Muskingum

Muskingum is a hydrologic-routing method which employs the equation of continuity to predict magnitude, volume and temporal patterns of flow as it translates downstream of a channel.

\[ I - Q = \frac{dS}{dt} \]

Channel routing functions of inflow, outflow and storage where storage can be considered as two parts, prism & wedge storage.

\[ S = K \cdot [x \cdot I^{m} + (1 - x) \cdot Q^{m}] \]

Where k is the travel time constant and x are weighting coefficient to determine the linearity of the water surface, and it ranges between 0 & 0.5, and m is an exponential constant varies from 0.6 for rectangle channel to 1.

For Muskingum version of the channel routing equation m equals one which made the relation between S and I, Q. Using coefficient k & x three weights can be calculated as follow:

\[ C_1 = \frac{\Delta t - 2KX}{2K(1 - X) + \Delta t} \]

\[ C_2 = \frac{\Delta t + 2KX}{2K(1 - X) + \Delta t} \]

\[ C_3 = \frac{2K(1 - X) - \Delta t}{2K(1 - X) + \Delta t} \]

To route the inflow hydrograph:

\[ Q = C_1 \cdot I_{j+1} + C_2 \cdot I_{j} + C_3 \cdot Q_{j} \]