Chap. 9] 



SEISMIC METHODS 



523 



Since Z = H cos <p and z = h cos <p, 



2H cos <p cos i 



hu — 



Vi 



-| — sin {i — ip) 



Vi 



and 



2H cos ip cos ^ , s , / . , ^ 

 hd = ^^ h - sin (z + <p). 



Vi Vi 



(9-51) 



The apparent velocities for the up- and down-dip cases are obtained by 

 differentiation of eq. (9-51): 



V2« = 



Vi 



sin (i — <p) 

 Hence, eqs. (9-51) become 



and 



Vm = 



Vi 



sin (i -{- (f) 



(9-52) 



and 



2H cos (p cos t s 



hu = H 



Vi V2„ 



2H cos v? cos I s 



hd — + — 



Vl V2d 



■ (9-53a) 



The true underlayer velocity is not the arithmetic mean of the two 

 apparent velocities. From eq. (9-52) 



V2 = 2 COS <p 



(9-536) 



Vu + Vd 



The true velocity has the same relation to the up-dip and down-dip 

 velocities as a resultant resistance has to its component resistances in 

 parallel. In the example of Fig. 9-63 (where V2„ = 6190 and v^^a = 2790), 

 the above formula gives V2 = 3795, whereas the arithmetic mean is 4490. 

 From eq. (9-52) the critical angle i and the dip are calculated as follows: 





1 Vl , . -i .X 



f I sm f- sm ^— 



V2„ 



1 Vl 



sm — — sm 



V2d 



1 Vi^\ 

 V2d/ 



. -1 vA 



V2u/ 



(9-54) 



and the true underlayer velocity is obtained from V2 = Vi/sin i. Fig. 9-62 

 shows apparent up- and down-dip velocities for dip angles up to ±40° and 

 for velocity ratios of four {i = 15°) and two {i — 30°). The variation is 

 small down dip. On shooting up dip, the velocities change rapidly when 

 the dip angle approaches the critical angle, and they become plus and 



