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

1. wave number $k =\frac{2\pi}{\lambda}$

 

2. Angular frequency $(\omega)= 2\pi f$

 

3. $\frac{\omega}{k} = \frac{2\pi f }{\frac{2\pi}{\lambda}} = \lambda f =c$

 

4. Displacement $(u)=A sin(kx-\omega t)$

 

5. Displacement $(u)=A sin k(x-c t)$

 

6. The intensity (or energy density,$I_b$)of the body waves is the energy per unit area of the wavefront, and at distance r is

 

$I_b =\frac{E_b}{2\pi r^2}$

 

7.The surface wave is constricted(shrinking) to spread out laterally. When the wavefront of a surface wave reaches a distance r from the source, the initial energy $(E_s)$ is distributed over a circular cylindrical surface with area $2 \pi r h$. Intensity of the surface wave from a distance r is

 

$I_s =\frac{E_s}{2\pi r h}$

 

8. equations 6 and 7 shows that the decrease in intensity of body waves is proportional to $\frac{1}{r^2}$ while the decrease in surface wave intensity is proportional to $\frac{1}{r}$.

 

9. The mean intensity (energy density) of the wave is proportional to the square of the amplitude

 

$I_{av} = \frac{1}{2} \rho \omega^2 A^2$

 

From above equation, Intensity of the waveform, or harmonic vibration, is proportional to the square of its amplitude.

 

10. Body wave amplitude is related with the distance

$A =\sqrt{\frac{E_b}{2\pi r^2}}$

 

$A =\sqrt{\frac{1}{r^2}} =\frac{1}{r}$

 

11. surface wave amplitude is related with the distance

$A_s =\sqrt{\frac{E_s}{2\pi r h}}$

 

$A_s =\sqrt{\frac{1}{r}} =\frac{1}{\sqrt{r}}$

 

From eq 10 and eq 11 , seismic body waves are attenuated more rapidly than surface waves with increasing distance form the source.

 

12. The damping of seismic waves is described by a parameter called the quality factor (Q), It is defined as the fractional loss of energy per cycle.

 

$\frac{2\pi}{Q}=\frac{-\triangle E}{E}$

 

 $\triangle E$ - Energy lost in one cycle and E is the total elastic energy stored in the wave.

 

13. D is the distance within which the amplitude falls to 1/e (36.8 percentage , or roughly a third) of its original value.

 

The inverse of this distance $(D^{-1})$ is called the absorption coefficient.

 

For a given wavelenght, D is proportional to the Q-factor of the region through which the wave travels.

 

A rock with a high Q-factor transmits a seismic wave with relatively little energy loss by absorption, and the distance D is large.

 

For body waves D is generally of the order of 10,000 km and damping of the waves by absorpion is not a very strong effect. It is slightly stronger for seismic surface waves, for which D is around 5000 km.

 

14. For Rayleigh waves with wavelength $\lambda$ the characteristic penetration depth is about 0.4$\lambda$.

 

15. This dependence of velocity on wavelength is termed dispersion. Love waves are always dispersive, because they can only propagate in a velocity-layered medium.

 

16. This means that the S-waves generated by the incident P-wave cannot be SH-waves and must be SV-waves. Similarly, an incident SV-wave can generate reflected and refracted SV- and Pwaves, but, in the case of an incident SH-wave, only SH-waves can be transmitted and reflected.

 

 

17. The constant along each ray path, sin i/velocity, is often called p,  the ray parameter. For the case of the incident P-wave (from above fig), the angles for  the reflected and transmitted P- and SV-waves are therefore determined from

 

18. The coverage obtained by any profile is

$ coverage = \frac{number of receivers}{twice the shot spacing}$

 

19. The signal-to-noise ratio of the stacked traces is increased by a factor of √n over the signal-to-noise ratio of the n individual traces.

 

 

 

Reference:

1. Fundamentals of Geophysics William Lowrie 

2.An Introduction to Geophysical Exploration by Philip Kearey, Michel Brooks and Ian Hill.