Calculation of true thickness of the bed

GATE-2017

22) The travel time difference between arrival times of a shear wave(s) and primary wave(p) observed on a seismogram recorded at an epicenter distance of 100 km from a near surface earthquake is _______ seconds(Assume the average P and S wave velocities to be 6 km/sec and 3.5 km/sec respectively)

(Thanks to Chandrasekhar, ANU)

Solution:

Given that the epicentral distance(D) = 100 km

Velocity of P- wave($V_{p}$)= 6 km/sec

Velocity of S- Wave($V_{s}$)= 3.5 km/sec

$t_{s}-t_{p}$=?

The travel time difference between arrival time of shear wave and Primary wave is

$t_{s}-t_{p}=D(\frac{1}{V_{s}}-\frac{1}{V_{p}})$-------------------------------(1)

Substitute these values in the equation (1) we get

$t_{s}-t_{p}=100(\frac{1}{3.5}-\frac{1}{6})$

$t_{s}-t_{p}=100(0.285-0.166)$

$t_{s}-t_{p}=100(0.1184)$

$t_{s}-t_{p}=11.84$

Extra information:

If D is the distance travelled from the focus to the receiver then

$t_{p}= \frac{D}{V_{p}}$

$t_{s}= \frac{D}{V_{s}}$ then

$t_{s}-t_{p}=D(\frac{1}{V_{s}}-\frac{1}{V_{p}})$-

The intercept to the horizontal axis (in wadati diagram ) gives the time of occurrence of the earthquake ($t_{0}$)

i.e $D=V_{p}(t_{p}-t_{0})$

if $\triangledown_{km}$ is the epicentral distance and D is the distance travelled by the wave , d is the focal depth then

$d = \sqrt{D^{2}-\triangledown_{km}^{2}}$

Reference : Fundamentals of Geophysics by William Lowrie.