## 1. Introduction

[2] Bed load transport, the water-driven rolling, sliding, or hopping motion of coarse particles in a stream, is an important mode of sediment 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 mitigation and in pure science. Bed load transport rates are known to fluctuate strongly both in nature [e.g., *Gomez and Church*, 1989; *Hassan and Church*, 2000] and in the laboratory [e.g., *Kuhnle and Southard*, 1988; *Ancey et al.*, 2008] even under steady flow conditions. Several different causes for these fluctuations have been identified [*Gomez et al.*, 1989; *Hoey*, 1992], including variations in sediment supply [e.g., *Benda and Dunne*, 1997], spatially and temporally varying distribution of grain sizes, relative grain arrangement, and grain sorting processes [e.g., *Kirchner et al.*, 1990; *Chen and Stone*, 2008], and the passage of bed forms [e.g., *Lisle et al.*, 2001; *Recking et al.*, 2009].

[3] Several models 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 random length, interrupted by short periods of motion of random distances. He assumed that both step lengths and rest times are exponentially distributed and derived distribution functions for the amount of sediment transported over a cross section. However, it is difficult to measure the distribution functions of rest periods and transport distances directly in the field or laboratory and many of the assumptions underlying *Einstein*'s [1937] model have not yet been validated. Guided by laboratory observations, *Hamamori* [1962] developed a distribution function of bed load transport rates for cases when bed form motion is dominant. In his model, secondary dunes are responsible for the total transport. These secondary dunes entrain material and grow linearly with distance while moving up the stoss slope of primary dunes. Thus, the assumptions underlying his distribution are physically restrictive and apply only in limited circumstances. More recent stochastic models of bed load transport often describe the entrainment and deposition of particles in a control volume using Markov birth-death models [e.g., *Lisle et al.*, 1998; *Papanicolaou et al.*, 2002; *Ancey et al.*, 2006, 2008; *Turowski*, 2009]. These models typically feature a large number of parameters that need to be calibrated on data but are hard to measure even under controlled laboratory conditions [cf. *Ancey et al.*, 2008]. Thus, it is challenging to test and validate such models 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 waiting times between particle arrivals (interarrival times) at a cross section to derive probability distributions for bed load transport rates. Using modern equipment such as light tables [*Frey et al.*, 2003; *Zimmermann et al.*, 2008] or video cameras [*Drake et al.*, 1988], it should be possible to directly measure this distribution in the laboratory and the field. The Birnbaum-Saunders distribution [*Birnbaum and Saunders*, 1968] arises as a general approximation when no explicit assumptions about the distribution of waiting times are made. By assuming exponentially or Poisson distributed waiting times, one arrives at the Poisson and Gamma distributions, respectively. The distribution functions are compared to a large field data set from the Pitzbach, Austria [*Rickenmann and McArdell*, 2008; *Turowski and Rickenmann*, 2009].