In TI media, exact phase-velocity equations ν(θ) are complex and difficult to exploit, in particular when derivatives with respect to some parameters have to be calculated. Moreover, traveltimes are not equally sensitive to the different parameters in the phase equations. Thus, to find simpler but still accurate equations for ν(θ), with relevant (sensitive) parameters, we examine different formulations found in the literature, and we also derive new original approximations of the phase-velocity equations. Thomsen's weak-anisotropy approximation yields simple formulas but breaks down for moderate but realistic anisotropy. We derive an approximation we call weak-anisotropy-squared approximation, which is more accurate than the former one (it respects horizontal velocities) but is still valid only for weak-anisotropy. Muir's double elliptical approximation is quite accurate, but very difficult to exploit. We derive a new empirical approximation based on Alkhalifah's ideas. This approximation is as accurate as Muir's approximation but requires fewer parameters, and it has a form allowing “easy” computation, in particular, the calculation of derivatives. Moreover, this approximation is valid for a much wider range of anisotropy parameters than the weak-anisotropy approximation. Thus we suggest using this approximation for ray-tracing and reflection tomography purposes.
Figures & Tables
“This volume contains 25 papers that represent most of the best work in seismic anisotropy in 1998 and 1999. Fracture characterizations and processing of converted waves are the two main topics covered in this volume. They are addressed from both theoretical and practical viewpoints. Also included are papers describing the historical roots of seismic anisotropy.”