If this gather is migrated with a different velocity model , the image obtained will be distorted from the true image of the subsurfaces.
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,xr) from equation (1) into equation (3) yields
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:
Therefore the relationship between (x,z) and ,is implicitly expressed by a pair of equations:
For constant offset sections,
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 xr 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.