** Next:** Dynamics of Zero-Offset Velocity
** Up:** Fomel: Velocity continuation
** Previous:** Kinematics of Residual DMO

The theory of characteristics Courant (1962) states that if a partial
differential equation has the form
| |
(46) |

where F is some arbitrary function, and if the eigenvalues of the matrix
are nonzero, and one of them is different in sign from the
others, then equation (46) describes a wave-type process, and its
kinematic counterpart is the characteristic equation
| |
(47) |

with the characteristic surface
| |
(48) |

corresponding to the wavefront. In velocity continuation problems,
it is appropriate to choose the variable to denote the time
*t*, to denote the velocity *v*, and the rest of the
-variables to denote one or two lateral coordinates *x*. Without
loss of generality, we can set the characteristic surface to be
| |
(49) |

and use the theory of characteristics to reconstruct the main
(second-order) part of the dynamic differential equation from the
corresponding kinematic equations. As in the preceding section, it is
convenient to consider separately the three different components of the prestack
velocity continuation process.

** Next:** Dynamics of Zero-Offset Velocity
** Up:** Fomel: Velocity continuation
** Previous:** Kinematics of Residual DMO
Stanford Exploration Project

11/12/1997