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The transformation
to DDHOCIG
of an image is defined as
| |
(10) |

The transformation
to DDHOCIG
of an impulse located at is thus (after inverse Fourier transforms):
| |
(11) |

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

| |
(12) |

The stationary path is defined by the solutions of the following
system of equations:
| |
(13) |

| (14) |

| |

By moving both and
on the right of
equations (13) and (14),
and then dividing equation (13)
by equation (14),
we obtain the following relationships between
and
:
| |
(15) |

Furthermore, by multiplying equations (13) by *k*_{z}
and equation (14) by *k*_{x},
and then substituting
them appropriately in the phase function (12),
we can evaluate the phase function along the stationary path as
| |
(16) |

that becomes, by substituting
equation (15),
| |
(17) |

Notice that the minus sign comes from the function
in expression (7).
By substituting expression (17)
in equation (11) it is immediate
to evaluate the kinematics of the impulse response as
| |
(18) |

** Next:** Evaluation of the image
** Up:** Application to a synthetic
** Previous:** Application to a synthetic
Stanford Exploration Project

11/11/2002