Nonhyperbolic reflection moveout of -waves: An overview and comparison of reasons |

Two important equations derived in Appendix A are:

where is the one-way traveltime of the direct wave propagating from the reflection point to the point at the surface . All derivatives in equations (66) and (67) are evaluated at the zero-offset ray. Both equations are based solely on Fermat's principle and, therefore, remain valid in any type of media for reflectors of an arbitrary shape, assuming that the traveltimes possess the required order of smoothness. It is especially convenient to use equations (66) and (67) in homogeneous media, where the direct traveltime can be expressed explicitly.

To apply equations (66) and (67) in VTI
media, we need to start with tracing the zero-offset ray. According to
Fermat's principle, the ray trajectory must correspond to an extremum of
the traveltime. For the zero-offset ray, this simply means that the
one-way traveltime satisfies the equation

Here, the function describes the reflector shape, and is the ray angle given by the trigonometric relationship (Figure 5)

Substituting approximate equation (5) for the group velocity into equation (69) and linearizing it with respect to the anisotropic parameters and , we can solve equation (68) for , obtaining

or, in terms of ,

where is the local dip of the reflector at the reflection point . Equation (72) shows that, in VTI media, the angle of the zero-offset ray differs from the reflector dip (Figure 5). As one might expect, the relative difference is approximately linear in Thomsen anisotropic parameters.

nmoray
Zero-offset reflection from a
curved reflector beneath a VTI medium (a scheme). Note that the ray angle
is not equal to the local reflector dip .
Figure 5. |
---|

Now we can apply equation (66) to evaluate the second term of the
Taylor series expansion (22) for a curved
reflector. The linearization in anisotropic parameters leads to the expression

Finally, using equation (67), we
determine the third coefficient of the Taylor series. After
linearization in anisotropic parameters and lengthy algebra, the
result takes the form

and the coefficient is defined by equation (58). For zero curvature (a plane reflector) , and the only term remaining in equation (75) is

If the reflector is curved, we can rewrite the isotropic equation (63) in the form

where the normal-moveout velocity and the quantity are given by equations (73) and (75), respectively. Equation (77) approximates the nonhyperbolic moveout in homogeneous VTI media above a curved reflector. For small curvature, the accuracy of this equation at finite offsets can be increased by modifying the denominator in the quartic term similarly to that done by Grechka and Tsvankin (1998) for VTI media.

Nonhyperbolic reflection moveout of -waves: An overview and comparison of reasons |

2014-01-27