Nonlinear least-squares reverse time migration of prismatic waves for delineating steeply dipping structures

Prismatic reflections in seismic data carry abundant information about subsurface steeply dipping structures, such as salt flanks or near-vertical faults, playing an important role in delineating these structures when effectively used. Conventional linear least-squares reverse time migration (L-LSRTM) fails to use prismatic waves due to the first-order Born approximation, resulting in a blurry image of steep interfaces. We propose a nonlinear LSRTM (NL-LSRTM) method to take advantage of prismatic waves for the detailed characterization of subsurface steeply dipping structures. Compared with current least-squares migration methods of prismatic waves, our NL-LSRTM is nonlinear and thus avoids the challenging extraction of prismatic waves or the prior knowledge of L-LSRTM result. The gradient of NL-LSRTM consists of the primary and prismatic imaging terms, which can accurately project both observed primary and prismatic waves into the image domain for the simultaneous depiction of near-horizontal and near-vertical structures. However, we find that the full Hessian based Newton normal equation has two similar terms, which prompts us to make further comparison between the Newton normal equation and the proposed NL-LSRTM. We demonstrate that the Newton normal equation is problematic when applied to the migration problem because the primary reflections in the seismic records will be wrongly projected into the image along the prismatic wavepath, resulting in an artifact-contaminated image. In contrast, the nonlinear data-fitting process included in the NL-LSRTM contributes to balancing the amplitudes of primary and prismatic imaging results, thus making NL-LSRTM produce superior images compared to the Newton normal equation. Several numerical tests validate the applicability and robustness of NL-LSRTM for the delineation of steeply dipping structures, and illustrate that the imaging results are much better than the conventional L-LSRTM.