  We introduce a novel approach to waveform inversion, based on a data driven
reduced order model (ROM) of the wave operator. The presentation is for the
acoustic wave equation, but the approach can be extended to elastic or
electromagnetic waves. The data are time resolved measurements of the pressure
wave gathered by an acquisition system which probes the unknown medium with
pulses and measures the generated waves. We propose to solve the inverse
problem of velocity estimation by minimizing the square misfit between the ROM
computed from the recorded data and the ROM computed from the modeled data, at
the current guess of the velocity. We give the step by step computation of the
ROM, which depends nonlinearly on the data and yet can be obtained from them in
a non-iterative fashion, using efficient methods from linear algebra. We also
explain how to make the ROM robust to data inaccuracy. The ROM computation
requires the full array response matrix gathered with collocated sources and
receivers. However, we show that the computation can deal with an approximation
of this matrix, obtained from towed-streamer data using interpolation and
reciprocity on-the-fly.
  While the full-waveform inversion approach of nonlinear least-squares data
fitting is challenging without low frequency information, due to multiple
minima of the data fit objective function, we show that the ROM misfit
objective function has a better behavior, even for a poor initial guess. We
also show by an explicit computation of the objective functions in a simple
setting that the ROM misfit objective function has convexity properties,
whereas the least squares data fit objective function displays multiple local
minima.
