Towards Computing Full 3D Seismic Sensitivity: The Axisymmetric Spectral Element Method

Finite frequency tomography has recently provided detailed images of the Earth's deep interior. However, the Fréchet sensitivity kernels used in these inversions are calculated using ray theory and can therefore not account for D''-diffracted phases or any caustics in the wavefield, as e.g. occurring in phases used to map boundary layer topography. Our objective is to compile an extensive set of full sensitivity kernels based on seismic forward modeling to allow for inversion of any seismic phase. The sensitivity of the wavefield due to a scatterer off the theoretical ray path is generally determined by the convolution of the source-to-scatterer response with, using reciprocity, the receiver-to-scatterer response. Thus, exact kernels require the knowledge of the Green's function for the full moment tensor (i.e., source) and body forces (i.e., receiver components) throughout the model space and time. We develop an axisymmetric spectral element method for elastodynamics to serve this purpose. The axisymmetric approach takes advantage of the fact that kernels are computed upon a spherically symmetric Earth model. In this reduced dimension formulation, all moment tensor elements and single forces can be included and collectively unfold in six different 2D problems to be solved separately. The efficient simulations on a 2D mesh then allow for currently unattainable high resolution at low hardware requirements. The displacement field {u} for the 3D sphere can be expressed as {u}( {x}, {t})= {u}( {x}_{φ =0}, {t}) {f}(φ ), where φ =0 represents the 2D computational domain and {f}(φ ) are trigonometric functions. Here, we describe the variational formalism for the full multipole source system and validate its implementation against normal mode solutions for the solid sphere. The global mesh includes several conforming coarsening levels to minimize grid spacing variations. In an effort of algorithmic optimization, the discretization is acquired on the basis of matrix-matrix operations and performed via unit-stride cache access.