seismology – lecture 2 normal modes and surface waves barbara romanowicz univ. of california,...
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![Page 1: Seismology – Lecture 2 Normal modes and surface waves Barbara Romanowicz Univ. of California, Berkeley CIDER Summer 2010 - KITP](https://reader035.vdocuments.mx/reader035/viewer/2022062309/56649e795503460f94b788e9/html5/thumbnails/1.jpg)
Seismology – Lecture 2Normal modes and surface waves
Barbara RomanowiczUniv. of California, Berkeley
CIDER Summer 2010 - KITP
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From Stein and Wysession, 2003CIDER Summer 2010 - KITP
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P S SS
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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Direction of propagation along the earth’s surface
L
Z
T
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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 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
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After Park et al, 2005After Park et al, 2005CIDER Summer 2010 - KITP
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Free oscillations
CIDER Summer 2010 - KITP
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CIDER Summer 2010 - KITP
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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
CIDER Summer 2010 - KITP
Free Oscillations (Standing Waves)
€
−0ω2u = L(u)
In the frequency domain:
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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
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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
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Fundamentalmode
overtones
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Spheroidal modes
n=0
nSl
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Spatial shapes:
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Depth sensitivity kernels of earth’s normal modes
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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
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• Rotation, ellipticity, 3D heterogeneity removes the degeneracy:
– -> For each (n, l) there are 2l+1 singlets with different frequencies
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0S2 0S3
2l+1=5 2l+1=7
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mode 0S3 7 singlets
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Geographical sensitivity kernel K0()
0S45
0S3
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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
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S
R
P(θ,Φ)
Masters et al., 1982
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Anomalous splitting of core sensitive modes
Data
Model
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Mantle mode
Core mode
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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
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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
n T l
l : angular order, horizontal nodal planes
n : overtone number, vertical nodes
n=0n=1
CIDER Summer 2010 - KITP
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CIDER Summer 2010 - KITP
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P S SS
Surface waves
Loma Prieta (CA) 1989 M 7 earthquake observed at KEV, Finland
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€
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
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€
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
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-> 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:
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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
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Rayleigh phase velocity maps
Reference: G. Masters – CIDER 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 can see into the transition zone and the top of the lower mantle.
from Ritsema et al, 2004
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Ritsema et al.,2004
FundamentalModeSurfacewaves
Overtone surface 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 with orientation
• Anisotropy causes seismic waves to propagate at different speeds– in different directions– If they have different polarizations
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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
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Montagner andNataf, 1986
RadialAnisotropy
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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
• => = (Vsh/Vsv) 2
ln =2(ln Vsh – lnVsv)
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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
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Montagner and Anderson, 1989
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• 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
CIDER Summer 2010 - KITP
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Montagner, 2002
= (Vsh/Vsv)2
RadialAnisotropy
Isotropic velocity
Azimuthal anisotropy
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Depth= 100 km
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%93black: Fo%92
Yuan and Romanowicz, in press
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Layer 1 thickness
Mid-continental rift zone
Trans HudsonOrogen
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“Finite frequency” effects
CIDER Summer 2010 - KITP
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Structure sensitivity kernels: path average approximation (PAVA)versus Finite Frequency (“Born”) kernels
SR
M
SR
M
PAVA
2DPhasekernels
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Panning et al., 2009
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Waveform tomography
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observed
synthetic
Waveform Tomography