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 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

4. 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

5. Modified K-2 model （ Hisada, 2000 ） k2 slip distributionk2 rupture time Kostrov-type slip velocity with 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/k 2 0 k (wavenumber) amplitude Phase: coherent random Source spec.: ω 2 model 1/ω 2 frequency k2 model

6. 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

7. 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

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. Moment Rate Function for ω 2 model Slip Velocity for ω 2 model Representation Theorem for ω 2 model For Point Dislocation Source

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

11. Boore ’ s Source Model with Zero Phases (Coherent Phases) ω 2 Amplitude ＋ Zero Phases fc=1 Hz fmax=10 Hz Moment Rate Function （ Slip Velocity Function ） Moment Rate + 1/fc sec delay Moment Function ・ Smoothed Ramp Function → ×Acceleration ○ Directivity Pulses ○ Fling Step ○ Seismic Moment FIT with No Time Window

12. Boore ’ s Source Model with Zero and Random Phases (Introduction of fr) ω 2 Amp. ＋ Zero and Random Phases fc=1 Hz fmax=10 Hz Moment Rate Function （ Slip Velocity Function ） fr=1 Hz Moment Function Moment Rate + 1/fc sec delay ・ Ramp Function with high freq. ripples → ○ Acceleration ○ Directivity Pulses ○ Fling Step ○ Seismic Moment FIT with Time Window

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

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

15. Summarized Results (Three Models) Displacement Velocity Acceleration Triangle Slip Velocity ω 2 Model + 0 Phases ω 2 Model + 0 & Random Phases

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.