Why is phase rotation needed to produce the colored inversion operator?
When performing "imaging" we are generally attempting to recover the reflectivity of the earth.  At zero offset the reflectivity is the difference in Acoustic Impedance (AI) normalized by the sum. When performing colored inversion we are seeking to recover a layered representation of the band limited AI and not the reflectivity.  As we go from an AI log to a reflectivity log by differentiating we need to integrate to go the other way.  In the frequency domain integration is a -90 degree phase term and an amplitude weight term.  The amplitude term (1/freq) is taken care of when matching the spectra to that of the well log in the amplitude term.