Seismic waves

1
Seismic waves

• Wave propagation
• Hookes law
• Newtons law
• ? wave equation
• Wavefronts and Rays
• Interfaces
• Reflection and Transmission coefficients

2
Seismic Waves
body waves
P-waves (longitudinal, compressional)
S-waves (shear, transverse)
S
-wave
V
S
-wave
H
3
Body waves
4
Different kind of waves

• Longitudinal waves (P-waves)

• Transversal waves
• (S-waves)

5
Examples of different waves
6
Surface waves
Rayleigh-waves
Love-waves
7
Newtons law
P is the acoustic pressure Uz is the displacement
8
Newtons law
P(z?z) - P(z)
? is the massdensity
9
Newtons law
10
Hookes law
Uz(z?z) - Uz (z)
? is the compressibility
11
Hookes law
P
12
Acoustic Wave equation
13
Propagation of seismic waves
(Roth et al., 1998)
14
Object detection using WAVES
15
Object detection using WAVES
16
Wavefronts versus Rays

• Wavefronts indicate the boundary of the material
  which already moves and the material which is
  still undisturbed.
• Rays are plotted perpendicular with respect to
  the wavefronts and describe the dominant
  propagation
• of the seismic energy between two locations

17
Geometrical Wave propagation
Source
Receiver
Source
Rays are perpendicular to the wavefronts,
18
Interface reflection
a
a
1
2
v
Angle of incidence
angle of reflection
1
v
2
a
a

1
2
19
Interface Refraction
a
1
v
a
sin
v
1
1
1
-----

------- -----
a
sin
v
v
2
2
2
20
Special case critical angle
a
a
90


1
2
v
1
a
sin
v
v
1
1

----------
a
sin
-----

2
1
90

sin
v
2
a
2
21
Interface Conversion from P wave to S wave
b
1
v
a
sin
p
a
1
-------- ---
1
--------

1
b
sin
v
1
s
1
v
v
p1,
s1
v
v
p2,
s2
v
a
sin
b
p
1
2
------- ---
1
--------

b
sin
v
s
2
2
22
Snells law
v
v
p
s
1
1
v
v
p
s
2
2
a
a
2
2
v
v
a
p
s
a
3
3
3
3
a
sin
b
sin
a
sin
b
sin
b
sin
1
1
2
2
n
------ --
------ --
-------- -
--------------
--------- p constant
v
v
v
v
v
p
s
p
s
s
1
1
2
2
n
p Slowness
23
Propagation of seismic waves
(Roth et al., 1998)
24
Transmission- and Reflection coefficients
E
E
R
E E
E
R
T
, r
v
1
1
, r
v
2
2
R T 1
E
T
E Energy
R Reflection coefficient
T Transmission coefficient
25
Zoeppritzs equations at normal incidence
Reflection coefficient
R
v
r
v
r
Z
Z


2
2
1
1
2
1
--------------- ------------
--------------------

R
, r
v
v
r
v
r

Z
Z

1
1
2
2
1
1
2
1
, r
v
2
2
Transmission coefficient
T
2v
r
2Z
1
1
1
---------------- -----------
--------------------

T
v
r
v
r

Z
Z

2
2
1
1
2
1
r
with Z v
acoustic Impedance