Chap. 9] SEISMIC METHODS 541 



In the following analysis it is assumed (see Fig, 9-72a) that the velocity 

 increases linearly from Vo at the surface to Vi at the bottom of the top 

 layer, changes abruptly to Va , and remains constant in the second layer. 

 Then the upper velocity as function of depth is 



Va = Vo + kh. (9-60a) 



In the upper layer the rays travel in circular paths; their radius of 

 curvature depends on the vertical velocity increase k. The locus for the 



centers of curvature is a plane whose distance from the surface is given by 



« 



y=-^{' (9-60&) 



The travel-time curve is no longer straight and the surface velocity is not 

 constant. By differentiation with respect to distance, an "apparent" 

 velocity is obtained (eq. [9-62]). 



For an arbitrary number i of thin parallel horizontal beds, the paths for 

 the incident and emerging rays are identical in the same stratum. The 

 horizontal displacement of the ray due to refraction is 



i=n 



X = 2 ^ hi tan ai , (9-61a) 



i-l 



since in each bed the distance is decreased by the amount hi tan ai . The 

 travel time for the downward and upward (curved) paths is therefore 



i=n 7 



f = 2 E — ^^ , (9-616) 



i=l Vi cos Q!i 



where /li/cos ai represents the oblique . ^ 



path within each layer. According to 



Snell's law, sin ai/sin ai+i = Vi/Vi+i or 



sin ai/vi = sin ai+i/Vi+i = constant 



= C for each ray. Since sin a/v = 



sin io/vo (where io is the angle of 



emergence as indicated in Fig. 9-726, /^!\ /■ > 



° o ) / I \^\/\^ove front 



and Vo is the surface velocity), the p^^ 9.72^. Apparent and true 



constant C may be determined at the velocity. 



surface by graphical differentiation of 



the travel-time curve. If v is the apparent velocity, it is seen from the 



figure that sin z'o = Vo/v; therefore, 



«i!^ = C = ^. (9-62) 



117-120 (March, 1935). M. Ewing and Don L. Leet, A.I.M.E. Geophys. Pros., 245- 

 270 (1932). M. Ewing and A. P. Crary, Soc. Petrol, Geophys. Trans., V, 154-160 

 (March, 1935). M. M. Slotnick, Geophysics, 1(1), 9-22 (Jan., 1936). 



