Traveltime approximations for transversely isotropic media with an inhomogeneous background |

For an expansion in and , simultaneously, I use the following trial solution:

which is simply the eikonal formula for elliptical anisotropy. By equating the coefficients of the powers of the independent parameter and , in succession starting with first powers of the two parameters, we end up first with the coefficients of first-power in and zeroth power in , simplified by using equation B-2, and given by

which is a first-order linear partial differential equation in . The coefficients of zero-power in and the first-power in is given by

The coefficients of the square terms in , with some manipulation, results in the following relation

which is again a first-order linear partial differential equation in with an obviously more complicated source function given by the right hand side. The coefficients of the square terms in , with also some manipulation, results in the following relation

which is again a first-order linear partial differential equation in with a again complicated source function.

Finally, the coefficients of the first-power terms in both and
results also in a first-order linear partial differential equation in
given by

Though the equation seems complicated, many of the variables of the source function (right hand side) can be evaluated during the evaluation of equations B-3 and B-4 in a fashion that will not add much to the cost.

Using Shanks transforms (Bender and Orszag, 1978) we can isolate and remove the most transient behavior of the expansion B-1
in (the expansion did not improve with such a treatment) by first defining the
following parameters:

The first sequence of Shanks transforms uses , , and , and thus, is given by

Traveltime approximations for transversely isotropic media with an inhomogeneous background |

2013-04-02