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We will illustrate the theory presented above using the multiple
reflection event *S*1010201*G* (the zeros denote the Earth surface). For
this event, *n*=6, 1 less than the total number of bounces in
the earth. The first step is generating sequences and *l*_{i}, according
to (5) and (6):
| |
(39) |

and
| |
(40) |

We prepend
to the
sequence of angles, then we compute the sequence:
| |
(41) |

| (42) |

| (43) |

| (44) |

| (45) |

| (46) |

| (47) |

It may be useful to notice the regularities in signs and indices. The
summation and trigonometric operators in (25) and (26)
can be written in matrix form to verify the correctness of their numerical
implementation. We then
compute the auxiliary vectors given by (29) and (33):
| |
(48) |

| |
(49) |

For our very particular case in which some of the bounces are with the
surface (*d*_{0}=0, ),
| |
(50) |

| |
(51) |

and the traveltime for each offset *h* can now be computed by plugging
these vectors directly into (35).
By performing trigonometric operations, we may find that the expression for the
distance is the same as that in Equation (A-14) of
Levin and Shah (1977).

** Next:** Angles of departure and
** Up:** Liner and Vlad: Multiple
** Previous:** Computing the traveltime: the
Stanford Exploration Project

10/23/2004