I didn't say there is no advection, I said that its effect is negligible. That is, the total distance traveled by the sound is almost the same than if we base the calculation solely on the speed of sound c. Sure there is advection.
You know, people of fluid dynamics have a particular way of talking, when we say negligible we mean that we know it exists, but the problem on this board the effect is so small compared with others (i.e. dissipation) that it is not worthy to include it in a calculation. I have found myself in hard times trying to explain this things to people of other fields of science (and I have had hard times trying to understand another concepts of their respective areas), but that's the way we discard things (there' s so many effects out there!).
Maybe the engineer in charge of designing that sensor for weather forecast needed an extraordinary accuracy for measuring the doppler effect and it is sensible to do that btw, but that's another problem, is not the problem stated by the opener.
Advection is proportional to the inertia of the fluid. Low speeds mean low advective effects, low Mach Numbers, large Mach angles (talking about Doppler effects) and behavior nearly similar to Incompressible=Linear Acoustics. When the velocity of the flow increases, the bulk inertia of the fluid transports information in a non linear way (v^2), that's what happens in sound waves and Compressible=Non Linear Acoustics (Large Mach Numbers). In the latter case, a sound wave always travels at the local speed of sound respect to the fluid (i.e. with a laboratory velocity that is the sum of the local fluid velocity and the local speed of sound). In that range both of them are of the same order of magnitude, and as you know from your courses of PDE's, the slope of the characteristic lines is exactly the inverse of the sum of both speeds. In our problem, the best approximation is the linear acoustics and consider that the sounds waves travel at the speed of sound regardless of the advection. In the problem of the sensor makes sense to consider the small effect of the doppler shift, because it is the main outcome of the calculation.