## Toward an exact adjoint:\\ semicircle versus hyperbola

#### Jun Ji

The correctness of an adjoint operator entirely depends
on the correctness of the corresponding forward modeling operator.
In migration, the input-oriented algorithm implies
more correct forward modeling than the output-oriented algorithm does.
One big difference between the two algorithms
is the domain of interpolation,
and this is one of the major sources of migration artifacts.
Whereas the input-oriented algorithm
requires interpolation in the model space,
the output-oriented algorithm
requires interpolation in the data space.
For Kirchhoff migration, the method of semicircle superposition
and the method of hyperbola summation correspond to
the input-oriented and output-oriented algorithms, respectively.
The same analogy can be applied to Stolt migration.
Levin (1994) suggested the input-oriented algorithm
that is the counterpart to the output-oriented algorithm
suggested by the SEP ``in-time" group
(Popovici et al., 1993; Lin et al.,1993; Blondel and Muir, 1993)

#### SEP-report, vol.80, 499-511 (1994).

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