The direct spectral element method for the calculation of synthetic seismograms in self-gravitating, spherically symmetric planets

This paper describes the implementation of the direct solution method (DSM) using radial spectral elements for the calculation of synthetic seismograms in self-gravitating, spherically symmetric, non-rotating, anelastic, and transversely isotropic Earth models. In contrast to previous implementations of the DSM that used a potential formulation within fluid regions, we use a displacement formulation throughout. It is this feature that allows us to extend the DSM to account fully for self-gravitation along with arbitrary fluid stratification. Our code, DSpecM1D, is benchmarked against the normal mode summation code specnm as well as the direct radial integration code YSpec. Agreement between the codes is excellent for both elastic and anelastic models.