Full-waveform inversion (FWI) solves model optimization problems by fitting simulated data to the observed data. The implementation of FWI requires involving as many physical features of the subsurface medium in the simulation as possible. The computation can be extremely costly and complex because the FWI algorithm usually deploys a consistent discretization over the entire model space, whereas a high-resolution analysis (and the accompanying complex physics) is often only required in the reservoir region. As an alternative, we have developed an FWI optimization scheme based on a convolution type of modeling from the datum. The solution of such an inversion consists of the overburden, which includes the medium above a datum level, and the virtual data at the datum, which represent the underlying medium, i.e., the reservoir. We formulate the redatuming operation using a modified expression of the extended Born representation. Based on that, the virtual data can be retrieved by using the subsurface-scattering imaging condition. By measuring the data misfit at the surface acquisition, we implement a simultaneous inversion for the overburden velocity and the virtual data at that datum. This velocity inversion is crucial to the redatuming, but we can rely on fairly simple physics compared to the complex reservoir region. The redatumed data, in turn, are involved in resolving the overburden velocity. We develop a robust estimate of the velocity using low-wavenumber updates along the reflection wavepaths generated by our modeling process including the overburden scattering and those coming from the datum. Using numerical examples, we determine that our optimization is capable of mitigating the complex wave-propagation effects of the overburden medium and output redatumed data with plausible relative amplitude, which is achieved given little knowledge of the model.