762 EXPLORATION GEOPHYSICS 



Hence 



slope = 77- sin fi 



(down dip) ^ 1 



Similarly 



slope = 77- sin a 



(up dip) yi 



The equations just derived permit the calculation of yS and a and so of 6 

 [which is equal to Vi ()8 — a)]. Also, F2 may be obtained from the 



relation: tt" = sin (a+ ^). In practice, a good approximation is V2 — 



V2{V2' + vh' 



Evidently, when the values of 6, Vi, and Vo are known, the depth h 



may be calculated from the intercept formula 



To= 2h cos 6 y y-^- Y^ 



and h' may be calculated from the relation 



A' = A + L tan e 



For successively deeper layers the calculation can proceed in a similar 

 way; that is, formulas may be derived for the depth, dips, and velocities 

 for the multi-layer, dipping cases by methods analogous to those already 

 described. Another procedure favored by many investigators is to re- 

 duce the problem at each step to the two layer case. Figure 461 shows 

 how this may be done. The dip and depth of the first interface and 

 the velocities Vx and V2 are determined by the methods outlined above.* 



From the slopes -ryj and 77-77 of the third segments of the travel-time 



K 3 K 3 



curves and the fact that the slopes are equal to the sines of the emergence 

 angles divided by Vi, the emergence angles a and /? can be found. Choose 

 any two points Si and S2 between the two shot-points. By simple trigo- 

 nometry the lengths 0/02', OiS\ and O2S2 can be found. Also, the times 

 along the segments 0i'0/'^/'5"/, 0/0r02"02', and 02'02"S2"S2' are readily 

 determined. For example, the total time along the path OiOi' Ox'S\'S\S\ 

 is read from the travel-time curve, the time along 0i'0i'''6'i'''6'i' is obtained 

 by subtracting the times along the segments OiOi' and S\Sx from the 

 total time. This gives two points on each of the second segments of the 

 new travel-time curve (Figure 461 B) so that these two segments may 

 be drawn. The first segments correspond to direct waves in the second 

 layer with velocity V2 and hence are straight lines with slope I/F2 through 

 Oi' and O2' respectively. The depth, dip, and velocity are now computed 

 as before. (It should be noted that the "depths" determined by this pro- 

 cedure are directed along the perpendiculars to the second boundary at 



* The numerical computations are indicated on p. 763. 



