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I compare the computational costs of full migration and residual
migration under the assumption that they are implemented with the Kirchhoff
integral in the common shot geometry.
I omit those costs that do not depends on the dimensions of data.

A Kirchhoff migration can be done in two steps: (1) calculating operators and
(2) applying operators. Suppose a dataset consists of *N*_{s} shot gathers.
In each shot gather, there are *N*_{r} receivers.
The image we try to obtain from this dataset has a
dimension of . Then, the computational cost of full migration
is

where *N*_{l}< *N*_{s}+*N*_{r} is the number of surface locations sampled
by the experiment, and the small letter *o* stands for *o*rder.
The computational cost of residual migration is

where *N*_{a} is the number of samples required to represent
residual-migration operators.
Because the aperture of a residual-migration operator is usually much smaller than
that of a full-migration operator, *N*_{a} is much smaller than *N*_{r},
which makes the application of residual migration efficient. From
this comparison, we see that, in full
migration, applying operators is the major consumer of the computation-time.
In residual migration, however, calculating
residual-migration operators is computationally more expensive than applying operators.
Therefore, efforts should be made on reducing the cost of calculating
residual-migration operators.

** Next:** Kinematic relations
** Up:** RESIDUAL MIGRATION OPERATOR
** Previous:** RESIDUAL MIGRATION OPERATOR
Stanford Exploration Project

1/13/1998