Waveform inversion via reduced order modeling

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 at the sensors in an active array, which probe the unknown medium with pulses and measure the generated waves. The ROM depends nonlinearly on the data but it can be constructed from them using numerical linear algebra methods. We show that the ROM can be used for the inverse problem of velocity estimation. While the full-waveform inversion approach of {nonlinear least-squares} data fitting is challenging without low frequency information, due to multiple minima of the objective function, the minimization of the ROM misfit function has a better behavior, even for a poor initial guess. In fact, the ROM misfit function is demonstrably a convex function for low-dimensional parametrizations of the unknown velocity. We give the construction of the ROM, introduce the inversion approach based on the ROM misfit and assess its performance with numerical simulations.