  The focus of this work is on rigorous mathematical analysis of the
topological derivative based detection algorithms for the localization of an
elastic inclusion of vanishing characteristic size. A filtered quadratic misfit
is considered and the performance of the topological derivative imaging
functional resulting therefrom is analyzed. Our analysis reveals that the
imaging functional may not attain its maximum at the location of the inclusion.
Moreover, the resolution of the image is below the diffraction limit. Both
phenomena are due to the coupling of pressure and shear waves propagating with
different wave speeds and polarization directions. A novel imaging functional
based on the weighted Helmholtz decomposition of the topological derivative is,
therefore, introduced. It is thereby substantiated that the maximum of the
imaging functional is attained at the location of the inclusion and the
resolution is enhanced and it proves to be the diffraction limit. Finally, we
investigate the stability of the proposed imaging functionals with respect to
measurement and medium noises.
