@article{fdi:010089130,
  title = {{E}xtending the search space of full-waveform inversion beyond the single-scattering {B}orn approximation : a tutorial review},
  author = {{O}perto, {S}. and {G}holami, {A}. and {A}ghamiry, {H}. and {G}uo, {G}. and {B}eller, {S}. and {A}ghazade, {K}. and {M}amfoumbi, {F}. and {C}ombe, {L}. and {R}ibodetti, {A}lessandra},
  editor = {},
  language = {{ENG}},
  abstract = {{F}ull-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. {T}o 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.  {T}his 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. {T}hen, 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.  {A}lthough 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. {T}he goal of this tutorial is to review these principles in the framework of inverse scattering theory whose governing forward equation is the {L}ippmann-{S}chwinger equation. {F}rom 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 {B}orn approximation. {W}e 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 {N}ewton {H}essian of the parameter-estimation problem. {T}he theory is finally illustrated with numerical examples. {U}nderstanding the physical principles governing these methods is a necessary prerequisite to assess their potential and limits and design relevant heuristics to manage the latter.},
  keywords = {},
  booktitle = {},
  journal = {{G}eophysics},
  volume = {88},
  numero = {6},
  pages = {{R}671--{R}702},
  ISSN = {0016-8033},
  year = {2023},
  DOI = {10.1190/geo2022-0758.1},
  URL = {https://www.documentation.ird.fr/hor/fdi:010089130},
}