2.1. Sediment Cover Continuum
 The development of, and dynamic change in, sediment cover in a bedrock channel are driven by grain entrainment (E), translation (T) and deposition (D). These processes vary depending on whether a grain is on a bedrock or alluvial surface (e.g., entrainment from an alluvial surface, Ea, or a bedrock surface, Eb). A single grain movement, comprising entrainment, translation via saltation, and deposition, can therefore be described by one of eight possible combinations. The combination of E − T − D that a grain experiences depends on the extent of sediment cover in the channel, ranging from Ea − Ta − Da at the fully alluvial end-member to Eb − Tb − Db in the pure bedrock case (Figure 1). Given the proposed distributions of sediment cover, three of the eight combinations are unlikely to occur. To understand how the extent of sediment cover affects both the total flux and its grain size distribution in rivers across the spectrum in Figure 1, we extend established theory to predict how each of entrainment, translation, deposition and net mass continuity will vary between the bedrock and fully alluvial cases.
2.2. Grain Entrainment
 Except in the rare case of extremely low sediment cover, sediment forms patches on the bed of a bedrock river rather then being multiple isolated grains. Such grouping occurs because grains accumulate in areas of the channel where the bed roughness, hydraulics and the sheltering influence of other grains increase grain stability [Finnegan et al., 2007; Johnson and Whipple, 2007; Chatanantavet and Parker, 2008]. Consequently, initial grain entrainment primarily occurs from alluvial surfaces, i.e., Ea is far more probable than Eb, regardless of the extent of sediment cover. During a transporting event, a mobile grain may undertake multiple steps before being deposited in a stable position [Einstein 1950; Drake et al., 1988], where each step itself contains multiple saltation hops [Sklar and Dietrich, 2001; Attal and Lavé, 2009]. In a river with low alluvial cover, the initial step for a grain in an event is most likely to involve entrainment from an alluvial surface, but subsequent (and hence the majority of) entrainments are more likely to be from bedrock surfaces.
 Differences between Ea and Eb reflect the effect of the underlying surface geometry on the grain position and hence critical entrainment shear stress (τc). We quantify this effect on τc by using a Monte Carlo application of the model of Kirchner et al. , which calculates τc for a grain by solving the force balance at the threshold of motion:
FD and FL are respectively drag and lift forces, FW is the immersed weight of the grain, Φ is the grain pivoting angle which describes pocket geometry, ρs is the density of sediment (taken as 2650 kg m−3), ρ is the density of water, g is acceleration due to gravity and d is the grain diameter. FD and FL are calculated assuming a logarithmic flow velocity profile and incorporate both grain exposure and elevation with respect to the local velocity profile. The boundary shear stress at the threshold of motion, τc, is calculated by expressing FD and FL in terms of τc and substituting into and rearranging equation (1); for the full derivation see Kirchner et al. .
 In Kirchner et al.'s  model, the underlying surface is assumed to be a granular material with grain size K (Figure 2), and its effect on τc is expressed through the values of Φ, FD and FL. By applying idealized geometrical relationships [Kirchner et al., 1990] and relationships derived from field data [Johnston et al., 1998], Φ, FD and FL can be expressed as functions of the relative sizes of the overlying grain (d) and the representative grain size of the underlying surface (K). When a grain is entrained from an alluvial surface (Ea), both d and K are drawn from the same grain size distribution (GSD) (Figure 2a). For illustration we use a measured lognormal sediment GSD from the River Calder (mean 5.56 ψ; standard deviation 0.69 ψ, where ψ = log2(d) and d is measured in mm).
Figure 2. Grain geometry in (a) alluvial and (b) bedrock cases. Bedrock is approximated by an alluvial surface with low surface roughness. Overlying grains (with diameter d) are black, underlying grains (with diameter K) are grey. (c) For each pair of d and K on an alluvial surface, Φ is calculated as Φ = 51.6(d/K)−0.30; coefficient values were calculated from field measurements by Johnston et al.  in Sagehen Creek (steep mountain stream, d50 = 58 mm), which is the most similar of their study rivers to the River Calder. On a bedrock surface, Φ = tan−1(γ/ ), where γ = 1/√3, which represents pivoting from a pocket formed by three grains [Eagleson and Dean, 1961]. The former alluvial relationship for Φ incorporates the effect of surrounding grains on Φ, whereas the latter bedrock relationship better represents grains pivoting on a solid surface. Grain exposure e is measured from local upstream maximum elevation and calculated as e = 0.5[d − K + (d + K) cos Φ]. Grain protrusion p is measured from local mean bed surface elevation as p = e + (π/12)K. When calculating p and e, all underlying grains are assumed to be the same size.
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 For entrainment from a bedrock surface (Eb), the local roughness of the bedrock surface is represented as an appropriate value of K. Bedrock surface roughness is very variable, from relatively smooth abraded surfaces to irregular plucked and jointed surfaces [e.g., Goode and Wohl, 2010b]. However, at the scale of an individual grain, most bedrock surfaces will be locally smooth so we omit the effect of macroscale roughness on local velocity profiles and represent these as a surface composed of roughness elements that are finer than the GSD of the overlying grains (Figure 2). To estimate the representative GSD of the roughness elements, bedrock roughness was calculated from high-resolution topographic profiles ∼3m long in total, measured in the River Calder. Roughness was quantified as the standard deviation of elevations (σz) within multiple 50 mm lengths of the profiles; 50 mm is the median alluvial grain size (d50), and so roughness is measured at a scale relevant to individual grains. Each value of σz was converted to an equivalent grain size. All together, these equivalent grain sizes follow a lognormal GSD with a mean of 4.42 ψ and standard deviation of 0.99 ψ. Trends in the model results are not sensitive to the particular values of these parameters.
 Our Monte Carlo application of Kirchner et al.'s  model predicts τc for 1000 grains entrained from each of the alluvial (Figure 2a) and bedrock (Figure 2b) surfaces. In each case, 1000 pairs of an overlying and an underlying grain size are drawn at random from the respective distributions. d/K is used to calculate grain pivoting angle (Φ), exposure (e) and projection (p) (see Figure 2c) [see Kirchner et al., 1990]. τc and dimensionless τc (τc*) are subsequently calculated for each grain, and the results are shown in Figure 3.
Figure 3. Distributions of (a) τc and (b) τc* for 1000 grains in each of an alluvial and bedrock setting. Markers are shaded according to K, the size of the underlying grains. Values for K = K50 are shown in black. Note that in Figure 3a grain size is plotted relative to the median size of the overlying grains (d50), whereas in Figure 3b grain size is plotted relative to the median size of the underlying grains (K50). The line in Figure 3b shows equal mobility, i.e., τc* is proportional to 1/d; this line was fitted to theoretical and field data by Johnston et al. . Comparison between the gradients of the line and the model results are instructive; the difference in absolute values of τc* is less instructive because of the dependency of τc* on model components, e.g., the pivoting angle relationship.
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 τc for grains on a bedrock surface is about an order of magnitude lower than for the same size grains on an alluvial surface (Figure 3); when d = d50 and K = K50, τc = 2.5 and 21.9 Pa for the bedrock and alluvial cases, respectively. Dancey et al.  also observed an order of magnitude increase in τc* when grain packing density increased from 3 to 91%, which is comparable to the difference between a bedrock and an alluvial surface. Re-entrainment during movement, Eb, therefore requires a significantly lower shear stress than the initial entrainment Ea. Consequently, for the same shear stress, excess shear stress (τ − τc) is greater for a grain in transport across a bedrock surface than for a grain on an alluvial surface.
 Entrainment from a bedrock surface shows a weak dependence of τc on grain size; fixing K at K50, τc decreases from 3.5 Pa for d = d05 to 2.1 Pa for d = d95. Model results are parallel to a line fitted by Johnston et al.  to both theoretical and field data (Figure 3b). Along this line, τc* is proportional to 1/d, i.e., grains are equally mobile. The small dependence of τc on d is because changes in grain geometry as grain size increases mean that grains have lower pivoting angles, larger exposure and protrude higher into faster flow. Entrainment from a bedrock surface will therefore not be a major cause of size-selective transport. However, as entrainment is only one component of grain movement, other aspects may produce or inhibit size-selectivity.
 In a bedrock river, Ea may be affected by the shallow sediment depths associated with low sediment cover. In an alluvial channel, surface coarsening acts to equalize τc for different grain sizes because smaller grains having relatively less exposure and higher pivoting angles than larger grains [Wiberg and Smith, 1987; Kirchner et al., 1990; Parker and Sutherland 1990]. Thin sediment layers cannot develop surface coarsening thus reducing or eliminating this equalizing effect, such that entrainment shear stress is primarily a function of grain size. Consequently, the exact functioning of Ea in a bedrock river may vary according to sediment depth.
 Bedrock channels have slopes up to ten times greater than alluvial channels for a given drainage basin area [Howard and Kerby, 1983; Montgomery et al., 1996]. Critical entrainment shear stress is usually assumed to be independent of channel slope. However, Shvidchenko et al. , Lamb et al.  and Recking  have demonstrated a significant effect of slope with the same size of grain being more stable on a steeper slope. This effect is due to slope dependence of the velocity profiles for a given discharge, and to the changes in the grain force balance due to relative roughness and partial grain emergence. Thus, the typically steeper nature of bedrock channels implies that grain mobility from alluvial patches in bedrock rivers will be reduced compared to alluvial cases.
2.3. Grain Translation
 As excess shear stress (τ − τc) is higher over bedrock surfaces than alluvial ones grain dynamics differ over the two surfaces (Ta and Tb). For example, saltation height, length and downstream velocity are each functions of (τ* − τc*/τc*)a, where a < 1 [Sklar and Dietrich, 2004]. Lajeunesse et al.  showed that grain velocities and step lengths are larger over non-erodible (i.e., bedrock) surfaces than over mobile alluvial surfaces. The properties of a bedrock surface also affect saltation dynamics as there will be few, if any, particles to be entrained by an impacting grain. Further, grain rebound is affected by the coefficient of restitution and the distribution of angles of the bedrock surface impacted by mobile grains. Hence, grains moving over a bedrock surface will undergo longer, more frequent, translation steps than grains on an otherwise equivalent alluvial bed. Consequently, for the same shear stress, sediment transport capacity will be higher over bedrock surfaces.
2.4. Grain Deposition
 Grains are more likely to be deposited on an alluvial surface (Da) than on a bedrock surface (Db); the higher grain pivoting angles and lower exposures predicted by the Kirchner et al.  model for the alluvial surface enhance deposition as well as impeding entrainment. In addition, grain roughness could also affect the local flow profile, decreasing shear stress and enhancing deposition. Consequently, in a bedrock channel with low sediment cover, deposition is mainly determined by sediment patch location. Deposition on a bedrock surface may occur as flow recedes, when translation is halted as shear stress falls before grains have reached an alluvial area.
 Sediment patch location is controlled by the interaction between local channel morphology and flow hydraulics. For example, patches may develop where the macroscale topography reduces local flow velocities, such as between bedrock ribs and within potholes [Goode and Wohl, 2010a, 2010b]. Sediment patches are therefore unlikely to exhibit the regular spacing of alluvial bed forms and bars. Once areas of sediment are established, the decreased probability of re-entrainment of grains from these areas provides positive feedback promoting their maintenance [Johnson and Whipple, 2007; Finnegan et al., 2007].
 Under low sediment cover, grains will mainly travel between areas of sediment, and so grain transport lengths will be determined by the inter-patch spacing. As sediment cover increases, the probability of deposition becomes more spatially uniform and the location of deposition is increasingly driven by processes identified in alluvial rivers [Pyrce and Ashmore, 2003b; Hassan et al., 1991]. The greater depth of sediment in alluvial rivers, and in bedrock rivers with higher sediment cover, also increases the potential for grains to be deposited at depth within the bed, thus reducing their probability of subsequent entrainment [Ferguson and Hoey, 2002].
2.5. Sediment Continuity
 All components of E − T − D show important differences between bedrock and alluvial surfaces, which will affect event-scale sediment movement. We now assess how these differences affect the long-term relative behavior of different size fractions and the long-term conditions under which a bedrock river could have steady state sediment cover. While our previous analysis treated sediment transport as a discrete process, we now turn to analysis of sediment continuity.
 The presence (or absence) of size selectivity in sediment transport is important because it links the volumes and GSDs of the incoming sediment, the sediment on the channel bed and the sediment transported out of the channel, and can result in aggradation or degradation. Size selectivity can occur in different components of sediment transport, including grain entrainment, velocities, frequency of motion and deposition. Here we focus on grain size dependence in grain travel distances, which is consistent with our use of gravel tracers.
 We extend the standard continuity (Exner) equation expressed for individual size fractions of bed sediment [Parker, 1991] to the case of a bedrock channel with fraction of bedrock exposure Fe, sediment cover 1 − Fe and uniform depth sediment deposits:
where λ is sediment porosity, La is the thickness of the active (surface) layer, η is bed surface elevation above bedrock, qbT is the volumetric bed load transport rate per unit width (m2 s−1). Fi, εi, and pi are the fractional abundances of sediment in the ith size class in the surface layer, the sediment that is exchanged between the surface and sub-surface layer during aggradation or degradation and the bed load, respectively. Equation (2) is consistent with that given by Chatanantavet et al. , although it assumes that abrasion and lateral sediment inputs are zero. Changes in sediment storage in a bedrock channel can arise via changes in La, sediment GSD and Fe. If the mean sediment thickness (z = 〈η〉) is greater than La, then a sub-surface layer and the potential for vertical variations in sediment GSD develop.
 We initially consider the maintenance of steady state in a bedrock river with very thin alluvial deposits, so that z is less than the active layer thickness, La, and there is no sub-surface layer. Averaging over several flood events to eliminate the impacts of stochastic upstream sediment supply, the following simplifications apply:
Under steady state, the first three conditions maintain the sediment volume in the reach. While an increase in η − La could be offset by a decrease in 1 − Fe, this is unlikely because η − La and 1 − Fe will normally be positively correlated. Furthermore, in any reach the bedrock morphology is likely to dictate an optimum storage configuration (i.e., values of 1 − Fe and η − La) for a given sediment volume. The second condition also applies because there is no subsurface layer if z < La. The final condition is satisfied if there is no change in the active layer GSD; La is generally assumed to scale with grain size, and in alluvial rivers is approximately 2d90 [e.g., Parker, 1991; DeVries, 2002].
 Under the limits from (3), (2) reduces trivially to
 Any change in the stored sediment grain size (Fi) thus is achieved by a spatial gradient in pi, indicating the operation of size-selective entrainment and deposition. If a monotonic change in grain size persists for long time periods, Fi changes to produce coarsening (so reducing transport, inducing aggradation and development of full alluvial cover) or fining (increasing transport and transition to full bedrock exposure). To maintain steady state, both ∂Fi/∂t and ∂pi/∂x must tend to zero.
 The form of steady state in a bedrock river with z < La will depend on the relationship between Fi and pi. We envisage two possible forms. In the first, Fi is equal to pi for all i, and so all grain sizes have the same probability of entrainment and deposition. This is broadly consistent with the Kirchner et al.  model predictions of τc for grains on a bedrock surface. In the second form, the stored sediment in the reach is coarse relative to bed load entering the reach, however bed load entering and leaving the reach have the same GSD. The typical lower probability of entrainment for larger grains is offset by their higher abundance in the active layer and vice versa, producing no aggradation. For larger grains therefore Fi > pi, with Fi < pi for smaller grains.
 In alluvial rivers, a coarse surface layer enables grain-size dependent probabilities of entrainment to exist under steady state [Parker and Sutherland, 1990; Allan and Frostick, 1999]. However, the development of such a layer requires z > La, which can only occur with relatively high sediment cover. Where sediment cover increases and Fe is low, a coarse surface layer may develop enabling grain-size dependent entrainment to operate in a bedrock river at steady state. If entrainment is correlated with grain size, over a given time period smaller grains will be entrained more frequently and will travel further than larger grains, subject to the relationship between grain size and translation distance.
 We consider total travel distance, rather than solely entrainment, as our measurements are of total travel distances. We hypothesize that size-selectivity of travel distances in a bedrock river is caused by sorting taking place in alluvial areas of the bed, and thus that the degree of size selectivity is a function of the sediment storage volume in the reach, and hence of sediment cover 1 − Fe. This hypothesis is illustrated in Figure 4, which shows the possible range of size-selectivity of sediment transport distances from a minimum value of 0 to a theoretical maximum for a fully alluvial river. As 1 − Fe increases, rivers are more likely to exhibit size-selectivity closer to the theoretical maximum, as shown by the increasing white area.
Figure 4. The dimensional space between proportional sediment cover and degree of size selectivity of sediment transport distances. Labeled points in this space are Oak Creek, an alluvial river with no size-selectivity [Parker et al., 1982] and the Allt Dubhaig, an alluvial river in which size-selectivity varies as a function of excess shear stress [Ferguson et al., 2002]. The difference in size selectivity between these two is because Oak Creek has no divergence in sediment transport (∇qbT) so is in steady state, whereas the Allt Dubhaig has divergence and is hence aggrading. It is hypothesized that bedrock rivers with low sediment cover cannot be size-selective, and thus no rivers should plot within the shaded area. However, little data have previously been collected from such rivers; the bounds of this area are therefore indicative with darker shading indicating a lower probability of rivers falling within this area.
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2.6. Use of Tracers
 Our theoretical analysis of the effect of sediment cover on transport dynamics in a bedrock river has considered both the short-term movement of individual grains and long-term, reach-averaged behavior. Both analyses suggest that the volume of sediment in a bedrock reach, which is correlated with 1 − Fe determines grain dynamics and size-selectivity. Two new sets of tracer data from bedrock rivers are used here to evaluate the theoretical analyses.
 Repeat mapping of magnetically tagged sediment grains quantifies sediment dynamics in alluvial rivers over the timescale of multiple events [Hassan et al., 1984; Ferguson et al., 2002]. Such data quantify locations and probability of entrainment and deposition, transport distances and the influence of grain size and flow magnitude; but within-event dynamics cannot be resolved. Tracer recovery rate affects data quality; in alluvial rivers recovery varies considerably due to the time intervals between searches and system scale. Low sediment volumes and low probabilities of deep tracer burial in small bedrock rivers facilitate tracer recovery [Goode and Wohl, 2010b].
 It is easier to use field data to test predictions of grain-scale dynamics than to test the longer-term predictions from the continuity analysis. Collecting longer-term data is more problematic, with a higher probability of changing boundary conditions. But, focusing on short-term data assumes that we do not need explicitly to consider stochastic forcing of discharge and sediment supply [e.g., Lague, 2010], and that extreme events, which are less likely in short-term data, do not contribute significantly to the long-term dynamics.