FUNDAMENTALS of ENGINEERING SEISMOLOGY

GROUND MOTIONS FROM SIMULATIONS

1

Observed data adequate for regression exceptclose to large ‘quakes

Observed data not adequate for regression, use simulated data

2

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

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

5

Representation Theorem

d

txGcudtxu

q

npjijpqin

0,;,,,

Model for the slip on the fault

6

Representation Theorem

d

txGcudtxu

q

npjijpqin

0,;,,,

Green’s function

7

Types of simulations• Deterministic

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

• Stochastic– Not purely theoretical– 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– Meaning 1: Deterministic at low frequencies,

stochastic at high frequencies– Meaning 2: Combine empirical ground-motion

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

– Meaning 3: Stochastic source, empirical Green’s function for path and site

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)– Usually use point-source stochastic model– Can be theoretical (many types: deterministic and/or

stochastic, and can also use empirical Green’s functions)

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 waves not possible (lack of Earth detail and computational limitations)

• Treat high-frequency motions as filtered white noise (Hanks & McGuire , 1981).

• combine deterministic target amplitude obtained from simple seismological model and quasi-random phase to obtain high-frequency motion. Try to capture the essence of the physics using simple functional forms for the seismological model. Use empirical data when possible to determine the parameters.

Stochastic modelling of ground-motion: Point Source

12

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 (4, 6, and 8)

13

-200

200

-10

10

-1000

1000

-200

20

-0.50

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)

File:

C:\m

etu_

03\s

imula

tion\

AS47

WA_

4ppt

.dra

w;Da

te:

2003

-09-

18;

Time:

22:1

6:44Acceleration,

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

14

• Ground motion and response parameters can be obtained via two separate approaches:

– Time-series simulation:• Superimpose a quasi-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

15

“The Stochastic method has a long history of performing better than it should in terms of matching observed ground-motion characteristics. It is a simple tool that combines a good deal of empiricism with a little seismology and yet has been as successful as more sophisticated methods in predicting ground-motion amplitudes over a broad range of magnitudes, distances, frequencies, and tectonic environments. It has the considerable advantage of being simple and versatile and requiring little advance information on the slip distribution or details of the Earth structure. For this reason, it is not only a good modeling tool for past earthquakes, but a valuable tool for predicting ground motion for future events with unknown slip distributions.”

--Motazedian and Atkinson (2005)

16

Time-domain simulation

17

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

18

Step 2: Windowing the noise

• Aim: produce time-series that look realistic

• Windowing function– Boxcar– Cosine-tapered

boxcar– Saragoni & Hart

(exponential)19

Step 3: Transformation to frequency-domain

• FFT algorithm

20

Step 4: Normalisation of noise spectrum

• Divide by rms integral

• Aim of random noise generation = simulate random PHASE onlyNormalisation required to keep energy content dictated by deterministic amplitude spectrum

21

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 response

Instrument or ground motion

TARGETFOURIER

AMPLITUDESPECTRUM

=Y(M0,R,f) E(M0,f) P(R,f) G(f) I(f)X X X

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 (N large enough to yield an accurate value of ground-motion intensity measure)

23

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

24

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

eler

atio

n(c

m/s

2 )

Using box window

Using exponential window

File:

C:\m

etu_

03\s

imula

tion\

box_

exp_

4ppt

.dra

w;Da

te:

2003

-09-

18;

Time:

21:4

7:27

Effect of shaping window on response spectra

25

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

26

Random Vibration Simulation

27

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

• yrms is easy to obtain from amplitude spectrum:

Is ground-motion time series (e.g., accel. or osc. response))(tu

2 22

0 0

1 2( ) ( )dT

rmsrms rms

y u t dt U f dfD D

rmsy is root-mean-square motion

is a duration measurermsDis Fourier amplitude spectrum of ground motion

2( )U f

• But need extreme value statistics to relate rms acceleration to peak time-domain ground-motion intensity measure (ymax)

Parseval's theorem

29

1 2max 2ln Zrms

y Ny

2Z ZN f D

1 22 0

12Zf m m

Peak parameters from random vibration theory:

For long duration (D) this equation gives the peak motion given the rms motion:

where

m0 and m2 are spectral moments, given by integrals over the Fourier spectra of the ground motion

30

Random Vibration Simulation – Possible Limitations

• 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

31

time

02 04 06 08 0 100

27

-210.01

-0.04420

-51013

-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.

32

0.1 1 100.01

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

File:

C:\m

etu_

03\s

imula

tion\

nsim

s_m

4_4p

pt.d

raw;

Date

:20

03-0

9-19

;Tim

e:10

:46:

42

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

For M = 4, R = 10 km

33

0.1 1 101

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

File:

C:\m

etu_

03\s

imula

tion\

nsim

s_m

7_4p

pt.d

raw;

Date

:20

03-0

9-19

;Tim

e:10

:57:

27

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

For M = 7, R = 10 km

34

Recent improvements on determining Drms (Boore and Thompson, 2012):

Contour plots of TD/RV ratios for an ENA SCF 250 bar model for 4 ways of determining Drms:

1. Drms = Dex

2. BJ843. LP994. BT12

35

Target amplitude spectrum

Deterministic function of source, path and site characteristics represented by separate multiplicative filters

Earthquake source

Propagation path

Site response

Instrument or ground

motion

0 0( , , ) ( , ) ( , ) ( ) ( )Y M R f E M f P R f G f I f

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

Source Function

39

Source function E(M0, f)

),(),( 000 fMSMCfME

Source DISPLACEMENT

Spectrum

Scaling of amplitude spectrum with earthquake size

Scaling constant• near-source crustal

properties• assumptions about wave-

type considered (e.g. SH)

Seismic momentMeasure of

earthquake size

Scaling constant C (frequency independent)

• 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)(

ff

fS

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 e determined empirically (visual inspection & best-fit)

32310 Mahf

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)

ru

u = average slipr = characterisic fault dimension

32310 Mahf

Stress parameter - Values

• SMSIM stress parameter– California: 50 - 200

bar– ENA: 150 – 1000 bar

(greater uncertainty)

45

0.01 0.1 1 10 1000.1

1

10

100

1000

10000

Frequency (Hz)

Four

ierA

ccel

erat

ion

Spe

ctru

m(c

m/s

)

AB95H96Fea96 (no site amp)BC92J97

M = 7.5

M = 4.5

File:

C:\m

etu_

03\re

c_pr

oc_s

trong

_mot

ion\F

AS_

XCA.

draw

;Da

te:

2003

-09-

15;

Time:

14:4

9:29

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).

46

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)

File:

C:\m

etu_

03\re

c_pr

oc_s

trong

_mot

ion\S

PCTS

CL_a

ki_fig

2_4p

pt.d

raw;

Date

:20

03-0

9-15

;Tim

e:14

:58:

22

Source Scaling • Low frequency

• A≈ M0, but log M0 ≈ 1.5M, so A ≈ 101.5M. This is a factor of 32 for a unit increase in M

• High frequency

• A ≈ M0(1/3), but log M0 ≈

1.5M, so A ≈ 100.5M. This is a factor of 3 for a unit increase in M

• Ground motion at frequencies of engineering interest does not increase by 10x for each unit increase in M

• 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 47

Source duration

• Determined from source scaling model via:

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

b

f

a

fsource f

wf

wT ba

Path Function

49

Path function P(R, f)

Point-source => spherical wave

P(R,f) = GeometricalSpreading (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

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)

51

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

52

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

Four

ier

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

File:

C:\m

etu_

03\s

imula

tion\

Fas0

1_16

_4pp

t.dra

w;Da

te:

2003

-09-

18;

Time:

10:2

2:23

53

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/Ra with a <1

• 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)G

eom

etric

alS

prea

ding

1/R

1/70

1/70 (130/R)0.5

File

:C:

\met

u_03

\sim

ulatio

n\G

SPR

DFIG

.dra

w;

Dat

e:20

03-0

9-18

;Ti

me:

10:2

4:17

54

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( )( ) q

π f RQ f c

55

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

File:

C:\m

etu_

03\s

imula

tion\

aki_q

_4pp

t.dra

w;Da

te:

2003

-09-

18;

Time:

10:2

1:39

56

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)

0 0( ) nQ f Q f f

0.01 0.1 1 10 10010-4

0.001

0.01

0.1

Frequency (Hz)

Q-1

ft1 ft2(fr1, Qr1)

(fr2, Qr2)

slope

=s1

slope= s2

File:

C:\m

etu_

03\s

imula

tion\

IQ_F

IG_4

ppt.d

raw;

Date

:20

03-0

9-18

;Tim

e:10

:23:

12

57

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)

File:

C:\m

etu_

03\s

imula

tion\

DURN

2_4p

pt.d

raw;

Date

:20

03-0

9-18

;Tim

e:10

:35:

40

59

Site Response Function

60

Site response

• Form of filter:( ) ( ) ( )G f A f D f

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)

62

Site attenuation• Form of filter:

• Reflects lack of consensus about representation– fmax (Hanks, 1982) : high-frequency cut-off– k0 (Anderson & Hough, 1984) : high-frequency decay– Both factors seldom used together

)exp(

1

1)( 08

max

f

ff

fD k

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 k0

• Anderson & Hough (1984) – empirical spectra plotted in semi-log axes exhibit exponential HF

decay– rate of this decay = k varies with distance)

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

and site amplification– parameter used = k0 = 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))

File:

C:\m

etu_

03\s

imula

tion\

A_K4

PAPR

_4pp

t.dra

w;Da

te:

2003

-09-

18;

Time:

10:4

5:55

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

66

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

File:

C:\m

etu_

03\s

imula

tion\

AMPK

_FAS

_4pp

t.dra

w;Da

te:

2003

-09-

18;

Time:

11:3

1:37

Combined amplification and diminution filter for various average site classes

67

Instrument/Ground motion filter

• For ground-motion simulations:

• For oscillator response:

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

rr iffffVffI

2)()( 22

2

fr = undamped natural frequency

ξ = damping V = gain (for response spectra, V=1).

i² = 1

Summary of Parameters Needed for Stochastic Simulations

69

Parameters needed for Stochastic Simulations

• Frequency-independent parameters– Density near the source– Shear-wave velocity near the source– Average radiation pattern– Partition factor of motion into two components

(usually )– Free surface factor (usually 2)

1 2

70

Parameters needed for Stochastic Simulations

• Frequency-dependent parameters– Source:

• Spectral shape (e.g., single corner frequency; two corner frequency)

• Scaling of shape with magnitude (controlled by the stress parameter Δσ for single-corner-frequency models)

71

Parameters needed for Stochastic Simulations

• Frequency-dependent parameters– Path (and site):

• Geometrical spreading (multi-segments?)• Q (frequency-dependent? What shear-wave and

geometrical spreading model used in Q determination?)• Duration• Crustal amplification (can include local site

amplification)• Site diminution (fmax? κ0?)

correlated

72

Parameters needed for Stochastic Simulations

• RV or TD parameters– Low-cut filter– RV

• Integration parameters• Method for computing Drms

• Equation for ymax/yrms

– TD• Type of window (e.g., box, shaped?)• dt, npts, nsims, etc.

73

Parameters that might be obtained from empirical analysis of small earthquake data

• Focal depth distribution• Crustal structure

– S-wave velocity profile– Density profile

• Path Effects– Geometrical spreading– Q(f)– Duration– κ0

– Site characteristics

74

Parameters difficult to obtain from small earthquake data

• Source Spectral Shape• Scaling of Source Spectra

(including determination of Δσ)

75

Some points to consider

• UncertaintySpecify parametric uncertainty when giving an estimate from empirical data

• Trade-offs between parameters– and k0 (or similar)– Geometrical spreading & anelastic attenuation– Site amplification and k0

always state assumptions • Consistency more important than

uniqueness76

Applicability of Point Source Simulations near Extended Ruptures

• Modify the value of Rrup used in point source, to account for finite fault effects– Use Reff (similar to Rrms) for a particular source-

station geometry)– Use a more generic modification, based on finite-

fault modeling (e.g., Atkinson and Silva, 2000; Toro, 2002)

77

(note: No directivity---EXSIM results are an average of motions from 100 random hypocenters)

Modified from Boore (2010)

Use a scenario-specific modification to Rrup

78

Using generic modifications to Rrup. For the situation in the previous slide (M 7, Rrup = 2.5):

Reff = 10.3 km for AS00Reff = 8.4 km for T02

Compared to Reff = 10.3 (off tip) and 6.7 (normal) in the previous slide

79

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 , k, 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

File:

C:\m

etu_

03\s

imula

tion\

mag

_sca

ling_

t0p1

_0p3

_1p0

_2p0

.dra

w;Da

te:

2003

-09-

19;

Time:

12:1

9:58

R = 20 km

83

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

3

4

5

6

Mev

ent_

term

4 5 6 7 81

2

3

4

5

6

M

even

t_te

rm

4 5 6 7 81

2

3

4

5

6

M

even

t_te

rm

4 5 6 7 80

1

2

3

4

5

Mev

ent_

term

T=0.1, 0.3, 1.0, 2.0 secrjb _< 80: buried eventsrjb _< 80: events with surface slip

File

:C

:\pe

er_n

ga\t

eam

x\st

age1

_eve

nt_t

erm

s_al

l_pe

r_no

bs_g

t_1_

rle_8

0_su

rfsl

ip_s

msi

m.d

raw

;D

ate:

2005

-05-

24;

Tim

e:10

:49:

37

84

0.1 0.2 1 2 10 200.1

1

10

100

1000

Period (sec)

PS

V(c

m/s

ec)

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

M 7.5

File:

C:\m

etu_

03\s

imula

tion\

SEM

S2AK

I_4p

pt.d

raw;

Date

:20

03-0

9-19

;Tim

e:12

:32:

55

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)

85

1 10 100 1000

5

6

7

8

Mom

entM

agni

tude

Used by BJF93 for pga

Western North America

1 10 100 1000

5

6

7

8

Distance (km)

Mom

entM

agni

tude

AccelerographsSeismographic Stations

Eastern North America

File

:C:\m

etu_

03\r

egre

ss\m

_d_w

na_e

na_p

ga.d

raw

;D

ate:

2003

-09-

05;

Tim

e:14

:40:

01

Observed data adequate for regression exceptclose to large ‘quakes

Observed data not adequate for regression, use simulated data

This is a major application of stochastic simulations

86

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

Some Limitations to Point-Source Model (“A kindergarten model”—A. Frankel)

• Faults are not points• No spatial coherence• No correlation between components• Inadequate period-to-period correlation• Etc.

Finite Faults• Stochastic

– Silva– Zeng et al.– Papageorgiou & Aki– Beresnev & Atkinson (FINSIM)– Motazedian & Atkinson (EXSIM)– Pacor, Cultrera, Mendez, & Cocco (DSM)– Irikura et al.– Etc.

• Deterministic + Stochastic– Graves et al.– Frankel– Pacor et al. ? (simple Green’s function, does not include wave propagation

effects)– Etc.

Extension of stochastic model

to finite faults (Silva; Beresnev

and Atkinson; Motazedian and

Atkinson, Pacor et al.)

90

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

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 an adjustment is made to the distance (and perhaps to the shape of the source spectrum

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

Case Study

• M scaling-go over each input. Don’t forget FFF, Rjb—Rrup calculation, discuss durpath, generalized source spectrum

• Use figure from Erice paper showing reff• TD vs RVT computations

93

94

95

96

97

Another Case Study—PSA

vs R

98

The future (is here?): Numerical simulations,

deterministic plus stochastic

Continue to Rob Grave’s PPT slides

99