  The inverse scattering problems have been popular for the past thirty years.
While very successful in many cases, progress has lagged when only {\em
limited-aperture} measurement is available. In this paper, we perform some
elementary study to recover data that can not be measured directly. To be
precise, we aim at recovering the {\em full-aperture} far field data from {\em
limited-aperture} measurement. Due to the reciprocity relation, the
multi-static response matrix (MSR) has a symmetric structure. Using the Green's
formula and single layer potential, we propose two schemes to recover {\em
full-aperture} MSR. The recovered data is tested by a recently proposed direct
sampling method and the factorization method. The numerical results show the
possibility to recover, at least partially, the missing data and consequently
improve the reconstruction of the scatterer.
