There are a lot of complications to measuring flow properly. You need a reliable way to measure the depth. To do it accurately you install a stilling-well so there is no current where you make the measurement. Then you need a level recorder. Different kinds but I think the USGS usually uses a float and shaft encoder. Then you need a way to log the data or transmit it.
But the biggest issue is how you relate the level to flow in cfs (or metric units if you prefer). To do that you need to develop a rating curve. You can measure the velocity across a transect of the river at many different levels and come up with a relationship.
If you don't care about the actual flow value, then you can just measure the level and use your experience to figure out how good the kayaking will be at different levels. For that, you can even use a stick gauge (basically a stick or board with marks) at the put in or somewhere and keep notes.
Deck Rigging Gear
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