Extending the search space of full-waveform inversion beyond the single-scattering Born approximation: A tutorial review

Abstract

Full Waveform Inversion can be made immune to cycle skipping by matching the recorded data arbitrarily well 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 aiming 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 feedback term to the data added to the source term. Then, the subsurface parameters are updated by canceling out these additional source terms, sometime called unwisely wave-equation errors, to push the background model towards the true model in the left-hand side wave-equation operator. Although many studies were 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 show 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.