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In the triangle (*PSG*) in Figure (1), the following
expression relates the horizontal traveling of the source and
geophone rays with the offset:

| |
(3) |

The vectors and
are related to the table that gives
the lateral distance traveled by the ray as a function of the ray
parameter and the travel time, ^{}.
The projection on the earth's surface of a
ray path (of parameter *p* and traveltime *t*) is the 2-D vector
. Thus, relation (3) yields
| |
(4) |

where and are the 2-D vectors of the source and
geophone coordinates.
A second relation expresses the coordinates of the point of emergence
of the zero-offset ray, *E*. In the triangle (*PEM*), we have :

| |
(5) |

Again, the vector is related
to the table of lateral distances, , yielding
| |
(6) |

where is the 2-D vector of the emergence point coordinates.
**Rayp3d
**

Figure 1 Three-dimensional view of the
source, receiver, and reflection points for 3-D v(z) DMO. The dashed
lines represent the ray paths in the earth, and the bold solid lines
represent the distances on the surface of the earth.

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Stanford Exploration Project

11/17/1997