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  

$${{\partial P} \over {\partial v}} + {{h^2} \over {v^3\,t}}\,{{\partial P} \over {\partial t}}\;,$$ (58)

whose general solution is easily found to be  

$$P(t,x,v) = \phi\left(t^2 + {h^2 \over v^2}\right)\;,$$ (59)

where $\phi$ is an arbitrary smooth function. The combination of dynamic equations (58) and (55) leads to an approximate prestack velocity continuation with the residual DMO effect neglected. To accomplish the combination, we can simply add the term ${{h^2} \over {v^3\,t}}\,{{\partial^2 P} \over {\partial t^2}}$ to the left-hand side of equation (55). This addition changes the kinematics of velocity continuation, but doesn't change the amplitude properties embedded in the transport equation (56).