Least-Squares Reverse Time Migration for Reflection-Angle-Dependent Reflectivity

Least-squares reverse time migration (LSRTM) can estimate high-quality reflectivity of subsurface medium from seismic data. However, the subsurface reflectivity depends on reflection angles, and its variations over reflection angles are extremely important because they can be used to estimate subsurface physical properties for seismic interpretation. We present a new formulation of the LSRTM method that can estimate reflection-angle-dependent reflectivity from seismic data. We derive a forward modeling operator which predicts the reflection data without calculating the reflection angles, and verify that it approximately equals to the reflection-angle-dependent wave-equation-based Kirchhoff modeling operator under the assumption that the velocity perturbation is small and the reflection angle is smaller than the critical angle. Based on the proposed modeling operator associated with the adjoint of the angle-dependent wave-equation-based Kirchhoff modeling operator, we reformulate LSRTM as an inverse problem to invert for reflection-angle-dependent reflectivity using a preconditioned conjugate gradient algorithm. The algorithm uses a low-rank filter as the preconditioner to attenuate migration artifacts. Imaging tests on synthetic and field seismic data are used to verify validity and superiority of the proposed method. The tests illustrate that the proposed method can produce the reflection-angle-dependent reflectivity with much higher signal-to-noise ratio (SNR), resolution and amplitude fidelity than reverse time migration. Compared with conventional LSRTM, it can produce more focused stacked image when the migration velocity contains errors. Moreover, conventional LSRTM only produces the angle-independent reflectivity, whereas the proposed method has the feasibility to produce the reflection-angle-dependent reflectivity.