The stage-discharge relation is defined by the complex interaction of channel characteristics, including area, shape, slope, and roughness of the cross-section. The combination of these effects has been designated control. The control is permanent if the stage-discharge relation it defines does not change with time, otherwise it is impermanent. The latter is designated a shifting control. The stage-discharge relation for a permanent control is established by simultaneous measurements of the two parameters stage and discharge, the relation is often described mathematically by a parabolic formula.
Controls may change because of the effects of a changing channel, backwater, rapidly changing stage, variable channel storage, and the freezing and breaking of ice. Backwater is produced when the normal water slope is decreased as the result of the normal stage being increased at some point downstream. This can result from the operation of a dam, an increase in discharge in a downstream tributary, or a rise in the stream into which the gaged stream empties. The effect of backwater is determined by introducing the slope or the fall through a reach of the channel downstream from the hydrometric station as a third parameter. This is graphically incorporated into a three-dimensional stage-fall-discharge relationship.
In Norway this relation is established by using Manning's formula and simultaneous measurements of discharge, stage, and fall (Hansson n. d., Westerberg 1920). In other methods a computational conversion of the measurements to a normal fall is made (Linsley, Kohler & Paulhus 1958). These methods give a graphical representation of the stage-fall-discharge relationship. They demand much work, are cumbersome in use, and badly adapted for conversion of stage measurements to discharges on digital computers.
The following describes a method well fitted for such conversions on computers. An eventual correction of the formulae used is also possible in a simple manner.