  The recently introduced non-iterative imaging method entitled \enquote{direct
sampling method} (DSM) is known to be fast, robust, and effective for inverse
scattering problems in the multi-static configuration but fails when applied to
the mono-static one. To the best of our knowledge no explanation of this
failure has been provided yet. Thanks to the framework of the asymptotic and
the far-field hypothesis in the 2D scalar configuration an analytical
expression of the DSM indicator function in terms of the Bessel function of
order zero and sizes, shapes and permittivities of the inhomogeneities is
obtained and the theoretical reason of the limitation identified. A modified
version of DSM is then proposed in order to improve the imaging method. The
theoretical results are supported by numerical results using synthetic data.

Numerical Analysis