Traveltime approximations for transversely isotropic media with an inhomogeneous background

Next: A homogeneous model test Up: Alkhalifah: TI traveltimes in Previous: The TI eikonal and

Expansion in terms of and

Though the expansion in terms of in the previous section allowed us to estimate traveltimes for a tilted symmetry axis, it also required that we solve the eikonal equation for a VTI medium, which is relatively challenging. For inversion purposes, it also required knowledge of , which might not be possible in TI media using initially a VTI approximation, especially if the tilt is large. However, an expansion in , in addition to (from their zero values), will result in an elliptically anisotropic background medium and it will allow us to search for both and , simultaneously, considering that the elliptical anisotropy model is known.

The two-parameter expansion can be obtained by substituting the following trial solution:

 (6)

into equation 3 resulting in linear first-order partial differential equations having the following general form:
 (7)

with , and satisfies the eikonal equation for an elliptical anisotropic background model. Again, the function gets more complicated for corresponding to the second-order term and it depends on terms for the first order and background medium solutions. Therefore, these linear partial differential equations also must be solved in succession starting with and . As soon as the , and coefficients are evaluated, they can be used, as Alkhalifah (2010) showed, to estimate the traveltime using the first-sequence of Shanks transform (Bender and Orszag, 1978), and as shown in Appendix B, has the form:
 (8)

The expansion does not adapt well to the Shanks transform requirements for predicting the behavior of the higher-order terms in . In this case, the second-order approximation in the expansion is sufficient.

For and scan applications, the coefficients (, , , , , and ) need to be evaluated only once and can be used with equation 8 to search for the best traveltime fit to those traveltimes extracted from the data.

 Traveltime approximations for transversely isotropic media with an inhomogeneous background

Next: A homogeneous model test Up: Alkhalifah: TI traveltimes in Previous: The TI eikonal and

2013-04-02