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![]() | Traveltime approximations for transversely isotropic media with an inhomogeneous background | ![]() |
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In 3D, the tilt of the symmetry axis is defined by an angle, , measured from vertical, and the azimuth,
, of the vertical plane that
contains the symmetry axis. Thus,
is an angle measured in the horizontal plane from a given axis
within that plane. To implement
an expansion with respect to
, we must consider
to be generally small. Since seismic acquisition is often performed
in the dip direction of the structure, and we anticipate that the tilt is influenced by the presumed subsurface structure (folding), it would be
reasonable to measure
from the acquisition direction. In this case, I can consider
to be small, and thus,
approximate the traveltime
solution of the eikonal equation with the following expansion:
Substituting the trial solution, equation 20, into the eikonal equation for 3D
TI media, equation 21, yields a polynomial expansion in the powers of . The zero-order term of this polynomial
represents the eikonal equation for TI media for the zero-azimuth case (the 2-D result). The coefficient of the
term yields
a first-order PDE for
. For simplicity, it is shown here for the case of elliptical anisotropy (
) as
A more realistic implementation is achieved by expanding from an elliptically anisotropic background with a vertical symmetry axis background
in terms of ,
, and
, simultaneously. However, since a vertical symmetry axis has no particular
azimuth (a singularity), I replace the tilt and azimuth by
and
(
and
).
For simplicity and to be able to include the resulting equations in this paper, I only consider first-order terms of the Taylor's series expansion,
and thus, consider the following trial solution:
Assuming a homogeneous-medium background yields an
analytic relation, as shown in Appendix D [equation D-13]:
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![]() | Traveltime approximations for transversely isotropic media with an inhomogeneous background | ![]() |
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