    Generalized nonhyperbolic moveout approximation  Next: Appendix E: HOMOGENEOUS VTI Up: Fomel & Stovas: Generalized Previous: Appendix C: REFLECTION FROM

# Appendix D: REFLECTION FROM A CIRCULAR REFLECTOR IN A HOMOGENEOUS VELOCITY MODEL

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
Figure 10.
Reflection from a circular reflector in a homogeneous velocity model (a scheme).   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   (67)   (68)

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   (69)   (70)

and the reflection traveltime can be expressed as     (71)

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   (72)   (73)

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   (74)   (75)   (76)

The behavior of the moveout at infinitely large offsets is controlled by and (77)

After substitution in equations 25-26, we obtain somewhat complicated but analytical expressions for parameters and .    Generalized nonhyperbolic moveout approximation  Next: Appendix E: HOMOGENEOUS VTI Up: Fomel & Stovas: Generalized Previous: Appendix C: REFLECTION FROM

2013-07-26