CIDER

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CIDER08-Seismology Lectures

• Lecture 1 Equations and Waves (BR)
• Lecture 2 Numerical Simulations (JT)
• Lecture 3 Surface waves (GM)
• Lecture 4 Geophysical Inverse Problem (GM)
• Lecture 5 Receiver Functions (AL)
• Lecture 6 Array Techniques (AL)

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Equations and Waves

• Barbara Romanowicz
• U.C. Berkeley
• KITP

CIDER Summer08
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Seismology and the earths interior

• Snapshot of present structure of the earth
• Defined the 1D layering of the earth
• 3D variations of seismic parameters as proxies
  for lateral variations of temperature and
  composition
• Isotropic velocities
• Anisotropy
• Anelastic attenuation

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x (Vsh/Vsv)2
Montagner, 2002
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Motivation for seismic Q tomography
Faul and Jackson, 2005
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From Stein and Wysession, 2003
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From Stein and Wysession, 2003
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Rayleigh
SS
P
S
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Seismic wave equation

• We apply F ma
• Stresses
• Force on plane normal to x1
• -Total force due to stresses

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Seismic wave equation

• We apply F ma
• Stresses
• Force on plane normal to x1
• -Total force due to stresses

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Body forces F(2)
Mass is
ma F is then
i1,2,3
In general, body forces include source term and
gravity term
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Body forces F(2)
Mass is
ma F is then
i1,2,3
In general, body forces include source term and
gravity term
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Body forces F(2)
Mass is
ma F is then
i1,2,3
In general, body forces include source term and
gravity term
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Elasticity Linear stress-strain relationship

• cijkl is the elastic tensor
• 21 independent elements (symmetries)
• Isotropic material properties are the same in
  all directions
• there are only 2 independent elements, l and m
  (Lamé Parameters)

Hookes law Strains lt10-4
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Elasticity Linear stress-strain relationship

• cijkl is the elastic tensor
• 21 independent elements (symmetries)
• Isotropic material properties are the same in
  all directions
• there are only 2 independent elements, l and m
  (Lamé Parameters)

Hookes law Strains lt10-4
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Isotropic medium
Described by distribution of (r, l, m) or (r,
k, m) or (r ,a ,b)
Shear modulus m
Bulk modulus
P velocity
S velocity
l, m, k have units of stress
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Isotropic medium
Described by distribution of (r, l, m) or (r,
k, m) or (r ,a ,b)
Shear modulus m
Bulk modulus
P velocity
S velocity
Bulk sound velocity
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Isotropic medium
Described by distribution of (r, l, m) or (r,
k, m) or (r ,a ,b)
Shear modulus m
Bulk modulus
P velocity
S velocity
Bulk sound velocity
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Isotropic medium
Described by distribution of (r, l, m) or (r,
k, m) or (r ,a ,b)
Shear modulus m
Bulk modulus
P velocity
S velocity
Bulk sound velocity
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Homogeneous wave equation for Isotropic medium
(no body forces)
Substituting expression of stress as a function
of Lame parameters
Gradients of structure, non zero for
inhomogeneous medium
?If structure depends only on radius Consider
a stack of homogeneous layers linked through
reflection/transmission coefficients
homogeneous layer methods ?gradient terms vary
as 1/ ?, where ? is angular frequency -gt neglect
them at high frequency ray methods ,
infinite frequency approximation (f gt1 hz)
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Homogeneous wave equation for Isotropic medium
(no body forces)
Substituting expression of stress as a function
of Lame parameters
Gradients of structure, non zero for
inhomogeneous medium
?If structure depends only on radius Consider
a stack of homogeneous layers linked through
reflection/transmission coefficients
homogeneous layer methods ?gradient terms vary
as 1/ ?, where ? is angular frequency -gt neglect
them at high frequency ray methods ,
infinite frequency approximation (f gt1 hz)
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Homogeneous wave equation for Isotropic medium
(no body forces)
Substituting expression of stress as a function
of Lame parameters
Gradients of structure, non zero for
inhomogeneous medium
?If structure depends only on radius Consider
a stack of homogeneous layers linked through
reflection/transmission coefficients
homogeneous layer methods ?gradient terms vary
as 1/ ?, where ? is angular frequency -gt neglect
them at high frequency ray methods ,
infinite frequency approximation (f gt1 hz)
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Homogeneous wave equation for Isotropic medium
(no body forces)
Substituting expression of stress as a function
of Lame parameters
Gradients of structure, non zero for
inhomogeneous medium
?If structure depends only on radius Consider
a stack of homogeneous layers linked through
reflection/transmission coefficients
homogeneous layer methods ?gradient terms vary
as 1/ ?, where ? is angular frequency -gt neglect
them at high frequency ray methods ,
infinite frequency approximation (f gt1 hz)
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• In a homogeneous medium (or high frequency
  approximation)
• Taking divergence and curl, the equation of
  motion separates in two parts

S waves
a2
P waves
Note
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• In a homogeneous medium(or high frequency
  approximation)
• Taking divergence and curl, the equation of
  motion separates in two parts

S waves
a2
P waves
Note
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• In a homogeneous medium(or high frequency
  approximation)
• Taking divergence and curl, the equation of
  motion separates in two parts

S waves
a2
P waves
Note
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Solutions are waves u(x,t) f(xvt) where v
velocity In Cartesian coordinates, plane
waves
k is the wave number
(Figures from Stein and Wysession, 2003)
In spherical coordinates
Amplitudes decay as 1/r geometrical spreading
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Solutions are waves u(x,t) f(xvt) where v
velocity In Cartesian coordinates, plane
waves
k is the wave number
(Figures from Stein and Wysession, 2003)
In spherical coordinates
Amplitudes decay as 1/r geometrical spreading
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Solutions are waves u(x,t) f(xvt) where v
velocity In Cartesian coordinates, plane
waves
k is the wave number
(Figures from Stein and Wysession, 2003)
In Spherical coordinates
Amplitudes decay as 1/r geometrical spreading
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Polarisation of P and S waves
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From Stein and Wysession, 2003
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Example Particle motion plots for SKSSKKS and
Sdiff
From Stein and Wysession, 2003
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Ray theory

• Simple and fast
• Used extensively in earthquake location, focal
  mechanisms, inversion for structure in crust and
  mantle
• Shortcomings
• High frequency approximation fails at long
  periods
• Does not predict non geometrical effects i.e.
  diffracted waves, head waves
• Limitations in predicting effects of
  heterogeneity on waveforms

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?x
p ray parameter apparent horizontal slowness
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Snells law
At the interface between two media
v1 lt v2
Ray angle at the interface must change to
preserve the timing of the wavefronts across the
interface
p sin ?1/v1 sin ?2/ v2
?Ray parameter is a constant of the ray
?Ray bottoms for a critical incidence angle
no transmitted wave for larger incidence angles
(post-critical or total reflection)
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In general, some of the energy is transmitted,
some reflected, and, in the P-SV case, some
converted
SH case
P-SV case
Angles of reflection/transmission depend only on
velocities Amplitudes depend on impedance (? v)
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Ray paths in spherical geometry, when velocity
increases with depth
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Steep gradients, such as upper-mantle
discontinuities, create triplications
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Low velocity layers create shadow zones
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Van der Hilst et al., 1998
P wave tomography
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P travel time (tranmission) tomography
Courtesy of D. Vasco
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Travel time kernels
D 60o T 20 s
P waves
S waves
Montelli et al., 2006
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Montelli et al., 2004
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Karason and van der Hilst, 2000
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Astiz, Earle and Shearer, 1996
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Predictions from IASP91 model
Shearer, 1996
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Shallow earthquake
From Stein and Wysession, 2003
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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 (gt30 s) radiated
  from earthquakes propagates as surface waves

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After Park et al, 2005
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Sumatra 12/26/04
(0.8 to 2.2 mHz)
CAN
1S5
3S2
1S4
UNM
0S9
0S12
0S8
0S7
0S10
0S11
0S0
0S5
1S3 /3S1
0S6
Frequency
Fourier spectrum200 hours starting 10 hours
before origin time
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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)
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The kth free oscillation
satisfies
SNREI model Solutions of the form
k (l,m,n)
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Free Oscillations (standing waves)
gt Frequency domain
The kth free oscillation
satisfies
SNREI model Solutions of the form
k (l,m,n)
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Free Oscillations (standing waves)
The kth free oscillation
satisfies
SNREI model Solutions of the form
k (l,m,n)
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Free Oscillations (standing waves)
The kth free oscillation
satisfies
SNREI model Solutions of the form
k (l,m,n)
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oS2
oSo
oS3
After Park et al, 2005
(53.9 min) (25.7 min) (20.5
min)
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Seismograms by mode summation
? Mode Completeness
? Orthonormality (L is an adjoint operator)
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Seismograms by mode summation
? Mode Completeness
? Orthonormality (L is an adjoint operator)
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Seismograms by mode summation
? Mode Completeness
? Orthonormality (L is an adjoint operator)
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The solution for the displacement then has the
form
Where Fk is the distribution of body forces, and
H(t) is the Heaviside function, and we
assume that the motion is zero before t0.
In terms of moment tensor
Where M is moment Tensor, e strain tensor At the
source
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The solution for the displacement then has the
form
Where Fk is the distribution of body forces, and
H(t) is the Heaviside function, and we
assume that the motion is zero before t0.
In terms of moment tensor
Where M is moment Tensor, e strain tensor At the
source
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The solution for the displacement then has the
form
r0source location
Where Fk is the distribution of body forces, and
H(t) is the Heaviside function, and we
assume that the motion is zero before t0.
In terms of moment tensor
Where M is Moment Rate Tensor, e strain
tensor eval. at the source
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Spheroidal modes Vertical Radial component
Toroidal modes Transverse component
n0
n1
n T l
l angular order, horizontal nodal planes n
overtone number, vertical nodes
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Normal mode summation 1D

• A excitation
• w eigen-frequency
• Q Quality factor ( attenuation )

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From ray theory to finite frequency effects

• Goal is to use as much information as
• possible from seismograms

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Standard tomographic ingredients

• Body waves
• Travel times of well separated phases
• Ray theory
• Surface waves
• Group/phase velocities or waveforms
• Path average approximation (PAVA)

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• c is the phase velocity velocity at which the
  peaks and troughs travel
• Energy travels with group velocity (along the
  actual ray paths)

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Towards Waveform Tomography
observed
synthetic
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SS
Sdiff
PAVA
NACT
Ray theory versus two approximations to first
order perturbation theory In the 2D vertical
plane
Li and Romanowicz, 1995
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Yoshizawa and Kennett, 2004
Looking from the top (horizontal plane)
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After Montelli et al., 2006
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Fundamental Mode Surface waves
Body waves
Overtone surface waves
Ritsema et al., 2004
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Towards Waveform Tomography
observed
synthetic
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Preliminary Level 4 radially anisotropic model
courtesy of Ved Lekic
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Seismic Wave attenuation

• Amplitudes of seismic waves are affected by
• Geometrical spreading
• Scattering/focusing (total energy in the
  wavefield conserved)
• Intrinsic attenuation (energy loss due to
  friction on anelastic processes).

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Intrinsic attenuation is described by the
quality factor Q
E peak strain energy ?E energy loss, per cycle
In ray theory
hence
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Intrinsic attenuation is described by the
quality factor Q
E peak strain energy ?E energy loss per cycle
In ray theory
hence
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Intrinsic attenuation is described by the
quality factor Q
E peak strain energy ?E energy loss per cycle
In ray theory
hence
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Examples of focusing
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• Dispersion due to attenuation
• Causality
• Physical mechanisms for attenuation
• In a frequency band where Q is constant

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Romanowicz and Gung, 2002
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Anisotropy

• In general elastic properties of a material vary
  with orientation
• Anisotropy causes wave splitting

From E. Garneros website
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SKS Splitting Observations
Interpreted in terms of a model of a layer of
anisotropy with a horizontal symmetry axis
Dt time shift between fast and slow
waves Yo Direction of fast velocity
axis
Montagner et al. (2000) show how to relate
surface wave anisotropy and shear wave splitting
Huang et al., 2000
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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, to
  first order (13) of which only a small number can
  be resolved
• Radial anisotropy
• Azimuthal anisotropy

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Radial Anisotropy

• Love/Rayleigh wave discrepancy
• Vertical axis of symmetry
• A,C,F,L,N (Love, 1911)
• Long period S waveforms can only resolve
• L r Vsv2
• N r Vsh2
• gt x (Vsh/Vsv) 2
• dln x 2(dln Vsh dlnVsv)

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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 -gt Bc,Gc, Hc
• Sin 2y -gt Bs,Gs, Hs
• Cos 4y-gt Ec
• Sin 4y -gt Es
• long period waveforms can resolve Gc and Gs

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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, Y, Q

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• Azimuthal anisotropy

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Study of Structure near the CMB
From Ed Garneros website
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Sharp boundary of the African Superplume
Ni et al., 2002
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Indian Ocean Paths - Sdiffracted
Toh et al.,2005
2sec, 5sec, 18 sec (corner
frequencies)
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Toh et al., EPSL, 2005
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Thank You