Seismic images under complex overburdens like salt are strongly affected by illumination variations due to overburden velocity variations and imperfect acquisition geometries, making it difficult to obtain reliable image amplitudes. Least-squares reverse-time migration (LSRTM) addresses these issues by formulating full wave-equation imaging as a linear inverse problem and solving for a reflectivity model that explains the recorded seismic data. Because subsurface reflection coefficients depend on the incident angle, and possibly on azimuth, quantitative interpretation under complex overburdens requires LSRTM with output in terms of image gathers, e.g., as a function of reflection angle or angle and azimuth. We present a reflection angle- or angle/azimuth-dependent LSRTM method aimed at obtaining physically meaningful image amplitudes interpretable in terms of angle- or angle/azimuth-dependent reflection coefficients. The method is formulated as a linear inverse problem solved iteratively with the conjugate gradient method. It requires an adjoint pair of linear operators for reflection angle/azimuth-dependent migration and demigration based on full wave-equation propagation. We implement these operators in an efficient way by using a mapping approach between migrated shot gathers and subsurface reflection angle/azimuth gathers. To accelerate convergence of the iterative inversion, we apply image-domain preconditioning operators computed from a single de-remigration step. An angle continuity constraint and a structural dip constraint, implemented via shaping regularization, are used to stabilize the solution in the presence of limited illumination and to control the effects of coherent noise. We demonstrate the method on a synthetic data example and on a wide-azimuth streamer dataset from the Gulf of Mexico, where we show that angle/azimuth-dependent LSRTM can achieve significant uplift in subsalt image quality, with overburden- and acquisition-related illumination variation effects on angle/azimuth-dependent image amplitudes largely removed.
Solving the Inverse Problem in OCT using Full-Wave Adjoint Models