SEISMIC METHODS 687 



It has been shown that the angle of dip of the reflecting interface is 

 given by sin ^ == F ^^ (20) 



When the depth of the interface has been determined, the velocity at 

 that depth may be obtained from the velocity-depth function V = Vi + aZ. 

 and the angle of dip can be computed from Equation 20. In practice this 

 process is reduced to one step by the use of a chart from which Q may be 

 read when sin ai and the depth Z are known. When this is done, all of the 

 data necessary to completely describe the position and orientation of the 

 reflecting interface are known. 



Computation Charts 



It has been shown that if the quantities p and T of a reflection are 

 measured at the shot-point, a complete solution for the position and attitude 

 of the reflecting interface is given by the three Equations 37, 39 and 20. 

 In the application of these equations to any arbitrary velocity-depth func- 

 tion, the integrations would probably have to be carried out by graphical 

 or numerical methods. A description of these procedures may be found 

 in any textbook of advanced calculus. 



In dealing with a non-linear velocity-depth function the results cannot 

 be expressed in a form in which H and Z can be separately expressed in 

 terms of the measured values oi p and T. Fortunately, however, it has 

 been found that in the majority of cases the velocity-depth function can 

 be satisfactorily described as a simple linear increase of velocity, V, with 

 depth, Z. With this type of mathematical function for the velocity, the 

 integrations of Equations 37 and 39 have been given as Equations 42 and 44. 



From an inspection of Figure 422 (page 685) it can be seen that the 

 depth is given hy Z = D + R cos 6, 



or Z = — -\ cosh -^ — 1 -f cos ^ sinh — • (54) 



This equation can be reduced to the more convenient expression, 



V-, r «r fjTl 



Z = ^\e~ -1 - (1- cos ^) sinh ^J (55) 



The horizontal distance is given by 



H=Rsme or H^^sin^sinh^ (56) 



a 2 



It is noted that the solution of these two equations requires knowledge of 



