Tarje Nissen-Meyer

Abstract. We outline a method that enables the efficient computation of exact Fréchet 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 diffractd waves, at relatively low computational cost.

Traveltime sensitivity kernels in PREM (Dziewonski & Anderson 1981), calculated using the ray-based approach of Dahlen et al. (2000). Question marks indicate regions where the sensitivity is not computed accurately due to the deficiencies of ray theory. (a) Sensitivity of a P wave grazing the core and emerging at an epicentral distance of 100°; the characteristic period is 5 s. An improved method is needed to compute the sensitivity in the D" region. (b) Sensitivity of a 10 s P wave that propagates to an epicentral distance of 40°. Turning P and S waves at this distance and less are affected by upper-mantle triplications. (c) Sensitivity of a 20 s SS surface reflection that propagates to an epicentral distance of 120°. Question marks cover the source-to-receiver and receiver-to-source caustics, where ray theory incorrectly approximates the sensitivity. In all of these instances, the expectation is that the ray-based kernels will be an excellent approximation to the exact sensitivity kernels, except where covered by the question marks.

Sketch illustrating the 2-D computation of the 3-D forward- and backward-propagating responses at an arbitrary scatterer r. The points r_s and r_r are the locations of the source and receiver, where waves are excited by a moment tensor MH(t) and a point force

, respectively. Shading is suggestive of the radial velocity variation within the spherically symmetric model, with a solid inner core and fluid outer core, overlain by a solid mantle and crust. Numerical computation of the forward- and backward-propagating wavefields is conducted within the two intersecting, semicircular 2-D domains, denoted by the darker shading. The full 3-D moment-tensor and point-force responses can be reconstructed from six 2-D solutions, due to the known azimuthal dependence of the wavefields in a spherically symmetric earth.

This work describes a method to construct global Fréchet sensitivity kernels to numerical precision for arbitrary resolutions and any portion of a seismogram. This is facilitated by a 2-D spectral-element method which produces time-space wavefields for each element of the moment tensor and single forces upon spherically symmetric earth models, utilizing radiation symmetry patterns such that all storage and simulation issues are confined to a 2-D semicircular domain. We show how these wavefields take on, upon performing rotations and accommodating azimuthal dependencies, the central role in time-dependent sensitivity kernels. These spatio-temporal, frequency-dependent sensitivity constructs are a valuable resource in assessing which fraction of the seismogram is sensitive to a given region of the 3-D earth, i.e. an a priori tool to reliably choose potential inversion parameters and phases. We present expressions for waveform sensitivity kernels from which we extract traveltime and amplitude kernels, and explore some of their dependencies such as moment tensor elements, source time function and depth, frequency, and azimuth via several examples of waveform, traveltime and amplitude kernels. These combined spectral-element and kernel calculation methods shall form the basis for any tomographic needs for arbitrary-resolution accounts of e.g. diffracted arrivals.

Reconstruction of 3-D wavefields from 2-D wavefields saved from the spectral-element method in a 2-D semi-disk. Shown is a sample snapshot of the z component for an M_xy source at 650 km (top panels) and for a P_z single-force source at the surface (bottom panels) at a dominant source period of 20 s.

λ waveform kernel snapshots for an epicentral distance of θ=60^o, a source period of T₀ =20 s for an explosion source at 545 km depth. For simplicity and symmetry reasons, the back-propagating wavefield denoting the receiver-to-scatterer path is the same as the forward-propagating wavefield for the explosion at depth. The core-mantle boundary is indicated by the dotted line. The times between 550 s and 825 s encompass the direct P wave, pP, PcP, PP and PPP.