758 



EXPLORATION GEOPHYSICS 



Hence, 



X = 



2Fo 



, aT 

 a seen ^r- 



V 



1 — sech^ 



(?) = ; 



2Ffl 



sech 



aT 



tanh 



aT 



or 



2Fo . ^ aT 

 X = sinh -77- 



(113) 



Thus, the travel-time curve is smooth and concave downward. (Figure 459.) 



Sloping Interfaces Between Strata. — While the cases of horizontal 

 interfaces are useful in formulating ideas concerning seismic wave paths, 

 horizontal interfaces are seldom encountered in practice. Consider, there- 

 fore, a simple two-layer problem in which the velocity in the upper layer 

 is Vx and the velocity in the lower layer is F2 and assume that F2 and V\ 

 differ by a finite amount. Referring to Figure 460, the vertical depth 

 below the shot-point is h, and the boundary dips down to the right so 

 as to form an angle 6 with the horizontal. (It is assumed that the strike 

 of the layer is perpendicular to the plane of the figure.) x is, as usual, the 

 distance between the shot-point and a seismometer station. 0' is a point 



Fig. 460. — Seismic ray paths through an inclined layer. 



on the bed from which a normal to the bed passes through 0; S' is the 

 corresponding point below 5" ; .^ is the point where the ray strikes the bed ; 

 and B the point where the ray leaves the bed. The angle a is the angle 

 between the ray and the vertical at 0; /? is the corresponding angle at 6^. 

 The travel-time for the path OABS is 



In triangle 00"A, 



OA+BS AB 

 Vi V2 



OA ^ sin (90 + 6) 

 h "sin [90- (a + e)] 



(114) 



