Jeroen Tromp Computational Seismology

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

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

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

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

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

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

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:

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

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:

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:

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

Southern California Simulations

June 12, 2005, M=5.1 Big Bear

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

Periods > 2 s Komatitsch et al. 2004

Soil-Structure Interaction railway bridge Stupazinni 2007

Soil-Structure Interaction (0.33 Hz) Stupazini 2007

Soil-Structure Interaction (1 Hz) Stupazini 2007

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

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

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

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

SEM Seismograms Nissen-Meyer & Luo Komatitsch & Le Goff

Global Simulations PREM benchmarks Dziewonski & Anderson 1981

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

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)

Effect of Attenuation

Attenuation

Effect of Anisotropy

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

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

PKP Phases 15 s s