A significant improvement of the Muskingum model was developed by Cunge (1969) known as the Muskingum-Cunge model. This increased the Muskingum
model's accuracy and made it applicable in situations where observed inflow and outflow hydrographs were not available for calibration and enabled it to be changed from a lumped to a distributed flow routing model. Cunge derived Eq. (10) using the assumption of a single-valued Q(h) relation, the classical kinematic wave equation [see Eq. (25)], and applying a four-point implicit finite-difference approximation technique. Equation (10) is rewritten where the flows /(/), I(t + At), 0(t), and 0(t + At) are replaced by Qj, Qj+X, Qj+l, and respectively, i.e.,
Equation (15) has been expanded to include effects of lateral flow (q) along the AX routing reach; and where the following expressions for K and X are determined:
in which c is the kinematic wave speed, Ax is the reach length, and Se is the energy slope approximated by evaluating Sa in Eq. (3) for the initial flow condition. The overbar indicates the variable is averaged over the Ax reach and over the At time step. Equation (19) may be expressed in an alternative form, i.e., c = K'Q/A
in which A is the cross-sectional area, B is the channel width at the water surface, h is the water-surface elevation of the flow, and the Manning equation is used to relate discharge (0 and depth or water-surface elevation (h). Depending on the cross-section shape, K' may have values in the range | < K' < |; the upper value is associated with either a very wide or rectangular channel. Selection of the appropriate time step At in sees is given by:
where Tr is the time of rise in hours of the inflow hydrograph and M is an integer (10 < M < 20) whose value depends on the extent of variation in the inflow hydro-
graph. The selection of Ax affects the accuracy of the solution. It is related to At and is limited by the following inequality (Jones, 1981):
While the Muskingum -Cungc model does not require observed inflow-outflow hydrographs to establish the routing coefficients as required in the Muskingum model, best results are obtained if the wave speed (c) is determined from actual flow data. Also, the model is restricted to applications where backwater is not significant and discharge-water elevation rating curves do not have significant loops and discharge hydrographs are not rapidly changing with time such as dam-break floods. Nonetheless, the Muskingum-Cunge model (Miller and Cunge, 1975; Ponce and Yevjevich, 1978) is a highly versatile simplified routing model.
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