Numerical Methods in Spherical Geometry
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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?
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
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