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## Vectorial relationships on the earth's surface

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