Meshing the Globe: A SEM, like a FEM, employs the technique of splitting up the volume of Earth into finite elements and then solving the seismic wave equation within these elements. We use a 'cubed sphere' mapping to split the globe into six 'chunks' (left). Each of the six chunks is further subdivided into 5 x 5 = 25 slices, for a total of 150 slices (right).

The Elements: To construct a mesh in a SEM, the sample volume is subdivided into a number of elements which are non-overlapping, nonconforming hexahedra (left). Each deformed cube is described by 27 anchor points. The points can be described in terms of their location in the element (the local description), as well as in terms of the entire volume (the global description). A local-to-global mapping is essential for the assembly of the system, when adjacent elements must communicate between shared points on borders (right).

 

Doubling Elements: To achieve accurate results, it is best to    have a fairly uniform gridpoint to wavelength ratio within each element. To handle this problem, we used a doubling technique in which the grid doubles first in one direction and then, at a greater depth, in the other direction in order to densify the elements toward the surface. One layer of elements is confined entirely to the crust (left). The doubling is implemented once below the Moho, a second time below the 670 km discontinuity, and a last time just above the ICB (right).

 

Central Cube: To avoid a singularity at the Earth's center, we place a small cube within the inner core that matches up perfectly with the cubed sphere (left). The central cube touches all the 150 slices that make up the rest of the mesh (right). This poses a significant parallel programming challenge, which calls for a master-slave programming methodology.

Mantle model: S20RTS (Ritsema et al. 1999) is superimposed on the mesh. The figure shows lateral variations in shear velocity projected onto the four sides of the six chunks that constitute the cubed sphere mesh. Blue colors denote faster than average shear-wave velocities, and red colors denote slower than average shear-wave velocities.

 

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Crustal model: Crust 5.2 (Bassin et al. 2000) is superimposed on the mesh. The figure shows Moho depth (which varies between 6.65 km and 75 km in the model). Red represents thicker than average crust, and blue thinner than average crust.

 

 

 



    Topography and bathymetry: The mesh is stretched or squished according to ETOPO5.

 

 

 

 

 

Topography and bathymetry: Close-up of Mexico and the Southern United States showing the spectral elements in the mesh at the surface (grey squares).

 

 

 

Ellipticity: As a result of its rotation, the Earth is slightly flattened at the poles (blue colors) and elongated at the equator (red colors). The ellipticity at the surface is small (approx. 1/300).

 

 

 

Oceans: The bathymetry map is taken from model ETOPO5 (NOAA 1988). The effect of the oceans can be quite significant and is incorporated in the SEM.

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