Davy, I'm going to have to look up some info to give you a better answer. Some 50 years ago I worked part time for the Texas Board of Water Engineers and U.S. Geological Survey, Surface Water Division. We did a lot of stream measurements, and as I recall we took average water velocity (water depth 4 feet, velocity taken at 2 foot depth) and this was averaged for the width of the stream. In other words, you essentially took a profile of the stream bed to calculate the cross sectional area which would be in square feet. Then using an average velocity, in feet per second, the discharge volume is a product of the area in square feet multiplied by the velocity in feet per second, and the number is quantified in cubic feet per second, cfs. Then taking this even further, stream reporting came in as the gage height at a particular point in the stream. Further refinements would establish a permanent gage mark on a particular stream, and with a set water level a known discharge was calculated. Then, with fluctuations in the water level the changes in discharge volumes would be known. The government does an awful lot of measuring and recording that sounds to be superfluous, but in overall watershed management becomes pretty important.
If I can shed anything further, just let me know.