Chap. 9] ^ SEISMIC METHODS 513 



and shales of increasing velocity, and the lowest layer is igneous material, 

 probably basalt and basaltic agglomerates. The depth determination 

 based on the travel-time curve gives 1200 feet for this igneous material; 

 a well drilled a mile away from the profile encountered it at 1140 feet. 



7. Multilayer case. If d^ is the depth to the lower surface of an n*^ 

 layer, h^ = dj^ — dn_i its thickness, v^ its velocity, and a a refraction angle, 

 then the travel time corresponding to the next (n + 1) layer is at the 

 distance s: 



Wi = ^ + 2 i: ^" ~ ^°-' . cos a„ . (9-47a) 



Vn+l X Vn 



For depth calculations it is more convenient to use eq. (9-47a), as applied 

 to the intercept travel time on the ordinate instead of the distance (x) 

 intercepts. Hence, with T as (ordinate) travel time for x = 0: 



Tn+i = 2Z;^cos«^. (9-476) 



1 Vn 



Likewise convenient for depth calculations is the use of travel times 

 corresponding to the distance intercepts x^ . Substituting Xj^ for s and 

 solving for the thickness of the deepest layer, 



2 ^ cos i^ = t^^^-(^ +2 e' -^ cos a^) , (9-47c) 



Vn \Vn+l k=l Vk / 



where i^ is the critical angle in the nth layer. 



By comparison with eqs. (9-43), (9-45a), and (9-46a), this is seen to 

 give for the successive thicknesses: 



2di . ^ xi 



— cos ti = 12 — - ; 



Vl V2 



2/l2 . , /X2 . 2hi \ 



— cos t2 = ^3 — I r — COS ai I 



V2 \V3 Vl / 



Ihs . ^ fxs , 2hi . 2h2 \ 



— cos ta = i4 — I h — COS ai + — cos az I , 



V3 \V4 Vl V2 / 



and so on. Application of these formulas to depth calculations of five 

 layers is shown in Fig. 9-51 and the schematic on page 515. Results 

 of a seven-mile shot are shown there, evaluated in the form of velocities, 

 intercept distances, and travel times corresponding to them. These data 

 are entered in columns ii, iii, and iv. Next is the calculation of all angles of 

 total refraction ^n in columns v through vii. Similarly, the refraction angles 

 ttk (k varies from to n — 1) are calculated in columns viii through x. 

 The procedure is seen to be in accordance with formulas preceding (9 -45a) 



2h. 



