16 



EWINO 



[chap. 1 



approaches zero slope (infinite velocity) at large distances and the Rs curve 

 approaches Rj. (The Rs curve will be asymptotic to Ri, asymptotic to a line 

 parallel to Rj or will cross Ri depending on whether the velocity in the upper 

 sediments is equal to, less than or greater than the water velocity.) 



In most instances where gradients in oceanic sediments have been studied, 

 it has been found that velocity-depth relationships of the type 



or 



C = C^^K^^Z (7) 



C = Co + A'Z + A^'(l-f-io2) . (8) 



fit the observational data better than the linear relationshii? given by (6). 



Time 



Fig. 10. Time-distance graph for velocity gradient above a discontinuity. 



Both (7) and (8) define velocity-depth relationships in which the gradient 

 decreases with depth in the sediments. Rays in sediments for which these 

 equations are appropriate will deviate from circular paths, i.e. the curvature 

 of the path decreases with depth in the sediments. 



If the structural model shown in Fig. 9 is altered to include a boundary 

 within the sedimentary material where the velocity increase is discontinuous, 

 we have a model which closely approximates the upper sediments in many 

 ocean-basin areas. Layer 2 would have a parabolic or exponential gradient and 

 Layers 1 and 3 negligible gradients by comparison. The time-distance graph 

 in Fig. 10 shows the relationship of the reflected and refracted waves for such 

 a model. The lower part of the Rn curve is determined by waves reflected from 

 the first sub-bottom interface. Beyond a certain range, determined by the 

 thickness of Layer 2 and by the velocity gradient, the Rn curve ends. If 

 Layer 2 is thick enough or has a sufficiently high velocity gradient, Rn termi- 

 nates in a cusp where it joins the Rd curve, as shown in the diagram. In the 



