SEISMIC METHODS 



741 



In triangle OhSi the angle 6'iO/i is equal to 90 + 0i, the distance O/i = 2//a, 

 and the distance hSi = OCSi = ViTi. (The velocity is assumed to vary with depth 

 only and Vi is defined as the "equivalent" or "average" vertical velocity to depth Hi. 

 Ti is the time interval between point 41 and 20 on the seismogram.) From the 

 cosine law 



Vi'Ti' = ;ri== + 4Hi'' + 4HiXi sin 0, 



Similarly, from triangle O/iS'a 



Fi'T/ = a:/ + AHi' + 4HiX2 sin <^i 



and from triangles OhSi and 0/25'2 



F^Ta' = ;iri=' + 4H2^ + 4H2.ri sin ^2 

 Vi'T," = X2^ + 4//2'' + 4H2X2 sin ^2 



(91) 



(92) 



(93) 

 (94) 



Fig. 452. — Diagrammatic sketch of ray paths corresponding to 

 Figure 450. (McCollum, U. S. Patent 2,118,441.) 



In the above equations, the quantities jTi and X2 are known. The quantities T2, 

 Ti and Ti are obtained from the seismogram as was T\. (Figure 451.) The quantity 

 V2 is an average velocity in the material above bed 5 (L2). Both Vi and V2 may be 

 determined by methods outlined in the section on Velocity Shooting. The unknown 

 quantities, therefore, are ^i, 02, H\ and H2. 



It is evident from Figure 452 that 



<p2 — (pi = a (95) 



Elimination of Hi between Equations 91 and 92 gives 



( Fi^r/ — .r2^ cos" 0i) '-^ — X2 sin ^i = ( Fi^Pi* — Xi" cos" <f>i)'^—xi sin <pi 



For the gentle dips usually treated, it is generally sufficiently accurate to replace 

 cos" (pi by 1. On rearranging terms, the last equation becomes : 



(x2 — xi) sin 01 = V1T2 ( 1 — J^f^iJ ~ V^Ti ( 1 — J^fl) 



