Numerical Methods in Spherical Geometry

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Kelly Long
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1 Numerical Methods in The wave equation in spherical coordinates The axi-symmetric approximation Single domain - multi-domain methods Sources in the axi-symmetric approach - Spherical sections - Cubed Sphere - Hybrid approach Examples

2 Wave equation - spherical coordinates Equations of motion in spherical coordinates velocity - stress formulation

3 Wave equation - stress - strain Equations of motion in spherical coordinates strain-displacement relation

4 Wave equation - Hooke s Law Equations of motion in spherical coordinates stress-strain relation (isotropic media)

5 Axisymmetric approximation When all fields (and the model) are axisymmetric the terms in the red squares vanish.

6 Axisymmetric grids - SH case Two-domain method SH case

7 Axisymmetric grids - sources Implementing sources in axisymmetric methods

8 Axisymmetric grids - SH waves SH waves - FD high order - Source at 600km depth PREM model - dominant period 25 seconds

9 Axisymmetric grids - SH waves SH waves - FD high order - Source at 600km depth PREM model - dominant period 25 seconds

10 Axisymmetric grids - SH waves SH waves - FD high order - Source at 600km depth PREM model - dominant period 25 seconds

11 Axisymmetric grids - seismograms Comparison of exact method (DSM) with FD method. Spherically symmetric model PREM.

12 Axisymmetric grids - whole Earth Single domain vs. multidomain

13 Axisymmetric grids - center What should we do with the center?

14 Axisymmetric grids - stability Top: lateral grid spacing as a function of depth multidomain and single domain (dashed). Bottom: Required time step for PREM for a stable calculation for multidomain and single domain (dashed).

15 Axisymmetric grids - staggering Staggering in the acoustic approximation. Multi-domain approach. Domain connection through interpolation.

16 Axisymmetric grids - centre The stress at the center is calculated from the radial displacements at r=dr c /2.

17 Acoustic case - homogeneous Acoustic waves in a homogeneous sphere. Multidomain approach.

18 Acoustic case - PREM model Acoustic waves in a sphere. Multidomain approach. PREM model.

19 Acoustic case - PREM model Acoustic waves in a sphere. Multidomain approach. PREM model.

20 Acoustic case - PREM model Acoustic waves in a sphere. Multidomain approach. PREM model.

21 Acoustic case - PREM model Acoustic waves in a sphere. Multidomain approach. PREM model.

22 Acoustic case - seismograms Seismograms for source at 200km depth. Dominant period 10s.

23 Acoustic case - seismograms Seismograms for source at 200km depth. Dominant period 10s. PKP phase.

24 Spherical sections Chebyshev grid - spherical section

25 Spherical sections - snaps

26 Spherical sections - snaps... seen this one before?

27 Spherical sections - seismograms Azimuthal effects of a slab.

28 Spherical sections - seismograms Azimuthal effects of a slab. Single seismogram.

29 The COSY Project 3D model pertubations, COSY Project

30 The COSY Project SH seismograms from different methods. PREM model.

31 The COSY Project 3-D effects with different techniques.

32 The whole Earth - Future Voronoi cells - natural neighbours

33 The whole Earth - Future Cylinder Waves in a Sphere

34 Cubed Sphere gnomic projection of a cube to the surface of a sphere 6 equivalent chunks 6 different coordinate transformations 4 coordinates : chunk, ξ, η, r spherical shell mesh

35 Cubed Sphere gnomic projection of a cube to the surface of a sphere 6 equivalent chunks 6 different coordinate transformations 4 coordinates : chunk, ξ, η, r

36 coordinate transformation Cubed Sphere <=> Cartesian

37 Chunk overlap fitting chunks together

38 Solid Sphere extension

39 Special Issues of the Spherical Code Cubed Sphere Chunk Partitioning

40 Special Issues of the Spherical Code The Mesh Doubling of Element Size

41 Status Quo

42 Hybrid approach Combine axisymmetric approach with spherical section SH 5s Can we have a teleseismic wavefield flow into a local 3D box?

Global seismology hybrid approach Combining Combining axisymmetric axisymmetric approach approach with with3d 3D spherical spherical section section Allows Allows modelling modelling higher

43 Global seismology hybrid approach Combining Combining axisymmetric axisymmetric approach approach with with3d 3D spherical spherical section section Allows Allows modelling modelling higher higher frequencies frequencies Localized Localized3D 3D structure structure (e.g. (e.g. plumes, plumes, subduction subduction zones) zones) Phenomenological Phenomenological studies studies

44 Global seismology scattering from a plume

Seismogram Interpretation. Seismogram Interpretation

Seismogram Interpretation. Seismogram Interpretation Travel times in the Earth Ray paths, phases and their name Wavefields in the Earth: SH waves, P-SV waves Seismic Tomography Receiver Functions Seismogram Example Long-period transverse displacement for