Chap. 9] SEISMIC METHODS 563 



With the ratio R = z/x, formulas (9 -75a) and (9-756) may be written: 



~ = V(^Ry + 2/2 sin ^ + 1 (9-76a) 



- = Vmy-'^R sin ip~-fl. (9-766) 



The depth under the down-dip shot point is approximately equal to 

 the arithmetic mean of the down-dip and up-dip travel times, multiplied 

 by one-half the velocity. Eqs. (9-75) hold only for equal distances on 

 opposite sides of the shot point. For different distances Xd (down-dip) 

 and Xu (up-dip), we have 



v^ ^d = 42 -f izxd sin (p -\- Xd 



v^ iu = 42^ — 4:ZXu sin cp -\- xl, 



2/,2 /2 \ _, 2 2 



— iZxT+n — ^ ''" '^ ^^~^^^^ 



^C\^d \^ ^u) 



2/,2 _J_ /2 \ 2 2 

 V \ldXu ~r luXd) XuXd XuXd _2 



4(a;d -\- Xu) 



so that 



and 



(9-776) 



If dips are determined in two or more distances on the same side of the 

 shot point, 



v^ h = ^z + 42a:i sin ip -\- x\ 



V <2 = 42 -\- 4zx2 sin <p -{- Xi, 



2/,2 ,2n 2,2 



V (^2 - ^i) - a:2 + a-i 



or 



and 



42(a;2 - a-i) 



2/,2 j2 \ 2,2 



V {tiXz — t2Xi) — a:2a;i i- a;2.'Ci 



sin (f (9-7 7c) 



= z\ (9-77d) 



4(a;2 — xi) 



if X2 > Xi and t2 > ti . In eqs. (9-77a) and (9-77c), an approximate value 

 for 2 calculated under the assumption of horizontal bedding gives sufficient 

 accuracy for small dips. Formula (9-77c) may be written 



v' /t2 + tiV k - k 

 sm <^ = ^ I 



2z\ 2 



Ax2~xJ 22 \ 2 7 



