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Seismology – Lecture 2 Normal modes and surface waves. Barbara Romanowicz Univ. of California, Berkeley. From Stein and Wysession, 2003. Loma Prieta (CA) 1989 M 7 earthquake observed at KEV, Finland. Surface waves. SS. S. P. Shallow earthquake. From Stein and Wysession, 2003. one hour.

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seismology lecture 2 normal modes and surface waves

Seismology – Lecture 2Normal modes and surface waves

Barbara Romanowicz

Univ. of California, Berkeley

CIDER Summer 2010 - KITP

slide2

From Stein and Wysession, 2003

CIDER Summer 2010 - KITP

slide4

Shallow earthquake

CIDER Summer 2010 - KITP

From Stein and Wysession, 2003

one hour

surface waves
Surface waves
  • Arise from interaction of body waves with free surface.
  • Energy confined near the surface
  • Rayleigh waves: interference between P and SV waves – exist because of free surface
  • Love waves: interference of multiple S reflections. Require increase of velocity with depth
  • Surface waves are dispersive: velocity depends on frequency (group and phase velocity)
  • Most of the long period energy (>30 s) radiated from earthquakes propagates as surface waves

CIDER Summer 2010 - KITP

slide8

After Park et al, 2005

CIDER Summer 2010 - KITP

free oscillations
Free oscillations

CIDER Summer 2010 - KITP

slide11

Free Oscillations (Standing Waves)

The k’th free oscillation satisfies:

In the frequency domain:

SNREI model; Solutions of the form

k = (l,m,n)

CIDER Summer 2010 - KITP

slide12

Free Oscillations

In a Spherical, Non-Rotating, Elastic and Isotropic Earth model,

the k’th free oscillation can be described as:

l = angular order; m = azimuthal order; n = radial order

k = (l,m,n) “singlet”

Degeneracy:

(l,n): “multiplet” = 2l+1 “singlets ” with the same eigenfrequency nwl

slide13

Spheroidal modes : Vertical & Radial component

Toroidal modes : Transverse component

overtones

Fundamental

mode

n=0

n=1

nTl

l : angular order, horizontal nodal planes

n : overtone number, vertical nodes

CIDER Summer 2010 - KITP

slide14

n=0

nSl

Spheroidal modes

slide17

Sumatra Andaman earthquake 12/26/04 M 9.3

0S2

0S3

0S0

20.9’ dr=0.05m

53.9’

3S1 2S2

1S3

0S4

44.2’

0S5

1S2

2S1

0T2

0T3

0T4

slide18

Rotation, ellipticity, 3D heterogeneity removes the degeneracy:

    • -> For each (n, l) there are 2l+1 singlets with different frequencies
slide19

0S2

0S3

2l+1=5

2l+1=7

slide22

Mode frequency shifts

Δω

SNREI->

ωo

frequency

Frequency shift depends only on the average structure along the vertical plane

containing the source and the receiver weighted by the depth sensitivity of

the mode considered:

slide23

P(θ,Φ)

S

R

Masters et al., 1982

slide24

Data

Model

Anomalous splitting of core sensitive modes

slide25

Mantle mode

Core mode

slide26

Seismograms by mode summation

 Mode Completeness:

Depends on source excitation f

 Orthonormality (L is an adjoint operator):

* Denotes complex conjugate

normal mode summation 1d
Normal mode summation – 1D

A : excitation

w : eigen-frequency

Q : Quality factor ( attenuation )

CIDER Summer 2010 - KITP

slide28

Spheroidal modes : Vertical & Radial component

Toroidal modes : Transverse component

n=0

n=1

nTl

l: angular order, horizontal nodal planes

n : overtone number, vertical nodes

CIDER Summer 2010 - KITP

slide31

Standing waves and travelling waves

Ak ---- linear combination of moment tensor elements and

spherical harmonics Ylm

When l is large (short wavelengths):

Replace x=a Δ, where Δ is angular distance and x linear distance along the earth’s

surface

Jeans’ formula : ka = l + 1/2

slide32

Hence:

Plane waves

propagating

in opposite

directions

slide33

-> Replace discrete sum over l by continuous sum over frequency (Poisson’s formula):

With k=k(ω) (dispersion)

Phase velocity:

S is slowly varying with ω ; The main contribution to the integral is when

the phase is stationary:

slide34

S is slowly varying with ω ; The main contribution to the integral is when the phase is stationary:

For some frequency ωs

The energy associated with a particular group centered on ωs travels with the group velocity:

slide36

Rayleigh phase velocity maps

Period = 50 s

Period = 100 s

Reference: G. Masters – CIDER 2008

slide37

Group velocity maps

Period = 50 s

Period = 100 s

Reference: G. Masters CIDER 2008

slide39

Importance of overtones for constraining structure

in the transition zone

overtones

n=1

n=2

n=0: fundamental mode

overtones
Overtones

By including overtones, we can see into the transition zone and the top of the lower mantle.

from Ritsema et al, 2004

slide41

120 km

Fundamental

Mode

Surface

waves

325 km

600 km

Body waves

1100 km

Overtone

surface waves

1600 km

2100 km

2800 km

Ritsema et al.,

2004

anisotropy
Anisotropy
  • In general elastic properties of a material vary with orientation
  • Anisotropy causes seismic waves to propagate at different speeds
    • in different directions
    • If they have different polarizations
types of anisotropy
Types of anisotropy
  • General anisotropic model: 21 independent elements of the elastic tensor cijkl
  • Long period waveforms sensitive to a subset (13) of which only a small number can be resolved
    • Radial anisotropy
    • Azimuthal anisotropy

CIDER Summer 2010 - KITP

slide44

Radial

Anisotropy

Montagner and

Nataf, 1986

radial polarization anisotropy
Radial (polarization) Anisotropy
  • “Love/Rayleigh wave discrepancy”
    • Vertical axis of symmetry
      • A=r Vph2,
      • C=r Vpv2,
      • F,
      • L= r Vsv2,
      • N= r Vsh2 (Love, 1911)
    • Long period S waveforms can only resolve
      • L , N
      • => x = (Vsh/Vsv) 2
      • dln x =2(dln Vsh – dlnVsv)
azimuthal anisotropy
Azimuthal anisotropy
  • Horizontal axis of symmetry
  • Described in terms of y, azimuth with respect to the symmetry axis in the horizontal plane
    • 6 Terms in 2y (B,G,H) and 2 terms in 4y (E)
      • Cos 2y -> Bc,Gc, Hc
      • Sin 2y -> Bs,Gs, Hs
      • Cos 4y-> Ec
      • Sin 4y -> Es
    • In general, long period waveforms can resolve Gc and Gs
slide48

x

y

Axis of symmetry

z

  • Vectorial tomography:
    • Combination radial/azimuthal (Montagner and Nataf, 1986):
    • Radial anisotropy with arbitrary axis orientation (cf olivine crystals oriented in “flow”) – orthotropic medium
    • L,N, Y, Q

CIDER Summer 2010 - KITP

slide49

x = (Vsh/Vsv)2

Isotropic

velocity

Radial

Anisotropy

Azimuthal

anisotropy

Montagner, 2002

slide50

Depth= 100 km

Pacific ocean radial anisotropy: Vsh > Vsv

Ekstrom and Dziewonski, 1997

Montagner, 2002

slide52

Absolute Plate Motion

Marone and Romanowicz, 2007

slide53

Continuous lines: % Fo (Mg) from

Griffin et al. 2004

Grey: Fo%93

black: Fo%92

Yuan and Romanowicz, in press

slide54

Layer 1 thickness

Trans Hudson

Orogen

Mid-continental rift zone

finite frequency effects
“Finite frequency” effects

CIDER Summer 2010 - KITP

slide56

Structure sensitivity kernels: path average approximation (PAVA)

versus Finite Frequency (“Born”) kernels

2D

Phase

kernels

PAVA

M

M

S

R

S

R

slide60

Waveform Tomography

observed

synthetic