  Approximated numerical techniques, for the solution of the elastic wave
scattering problem over semi-infinite domains are reviewed. The approximations
involve the representation of the half-space by a boundary condition described
in terms of 2D boundary element discretizations. The classical BEM matrices are
initially re-written into the form of a dense dynamic stiffness matrix and
later approximated to a banded matrix. The resulting final banded matrix is
then used like a standard finite element to solve the wave scattering problem
at lower memory requirements. The accuracy of the reviewed methods is
benchmarked against the classical problems of a semi-circular and a rectangular
canyon. Results are presented in the time and frequency domain, as well as in
terms of relative errors in the considered approximations. The main goal of the
paper is to give the analyst a method that can be used at the practising level
where an approximate solution is enough in order to support engineering
decisions.
