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Constant vertical velocity gradient media

Let us consider a velocity field of the form v = v0 + az, where a is a constant. Both the ray and the wavefront are circles as illustrated in Figure [*]. If the ray from a source located at the origin surfaces at (xout,0), then according to Equation (15) in Slotnick (1936), the depth of penetration of the ray is:  
 \begin{displaymath}
z_0 = \frac{v_0}{a}
\left[\sqrt{1+\left(\frac{a x_{out}}{2v_0}\right)^2}-1\right].\end{displaymath} (45)

 
wavefront
Figure 5
Geometry of a wavefront and a ray in a constant vertical velocity gradient media.
wavefront
view

The ray takes the time 2t to travel from origin to (xout,0), and only half of that - t - to get to the maximum depth point $(\frac{x_{out}}{2},z_0)$. According to Equation (14) in Slotnick (1936),  
 \begin{displaymath}
t = \frac{1}{a}\sinh^{-1}\left(\frac{a x_{out}}{2 v_0} \right).\end{displaymath} (46)
Because wavefronts are orthogonal to rays, and because the ray is horizontal at the location where it reaches penetration depth, the equation of the only wavefront that passes through $(\frac{x_{out}}{2},z_0)$ is:  
 \begin{displaymath}
x^2+(z-z_0)^2=\left(\frac{x_{out}}{2}\right)^2.\end{displaymath} (47)
By plugging (45) and (46) into (47) we get the general equation of a wavefront:  
 \begin{displaymath}
x^2+\left[z-\frac{v_0}{a}\left(\sqrt{1+\sinh^2(at)}-1\right)\right]^2=
\left[\frac{v_o}{a} \sinh(at)\right]^2 = R^2.\end{displaymath} (48)
There is a single ray that departs from the origin and arrives at the point (x,z). Its ray parameter is p and the time it takes to get to (x,z) is the solution of the integral in Equation (8) of Slotnick (1936) for the particular case of v = v0 + az:  
 \begin{displaymath}
t=\frac{1}{a}\log F,\end{displaymath} (49)
where we use the notation  
 \begin{displaymath}
F=\left(1+\frac{az}{v_0}\right)
\frac
{1+\sqrt{1-p^2v_0^2}}
{1+\sqrt{1-p^2(v_0+az)^2}}.\end{displaymath} (50)
We can write p as a function of x and z by writing Equation (12) in Slotnick (1936) as:  
 \begin{displaymath}
p^2=\frac{1}{v_0^2+\left[\frac{\frac{a}{2}\left(x^2+z^2\right)+ zv_0}{x} \right]^2}.\end{displaymath} (51)
By combining (48) and (49) we can write R as a function of p instead of t:  
 \begin{displaymath}
R=\frac{v_0}{2a}\left(F-\frac{1}{F}\right).\end{displaymath} (52)
To compute the wavefront radius R as a function only of x, z, v0 and a, we have to plug (51) into (50), and then (50) into (52).

 


next up previous print clean
Next: About this document ... Up: Solving equation 21 Previous: Constant velocity media
Stanford Exploration Project
7/8/2003