Ground-Motions for Regions Ground-Motions for Regions Lacking Data from Earthquakes in Lacking Data from Earthquakes in

M-D Region of Engineering InterestM-D Region of Engineering Interest• Most predictions based on the stochastic

method, using data from smaller earthquakes to constrain such things as path and site effects

Simulation of ground motions

• representation theorem

• source– dynamic– kinematic

• path & site– wave propagation– represent with simple functions

Representation Theorem

d

txGcudtxu

q

npjijpqin

0,;,,,

Computed ground displacement

Representation Theorem

d

txGcudtxu

q

npjijpqin

0,;,,,

Model for the slip on the fault

Representation Theorem

d

txGcudtxu

q

npjijpqin

0,;,,,

Green’s function

Types of simulations

• Deterministic– Deterministic description of source– Wave propagation in layered media– Used for lower frequency motions

• Stochastic– Random source properties– Capture wave propagation by simple functional

forms– Can use deterministic calculations for some parts– Primarily for higher frequencies (of most engineering

concern)

Types of simulations

• Hybrid– Deterministic at low frequencies, stochastic

at high frequencies– Combine empirical ground-motion

prediction equations with stochastic simulations to account for differences in source and path properties (Campbell, ENA)

• Empirical Green’s function

Stochastic simulations

• Point source– With appropriate choice of source scaling,

duration, geometrical spreading, and distance can capture some effects of finite source

• Finite source– Many models, no consensus on the best (blind

prediction experiments show large variability)– Often incorporate point source stochastic model

The stochastic method

• Overview of the stochastic method– Time-series simulations– Random-vibration simulations

• Target amplitude spectrum– Source: M0, f0, Ds, source duration– Path: Q(f), G(R), path duration– Site: k, generic amplification

• Some practical points

• Deterministic modelling of high-frequency modelling not possible except in the far-field (lack of Earth detail and computational limitations)

• Hanks & McGuire (1981): Incoherent character of HF ground motion can be captured by representing it as band-limited, finite-duration Gaussian white noise (captures the essence of the physical process)

• Several methods use stochastic representation– D. Boore (1983 -)– Papageorgiou & Aki (1983)– Zeng et al. (1994)

• NB: Stochastic method = stochastic model

Stochastic modelling of ground-motion

SMSIM Programs (1983-)

• SMSIM = Stochastic Model SIMulation• Series of FORTRAN programs based on :

– Hanks & McGuire (1981) – Brune's point-source model (1970)

• Idea: combine deterministic target amplitude obtained from simple seismological model and random phase to obtain realistic high frequency motion. Try to capture the essence of the physics using simple functional forms for the seismological model

0.1 1.0 10 1000.01

0.1

1.0

10

100

Frequency (Hz)

Fou

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acce

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spec

trum

(cm

/sec

)

f0 f0

M = 7.0

M = 5.0

-200

0

200

20 25 30 35 40 45

-200

0

Time (sec)

Acc

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ec2)

M = 7.0

M = 5.0

R=10 km; =70 bars; hard rock; fmax=15 Hz

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Basis of stochastic method

Radiated energy described by the spectra in the top graph is assumed to be distributed randomly over a duration given by the addition of the source duration and a distant-dependent duration that captures the effect of wave propagation and scattering of energy

These are the results of actual simulations; the only thing that changed in the input to the computer program was the moment magnitude (5 and 7)

SMSIM Programs (1983-) - continued

• Ground motion and response parameters can then be obtained via two separate approaches:– Time-series simulation:

• Superimpose a random phase spectrum on a deterministic amplitude spectrum and compute synthetic record

• All measures of ground motion can be obtained

– Random vibration simulation:• Probability distribution of peaks is used to obtain peak parameters

directly from the target spectrum• Very fast• Can be used in cases when very long time series, requiring very large

Fourier transforms, are expected (large distances, large magnitudes)• Elastic response spectra, PGA, PGV, PGD, equivalent linear

(SHAKE-like) soil response can be obtained

Time-domain simulation

Step 1: Generation of random white noise

• Aim: Signal with random phase characteristics

• Probability distribution for amplitude– Gaussian (usual choice)– Uniform

• Array size from– Target duration

– Time step (explicit input parameter)

pathsourcegm TTT

Step 2: Windowing the noise

• Aim: produce time-series that look realistic

• Windowing functionWindowing function– Boxcar– Cosine-tapered

boxcar– Saragoni & Hart

(exponential)

Step 3: Transformation to frequency-domain

• FFT algorithm

Step 4: Normalisation of noise spectrum

• Divide by rms integral

• Aim of random noise generation = simulate random PHASE only

Normalisation required to keep energy content Normalisation required to keep energy content dictated by deterministic amplitude spectrumdictated by deterministic amplitude spectrum

Step 5: Multiply random noise spectrum by deterministic target amplitude

spectrum

Normalised amplitude spectrum

of noise with random phase characteristics

Earthquake source

Propagation path

Site respons

e

Instrument or ground

motion

TARGETTARGETFOURIERFOURIER

AMPLITUDEAMPLITUDESPECTRUMSPECTRUM

==Y(MY(M00,R,f),R,f) E(ME(M00,f),f) P(R,f)P(R,f) S(f)S(f) I(f)I(f)

Step 6: Transformation back to time-domain

• Numerical IFFT yields acceleration time series

• Manipulation as with empirical record

• 1 run = 1 realisation of random process

– Single time-history not necessarily realistic

– Values calculated =average over N simulations (50 < N< 200)

Steps in simulating time series

• Generate Gaussian or uniformly distributed random white noise

• Apply a shaping window in the time domain

• Compute Fourier transform of the windowed time series

• Normalize so that the average squared amplitude is unity

• Multiply by the spectral amplitude and shape of the ground motion

• Transform back to the time domain

0.1 1 101

2

10

20

100

Period (sec)

psv

(cm

/s)

box windowexponential window M = 7, R = 10 km

-200

0

200

20 30 40 50

-200

0

200

Time (sec)

Acc

ele

ratio

n(c

m/s

2)

Using box window

Using exponential window

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Effect of shaping window on response spectra

-200

200

-10

10

-1000

1000

-200

20

-0.5

0

0.5

0 10 20 30 40

-505

Time (sec)

M=7,R=10

Acceleration (cm/s2)

Velocity (cm/sec)

Response of Wood-Anderson seismometer (cm)

M=4,R=10

Acceleration (cm/s2)

Velocity (cm/sec)

Response of Wood-Anderson seismometer (cm)

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Acceleration, velocity, oscillator response for two very different magnitudes, changing only the magnitude in the input file

Warning: the spectrum of any one simulation may not closely match the specified spectrum. Only the average of many simulations is guaranteed to match the specified spectrum

Random Vibration Simulation

Random Vibration Simulations - General

• Aim: Improve efficiency by using Random Vibration Theory to model random phase

• Principle:– no time-series generation– peak measure of motion obtained directly

from deterministic Fourier amplitude spectrum through rms estimate

• Predicts peak time domain motion from frequency domain

• Makes use of Parseval’s Theorem

dFdttf22 )(

2

1)(

)(F is Fourier Transform of )(tf

• arms is easy to obtain from amplitude spectrum:

is acceleration time series)(tu

0

2

0

22 )(2

)(1

dffAT

dttuT

ad

T

drms

d

rmsa is root-mean-square acceleration

is ground motion durationdT

is Fourier amplitude spectrum of ground motion

2)( fA

• But need extreme value statistics to relate rms acceleration to peak time-domain acceleration

m2, m0 are spectral moments

time

02 04 06 08 0 100

27

-21

0.01

-0.04

420

-510

13

-13

Dec1 6, 2000 9:08:37a mC:\AKI_SYMP\RDTSM4M7.GRAC:\AKI_SYMP\RDTSM4M7.DT

M= 4

M= 7

acceleration (low-cut filtered, fc=0.1 Hz)

acceleration (low-cut filtered, fc=0.1 Hz)

T=10 sec, 5% damped oscillatorr esponse

T=10 sec, 5% dampedo scillatorr esponse

Special consideration needs to be given to choosing the proper duration T to be used in random vibration theory for computing the response spectra for small magnitudes and long oscillator periods. In this case the oscillator response is short duration, with little ringing as in the response for a larger earthquake. Several modifications to rvt have been published to deal with this.

0.1 1 10

0.01

0.02

0.1

0.2

1

2

Period (sec)

psv

(cm

/s)

Random Vibration, Boore & JoynerRandom Vibration, Liu & PezeshkTime Domain: 10 runsTime Domain: 40 runsTime Domain: 160 runsTime Domain: 640 runs

M = 4.0, R = 10 km

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Comparison of time domain and random vibration calculations, using two methods for dealing with nonstationary oscillator response.

For M = 4, R = 10 km

0.1 1 10

1

2

10

20

100

Period (sec)

psv

(cm

/s)

Random Vibration, Boore & JoynerRandom Vibration, Liu & PezeshkTime Domain: 10 runsTime Domain: 40 runsTime Domain: 160 runsTime Domain: 640 runs

M = 7.0, R = 10 km

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Comparison of time domain and random vibration calculations, using two methods for dealing with nonstationary oscillator response.

For M = 7, R = 10 km

Target Amplitude Spectrum

Basic Assumption

• Ground motion can be modeled as finite-duration bandlimited Gaussian noise, whose underlying spectrum can be specified (based on a seismological model)

THE KEY TO THE SUCCESS OF THE MODEL LIES IN BEING ABLE TO DEFINE FOURIER ACCELERATION SPECTRUM AS F(M, DIST)

Parameters required to specify Fourier accn as f(M,dist)

• Model of earthquake source spectrum

• Attenuation of spectrum with distance

• Duration of motion [=f(M, d)]

• Crustal constants (density, velocity)

• Near-surface attenuation (fmax or kappa)

Stochastic method

• To the extent possible the spectrum is given by seismological models

• Complex physics is encapsulated into simple functional forms

• Empirical findings can be easily incorporated

Target amplitude spectrumDeterministic function of source, path and site characteristics represented by separate multiplicative filters

Earthquake source

Propagation path

Site response

Instrument or ground

motion

)()(),(),(),,( 00 fIfGfRPfMEfRMY

Source Function

Source function E(M0, f)

),(),( 000 fMSMCfME

Source Source DISPLACEMENTDISPLACEMENT

SpectrumSpectrum

Scaling of amplitude spectrum with earthquake size

Scaling constantScaling constant• near-source crustal properties• assumptions about wave-type considered (e.g. SH)

Seismic momentSeismic moment

Measure of earthquake size

Scaling constant C

•βs = near-source shear-wave velocity

•ρs = near-source crustal density

•V = partition factor

•(Rθφ) = average radiation pattern

•F = free surface factor

•R0 = reference distance (1 km).

034 R

VFRC

ss

Brune source model• Brune’s point-source model

– Good description of small, simple ruptures – "surprisingly good approximation for many large events".

(Atkinson & Beresnev 1997)

• Single-corner frequency model

• High-frequency amplitude of acceleration scales as:

20

2

20

2

1

1

1

1)(

f

ffS

32310 Mahf

Semi-empirical two-corner-frequency models

• Aim: incorporate finite-source effects by refining the source scaling

• Example: AB95 & AS00 models

• Keep Brune's HF amplitude scaling

2

2

2

2

11

1)(

ba ff

ff

fS

fa, fb and determined empirically (visual inspection & best-fit)

32310 Mahf

0.01 0.1 1 10 1000.1

1

10

100

1000

10000

Frequency (Hz)

Fou

rier

Acc

eler

atio

nS

pect

rum

(cm

/s)

AB95H96Fea96 (no site amp)BC92J97

M = 7.5

M = 4.5

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The spectra can be more complex in shape and dependence on source size. These are some of the spectra proposed and used for simulating ground motions in eastern North America. The stochastic method does not care which spectral model is used. Providing the best model parameters is essential for reliable simulation results (garbage in, garbage out).

Scaling of the source spectrum• Based on Aki’s (1967) ω²-model

– Single corner frequency – For acceleration:

• LF: ω² increase

proportional to M0

• HF: constant amplitude,

depends on M0, as shown

– Self-similar scaling

• The key is to describe how the corner frequencies vary with M. Even for more complex sources, often try to relate the high-

frequency spectral level to a single stress parameter

cstfM 300

0.01 0.10 1.0 101024

1025

1026

1027

Frequency (Hz)

Acc

eler

atio

nS

ourc

eS

pect

rum

= 100 bars= 200 bars

M 6.5

M 7.5

M0

(M0)(1 / 3)( )(2 / 3)

-square spectra: Aki (1967)

M0 f03 = constant (similarity: Aki, 1967)

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Stress parameter: introduction• “Seismologists argue over everything about this parameter,

including its name, meaning, measurement, and scaling with magnitude”.

(Atkinson & Beresnev 1997 – "Don't call it stress drop")

• Different notions covered by single term/notation• Non-unique determination even for Brune model

(arbitrary assumptions required to represent finite fault by a point-source model)

• Physical meaning for 2-corner-frequency models not straightforward

Stress parameter: definitions• "Stress drop" should be reserved for static measure of

slip relative to fault dimensions

• "Brune stress drop" = change in tectonic (static) stress due to the event

• "SMSIM stress parameter" = “parameter controlling strength of high-frequency radiation” (Boore 1983)

r

u

u = average slip

r = characterisic fault dimension

Stress parameter:treatment in SMSIM

• Allowed to vary linearly with magnitude – Self-similar scaling breaks down at higher magnitudes

– Possibly trade-off with upper-crust attenuation ()

• Most applications use constant – Required for single-corner Brune model

– Trial & error approach: all other parameters fixed, multiple runs to find best estimate

– Characteristic of regional geology (shattered vs. competent rock)

– Brune stress drop used as guideline for trial & error

Stress parameter - Values• Brune stress drop

– Hanks & Kanamori (1975) • 10 to 100 bar• 30 bar for inter-plate events• 100 bar for intra-plate events • overall average of 60 bar

– Boore & Atkinson (1987) • 1 to 200 bar for moderate to

large earthquakes• literature review compiling the

opinion of several authors• is “apparently independent

of source strength over 12 orders of magnitude in seismic moment”.

• SMSIM stress parameter– California: 50 - 100 bar

– ENA: 150 – 200 bar(greater uncertainty)

• High values due to finite-fault & other effects (asperity) not included in point-source model

Source duration• Required to define array size (both TD & RV)• Determined from source scaling model via:

– For single-corner model, fa= fb = f0

– Weights wa and wb should add up to the distance-indepent coefficient in the expression giving total duration

b

f

a

fsource f

w

f

wT ba

Path Function

Path function P(R, f)

Point-source => spherical wave

P(R,f)P(R,f) =Geometrical

Spreading (R)Anelastic

Attenuation (R,f)

Loss of energy through spreading of the wavefront

Loss of energy through material damping & wave scattering by

heterogeneities

Propagation medium is neither perfectly elastic

nor perfectly homogeneous

The overall behavior of complex path-related effects can be captured by simple functions, leaving aleatory scatter. In this case the observations can be fit with a simple geometrical spreading and a frequency-dependent Q operator

In eastern North America a more complicated geometrical spreading factor is needed (here combined with the Q operator)

1 1.5 2 2.5 3-4

-3

-2

-1

0

1

log distance (km)

log

Fou

rier

ampl

itude

(cm

/sec

)

3.0<= mbLg <= 3.4vertical component

f = 1.0 Hz

1 1.5 2 2.5 3-4

-3

-2

-1

0

1

log distance (km)

3.0<= mbLg <= 3.4vertical component

f = 16 Hz

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Geometrical Spreading Function• Often 1/R decay (spherical

wave), at least within a few tens of km

• At greater distances, the decay is better characterised by 1/Rwith

• SMSIM allows n segment piecewise linear function in log (amplitude) – log (R) space

• Magnitude-dependent slopes possible : allows to capture finite-source effect (Silva 2002) Example: Boore & Atkinson 1995

Eastern North America model

10 20 30 100 200 300

0.01

0.02

0.03

0.1

Distance (km)

Geo

met

rical

Spr

eadi

ng

1/R

1/70

1/70 (130/R)0.5

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Path-dependent anelastic attenuation

Form of filter:

– Q(f) = « quality factor » of propagation medium in terms of (shear) wave transmission

– cq= velocity used to derive Q; often taken equal to s (not strictly true – depends on source depth)– N.B. Distance-independent term (kappa) removed since

accounted for elsewhere

))(

exp(qcfQ

Rf

Observed Q from around the world, indicating general dependence on frequency

SurfaceWaves

0.01 0.1 1 10 10010-4

0.001

0.01

0.1

Frequency (Hz)

Q-1

Imperial Fault, California

Central U.S.

Garm, Central Asia

Alaska

California

HinduKush

South Kurile

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Wave-transmission quality factor Q(f)

• Form usually assumed:

• Study of published relations led Boore to assign 3-segment piecewise linear form (in log-log space)

nfQfQ 0)(

0.01 0.1 1 10 10010-4

0.001

0.01

0.1

Frequency (Hz)

Q-1

ft1 ft2(fr1, Qr1)

(fr2, Qr2)

slop

e=

s1

slope=

s2

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Empirical determination of Q(f)

• Assuming simple functional forms for source function and geometrical spreading function (e.g. Brune model & 1/R decay)– Source-cancelling (for constant Q)– Best fit for assumed functional form (Q=Q0fn)

• Simultaneous inversion of source, path and site effects– One unconstrained dimension => additional assumption

required (e.g. site amplification)

• Trade-off problems

Path duration

• Required for array size• Usually assumed linear

with distance• SMSIM allows

representation by a piecewise linear function

• Regional characteristic, should be determined from empirical data

• Example: AB95 for ENA

0 100 200 300 400 5000

5

10

15

20

25

30

distance (km)

dura

tion

(sec

)

observed - sourcemean in 15-km wide binsused by Atkinson & Boore (1995)

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Site Response Function

Site response

• Form of filter:)()()( fDfAfG

Linear amplification for GENERIC site

Regional distance-independent attenuation (high frequency)

• Near-surface anelastic attenuation?• Source effect?• Combination?

Site amplification• Attenuation function for

GENERIC site• Modelled as a piecewise linear

function in log-log space• Soil non-linearity effects not

included• Determined from crustal velocity

& density profile via SITE_AMP– Square-root of impedance

approximation– Quarter-wave-length approximation

(f – dependent)

Site attenuation• Form of filter:

• Reflects lack of consensus about representation– fmax (Hanks, 1982) : high-frequency cut-off

– (Anderson & Hough, 1984) : high-frequency decay

)exp(

1

1)( 08

max

f

f

ffD

Cut-off frequency fmax

• Hanks (1982)– Observed empirical spectra exhibit cut-off in log-log space– Value of cut-off in narrow range of frequency– Attributed to site effect

• Other authors (e.g. Papageorgiou & Aki 1983) consider fmax to be a source effect

• Boore’s position: – Multiplicative nature of filter allows for both approaches– Classification as site effect = « book-keeping » matter– Often set to a high value (50 to 100 Hz) when preference

is given to the kappa filter

Kappa factor • Anderson & Hough (1984)

– empirical spectra plotted in semi-log axes exhibit exponential HF decay

– rate of this decay = varies with distance)

• Treatment in SMSIM– similar determination, but with records corrected for path

effects and site amplification– parameter used = = zero-distance intercept– allowed to vary with magnitude

• source effect (at least partly)• trade-off with source strength (characteristic of regional surface

geology)

0.1 1 10 100

1

2

3

4

Frequency (Hz)

Am

plifi

catio

n,re

lativ

eto

sour

ce

0 = 0.0 s

0 = 0.01

0 = 0.02

0 = 0.04

0 = 0.08

Combined effect of generic rock amplification and diminution

(diminution = e(- 0 f))

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Site response is represented by a table of frequencies and amplifications. Shown here is the generic rock amplification for coastal California ( = 0.04), combined with the diminution factor for various values of

0.01 0.1 1 100.4

1

2

3

4

Frequency (Hz)

Am

ps*

e(-

0f)

V30 = 255 m/s (NEHRP class D)

V30 = 310 m/s (generic soil)

V30 = 520 m/s (NEHRP class C)

V30 = 620 m/s (generic rock)

V30 = 2900 m/s (generic very hard rock)

0 = 0.035 s, except = 0.003 s for very hard rock

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Combined amplification and diminution filter for various average site classes

Instrument/Ground motion filter

• For ground-motion simulations:

• For oscillator response:

niffI )2()( n=0 displacementn=1 velocityn=1 acceleration

rr iffff

VffI

2)()(

22

2

fr = undamped natural frequency

ξ = damping

V = gain

(for response spectra, V=1).

i² = 1

A few applications

• Scaling of ground motion with magnitude

• Simulating ground motions for a specific M, R

• Extrapolating observed ground motion to part of M, R space with no observations

Other applications of the stochastic method

• Generate motions for many M, R; use regression to fit ground-motion prediction equations (used in U.S. National Seismic Hazard Maps)

• Design-motion specification for critical structures

• Derive parameters such as ,, from strong-motion data

Other applications of the stochastic method

• Studies of sensitivity of ground motions to model parameters

• Relate time-domain and frequency-domain measures of ground motion

• Generate time series for use in nonlinear analysis of structures and site response

• Use to compute motions from subfaults in finite-fault simulations

Application: study magnitude scaling for a fixed distance

Note that response of long period oscillators is more sensitive to magnitude than short period oscillators, as expected from previous discussions. Also note the nonlinear magnitude dependence for the longer period oscillators.

4 5 6 7 8

1

10

100

1000

M

PS

A(M

)/(P

SA

(4.0

)

T = 0.1 s

T = 0.3 s

T = 1.0 s

T = 2.0 s

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R = 20 km

Magnitude scaling

Expected scaling for simplest self-similar model. Does this imply a breakdown in self similarity? The simulations were done for a fixed close distance, and more simulations are needed at other distances to be sure that a M-dependent depth term is not contributing.

4 5 6 7 81

2

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6

Me

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Observed (M 5.6)Observed (M 5.6), scaled to M 7.5 using SMSIMObserved, from S wave only (no basin waves)Regression: Boore etal, 1997, V30 = 216 m/sRegression: Abrahamson & Silva, 1997, corrected to 216 m/sStochastic-method simulation: AB98 model

M 5.6

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Applications:

1) extrapolating small earthquake motion to large earthquake motion (makes use of path and site effects in the small ‘quake recording)

2) predict response spectra for two magnitudes without use of the observed recording

Note comparison with empirical prediction equations and importance of basin waves (not in simulations)

1 10 100 1000

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Used by BJF93 for pga

Western North America

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Eastern North America

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Observed data adequate for regression exceptclose to large ‘quakes

Observed data not adequate for regression, use simulated data

Atkinson and Boore (1995)

stochastic ground-motion relations (used in 2003 GSC

national hazard maps)

Some observations about predicting ground motions

• Empirical analysis:– Adequate data in some places for M around 7– Lacking data at close distances to large earthquakes– Lacking data in most parts of the world– May be possible to “export” relations from data-rich areas to

tectonically similar regions– Need more data to resolve effects such as nonlinear soil response,

breakdown of similarity in scaling with magnitude, directivity, fault normal/fault parallel motions, style of faulting, etc.

• Stochastic method:– Very useful for many applications– Source specification for one region might be used for predictions in

another region– Can use motions from more abundant smaller earthquakes to define

path, site effects– Site effects, including nonlinear response, might be exportable from

data-rich regions • Hybrid method combining empirical and theoretical predictions can be

very useful

Stochastic model shown to this point is based on a point source model of the earthquake

source. But we know finite-fault effects are important for large earthquakes…..

Therefore need to extend stochastic model to account for main finite fault effects, such as

geometric effects and directivity

Self-healing model: in the lower

panel, only part of the

fault is slipping at

any one time

Finite-fault model reproduces 2-corner

shape of source spectrum that is

observed empirically (thus providing an

explanation as to why we obtain a 2-corner

shape)

Summary• Stochastic models are a useful tool for interpreting

ground motion records and developing ground motion relations

• Model parameters need careful validation• Point-source models appropriate for small

magnitudes, and work pretty well on average for large magnitudes IF a 2-corner source spectrum is used to mimic overall finite fault effects

• Finite-fault models preferable for large earthquakes (M>6) because they can model more realistic fault behaviour

The future: Numerical simulations, deterministic

plus stochastic

•Show southeast epicenter svx.mpeg•Show Chuetsu Tokyo simulation’•Show Tottori observed, simulated (discuss what is explained, what is not explained)

End

Use Random Vibration Theory (RVT) to obtain the peak factor

• Cartwright & Longuet-Higgins (1956)

dzzN

N

y

yeN

e

z

rms

0

2max exp112

Ne = number of peaksNz = number of zero-crossings

max / rmsy y

Applies to motions in general, not just acceleration (hence “y” rather than “a”)

Random Vibration Theory - continued

• Assuming that the peaks are uncorrelated and the signal is stationary, Ne

and Nz depend only on– Trms (obtained from Tgm after oscillator correction)– Statistical moments mk of squared amplitude spectrum

• Yrms is also obtained from the amplitude spectrum using Parseval's theorem:

• Replacing in CLH equation yields Ypeak

dffYfm kk

2

0)()2(2

rmsrms Tmy 0

Random Vibration Simulation - Results• Neither stationarity nor

uncorrelated peaks assumption true for real earthquake signal

• Nevertheless, RV yields good results at greatly reduced computer time

• Problems essentially with– Long-period response– Lightly damped oscillators– Corrections developed

Seismic moment M0

• Definition

• Use– Provides measure of seismic energy for a

double-couple point source model– Can be determined independently from strong-

motion records => source scaling

UAM 0

= average rigidity of the crust

A= rupture area

U=average slip

Determination of seismic moment

• Related to moment magnitude Mw via

• Harvard CMT solution for empirical records• In the absence of CMT solution, can be

estimated from magnitude (converted to Mw if

necessary)

7.10log3

20 MM w (Hanks & Kanamori,1979)

N.B. M0 in dyne.cm

Wave propagation produces changes of amplitude (not shown here) and increased duration with distance. Not included in these simulations is the effect of scattering, which adds more increases in duration and smooths over some of the amplitude changes (as does combining propagation over various profiles with laterally changing velocity)

Some practical points…

Parameters to get from empirical data

• Focal depth distribution

• Crustal structure– S-wave velocity profile

– Density profile

– Q(f)

– Geometrical spreading

– Kappa

• Duration– Total duration

– Path duration

• Source scaling– Corner frequency/ies

– High-frequency spectral amplitude

– Seismic moment (independent determination)

– Brune stress drop for CALIBRATION => catalogue

• Information about site

Some points to consider• Uncertainty

Specify parametric uncertainty when giving an estimate from empirical data

• Trade-offs between parameters– and (or similar)– Geometrical spreading & anelastic attenuation– Site amplification and

always state assumptions

• Consistency more important than uniqueness

Recent ground motion relations compared to ENA data (rock sites): M 5 at 1 Hz

S02

Recent ground motion relations compared to ENA data (rock sites): M 5 at 5 Hz

Recent ground motion relations compared to ENA data (rock sites): M 5 at 10 Hz

No data for validation at M7: alternatives diverse at low frequency

No data for validation at M7: alternatives agree at high frequency!

2-corner stochastic California relations are in good agreement with empirical relations (Atkinson and

Silva, 2000)

Four realizations of a composite source.

400 bars 25 bars

100 bars 100 bars

Recap on Composite Source Model

• Much more physics than ground motion prediction equations, so should be able to do better.

• Source is still kinematic, not realistic.– Real physics of the source are not known.

– Current dynamic models can only generate low frequency seismograms, based on ad-hoc assumptions.

– May be a long time before we can make the source “realistic”.

– Meanwhile, the composite source is satisfactory.

Example of observed directivity effects in the Landers earthquake ground motions near the fault.

Directivity was a key factor in causing large ground motions in Kobe, Japan, and a major damage factor.

Extension of stochastic model

to finite faults (Silva; Beresnev and Atkinson;

Motazedian and Atkinson, Pacor et

al.)

Parameters needed to apply stochastic finite-fault model

• All parameters needed for stochastic point source model

• Geometry of source (can assume fault plane based on empirical relations such as Wells and Coppersmith on fault length and width vs. M)

• Direction of rupture propagation (can assume random or bilateral)

• Slip distribution on fault (can assume random)• Can also incorporate ‘self-healing’ slip behaviour

Other stochastic finite-fault models (Zeng & Anderson, Papageorgiou &

Aki)

• Kinematic model• Power-law distribution of circular subevents

– Maximum radius– Fractal dimension

• Random placement on fault• Visualize as Eshelby crack

– Subevent stress drop– Radiated pulse

Green’s Function

• Flat-layered half space– Luco-Apsel (1982) approach to compute– Use model for

• Adjustments for scattering on path, at source.