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Y. Hisada and J. Bielak

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

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Y. Hisada and J. Bielak

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  1. An Extension of Stochastic Green’s Function Method to Long-Period Strong Ground-motion Simulation Y. Hisada and J. Bielak

  2. 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)

  3. Broadband Strong Ground Motion Simulation (Hybrid Methods) short ←→ long period short ←→ long period • Short period（<１s）：Stochastic and empirical methods （ex., Stochastic Green’s function method）　　　　　　　→ omega-squared model, random phases • Long period（>１s）：Deterministic methods （FDM, FEM, Green’s functions for layered media） → coherent phases (e.g., directivity pulses), seismic moment M7 Eq. M8 Eq. 0 1 2 period (s) 0 1 2 4 period (s)

  4. Broadband Strong Ground Motion Simulation (Hybrid Methods) short ←→ long period short ←→ long period • 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. M7 Eq. M8 Eq. 0 1 2 period (s) 0 1 2 4 period (s)

  5. Phase: coherent random 1/ω2 frequency Modified K-2 model（Hisada, 2000） k2 slip distribution k2 rupture time Kostrov-type slip velocity with fmax 1/k2 amplitude k2 model 0 k (wavenumber) Source spec.: ω2 model ・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

  6. Broadband Strong Ground Motion Simulation (Hybrid Methods) short ←→ long period short ←→ long period • Short period（<１s）：Stochastic and empirical methods （ex., Stochastic Green’s function method）　　　　　　　→ omega-squared model, random phases • Long period（>１s）：Deterministic methods （FDM, FEM, Green’s functions for layered media） → coherent phases (e.g., directivity pulses), seismic moment M7 Eq. M8 Eq. 0 1 2 period (s) 0 1 2 4 period (s)

  7. Stochastic Green’s Function Method (Kamae et al., 1998) : Boore’s Source Model + Irikura’s Empirical Green’s Function Summation Method Observation Point Seismic Fault → Fast Computation: One source point per sub-fault → Green’s Functions of the Far-Field S Wave (1/r)

  8. 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

  9. Slip Velocity for ω2 model • Moment Rate Function for ω2 model • Representation Theorem for ω2 model For Point Dislocation Source

  10. Boore’s Source Model with Random Phases • Moment Rate Function （Slip Velocity Function） • ω2 Amplitude ＋Random Phases FIT with Time Window Example 1 fc=1 Hz fmax=10 Hz ・Unstable and Incoherent at Longer Periods → ○Acceleration ×Directivity Pulses ×Fling Step ×Seismic Moment Example 2

  11. Boore’s Source Model with Zero Phases (Coherent Phases) • Moment Rate Function （Slip Velocity Function） • ω2 Amplitude ＋ Zero Phases Moment Rate + 1/fc sec delay FIT with No Time Window fc=1 Hz fmax=10 Hz ・Smoothed Ramp Function → ×Acceleration ○Directivity Pulses ○Fling Step ○Seismic Moment Moment Function

  12. Boore’s Source Model with Zero and Random Phases (Introduction of fr) • Moment Rate Function （Slip Velocity Function） • ω2 Amp.＋ Zero and Random Phases Moment Rate + 1/fc sec delay FIT with Time Window fc=1 Hz fmax=10 Hz ・Ramp Function with high freq. ripples → ○Acceleration ○Directivity Pulses ○Fling Step ○Seismic Moment Moment Function fr=1 Hz

  13. Example (Boore’s Source with Zero Phases): r=20 km, Vp=5,Vs=3km/s Far Field Displacement S Wave P wave R=20ｋｍ P Wave S wave 45° S Wave R=20ｋｍ Moment Rate Function １．Triangle（τ=1s） ２．ω２Model（fc=1 Hz fmax=10 Hz, 0 phases） P Wave

  14. Far-Field Displacement S Wave Boore’s Source with Zero and Random Phases Proposed Model fc=1 Hz fmax=10 Hz fr=1 Hｚ S Wave Far-Field Acceleration

  15. Summarized Results (Three Models) Triangle Slip Velocity ω2 Model + 0 Phases ω2 Model + 0 & Random Phases DisplacementVelocityAcceleration

  16. 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.

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