  This paper introduces a novel matrix-free approach for full waveform
inversion in anisotropic elastic media, incorporating density variation through
the utilization of the distorted Born iterative method. This study aims to
overcome the computational and storage challenges associated with the
conventional matrix-based distorted Born iterative inversion method while
accurately capturing the subsurface's anisotropic properties and density
variations. An elastic integral equation is utilized to account for the
anisotropic nature of elastic wave propagation, enabling more precise modeling
of subsurface complexities. This integral equation is efficiently solved by a
fast Fourier transform accelerated Krylov subspace method. Leveraging the
integral equation with the distorted Born approximation, a linear relationship
between the scattered wavefield and the model parameter perturbation is
formulated for an integrated inversion scheme. To address the inherent
ill-posedness of each linear inversion step, we formulate the normal equation
with a regularization term. This is achieved by minimizing an objective
function using the generalized Tikhonov method. Therefore, we can find an
adequate solution for the inverse scattering problem by solving the normal
equation. Following the physical interpretation of Green's function, the
Fr{\'e}chet and adjoint operators within the normal equation can be employed in
a matrix-free manner, allowing for significant improvement of the computational
efficiency and memory demand without compromising accuracy. The proposed
matrix-free full waveform inversion framework is thoroughly validated through
extensive numerical experiments on synthetic datasets, showcasing its ability
to reconstruct complex anisotropic structures and accurately recover stiffness
parameters and density.
