  We study an inverse problem for the wave equation, concerned with estimating
the wave speed, aka velocity, from data gathered by an array of sources and
receivers that emit probing signals and measure the resulting waves. The
typical mathematical formulation of velocity estimation is a nonlinear least
squares minimization of the data misfit, over a search velocity space. There
are two main impediments to this approach, which manifest as multiple local
minima of the objective function: The nonlinearity of the mapping from the
velocity to the data, which accounts for multiple scattering effects, and poor
knowledge of the kinematics (smooth part of the wave speed) which causes
cycle-skipping. We show that the nonlinearity can be mitigated using a data
driven estimate of the internal wave field. This leads to improved performance
of the inversion for a reasonable initial guess of the kinematics.
