Approximating the Gauss–Newton Hessian Using a Space-Wavenumber Filter and its Applications in Least-Squares Seismic Imaging

The acquisition footprint, finite-frequency source, and unbalanced subsurface illumination make it difficult for traditional adjoint-based migration to produce a high-quality image for complex structures. By fitting reflection events with linearized simulation data, least-squares migration (LSM) can iteratively incorporate the effects of the Gauss–Newton Hessian (GNH) to produce high-quality depth profiles. However, high computational costs of forward and adjoint simulations limit the LSM applications in production. In this study, we present an efficient approximation approach for the GNH and utilize it as a preconditioner for the misfit gradient of the LSM to accelerate its convergence. The analytic solution of the GNH in homogeneous media reveals that the columns of the GNH are local spatial functions. Based on this observation, we design a space-wavenumber filter to approximate the GNH for heterogeneous media, which can be efficiently computed with the S-transform and spectral division. The mixed-domain property of this filter allows it to automatically take the wavenumber dependence of the GNH into account and, therefore, helps to improve spatial resolution. Numerical examples demonstrate that the approximated GNH can considerably improve the image quality and speed up the convergence of the LSM.