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,

where

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,

Using these two equations to solve for and and plugging them into equation D-1 yields an eikonal for TTI media in terms of and . Thus, inserting the following trial solution

where , , and are independent parameters and small, into the eikonal equation yields an extremely long equation. Again, setting the coefficients of the independent parameters (, , and ) to zero in the equation gives the eikonal equation for elliptical anisotropy with vertical symmetry axis. On the other hand, the coefficients of the first power of the independent parameters yield:

corresponding to , , and , respectively.

The coefficient of the term, for higher accuracy in , is given by

These first-order PDEs, when solved, provide traveltime approximations using equation D-9 for 3D TI media in a generally inhomogeneous elliptical anisotropic background.

For a homogeneous medium simplification, the traveltime is given by the following analytical relation in 3-D
elliptical anisotropic media:

respectively.

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 |

2013-04-02