This question comes from a realistic problem that how to obtain a time domain wind pressure from known wind spectrum. Or similarly, gets rail surface curve from spatial rail spectrum.
Obviously, this question does not have unique solution, since phase information is not given. Usually we assume that phase of the time/spatial domain data in uniformly distributed in ($-\pi$, $\pi$], and using a random distributed phase to do inverse Fourier transform to obtain the time/spatial domain data. However, I feel not so comfortable to assign such rapid changing phase to the spectrum. I wonder whether there are some more reasonable assumptions to get these jobs done?
BTW, question ifft of signal without phase information discussed some strategies on how to do this job. @LutzL suggested Human perception can be used to help determine the phase information. But I cannot understand how Human perception is working. @Hilmar said minimum phase may be a choice, but I do not know whether there is a physical meaning of minimum phase. Is there any discussion to show minimum phase is a good choice for physical phenomena, like wind.