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Unfortunately equation (59) is valid only for common source and common receiver
gathers. Let us take, for instance, the common offset pattern in the 2D case :

at fixed *l*. In this case we have datum
where is the arrival time of a reflected wave propagating from source *s* to
receiver *r*. Then
We can't separately determine values and
and, consequently, can't determine initial
conditions for eiconal .We see that the reverse eikonal's continuation with datum can't
reconstruct the true location of the seismic rays. The problem simplifies
greatly when *l*=0 (zero offset pattern). In this case incidence and
reflected rays coincide, and due to the principle of reciprocity,
so
and
| |
(61) |

This means that the solution of equation (61) with initial condition
| |
(62) |

will give the true ray pattern although the travel-times will be different.
This is not bad. Since the time of propagation from the reflector *R* to
surface along the normal ray equals , the condition
| |
(63) |

determines the location of the reflector.
We could obtain the same result by solving the half-velocity eikonal equation
| |
(64) |

with initial datum
| |
(65) |

Usually this scheme is obtained from some thought experiment (i.e., the exploding
reflector hypothesis). Let us find the forward eikonal's continuation
in the medium with velocity and initial condition
| |
(66) |

This corresponds to a thought experiment with an impulsive source that is distributed along
the surface *R* at *t*=0).
It is easy to understand that the solution of this
problem will coincide on the surface with . It is clear
that the reverse eikonal's continuation of in a medium with half-velocity has the isochron on the surface *R*.

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Stanford Exploration Project

1/13/1998