A two-dimensional spectral-element method for computing spherical-earth seismograms - I. Moment-tensor source

We develop a spectral-element method for computing the full 3-D moment-tensor and point-force response of a spherically symmetric earth model in a 2-D semi-circular computational domain. The full elastodynamic response to a six-component moment tensor at an earthquake hypocentre and a three-component point force at a seismic station can be determined by solving six independent 2-D problems, three for a monopole source, two for a dipole source, and one for a quadrupole source. This divide-and-conquer 3-D to 2-D reduction strategy provides a basis for the efficient computation of exact Frechet sensitivity kernels in a spherically symmetric earth, with all wavefield features accounted for. To focus on the novel inclusion of the full source in a cylindrical coordinate system, we describe the 2-D weak formulation of the set of elastodynamic equations, its discretization using spectral elements, and the associated axial boundary conditions and source representations for each of the excitation types in the case of a homogeneous, solid elastic sphere. The method is numerically validated against both analytical solutions and normal-mode summation.