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During World War II the Soviet geophysicist J. Riznichenko
proposed the following method for reconstruction of a reflector in a
medium with a given velocity .Let the source be located at point and let
be a surface that shows the location of a reflector *R*.
In a medium *D* between surfaces and *R* two eikonals can be
introduced:

- eikonal of the direct wave:
- eikonal of the wave reflected from
*R*: .

Both eikonals satisfy the eikonal equation

| |
(53) |

The initial condition for is
| |
(54) |

initial condition for is
| |
(55) |

where
| |
(56) |

are curvilinear coordinates on the surface *R*.
and are determined as the direct eikonal's continuations.
It is by definition that eikonals and have the same values on
the reflector *R*:

| |
(57) |

In fact the surface *R* is a geometrical collection of points that
satisfies equation
| |
(58) |

We suggest that travel-times for a common source are
given on the surface . Since the velocity-function is proposed
to be known, we can determine the reverse eikonal continuation .Both functions and :
- satisfy the same equation (53)
- coincide with on
- approach the surface from below.

This means that
| |
(59) |

and the reflector's location is determined by
| |
(60) |

In fact, Riznichenko proposed not only this theoretical scheme but also a
graphical scheme. Huygens principle founded method of reconstructing
and in the 2D case. Nowadays this second scheme has lost its
significance since we have a great deal of different computer-flexible
techniques to solve equation (53). Due to the principle
of reciprocity, absolutely the same scheme works when, instead of , we
have : travel-times for common receiver patterns with source location *s*.
It is easy to extend this technique to converted waves *PS* or *SP*: in this
case we must use different velocities for reconstructing and
.

** Next:** Zero-offset case
** Up:** 6: WHY SO MANY
** Previous:** 6: WHY SO MANY
Stanford Exploration Project

1/13/1998