542 SEISMIC METHODS [Chap. 9 



It is further seen that if sin a = 1, the apparent surface velocity is equal 

 to the true velocity in the deepest bed reached, that is, the bed corre- 

 sponding to the depth of penetration (see Fig. 9-72a), which is 



ro (1 - sip fo) (g_g3^^ 



k sin ?o 



This relation applies, of course, only to the part of the travel-time 

 curve ahead of the intercept, that is, to the ray in the overburden. The 

 travel time for a ray corresponding to the distance x with the angle zo is 



f = hog.l±-^ (9-636) 



k 1 — cos to 



and the distance 



X = ^I^^^^^. (9-63C) 



Since I'o may be determined by graphical differentiation of the travel- 

 time curve at any distance x, k may be obtained from eq. (9-63c). A 

 calculation of this kind is illustrated in Table 57-A. The observed data 

 in the first two columns are distances and travel times; next follow the 

 apparent velocities from which the angles of emergence at these distances 

 are calculated. Application of formula (9-63c) then yields the vertical 

 velocity increase k, which, for an ideal travel-time curve, has to have the 

 same value for any distance. The next set of columns contains a compu- 

 tation of penetration-depth from formula (9-63a), although for bedrock 

 or basement-depth determinations this part may be left out. 



The method followed in the calculation of the depth of a high-speed 

 layer of presumably constant velocity beneath an overburden of linear 

 velocity increase will be evident from the following discussion and reference 

 to Fig. 9-72c. In the latter D = AE = shot detector separation, Vo = 

 speed at surface, Vi = speed at base of the overburden, Va = speed of 

 underlayer, a = angle of incidence at contact, p = radius of arc AB, and 

 t = travel time of refraction wave from A to E. 



Since PB = Vo/k + A, Vi = Vo + kh, and h = (vi — Vo)/k, we have 

 PB = Vo/k + (vi — Vo)/k = Vi/k. Further, since sin a = V1/V2 = 

 PB/p, V1/V2 is Vi/kp, or p = V2/k. 



The travel time of the trajectory ABCE is f^B + tsc + tcs , where <as = 



tcD = j ds/v. The value of s is p<pi and (p is a — 7 ; p and a are given 

 Jo 



