534 



SEISMIC METHODS 



[Chap. 9 



formation members exists and where, therefore, the oblique overburden 

 path does not differ sufficiently from the vertical path. In weathered- 

 layer procedure in reflection shooting, interpretation is based almost ex- 

 clusively on vertical-ray propagation. The following paragraphs contain 

 derivations of vertical-ray formulas for the single and double horizontal 

 layer, and for the single and double incHned layer. 



In the case of a single horizontal layer, the wave is assumed to travel 

 vertically to the lower layer, to travel along the interface with the velocity 

 of the lower layer, and to come up vertically. Hence, ^2 = 2d/vi -|- s/v2 , 



Fig. 9-6 



W 20 30 ^0 X 60 70 00 SQ% 

 Error of vertical-ray calculation as a function of velocity ratio. 



so that the slope of the second part of the travel-time curve dsldU = Vj 

 For the intercept, s = a;, so that 



=i(-3 



d = ^ (1 — sin i) = 



whereas the application of Fermat's principle gives 



X (1 — sin i) 



(9-59a) 



d = 



cost 



which for an angle i of 30° is c^Fennat = dvert.1-15. With dr as depth 

 determined by the refraction interpretation and d« as depth from vertical- 

 ray interpretation, the error {dr — d^jdr = 1 — cos i. Fig. 9-68 shows 

 this error as a function of the velocity ratio Vi/v2 . 



