1. MUSKINGUM VS MUSKINGUMCUNGE METHODS The MuskingumCunge method is a method of flood routing that improves on the classical Muskingum method (Chow, 1959) by using physicalnumerical principles established by Cunge to calculate the routing parameters (Cunge, 1969). By way of comparison, it may be affirmed that while the classical Muskingum method is hydrologic in nature, the MuskingumCunge method has a distinct hydraulic basis. Table 1 compares both methods, describing their differences.
McCarthy, G. T. 1938. "The unit hydrograph and flood routing," unpublished manuscript, presented at a Conference of the North Atlantic Division, U.S. Army Corps of Engineers, June 24. Cunge. J. A. 1969. "On the subject of a flood propagation
computation method (Muskingum Method)," Journal of Hydraulic Research, Vol. 7, No. 2, 205230. Chow, V. T. 1959. Openchannel Hydraulics, McGraw Hill, New York, Ponce, V. M. 2014. "The MuskingumCunge method,"
Section 10.6 of
Fundamentals of Openchannel Hydraulics, online textbook. Hydrologic method, where the routing parameters
K and X are calculated by trialanderror calibration,
using a pair of inflowoutflow hydrographs measured in a gaged river
reach.
Hydraulic method, where the routing parameters K and X are based on the channel morphology, as represented by the
prevailing channel slope and
crosssectionalshape characteristics. The availability of measured flood hydrographs suited for parameter calibration.
The availability of geometric and hydraulic channel data,
to the extent that this data properly represents the
reach under consideration.
Good with any computational tool, including a spreadsheet, programming, and available commercial and government software.
Good with any computational tool, including a spreadsheet, programming, and available commercial and government software.
Good for the reach and event
used for calibration;
poor for any other reach or flood event.
Very good for all events within the reach under consideration; only
limited to the extent that the geometric and hydraulic channel data
must be representative of the reach. Very limited; usually not available due to large hydrologic data requirements.
Available at the cost of additional complexity. Dated method; limited in
accuracy when used for extensive watershed/basin routing.
Newer method, featuring stateoftheart hydraulic numerical modeling knowledge; ideally
suited for extensive watershed/basin routing. Given the facts described in Table 1, we posit that the MuskingumCunge method is altogether theoretically better than the Muskingum method. Thus, we encourage its wider use in U.S. and world hydrologic and hydraulic engineering modeling practice.
2. MUSKINGUMCUNGE AND FLOOD WAVES The key to the understanding of the theoretical basis of the MuskingumCunge method is the recognition that the diffusion wave applies through a wider range than competing waves such as the kinematic and dynamic waves (Ponce and Simons, 1977). Kinematic waves do not attenuate, while most flood waves attenuate at least a little bit; on the other hand, dynamic waves attenuate too much and, therefore, do not represent flood waves in typical cases. A diffusion wave, lying in the midrange of attenuation, is, by far, the most applicable wave model from the standpoint of practice.
This fact was recognized early by McCarthy in 1938 (Fig. 1)
and later by Cunge in 1969. However, unlike McCarthy,
Cunge tied the Muskingum method to the properties of the governing
diffusion wave. Fig. 1 The Muskingum river near Marietta, Ohio. Given the demonstrated simplicity of the MuskingumCunge method, particularly when compared to alternative hydraulic routing models, the former remains a strong candidate among the gamut of available routing models. This is particularly the case in view of the fact that hydraulic routing methods are generally unsuited for hydrologic applications involving intensive watershed/basin routing, where, clearly, the use of a much simpler method is advised. Thus, the MuskingumCunge method emerges as the only diffusionwave routing model which is simple and accurate enough to be suited for largescale hydrologic modeling applications. In the remainder of this paper, we andeavor to make the case for justifying this assertion.
3. MUSKINGUMCUNGE AND COMPUTATIONAL ACCURACY The strength of the MuskingumCunge method is clearly its theoretical basis as an analog of the diffusion wave equation. Cunge realized that the Muskingum method and the kinematic wave model shared the same theoretical basis. Furthermore, Cunge was able to calculate the error of the firstorder numerical scheme (i.e., the Muskingum method) and to tie this error to the hydraulic diffusivity of the diffusion wave (Hayami, 1951). This accomplishment paved the way for the calculation of routing parameters in terms of geometric and hydraulic variables, thus circumventing the need for the expensive and impractical streamgage measurements. The MuskingumCunge method is accurate because of its strong theoretical basis. Its numerical properties, including stability and convergence, have been extensively documented both in theory and in practice (Ponce and Vuppalapati, 2016). The method is strongly stable and with excellent convergence properties for values of Courant number in the neighborhood of 1. This property of strong stability and excellent convergence all but assures its grid independence, i.e., the property of a numerical scheme to reproduce the same result, regardless of grid resolution. Competing (numerical) methods, including the kinematic wave, can be shown to lack grid independence, casting a cloud on their theoretical correctness (Ponce, 1986). In summary, the MuskingumCunge method is the only numerical analog of the diffusion wave equation based on a straightforward, explicit, pointbypoint computation, allthewhile featuring grid independence. No other flood routing method can claim this particular set of properties at this time.
4. ROUTING EQUATIONS
The basic routing equation of the MuskingumCunge method is the following (Fig. 2):
Fig. 2 Definition sketch.
in which j is a spatial index, n is a temporal index and C_{0}, C_{1} and C_{2} are calculated as follows:
The parameters K and X are calculated as follows (Cunge, 1969; Ponce, 2014):
in which Δx = reach length (space interval); c = flood wave celerity; q =
unit width discharge; and
5. LINEAR VS NONLINEAR ROUTING
In Nature, flood waves exhibit a nonlinear behavior, i.e., their celerity and attenuation properties (tend to) vary with the flow. The magnitude of this effect varies with the crosssectional shape in predictable ways. Three asymptotic crosssectional shapes are recognized (Ponce, 2014):
The inherently stable channel is that for which the hydraulic radius is a constant in the overflow channel. The shape of the inherently stable channel has been calculated by Ponce and Porras (1995) (Fig. 3).
Fig. 3 The inherently stable crosssectional shape. It can be shown that the nonlinear effect is strongest for hydraulically wide channels, weak for triangular channels, and totally nonexistent for inherently stable channels. In practice, most crosssectional shapes are likely to be close to being hydraulically wide. Thus, the nonlinear effect may be marked in certain, if not typical, cases. Notwithstanding the nonlinear effect, there are two ways to calculate the routing parameters in MuskingumCunge routing (Ponce and Yevjevich, 1978):
In the linear approach, the routing parameters K and X are based on average (or representative) hydraulic variables (c and q) and kept constant throughout the computation. In the nonlinear approach, the routing parameters are allowed to vary with the flow, i.e., to vary for each computational cell as a function of local flow variables. The tradeoffs between linear and nonlinear MuskingumCunge routing are described in Table 2.
6. DISCUSSION The MuskingumCunge method represents a considerable improvement in computational accuracy when compared with the related Muskingum method. The only caveat is that the parameter calculation (Eqs. 5 and 6) should be based on values of flood wave celerity c and unitwidth discharge q that are representative of the reach under consideration. To acomplish this objective, it is recommended that GISsupported geometric and hydraulic data be used to better estimate the relevant input data and variables on which to base the calculation of the routing parameters.
7. SUMMARY The MuskingumCunge method is reviewed to further clarify its theoretical basis and encourage its wider acceptance and use in current hydrologic engineering practice. Its theoretical background and computational accuracy are reviewed and clarified. The method is recognized to be the only numerical analog of the diffusion wave equation based on a straightforward, explicit, pointbypoint computation, allthewhile featuring grid independence. We note that no other flood routing method can claim this particular set of properties at this time.
REFERENCES Chow, V. T. 1959. Openchannel Hydraulics. McGraw Hill, New York. Cunge, J. A. 1969. On the Subject of a Flood Propagation Computation Method (Muskingum Method), Journal of Hydraulic Research, Vol. 7, No. 2, 205230. Hayami, I. 1951. On the propagation of flood waves. Bulletin, Disaster Prevention Research Institute, No. 1, December. McCarthy, G. T. 1938. The Unit Hydrograph and Flood Routing. Unpublished manuscript, presented at a Conference of the North Atlantic Division, U.S. Army Corps of Engineers, June 24. Natural Environment Research Council. 1975. Flood Studies Report. Vol. 3: Flood Routing. London. England. Ponce, V, M., and D. B. Simons. 1977. Shallow wave propagation in openchannel flow. Journal of the Hydraulics Division, ASCE, Vol. 103, No. HY12, December, 14611476. Ponce, V. M., and V. Yevjevich. 1978. MuskingumCunge method with variable parameters. Journal of the Hydraulics Division, ASCE, Vol. 104, No. HY12, December, 16631667. Ponce, V. M. 1986. Diffusion Wave Modeling of Catchment Dynamics. Journal of Hydraulic Engineering, ASCE, Vol. 112, No. 8, August, 716727. Ponce, V. M., and P. J. Porras. 1995. Effect of crosssectional shape on freesurface instability. Journal of Hydraulic Engineering, ASCE, Vol. 121, No. 4, April, 376380. Ponce, V. M. 2014. Fundamentals of openchannel hydraulics. Online textbook. Ponce, V. M., and B. Vuppalapati. 2016. MuskingumCunge amplitude and phase portraits with online computation. Online article.
APPENDIX: Origin of the term MuskingumCunge
In a hydrologic context, the word "Muskingum" is taken from the Muskingum river,
in eastern Ohio.
The term "Cunge" gives the proper credit to Jean A. Cunge, a PolishFrench engineer
who, in 1969, published the equations used in the MuskingumCunge method.

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