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Denote the surface position vector by
| |
(1) |

the temporal frequency by , and the velocity by
| |
(2) |

We define the operator as
| |
(3) |

and the operator as
| |
(4) |

*Without* the amplitude correction, shot profile migration proceeds
by computing the upward going and the downward going wavefields at all
depths:
| |
(5) |

| (6) |

To solve these equations, we need boundary conditions (inputs into the
downward continuation). The recorded shot gather is taken as the upper boundary
condition for the upward going wavefield, and a Dirac, approximated by
a small wavelet, is taken as the upper boundary condition for
the downward going wavefield:
| |
(7) |

| (8) |

The reflectivity image is produced using the imaging condition:
| |
(9) |

According to Zhang et al. (2003a), to improve the amplitude and
phase of the image R, we only need to add the operator into
the downward continuation equations:
| |
(10) |

| (11) |

and while taking the same boundary conditions for the upward going
wavefield, to apply a correction to the
downward going wavefield boundary condition:
| |
(12) |

| (13) |

The imaging condition remains the same:
| |
(14) |

The only differences between the amplitude-preserving algorithm and
the traditional one are the use of the operator in
downward continuation and the application of the operator to the shot boundary condition. In the following sections, we will
describe implementations of the two operators in the
domain (finite-difference), as well as in the mixed domain. We will also show 2-D synthetic examples
of applying the operators.

** Next:** Implementing the operator
** Up:** Vlad et al.: Improving
** Previous:** Introduction
Stanford Exploration Project

7/8/2003