Introduction to the spectral element method for three-dimensional seismic wave propagation

We present an introduction to the spectral element method, which provides a n innovative numerical approach to the calculation of synthetic seismograms in 3-D earth models. The method combines the flexibility of a finite eleme nt method with the accuracy of a spectral method. One uses a weak formulati on of the equations of motion, which are solved on a mesh of hexahedral ele ments that is adapted to the free surface and to the main internal disconti nuities of the model. The wavefield on the elements is discretized using hi gh-degree Lagrange interpolants, and integration over an element is accompl ished based upon the Gauss-Lobatto-Legendre integration rule. This combinat ion of discretization and integration results in a diagonal mass matrix, wh ich greatly simplifies the algorithm. We illustrate the great potential of the method by comparing it to a discrete wavenumber/reflectivity method for layer-cake models. Both body and surface waves are accurately represented, and the method can handle point force as well as moment tensor sources. Fo r a model with very steep surface topography we successfully benchmark the method against an approximate boundary technique. For a homogeneous medium with strong attenuation we obtain excellent agreement with the analytical s olution for a point force.