Generalized nonhyperbolic moveout approximation |

In the case of a circular (cylindrical or spherical) reflector in a homogeneous velocity model, there is no closed-form analytical solution. However, the moveout can be described analytically by parametric relationships (Glaeser, 1999).

crefl
Reflection from a circular reflector in
a homogeneous velocity model (a scheme).
Figure 10. |
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Consider the geometry of the reflection shown in
Figure D-1. According to the trigonometry of the
reflection triangles, the source and receiver positions can be expressed as

where is the reflector radius, is the minimum reflector depth, is the reflector dip angle at the reflection point, and is the reflection angle. Correspondingly, the midpoint and offset coordinates can be expressed as

and the reflection traveltime can be expressed as

where is the medium velocity. Expressing the reflection angle from equation D-3 and substituting it into equations D-4 and D-5, we obtain a pair of parametric equations

which define the exact reflection moveout for the case of a circular reflector in a homogeneous medium.

The connection with parameters of equations 27-29 is given by

The behavior of the moveout at infinitely large offsets is controlled by and

After substitution in equations 25-26, we obtain somewhat complicated but analytical expressions for parameters and .

Generalized nonhyperbolic moveout approximation |

2013-07-26