Exact seismic velocities for TI media and extended Thomsen formulas for stronger anisotropies |

The work presented here has necessarily been restricted to VTI and HTI symmetries, because these correspond to the simplest and most studied cases in the anisotropy literature. It has sometimes been noted, however, that the HTI symmetry in particular is actually a fairly unrealistic model for seismic exploration problems (Schoenberg and Helbig, 1997; Tsvankin, 1997, 2005; Thomsen, 2002). The reason for this is that the earth, to a first approximation, is often horizontally layered and such horizontal layering is well-known to produce VTI symmetry (Postma, 1955; Backus, 1962). If aligned vertical fractures are superimposed on this already anisotropic background medium (unlike the isotropic background medium used in the models presented here), then the resulting symmetry is likely to be closer to orthorhombic (having nine independent elastic constants) than to HTI (having at most only five independent constants). This viewpoint no doubt provides a valid criticism of the work presented in the examples as far its value for practical applications. However, the author expects the present paper to be followed by others on this topic, and future efforts will be devoted to obtaining comparable results for such orthotropic systems (Mensch and Rasolofosaon, 1997), and thereby becoming more realistic for exploration purposes.

All the results presented here are specifically for phase velocities of the seismic waves.
In heterogeneous media, it is instead the ray (or group) velocities that are needed for
ray tracing applications, and in particular for wave equation migration. However, recent work
by Zhu *et al.* (2005; 2007) has reformulated the Gaussian beam migration approach to permit
direct use of phase velocities, with a corresponding reduction in the overall computational burden.
Although it is too soon to be certain which potential applications of the results contained herein may
prove to be of value, it is seems likely that this particular application will be one of the more
interesting ones for seismic data analysis.

Exact seismic velocities for TI media and extended Thomsen formulas for stronger anisotropies |

2007-09-15