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## The method of time fields (Common Source Pattern)

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