Traveltime approximations for transversely isotropic media with an inhomogeneous background |

To develop analytical traveltime representation for TI media, I start with a background velocity model that is homogeneous.
The expansion here will be with respect to and from a background elliptical anisotropic model.
In this case, the traveltime from a point source at and is given by the following simple relation in 2-D:

as well as insert them into equation B-4 and solve the equation to obtain

I now evaluate
and
and use them to solve equation B-5.
After some tedious algebra, I obtain

Finally, I solve equation B-7.
After some tedious algebra once again, I obtain

Using the first sequence of Shanks transform, equation B-9, applied to the Taylor's series expansion, we obtain an analytical equation that describes traveltime as a function of and .

For 3-D media, I include the azimuth angle as we will see next.

Traveltime approximations for transversely isotropic media with an inhomogeneous background |

2013-04-02