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

To exemplify the use of weak anisotropy, let us consider the
simplest model of a homogeneous VTI medium above a horizontal
reflector. For an isotropic medium, the reflection traveltime curve is
an exact hyperbola, as follows directly from the Pythagorean theorem
(Figure 2)

Substituting equation (10) into (5) and linearizing the expression

with respect to the anisotropic parameters and , we arrive at the three-parameter nonhyperbolic approximation (Tsvankin and Thomsen, 1994)

where the normal-moveout velocity is defined by equation (4). At small offsets , the influence of the parameter is negligible, and the traveltime curve is nearly hyperbolic. At large offsets , the third term in equation (12) has a clear influence on the traveltime behavior. The Taylor series expansion of equation (12) in the vicinity of the vertical zero-offset ray has the form

When the offset approaches infinity, the traveltime approximately satisfies an intuitively reasonable relationship

where the horizontal velocity is defined by equation (3). Approximation (12) is analogous, within the weak-anisotropy assumption, to the ``skewed hyperbola'' equation (Byun et al., 1989) which uses the three velocities , , and as the parameters of the approximation:

The accuracy of equation (12), which usually lies within 1% error up to offsets twice as large as reflector depth, can be further improved at any finite offset by modifying the denominator of the third term (Grechka and Tsvankin, 1998; Alkhalifah and Tsvankin, 1995).

nmoone
Reflected rays
in a homogeneous VTI layer above a horizontal reflector (a scheme).
Figure 2. |
---|

Muir and Dellinger (1985) suggested a different nonhyperbolic moveout
approximation in the form

Comparing equations (14) and (17), we can establish the correspondence

Taking this equality into account, we see that equation (16) is approximately equivalent to equation (12) in the sense that their difference has the order of .

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

2014-01-27