### Seismology – Lecture 2 Normal modes and surface waves

```Seismology – Lecture 2
Normal modes and surface waves
Barbara Romanowicz
Univ. of California, Berkeley
CIDER Summer 2010 - KITP
From Stein and Wysession, 2003
CIDER Summer 2010 - KITP
Loma Prieta (CA) 1989 M 7 earthquake observed at KEV, Finland
Surface waves
P
S
SS
Shallow earthquake
From Stein and Wysession, 2003
one hour
CIDER Summer 2010 - KITP
Direction of propagation along the earth’s surface
T
Z
L
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
CIDER Summer 2010 - KITP
After Park et al, 2005
Free oscillations
CIDER Summer 2010 - KITP
CIDER Summer 2010 - KITP
Free Oscillations (Standing Waves)
u


L
(u
)f
0
2

t
2
The k’th free oscillation


k
u

u
(
r
,
,
)
e
satisfies
:
k
it
In the frequency domain:


L
(
u
)
 u

0
k
2
0
kk
   u  L (u)
2
SNREI model; Solutions
of the form
0
k = (l,m,n)
CIDER Summer 2010 - KITP
Free Oscillations
In a Spherical, Non-Rotating, Elastic and Isotropic Earth model,
the k’th free oscillation can be described as:



u

u
(
r
,,)
e
k
ik
t
 k n  l
m
m
m
u k (r,  ,  )  rˆ n U l (r)Y l ( ,  )  n V l (r) 1 Y l ( ,  )  n W l (r) rˆ   1 Y l ( ,  )
Y l ( ,  )  X l ( )e
m
m
im 
l  m  l
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 nl
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
n=0
nSl
Spheroidal modes
Spatial shapes:
Depth sensitivity kernels of earth’s normal modes
Sumatra Andaman earthquake 12/26/04 M 9.3
53.9’
0S 2
0S 3
44.2’
0S 0
20.9’
dr=0.05m
3S 1
0S 4
2S 2
0S 5
T 2S1
0
2
1S 2
0T 3
0T 4
1S3
• Rotation, ellipticity, 3D heterogeneity
removes the degeneracy:
– -> For each (n, l) there are 2l+1 singlets
with different frequencies
0S2
2l+1=5
0S3
2l+1=7
mode 0S3
7 singlets
Geographical sensitivity kernel K0(,)
0
S3
0S45
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:
ˆk 
d 
1
2
 d ( s) ds
a
d ( ,  ) 
M
0
( r)d m ( r,  ,  ) r dr
2
kk
P(θ,Φ)
S
R
Masters et al., 1982
Data
Model
Anomalous splitting of core sensitive modes
Mantle mode
Core mode
Seismograms by mode summation
u


L
(u
)f
0
2

t
2
 Mode
u  Re 
Completeness:
a
k
(t )u k (r,  ,  )e
Depends on source excitation f
i k t
e
 k t

k
 Orthonormality (L is an adjoint operator):

 0 u k '  u k dV  d kk '
*
V
* Denotes complex conjugate
Normal mode summation – 1D
A : excitation
w : eigen-frequency
Q : Quality factor ( attenuation )
CIDER Summer 2010 - KITP
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
CIDER Summer 2010 - KITP
Loma Prieta (CA) 1989 M 7 earthquake observed at KEV, Finland
Surface waves
P
S
SS
Standing waves and travelling waves
u(t)  Re 
Ae
k
i k t
e
 k t

k
Ak ---- linear combination of moment tensor elements and
spherical harmonics Ylm
When l is large (short wavelengths):


1
 m
Y l ( ,  ) 
cos ( l  )   

2
4
2
 sin 
m
1
 im 
e

Replace x=a Δ, where Δ is angular distance and x linear distance along the earth’s
surface
Jeans’ formula : ka = l + 1/2

Y l ( ,  ) 


 m
cos kx  

4
2
sin 
1
m

 im 
e

 m  
 i( kx    m  )
 i( kx  
)
4
2
4
2
e
e

sin  

1
2
Hence:
u(t)  Re 

Ake
i k t
e
 k t

k

  e


i( k t  kx )
e
i( k t  kx )
Plane waves
propagating
in opposite
directions
-> Replace discrete sum over l by continuous
sum over frequency (Poisson’s formula):
u( x, t ) 
With
 S ( )e
i( t  kx )
d
k k=k(ω)
k ( ) (dispersion)
Phase velocity:

C ( ) 

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:
d
d
t
dk
d
x 0
For some frequency ωs
The energy associated with a particular group
centered on ωs travels with the group velocity:
U ( ) 
x
t

d
dk
Rayleigh phase velocity maps
Period = 50 s
Reference: G. Masters – CIDER 2008
Period = 100 s
Group velocity maps
Period = 50 s
Reference: G. Masters CIDER 2008
Period = 100 s
Importance of overtones for constraining structure
in the transition zone
overtones
n=2
n=1
n=0: fundamental mode
Overtones
By including overtones, we can
see into the transition zone and
the top of the lower mantle.
from Ritsema et al, 2004
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
• 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
– Azimuthal anisotropy
CIDER Summer 2010 - KITP
Anisotropy
Montagner and
Nataf, 1986
• “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 Vsh – dlnVsv)
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
Montagner and Anderson, 1989
• Vectorial tomography:
Nataf, 1986):
– Radial anisotropy with arbitrary axis
orientation (cf olivine crystals oriented in
“flow”) – orthotropic medium
– L,N, Y, Q
Y
y
x
Q
Axis of symmetry
z
CIDER Summer 2010 - KITP
x = (Vsh/Vsv)2
Isotropic
velocity
Anisotropy
Azimuthal
anisotropy
Montagner, 2002
Depth= 100 km
Pacific ocean radial anisotropy: Vsh > Vsv
Ekstrom and Dziewonski, 1997
Montagner, 2002
Gung et al., 2003
Absolute Plate Motion
Marone and Romanowicz, 2007
Continuous lines: % Fo (Mg)
from
Griffin et al. 2004
Grey: Fo%93
black: Fo%92
Yuan and Romanowicz, in press
Layer 1 thickness
Trans Hudson
Orogen
Mid-continental rift zone
“Finite frequency” effects
CIDER Summer 2010 - KITP
Structure sensitivity kernels: path average approximation (PAVA)
versus Finite Frequency (“Born”) kernels
2D
Phase
kernels
PAVA
M
M
S
S
R
R
Panning et al., 2009
Waveform tomography
Waveform Tomography
observed
synthetic
```