Accurate numerical methods for solving the elastic equation of motion for arbitrary source locations

Abstract

SUMMARY In our previous studies we derived a general criterion for optimally accurate numerical operators for computing synthetic seismograms. In this paper we extend our results by deriving source representations that are ‘tuned’ to match the errors of the numerical operators, so that optimally accurate synthetic seismograms are obtained. The same accuracy can be obtained whether the sources are at nodes of the numerical grid or between nodes. The accuracy of the synthetic seismograms is confirmed by numerical experiments. These results should facilitate inversion of seismic waveform data for 3‐D Earth structure and earthquake source parameters.

Complete synthetic seismograms up to 2 Hz for transversely isotropic spherically symmetric media

“…However, in most cases the source will be between nodes. For such cases we use the source representation of Takeuchi & Geller (2003), which allows us to calculate synthetics with the same accuracy as for a source at a node.…”

“… Toroidal (SH) velocity synthetic seismograms for the anisotropic PREM model (transverse component) computed up to 0.05 Hz and a maximum angular order of 2000 (about four times higher than required for practical calculations, to eliminate any possibility of errors due to truncation of the spherical harmonic expansion) using the source representation of Takeuchi & Geller (2003). The source is a 600‐km‐deep point double‐couple source, with M r θ = M θ r = 1 and all other components of the moment tensor zero.…”

“… Spheroidal ( P–SV ) velocity synthetic seismograms for anisotropic PREM model (radial component) computed using the source representation of Takeuchi & Geller (2003). Details are the same as Fig.…”

Finite boundary perturbation theory for the elastic equation of motion

“…However, we should newly design the operators for the P – SV problem and the 3‐D heterogeneous problem. With regard to the external force term, g in , Takeuchi & Geller (2003) proposed an accurate discretization. If we use these accurate discretization methods together with the finite boundary perturbation theory presented in this paper, we should be able to further improve the accuracy of crustal correction.…”

Theory and Obeservations - Forward Modeling and Synthetic Seismograms, 3D Numerical Methods

“…Complications due to the representation of the source term, which may not be representable in terms of these basis functions, are discussed in Takeuchi and Geller (2003). For spherically symmetrical Earth models, the DSM may be used to calculate high-frequency synthetic seismograms, as illustrated in Figure 7.…”