The question is examined whether there may be scattering media which could generate spectral frequency shifts in radiation from sources that are at rest relative to the observer and yet would imitate Doppler shifts. Scattering kernels of media whose physical properties fluctuate randomly in both space and time are presented which achieve this to a good approximation. The frequency shifts produced in this manner are not necessarily small. The results might be of particular interest in connection with the long-standing controversy about the origin of some of the large redshifts observed in the spectra of certain quasars.