On the instantaneous frequency and quality factor

We analyse the concepts of instantaneous frequency (IF) and quality factor (IQ). It is verified that the time-averaged IF is equal to the centroid of the signal energy of the spectrum and that the centroid of the signal spectrum is equal to the IF at the peak of the signal envelope. The latter property can be used to obtain the frequency shift required by tomographic methods. Then, we analyse the two-tone stationary Mandel signal in the lossless and lossy cases. The IQ is not infinite in the lossless case, although its reciprocal average vanishes, and the lossless and lossy IFs at the peak of the signal envelope are the same, whereas the IQ at this peak depends on the amplitudes and quality factors of the tones. The IQ of a propagating Ricker wavelet has a singularity at the peak of the envelope, which shows a shift in the lossy case, related to the velocity dispersion. We consider a lossy layer described by the Zener model. Varying its thickness implies a large variation in the IF, introducing unphysical spikes when the top and bottom reflections of the layer start to overlap. Finally, a practical application to real seismic data is presented.