Kinematic Representation TheoremKinematic Representation Theorem

€

Γin(r x ,t −τ ,

r ξ ,0) = cijpqnj

∂

∂ξq

Gnp(r x ,t −τ ,

r ξ ,0)

€

un(r x ,t) = dτ

−∞

+∞

∫ Δui(r ξ ,τ )Γin(

r x ,t −τ ,

r ξ ,0){ }

Σ

∫∫ dΣ = dτ−∞

+∞

∫ Δui ∗ΓindΣΣ

∫∫

KINEMATIC TRACTIONS

€

un (r x ,ω) = Δui(

r ξ ,ω) ⋅Γin (

r x ,ω,

r ξ ,0){ }

Σ

∫∫ dΣ

Time domain representation

Frequency domain representation

€

Gij(r x ,t) =

1

4πρ(3γ iγ j −δ ij )

1

r 3τ

r α

r β

∫ δ (t −τ )dτ

+1

4πρα 2γ iγ j

1

rδ (t −

r

α)−

1

4πρβ 2γ iγ j −δ ij( )

1

rδ ( t −

r

β)

Green FunctionGreen Function

Far Field caseFar Field case• Near Field vs Far Field

f = 1 Hz, r = 6 km f = 0.01 Hz, r =300 km, c==3.5 Km/s c==3.0 Km/s

In the Far Field we have r >>

If

Thus we have the far field – far source (point source)

IfThus we have the far field – near source (extended source)

€

L2 <<1

2λro

€

L2 ≥1

2λro

€

FFA

NFA∝

ωr

c

€

FFA

FNA≈10

€

FFA

FNA≈ 6,3

Numerical complete solution for elastodynamic Green Function

Numerical complete solution for elastodynamic Green Function

• Equation of motion

m = 0,±1,±2,±3,etc….Jm is the Bessel of order m

We develop the solution in a cylindrical coordinate system (r , z), in which z is the vertical axis.The elastic parameters vary only on the vertical axis z.

The dependence on r and results only superficial harmonics, which are orthogonal vectors

€

ρ ru = (λ + μ)∇(∇ ⋅

r u ) + μ∇ 2u +∇λ /(∇ ⋅

r u ) + 2 ∇μ( ) ⋅

r e

€

ℜkm,Sk

m ,Tkm

ℜ km (r,φ) = Yk

m (r,φ)ˆ e z

Skm (r,φ) =

1

k∂rYk

m (r,φ) ˆ e r +1

kr∂φYk

m (r,φ)ˆ e φ

Tkm (r,φ) =

1

kr∂φYk

m (r,φ)ˆ e r −1

k∂rYk

m (r,φ)ˆ e φ

Ykm (r,φ) = Jm (kr)exp imφ( )

Development of a generic vector in orthogonal functions

Development of a generic vector in orthogonal functions

A generic vector that is a function of the variable r and , can be written in terms

The Fourier transform of its components can be written as

€

vg (r,φ) = Gzkn

mr R kn

m (r,φ) + Grkn

mr S kn

m (r,φ) + Gφkn

mr T kn

m (r,φ)[ ]n= 0

∞

∑m= 0

∞

∑

€

vg (r,φ)

€

gαm (r,φ) =

1

2πgα

0

2π

∫ (r,φ)e−imφdφ

with α = z,r,φ

General solution General solution

€

rf (r,φ,z, t) = Fzkn

mr R kn

m (r,φ) + Frkn

mr S kn

m (r,φ) + Fφkn

mr T kn

m (r,φ)

€

ru (r,φ,z, t) = Uzkn

mr R kn

m (r,φ) + Urkn

mr S kn

m (r,φ) + Uφkn

mr T kn

m (r,φ)

Discrete wavenumber method

• Solution has the form

The solution is expressed in terms of Bessel functon

€

uzi (r,φ,z, t,zo) = Wn

i

n= 0

∞

∑ Uzkn

i (z, t,zo)J1(knr)cos(φ)

uri (r,φ,z, t,zo) = Wn

i

n= 0

∞

∑ .........

uφi (r,φ,z, t,zo) = Wn

i

n= 0

∞

∑ .........

€

Wn3 =

1

π RJ1(knR)[ ]2

Wn1 =

1

π RJo(knR)[ ]2

Waves

Amplitude Attenuation

• Wave amplitudes decrease during propagation

• Causes:– geometrical spreading (elastic)– Reflection and transmission coefficients– scattering (elastic)– Impedance contrast (elastic)– attenuation (anelastic)

THE GEOMETRICAL SPREADINGTHE GEOMETRICAL SPREADING

The geometrical spreading factor in inhomogeneous media describes the focusing and defocusing of seismic rays. In other words, the geometrical spreading can be seen as the density of arriving rays; high amplitudes are expected where rays are concentrated and low amplitudes where rays are sparse.

The geometrical spreading factor in inhomogeneous media describes the focusing and defocusing of seismic rays. In other words, the geometrical spreading can be seen as the density of arriving rays; high amplitudes are expected where rays are concentrated and low amplitudes where rays are sparse.

The focusing or defocusing of the rays can be estimated by measuring the areal section on the wave front at different times defined by four rays limiting an elementary ray tube.Each elementary area at a given time is proportional to the solid angle defining the ray tube at the source , but the size of the elementary area varies along the ray tube.

€

dσ (τ ) = ℜ(r x ,

r ξ )[ ]

2

dΩ

€

ℜ(r x ,

r ξ ) =

dσ

dΩ

Geometrical spreading of four rays at two different values of travel time (o, )

d(o) and d() are the two elementary surfaces describing the section of the ray tube on the wave front at different times.

d(o) and d() are the two elementary surfaces describing the section of the ray tube on the wave front at different times.

€

ru P(

r x ,t) = (u1,0,0)

u1 =ℑ P(e1,e2 )

ℜ(r x ,

r ξ )

U1( t − T P(r x ,

r ξ ))

Reflection & Transmission CoefficientThe reflection coefficient is used in physics and electrical engineering when wave propagation in a medium containing discontinuities is considered. A reflection coefficient describes either the amplitude or the intensity of a reflected wave relative to an incident wave. The reflection coefficient is closely related to the transmission coefficient.

€

ρ1,c1 €

ρ2,c2

€

u1(x, t) = Ae i ωt−k1x( ) + Be i ωt +k1x( )

u2(x, t) = Ce i ωt−k2x( )

€

u1(x = 0, t) = u2(x = 0, t)

Ae i ωt( ) + Be i ωt( ) = Ce i ωt( )

€

∂u1(x = 0, t)

∂x= τ

∂u2(x = 0, t)

∂xτk1(A − B) = τk2C

€

ki =ω

c i

,

c i =τ

ρ

€

R12 =B

A=

ρ1c1 − ρ 2c2

ρ1c1 + ρ 2c2

T12 =C

A=

2ρ1c1

ρ1c1 + ρ 2c2

Reflection & transmission coeff.Reflection & transmission coeff.

€

ρ c → acustic impedance

Some properties

See Aki & Richards (2002) chapter 5 for an extended presentation of R and T for more realistic waves€

R12 = −R21

T12 + T21 = 2

€

ω =c1k1 = c2k2 = c1

2π

λ1

= c2

2π

λ 2

SH wave transmission & reflection coefficientsSH wave transmission & reflection coefficients

€

R12 =B2

B1

=ρ1β1 cos j1 − ρ 2β 2 cos j2

ρ1β1 cos j1 + ρ 2β 2 cos j2

T12 =′ B

B1

=2ρ1β1 cos j1

ρ1β1 cos j1 + ρ 2β 2 cos j2

P-SV waves at the free surfaceP-SV waves at the free surface

€

RP =A2

A1

=4 p2ηαη β − η β

2 − p2( )

2

4 p2ηαη β + η β2 − p2

( )2

RSV =B2

A1

=4 pηα p2 −η β

2( )

4 p2ηαη β + η β2 − p2

( )2

Some more info on head wavesSome more info on head waves

€

TD (s) =s

vo

TR2(s) =

s2

vo2

+ 4ho

2

vo2

TH (s) =s

v1

+ 2ho

1

vo cosic

−tanic

v1

⎛

⎝ ⎜

⎞

⎠ ⎟=

s

v1

+ 2ho

1

vo2

−1

v12

⎛

⎝ ⎜

⎞

⎠ ⎟

1/ 2

Realistic complexity

Impedance contrastImpedance contrast• The impedance that a given

medium presents to a given motion is a measure of the amount of resistance to particle motion.

The product of density and seismic velocity is the acoustic impedance, which varies among different rock layers, commonly symbolized by Z. The difference in acoustic impedance between rock layers affects the reflection coefficient.

For a SH wave €

Z = ρc

€

Z SH =τ yz

˙ c = −ρβ cos j( )

Anelastic Attenuation – the Quality factor

If a volume of material is cycled in stress at a frequency ω, a dimensionless measure of the anelasticity (internal friction) is given by

where E is the peak strain energy stored in the volume and E is the energy lost in each cycle. We can transform this in terms of the amplitudes E = A2,

If a volume of material is cycled in stress at a frequency ω, a dimensionless measure of the anelasticity (internal friction) is given by

where E is the peak strain energy stored in the volume and E is the energy lost in each cycle. We can transform this in terms of the amplitudes E = A2,

€

1

Q(ω)= −

ΔE

2πE

€

1

Q(ω)= −

ΔA

πA

€

A(t) ≈ Ao exp −ωt

2Q

⎡

⎣ ⎢ ⎤

⎦ ⎥

ΔA =dA

dxλ ,

λ = 2πc

ω

€

A(x) ≈ Ao exp −ωx

2cQ

⎡

⎣ ⎢ ⎤

⎦ ⎥

If we consider a damped harmonic oscillator, we can write

where ωo is the natural frequency and is the damping factor

€

Q =ωo

γ

A numerical example

Body wave attenuationBody wave attenuation

Body wave attenuation is commonly parameterized through the parameter t*

€

t* =travel time

quality factor=

T

Q

t* =dt

Qpath∫

€

Mo =4πρc 3R

FFSℑc

ΩO

€

ER =4πρcR2

FFSℑc

˙ u ∫2( f )e

πfRcQ( f )df

€

Qc ( f ) = Qo f n

€

A( f ,R) = exp −πfR

βQ( f )

⎧ ⎨ ⎩

⎫ ⎬ ⎭

n =1

A(R) = exp −2πR

βQo

⎧ ⎨ ⎩

⎫ ⎬ ⎭

Comportamento anelastico: anisotropia

Olivine is seismically anisotropicOlivine is seismically anisotropic(mantello)(mantello)

Courtesy of Ben HoltzmanCourtesy of Ben Holtzman

Comportamento anelastico

Anisotropia nel mantello

• SKS splittingAnimation from the website of Ed Garnero

Comportamento anelastico

Anisotropia nel mantello

Fenomeno della birifrangenza

• SKS splitting

radial comp.

transversal comp.

SKS phases

fast

slow

fastslow

delay delay timetime

Courtesy of Ben Holtzman