Velocity continuation and the anatomy of residual prestack time migration |

Equation (22) does not depend on the midpoint . This fact indicates the one-dimensional nature of normal moveout. The general solution of equation (22) is obtained by simple integration. It takes the form

where is an arbitrary velocity-independent constant, and I have chosen the constants and so that . Equation (23) is applicable only for different from zero.

For the case of a point diffractor, equation (23) easily
combines with the zero-offset solution (16). The result
is a simplified approximate version of the prestack residual migration
summation path:

vlcve1
Summation paths of the simplified
prestack residual migration for a series of depth diffractors.
Residual slowness is 1.2; half-offset is 1 km. This
figure is to be compared with Etgen's Figure 2.4, reproduced in
Figure 8.
Figure 3. | |
---|---|

vlcve2
Summation paths of the simplified
prestack residual migration for a series of depth diffractors.
Residual slowness is 0.8; offset is 1 km. This figure is
to be compared with Etgen's Figure 2.5, reproduced in
Figure 9.
Figure 4. | |
---|---|

Neglecting the residual DMO term in residual migration is
approximately equivalent in accuracy to neglecting the DMO step in
conventional processing. Indeed, as follows from the geometric analog
of equation (1) derived in Appendix A
[equation (A-17)], dropping the residual
DMO term corresponds to the condition

where is the reflection traveltime, and and are the source and receiver coordinates: , . In geometric terms, approximation (26) transforms to

Taking the difference of the two sides of equation (27), one can estimate its accuracy by the first term of the Taylor series for small and . The estimate is (Yilmaz and Claerbout, 1980), which agrees qualitatively with (25). Although approximation (24) fails in situations where the dip moveout correction is necessary, it is significantly more accurate than the 15-degree approximation of the double-square-root equation, implied in the migration velocity analysis method of Yilmaz and Chambers (1984) and MacKay and Abma (1992). The 15-degree approximation

corresponds geometrically to the equation

Its estimated accuracy (from the first term of the Taylor series) is . Unlike the separable approximation, which is accurate separately for zero offset and zero dip, the 15-degree approximation fails at zero offset in the case of a steep dip and at zero dip in the case of a large offset.

Velocity continuation and the anatomy of residual prestack time migration |

2014-04-01