680 EXPLORATION GEOPHYSICS 



Equation 33 contains the differential equation of the ray 



dh . ^ - . 



-7- = tan a (35) 



dh PZ ,,,, 



If the ray meets a reflecting interface at the depth Z, the corresponding hori- 

 zontal distance, H, is obtained by integrating Equation 36. 



H 



= i dh 



„=( 







or z 



( pVds 

 H=\ / (37) 



Returning to Figure 421, it is seen that 



dz 



-7- = cos a 



ds 



This equation is the differential equation of motion of the wave by 

 making use of Equations 34 and 33. 



dt 1 1 



ds V cos a Fvi ~{pyy 



(38) 



The reflection time T to the origin is twice the travel-time to the reflect- 

 ing interface and is obtained by integrating Equation 38. 



T 



T=i dt 

 

 or 



2 



-f 



dz 

 T=2\ , (39) 







When F is a known function of Z, the integrations can be performed 

 and Equations 27 and 39 are sufficient to determine H and Z from the 

 measured values of T and p. H and Z cannot in general be expressed ex- 

 plicitly in terms of T and p. Even in those cases in which the velocity- 

 depth function, V — F{Z), can be expressed in simple enough form to 



