SECT. 1] THE UNCONSOLIDATED SEDIMENTS 77 



2 



-Rs = i?io sec di + — (sinh-i cot ^2 — sinh-i cot ^3) 



Bi = i?io sec 64 



Co 



D 4- a , -^2 / COS 02 - COS 62 



Rio tan ^1 + -— ^ : — 



ACi\ sin 62 



(2) 



where 7?io is the time for the bottom-reflected wave at zero shot-receiver 

 distance, K is the gradient, Co is the velocity in the surface channel, Ci is the 

 mean velocity in the water, and C'2 is the velocity in the upper part of the sedi- 

 ments. The ray paths, with angles di-d^ indicated, are shown in Fig. 1. The 

 theoretical curves of Fig. 2 are those corresponding to the upper branch, Rs, 

 of the refraction curve shown in Fig. 9, Chapter 1. Each curve begins at the 

 "critical range" corresponding to the minimum distance at which a ray can be 

 returned to the surface by refraction under the conditions of velocity gradient 

 and water depth indicated. The Rd branches of the curves, corresponding to 

 the deeper penetrating rays, are not shown. They join the Rs branches in 

 cusps at the critical range and are curved in the opposite direction. For simpli- 

 city it was assumed that the water has uniform velocity equal to that in the 

 upper sediments. Hence, in equations (2), Co = Ci = C2, 61 = 62 and, since we 

 are dealing only with refracted waves, ^3 = 7r/2. For the ranges involved, it can 

 be shown that errors introduced by the assumption that Co = Ci are small, 

 and as discussed earlier, we have much evidence that the velocity in the upper 

 sediments is approximately equal to that in the water. 



These same formulae can also be used for the cases in which the upper 

 sediment velocity is either lower than or higher than water velocity. For 

 Ci > C2, the curves corresponding to those in Fig. 2 would be displaced down 

 and to the right. For Ci<C2, they would be displaced upward and toward 

 the left. In the first case there is no critical angle for reflection, even at grazing 

 incidence, hence the Rs arrival is always later than Ri. In the second case, 

 Rs will cross Rj. In the case illustrated, i.e. C'i = C*2, Rs joins Ri at the range 

 corresponding to a grazing ray. 



On these theoretical curves are plotted squares and triangles representing 

 the observational data from two reflection profiles, A and B, with water depths 

 corresponding to Riq = 5 and 7 see respectively. The data are plotted for 

 ranges from to 20 sec, hence both reflected, Rn, and refracted, Rs, arrivals 

 are shown. In profile A, the outer points best fit the refraction curve for Rio = 

 5 sec and A'^ 0.45 sec~i. At ranges beyond about 9.3 sec, the observed arrivals 

 are purely refracted. Since 9.3 sec is the minimum range at which a refracted 

 arrival can be returned to the surface under the conditions i?io = 5 sec and 

 K = 0A5 sec~i, the arrivals at shorter range must be reflected. The fact that 

 the reflected arrivals join smoothly to the refracted ones indicates that the 

 reflecting interface must lie at aiDproximately the def)th penetrated by the 

 critical ray, i.e. the ray at range 9.3 sec. If the interface had been deeper, 

 the reflection curve would have crossed the Rs-Ri curve above the cusp and 

 joined the Rd-Ri curve as indicated diagrammatically in Fig. 3. 



