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 that 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 a step by step computation of the ROM, which depends nonlinearly on the data and yet can be obtained from them in a noniterative 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 colocated sources and receivers. However, we find that the computation can deal with an approximation of this matrix, obtained from towed-streamer data using interpolation and reciprocity on-the-fly. Although 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 find that the ROM misfit objective function has better behavior, even for a poor initial guess. We also find by 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.