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Seismic waves
• Wave propagation• Hooke’s law• Newton’s law• wave equation
• Wavefronts and Rays• Interfaces • Reflection and Transmission coefficients
Seismic Waves
body wavesP-waves (longitudinal, compressional)
S-waves (shear, transverse)SV-waveSH-wave
Body waves:
Different kind of waves
• Transversal waves (S-waves)
• Longitudinal waves (P-waves)
1 3 542 2 41 3 5
Examples of different waves
Elektromagnetic spectrum
Frequency
AM, FM, Georadar, Visible, X-ray
Acoustic spectrumEarthquake, audible + seismic106100 1019 Hz
Surface waves
Rayleigh-waves
Love-waves
Newton’s law
P(z) P(z+z)Uz
P is the acoustic pressure
Uz is the displacement
Newton’s law
P(z) P(z+z)
P(z+z) - P(z) =
Uz
d2
dt2Uz- z
is the massdensity
Newton’s law
P(z) P(z+z)Uz
- Uz2
t2P =
z
Hooke’s law
P
- z PUz(z+z) - Uz (z) =
Uz(z) Uz(z+z)
is the compressibility
Hooke’s law
PUz(z) Uz(z+z)
Uz= - Pz
Acoustic Wave equation
2
z2 P
2
t2 P
1c2
= -w(t) (z)
w(t) = q(t) (sourcesignal)2
t2
c = ()-1/2 (wavespeed)
Propagation of seismic waves
(Roth et al., 1998)
Object detection using WAVES:
Object detection using WAVES
BSource
OReceiver
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
Geometrical Wave propagation
Source ReceiverSource
Rays are perpendicular to the wavefronts,
1 2
v1
v2
Angle of incidence = angle of reflection
1 = 2
Interface: reflection
1
v1
v2
1sin2sin------- -----
v1v2-----=
Interface: Refraction
v1v2 >
2 v1v2 <
2
1
v1
v21sin90sin
---------- 1sinv1v2-----=
2 90=
Special case: critical angle
=
2
1
vp1,vs1
1sin
1sin-------- ---
vp1
vs1
--------=
1
21sin
2sin------- ---
vp1
vs2
--------=
vp2,vs2
Interface: Conversion from P wave to S wave
1sin
vp1
------ = --1sin
vs1
------ = --2sin
vp2
-------- = -2sin
vs2
-------------- = nsin
vsn
--------- = p = constant
p = Slowness
1
2 2
33
1
Snell’s lawvp1
vs1
vp2vs2
vp3vs3
Propagation of seismic waves
(Roth et al., 1998)
E ER
ET
v11
v22
E = ER + ET
R + T = 1
R = Reflection coefficient
T = Transmission coefficient
E = Energy
Transmission- and Reflection coefficients
R
T
v11
v22
Reflection coefficient
Transmission coefficient
Rv2 2 v1 1–
v2 2 v1 1+--------------- = ------------
Z2 Z1–
Z2 Z1+--------------------=
T2v1 1
v22 v1 1+---------------- = -----------
2Z 1
Z2 Z1+--------------------=
with Z = v = acoustic Impedance
Zoeppritz’s equations at normal incidence