## 1. Introduction

[2] In the past decade a number of researchers have attempted to explore the probabilistic relationship between what is referred to as ‘pickup probability’ or ‘probability of entrainment’ and both turbulent flow conditions and bed texture properties that characterize the resistance to motion of sediments. *Papanicolaou et al.* [2002] explained that the initial development of stochastic models was primarily based on the assumption that hydrodynamic forces were statistically represented by a normal distribution. Other experimental findings [e.g., *Wang and Larsen*, 1994; *Jiménez*, 1998] had shown that the Reynolds normal stress in natural streams can be reasonably approximated by gamma or exponential distributions. *Cheng and Chiew* [1998] and *Wu and Chou* [2003]developed theoretical formulations of the entrainment probability, where the variability of instantaneous velocities was described adopting Gaussian and lognormal distributions, respectively. Higher-order distributions were presented by*Papanicolaou et al.* [2002] and *Wu and Yang* [2004]. Despite slightly different approaches presented in these studies, it is generally assumed that the near-bed velocity*U*_{f} (or the fluid shear stress T_{f}) is represented by a cumulative distribution *F*_{Uf} with probability density function *f*_{Uf}, and that the probability of entrainment *P*_{E} can be expressed as

where *u*_{g} denotes the threshold velocity or critical velocity for grain entrainment (equivalent to the critical shear stress *τ*_{g} for individual particles). Noting that *U*_{g} is a random variable represented by a cumulative distribution *F*_{Ug} with probability density function *f*_{Ug}, the mean entrainment probability is the expected value of equation (1) over the full range of *U*_{g} [e.g., *McEwan et al.*, 2004]

with *F*_{Ug}(*u*_{f}) = *P*(*U*_{g} < *u*_{g} = *u*_{f}). Herein, capital letters indicate random variables, and lower-case letters indicate single occurrences of these variables.

[3] Equation (2) reflects the conclusions by *Kirchner et al.* [1990, p. 668] who suggested that “it may be possible to predict the bed-load transport rate from a ‘convolution’ of the probability distributions of the instantaneous local shear stress (…) and the local critical shear stress.” A. J. Grass first demonstrated that the entrainment of grains could be linked with turbulent flow features and combined with the probabilistic distribution of the critical shear stresses [*Grass*, 1970]. Solutions for the entrainment probability in the form of equation (2) are here referred to as following the ‘Grass approach’. A schematic representation of this method is reported in Figure 1, where the larger the overlapping of the shear stress distributions, the higher the probability that individual grains are entrained.

[4] A crucial shortcoming of existing models based on equation (2) is that the underlying approach requires T_{f} and T_{g} (or *U*_{f} and *U*_{g}) to be statistically independent. This constraint is not generally satisfied as we suggest that both depend at least on the statistical distribution of the surface grain elevations, *Z*_{g}and that this dependence has a significant impact on the probability of entrainment. Evidence indicates that the higher the exposure of a particle to the near-bed fluid, the higher the forces exerted by the flow (as the local mean velocity increases) and the weaker the resistance to motion of the particle (as the ratio of the lever arms of the submerged weight and fluid force decreases). Particles that rest in deep bed pockets are harder to move than particles in shallow pockets not only due to the higher forces required to begin motion, but also because the occurrence of the very high fluid forces is less frequent. A general form of the entrainment probability must account for such combined likelihood of occurrence of T_{f}, and T_{g} as follows

where *f*_{Tf},_{Tg} is the joint probability density function of T_{f} and T_{g}, and the notation *R*_{E} stands for risk of entrainment [e.g., *Lopez and Garcia*, 2001; *McEwan et al.*, 2004]. The symbols *τ*_{f} and *τ*_{g} are used here to refer to the fluid and critical shear stress of a single grain, while the spatially averaged boundary shear stress and the threshold shear stress of the grain population are represented by *τ*_{0} and *τ*_{0c}.

[5] This study is the first of two companion papers. This paper seeks to extend the original Grass approach by including the correlation between fluid forces and grain resistance and to make the problem tractable by introducing the hypothesis of conditional independence (Section 2). A general derivation of equation (3) is provided along with the analysis of the conditions that applies to the present model (Section 3). Results are used to illustrate the implications that the new theoretical development has for sediment transport (Section 4). The related companion paper [*Tregnaghi et al.*, 2012] describes how detailed statistical information can be obtained to test the influence of the constituent sub-processes leading toequation (3) on sediment entrainment. This is achieved in a series of laboratory experiments using particle imaging velocimetry (PIV) techniques, and numerical simulations performed with discrete particles models (DPM).