The convergence of full-waveform inversion can be improved by extending the velocity model along either the subsurface-offset axis or the time-lag axis. The extension of the velocity model along the time-lag axis enables us to linearly model large time shifts caused by velocity perturbations. This linear modeling was based on a new linearization of the scalar wave equation in which perturbation of the extended slowness squared was convolved in time with the second time derivative of the background wavefield. The linearization was accurate for reflected events and transmitted events. We determined that it can effectively model conventional reflection data as well as modern long-offset data containing diving waves. It also enabled the simultaneous inversion of reflections and diving waves, even when the starting velocity model was far from being accurate. We solved the optimization problem related to the inversion with a nested algorithm. The inner iterations were based on the proposed linearization and on a mixing of scales between the short- and long-wavelength components of the velocity model. Numerical tests performed on synthetic data modeled on the Marmousi model and on the “Caspian Sea” portion of the well-known BP model demonstrated the global-convergence properties as well as the high-resolution potential of the proposed method.

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