Traveltime approximations for transversely isotropic media with an inhomogeneous background

# Appendix D: Expansion in 3D

The eikonal equation for -waves in a TI medium with a tilt in the symmetry axis satisfies the following relation,

 (45)

where
 (46) (47) (48) (49) (50)

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,

 (51)

and
 (52)

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
 (53)

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:
 (54)

corresponding to , , and , respectively.

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

 (55)

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:

 (56)

which satisfies the eikonal equation B-2 in 3D. Using equation D-12, I evaluate , and and insert them into equations D-10 to solve these first-order linear equations to obtain
 (57)

respectively.

I now evaluate , , and and use them to solve equation D-11. After some tedious algebra, I obtain

 (58)

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

 (59)

 Traveltime approximations for transversely isotropic media with an inhomogeneous background