Muskingum Channel Storage

Routing a flood down a river reach needs a storage relation, and the Muskingum method supplies one by splitting the water in the reach into two pieces. The prism is the volume that would be there under a level surface set by the outflow. The wedge is the extra volume the sloping surface holds during the rise, and it is negative during the recession, when the reach is emptying faster than it is filling. Weighting the two gives $S = K[XI + (1-X)O]$. Twelve hours of travel time with X at 0.2, 50 m³/s in and 30 out, holds $43200 \times 34 = 1.47$ million m³.

The two parameters have honest physical readings. K is the travel time of the flood wave through the reach, close to the reach length divided by the wave celerity, which is itself roughly 1.5 times the mean water velocity in a wide channel. X measures how much wedge storage the reach carries: 0 is a level-pool reservoir with no wedge at all, and 0.5 is pure translation with no attenuation whatsoever. Natural streams calibrate to 0.2 or 0.3, and the fact that the method was named after the Muskingum River in Ohio, where the US Army Corps first fitted it in the 1930s, tells you what kind of channel it was built for.

The calibration trick is worth knowing and is not obvious. Plot the weighted discharge against storage for a measured flood and the loop you get closes into a single line only at the correct X, so you try values until the hysteresis disappears. There is also a numerical constraint people violate constantly: the routing step must satisfy $2KX < \Delta t < 2K(1-X)$, and stepping outside it produces negative outflows at the start of the hydrograph. Those negative flows are a signature of a bad time step, not of bad data, and the fix is the step rather than the parameters.