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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:
 (45)

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

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 . According to Equation (14) in Slotnick (1936),
 (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 is:
 (47)
By plugging (45) and (46) into (47) we get the general equation of a wavefront:
 (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:
 (49)
where we use the notation
 (50)
We can write p as a function of x and z by writing Equation (12) in Slotnick (1936) as:
 (51)
By combining (48) and (49) we can write R as a function of p instead of t:
 (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: About this document ... Up: Solving equation 21 Previous: Constant velocity media
Stanford Exploration Project
7/8/2003