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![]() | Traveltime approximations for transversely isotropic media with an inhomogeneous background | ![]() |
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The main objective of the newly developed expressions is parameter estimation in complex media.
Specifically, the perturbation PDEs developed here are with respect to a background generally inhomogeneous, and
possibly anisotropic, medium. If a generally inhomogeneous isotropic velocity
field is available (for example from conventional
migration velocity analysis), in addition to a map of the well-to-seismic misties, which
can be used to develop a vertical velocity field, then an elliptical
anisotropic model with a vertical symmetry axis can be constructed.
We can use this model to solve for traveltimes in elliptically
anisotropic media as a background model, as well as to solve for the expansion coefficients using equations 7.
These coefficients can be used with, for example, equation 23 to search
explicitly for the , and tilt angles
and
in 3D that provides the best traveltime fit to the data.
This process can be implemented in a semblance-type search or incorporated as part of a
tomographic inversion. Though the scans are based on an underline factorized assumption in the perturbation parameters,
and the tilt angles, we can allow them to vary
smoothly with location, and thus, produce effective values. The conversion of these
effective values to interval ones in generally inhomogeneous media is not trivial and might require a tomographic treatment
of its own.
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etaTheta
Figure 7. The traveltime difference between the TTI model computed using equation 9 and the elliptically anisotropic with a vertical symmetry axis background model for (a) an offset of 1 km, (b) an offset of 2 km, and (c) an offset of 4 km. The medium has ![]() ![]() ![]() ![]() |
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The availability of multi-offset data will increase our chances in resolving both and the tilt angle in 2D. The addition
of multi azimuth should help resolve the tilt in 3D.
Of course, the accuracy of resolving these parameters will depend
mainly on how well we estimate the original elliptically anisotropic background medium. However, we can always go back
and improve on our velocity picks once an approximate effective
and tilt-angle fields are estimated. There are probably
many other more sophisticated ways to explore this parameter matrix, however, the equations introduced here provides the basis for doing
so.
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![]() | Traveltime approximations for transversely isotropic media with an inhomogeneous background | ![]() |
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