Imaging/Inversion Formulas with Bounded Integrands

We collect recent ideas on transforming integral operators for imaging/inversion that use quotients of wave fields into operators that use products of wave fields; that is, transforming deconvolution-type operators into correlation-type operators. The transformation avoids evaluation of extremely large or extremely small numbers; hence, the designation “computer-friendly.” For this author, “inversion” means producing an output where the peak amplitude on each reflector is asymptotically proportional to the ray-theoretic reflection coefficient at a determinable incidence angle, consistent with the model wave equation and background wave speed. We describe the process for the classical imaging condition of wave equation migration—integral over frequency of the quotient of downward propagated wave fields—and for common-angle Kirchhoff inversion in 3D. For the former, we consider the application to common-shot data and to synthesized plane wave data. These are two cases for which the wave fields are solutions of a single wave equation. For such cases, Kirchhoff inversion is the asymptotic limit of the imaging operator.