Instantaneous frequency Instantaneous frequency computation: theory and practice Matthew J. Yedlin 1 , Gary F. Margrave 2 , and Yochai Ben Horin 3 ABSTRACT We present a review of the classical concept of instantaneous frequency, obtained by dif- ferentiating the instantaneous phase and also show how the instantaneous frequency can be computed as the first frequency moment of the Gabor or Stockwell transform power spectrum. Sample calculations are presented for a chirp, two sine waves, a geostationary reflectivity trace and a very large quarry blast. The results obtained clearly demonstrate the failure of the classical instantaneous frequency computation via differentiation of the instantaneous phase, the necessity to use smoothing and the advantage of the first moment computation which always results in a positive instantaneous frequency as a function of time. This research points to the necessity of devising an objective means to obtain optimal smoothing parameters. Future work will focus on using linear and nonlinear inverse theory to achieve this goal. INTRODUCTION While the concept of frequency is very old, dating back to Pythagoras,( 529-570 B.C.), creator of the Pythagorean scale, a system of tuning that was used until the early 1500’s, the concept of instantaneous frequency is relatively new, originating with Nobel Laureate, Dennis Gabor (Gabor, 1946). In exploration seismology, instantaneous frequency has been used as one of number of seismic attributes, beginning with the work of Taner (Taner et al., 1979) and extended by Barnes (1992, 1993), In electrical engineering, an excellent re- view of the theory and application of the instantaneous frequency is provided by Boashash (1992a,b). In this research work, we will first review the basic theory of the classical method of computing the instantaneous frequency in the continuous domain. Then we will review an application of a technique developed by Fomel (Fomel, 2007b) to address the smoothing issues arising in the computation of the instantaneous frequency. We will then present the method using the moments of the linear time-frequency transforms as applied by Margrave (Margrave et al., 2005) and Stockwell (Stockwell et al., 1996). In addition to the linear time-frequency transform analysis, there exists the bi-linear Wigner-Ville transform Wigner (1932); Ville et al. (1948), which has also been applied to dispersive signal detection in earthquake records (Prieto et al., 2005). This bi-linear transform will not be considered in the present research scope. 1 Department of Electrical and Computer Engineering, University of British Columbia 2 Geoscience, University of Calgary 3 SOREQ, Yavneh, Israel CREWES Research Report — Volume 25 (2013) 1