Jeroen Tromp

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Computational Seismology
Jeroen Tromp
2
Governing Equations
Equation of motion
Constitutive relationship (Hookes 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

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Strong Form

• Velocity-stress formulation (first-order PDEs,
  )
• Second-order finite-difference method
• Fourth-order finite-difference method
• Challenges
• Boundary conditions
• Numerical dispersion anisotropy

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Strong Form

• Pseudospectral methods
• Challenges
• Boundary conditions
• Parallel implementation

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Weak Form

• Weak form valid for any test vector
• Boundary conditions automatically included
• Source term explicitly integrated

Finite-fault (kinematic) rupture
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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

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Spectral Elements
Degree 4 GLL points
The 5 degree 4 Lagrange polynomials
GLL points are n1 roots of
General definition
Note that at a GLL point
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Interpolation
Representation of functions on an element in
terms of Lagrange polynomials
Gradient on an element
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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
Parallel Implementation
Regional mesh partitioning
Global mesh partitioning
n x m mesh slices
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Southern California Simulations
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June 12, 2005, M5.1 Big Bear
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3D Regional Forward Simulations
Periods gt 6 s
379 km
100km
June 12, 2005, M5.1 Big Bear
Qinya Liu
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Periods gt 2 s
Komatitsch et al. 2004
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Soil-Structure Interaction
railway bridge
Stupazinni 2007
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Soil-Structure Interaction (0.33 Hz)
Stupazini 2007
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Soil-Structure Interaction (1 Hz)
Stupazini 2007
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Recent Current Developments

• Switch to a (parallel) GEOCUBIT 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
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2D SEG model
Model Construction Meshing
Nissen-Meyer Luo
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2D SEG mesh
Model Construction Meshing
Nissen-Meyer Luo
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SEM Simulation
2D SEG model (deep explosive source)
Komatitsch Le Goff
Nissen-Meyer Luo
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SEM Seismograms
Komatitsch Le Goff
Nissen-Meyer Luo
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Global Simulations
PREM benchmarks
Dziewonski Anderson 1981
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New V4.0 Mesh

• Four doublings
• below the crust
• in the mid mantle
• two in the outer core
• Note two-layer crust

Michea Komatitsch
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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
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Effect of Attenuation
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Attenuation
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Effect of Anisotropy
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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

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PKP Phases15 s - 500 s