Wave-equation, finite-frequency imaging and inversion still face many challenges in addressing the inversion of highly complex velocity models as well as in dealing with nonlinear imaging (e.g., migration of multiples, amplitude-preserving migration). Extended images (EIs) are particularly important for designing image-domain objective functions aimed at addressing standing issues in seismic imaging, such as two-way migration velocity inversion or imaging/inversion using multiples. General one- and two-way representations for scattered wavefields can describe and analyze EIs obtained in wave-equation imaging. We have developed a formulation that explicitly connects the wavefield correlations done in seismic imaging with the theory and practice of seismic interferometry. In light of this connection, we define EIs as locally scattered fields reconstructed by model-dependent, image-domain interferometry. Because they incorporate the same one- and two-way scattering representations usedfor seismic interferometry, the reciprocity-based EIs can in principle account for all possible nonlinear effects in the imaging process, i.e., migration of multiples and amplitude corrections. In this case, the practice of two-way imaging departs considerably from the one-way approach. We have studied the differences between these approaches in the context of nonlinear imaging, analyzing the differences in the wavefield extrapolation steps as well as in imposing the extended imaging conditions. When invoking single-scattering effects and ignoring amplitude effects in generating EIs, the one- and two-way approaches become essentially the same as those used in today's migration practice, with the straightforward addition of space and time lags in the correlation-based imaging condition. Our formal description of the EIs and the insight that they are scattered fields in the image domain may be useful in further development of imaging and inversion methods in the context of linear, migration-based velocity inversion or in more sophisticated image-domain nonlinear inverse scattering approaches.

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