What assumptions should be used to invert spectrum into time domain data? 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.
If you know anything at all about the time-domain signal, for example that it's piecewise linear, or should never be negative, or should look like a series of spikes with exponentially decaying tails, you could define a cost function that measures how well a proposed signal meets those criterea.  How to do that is hard to describe in a generic way.  Then feed that cost function into your favorite multi-variable optimizer.  The phases of the Fourier amplitudes are the variables to be optimized.   Try a variety of initial values, beware local minima, and all the other usual cautions for optimization apply.