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, $\mathtt{\text{DSpecM1D}}$, is benchmarked against the normal mode summation code $\mathtt{\text{MINEOS}}$ as well as the direct radial integration code $\mathtt{\text{YSpec}}$. Agreement between the codes is excellent for both elastic and anelastic models.