Traveltime approximations for transversely isotropic media with an inhomogeneous background |

In VTI media, the eikonal equation (Alkhalifah, 1998)
in the acoustic approximation has the form:

where is the traveltime (eikonal) measured from the source to a point with the coordinates , and and are the velocity and NMO velocity (= ), respectively, described with respect to the symmetry direction at that point. To formulate a well-posed initial-value problem for equation 1, it is sufficient to specify at some closed surface and to choose one of the two solutions: the wave going from or toward the source. The level of nonlinearity in this quartic (in terms of ) equation is higher than that for the isotropic or elliptically anisotropic eikonal equations. This results in much more complicated finite-difference approximations of the VTI eikonal equation.

For a tilted TI medium, the traveltime derivatives in equation 1 are taken with respect to the tilt direction, and
thus, we have to
rotate
the derivatives in equation 1 using the following Jacobian in 3D:

The full 3D version of this equation is stated in Appendix D.

scanTheta
A schematic plot showing the relation between a background traveltime field for =0
and that when is larger than zero. The round dot at the top of the plane represents a source.
Figure 1. | |
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Solving equation 3 numerically requires
solving a quartic equation (instead of the quadratic in the isotropic and elliptical
anisotropic case) at each computational step. Alternatively, it can be solved using perturbation theory (Bender and Orszag, 1978)
by approximating equation 3 with a series of simpler linear equations. Considering
constant and small, we can represent the traveltime solution as a series expansion in . This will result in a solution that is globally representative in the
space domain and, despite the approximation of small ,
the accuracy for even large , as we will see later, is high. The constant- assumption
assumes a factorized medium (Alkhalifah, 1995)
in (useful for smooth estimation applications). However,
all other velocities and parameters including , (or ) and are allowed to vary.
Figure 1 illustrates
the concept of the global expansion as we predict the traveltime for
any from its behavior at for the full traveltime
field
using, in this case, a quadratic approximation. Specifically, we
substitute the following trial solution,

with . The functions get more complicated for larger and depend on terms that can be evaluated only sequentially. Therefore, these linear partial differential equations must be solved in the order of increasing starting with .

Traveltime approximations for transversely isotropic media with an inhomogeneous background |

2013-04-02