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Downward continuation and imaging in the isotropic
case (Gazdag, 1978) can be written as
| |
(1) |

where
| |
(2) |

In equation (1), represents the
Fourier transform of the seismic field *p*(*t*,*x*) recorded at
the surface, following Claerbout's (1985) sign convention
and *v* represents half the velocity, as
used in the exploding reflectors model.
We can rewrite equation (1) as time migration replacing
the depth steps by equivalent time steps :

| |
(3) |

where we define
For a constant velocity medium, Stolt (1978) transforms the integral
in using a Fourier transform, which can be computed rapidly
via a Fast Fourier Transform (FFT) algorithm

| |
(4) |

where represents the Jacobian of the transformation
from to
and represents the initial data
as function of the new variable .

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Stanford Exploration Project

11/16/1997