Three-dimensional (3-D) elastic wave propagation and seismic tomography is computationally challenging in large scales and high-frequency regime. In this talk, we propose the frozen Gaussian approximation (FGA) to compute the 3-D elastic wave equation and use it as t he forward modeling tool for seismic tomography with high-frequency data. 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. 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. With a parallel FGA solver built as an computational engine, we explore various applications in 3-D seismic tomography, including seismic traveltime tomography, full waveform inversion, and optimal transport theory-based seismic tomography (using Wasserstein distance), respectively. Global minimization for seismic tomography is investigated based on particle swarm algorithm. We also apply the FGA algorithm to train a neural network to learn simple subsurfaces structures.