Traveltime approximations for transversely isotropic media with an inhomogeneous background |
The eikonal equation for -waves in a TI medium with a tilt in the symmetry axis satisfies the following relation,
To develop equations for the coefficients of a traveltime expansion in 3D from a background elliptical anisotropy with
a vertical symmetry axis I use vector notations ( and ) to describe the tilt angles, where the components of this 2D
vector describe the projection of the symmetry axis on each of the and planes, respectively. As a result,
The coefficient of the term, for higher accuracy in , is given by
For a homogeneous medium simplification, the traveltime is given by the following analytical relation in 3-D
elliptical anisotropic media:
I now evaluate
,
, and
and use them to solve equation D-11.
After some tedious algebra, I obtain
The application of Pade approximation on the expansion in , by finding a first order polynomial representation
in the denominator,
yields a TI equation that is accurate for large (Alkhalifah, 2010),
as well as small tilt, given by
Traveltime approximations for transversely isotropic media with an inhomogeneous background |