  The purpose of this work is to generalize the frozen Gaussian approximation
(FGA) theory to solve the 3-D elastic wave equation and use it as the forward
modeling tool for seismic tomography with high-frequency data. FGA has been
previously developed and verified as an efficient solver for high-frequency
acoustic wave propagation (P-wave). The main contribution of this paper
consists of three aspects: 1. We derive the FGA formulation for the 3-D elastic
wave equation. Rather than standard ray-based methods (e.g. geometric optics
and Gaussian beam method), the derivation requires to do asymptotic expansion
in the week sense (integral form) so that one is able to perform integration by
parts. Compared to the FGA theory for acoustic wave equation, the calculations
in the derivation are much more technically involved due to the existence of
both P- and S-waves, and the coupling of the polarized directions for SH- and
SV-waves. In particular, we obtain the diabatic coupling terms for SH- and
SV-waves, with the form closely connecting to the concept of Berry phase that
is intensively studied in quantum mechanics and topology (Chern number). The
accuracy and parallelizability of the FGA algorithm is illustrated by comparing
to the spectral element method for 3-D elastic wave equation in homogeneous
media; 2. We derive the interface conditions of FGA for 3-D elastic wave
equation based on an Eulerian formulation and the Snell's law. We verify these
conditions by simulating high-frequency elastic wave propagation in a 1-D
layered Earth model. In this example, we also show that it is natural to apply
the FGA algorithm to geometries with non-Cartesian coordinates; 3. We apply the
developed FGA algorithm for 3-D seismic { wave-equation-based traveltime
tomography and full waveform inversion, respectively
