Regional seismic hazard scenarios (ground motion at bedrock) Multiscale Hazard Scenarios
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
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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
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]
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
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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
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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