Velocity continuation and the anatomy of residual prestack time migration |

It is more convenient to consider the residual dip-moveout process coupled with residual normal moveout. Etgen (1990) describes this procedure as the cascade of inverse DMO with the initial velocity , residual NMO, and DMO with the updated velocity . The kinematic equation for residual NMO+DMO is the sum of the two terms in (1):

The derivation of the residual DMO+NMO kinematics is detailed in Appendix B. Figure 5 illustrates it with the theoretical impulse response curves. Figure 6 compares the theoretical curves with the result of an actual cascade of the inverse DMO, residual NMO, and DMO operators.

vlcvcp
Theoretical
kinematics of the residual NMO+DMO impulse responses for three
impulses. Left plot: the velocity ratio is . Right
plot: the velocity ratio is . In both cases the
half-offset is 1 km.
Figure 5. |
---|

vlccps
The result of
residual NMO+DMO (cascading inverse DMO, residual NMO, and DMO) for
three impulses. Left plot: the velocity ratio is
. Right plot: the velocity ratio is . In both
cases the half-offset is 1 km.
Figure 6. |
---|

Figure 7 illustrates the residual NMO+DMO velocity continuation for two particularly interesting cases. The left plot shows the continuation for a point diffractor. One can see that when the velocity error is large, focusing of the velocity rays forms a distinctive loop on the zero-offset hyperbola. The right plot illustrates the case of a plane dipping reflector. The image of the reflector shifts both vertically and laterally with the change in NMO velocity.

vlcvrd
Kinematic
velocity continuation for residual NMO+DMO. Solid lines denote
wavefronts: zero-offset traveltime curves; dashed lines denote
velocity rays. a: the case of a point diffractor; the velocity
ratio changes from to . b:
the case of a dipping plane reflector; the velocity
ratio changes from to . In both cases, the
half-offset is 2 km.
Figure 7. |
---|

The full residual migration operator is the chain of residual zero-offset migration and residual NMO+DMO. I illustrate the kinematics of this operator in Figures 8 and 9, which are designed to match Etgen's Figures 2.4 and 2.5 (Etgen, 1990). A comparison with Figures 3 and 4 shows that including the residual DMO term affects the images of objects with the depth smaller than the half-offset . This term complicates the residual migration operator with cusps.

vlcve3
Summation paths of prestack
residual migration for a series of depth diffractors. Residual
slowness is 1.2; half-offset is 1 km. This figure
reproduces Etgen's Figure 2.4.
Figure 8. | |
---|---|

vlcve4
Summation paths of prestack
residual migration for a series of depth diffractors. Residual
slowness is 0.8; half-offset is 1 km. This figure
reproduces Etgen's Figure 2.5.
Figure 9. | |
---|---|

Velocity continuation and the anatomy of residual prestack time migration |

2014-04-01