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This paper concerns an inverse elastic scattering problem which is todetermine the location and the shape of a rigid obstacle from the phased orphaseless far-field data for a single incident plane wave. By introducing theHelmholtz decomposition, the model problem is reduced to a coupled boundaryvalue problem of the Helmholtz equations. The relation is established betweenthe compressional or shear far-field pattern for the elastic wave equation andthe corresponding far-field pattern for the coupled Helmholtz equations. Anefficient and accurate Nystr\"{o}m type discretization for the boundaryintegral equation is developed to solve the coupled system. The translationinvariance of the phaseless compressional and shear far-field patterns areproved. A system of nonlinear integral equations is proposed and two iterativereconstruction methods are developed for the inverse problem. In particular,for the phaseless data, a reference ball technique is introduced to thescattering system in order to break the translation invariance. Numericalexperiments are presented to demonstrate the effectiveness and robustness ofthe proposed method.

Inverse obstacle scattering problem for elastic waves with phased or phaseless far-field data