Erratum: Broadband Q factor imaging by seismic tomography and instantaneous frequency (Geophys. J. Int. DOI: 10.1093/gji/ggy168)

We received a very kind message form Martin Ackroyd about the Appendix 2 of our paper 'Broadband visco-acoustic Q factor imaging by seismic tomography and instantaneous frequency', which was published in this journal (Lin et al. 2018). In that Appendix, we used a formula from his paper (Ackroyd 1970) that unfortunately had a typographical error. The spectral centroid fc(t), expressed in that paper as: (Equationp presented) where m(t) is a complex trace, the upper dot indicates a time derivative and the asterisk a complex conjugation. Introducing the correct formula in our Appendix 2, the final result remains the same as presented by Ackroyd (1970): (Equationp presented) i.e. the instantaneous frequency is proportional to the centroid of the instantaneous spectrum of the analytic signal m(t). However, although the practical results of our paper remain correct, the theoretical background becomes stronger. Indeed, when using equation (1), we and Saha (1987) could claim that equation (3) is correct at the envelope maxima, but only an approximation elsewhere. Instead, using the corrected version (2), equation (3) holds for any value of time t. In other words, equation (3) is an exact link between the instantaneous frequency and the centroid of the instantaneous spectrum, and not only at a few sparse points in time. In the latter limiting case, the use of the instantaneous frequency had to be viewed as a heuristic indicator of very local spectral changes, but lifting the limits with the amended derivation, we can rely on it as a continuous indicator. This does not change the algorithm applicability to real data, which is validated in the paper, but strengthens its physical background. In the amended Appendix 2 below, ideally replacing the old one, we added a few intermediate steps to our derivation, as clarity seems particularly needed here. We apologize for our error and thank Martin Ackroyd for his correction.