kinematic representation theorem kinematic tractions time domain representation frequency domain...
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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