  In this work we consider the inverse elastic scattering problem by an
inclusion in two dimensions. The elastic inclusion is placed in an isotropic
homogeneous elastic medium. The inverse problem, using the third Betti's
formula (direct method), is equivalent to a system of four integral equations
that are non linear with respect to the unknown boundary. Two equations are on
the boundary and two on the unit circle where the far-field patterns of the
scattered waves lie. We solve iteratively the system of integral equations by
linearising only the far-field equations. Numerical results are presented that
illustrate the feasibility of the proposed method.
