Seismology – Lecture 2 Normal modes and surface waves Barbara Romanowicz Univ. of California,...

Seismology – Lecture 2Normal modes and surface waves

Barbara RomanowiczUniv. of California, Berkeley

CIDER Summer 2010 - KITP

From Stein and Wysession, 2003CIDER Summer 2010 - KITP

P S SS

Surface waves

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

From Stein and Wysession, 2003

Shallow earthquake

CIDER Summer 2010 - KITPone hour

Direction of propagation along the earth’s surface

L

Z

T

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


After Park et al, 2005After Park et al, 2005CIDER Summer 2010 - KITP

Free oscillations


The k’th free oscillation satisfies:

SNREI model; Solutions of the form

k = (l,m,n)

fLt

)(2

2

0 uu

0)( 20 kkk uuL

tik

keru ),,(u


Free Oscillations (Standing Waves)

€

−0ω2u = L(u)

In the frequency domain:

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 orderk = (l,m,n) “singlet” Degeneracy:(l,n): “multiplet” = 2l+1 “singlets ” with the same eigenfrequency nl

tik

keru ),,(u

€

uk (r,θ ,φ) =ˆ r nU l (r)Ylm (θ ,φ) +n Vl (r)∇1Yl

m (θ ,φ) −n W l (r)ˆ r ×∇1Ylm (θ ,φ)

€

k =n ω l

€

−l ≤ m ≤ l

€

Ylm (θ ,φ) = X l

m (θ )e imφ

Spheroidal modes : Vertical & Radial component

Toroidal modes : Transverse component

n T l

l : angular order, horizontal nodal planes

n : overtone number, vertical nodes

n=0n=1


Fundamentalmode

overtones

Spheroidal modes

n=0

nSl

Spatial shapes:

Depth sensitivity kernels of earth’s normal modes

53.9’

44.2’

20.9’ r=0.05m

0T22S1

0S30S2

0T4

1S2

0S5

0S0

0S43S1

2S2

1S3

0T3

Sumatra Andaman earthquake 12/26/04 M 9.3

• Rotation, ellipticity, 3D heterogeneity removes the degeneracy:

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

0S2 0S3

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

mode 0S3 7 singlets

Geographical sensitivity kernel K0()

0S45

0S3

ωo

Δω

frequency

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

Mode frequency shifts

SNREI->

€

ˆ ω k ≈1

2πδω(s)ds∫

δω(θ ,φ) = Mkk (r)δm0

a

∫ (r,θ ,φ)r2dr

S

R

P(θ,Φ)

Masters et al., 1982

Anomalous splitting of core sensitive modes

Data

Model

Mantle mode

Core mode

Seismograms by mode summation

Mode Completeness:

€

u = Re{ akk

∑ (t)uk (r,θ ,ϕ )e iω k t e−α k t}

Orthonormality (L is an adjoint operator):

€

0uk'* ⋅ ukdV = δ kk '

V

∫

fLt

)(2

2

0 uu

* Denotes complex conjugate

Depends on source excitation f

Normal mode summation – 1D

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


Spheroidal modes : Vertical & Radial component

Toroidal modes : Transverse component

n T l

l : angular order, horizontal nodal planes

n : overtone number, vertical nodes

n=0n=1


P S SS

Surface waves

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

€

u(t) = Re{ Akk

∑ e iω k t e−α k t}

Standing waves and travelling waves

Ak ---- linear combination of moment tensor elements and spherical harmonics Yl

m

When l is large (short wavelengths):

€

Ylm (θ ,ϕ ) ≈

1

π sinΔcos (l +

1

2)Δ −

π

4+

mπ

2

⎡ ⎣ ⎢

⎤ ⎦ ⎥e

imϕ

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

Jeans’ formula : ka = l + 1/2

€

Ylm (θ ,ϕ ) ≈

1

π sin Δcos kx −

π

4+

mπ

2

⎡ ⎣ ⎢

⎤ ⎦ ⎥e

imϕ

≈1

2π sinΔe

i(kx −π

4+

mπ

2)+ e

−i(kx −π

4+

mπ

2) ⎡

⎣ ⎢

⎤

⎦ ⎥

Hence:

€

u(t) = Re{ Akk

∑ e iω k t e−α k t}

⏐ → ⏐ ∝ e i(ω k t −kx )

⏐ → ⏐ e i(ω k t +kx )

Plane wavespropagatingin opposite directions

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

€

u(x, t) = S(ω)e i(ωt −kx )∫ dω

With k=k(ω) (dispersion)

€

k = k(ω)

Phase velocity:

€

C(ω) =ω

k

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

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

€

dΦ

dω= t −

dk

dωx = 0 For some frequency ωs

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

€

U(ω) =x

t=

dω

dk

Rayleigh phase velocity maps

Reference: G. Masters – CIDER 2008

Period = 50 s Period = 100 s

Group velocity maps

Period = 100 sPeriod = 50 s

Reference: G. Masters CIDER 2008

Importance of overtones for constraining structurein the transition zone

n=0: fundamental mode

n=1n=2

overtones

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

from Ritsema et al, 2004

Ritsema et al.,2004

FundamentalModeSurfacewaves

Overtone surface waves

Body waves

120 km

325 km

600 km

1100 km

1600 km

2100 km

2800 km

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

• 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


Montagner andNataf, 1986

RadialAnisotropy

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

• => = (Vsh/Vsv) 2

ln =2(ln Vsh – lnVsv)

Azimuthal anisotropy

• Horizontal axis of symmetry• Described in terms of , azimuth with

respect to the symmetry axis in the horizontal plane– 6 Terms in 2 (B,G,H) and 2 terms in 4 (E)

• Cos 2 -> Bc,Gc, Hc• Sin 2 -> Bs,Gs, Hs• Cos 4-> Ec• Sin 4 -> Es

– In general, long period waveforms can resolve Gc and Gs

Montagner and Anderson, 1989

• 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, ,

x

y

z

Axis of symmetry


Montagner, 2002

= (Vsh/Vsv)2

RadialAnisotropy

Isotropic velocity

Azimuthal anisotropy

Depth= 100 km

Montagner, 2002

Ekstrom and Dziewonski, 1997

Pacific ocean radial anisotropy: Vsh > Vsv

Gung et al., 2003

Marone and Romanowicz, 2007

Absolute Plate Motion

Continuous lines: % Fo (Mg) fromGriffin et al. 2004Grey: Fo%93black: Fo%92

Yuan and Romanowicz, in press

Layer 1 thickness

Mid-continental rift zone

Trans HudsonOrogen

“Finite frequency” effects


Structure sensitivity kernels: path average approximation (PAVA)versus Finite Frequency (“Born”) kernels

SR

M

SR

M

PAVA

2DPhasekernels

Panning et al., 2009

Waveform tomography

observed

synthetic

Waveform Tomography

Seismology – Lecture 2 Normal modes and surface waves Barbara Romanowicz Univ. of California,...

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Transcript of Seismology – Lecture 2 Normal modes and surface waves Barbara Romanowicz Univ. of California,...