Nonlinear extended wave-equation imaging by image-domain seismic interferometry

Wave-equation, finite-frequency imaging and inversion still faces considerable challenges in addressing the inversion of highly complex velocity models as well as in dealing with nonlinear imaging (e.g., migration of multiples, amplitudepreserving migration). Extended images (EI’s), as we present here, 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. Using general twoand one-way representations for scattered wavefields, we describe and analyze EI’s obtained in wave-equation imaging. The presented formulation explicitly connects the wavefield correlations done in seismic imaging with the theory and practice of seismic interferometry. We define extended images as locally scattered fields reconstructed by model-dependent, image-domain interferometry. Because we use the same twoand one-way scattering representations that are used for seismic interferometry, the reciprocity-based EI’s can in principle account for all possible nonlinear effects in the imaging process, i.e., migration of multiples, amplitude corrections, etc. In that case, the practice of two-way imaging departs considerably from that of the one-way approach. Here we elaborate on the differences between these approaches in the context of nonlinear imaging, describing these differences both in the wavefield extrapolation steps as well as in imposing the extended imaging conditions. When invoking single-scattering and ignoring amplitude effects in generating EI’s, the oneand two-way approaches become essentially the same as those employed in today’s migration practice, with the straightforward addition of spaceand time-lags in the correlationbased imaging condition. Our formal description of the EI’s and the insight that they are scattered fields in the image-domain may be useful in further development of imaging and inversion methods: either in the context of linear, migration-based velocity inversion, or in more sophisticated image-domain nonlinear inverse scattering approaches.