510 SEISMIC METHODS [Chap. 9 



differentiation of eq. (9-45a), tan 7 = dtz/ds = I/V3 . Neither the first 

 nor the second layer is effective after the break x-^, is passed. For deter- 

 mining the depth d^ . let h = h . By equating eqs. (9-42) and (9-45a), 

 substituting xi for s, and proceeding as before, 



4/^^' + rfiTl + -^ ^ (cos i - cos a) 



V V3 + V2 L sm i cos /3 



= di. (9-45b) 



All data for the computation of the depth of the lower surface of the 

 second layer are obtainable from the travel-time curve, since sin i = V1/V2 , 

 sin j8 = V2/V3 , and sin a = V1/V3 . The depth to the bottom of the second 

 layer may also be calculated by extending the Vi and V3 portions of the 

 travel-time curve to the intersection at the abscissa 2:13 (Fig. 9-48). Then 



J , 3:13(1 — sm a) — 2di cos a m ai: \ 



02 - rfi = — ^r-^ — : . (9-45c; 



2 sm I cos )3 



Another convenient depth-calculation method uses the time obtained on 

 the ordinate by extending back the third part of the travel-time curve. 

 Then from eq. (9-45a), with T-i = ^3 for s = and hi for di — di , 



Tz = — ' cos iS + — ' cos a. (9-45d) 



V2 Vi 



As an example of depth calculations in the single- and two-layer case, 

 the curves in Fig. 9-48 furnish: xi{= X12) = 1000 m; a:23(= 3^2) = 3000 m; 

 a-13 = 1750 m; Vi = 1000 m-sec'^ V2 = 2000 m-sec~^ V3 = 5000 m-sec~\ 

 Hence, sin a = V1/V3 = 11.5°; sin jS = V2/V3 = 23.5°; sin i = V1/V2 = 30°. 



x.(l- sin 0^50^5^ 

 2 cost 0.866 



rf, - rf, = I ,,/^^ + A 52ii 

 2 Y V3 + V2 sm 



— cos a 



Also, 



i cos /3 



, -^ , /^5000 - 2000 , „Qo 0.866 - 0.979 „ . . _ . 

 = ^^-<^ 5000 + 2000 + '^' 0.5.0.916 = '^^ "^^ 



3:13(1 — sin ot) — 2di cos- a 

 rfi = — 



2 sin i cos /3 



1750-0.8 - 578.0.979 

 0.916 



= 911 m. 



52 Schmidt, op. cit., 7(1/2), 37-56 (1931). 



