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weight of the water (pghS). In this case, Sf is estimated by solving the one-dimensional equation for gradually varied flow: z Sf = dH = - d (z + h + °- (6) dx dx t 2g where H is the total energy, z is the bed elevation, u is the mean velocity in the downstream (x) direction. Equation 6 is solved using the step-backwater method, a trial-and-error procedure that is described in many hydraulics texts (e.g. Henderson, 1966; Dingman, 1984), and can be easily programmed on spreadsheets. In this study, the step-backwater method was used with cross section and water-surface elevation data from short (--500 m) reaches to develop relations between r" and Q, which were then used to estimate discharges for initial motion and bankfull flow. At scales much greater than individual pools and riffles (> 1000 m), the variations in channel form have much less of an effect on the boundary shear stress. In this case, the flow can be considered uniform, and the water-surface slope S, can be used in place of Sf in (6). This assumption was adopted herein to calculate reach-average values of r*. To estimate the discharge required to initiate bed load transport (termed the critical discharge, Q,) we assume uniform flow, and combine the continuity equation (Q = w h u, where w is the flow width), with the Manning equation (u = h73 Sf 1'2/n, where n is a resistance coefficient), to get w h5/3 S112 Q = C f- n (7) Here, h, is the depth required to produce the assumed critical dimensionless shear stress at each cross section, calculated from (4) and (5) using the reach-average D50. To estimate the bankfull discharge Qb, we used the same equation (7) with the bankfull depth hb and slightly lower values of Manning's n. On the basis of the results obtained in calibrating the step-backwater model, we selected values of n ranging from 0.035 in upstream reaches to 0.025 in downstream reaches. Equation (7), after some algebraic manipulation involving the term S, can be rearranged to give a uniform-flow equation for the average boundary shear stress as a function of discharge: r = PghS = Pg(Qn)0.6S0.7 W (8) The above equation indicates that for constant n and w, r increases to the 0.6 power of Q and the 0.7 power of S; graphically, eqn. 8 plots as a concave-down curve, with a slope that gradually decreases as Q increases. However, if any of the variables in (8) change with discharge, then the relation between r and Q may differ from the uniform-flow case. For example, if S increases with Q, the relation formed by (8) will be steeper and more nearly linear; conversely, if S or n decrease with Q, or if w increases, the relation will be flatter and more concave. Reach characteristics determine whether S, n or w change much with discharge, but whatever the case, the shape of the Q- r relation gives an indication of the potential change in shear stress for a given increment of water discharge: a near-linear relation indicates that r increases in proportion to Q, whereas a concave relation indicates that r increases more slowly as Q increases. Equation 8 thus represents a base case against which we can compare modeled discharge-shear stress relations in short reaches where the assumptions of uniform flow, or constant w, n and S, generally do not hold. 18