  This paper concerns the reconstruction of multiple elastic parameters (Lam\'e
parameters and density) of an inhomogeneous medium embedded in an infinite
homogeneous isotropic background in $\mathbb{R}^2$. The direct scattering
problem is reduced to an equivalent system on a bounded domain by introducing
an exact transparent boundary condition and the wellposedness of the
corresponding variational problem is established. The Fr\'{e}chet
differentiability of the near-field scattering map is studied with respect to
the elastic parameters. Based on the multi-frequency measurement data and its
phaseless term, two Landweber iterative algorithms are developed for the
reconstruction of the multiple elastic parameters. Numerical examples,
indicating that plane pressure incident wave is a better choice, are presented
to show the validity and accuracy of our methods.
