GATE-2018 (24) :Wyllie – Time average equation

CSIR-NET JUNE 2018

98)In a seismic survey over a refractor/reflector with a P-wave velocity twice that of the overburden, both the reflected and refracted waves are recorded simultaneously at an offset distance of 2.31km from the shot point at 2.3s, respectively, after the shot. The thickness of the overburden and the velocity of the refractor are:

(Thanks to Suresh Sir ANU) 

(Thanks to Neeraja AU)

a) 2km and 2km/s

b) 2km and 4km/s

c) 3km and 2km/s

d) 3km and 4 km/s

sol:

$V_1=x km/s$

$V_2=2x km/s$

$X_{crit}=2.31 km$

t=2.3s

$critical distance(X_{crit})=2h\tan i_c-(1)$

$2.31=2h\tan i_c$

$sini_c=\frac{V_1}{V_2}$

$\cos i_c=\sqrt{1-\frac{V_1 ^2}{V_2 ^2}}$

$\tan i_c=\frac{V_1}{\sqrt{V_2^2-V_1^2}}----------(2)$

$2.31=2h\frac{V_1}{\sqrt{V_2^2-V_1^2}}$

$V_1=x$

$V_2=2x$

$2.31=2h\frac{x}{\sqrt{4x^2-x^2}}$

$2.31=2h\frac{x}{\sqrt{3}x}$

$2.31\times\sqrt{3}=2h$

4.001=2h

h=2km

thickness(h)=2km 

Refraction -Travel time equation for two layer case 

$t_x=\frac{x_{crit}}{V_2}+\frac{2h\sqrt{V_2^2-V_1^2}}{V_1V_2}----------(3)$

$t_x=2.3s$

$x_{crit}=2.31km$

h=2km

$V_1=x$

$V_2=2x$

$2.3=\frac{2.31}{2x}+\frac{2\times2\sqrt{4x^2-x^2}}{2x^2}$

$2.3=\frac{2.31}{2x}+\frac{4\sqrt{3}x}{2x^2}$

$2.3=\frac{2.31}{2x}+\frac{4\sqrt{3}}{2x}$

$2.3\times2x=2.31+4\sqrt{3}$

4.6x=2.31+6.92

4.6x=9.23

$x=\frac{9.23}{4.6}=2km/s$

$V_1=x=2km/s$

$V_1=x=2km/s$

$V_2=2x=4km/s$