The kinematic representation of seismic source

The kinematic representation of seismic source

€

ui(r x , t) = dτ

−∞

+∞

∫ Δu j (r ξ ,τ )c ikpqnk

∂

∂ξ q

Gip (r x , t − τ ,

r ξ ,0)

⎧ ⎨ ⎩

⎫ ⎬ ⎭Σ

∫∫ dΣ(ξ )

€

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)

The double-couple solution

double-couple solution in an infinite, homogeneous isotropic medium.

€

un(r x ,t) = Mpq ∗Gnp, q =

ℑ(γ n ,γ p ,γ q )

4πρ

1

r 4τ

r α

r β

∫ Mpq(t −τ )dτ +

ℑ P(γ n ,γ p ,γ q )

4πρα 2

1

r 2Mpq( t −

r

α)−

ℑ S (γ n ,γ p ,γ q )

4πρβ 2

1

r 2Mpq(t −

r

β)

+γ iγ jγ q

4πρα 3

1

r˙ M pq(t −

r

α)−

γ nγ p −δnp( )

4πρβ 3γ q

1

r˙ M pq(t −

r

β)

€

ℑ(γ n ,γ p ,γ q )

€

ℑP(γ n ,γ p ,γ q )

€

ℑS (γ n ,γ p ,γ q )

RadiationPattern

€

˙ M pq( t) moment rate function

NF

IT

FF

Far Field representationFar Field representation

€

ui(r x , t) = dτ

−∞

+∞

∫ Δu j (r ξ ,τ )c ikpqnk

∂

∂ξ q

Gip (r x , t − τ ,

r ξ ,0)

⎧ ⎨ ⎩

⎫ ⎬ ⎭Σ

∫∫ dΣ(ξ )

€

ui(r x , t) = −

1

4πρα 2

∂

∂xq

c ikpq

γ iγ p

rΔu j (

r ξ , t −

r

α)nk

⎧ ⎨ ⎩

⎫ ⎬ ⎭Σ

∫∫ dΣ(ξ )

+1

4πρβ 2

∂

∂xq

c ikpq

(γ iγ p −δ ip )

rΔu j (

r ξ , t −

r

β)nk

⎧ ⎨ ⎩

⎫ ⎬ ⎭Σ

∫∫ dΣ(ξ )

€

∂∂ξq

= −∂

∂xq

Far Field representationhomogeneous, isotropic, elastic mediumFar Field representationhomogeneous, isotropic, elastic medium

€

ui(r x , t) =

c ikpq

4πρα 3rγ iγ pΔ˙ u j (

r ξ , t −

r

α)γ qnk

⎧ ⎨ ⎩

⎫ ⎬ ⎭Σ

∫∫ dΣ(ξ )

−c ikpq

4πρβ 3

(γ iγ p −δ ip )

rΔ˙ u j (

r ξ , t −

r

β)γ qnk

⎧ ⎨ ⎩

⎫ ⎬ ⎭Σ

∫∫ dΣ(ξ )

€

∂r

∂xq

= γ q

Neglecting all terms that attenuatewith distance more rapidly than 1/r

Neglecting all terms that attenuatewith distance more rapidly than 1/r

Far Field representationhomogeneous, isotropic, elastic mediumFar Field representationhomogeneous, isotropic, elastic medium

€

ui(r x , t) =

γ i

4πρα 3ro

c ikpqγ pγ qnk Δ˙ u j (r ξ , t −

r

α)

⎧ ⎨ ⎩

⎫ ⎬ ⎭Σ

∫∫ dΣ(ξ )

+(δ ip − γ iγ p )

4πρβ 3ro

c ikpqγ qnk Δ˙ u j (r ξ , t −

r

β)

⎧ ⎨ ⎩

⎫ ⎬ ⎭Σ

∫∫ dΣ(ξ )

If the receiver is far enough away with respect to the linear dimension of fault L, we can assume that the distance and the direction cosines are approximately constant, independent of ξ Thus, the constant or slowly variable factors can be moved outside the integral

If the receiver is far enough away with respect to the linear dimension of fault L, we can assume that the distance and the direction cosines are approximately constant, independent of ξ Thus, the constant or slowly variable factors can be moved outside the integral

€

Mo = μ Δu(r ξ , t → ∞)

Σ

∫∫ dΣ = μ Δu Σ

Far Field Displacement pulse

€

Ω(r x , t) = Δ˙ u (

r ξ , t −

r

c)

⎧ ⎨ ⎩

⎫ ⎬ ⎭Σ

∫∫ dΣ(ξ )

€

Δ˙ u j (r ξ , t −

r

α) = ν jΔ˙ u (

r ξ , t −

r

c)

€

Ω(r x ,ω) = Δ˙ u (

r ξ ,ω)exp

iω ro − ( ˆ ξ ⋅ ˆ γ )[ ]

c

⎧ ⎨ ⎪

⎩ ⎪

⎫ ⎬ ⎪

⎭ ⎪

⎧ ⎨ ⎪

⎩ ⎪

⎫ ⎬ ⎪

⎭ ⎪Σ

∫∫ dΣ(ξ )

€

Ω(r x ,ω → 0) = Δ˙ u (

r ξ ,ω → 0)

Σ

∫∫ dΣ

€

Ω(r x ,ω → 0) = Δu(

r ξ , t → ∞)

Σ

∫∫ dΣ

€

Δ˙ u (r ξ ,ω) = Δ˙ u (

r ξ , t)exp(iωt)dt∫

€

Δ˙ u (r ξ ,ω → 0) = Δ˙ u (

r ξ , t)dt∫ =

Δu(r ξ , t → ∞)

Final slipFinal slip

€

ˆ ξ

€

ˆ γ

€

dΣ

€

ϑ =Ψ

in followingslides

Fraunhofer ApproximationFraunhofer Approximation

€

r =r x −

r ξ = ro 1+

ξ 2

ro2

−2

r ξ ⋅ ˆ γ ( )

ro

= ro −r ξ ⋅ ˆ γ ( ) +

1

2

ξ 2

ro

−

r ξ ⋅ ˆ γ ( )

2

2ro

€

r ≈ ro −r ξ ⋅ ˆ γ ( )

The error in this approximation is

€

∂r =1

2

1

ro

ξ 2

−r ξ ⋅ ˆ γ ( )

2 ⎡ ⎣ ⎢

⎤ ⎦ ⎥<<

λ

4

€

L2 <<1

2λro

€

ro >> L

DISPLACEMENT FOURIER SPECTRUM

The ground displacement Fourier spectrum is nearly flat at the origin

€

ω 2

€

ω−2

Corner frequency

AccelerationAcceleration

displacementdisplacement

Far Field representationinhomogeneous, isotropic, elastic mediumFar Field representationinhomogeneous, isotropic, elastic medium

€

vu P (

r x , t) =

ℑ P ˆ t

4π ρ ξ o( )ρ x( )α ξ o( )α x( )

1

α 2 ξ o( )

1

ℜ P (r x ,

r ξ o)

Δ˙ u j (r ξ , t − T P (

r x ,

r ξ )){ }

Σ

∫∫ dΣ(ξ )

v u SV (

r x , t) =

ℑ SV ˆ n

4π ρ ξ o( )ρ x( )β ξ o( )β x( )

1

β 2 ξ o( )

1

ℜ S (r x ,

r ξ o)

Δ˙ u j (r ξ , t − T S (

r x ,

r ξ )){ }

Σ

∫∫ dΣ(ξ )

v u SH (

r x , t) =

ℑ SH ˆ b

4π ρ ξ o( )ρ x( )β ξ o( )β x( )

1

β 2 ξ o( )

1

ℜ S (r x ,

r ξ o)

Δ˙ u j (r ξ , t − T S (

r x ,

r ξ )){ }

Σ

∫∫ dΣ(ξ )

If the receiver is far enough away with respect to the linear dimension of fault L, we can assume that the distance and the direction cosines are approximately constant, independent of ξ Thus, the constant or slowly variable factors can be moved outside the integral

If the receiver is far enough away with respect to the linear dimension of fault L, we can assume that the distance and the direction cosines are approximately constant, independent of ξ Thus, the constant or slowly variable factors can be moved outside the integral

Unilateral Rupture propagation

Source time function

Ψ is the angle between the direction of rupture propagation and the direction of the receiver

€

Δu(r ξ , t) = f t −ξ1

vr

⎛ ⎝ ⎜ ⎞

⎠ ⎟

€

Ω(r x , t) = ˙ f t −

ro

c−

ξ1

vr

+ξ1γ1 + ξ 2γ 2

c

⎛

⎝ ⎜

⎞

⎠ ⎟dξ1dξ 2

0

L

∫0

W

∫ =

W ˙ f t −ro

c−

ξ1

vr

+ ξ1

1

vr

−cos(Ψ)

c

⎛

⎝ ⎜

⎞

⎠ ⎟

⎛

⎝ ⎜

⎞

⎠ ⎟dξ1

0

L

∫

€

ξ1

€

ξ2

Aki – Richards, 2002, p.499

The integrand of this equation ranges between and

The pulse is proportional to a moving average of taken over a time interval of duration €

˙ f (t − roc )

€

˙ f [t − roc − L( 1

vr− cos(Ψ)

c )]

€

˙ f (t − roc )

€

T = L( 1vr

− cos(Ψ)c )

Unilateral Rupture propagation

• Taking Fourier Transform

€

Ω(r x ,ω) = −iωWf (ω)e

iωro

c exp iωξ1

1

vr

−cos(Ψ)

c

⎛

⎝ ⎜

⎞

⎠ ⎟

⎧ ⎨ ⎩

⎫ ⎬ ⎭dξ1

0

L

∫ =

ωf (ω)WLsin(X)

Xexp i

ωro

c−

π

2+ X

⎛

⎝ ⎜

⎞

⎠ ⎟

⎧ ⎨ ⎩

⎫ ⎬ ⎭

.

X = ωL

2

1

vr

−cos(Ψ)

c

⎛

⎝ ⎜

⎞

⎠ ⎟

The term sin(X)/X expresses the effect of fault finiteness on the amplitude spectrum.At high frequency this term is proportional to ω-1. The smoothing effect is weakest in the direction of propagation (=0) and strongest in the opposite direction (=). Thus, we observe more high-frequency in the direction of rupture propagation: that is DIRECTIVITY

The term sin(X)/X expresses the effect of fault finiteness on the amplitude spectrum.At high frequency this term is proportional to ω-1. The smoothing effect is weakest in the direction of propagation (=0) and strongest in the opposite direction (=). Thus, we observe more high-frequency in the direction of rupture propagation: that is DIRECTIVITY

The effect of finite rise time

t < tr = ξ vr

Tr < t < T + tr

t T + tr

)(tD

t

Tr = rise time

maxD

€

f (t) =

0

Dmax t T

Dmax

⎧

⎨ ⎪

⎩ ⎪

€

Ω(r x ,ω) = WLDmax

sin(X)

X

1− e iωT

ωT.

The effect of finite rise time introduces an additional smoothing of the waveform: forhigh frequency it attenuates the spectrum proportional to ω-1. Together with the effect of the term sin(X)/X, the spectrum decays asto ω-2.

The effect of finite rise time introduces an additional smoothing of the waveform: forhigh frequency it attenuates the spectrum proportional to ω-1. Together with the effect of the term sin(X)/X, the spectrum decays asto ω-2.

Some properties

At ω = 0 it is proportional to WLDmax, which is the seismic moment

At frequency larger than the characteristic frequency given by 1/T or 1/L(1/v – cos(Ψ)/c) the spectrum attenuates as ω-2

If the effect of finite width is taken into account, we have a high frequency spectral decay proportional to as ω-3

€

Ω(r x ,ω) = WLDmax

sin(X)

X

1− e iωT

ωT.

An example from the 1997 Colfiorito earthquake sequence

A brief note on earthquake dynamicA brief note on earthquake dynamic

Slip, Slip velocity & Traction evolution

A brief note on earthquake dynamicA brief note on earthquake dynamic

The Slip Weakening mechanism

A case study: The 1997 Colfiorito Earthquake

Normal faulting earthquakes

Multiple main shocks of similar size

Moderate magnitudes

Peak ground motionattenuation

a) Colfiorito eventunilateral NW rupture

b) Sellano eventnearly unilateralSE propagation

Colfiorito earthquake • Some spectra

Comparison between predicted and observed PGAColfiorito earthquake

PREDICTED PGA comparison with data & empirical law

• Azimuthal variation

Comparison between predicted and observed data with empirical regression law

The 2007 Niigata-ken Chuetsu-oki earthquake KKNPP is the nuclear power plant

Waveform inversion to infer seismic sourceWaveform inversion to infer seismic source

Seismic source models obtained by inverting seismograms and GPS displacements

Ground Motion Predictionthrough the inferred model

Some numbers

MAGNITUDE FAULT LENGHT

[Km]

DISLOCATION [m]

RUPTURE DURATION

[s]

4 1 0.02 0.3

5 5 0.05 1

6 10 0.2 3

7 50 1 15

8 250 5 85

9 800 8 250

Spectral modelsSpectral models

Omega cube model

Omega square model

€

S( f ) =Ωo

1+f

fc

⎛

⎝ ⎜

⎞

⎠ ⎟

2€

S( f ) =Ωo

1+f

fc

⎛

⎝ ⎜

⎞

⎠ ⎟

2 ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

3 / 2

Computing earthquake magnitudeComputing earthquake magnitude

M = log (A/T) + F(h,R) + CA – amplitudeT – dominant periodF – correction for depth & distanceC – regional scale factor

M = log (A/T) + F(h,R) + CA – amplitudeT – dominant periodF – correction for depth & distanceC – regional scale factor

€

ML = log(A) + 2.76log(R) − 2.48

MS = log(A20) +1.66log(R) + 2.0

Seismic Moment & MagnitudeSeismic Moment & Magnitude

From seismic moment we can compute an equivalent magnitude called the moment magnitude

€

MW =2

3log(Mo) −10.73

Mo = μ Dmax Σ = μ Dmax (LW )

Corner frequency shift with magnitudeCorner frequency shift with magnitude

€

Mo ∝ L2

TR =L

vr

∝ L

Mo ∝ L3 ≈ fc−3

fc = 2.34β

L

fcS = CS

β

R

fcP = CP

α

R

Scaling of final slip with fault lengthScaling of final slip with fault length

Wells & Coppersmith 1994

STRESS DROP SCALINGSTRESS DROP SCALING

is a factor depending on fault’s shape

For a circular fault with radius R

€

Δσ ≅μD

L

D =χMo

μL2

Δσ ≅χMo

L3=

χMo

Σ3 / 2

€

Δσ =7

16

Mo

R3∝ Mo fc

3

Magnitude & Energy

Stress and Radiated EnergyStress and Radiated Energy

Strain energy release

Seismic efficiency

Apparent stress€

W = σ D Σ

Δσ = σ o −σ 1

E = W − H = σ D Σ −σ f D Σ

σ = σ 1 +1

2Δσ

E =1

2ΔσD Σ + σ 1 −σ f( )D Σ

Eo =1

2ΔσD Σ

€

η =E

W=

1

2

Δσ

σ

σ a = μE

Mo

=1

2Δσ + (σ 1 −σ f )

A slip weakening model

Energy loss

€

τ =τy − τ y −τ f( ) Δu

Dc,⇒ Δu < Dc

τ f ,,⇒ Δu > Dc

⎧ ⎨ ⎪

⎩ ⎪

€

σ f ⋅Dtot ⋅Σ

€

σ f =1

Dtot

⋅ τ (D )dDo

Dtot∫

€

EΣ = G + Q ≅ τ (D)dDo

Dtot∫