SEISMIC METHODS 683 



purely empirical distribution of velocity with depth, though such a process 

 may become somewhat burdensome. 



The use of simple mathematical relationships between the velocity and 

 time or depth Z gives a very convenient means of extrapolating data to 

 depths greater than that of actual velocity measurements, an ever-present 

 necessity since the velocity data are often not available at depths from which 

 reflections can be consistently obtained. 



Linear Increase of Velocity with Depth 



In the case when the seismic velocity is proportional to the depth, the 

 basic equations 37 and 39 can be integrated to obtain the time, T, and the 

 horizontal displacement H. The velocity at any point may be written then 

 as function of the first power of the depth Z. 



V=Vi + aZ (40) 



Considering the first Equation 37, we may transform it by the substi- 

 tution 



dV 



as = 



a 



// = — \- ^ (41) 



ap) 



This is a standard integral form readily found in integrating handbooks, 

 and the solution is 



H = j^{y/'^-(pV,y-Vl-(pyr) (42) 



Since F is a known function of Z, Equation 42 gives H as a function of 



p and Z. 



dV 

 Similarly by the substitution, ds = — , Equation 39 may be converted 



into the standard form : 



d(pV) 

 T = - \ / (43) 





which, on being integrated, gives : 



where cosh ~^\ — ) = sech ~'^x = log ( — H ) 



\x J ^\x X ) 



