684 EXPLORATION GEOPHYSICS 



The desired equation for the surface of constant reflection time is obtained 

 by eliminating p from Equations 42 and 44. 

 From Equation 44 



^=cosh(cosh-i^+^) (45) 



= ^coshf +Vpr).-lsinh(f) 

 where cosh {x -\- y) = cosh x cosh 3; + sinh x sinh y 

 and sinh [ cosh-i ^] = V(^J-1 

 Multiplying by pV : 



-y- = l+^ = cosh Y + VI -(/'n' sinh ^ (46) 



pV 

 From this equation : pVi = ;= ^ /'47>v 



cosh ^ + VI -(/'H' sinh -^ ^ 



VI -C/'^)'' cosh ^ + sinh -^ 



and ^l-(pv,y= — ^ ^ (48) 



cosh -^ + Vl-(/'f^)^ sinh "Y 



now: H = -^(vi " (pViV " VI " (/'^)') (42) 



Substituting Equation 48 into Equation 42 : 



(/.F)2sinhf^ 



H = -^ ^^ ^ (49) 



''^ cosh -^ + Vl^^IpFFsinh-Y" 



c J 7 J/ T^ 



or, from Equation 46: H = ^ sinh— (50) 



it ^ 



From Equation 46 : 



[2_Zl(,„3hf-l)]^(^)^l-(,F)^lsi„h=f (52) 

 adding Equations 51 and 52: 



H^+[z-^(cosHf-:J=(^)%i„H^f (53) 



