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