(1) |

(2) |

If this gather is migrated with a different velocity model , the image obtained will be distorted from the true image of the subsurfaces.

(3) |

Residual migration is a transformation from image to image *Q*(*x*,*z*). Therefore, we want to find out the relationship
between (*x*,*z*) and that defines the kinematic operators
of this transformation.
Substituting *P*(*t*,*x*_{r}) from equation (1) into equation
(3) yields

(4) |

It is well known that the kinematics of the summation operator that does full migration is defined by the trajectory of the reflection event from a scatterer. Similarly, the kinematics of the summation operator that does residual migration can be determined from , or more specificly from the argument of the -function in equation (4). Let

Recall the properties of -function:(5) |

For each point (*x*,*z*), equation (5) defines a curve in
. This curve is exactly the kinematics of the
residual-migration operator at point (*x*,*z*).
For general velocity models, the traveltimes must be computed by
some numerical methods. These methods generate traveltime tables
rather than continuous functions. Therefore, it is
natural to solve equation (5) numerically. A straight forward
method is searching. For
each *x*_{r} and (*x*,*z*), all points around (*x*,*z*) are checked to find
the that satisfies equation (5).
But the this algorithm is time-consuming when the dimensions
of images are large. Motivated by the results of the finite-difference
calculation of
traveltimes (Van Trier, 1990), I begin to explore the possibility to calculate
residual-migration operators with finite-difference techniques.

1/13/1998