Traveltime approximations for transversely isotropic media with an inhomogeneous background

Next: Conclusions Up: Alkhalifah: TI traveltimes in Previous: The symmetry-axis azimuth and

# Discussion

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.

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 =2 km/s, =2 km/s (), and a reflector depth, =2 km.

In 3D, the search for and the symmetry direction angles can be applied either sequentially or to all the parameters at once. A sequential search, though faster and easier, may propagate some of the errors of an initial (wrong) tilt into the estimation of the parameter (Behera and Tsvankin, 2009). The search for all three parameters simultaneously would reduce such errors, but it will suffer from a null space based on the tradeoff between and the tilt angles. Conventionally, the information for could be extracted, especially for small tilt angles from vertical, which is assumed here, from long offsets and dipping reflectors. The tilt information resolution in 3D requires a 3D coverage (i.e. wide azimuth or even narrow azimuth for small tilt azimuth). There is also a general tradeoff, even in 2D media, between and the tilt angle, which may require some a priori information for or constraining the tilt angle to be normal to the reflector dip (Alkhalifah and Bednar, 2000). Figure 7 shows the dependence of traveltime in the 2D case, based on equation 9, on the parameters and . For a single offset, clearly there are combinations of and (given by the contour lines) that provide equal traveltimes. Nevertheless these curves clearly vary from one offset to another. Specifically, near offsets (Figure 7a, where ), for mostly small tilt angles, show little dependence of traveltime on , and more dependence on . On the other hand, as the tilt angle increases the resolution of increases as its influence starts to affect even the shorter offsets. The dependence of the traveltime on increases for (Figure 7b) and (Figure 7c), which implies that is better resolved at large offsets for small tilt angles. However, it is resolved even better for large tilt angles in all cases. Meanwhile, larger offsets with reasonable values result in less dependence of the traveltime on tilt angle.

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.

 Traveltime approximations for transversely isotropic media with an inhomogeneous background

Next: Conclusions Up: Alkhalifah: TI traveltimes in Previous: The symmetry-axis azimuth and

2013-04-02