Velocity continuation and the anatomy of residual prestack time migration

Dynamics of Residual NMO

According to the theory of characteristics, described in the beginning of this section, the kinematic residual NMO equation (22) corresponds to the dynamic equation of the form

$${{\partial P} \over {\partial v}} + {{h^2} \over {v^3\,t}}\,{{\partial P} \over {\partial t}} + F(h,t,v,P) = 0$$ (44)

with the undetermined function $F$. In the case of $F=0$, the general solution is easily found to be

$$P(t,h,v) = \phi\left(t^2 + {h^2 \over v^2}\right)\;.$$ (45)

where $\phi$ is an arbitrary smooth function. The combination of dynamic equations (44) and (41) leads to an approximate prestack velocity continuation with the residual DMO effect neglected. To accomplish the combination, one can simply add the term ${{h^2} \over {v^3\,t}}\,{{\partial^2 P} \over {\partial t^2}}$ from equation (44) to the left-hand side of equation (41). This addition changes the kinematics of velocity continuation, but does not change the amplitude properties embedded in the transport equation (42).

Dunkin and Levin (1973) and Hale (1983) advocate using an amplitude correction term in the NMO step. This term can be easily added by selecting an appropriate function $F$ in equation (44). The choice $F=\frac{h^2}{v^3\,t^2}\,P$ results in the equation

$${{\partial P} \over {\partial v}} + {{h^2} \over {v^3\,t^2}}\,\left(t\,{{\partial P} \over {\partial t}} + P\right) = 0$$ (46)

with the general solution

$$P(t,h,v) = \frac{1}{t}\,\phi\left(t^2 + {h^2 \over v^2}\right)\;,$$ (47)

which has the Dunkin-Levin amplitude correction term.

Velocity continuation and the anatomy of residual prestack time migration