544 



SEISMIC METHODS 



[Chap. 9 



or 



h = l |[C0S h-' (V2/V0) - Vl - MV2Y] 



- [cos hr' (V2/V:) - Vl - (V1/V2)']) + - . (9-65) 



J V2 



Vl— 7, with r = V0/V2 and r' = V1/V2 , 



If we let /(r) = cosh~' (l/r) - 

 respectively, eq. (9-65) becomes 



t2 = § + l[fiT)-f(r% 



V2 K 



(9-66a) 



Fig. 9-73a shows the variation of r with /(r) and is valid for any condi- 

 tions. If 



2 



[/(V0/V2) 



/(V1/V2)] ^ T, 

 then 



(9-666) 



0.1 0.2 0.3 0.4 0.5 0.6 0.7 0j6 OS tj} 



r 



Fig. 9-73a. Graph of /(r) for curved-ray inter- 

 pretation. 



D 



T = h-~. (9-66c) 

 V2 



The application of these re- 

 lations is as follows: Assume 

 k to have been determined 

 from the first part of the 

 travel-time curve, as shown 

 in Table hi- A. If, further, V2 

 has been measured in the 

 second part of the travel 

 curve (past the 3000-m inter- 

 cept in the example), the 

 value of the time function T 

 may be calculated from form- 

 ula (9-666) for various depths 

 (and therefore for various 

 values of Vi). In this manner 

 a curve T = /(A) is obtained, 

 as shown in Fig. 9-736. For 

 the depth determination, T is 

 calculated from the travel- 

 time curve by subtracting, in 

 accordance with formula 

 (9-66c), the value of D/V2 for 

 each distance as shown in the 

 last set of columns in Table 



