CIDER Seismo2 Romanowicz b

8/13/2019 CIDER Seismo2 Romanowicz b

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SeismologyLecture 2

Normal modes and surface waves

Barbara Romanowicz

Univ. of California, Berkeley

CIDER Summer 2010 - KITP


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From Stein and Wysession, 2003CIDER Summer 2010 - KITP


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P SSS

Surface waves

Loma Prieta (CA) 1989 M 7 earthquake observed at KEV, Finland


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From Stein and Wysession, 2003

Shallow earthquake

CIDER Summer 2010 - KITPone hour


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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 onfrequency (group and phase velocity)

Most of the long period energy (>30 s) radiated fromearthquakes propagates as surface waves



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After Park et al, 2005CIDER Summer 2010 - KITP


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Free oscillations



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The kth free oscillation satisfies:

SNREI model; Solutions of the form

k = (l,m,n)

Lt

)(2

2

0 u

u

)( 20 kkk uuL

i

kru),,(u


Free Oscillations (Standing Waves)

0

2uL(u)

In the frequency domain:


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Free Oscillations

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

the kth 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 nl

i

kru),,(

u

uk(r,,)r

nU

l(r)Y

l

m(,)nV

l(r)

1Yl

m(,) nW

l(r)r

1Yl

m(,)

knl

lmlYlm

(,)Xlm

()eim


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Spheroidal modes : Vertical & Radial component

Toroidal modes : Transverse component

nTl

l: angular order, horizontal nodal planes

n: overtone number, vertical nodes

n=0

n=1


Fundamentalmode

overtones


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n=0

nSl

S i l h


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Spatial shapes:

D th iti it k l f th l d


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Depth sensitivity kernels of earths normal modes


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53.9

44.2

20.9

dr=0.05m

0T22S1

0S30S2

0T4

1S2

0S5

0

S0

0S43S1

2S2

1S3

0T3

Sumatra Andaman earthquake 12/26/04 M 9.3


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Rotation, ellipticity, 3D heterogeneityremoves the degeneracy:

-> For each (n, l) there are 2l+1 singletswith different frequencies

0S2 0S3


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0S2 0S3

2l+1=5 2l+1=7

mode S 7 singlets


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mode 0S3 7 singlets

G hi l iti it k l K ( )


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Geographical sensitivity kernel K0(,)

0S45

0S3

Mode frequency shifts


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o

frequency

Frequency shift depends only on the average structure along the vertical planecontaining the source and the receiver weighted by the depth sensitivity ofthe mode considered:

Mode frequency shifts

SNREI->

dk

1

2d(s)ds

d(,) Mkk

(r)dm0

a

(r,,)r2dr


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Anomalous splitting of core sensitive modes

Data

Model


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Mantle mode

Core mode

S i b d ti


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Seismograms by mode summation

Mode Completeness:

u Re akk

(t)uk(r,,)eiktekt

Orthonormality (L is an adjoint operator):

0uk'* u

kdVd

kk'

V

Lt

)(2

2

0 u

u

* Denotes complex conjugate

Depends on source excitation f


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Normal mode summation 1D

A : excitationw : eigen-frequencyQ : Quality factor ( attenuation )



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Spheroidal modes : Vertical & Radial component

Toroidal modes : Transverse component

nTl

l: angular order, horizontal nodalplanes

n: overtone number, vertical nodes

n=0

n=1



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P SSS

Surface waves

Loma Prieta (CA) 1989 M 7 earthquake observed at KEV, Finland


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u(t) Re Akk

eiktektStanding waves and travelling waves

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

spherical harmonics Ylm

When l is large (short wavelengths):

Yl

m(,) 1

sincos (l

1

2)

4m

2

eim

Replace x=a , where is angular distance and x linear distance along the earths

surface

Jeans formula : ka = l + 1/2


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Yl

m(,)

1

sin cos kx

4m

2

eim

12 sin

ei(kx

4m

2)

ei(kx

4m

2)

Hence:

u(t) Re Akk

eiktekt

ei(ktkx)

ei(ktkx)Plane waves

propagating

in opposite

directions


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-> Replace discrete sum over lby continuous

sum over frequency (Poissons formula):

u(x,t) S()ei(tkx)d

With k=k() (dispersion)k k()

Phase velocity:

C()

k

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

the phase is stationary:


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S is slowly varying with ; The main contribution to the

integral is when the phase is stationary:

dd

t dk

dx 0 For some frequency s

The energy associated with a particular group

centered on stravels with the group velocity:

U()x

td

dk


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Rayleigh phase velocity maps

Reference: G. MastersCIDER 2008

Period = 50 s Period = 100 s


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Group velocity maps

Period = 100 sPeriod = 50 s

Reference: G. Masters CIDER 2008


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Importance of overtones for constraining structurein the transition zone

n=0: fundamental mode

n=1n=2

overtones


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Overtones By including overtones, we cansee into the transition zone andthe top of the lower mantle.

from Ritsema et al, 2004


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Ritsema et al.,

2004

FundamentalMode

Surfacewaves

Overtonesurface waves

Body waves

120 km

325 km

600 km

1100 km

1600 km

2100

km

2800 km


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Anisotropy

In general elastic properties of a material vary withorientation

Anisotropy causes seismic waves to propagate atdifferent speeds in different directions

If they have different polarizations

f


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Types of anisotropy

General anisotropic model: 21independent elements of the elastictensor cijkl

Long period waveforms sensitive to asubset (13) of which only a small numbercan be resolved

Radial anisotropy Azimuthal anisotropy



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Montagner andNataf, 1986

Radial

Anisotropy


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Radial (polarization) Anisotropy

Love/Rayleigh wave discrepancy Vertical axis of symmetry A=Vph2,

C=Vpv2,

F,

L= Vsv2,

N= Vsh2(Love, 1911)

Long period S waveforms can only resolve L , N

=> x= (Vsh/Vsv) 2

dln x=2(dln VshdlnVsv)

A i th l i t


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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 Gcand Gs


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Montagner and Anderson, 1989

Vectorial tomography:


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Vectorial tomography: Combination radial/azimuthal (Montagner and

Nataf, 1986):

Radial anisotropy with arbitrary axisorientation (cf olivine crystals oriented inflow) orthotropic medium

L,N, Y, Q

x

y

z

Axis of symmetry



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Montagner, 2002

x= (Vsh/Vsv)2

Radial

Anisotropy

Isotropic

velocity

Azimuthal

anisotropy

Depth= 100 km


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Montagner, 2002

Ekstrom and Dziewonski, 1997

Pacific ocean radial anisotropy: Vsh > Vsv


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Gung et al., 2003


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Marone and Romanowicz, 2007

Absolute Plate Motion


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Continuous lines: % Fo (Mg)fromGriffin et al. 2004Grey: Fo%93

black: Fo%92 Yuan and Romanowicz, in press

Layer 1 thickness


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y

Mid-continental rift zone

Trans HudsonOrogen


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Finite frequency effects


Structure sensitivity kernels: path average approximation (PAVA)versus Finite Frequenc (B rn) kernels


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versus Finite Frequency ( Born) kernels

SR

M

SR

M

PAVA

2DPhase

kernels


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Panning et al., 2009


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Waveform tomography


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Waveform Tomography


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observed

synthetic

Wa form omography

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