For example, when I was designing a home weather station for myself, I wondered if the anemometer and wind vane could be replaced with ultrasonic sensors and emitters. If you take four ultrasonic emitter/ultrasonic sensor pairs and arrange them in a square, and measure the time of flight of four pulses, one from each corner emitter to the diagonally opposite sensor, you can determine the component of the wind vector along that diagonal. Combine the components from the diagonals, and you get the wind vector.
A bit of research turned up that this was not an original idea, and it does work. There are commercial anemometers and wind vanes (are they still called wind vanes when they don't have a vane?) for terrestrial use that work this way. There have also been many home made ones, such as this one [1].
Offhand, I don't see why such a thing would not work for an airplane.
[1] http://embedded-lab.com/blog/making-an-ultrasonic-anemometer...