  Full-Waveform Inversion (FWI) is a nonlinear iterative seismic imaging
technique that, by reducing the misfit between recorded and predicted seismic
waveforms, can produce detailed estimates of subsurface geophysical properties.
Nevertheless, the strong nonlinearity of FWI can trap the optimization in local
minima. This issue arises due to factors such as improper initial values, the
absence of low frequencies in the measurements, noise, and other related
considerations. To address this challenge and with the advent of advanced
machine-learning techniques, data-driven methods, such as deep learning, have
attracted significantly increasing attention in the geophysical community.
Furthermore, the elastic wave equation should be included in FWI to represent
elastic effects accurately. The intersection of data-driven techniques and
elastic scattering theories presents opportunities and challenges. In this
paper, by using the knowledge of elastic scattering (Physics of problem) and
integrating it with deep learning techniques, we propose methods for the
solution of time-harmonic FWI to enhance accuracy compared to pure data-driven
approaches. Moreover, by modifying the structure of the Variational
Autoencoder, we introduce a probabilistic deep learning method based on the
physics of the problem that enables us to explore the uncertainties of the
solution. According to the limited availability of datasets in this field and
to assess the performance and accuracy of the proposed methods, we create a
comprehensive dataset close to reality and conduct a comparative analysis of
the presented approaches to it.
