus not currently possible; hoever, considering the large nuber of arriving particles, the easured ean should be fairly close to the population ean. [23] In addition to these statistical errors, at sall transport rates the instruents are unreliable, hich affects goodness of fit. Often sall volue changes are issed by the equipent and data in this range is underrepresented in the distribution. One can get around these probles by just extending the easureent intervals. Hoever, again it ay be probleatic to find a suitable interval in hich the variation of discharge is slo enough such that the hydraulic conditions can be assued to be constant. [24] In the Birnbau Saunders distribution the ean transport rate decreases ith increasing sapling interval (see equation (8)). Siilarly, both in the field [Bunte and Abt, 2005] and in the laboratory [Singh et al., 2009], it as found that at lo transport rates the estiated ean transport rates decrease as the sapling interval increases, hile at high transport rates the trend reverses. Singh et al. [2009] related this trend reversal to the presence of large bed fors and their doinance in the transport process at high transport rates. As discussed above, the assuptions underlying the Birnbau Saunders distribution are not valid hen transport is doinated by bed for otion, and thus the predictions can be considered to be in line ith observations. The Pitzbach data span 2 years, and during this tie no exceptional flood occurred in the catchent. Sufficient data for the construction of a distribution function is available for discharges up to 10 3 /s, hereby the estiated peak discharge of a 2 year flood is /s. Consequently, the Birnbau Saunders distribution cannot currently be tested under high discharge conditions. Singh et al. [2009] also described ultiscaling of the oents of bed load transport for different sapling intervals, i.e., different oents such as ean and standard deviation are dependent on the sapling interval in different ays. This behavior is predicted by the Birnbau Saunders distribution (cf. equations (8) and (9)) The Pitzbach Data and Coparison With Other Distribution Functions [25] Bed load volues easured at the Pitzbach ithin 15 in periods scatter idely, over up to four orders of agnitude at a given discharge (Figure 2). It is clear that nonlinear regression ith a poer la, as is often used to analyze field data of sedient transport, is inadequate to describe observations. First, regression alays gives a global optiization and can lead to strong local deviations. Second, the large scatter in the y direction is treated as a easureent error and not as a genuine signal. Herein, a binning procedure as used to analyze the data. This has the advantage that statistics for transport rates at a given discharge can be calculated for each bin, and one can obtain inforation not only on the ean behavior but also on the standard deviation and the shape of the probability distributions. [26] Fro the tested distributions (Birnbau Saunders, continuous Poisson type distribution, gaa, Haaori, Carey Hubbell), the gaa distribution gives the best fit to the Pitzbach data. Both Birnbau Saunders and continuous Poisson distributions provide reasonable fits in the righthand tail, hile the Haaori and Carey Hubbell distributions fit ell for lo transport rates. Fe other data sets exist ith a sufficient size and quality to ake siilar calculations as done here for the Pitzbach. Kuhnle and Willis [1998] tested the exponential, the noral, the gaa, and the Haaori distributions for bed load transport data fro Goodin Creek and likeise found that the gaa distribution gives the best description of the data. Thus, the gaa distribution has been observed to ork ell at to field sites and can be recoended for use in field applications at the current state of knoledge. Hoever, the Birnbau Saunders distribution ay be ore suitable in certain applications, because its derivation is ore general. After all, the gaa distribution as derived fro the assuption of Poisson distributed interarrival ties. The precise for of the distribution of interarrival ties has not been easured in the field and is debated fro a theoretical point of vie. For the Pitzbach data, the tail behavior of sedient transport rates at a given discharge is ell described by the Birnbau Saunders distribution for the upper tentieth percentile, at any discharges the good fit region reaches to the upper thirtieth percentile or further. The upper tentieth percentile corresponds to 60% 70% of the total load transported for a given discharge. The distribution ay therefore provide an adequate approxiation in any instances. [27] The Birnbau Saunders and the gaa distribution decline exponentially in Q s in the liit of large transport rates. Hence, large events occur ore coonly as ould be expected hen using a noral distribution. Environental paraeters often sho a poer la tail (heavy tail), for exaple landslide sizes or strea discharge [e.g., Stark and Hovius, 2001; Lague et al., 2005], and it is soe 8of10
9 ties assued that sedient transport rates likeise are heavy tailed. On the contrary, in the Pitzbach bed load transport rates sho an exponential rather than a poer la tail. Large events are thus not as doinant for the total transport rate as is observed for any other environental processes. Note, hoever, that this is only true for transport rates at a given discharge. The long ter sedient budget of a catchent ay be ore dependent on the distribution of flood sizes (often heavy tailed) than on the properties of the distributions considered herein. 5. Conclusions [28] Bed load transport rates are knon to fluctuate strongly even under steady hydraulic conditions. Here, I have shon a ne ay of deriving probability distributions of bed load transport rates by considering the aiting ties beteen the arrivals of individual particles (interarrival ties). The assuptions underlying the derivation should ake the distributions applicable for bed load transport not doinated by bed for otion. Without any further assuptions about the distribution of interarrival ties, the foralis yields the Birnbau Saunders distribution as a general approxiation. A continuous version of the Poisson distribution and the gaa distribution can be derived if the interarrival ties are assued to be exponentially or Poisson distributed, respectively. The approach circuvents the need for calibration of several paraeters that are hard to observe directly, as is often necessary in odern stochastic theories of sedient transport. In fact, ith odern equipent such as light tables and video based observation [Drake et al., 1988; Frey et al., 2003; Zierann et al., 2008], it should be possible to directly easure the distribution of interarrival ties in the field or the laboratory. [29] Together ith the previously proposed Haaori and Carey Hubbell distributions [Haaori, 1962; Carey and Hubbell, 1986], the derived functions ere tested on a large data set fro the Pitzbach, Austria. Out of the tested distributions, the gaa distribution perfors best, ith axial deviations fro the data of around 5%. The Birnbau Saunders distribution provides a reasonable approxiation especially in the right hand tail. The Poisson distribution fits ell over the hole range of the data. The Haaori and Carey Hubbell distributions give reasonable fits especially for lo transport rates. The gaa distribution has previously been found to also describe the Goodin Creek data ell [Kuhnle and Willis, 1998]. At the current state of knoledge, the gaa distribution can thus be recoended for use in field applications. Hoever, the Birnbau Saunders distribution is a reasonable approxiation, hich has been derived ithout aking any debated physical assuptions. Thus, its use ay be advantageous in certain applications. Notation Variables M Stochastic variable of ass, kg Total ass of N particles, kg Average ass of a particle, kg N Nuber of particles n Carey Hubbell exponent Q Mean bed load ass flux, kg/s Q ax Maxiu bed load ass flux in the Carey Hubbell distribution, kg/s Q s Bed load ass flux, kg/s T Stochastic variable of aiting tie, s T N Tie until the arrival of the Nth particle, s t Length of easureent interval, s t N Stochastic variable of the tie until the arrival of the Nth particle, s x, y, z Variables, used variously b, b Scale factor in the Birnbau Saunders distribution, kg (unpried), kg/s (pried) g Shape factor in the Birnbau Saunders distribution Mean aiting tie beteen individual particle arrivals, s s Standard deviation of aiting ties beteen individual particle arrivals, s Functions cdf(x) Cuulative distribution function of x cva(x) Coefficient of variation of x E(x) Expectation values of x pdf(x) Probability density function of x std(x) Standard deviation of x var(x) Variance of x G(x), G(x, y) Coplete and upper incoplete Gaa function F((x )/s) Cuulative noral distribution of x ith ean and standard deviation s [30] Acknoledgents. Many people have coented on the ideas presented herein and have helped to shape the in discussions. I thank especially D. Rickenann, A. Badoux, J.W. Kirchner, B.W. McArdell, M. Nitsche, D. Lague, and M. Stähli for support, encourageent, stiulating discussions, and coents on earlier versions of the anuscript. TIWAG supported the easureent capaign at the Pitzbach and ade the data available. A. Recking, C. Ancey, and an anonyous revieer coented on an earlier version of the anuscript. References Ancey, C., T. Böh, M. Jodeau, and P. Frey (2006), Statistical description of sedient transport experients, Phys. Rev. E, 74, , doi: /physreve Ancey, C., A. C. Davison, T. Böh, M. Jodeau, and P. Frey (2008), Entrainent and otion of coarse particles in a shallo ater strea don a steep slope, J. Fluid Mech., 595, , doi: / S Benda, L., and T. Dunne (1997), Stochastic forcing of sedient routing and storage in channel netorks, Water Resour. Res., 33(12), , doi: /97wr Berry, A. C. (1941), The accuracy of the Gaussian approxiation to the su of independent variates, Trans. A. Math. Soc., 49, Birnbau, Z. W., and S. C. Saunders (1968), A probabilistic interpretation of Miner s rule, SIAM J. Appl. Math., 16(3), Bunte, K., and S. R. Abt (2005), Effect of sapling tie on easured gravel bed load transport rates in a coarse bedded strea, Water Resour. Res., 41, W11405, doi: /2004wr Carey, W. P. (1985), Variability in easured bed load transport rates, Water Resour. Bull., 21, Carey, W. P., and D. W. Hubbell (1986), Probability distributions for bed load transport, in Proceedings of the 4th Federal Inter Agency Sedient Conference, US Geological Survey, vol. 2, Chen, L., and M. C. Stone (2008), Influence of bed aterial size heterogeneity on bed load transport uncertainty, Water Resour. Res., 44, W01405, doi: /2006wr Desond, A. (1985), Stochastic odels of failure in rando environents, Can. J. Stat., 13(2), Drake, T. G., R. L. Shreve, W. E. Dietrich, P. J. Whiting, and L. B. 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