## A threeparameter Muskingum model

Both the original Muskingum method of Section 14.2.2 and the modified Muskingum-Cunge method of Section 14.2.3 apply to situations where there is no lateral inflow to the river reach between the upstream and downstream gauging stations. In most rivers, this constrains the routing reaches to be rather short, generally terminating at tributaries, and requires gauged or estimated tributary inflows to be added to the

1985.)

main channel inflow term. In turn, this means using many reaches in the total routing procedure. A second modified Muskingum method has been developed by Terrence O'Donnell that incorporates a simple lateral inflow model. This modified method has two further advantages: (a) it provides a numerical and direct best-fit solution technique to obtain the parameters K and x; and (b) by treating the whole river as one reach, it avoids the need for multiple routings (and multiple parameter determinations) over many sub-reaches.

The lateral inflow model used is shown in Fig. 14.12 (O'Donnell, 1985) and assumes that the total rate of lateral inflow over the whole reach is directly proportional to the upstream inflow rate. The proportionality constant, a, is taken to be fixed for any one event but takes different values for different events. The original two-parameter Muskingum model (K, x) is thus extended to a three-parameter model (K, x, a).

The three coefficients, ci of the routing equation (14.11) can be related to K, x and a (O'Donnell, 1985) and vice versa. A direct least-squares solution by a matrix inversion technique yields a set of best-fit values for the ci coefficients from the set of equations formed via equation (14.11) applied to all the At intervals in an observed event. The three coefficients no longer sum to unity as in the original two-parameter Muskingum method since it is expected that the mass outflow will be greater than upstream inflow because of the lateral inputs.

O'Donnell et al. (1987) applied this extended Muskingum procedure in a split sample test to a number of flood events over a single 50 km reach on the Grey River, New Zealand. Half the events were used for calibration, i.e. to establish average values for K and x for the reach. (The latter parameters are postulated to be fixed properties of the reach, whereas each event has its own a value, a property of the causative storm.) The average K and x values were then applied to reconstruct the outflow hydrographs for the events not used in the calibration from their individual inflow hydrographs and a values. Fig. 14.13 shows such a reconstruction for an event in which the value of a was 6.92. The volume of outflow at Dobson for this event was nearly seven times the volume of inflow at Waipuna due to the very substantial lateral inflow between the two stations.

Fig. 14.13 Grey River flood event hydrographs at Waipuna (upstream) and Dobson (downstream). K = 5.06 h, x = 0.158, a = 6.92. (Reproduced from O'Donnell et al., 1987.)

Fig. 14.13 Grey River flood event hydrographs at Waipuna (upstream) and Dobson (downstream). K = 5.06 h, x = 0.158, a = 6.92. (Reproduced from O'Donnell et al., 1987.)

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