In this paper, we extend a recently proposed approach for inverse scattering with Neumann boundary conditions [Druskin et al., Inverse Probl. 37, 075003 (2021)] to the 1D Schrödinger equation impedance (Robin) conditions. This method approaches in two steps: first, extract reduced order model (ROM) directly from data and, subsequently, potential ROM. We also propose novel data-assimilation (DA)...