  This paper concerns an inverse elastic scattering problem which is to
determine the location and the shape of a rigid obstacle from the phased or
phaseless far-field data for a single incident plane wave. By introducing the
Helmholtz decomposition, the model problem is reduced to a coupled boundary
value problem of the Helmholtz equations. The relation is established between
the compressional or shear far-field pattern for the elastic wave equation and
the corresponding far-field pattern for the coupled Helmholtz equations. An
efficient and accurate Nystr\"{o}m type discretization for the boundary
integral equation is developed to solve the coupled system. The translation
invariance of the phaseless compressional and shear far-field patterns are
proved. A system of nonlinear integral equations is proposed and two iterative
reconstruction methods are developed for the inverse problem. In particular,
for the phaseless data, a reference ball technique is introduced to the
scattering system in order to break the translation invariance. Numerical
experiments are presented to demonstrate the effectiveness and robustness of
the proposed method.
