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The transformation
to GOCIG
of an image is defined as
| |
(26) |

The transformation
to GOCIG
of an impulse located at is thus (after inverse Fourier transforms):
| |
(27) |

We now approximate by stationary phase the inner double integral.
The phase of this integral is:

| |
(28) |

The stationary path is defined by the solutions of the following
system of equations:
| |
(29) |

| (30) |

| |

By moving both and
to the right of
equations (29) and (30),
and then dividing equation (29)
by equation (30),
we obtain the following relationship between
and
:
| |
(31) |

Furthermore, by multiplying equation (29) by *k*_{z}
and equation (30) by *k*_{x},
and then substituting
them appropriately in the phase function (28),
we can evaluate the phase function along the stationary path as follows:
| |
(32) |

which becomes, by substituting
equation (31),
| |
(33) |

Notice that the minus sign comes from the function
in expression (23).
By substituting expression (33)
in equation (27) it is immediate
to evaluate the kinematics of the impulse response as
follows:
| |
(34) |

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Stanford Exploration Project

7/8/2003