where t and t + At indicate that the values of x and K are evaluated at the start and end of a time step. Since at the start of the time step, the values of c and Q and therefore x and K will not be known, this method requires an iterative solution at each time step. Ezio Todini shows that the method gives a very good approximation to a full solution of the 1D dynamic equations (see Section 14.3) below, at least for simple channel shapes.
14.2.7 Practical application of the Muskingum routing method
The Muskingum routing method has been very widely applied since it was first developed in the 1930s. It is still an option in a number of river-modelling software packages currently in use, such as in the Infoworks software package from Wallingford Software. It must be used, however, with some care. It was noted earlier that, mathematically, the Muskingum equations have the form of a linear transfer function (see Section 12.8.3) with one a parameter, two b parameters and no time delay. The values of K, x and a will determine the equivalent values of a and b coefficients in the general linear transfer function equation. While the values of K and x can determine the expected attenuation of a flood wave in moving downstream, they can only indirectly account for any pure time delay between the start of a rise in discharge at the upstream site and the start of the rise at a downstream site. If this time delay is long enough, then fitting the Muskingum parameters will often give rise to parameter values that correspond to a transfer function that initially has a negative response to an upstream rise, before the expected positive impulse response (see Fig. 14.14). This provides the delay necessary, but clearly only in a non-physical way.
There are two ways around this in practice. The first is to subdivide the reach into smaller reaches, all with the same fitted values of K,x and a. Output at a time step from the first reach then becomes the input at the next time step to the next reach. If there are at least as many reaches as time steps required to represent a pure delay, then the Muskingum transfer function for each reach should remain positive. Routing the input through multiple Muskingum models in this way will, however, also have an effect on the attenuation of the flood wave (as with the Nash cascade model of the unit hydrograph discussed in Section 12.5.1, when the higher the number of stores in series, the greater the smoothing of the impulse-response).
The second is to recognise that the Muskingum model is only a specific example of the more general family of linear transfer functions and take advantage of the routines available to fit transfer functions with time delays (such as in the DBM models discussed in Section 12.8.2). This is discussed in the next section.
Fig. 14.14 Comparison of extended Muskingum and DBM Flood routing results, River Wyre, Lancashire: (a) predicted flows against time for Muskingum and best-fit DBM model; (b) fitted transfer functions (after Young, 1986).
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