Full-waveform inversion (FWI) can be made immune to cycle skipping by matching the recorded data with traveltime errors smaller than one-half period from inaccurate subsurface models. To achieve this goal, the simulated wavefields can be computed in an extended search space as the solution of an overdetermined problem aimed at jointly satisfying the wave equation and fitting the data in a least-squares sense. This leads to data-assimilated wavefields that are computed by solving the wave equation in the inaccurate background model with a data-dependent source extension added to the source term. Then, the subsurface parameters are updated by canceling out these additional source terms, sometimes inaccurately called wave equation errors, to push the background model toward the true model in the left-side wave equation operator. Although many studies are devoted to these approaches with promising numerical results, their governing physical principles and their relationships with classical FWI do not seem to be understood well yet. The goal of this tutorial is to review these principles in the framework of inverse scattering theory whose governing forward equation is the Lippmann-Schwinger equation. From this equation, we find how the data-assimilated wavefields embed an approximation of the scattered field generated by the sought model perturbation and how they modify the sensitivity kernel of classical FWI beyond the Born approximation. We also clarify how the approximation with which these wavefields approximate the unknown true wavefields is accounted for in the adjoint source and in the full Newton Hessian of the parameter-estimation problem. The theory is finally illustrated with numerical examples. Understanding the physical principles governing these methods is a necessary prerequisite to assess their potential and limits and design relevant heuristics to manage the latter.
