They are not woolly.
When I first looked at the design I thought the same about doppler.
After some research I found:
http://www.circuitcellar.com/library/print/0106/Cyliax-186/2.htm
I will quote from his site:
You might have noticed that you aren't measuring the Doppler shift of the sound burst. A common misconception about using a sonic anemometer is that you can measure the wind speed by measuring the Doppler frequency shift of the ultrasonic signal.
Recall the high school physics experiment relating to train whistles changing pitch as they approach and recede from a stationary listener. Of course this is true, but in the sonic anemometer setup, both the listener and the train whistle appear to be moving at the same speed. So, it's more like listening to the train whistle while you're on a car in the back of the train (i.e., the pitch doesn't change) even though the train may be moving at great speed.
The Doppler signal in a sonic anemometer measures the change in wind speed, which can give an indication of the wind speed's stability at the moment it's measured. You can use this to measure turbulence and fluctuations in wind speed. Of course, this involves being able to measure the changes in the received signal's frequency. You can use a fast Fourier transform (FFT) to look at the purity of the spectrum.
Phaseshift on a waveform is equilivant to a time delay - The concept of "group delay" is related to phase shift of the wave.
Yes the tof is measured on north-south then east-west direction. From that the actual wind speed and direction is calculated.