An Extension of Stochastic Green’s Function Method to Long-Period Strong

Ground-motion Simulation

Y. Hisada and J. Bielak

Purpose: Extension of Stochastic Green’s Function to Longer Periods

Realistic Phases

　・ Random Phases at Shorter Period

　・ Coherent Phases at Longer Periods

→ Directivity Pulses, Fling Step, Seismic Moment Realistic Green’s Functions (e.g., surface wave)

　・ Green’s Functions of Layered Half-Space

→ Easy to compute them at shorter periods

(e.g., Hisada, 1993, 1995)

Broadband Strong Ground Motion Simulation (Hybrid Methods)

Short period （ < １ s ）： Stochastic and empirical methods （ ex., Stochastic Green’s function method ）　　　　　　　→ omega-squared model, random phases

Long period （ > １ s ）： Deterministic methods （ FDM, FEM, Green’s functions for layered media ） → coherent phases (e.g., directivity pulses), seismic moment

0 　 1 2 period (s)

short ←→ long period

M7 Eq.

0 　 1 2 　 4 period (s)

M8 Eq.

short ←→ long period

Broadband Strong Ground Motion Simulation (Hybrid Methods)

The crossing period is around 1 sec. Ok for M7 eq., but not for M8 eq. → Resolution for M8 eq. is not fine enough at 1 sec (e.g., Size of sub-faults is 10 – 20 km.) Extension of deterministic methods to shorter periods.

0 　 1 2 period (s)

short ←→ long period

M7 Eq.

0 　 1 2 　 4 period (s)

M8 Eq.

short ←→ long period

Modified K-2 model （ Hisada, 2000 ）

0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2

time (sec)

Slip

Vel

ocity

k2 slip distribution k2 rupture time Kostrov-type slip velocitywith fmax

・ Slip and rupture time are continuous on a fault plane・ Large number of source points at shorter periods・ Ok for FEM (FDM), but not for theoretical methods using Green’s Function of Layered half-space

1/k2

0 k (wavenumber)

ampl

itude

Phase: coherent random

Source spec.: ω2 model

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

0.01 0.1 1 10 100

frequency (Hz)

Sour

ce S

pect

ra 1/ω2

frequencyk2 model

Broadband Strong Ground Motion Simulation (Hybrid Methods)

Short period （ < １ s ）： Stochastic and empirical methods （ ex., Stochastic Green’s function method ）　　　　　　　→ omega-squared model, random phases

Long period （ > １ s ）： Deterministic methods （ FDM, FEM, Green’s functions for layered media ） → coherent phases (e.g., directivity pulses), seismic moment

0 　 1 2 period (s)

short ←→ long period

M7 Eq.

0 　 1 2 　 4 period (s)

M8 Eq.

short ←→ long period

Stochastic Green’s Function Method (Kamae et al., 1998) : Boore’s Source Model + Irikura’s Empirical Green’s Function Summation Method

→ Fast Computation: One source point per sub-fault→ Green’s Functions of the Far-Field S Wave (1/r)

Observation Point

Seismic Fault

Moment Rate Function with ω2 Model (Ohnishi and Horike, 2000)

Far-Field S-waves from a point source

Far-Field S-waves from Boore’s source model

Moment Rate Function for ω2 model

)()()(

)( max00

0 fPfSMdt

MdMi c

Slip Velocity for ω2 model

Representation Theorem for ω2 model

For Point Dislocation Source

)()()(

)( maxfPfDSdt

DdDi c

deUneneDU rtijikijjik

)())(()( *,

)())(()( *,0 jikijjik UneneMU

Boore’s Source Model with Random Phases

ω2 Amplitude

＋ Random Phases

fc=1 Hz

fmax=10 Hz

Moment Rate Function

（ Slip Velocity Function ）FIT withTime Window

・ Unstable and Incoherent at Longer Periods→ ○Acceleration×Directivity Pulses ×Fling Step×Seismic Moment

Example 1

Example 2

Boore’s Source Model with Zero Phases (Coherent Phases)

ω2 Amplitude ＋ Zero Phases

fc=1 Hz

fmax=10 Hz

Moment Rate Function

（ Slip Velocity Function ）

-0.5

0

0.5

1

1.5

0 2 4 6 8 10

time (sec)

slip

vel

octy

Moment Rate +1/fc sec delay

Moment Function

・ Smoothed Ramp Function→ ×Acceleration ○Directivity Pulses○Fling Step ○Seismic Moment

FIT withNo Time Window

Boore’s Source Model with Zero and Random Phases (Introduction of fr) ω2 Amp. ＋ Zero and

Random Phases

fc=1 Hz

fmax=10 Hz

Moment Rate Function

（ Slip Velocity Function ）

fr=1 HzMoment Function

Moment Rate +1/fc sec delay

・ Ramp Function with high freq. ripples→ ○Acceleration ○Directivity Pulses ○Fling Step○Seismic Moment

FIT withTime Window

Example (Boore’s Source with Zero Phases) : r=20 km, Vp=5,Vs=3km/s

R=20 ｋｍ

R=20 ｋｍ

45° S wave

P wave

Moment Rate Function１． Triangle （ τ=1s ）２． ω ２ Model （ fc=1 Hz fmax=10 Hz, 0 phases ）

Point Source at 20 km

0.00001

0.0001

0.001

0.01

0.1

1

10

0.01 0.1 1 10 100

frequency (Hz)

Fo

uri

er

Dis

pla

ce

me

nt

Am

plit

ud

e

(cm

*se

c)

x11(20,0)-no rand

x11(14.1,14.1)-1/r & no rand

x11(20,0)-grfltHF1

x11(14.1,14.1)-grfltHF2

Far Field Terms at r = 20 km (Point Source)

0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6 7 8 9 10 11 12

time (sec)

dis

pla

ce

me

nt

(X)

Current Model (S wave)

Current Model (P wave)

Triangle Slip Model (S wave)

Triangle Slip Model (P wave)

Far Field Displacement

P Wave

S Wave

S Wave

P Wave

Boore’s Source with Zero and Random Phases

acc

-400

-300

-200

-100

0

100

200

300

400

0 1 2 3 4 5 6 7 8 9 10 11 12

acc

point source and far field term at (20,0) km

-2

-1

0

1

2

3

4

5

6

7

8

9

0 1 2 3 4 5 6 7 8 9 10 11 12

time (sec)

dis

p

x11(20,0)-1/r & idum=1 &fran=1 HzFar-Field

Displacement

S Wave

0.0001

0.001

0.01

0.1

1

10

0.01 0.1 1 10 100

frequency (Hz)

Mo

men

t R

ate

original

result

Far-Field Acceleration

Proposed Modelfc=1 Hzfmax=10 Hzfr=1 H ｚ

S Wave

Summarized Results (Three Models)

- 3

2

7

12

17

22

27

0 10 20time (s)

Dis

plac

emen

t (c

m)

TriangleSlipVelocity

Boore'sAmplitude+ 0 Phases

Boore'sAmplitude+ fr=1 Hz

- 40

- 20

0

20

40

60

80

100

120

140

0 10 20

time (s)

Vel

ocity

(cm

/s)

-600

-100

400

900

1400

1900

2400

0 10 20

time (s)

acce

lera

tion

(gal

) Displacement 　　 Velocity 　 　 Acceleration

Triangle Slip Velocity

ω2 Model + 0 Phases

ω2 Model + 0 & Random Phases

Summary

We extended a stochastic Green’s function method to longer periods in order to simulate coherent waves, by introducing zero phases at frequencies smaller than fr (a corner frequency).

We can easily incorporate this method with more realistic Green’s functions, such as those of layered half-spaces.