1. Jeroen Tromp Computational Seismology

2. Governing Equations Equation of motion: Boundary condition: Initial conditions: Earthquake source: Constitutive relationship (Hooke’s law):

3. Classic Tools Semi-analytical methods –Layer cake models: reflectivity/discrete wave number –Spherically symmetric models: normal modes Asymptotic methods –Ray theory –WKBJ theory –Coupled modes

4. Strong Form Velocity-stress formulation (first-order PDEs, ): Second-order finite-difference method: Fourth-order finite-difference method: Challenges: –Boundary conditions –Numerical dispersion & anisotropy

5. Strong Form Pseudospectral methods: Challenges: –Boundary conditions –Parallel implementation

6. Weak Form Weak form valid for any test vector Boundary conditions automatically included Source term explicitly integrated Finite-fault (kinematic) rupture:

7. Finite Elements Mapping from reference cube to hexahedral elements: Volume relationship: Jacobian of the mapping: Jacobian matrix: Challenges: – – Numerical anisotropy & dispersion – – Mass lumping/implicit time stepping

8. Spectral Elements Degree 4 GLL points: GLL points are n +1 roots of: The 5 degree 4 Lagrange polynomials: Note that at a GLL point: General definition:

9. Interpolation Representation of functions on an element in terms of Lagrange polynomials: Gradient on an element:

10. Integration Integration of functions over an element based upon GLL quadrature: Integrations are pulled back to the reference cube In the SEM one uses: interpolation on GLL points GLL quadrature Degree 4 GLL points:

11. The Diagonal Mass Matrix Representation of the displacement: Representation of the test vector: Weak form: Diagonal mass matrix: Integrations are pulled back to the reference cube In the SEM one uses: interpolation on GLL points GLL quadrature Degree 4 Lagrange polynomials: Degree 4 GLL points:

12. Parallel Implementation Global mesh partitioning: Cubed Sphere: 6 n mesh slices 2 Regional mesh partitioning: n x m mesh slices

13. Southern California Simulations

14. June 12, 2005, M=5.1 Big Bear

15. 100km 379 km 3D Regional Forward Simulations Qinya Liu June 12, 2005, M=5.1 Big Bear Periods > 6 s

16. Periods > 2 s Komatitsch et al. 2004

17. Soil-Structure Interaction railway bridge Stupazinni 2007

18. Soil-Structure Interaction (0.33 Hz) Stupazini 2007

19. Soil-Structure Interaction (1 Hz) Stupazini 2007

20. Recent & Current Developments Switch to a (parallel) “GEO”CUBIT hexahedral finite-element mesher –Topography & bathymetry –Major geological interfaces –Basins –Fault surfaces Use ParMETIS or SCOTCH for mesh partitioning & load-balancing 2D mesher & solver are ready (SPECFEM2D) Currently developing SPECFEM3D solver (elastic & poroelastic) Add dynamic rupture capabilities SPECFEM3D Users Map

21. Model Construction & Meshing 2D SEG model Nissen-Meyer & Luo

22. Model Construction & Meshing 2D SEG mesh Nissen-Meyer & Luo

23. SEM Simulation 2D SEG model (deep explosive source) Nissen-Meyer & Luo Komatitsch & Le Goff

24. SEM Seismograms Nissen-Meyer & Luo Komatitsch & Le Goff

25. Global Simulations PREM benchmarks Dziewonski & Anderson 1981

26. New V4.0 Mesh Michea & Komatitsch Four doublings: below the crust in the mid mantle two in the outer core Note: two-layer crust

27. SEM Implementation of Attenuation Modulus defect: Unrelaxed modulus: Memory variable equation: Equivalent Standard Linear Solid (SLS) formulation: Anelastic, anisotropic constitutive relationship: Attenuation (3 SLSs) Physical dispersion (3 SLSs)

28. Effect of Attenuation

29. Attenuation

30. Effect of Anisotropy

31. 11/26/1999 Vanuatu shallow event 50 s - 500 s benchmark PREM benchmarks include: Attenuation Transverse isotropy Self-gravitation (Cowling) Two-layer crust

32. 6/4/1994 Bolivia Deep Event 10 s - 500 s benchmark PREM benchmarks include: Attenuation Transverse isotropy Self-gravitation (Cowling) Two-layer crust

33. PKP Phases 15 s - 500 s