Applications of Directional Wavefield Decomposition, Phase Space, and Path Integral Methods to Seismic Wave Propagation and Inversion

Comments on the article ?Applications of Directional Wavefield Decomposition, Phase Space, and Path Integral Methods to Seismic Wave Propagation and Inversion? [Pure Appl. Geophys. 159 (2002) 1637?1679]

“…where KðÁÞ is the refractive index field, H ðÁÞ is the Heaviside unit step function, K 1 , K AE1 , and ' are positive constants, and where the width of the central layer is 2', was also considered in FISHMAN (2002). Unfortunately, only a brief discussion of the 3-LPC profile was given and the figures illustrating the Weyl operator symbol surfaces corresponding to this profile (Figs.…”

“…The objectives of the present communication are the following: (I) To make clear the fact that the above-mentioned figures were generated using the approximation method of MILINAZZO et al (1997) and give details on how this was accomplished. (II) To present high-resolution color versions of several figures from FISHMAN (2002) which show details that are either difficult to observe or completely lost in the latter. (III) To illustrate how the mathematical connections that exist between the theory presented in FISHMAN (2002) and that of quantum mechanics can be exploited to aid seismic/acoustic propagation studies by determining the trapped modes associated with a special case of the 3-LPC profile.…”

“…(II) To present high-resolution color versions of several figures from FISHMAN (2002) which show details that are either difficult to observe or completely lost in the latter. (III) To illustrate how the mathematical connections that exist between the theory presented in FISHMAN (2002) and that of quantum mechanics can be exploited to aid seismic/acoustic propagation studies by determining the trapped modes associated with a special case of the 3-LPC profile. …”

“…(12) and (16) of FISHMAN, 2002), can be approximated using the formula where J ðp; q; n; sÞ ¼ 1 À X ðnþsLÞ À1 ðp; qÞ (FISHMAN, 1992), and where…”

“…9 0 a, 10 0 a, and 11 0 a, all of which were plotted for K À1 ¼ 2, K 1 ¼ 3, K þ1 ¼ 1, and ' ¼ 5, clearly, and very strikingly, illustrate the intricate, oscillatory structure exhibited by these surfaces, as well as the asymmetric nature of the two halfspaces jqj > ' both in terms of amplitude and frequency of oscillation. We also observe the ''square-root singularity'' (FISHMAN, 2002) running across each of these surfaces along the line p ¼ 3=2 (i.e., p ¼ 1 2 ðK þ1 þ K À1 Þ). In addition, we call attention to the fact that the number of oscillations exhibited by these surfaces increases, while their amplitudes decrease, as " k is increased.…”