Traveltime approximations for transversely isotropic media with an inhomogeneous background |

To derive a traveltime equation in terms of perturbations in , we first establish the form for the
governing equation for TI media given by the eikonal representation.
The eikonal equation for -waves in TI media in 2D (for simplicity) is
given by

To solve equation A-1 through perturbation theory, we assume that is small, and thus, a trial solution can be expressed as a series expansion in given by

where , and are coefficients of the expansion given in units of traveltime, and, for practicality, terminated at the second power of . Inserting the trial solution, equation A-2, into equation A-1 yields a long formula, but by setting , I obtain the zeroth-order term given by

which is the eikonal formula for VTI anisotropy. By equating the coefficients of the powers of the independent parameter , in succession, we end up first with the coefficients of first-power in , simplified by using equation A-3, and given by

which is a first-order linear partial differential equation in . The coefficient of , with some manipulation, has the following form

which is again a first-order linear partial differential equation in with an obviously more complicated source function given by the right-hand side. Though the equation seems complicated, many of the variables of the source function (right-hand side) can be evaluated during the evaluation of equations A-3 and A-4 in a fashion that will not add much to the cost.

Traveltime approximations for transversely isotropic media with an inhomogeneous background |

2013-04-02