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

Here

is the length of the zero-offset ray, and is the reflector dip. Formula (53) is inaccurate if the reflector is both dipping and curved. The Taylor series expansion for moveout in this case has the form of equation (22), with coefficients (Fomel, 1994)

where

is the reflector curvature [defined by equation (61)] at the reflection point of the zero-offset ray, and is the third-order curvature [equation (62)]. If the reflector has an explicit representation , then the parameters in equations (56) and (57) are

Keeping only three terms in the Taylor series leads to the approximation

As indicated by equation (61), the sign of the curvature is positive if the reflector is locally convex (i.e., an anticline-type). The sign of is negative for concave, syncline-type reflectors. Therefore, the coefficient expressed by equation (58) and, likewise, the nonhyperbolic term in (63) can take both positive and negative values. This means that only for concave reflectors in homogeneous media do nonhyperbolic moveouts resemble those in VTI and vertically heterogeneous media. Convex surfaces produce nonhyperbolic moveout with the opposite sign. Clearly, equation (63) is not accurate for strong negative curvatures , which cause focusing of the reflected rays and triplications of the reflection traveltimes.

In order to evaluate the accuracy of approximation (63), we
can compare it with the exact expression for a point diffractor, which
is formally a convex reflector with an
infinite curvature. The exact expression for normal moveout
in the present notation is

nmoerr
Relative error of the
nonhyperbolic moveout approximation (63) for a
point diffractor. The error corresponds to offset twice the diffractor depth and is plotted against the angle from vertical of
the zero-offset ray.
Figure 4. |
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Nonhyperbolic reflection moveout of -waves: An overview and comparison of reasons |

2014-01-27