in which k is the iteration counter; and f(hJk+l) is the left-hand side of Eq. (7) evaluated with the first estimate for h¡+] , which for k = 1 is either hJ or a linear extrapolated estimate of hJ+l; f'(hJk+i); is the derivative of Eq. (7) with respect to /z7+1. It can be approximated by using a numerical derivative as follows:
f\K+X) = mi+X + <0 -m+X - m(H+l +B)~ (h(+l - e)] (9)
in which e is a small value, say 0.1 ft (0.03 m). Using Eq. (8), only one or two iterations are usually required to solve Eq. (7) for hJ+K Initially, the reservoir pool elevation (h j) must be known to start the computational process. Once h/+1 is obtained, Q1'1 can be computed from the spillway discharge equation, Q = f(hJk+1).
Level-pool routing is less accurate as the reservoir length increases, as the reservoir mean depth decreases, and as the time of rise of the inflow hydrograph decreases (Fread, 1992). This inaccuracy can have significant economic effects on water control management practices (Sayed and Howard, 1983).
A widely used hydrologic flow routing model is the Muskingum model developed by using Eq. (5), with nonzero values for both K and X, for the storage relationship. Substituting this information into Eq. (4), the following is obtained for computing
where c(i = k-kx + At ¡2 C, = -(KX - At/2)/C0 C2 = (KX + At/2)/C0 C3 = (k-kX- At/2)/C0
and where C, + C2 + C3 = 1 and K/3 < At < K is usually the range for At.
Equation (10) is the widely used Muskingum routing model first developed by McCarthy (1938). The parameters K and X are determined from observed inflow-outflow hydrographs using least squares or its equivalent, the graphical method, or other techniques (Singh and McCann, 1980). Among the many descriptions and variations of the Muskingum model are Chow (1964); Chow et al. (1988), Strupc-zewski and Kundzewicz (1980), Dooge et al. (1982), and Linsley et al. (1986).
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