  We introduce a novel nonlinear imaging method for the acoustic wave equation
based on data-driven model order reduction. The objective is to image the
discontinuities of the acoustic velocity, a coefficient of the scalar wave
equation from the discretely sampled time domain data measured at an array of
transducers that can act as both sources and receivers. We treat the wave
equation along with transducer functionals as a dynamical system. A reduced
order model (ROM) for the propagator of such system can be computed so that it
interpolates exactly the measured time domain data. The resulting ROM is an
orthogonal projection of the propagator on the subspace of the snapshots of
solutions of the acoustic wave equation. While the wavefield snapshots are
unknown, the projection ROM can be computed entirely from the measured data,
thus we refer to such ROM as data-driven. The image is obtained by
backprojecting the ROM. Since the basis functions for the projection subspace
are not known, we replace them with the ones computed for a known smooth
kinematic velocity model. A crucial step of ROM construction is an implicit
orthogonalization of solution snapshots. It is a nonlinear procedure that
differentiates our approach from the conventional linear imaging methods
(Kirchhoff migration and reverse time migration - RTM). It resolves all
dynamical behavior captured by the data, so the error from the imperfect
knowledge of the velocity model is purely kinematic. This allows for almost
complete removal of multiple reflection artifacts, while simultaneously
improving the resolution in the range direction compared to conventional RTM.
