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 slices2

Regional mesh partitioning:

n x m mesh slices

Southern California Simulations

June 12, 2005, M=5.1 Big Bear

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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 bridgerailway bridge

Stupazinni 2007

Soil-Structure Interaction (0.33 Hz)

Stupazini 2007

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Soil-Structure Interaction (1 Hz)

Stupazini 2007

QuickTime™ and aYUV420 codec decompressor

are needed to see this picture.

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)Attenuation (3 SLSs)

Physical dispersion (3 SLSs)Physical dispersion (3 SLSs)

Effect of Attenuation

Attenuation

Effect of Anisotropy

11/26/1999 Vanuatu shallow event50 s - 500 s benchmark

PREM benchmarks include:• Attenuation• Transverse isotropy• Self-gravitation (Cowling)• Two-layer crust

6/4/1994 Bolivia Deep Event10 s - 500 s benchmark

PREM benchmarks include:• Attenuation• Transverse isotropy• Self-gravitation (Cowling)• Two-layer crust

PKP Phases15 s - 500 s