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Let us denote by *n* a value smaller 1 less than
the total number of bounces of the wave in the earth. Let us pretend
``not to know'' that the horizontal coordinates of *S* and *G* are
-*h* and *h*, respectively, and denote them with *s* and *g* instead,
since we will later need a more general expression that can be
differentiated with respect to these variables. We start by generating
the sequences and *l*_{i}, according to (5) and
(6), and keeping in mind the nonphysical prependix .
Using the fact that *n* is always even because the total number of
bounces inside the earth is always odd, and substituting into
(25), we find the image cascaded through *n* reflection
operations from the source to be

| |
(27) |

The receiver image is obtained from a single reflection
operation, through the last reflecting interface:
| |
(28) |

The traveltime is the distance between and divided by the velocity. This distance will be computed as the magnitude
of the vector . By making the notations
| |
(29) |

| |
(30) |

| |
(31) |

we can write:
| |
(32) |

In particular, for *s*=-*h* and *g*=*h* and
| |
(33) |

the traveltime can be written as
| |
(34) |

This vector magnitude can be computed using scalar products:
| |
(35) |

or it can be written as
| |
(36) |

which is the equation of a hyperbola with the apex at
| |
(37) |

| |
(38) |

** Next:** Computing the traveltime: an
** Up:** Liner and Vlad: Multiple
** Previous:** Cascading image-construction operations
Stanford Exploration Project

10/23/2004