Multiscale Hazard Scenarios Regional seismic hazard

Regional seismic hazard scenarios(ground motion at bedrock)

Multiscale Hazard Scenarios

Regional Scale - NDSHA

Regionalpolygons

Structural model

Earthquakecatalogue

Seismogeniczones

Magnitude discretizationand smoothing

Focalmechanisms

Choice of maximum magnitude for each

cell

Choice of focal mechanism for each seismogenic zone

Seismic sources

Sites considered for each source

Synthetic seismograms

Extraction of significantground motion parameters

1-D layeredanelastic

structures

• Seismic zonation based on the computation of synthetic seismograms on the nodes of a grid that covers the study area

• Average structural properties

• Simple source model (scaled point source)

• Cut-off frequency 1 Hz

• Maps of peak displacement, velocity and Design Ground Acceleration

Regionalpolygons

Structural model

Earthquakecatalogue

Seismogeniczones

Magnitude discretizationand smoothing

Focalmechanisms

Choice of maximum magnitude for each

cell

Choice of focal mechanism for each seismogenic zone

Seismic sources

Sites considered for each source

Synthetic seismograms

Extraction of significantground motion parameters

1-D layeredanelastic

structures

0710.por

6˚

6˚

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36˚ 36˚

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1

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16252

Regional Scale - Structural models

Regionalpolygons

Structural model

Earthquakecatalogue

Seismogeniczones

Magnitude discretizationand smoothing

Focalmechanisms

Choice of maximum magnitude for each

cell

Choice of focal mechanism for each seismogenic zone

Seismic sources

Sites considered for each source

Synthetic seismograms

Extraction of significantground motion parameters

1-D layeredanelastic

structures

0

10

20

30

40

50

60

70

80

0 2 4 6 8 10 12

0

10

20

30

40

50

60

70

80

0 2 4 6 8 10 12

Depth

[km

]

Density [g/cm^3]

Structure: svalp

V [km/s]

DensityVpVs

Regional Scale

Regionalpolygons

Structural model

Earthquakecatalogue

Seismogeniczones

Magnitude discretizationand smoothing

Focalmechanisms

Choice of maximum magnitude for each

cell

Choice of focal mechanism for each seismogenic zone

Seismic sources

Sites considered for each source

Synthetic seismograms

Extraction of significantground motion parameters

1-D layeredanelastic

structures

0710.ung

6˚

6˚

7˚

7˚

8˚

8˚

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20˚

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36˚ 36˚

37˚ 37˚

38˚ 38˚

39˚ 39˚

40˚ 40˚

41˚ 41˚

42˚ 42˚

43˚ 43˚

44˚ 44˚

45˚ 45˚

46˚ 46˚

47˚ 47˚

48˚ 48˚

3.5

5.0

5.5

6.0

6.5

7.0

7.4M

Regional Scale - Historical seismicity

Regionalpolygons

Structural model

Earthquakecatalogue

Seismogeniczones

Magnitude discretizationand smoothing

Focalmechanisms

Choice of maximum magnitude for each

cell

Choice of focal mechanism for each seismogenic zone

Seismic sources

Sites considered for each source

Synthetic seismograms

Extraction of significantground motion parameters

1-D layeredanelastic

structures

0710.gmt

6˚

6˚

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36˚ 36˚

37˚ 37˚

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41˚ 41˚

42˚ 42˚

43˚ 43˚

44˚ 44˚

45˚ 45˚

46˚ 46˚

47˚ 47˚

48˚ 48˚

251252

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901 902903

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908909

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914915916

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Regional Scale - Tectonic setting

Regionalpolygons

Structural model

Earthquakecatalogue

Seismogeniczones

Magnitude discretizationand smoothing

Focalmechanisms

Choice of maximum magnitude for each

cell

Choice of focal mechanism for each seismogenic zone

Seismic sources

Sites considered for each source

Synthetic seismograms

Extraction of significantground motion parameters

1-D layeredanelastic

structures

Regional Scale - Seismograms computation

Equivalent Forces

The scope is to develop a representation of the displacement generated in an elastic body in terms of the quantities that originated

it: body forces and applied tractions and displacements over the surface of the body.

The actual slip process will be described by superposition of equivalent body forces acting in space (over a fault) and time (rise time).

The observable seismic radiation is through energy release as the fault surface moves: formation and propagation of a crack. This complex

dynamical problem can be studied by kinematical equivalent approaches.

Final source representation

un(x,t)= [ui ]cijpqν j ∗

∂G np

∂ξqdΣ

Σ∫∫

mpq = [ui]cijpqν j un(x,t)= mpq ∗

∂Gnp

∂ξqdΣ

Σ∫∫

And if the source can be considered a point-source (for distances greater than fault dimensions), the contributions from different surface elements can be

considered in phase. Thus for an effective point source, one can define the moment tensor:

M pq = mpqdΣΣ∫∫

un(x,t)=M pq ∗G np,q

Far field term

Near field term

GF for double coupleAn important case to consider in detail is the radiation pattern expected when the source is a double-couple. The result for a moment time function M0(t) is:

Intermediate field term

�

u = ANF

4πρx 4 τx /α

x /β

∫ M0(t - τ)dτ +

+ APIF

4πρα2 x 2 M0(t - xα

) - ASIF

4πρβ2 x 2 M0(t - xβ

) +

+ APFF

4πρα3 x˙ M 0(t - x

α) - AS

FF

4πρβ3 x˙ M 0(t - x

β)

�

ANF = 9sin2θcosφˆ r −6 cos2θcosφ ˆ θ − cosθsinφ ˆ φ ( )AP

IF = 4sin2θcosφˆ r −2 cos2θcosφ ˆ θ − cosθsinφ ˆ φ ( )AS

IF = −3sin2θcosφˆ r + 3 cos2θcosφ ˆ θ − cosθsinφ ˆ φ ( )AP

FF = sin2θcosφˆ r AS

FF = cos2θcosφ ˆ θ − cosθsinφ ˆ φ

NF DC (static) Radiation pattern The static final displacement for a shear dislocation of strength M0 is:

�

u = M0 ∞( )4πρx 2 ANF 1

2β2 −1

2α2

⎛

⎝ ⎜

⎞

⎠ ⎟ +

APIF

α2 + ASIF

β2

⎡

⎣ ⎢

⎤

⎦ ⎥ =

=M0 ∞( )4πρx 2

32β2 −

12α2

⎛

⎝ ⎜

⎞

⎠ ⎟ sin2θcosφˆ r + 1

α2 cos2θcosφ ˆ θ − cosθsinφ ˆ φ ( )⎡

⎣ ⎢

⎤

⎦ ⎥

−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

Figure 7: Near-field Static Displacement Field From a Point Double CoupleSource (⌃ = 0 plane); � = 31/2, ⇥ = 1, r = 0.1, 0.15, 0.20, 0.25, ⇧ = 1/4⌅,M⇥ = 1; self-scaled displacements

The near-field term gives a static displacement as t⇥⇤

u =M0(⇤)4⌅⇧r2

�AN

2

�1⇥2� 1

�2

⇥+

AIP

�2+

AIS

⇥2

⇥(64)

=M0(⇤)4⌅⇧r2

�12

�3⇥2� 1

�2

⇥sin 2⇤ cos ⇤r̂ +

1�2

(cos 2⇤ cos ⌃⇤̂ � (cos ⇤ sin ⌃⌃̂)⇥

,

where M0(⇤) is the final value of the seismic moment. Interestingly, this ex-pression contains two terms with the same angular dependence as those for thefar-field, but decays as r�2. The strain field, which is the usual observable usedto study such permanent near field terms, will correspondingly decay as r�3.

16

Coseismic deformation

Co- & Post- seismic: Tohoku-oki

a, Coseismic displacements for 10–11 March 2011, relative to the Fukue site. The black arrows indicate the horizontal coseismic movements of the GPS sites. The colour shading indicates vertical displacement. The star marks the location of the earthquake epicentre. The dotted lines indicate the isodepth contours of the plate boundary at

20-km intervals28. The solid contours show the coseismic slip distribution in metres. b, Postseismic displacements for 12–25 March 2011, relative to the Fukue site. The red contours show the afterslip

distribution in metres. All other markings represent the same as in a.

From: Ozawa et al., 2011,, Nature, 475, 373–376.

Far field for a point d-c point source

From the representation theorem we have:

that, in the far field and in a spherical coordinate system becomes:

�

un(x,t) = Mpq ∗Gnp,q

�

u(x,t) = 14πρα3 sin2θcosφˆ r ( )

˙ M t − r /α( )r

+

14πρβ3 cos2θcosφ ˆ θ − cosθsinφ ˆ φ ( )

˙ M t − r /β( )r

and both P and S radiation fields are proportional to the time derivative of the moment function (moment rate). If the moment function is a ramp of duration τ (rise time), the propagating disturbance in the far-field will be a boxcar, with the same duration, and whose amplitude is varying depending on the radiation pattern.

FF DC Radiation pattern

receiverstructuresource

Methodology - Modal Summation Technique

Expression of the displacement generated by a double-couple point source in a flat layered halfspace

�

uyL x,z,ω( ) = e−i3π / 4

8πωe−ik mx

xχmL (hs,ω)( )cmvmIm

Fy (z,ω)( )vmImm=1

∞

∑

�

uxR x,z,ω( ) = e−i3π / 4

8πωe−ik mx

xχmR (hs,ω)( )cmvmIm

Fx (z,ω)( )vmImm=1

∞

∑

�

uzR x,z,ω( ) = e−iπ / 4

8πωe−ik mx

xχmR (hs,ω)( )cmvmIm

Fz (z,ω)( )vmImm=1

∞

∑

Source definition and examples of radiation pattern

vertical strike-slip

45° dipping strike-slip

45° dipping oblique slip

45° dip-slip (thrust)

45° dip-slip (normal)

vertical dip-slip

Love Rayleigh

Methodology - Modal Summation Technique

Expression of the source radiation pattern

�

χL = i(d1L sinϕ + d2L cosϕ) + d3L sin2ϕ + d4L cos2ϕχR = d0 + i(d1R sinϕ + d2R cosϕ) + d3R sin2ϕ + d4R cos2ϕ

�

A(hs) = −Fx * (hs)Fz(0)

B(hs) = − 3− 4 β 2(hs)α 2(hs)

⎛

⎝ ⎜

⎞

⎠ ⎟

Fx * (hs)Fz(0)

− 2ρ(hs) α

2(hs)σ zz * (hs)« F z(0) /c

C(hs) = − 1µ(hs)

σ zx (hs)« F z(0) /c

G(hs) = − 1µ(hs)

σ zy * (hs)« F y (0) /c

V(hs) =« F y (hs)

« F y (0) /c

=Fy (hs)

Fy (0) /c�

d1L = G(hs) cosλ sinδd2L = −G(hs) sinλ cos2δ

d3L = 12

V(hs) sinλ sin2δ

d4L = V(hs) cosλ sinδ

�

d0 = 12

B(hs) sinλ sin2δ

d1R = −C(hs) sinλ cos2δd2R = −C(hs) cosλ cosδd3R = A(hs) cosλ sinδ

d4R = − 12

A(hs) sinλ sin2δ

where

.

.

.

.

.

Methodology - Modal Summation Technique

Example of quantities associated with a structure

0

10

20

30

40

50

60

70

80

0 2 4 6 8 10 12

0

10

20

30

40

50

60

70

80

0 2 4 6 8 10 12

Depth

[km

]

Density [g/cm^3]

Structure: svalp

V [km/s]

DensityVpVs

Methodology - Modal Summation Technique

Phase velocity dispersion curve

2

3

4

5

6

78

9 10 11 12

4

not dispersed

dispersed

dispersed

Methodology - Modal Summation Technique

Eigenfunctions

44

not dispersed

dispersed

dispersed

0

5

10

15

20

25 0 2 4 6 8 10

0

5

10

15

20

25

-1.5 -1 -0.5 0 0.5 1

Depth

[km

]

Vp - Vs [km/s]

Modes: 4- 4 Freq: 1.500- 1.500 End: 0 (svalp.spr)

Rayleigh Eigenfunctions

u/w(0)VpVs

0

5

10

15

20

0 2 4 6 8 10

0

5

10

15

20

-15 -10 -5 0 5 10 15

Depth

[km

]

Vp - Vs [km/s]

Modes: 4- 4 Freq: 2.000- 2.000 End: 0 (svalp.spr)

Rayleigh Eigenfunctions

u/w(0)VpVs

Methodology - Modal Summation Technique

Synthetic seismograms

Radial Velocity

0 5 10 15 20 25 30 35 40 45 50

Time (s)

Vertical Velocity

22.3176

0 5 10 15 20 25 30 35 40 45 50

Time (s)

Transverse Velocity

0 5 10 15 20 25 30 35 40 45 50

Time (s)

�

uyL x,z,ω( ) = e−i3π / 4

8πωe−ik mx

xχmL (hs,ω)( )cmvmIm

Fy (z,ω)( )vmImm=1

∞

∑

�

uxR x,z,ω( ) = e−i3π / 4

8πωe−ik mx

xχmR (hs,ω)( )cmvmIm

Fx (z,ω)( )vmImm=1

∞

∑

�

uzR x,z,ω( ) = e−iπ / 4

8πωe−ik mx

xχmR (hs,ω)( )cmvmIm

Fz (z,ω)( )vmImm=1

∞

∑

Methodology - Modal Summation Technique

Synthetic seismograms

(s1f1) sre=168.00 dip=30.0 sde= 7.000 edi= 15.000 rde= 0.000 mod= 0- 0 int= 1 mag=6.5

0

5

10

15

20

25

90 100 110 120 130 140 150 160 170 180

cm

/s

Rake Angle (degrees)

rzt

Radial Velocity

0 5 10 15 20 25 30 35 40 45 50

Time (s)

Vertical Velocity

22.3176

0 5 10 15 20 25 30 35 40 45 50

Time (s)

Transverse Velocity

0 5 10 15 20 25 30 35 40 45 50

Time (s)

�

uyL x,z,ω( ) = e−i3π / 4

8πωe−ik mx

xχmL (hs,ω)( )cmvmIm

Fy (z,ω)( )vmImm=1

∞

∑

�

uxR x,z,ω( ) = e−i3π / 4

8πωe−ik mx

xχmR (hs,ω)( )cmvmIm

Fx (z,ω)( )vmImm=1

∞

∑

�

uzR x,z,ω( ) = e−iπ / 4

8πωe−ik mx

xχmR (hs,ω)( )cmvmIm

Fz (z,ω)( )vmImm=1

∞

∑Parametric tests

Methodology - Modal Summation Technique

Haskell, 1964sumatra

　　 Ishii et al., Nature 2005 doi:10.1038/nature03675

Rupture

Sumatra earthquake, Dec 28, 2004 　

Haskell dislocation model

Bulletin of the Seismological Society of America. Vol. 61, No. 1, pp. 221-223. February, 1971

MEMORIAL

NORMAN A. HASKELL (1905--1970)

Norman A. Haskell, a former Research Physicist and Branch Chief at Air Force Cambridge Research Laboratories, and President of the Seismological Society of America, died at Hyannis, Massachusetts on April 11 1970 after a long illness. He is survived by his wife and two children.

Dr. Haskell was one of the world's leading theoretical seismologists, perhaps best known for his development of the matrix method of computing the seismic effects of multiple horizontally-layered structures. His research interests spanned an extra- ordinarily broad geophysical range including the development of computational tech- niques in seismic, prospecting, the extension of seismic prospecting techniques to the mining industry, mechanics of the deformation of granitic rocks, underwater ballistics, atmospheric acoustics, blast phenomena, operations research, crustal structure and nuclear test detection.

NORMAN A. HASKELL

221

Haskell N. A. (1964). Total energy spectral density of elastic wave radiation from propagating faults, Bull. Seism. Soc. Am. 54, 1811-1841

Haskell source model: far field

�

ur (r,t) = ui ri,t − ri /α−Δti( )i=1

N∑ =

= RiPµ

4πρα3 W˙ D iri

t −Δti( )i=1

N∑ dx ≈

≈ RiPµ

4πρα3Wr

˙ D (t)∗δ t − xvr

⎛

⎝ ⎜

⎞

⎠ ⎟

i=1

N∑ dx ≈

≈ RiPµ

4πρα3Wr

˙ D (t)∗ δ t − xvr

⎛

⎝ ⎜

⎞

⎠ ⎟ dx

0

x∫ =

= RiPµ

4πρα3Wr

vr˙ D (t)∗B(t;Tr )

�

u(x,t) = 14πρα3 sin2θcosφˆ r ( )

˙ M t − r /α( )r

+

14πρβ3 cos2θcosφ ˆ θ − cosθsinφ ˆ φ ( )

˙ M t − r /β( )r

For a single segment (point source)

Haskell source model: far field

resulting in the convolution of two boxcars: the first with duration equal to the rise time and the second with duration

equal to the rupture time (L/vr)

�

ur (r,t)∝ ˙ D (t)∗vrH(z) t−x /vr

t = vr˙ D (t)∗B(t;Tr )

Haskell source model: directivityThe body waves generated from a breaking segment will arrive at a receiver before

than those that are radiated by a segment that ruptures later. If the path to the station is not perpendicular, the waves generated by different

segments will have different path lengths, and then unequal travel times.

Tr= L

vr

+ r − Lcosθc

⎛

⎝⎜

⎞

⎠⎟

⎡

⎣⎢⎢

⎤

⎦⎥⎥− r

c=

= Lv

r

− Lcosθc

⎛

⎝⎜

⎞

⎠⎟ =

Lv

r

1 −v

r

ccosθ

⎛

⎝⎜⎜

⎞

⎠⎟⎟

Rupture velocityEarthquake ruptures typically propagate at velocities that are in the range 70–90% of the S-wave velocity and this is independent of earthquake size. A small subset of earthquake ruptures appear to have propagated at speeds greater than the S-wave velocity. These supershear earthquakes have all been observed during large strike-slip events.

http://pangea.stanford.edu/~edunham/research/supershear.html

Directivity example

Ground motion scenarios

The two views in this movie show the cumulative velocities for a San Andreas earthquake TeraShake simulation, rupturing south to north and north to south. The crosshairs pinpoint the peak velocity

magnitude as the simulation progresses.www.scec.org

Source spectrum

�

U ω( ) = M0F ω( ) = M0

sin ωτ2

⎛ ⎝ ⎜

⎞ ⎠ ⎟

ωτ2

⎛ ⎝ ⎜

⎞ ⎠ ⎟

sin ωLvr 2⎛

⎝ ⎜

⎞

⎠ ⎟

ωLvr 2⎛

⎝ ⎜

⎞

⎠ ⎟

≈

M0 ω < 2Tr

2M0

ωTR 2

Tr< ω < 2

τ4M0

ω2τTR ω > 2

τ

⎧

⎨

⎪ ⎪ ⎪

⎩

⎪ ⎪ ⎪

�

u(t) = M0 B t;τ( )∗B t;TR( )( )

The displacement pulse, corrected for the geometrical spreading and the radiation pattern can be written as:

and in the frequency domain:

Source spectrum

�

U ω( ) ≈M0 ω < 2

Tr

2M0

ωTR

2Tr

< ω < 2τ

4M0

ω2τTR

ω > 2τ

⎧

⎨

⎪ ⎪ ⎪ ⎪

⎩

⎪ ⎪ ⎪ ⎪

Empirical source spectra

represent a set of average amplitude curves respect to:

Tectonic setting

Source mechanism

Directivity effects

13

15

17

19

21

23

1000 100 10 1 0.1

0.001 0.01 0.1 1 10

period (s)

frequency (Hz)

Log

FT[d

M/d

t] (N

m)

Empirical source spectra