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:  

$$\stackrel{\longrightarrow}{SG} = \stackrel{\longrightarrow}{SP} + \stackrel{\longrightarrow}{PG} .$$ (3)

The vectors $\stackrel{\longrightarrow}{PS}$ and $\stackrel{\longrightarrow}{PG}$ 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, $\xi(p,t)$. The projection on the earth's surface of a ray path (of parameter p and traveltime t) is the 2-D vector $\xi(p,t)\frac{{\bf p}}{p}$. Thus, relation (3) yields  

$${\bf x}_g - {\bf x}_s = \xi(p_g,t_g)\frac{{\bf p}_g}{p_g} - \xi(p_s,t_s)\frac{{\bf p}_s}{p_s} ,$$ (4)

where ${\bf x}_s$ and ${\bf x}_g$ 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 :  

$$\stackrel{\longrightarrow}{ME} = \stackrel{\longrightarrow}{MP} + \stackrel{\longrightarrow}{PE} .$$ (5)

Again, the vector $\stackrel{\longrightarrow}{PE}$ is related to the table of lateral distances, $\xi(p,t)$, yielding  

$${\bf x}_0 = - \frac{1}{2} \left( \xi(p_s,t_s)\frac{{\bf p}_... ...{{\bf p}_g}{p_g} \right) + \xi(p_0,t_0)\frac{{\bf p}_0}{p_0} ,$$ (6)

where ${\bf x}_0$ 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.