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Jeroen Tromp

Computational Seismology. Jeroen Tromp. Governing Equations. Equation of motion:. Constitutive relationship (Hooke’s law):. Boundary condition:. Initial conditions:. Earthquake source:. Classic Tools. Semi-analytical methods Layer cake models: reflectivity/discrete wave number

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Jeroen Tromp

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  1. Computational Seismology Jeroen Tromp

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

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

  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: Degree 4 Lagrange polynomials: 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 GLL points:

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

  13. Southern California Simulations

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

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

  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) Komatitsch & Le Goff Nissen-Meyer & Luo

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

  25. Global Simulations PREM benchmarks Dziewonski & Anderson 1981

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

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

  28. Effect of Attenuation

  29. Attenuation

  30. Effect of Anisotropy

  31. 11/26/1999 Vanuatu shallow event50 s - 500 s benchmark • PREM benchmarks include: • Attenuation • Transverse isotropy • Self-gravitation (Cowling) • Two-layer crust

  32. 6/4/1994 Bolivia Deep Event10 s - 500 s benchmark • PREM benchmarks include: • Attenuation • Transverse isotropy • Self-gravitation (Cowling) • Two-layer crust

  33. PKP Phases15 s - 500 s

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