  The data-driven reduced order models (ROMs) have recently emerged as an
efficient tool for the solution of the inverse scattering problems with
applications to seismic and sonar imaging. One specification of this approach
is that it requires the full square multiple-output/multiple-input (MIMO)
matrix valued transfer function as data for multidimensional problems. The
synthetic aperture radar (SAR), however, is limited to single input/single
output (SISO) measurements corresponding to the diagonal of the matrix transfer
function. Here we present a ROM based Lippmann-Schwinger approach overcoming
this drawback. The ROMs are constructed to match the data for each
source-receiver pair separately, and these are used to construct internal
solutions for the corresponding source using only the data-driven Gramian.
Efficiency of the proposed approach is demonstrated on 2D and 2.5D (3D
propagation and 2D reflectors) numerical examples. The new algorithm not only
suppresses multiple echoes seen in the Born imaging, but also takes advantage
of illumination by them of some back sides of the reflectors, improving the
quality of their mapping.
