Short Note The calculation of instantaneous frequency and instantaneous bandwidth

A number of ways have been offered to calculate instantaneous frequency, an important complex seismic trace attribute. The standard calculation follows directly from its definition and requires two differentiations (Taner et al., 1979). By avoiding these differentiations, three formulas that approximate instantaneous frequency are faster to compute. The first employs a two-point finite-impulse response (FIR) differentiator in place of the derivative filter (Scheuer and Oldenburg, 1988). The second is nearly the same as the first, except that it employs a three-point FIR differentiator (Boashash et al., 1991). The third takes a different approach and involves two approximations (Claerbout, 1976, p. 20; Yilmaz, 1987, p. 521). How do these formulas compare, and which is best?