How exactly do we define wind velocity?
Obviously, I suppose this is something like "speed of displacement of a quantity of air", but this seems to be too imprecise.
For example, a cup anemometer measures this velocity, but any anemometer depends on a calibration function that depends on size of cups, material used, and so on. So how do we know this function?
Before defining wind velocity, I think it is necessary to explain the continuum assumption. The basic idea is that even though gases are composed of discrete molecules occupying a small fraction of the total volume filled by the gas, gas flows are made up of many individual collisions between gas molecules. In most applications, the flow field is assumed to behave as a continuum and the discrete molecular nature of gases is ignored. By doing so, fluid properties (e.g., density, velocity) can be defined anywhere in the flow.
In order for the continuum assumption to be appropriate, the length scale of the application must be much larger than the mean gas particle path length. In most atmospheric conditions, including in anemometers, the mean path length is on the order of micrometers, while the anemometer size is much larger.
While most anemometers (cup, vane, laser, acoustic resonance) measure wind speed some measure the wind's pressure (e.g., tube, plate anemometers). In the case of cup anemometer, like any other measuring device, there is an error associated with the measurement that depends primarily on the cup size. In general, the anemometer factor (the ratio of wind speed to the rotation speed of the cups) has a value between two and a bit more than three and it depends on the size of the cups and arms.
You can calibrate anemometers from other anemometers, or loosely from a windsock if precision isn't that important. But per your question, how was the first anemometer calibrated?
See my answer about wind and you'll see that we have equations that relate fluid velocity to factors like pressure gradients, friction, coriolis force, advection, etc. We could control for these and neglect everything except the pressure gradient within a wind tunnel, and in this wind tunnel with known pressure differences we could derive the flow velocity at the anemometer and thus calibrate it.
You could use the same equations to determine the forces on an anemometers cups and derive or model the relation to flow velocity directly. At some point you'd probably like the wind tunnel to empirically verify the model though or at least another known good anemometer.
