SURFACE WAVES JEANNOT TRAMPERT

SURFACE WAVES Most seismograms are dominated by surface waves whose energy is concentrated near the earth’s surface. Geometric spreading (energy) is in 1/r (1/r2 for body waves) Rayleigh waves are trapped P-SV energy

Love waves are trapped SH energy

Because surface waves decay slowly they circle the Earth many times. Rayleigh Rn, Love Gn

Dispersion: waves with different frequencies travel with different speeds à spread out in time

SUMATRA EARTHQUAKE

RAYLEIGH WAVE IN A HOMOGENEOUS HALFSPACE Dispersion Equation

(C2

β 2− 2)(2− C

2

β 2)+ 4β 2 1− c

2

α 2 1− c2

β 2= 0

For a Poisson solid this has one root with phase velocity c=0.914β with elliptical particle motion and no dispersion (see exercise)

LOVE WAVE IN A LAYER OVER HOMOGENEOUS HALFSPACE

1β12∂2v∂t2

= (∂2v∂x2

+∂2v∂z2), 0 < z < h

1β22∂2v∂t2

= (∂2v∂x2

+∂2v∂z2), z > h

Equation of motion For displacement v In direction y

LOVE WAVE IN A LAYER OVER HOMOGENEOUS HALFSPACE Fourier Transform equation of motion

ν i2 =

ω 2

βi2 − k

2 =ω 2

βi2 −

ω 2

c2

Trial solution

v = [Aexp(−iν1z)+Bexp(iν1z)]expi(kx −ωt), 0 < z < hv = [C exp(−iν2z)+Dexp(iν2z)]expi(kx −ωt), z > h

LOVE WAVE IN A LAYER OVER HOMOGENEOUS HALFSPACE Boundary conditions give the constants A, B, C and D To have an inhomogeneous wave at the interface β1<=c<=β2 No energy at z=∞ à D=0

No traction at z=0 à A=B

Continuity of displacement and traction at z=h

tan[ωh 1β12 −

1c2]=

µ21c2−1β22

µ11β12 −

1c2

vn = 2Acos[ω1β12 −

1cn2 z]exp[i(knx −ωt)], 0 <= z <= h

vn = 2Acos[ω1β12 −

1cn2 h]exp[−ω

1cn2 −

1β22 (z− h)]− exp[i(knx −ωt)], z >= h

LOVE WAVE IN A LAYER OVER HOMOGENEOUS HALFSPACE The relative excitation of the different modes depends on the depth and nature of the seismic source. A way to separate the modes is to observe them at large distances where they arrive at different times due to the propagation with different group velocities The group velocity at a given frequency is the velocity at which an envelope of a wave packet is transported. The peaks, troughs and zeros are transported with the phase velocity.

STATIONARY PHASE APPROXIMATION The waveform for a single mode with spectral density F(ω) and initial phase ϕ(ω) can be written as

If non-dispersive f(x,t)=f(t-x/cn) (phase is constant)

If dispersive à stationary phase approximation

f (x, t) =1/ 2π | F | expi(Φ+ωt − knx)dω−∞

+∞

∫

ddω(ωt − knx) = 0 For fixed ω has solutions when t=x/U

STATIONARY PHASE APPROXIMATION

STATIONARY PHASE APPROXIMATION Taylor expansion of the phase around ω0 gives

And

If dU/dω=0: Airy phase à higher order terms

ωt − knx =ω0t − knx +x2d 2kndω 2 (ω −ω0 )

2

f (x, t) ~| F | /π 2πx | d 2kn / dω

2 |cos(ω0t − kn (ω0 )x ±π / 4)

VARIATIONAL PRINCIPLES Hamilton’s principle: every mechanical system is defined by a Lagrangian function Satisfying is minimum Or The equations of motion are obtained by For an linear elastic body

L(q1,...,qn, q1,..., qn, t)

S = L(q, q, t)dtt1

t2∫

dS = d Ldtt1

t2∫ = (∂L∂q

dq+ ∂L∂ qt1

t2∫ d q)dt = 0

ddt∂L∂ q

−∂L∂q

= 0

L = 12ρ ui ui −[

12λ(ekk )

2 +µeijeij ]

VARIATIONAL PRINCIPLE FOR LOVE WAVES

< L >= 14ρω 2l1

2 −14µ[k2l1

2 + (dl1dz)2 ]

u = (0, l1(k, z,ω)expi(kx −ωt), 0)

(∂L∂l10

∞

∫ dl1)dz = 0

ω 2dI1 − k2dI2 − dI3 = 0 I1 =1/ 2 ρl1

20

∞

∫ dz

I2 =1/ 2 µl12

0

∞

∫ dz

I3 =1/ 2 µ(dl1dz0

∞

∫ )2dz

For a displacement The average Lagrangian is Hamilton’s principle says that Or where

VARIATIONAL PRINCIPLE FOR LOVE WAVES

ω 2I1 − k2I2 − I3Which means that

Is stationary for perturbations of the eigenfunction l1 We can further show that for an eigenfunction l1

ω 2I1 − k2I2 − I3 = 0

VARIATIONAL PRINCIPLE FOR LOVE WAVES

Which leads to three interesting applications

k2 = ω2 (I1 + dI1)− (I3 + dI3)

(I2 + dI2 )

U =dωdk

=I2cI1

dcc= −

dkk=

{[k2l12 + (dl1 / dz)

2 ]dµ −ω 2l12 dρ}dz

0

∞

∫2k2I2

VARIATIONAL PRINCIPLE FOR RAYLEIGH WAVES

Similar, but a bit more algebra (see Aki and Richards)

FUNDAMENTAL MODE SENSITIVITY KERNELS

FUNDAMENTAL MODE SENSITIVITY KERNELS

FUNDAMENTAL MODE SENSITIVITY KERNELS

NUMERICAL INTEGRATION TO FIND EIGEN-VALUES AND -VECTORS Transform wave equation (2nd order differential equation) into a first order coupled differential equation. d/dz (motion, stress)t = matrix * (motion, stress)t

à  Propagator matrix f(z)=P(z,z0)f(z0)

à  Trial solution (ω,k) at infinity so that stress = 0 at the surface

Rayleigh-Ritz method which uses the variational principles

l1=ΣciBi(z) and Bi verifies the BC at z=0 and z=∞

MEASURING SURFACE WAVE DISPERSION Fourier transform

F(ω) = f (t)exp(−iωt)dt−∞

+∞

∫ = Aexp(−iφ)

whereφ = k(ω)r +φs +φi

f (t) = 12π

Aexp[i(ωt − kr)]dω−∞

+∞

∫

MEASURING SURFACE WAVE DISPERSION Velocity of propagation of monochromatic wave Velocity of propagation of maximum energy

Useful relations

ωt − kr = constω(dt / dr)− k = 0tph = r / c

ddω(ωt − kr) = 0

tgr = (dk / dω)r = r /U

U = c+ k dcdk

U = c−λ dcdλ

U =c

1+ TcdcdT

PHASE VELOCITY MEASUREMENTS Sato (1955) uses FT for the first time n obvious at long periods à smooth dispersion curve Inter-station method eliminates source Cross-correlation makes phase difference numerically more stable

kr = φ −φs −φi + 2nπ

c = ω(r2 − r1)φ2 −φ1 −φi2 +φi1 + 2nπ

PHASE VELOCITY MEASUREMENTS Single-station method on world-circling paths eliminates source and instrument, for l the difference in the number of polar passages

Auto-correlation makes phase difference numerically more stable

c =

12ωlL

φ2 −φ1 + 2π (n+14l)

GROUP VELOCITY MEASUREMENTS Analytical signal associated to the seismogram

where e(t) is the envelope and Φ(t) the instantaneous phase

The measurement of U is related to its definition: we evaluate

To first order, we can show that de(t)/dt=0 if t=r/U

the maximum of the envelope corresponds to the group arrival time

s(t) = s(t)− iH (s(t)) = e(t)exp(iφ(t))

hn (t) =12π

S(ω)H (ωn,ω)exp(iωt)dω−∞

+∞

∫

AUTOMATIC WAVEFROM INVERSION FOR PHASE VELOCITY Trampert and Woodhouse, 1995 Ekstrom, Tromp and Larson, 1997

RAYLEIGH WAVE GROUP VELOCITY AT 40 S

RAYLEIGH WAVE GROUP VELOCITY AT 125 S

GREEN’S FUNCTION FOR SURFACE WAVES Solution for surface waves generated by a point force with time dependence exp(-iωt) buried at depth h. It is most useful to use a cylindrical reference frame. The derivation follows Saito (1967)

GREEN’S FUNCTION FOR SURFACE WAVES The Helmholtz decomposition separates the displacement field into P-, SV- and SH-wave components

u =∇Φ+∇×∇(0, 0,Ψ)+∇× (0, 0,Χ)Φ(r,ω) =Yk

m[Aexp(−γz)+Bexp(γz)]exp(−iωt)Ψ(r,ω) =Yk

m[C exp(−νz)+Dexp(νz)]exp(−iωt)Χ(r,ω) =Yk

m[E exp(−νz)+F exp(νz)]exp(−iωt)whereYk

m = Im (kr)exp(imϕ )

γ = ω 2

α 2 − k2

ν = ω 2

β 2− k2

GREEN’S FUNCTION FOR SURFACE WAVES The Helmholtz decomposition separates the displacement field into P-, SV- and SH-wave components, All potentials satisfy the scalar wave equation, A, B, C, D, E and F are constants

Im is the mth order Bessel function

Ykm is a horizontal wave function

We will write the wave as coupled first order differential equation df/dz A = f For Love waves f=(l1,l2)t is the motion-stress vector

GREEN’S FUNCTION FOR LOVE WAVES

uSH = (1r∂Χ∂ϕ,−∂Χ

∂r, 0)

stress+BC...u = l1(k, z,ω)Tk

m (r,ϕ )exp(iωt)T = l2 (k, z,ω)Tk

m (r,ϕ )exp(iωt)where

Tkm 1kr∂Yk

m

∂ϕr + 1

k∂Yk

m

∂rϕ

GREEN’S FUNCTION FOR LOVE WAVES Love generated by a point force at r=0 and z=h The applied for is equivalent to a discontinuity in traction

Method:

①  Decompose the discontinuity in (k,m) components

②  Solve df\dz A = f for each (k,m) component where f is the z-dependent motion-stress vector with the discontinuity at z=h

③  The total solution is the sum of all (k,m) components

T (h+ )−

T (h− ) = −

F exp(−iωt)∂(x)∂(y)

GREEN’S FUNCTION FOR LOVE WAVES

−F exp(−iωt)∂(x)∂(y) = exp(−iωt) / 2π k[ fT0

∞

∫m∑ (k,m)Tk

m ]dk

fT (k,1) =1/ 2(Fy + iFx )fT (k,−1) =1/ 2(−Fy + iFx )

For all other m fT=0 which finally gives asymptotically

u = exp(−iωt)(Fy cosϕ −Fx sinϕ )l1(kn,h,ω)

8cUI1n∑ 2

πknr[l1(kn, z,ω)expi(knr +π / 4)

ϕ ]

GREEN’S FUNCTION FOR LOVE WAVES

G =

l1(kn,h,ω)l1(kn, z,ω)8cUI1n

∑sin2ϕ −sinϕ cosϕ 0

−sinϕ cosϕ cos2ϕ 00 0 0

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((((

2πknr

exp[i(knr +π / 4)]

u = l1(kn, z,ω)8cUI1n

∑ 2πknr

exp[i(knr +π / 4)]

{iknl1(h)[Mxx sinϕ cosϕ −Mxy cos2ϕ +Mxy sin

2ϕ −Myy sinϕ cosϕ ]

dl1(h)dz

[Mxz sinϕ −Myz cosϕ ]}−sinϕcosϕ0

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((((

Where I used ui=MpqGip,q