Use of scale and shift invariances of the Mellin‐Fourier transforms in seismics

Scale invariance in assocation with shift invariance play a major role in seismic image processing. This paper is to focus the importance of the utility of scale invariance property of the Mellin transform and combined scale and shift invariances of the Mellin-Fourier transform, not only for the resolution of seismic waveform but also extraction of more information from the seismic data. It is demonstrated that the resolution of Mellin-Fourier power spectra of the (a) waveform which is shifted and compressed and (b) angulardependent waveforms are remarkably uniform. Further, mathematics using the Mellin-Fourier transform for migration is discussed. The importance of the Mellin-Fourier transform has not yet at been noticed either by the exploration seismologists or earthquake seismologists (to the knowledge of the author+