Spherical‐earth Fréchet sensitivity kernels

We outline a method that enables the efficient computation of exact Frechet sensitivity kernels for a non-gravitating 3-D spherical earth model. The crux of the method is a 2-D weak formulation for determining the 3-D elastodynamic response of the earth model to both a moment-tensor and a point-force source. The sources are decomposed into their monopole, dipole and quadrupole constituents, with known azimuthal radiation patterns. The full 3-D response and, therefore, the 3-D waveform sensitivity kernel for an arbitrary source–receiver geometry, can be reconstructed from a series of six independent 2-D solutions, which may be obtained using a spectral-element or other mesh-based numerical method on a 2-D, planar, semicircular domain. This divide-and-conquer, 3-D to 2-D reduction strategy can be used to compute sensitivity kernels for any seismic phase, including grazing and diffracted waves, at relatively low computational cost.