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. The extension is based on a new linearization of the scalar wave equation where the extended-velocity perturbation is convolved in time with the Laplacian of the background wavefield. This linearization is accurate for both reflected events and transmitted events and, in particular, for diving waves recorded at large offsets. The modeling capabilities of the proposed linearization enable the simultaneous inversion of reflections and diving waves even when the starting velocity model is far from being accurate. We solve the resulting optimization problem with a nested algorithm. The inner iterations are 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 two synthetic data sets modeled on the Marmousi model and on the Caspian Sea portion of the well-known BP model show the global-convergence properties as well as the high-resolution potential of the method.