Short Course Spectral-element solution of the elastic wave equation Andreas Fichtner
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Short Course
Spectral-element solution of the
elastic wave equation
Andreas Fichtner
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OUTLOOK
1. Introduction recalling the elastic wave equation
2. The spectral-element method: General conceptdomain mapping
from space-continuous to space-discrete
time interpolation
Gauss-Lobatto-Legendre interpolation and integration
3. A special flavour of the spectral-element method: SES3D programme code description
computation of synthetic seismograms
long-wavelength equivalent models
4. Computation of sensitivity kernels using SES3Dconstruction of the adjoint source
5. On the sensitive nature of numerical methods: Calls to caution!
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Introduction
Recalling the elastic wave equation
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dt),(:)t,(t),()ρ()ρ,,( 2
t xuxCxuxCuL
0 0tt|t),(xu 0 0ttt |t),(xu t
Γ|d),(:)t,( 0xxuxCn
Elastic wave equation:
Subsidiary conditions:
f),( CuL ρ,
THE ELASTIC WAVE EQUATION
• No analytical solutions exist for Earth models with 3D heterogeneities.
• We use numerical methods:
- Finite-difference method (FDM)
- Finite-element method (FEM)
- Direct-solution method (DSM)
- Discontinuous Galerkin method (DGM)
- …
- Spectral-element method
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The spectral-element method
General concept
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SPECTRAL-ELEMENT METHOD: General Concept
see Bernhard‘s web site:
www.geophysik.uni-muenchen.de/~bernhard
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SPECTRAL-ELEMENT METHOD: General Concept
Subdivision of the computational domain into hexahedral elements:
(a) 2D subdivision that honours layer boundaries
(b) Subdivision of the globe (cubed sphere) (c) Subdivision with topography
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SPECTRAL-ELEMENT METHOD: General Concept
Mapping to the unit cube:
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SPECTRAL-ELEMENT METHOD: General Concept
Representation in terms of polynomials:
N
i
Nii xtutxu
0
)( )()(),(
:)()( xNi Nth-degree Lagrange polynomials
→ We can transform the partial differential equation into an ordinary differential equation
where we solve for the polynomial coefficients:
(within the unit interval [-1 1])
kikiiki fuKuM
::
ki
ki
KM mass matrix
stiffness matrix
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SPECTRAL-ELEMENT METHOD: General Concept
Choice of the collocation points:
Interpolation of Runge‘s function R(x)
using 6th-order polynomials and equidistant collocation points
211)(ax
xR
interpolant
Runge‘s
phenomenon
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SPECTRAL-ELEMENT METHOD: General Concept
Choice of the collocation points:
Interpolation of Runge‘s function R(x)
using 6th-order polynomials and Gauss-Lobatto-Legendre collocation points
[ roots of (1-x2)LoN-1= completed Lobatto polynomial ]
211)(ax
xR
interpolant
We should use the GLL points as collocation points for the Lagrange polynomials.
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SPECTRAL-ELEMENT METHOD: General Concept
Example: GLL Lagrange polynomials of degree 6
• collocation points = GLL points
• global maxima at the collocation points
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SPECTRAL-ELEMENT METHOD: General Concept
Numerical quadrature to determine mass and stiffness matrices:
Quadrature node points = GLL points
→ The mass matrix is diagonal, i.e., trivial to invert.
→ This is THE advantage of the spectral-element method.
Time extrapolation:
2
)()(2)()(t
ttututtutu
)()()()(2)( 12 tKutfMtttututtu
… this can be solved on a computer.
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A special flavour of the spectral-element method
SES3D
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SES3D: General Concept
• Simulation of elastic wave propagation in a spherical section.
• Spectral-element discretisation.
• Computation of Fréchet kernels using the adjoint method.
• Operates in natural spherical coordinates!
• 3D heterogeneous, radially anisotropic,
visco-elastic.
• PML as absorbing boundaries.
• Programme philosophy: Puritanism [easy to modify and
adapt to different problems, easy
implementation of 3D models,
simple code]
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SES3D: Example
Southern Greece
8 June, 2008
Mw=6.3
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SES3D: Example
Southern Greece
8 June, 2008
Mw=6.3
1. Input files [geometric setup, source, receivers, Earth model]
2. Forward simulation [wavefield snapshots and seismograms]
3. Adjoint simulation [adjoint source, Fréchet kernels]
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SES3D: Input files
• Par:
- Numerical simulation parameters
- Geometrical setup
- Seismic Source
- Parallelisation
• stf:
- Source time function
• recfile:
- Receiver positions
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SES3D: Parallelisation
• Spherical section subdivided into equal-sized subsections
• Each subsection is assigned to one processor.
• Communication: MPI
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SES3D: Source time function
Source time function
- time step and length agree with the simulation parameters
- PMLs work best with bandpass filtered source time functions
- Example: bandpass [50 s to 200 s]
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LONG WAVELENGTH EQUIVALENT MODELS
Single-layered crust that coincides with the upper layer of
elements …
… and PREM below
boundary between the upper 2 layers of elements
lon=142.74°lat=-5.99°d=80 km
SA08lon=150.89°lat=-25.89°
vertical displacement
Dalkolmo & Friederich, 1995. Complete synthetic seismograms for a spherically symmetric Earth …, GJI, 122, 537-550
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2-layered crust that does not coincide
with a layer of elements …
… and PREM below
boundary between the upper 2 layers of elements
lon=142.74°lat=-5.99°d=80 km
SA08lon=150.89°lat=-25.89°
verification
vertical displacement
Dalkolmo & Friederich, 1995. Complete synthetic seismograms for a spherically symmetric Earth …, GJI, 122, 537-550
LONG WAVELENGTH EQUIVALENT MODELS
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• Replace original crustral
model by a long-
wavelength equivalent
model …
• … which is transversely
isotropic [Backus, 1962].
• The optimal smooth model
is found by dispersion
curve matching.
Fichtner & Igel, 2008. Efficient numerical surface wave propagation through the optimisation of discrete crustal models, GJI.
long wavelength equivalent modelsLONG WAVELENGTH EQUIVALENT MODELS
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Minimisation of the phase velocity differences for the fundamental and higher modes in the frequency range of interest through simulated annealing.
long wavelength equivalent modelsLONG WAVELENGTH EQUIVALENT MODELS
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crustal thickness map (crust2.0)
• 3D solution: interpolation of long wavelength equivalent profiles to obtain 3D crustal model.
• Problem 1: crustal structure not well constrained (receiver function non-uniqueness)
• Problem 2: abrupt changes in crustal structure (not captured by pointwise RF studies)
long wavelength equivalent modelsLONG WAVELENGTH EQUIVALENT MODELS
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Computation of Fréchet kernels
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SES3D: Computation of Fréchet kernels
Vertical component velocity seismogram:
Adjoint source time function: [cross-correlation time shift]
taper target waveform
divide by L2 norm
invert time axis
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On the sensitive nature of numerical methods
Calls to caution!
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SES3D: Calls to caution!
1. Long-term instability of PMLs
- All PML variants are long-term unstable!
- SES3D monitors the total kinetic energy Etotal.
- When Etotal increases quickly, the PMLs are switched off and …
- … absorbing boundaries are replaced by less efficient multiplication by small numbers.
2. The poles and the core
- Elements become infinitesimally small at the poles and the core.
- SES3D is efficient only when the computational domain is sufficiently far from
the poles and the core.
3. Seismic discontinuities and the crust
- SEM is very accurate only when discontinuities coincide with element boundaries.
- SES3D‘s static grid may not always achieve this.
- It is up to the user to assess the numerical accuracy in cases where discontinuities run
through elements. [Implement long-wavelength equivalent models.]
- Generally no problem for the 410 km and 660 km discontinuities.
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Thanks for your attention!