Velocity continuation and the anatomy of residual prestack time migration

Next: FROM KINEMATICS TO DYNAMICS Up: KINEMATICS OF VELOCITY CONTINUATION Previous: Kinematics of Residual NMO

## Kinematics of Residual DMO

The partial differential equation for kinematic residual DMO is the third term in equation (1):
 (30)

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):
 (31)

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
Figure 5.
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.

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Figure 6.
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 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
Figure 7.
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.

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
Figure 8.
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.

vlcve4
Figure 9.
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.

 Velocity continuation and the anatomy of residual prestack time migration

Next: FROM KINEMATICS TO DYNAMICS Up: KINEMATICS OF VELOCITY CONTINUATION Previous: Kinematics of Residual NMO

2014-04-01